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crawled_repos/ABC-Exceptions.jsonl

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+{"id":"def:omega","label":"omega","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(\\omega (n)\\) denote the number of distinct prime factors of an integer \\(n\\).","doc_url":null,"latex_link":"sect0003.html#def:omega","lean_link":null,"lean_decl_name":"omega","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/SetTheory/ZFC/Basic.lean","lean4_code":"def omega : ZFSet :=\n  mk PSet.omega\n\n","line":383,"end_line":385,"proof_status":"statement_formalized"}
+{"id":"lem:omegaUpperBound","label":"omegaUpperBound","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["def:omega"],"nlp_text":"For any \\(n\\ge 2\\), we have \\(\\omega (n) \\ll \\log (3n)/\\log \\log (3n)\\).","doc_url":null,"latex_link":"sect0003.html#lem:omegaUpperBound","lean_link":null,"lean_decl_name":"omegaUpperBound","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"thm:BombierSchmidtforThueEqs","label":"BombierSchmidtforThueEqs","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["def:omega"],"nlp_text":"Given integers \\(n,a_1,a_2\\neq 0\\) and \\(p\\ge 3\\), there are at most \\(O(p^{1+\\omega (|n|)})\\) many solutions \\((x_1,x_2)\\in \\mathbb {Z}^2\\) such that \\(|x_i|\\le X_i\\) and \\begin{align*} n = a_1 x_1^p + a_2 x_2^p. \\end{align*}","doc_url":null,"latex_link":"sect0003.html#thm:BombierSchmidtforThueEqs","lean_link":null,"lean_decl_name":"BombierSchmidtforThueEqs","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:MixedThueMethods","label":"MixedThueMethods","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["lem:omegaUpperBound","thm:BombierSchmidtforThueEqs","thm:QuadraticFormsEstimate"],"nlp_text":"Let \\(\\varepsilon {\\gt}0\\) and \\(D\\geq 1\\) and assume that \\(p=q,r\\in [1,D]\\) are integers. Then \\[ N(X,Y,Z)\\ll _{\\varepsilon ,D} Z (|a_1a_2a_3|XYZ)^\\varepsilon . \\]","doc_url":null,"latex_link":"sect0003.html#lem:MixedThueMethods","lean_link":null,"lean_decl_name":"MixedThueMethods","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"prop:ThueEquations","label":"ThueEquations","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["lem:MixedThueMethods"],"nlp_text":"Let \\(d\\geq 1\\), and let \\[ X_1,\\ldots , X_d, Y_1,\\ldots , Y_d,Z_1,\\ldots , Z_d\\geq 1. \\]","doc_url":null,"latex_link":"sect0003.html#prop:ThueEquations","lean_link":null,"lean_decl_name":"ThueEquations","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:nu","label":"nu","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["def:BdDiophantineCount"],"nlp_text":"It will be convenient to henceforth define \\begin{equation*} \\nu =2\\varepsilon ^2+\\frac{\\log B_d(\\mathbf{c},\\mathbf{X},\\mathbf{Y},\\mathbf{Z})}{\\log X}. \\end{equation*}","doc_url":null,"latex_link":"sect0004.html#def:nu","lean_link":null,"lean_decl_name":"nu","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"prop:FourierBound","label":"FourierBound","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["def:nu","lem:Trivialaibici","prop:FourierAnalysis"],"nlp_text":"\\begin{align*} \\nu & {\\lt} \\frac{1}{2}\\Big(1+\\delta +\\sum _{i\\leq d} \\max (a_i,b_i) - \\max _{m{\\gt}1}(a_m,b_m)\\Big). \\end{align*}","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/FourierBound","latex_link":"sect0004.html#prop:FourierBound","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/FourierBound","lean_decl_name":"FourierBound","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section4.lean","lean4_code":"/--\nA statement of the Fourier bound. Note that this is _not_ saying the bound holds, but defining\nwhat it means for the bound to hold. In Section4.lean, we will take this as an assumption to many\nstatements in order to deduce bounds on `ν`.\nElsewhere we will show that the bound holds, and thus its proof can be fed in to those lemmas\nwhich have it as an assumption.\n-/\ndef FourierBound (d : ℕ) (δ ν : ℝ) (a b : ℕ → ℝ) : Prop :=\n  ν < 1/2 * (1 + δ + ∑ i ≤ d, max (a i) (b i) - sSup {max (a i) (b i) | i ∈ Set.Ioc 1 d})\n","line":92,"end_line":100,"proof_status":"statement_formalized"}
+{"id":"prop:DeterminantBound","label":"DeterminantBound","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["def:nu","prop:DeterminantMethod","def:aibici"],"nlp_text":"\\begin{align*} \\nu {\\lt} \\min _{p,q\\ge 1} \\left( 1+\\delta - a_p - b_q +\\min \\left(\\frac{a_p}{q}, \\frac{b_q}{p}\\right)\\right). \\end{align*}","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/DeterminantBound","latex_link":"sect0004.html#prop:DeterminantBound","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/DeterminantBound","lean_decl_name":"DeterminantBound","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section4.lean","lean4_code":"/--\nA statement of the Determinant bound. Note that this is _not_ saying the bound holds, but defining\nwhat it means for the bound to hold. In Section4.lean, we will take this as an assumption to many\nstatements in order to deduce bounds on `ν`.\nElsewhere we will show that the bound holds, and thus its proof can be fed in to those lemmas\nwhich have it as an assumption.\n-/\ndef DeterminantBound (d : ℕ) (δ ν : ℝ) (a b : ℕ → ℝ) : Prop :=\n  ν < sInf {1 + δ - a p - b q + min (a p / q) (b q / p) |\n    (p : ℕ) (q : ℕ) (_ : p ∈ Set.Ioc 1 d) (_ : q ∈ Set.Ioc 1 d)}\n","line":136,"end_line":145,"proof_status":"statement_formalized"}
+{"id":"prop:ThueBound","label":"ThueBound","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["def:nu","lem:Trivialaibici","prop:ThueEquations"],"nlp_text":"\\begin{align*} \\nu {\\lt} 1 +\\delta - \\max _{p\\ge 2}\\sum _{p\\mid i}(a_i+b_i). \\end{align*}","doc_url":null,"latex_link":"sect0004.html#prop:ThueBound","lean_link":null,"lean_decl_name":"ThueBound","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"./ABCExceptions/Section4.lean","lean4_code":"def ThueBound (d : ℕ) (δ ν : ℝ) (a b : ℕ → ℝ) : Prop :=\n  ν < 1 + δ - sSup {∑ i ≤ d with p ∣ i, (a i + b i) | p ∈ Set.Ioc 1 d}\n\nvariable (htab : ThueBound d δ ν a b) (htac : ThueBound d δ ν a c) (htbc : ThueBound d δ ν b c)\n\nsection\n\ninclude htab\n\n","line":208,"end_line":216,"proof_status":"statement_formalized"}
+{"id":"prop:GeometryBound","label":"GeometryBound","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["def:nu","lem:aibiciconstraints","prop:GeometryofNumbers","lem:Trivialaibici"],"nlp_text":"\\begin{align*} \\nu {\\lt} \\delta + \\min _{I,I',I''\\subset [d]} \\left( \\max \\left( 1 \\, ,\\, \\sum _{i\\in I} ia_i +\\sum _{i\\in I'} ib_i + \\sum _{i\\in I''} ic_i\\right) - \\sum _{i\\in I} a_i -\\sum _{i\\in I'} b_i - \\sum _{i\\in I''} c_i\\right). \\end{align*}","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/GeometryBound","latex_link":"sect0004.html#prop:GeometryBound","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/GeometryBound","lean_decl_name":"GeometryBound","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section4.lean","lean4_code":"/--\nA statement of the Geometry bound. Note that this is _not_ saying the bound holds, but defining\nwhat it means for the bound to hold. In Section4.lean, we will take this as an assumption to many\nstatements in order to deduce bounds on `ν`.\nElsewhere we will show that the bound holds, and thus its proof can be fed in to those lemmas\nwhich have it as an assumption.\n-/\ndef GeometryBound (d : ℕ) (ε ν : ℝ) (a b c : ℕ → ℝ) : Prop :=\n  ∀ Ia ⊆ Finset.Icc 1 d, ∀ Ib ⊆ Finset.Icc 1 d, ∀ Ic ⊆ Finset.Icc 1 d,\n    ν < ε +\n      ((∑ i ∈ Finset.Iic d \\ Ia, a i +\n        ∑ i ∈ Finset.Iic d \\ Ib, b i +\n        ∑ i ∈ Finset.Iic d \\ Ic, c i) +\n       max 0 (∑ i ∈ Ia, i * a i + ∑ i ∈ Ib, i * b i + ∑ i ∈ Ic, i * c i - 1))\n","line":273,"end_line":286,"proof_status":"statement_formalized"}
+{"id":"lem:Subcase2.6","label":"Subcase2.6","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["prop:FourierBound","lem:Subcase2.3","lem:Subcase2.5","lem:Subcase2.4"],"nlp_text":"\\(\\nu \\leq 0.66\\) in the case \\(s_2{\\lt}0.3\\) and \\(2s_1-s_2{\\gt}0.025\\).","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/case_2_subcase_6","latex_link":"sect0004.html#lem:Subcase2.6","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/case_2_subcase_6","lean_decl_name":"case_2_subcase_6","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section4.lean","lean4_code":"lemma case_2_subcase_6\n    (hd : 5 ≤ d)\n    (hν : 0.66 < ν)\n    (hs₂ : s 2 < 0.3)\n    (hδ : δ ≤ 0.001)\n    (hε₀ : 0 < ε)\n    (hε : ε ≤ 1 / 100000)\n    (hba : b 3 ≤ a 3) (hcb : c 3 ≤ b 3)\n    (h : 0.025 < 2 * s 1 - s 2) :\n    False := by\n  have hτ (τ) (hτ : τ ∈ SubSums 3 a b c) :\n      τ ∉ Set.Icc (0.34 - s 1 - s 2 + δ) (0.33 - 1 / 2 * δ) := by\n    intro h'\n    apply bound_4_point_18 h44 hg τ hτ (by omega) hν\n    simp only [Set.mem_union, Set.mem_Icc, h'.1, h'.2, true_and, and_true]\n    have : 0.34 - s 1 + δ ≤ 0.33 - 1 / 2 * s 2 - 1 / 2 * δ := by\n      linear_combination 1 / 2 * h + 3 / 2 * hδ\n    by_contra! h''\n    exact lt_irrefl (0 : ℝ) (by linear_combination this + h''.1 + h''.2)\n  have hs3 : 4 * s 1 + 3 * s 2 ≤ 0.73 := by\n    by_contra! h'\n    exact case_2_subcase_3 ha hb hc h43ab h43ac h43bc h44 h45b h45c hdab hdac htab htac htbc hg\n        (by omega) hν hs₂ (by linear_combination hδ) hε₀ (by linear_combination hε) hba hcb h'\n  have hs4 : 0.35 ≤ 4 * s 1 + s 2 := by\n    by_contra! h'\n    exact case_2_subcase_4 ha hb hc h43ab h43ac h43bc h44 h45b h45c hdab hdac htab htac htbc hg\n        hd hν hs₂ (by linear_combination hδ) hε₀ hε hba hcb h'\n  have hs1 : s 1 ≤ 0.1825 := by\n    linear_combination 1 / 4 * hs3 + 3 / 4 * s_nonneg ha hb hc 2\n  have hs2 : s 2 ≤ 0.19 := by\n    linear_combination 1 / 2 * (hs3 + hs4)\n  replace hs2 : s 2 ≤ 0.07 := by\n    by_contra! h'\n    exact case_2_subcase_5 ha hb hc h43ab h43ac h43bc h44 h45b h45c hdab hdac htab htac htbc hg\n        hd hν hs₂ (by linear_combination hδ) hε₀ hε hba hcb ⟨h'.le, by linear_combination hs2⟩\n  have ha3 : a 3 < 0.32 := by\n    by_contra! h'\n    exact case_2_subcase_1 ha hb hc h43bc h45b h45c hdab hdac htab htac hg hd hν hs₂\n      (by linear_combination hδ) hε₀ (by linear_combination hε) h'\n  have ha3' : a 3 < 0.34 - s 1 - s 2 + δ := by\n    by_contra!\n    refine hτ (a 3) (by simp [SubSums]) ⟨this, ?_⟩\n    linear_combination ha3 + 1 / 2 * hδ\n  have hbc3 : 0.34 - s 1 - s 2 + δ ≤ b 3 + c 3 := by\n    by_contra! h'\n    have : s 3 < 0.68 - 2 * s 1 - 2 * s 2 + 2 * δ := calc\n      s 3 = a 3 + b 3 + c 3 := by rw [s_apply]\n      _ < _ := by linear_combination ha3' + h'\n    have : 0.32 - 4 * δₛ - 2 * δ < s 1 := by\n      linear_combination this + bound_4_point_20 ha hb hc (by omega)\n    have : 0.1825 < s 1 := by\n      linear_combination this + 2 * hδ + 4 * hε +\n        4 * bound_4_point_10_upper hε₀ (by linear_combination hε) h43ab h43ac h43bc\n    exact hs1.not_lt this\n  have hbc3' : 0.33 - 1 / 2 * δ ≤ b 3 + c 3 := by\n    by_contra!\n    exact hτ (b 3 + c 3) (by simp [SubSums]) ⟨hbc3, this.le⟩\n  exact case_2_subcase_6_end_ec ha hb hc h45a h45b h45c hfab hfac hfbc hd hν hδ hε₀\n    hε hba hcb hbc3' hs2\n","line":1477,"end_line":1535,"proof_status":"formalized"}
+{"id":"lem:Subcase1.2","label":"Subcase1.2","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["prop:FourierBound","prop:SpecialThue","lem:SpecialGeometry","lem:SubSumGeometry","lem:deltaBounds"],"nlp_text":"\\(\\nu \\leq 0.66\\) in the case \\(s_2 \\geq 0.3\\) and \\(b_3 {\\gt} 0.34-s_1-s_2+\\delta \\).","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/subcase_1_point_2","latex_link":"sect0004.html#lem:Subcase1.2","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/subcase_1_point_2","lean_decl_name":"subcase_1_point_2","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section4.lean","lean4_code":"lemma subcase_1_point_2\n    (hd : 6 ≤ d) (hν : 0.66 < ν) (hba : b 3 ≤ a 3) (hs₂ : 0.3 ≤ s 2)\n    (hb₃ : 0.34 - s 1 - s 2 + δ < b 3) (hε₀ : 0 < ε) (hδ : δ ≤ 0.003) (hε : ε ≤ 1 / 10000) :\n    False := by\n  have h₁ : 6 * max (a 1) (b 1) + min (a 1) (b 1) ≤ 6 * s 1 := by\n    linear_combination min_add_max (a 1) (b 1)\n                     + 5 * max_le_add_of_nonneg (ha.nonneg 1) (hb.nonneg 1)\n                     - 6 * s_apply a b c 1 + 6 * hc.nonneg 1\n  have h₂ := bound_4_point_8 h43ab\n  have h₃ : ε ^ 2 ≤ 1 / 4000 := by nlinarith only [hε, hε₀]\n  obtain ⟨h₄, h₅, h₆⟩ :=\n    self_improve_bounds ha hb hc h44 h45a h45b htab htac htbc hg hd hν hb₃ hε₀ hδ hε hba\n  have : 2 * ν - 1 - δ < (5 / 2) * δ + 0.2761 := calc\n    2 * ν - 1 - δ < _ := bound_4_point_24 ha hb hfab hd hba\n    _ ≤ 0.334 + 1 / 5 * (4 * a 3 - b 3 + 6 * δ + 2 * ε ^ 2) - 1 / 2 * s 2 := by\n      linear_combination 1 / 5 * (h₁ + 3 * h₄ + 2 * h₅ + h₆ + 2 * h₂ + 7 / 2 * hs₂)\n    _ ≤ 0.276 + δ * (5 / 2) + ε ^ 2 * (2 / 5) := by\n      linear_combination 1 / 5 * subcase_1_point_2_aux ha hb h44 htab hg (by omega) hν hb₃\n    _ ≤ (5 / 2) * δ + 0.2761 := by linear_combination (2 / 5) * h₃\n  have : ν < 0.66 := by linear_combination 1 / 2 * this + (7 / 4) * hδ\n  exact this.not_le (by linear_combination hν)\n","line":1051,"end_line":1071,"proof_status":"formalized"}
+{"id":"lem:Subcase2.1","label":"Subcase2.1","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["prop:DeterminantBound","prop:GeometryBound","lem:deltaBounds","lem:Case2Basic1"],"nlp_text":"\\(\\nu \\leq 0.66\\) in the case \\(s_2{\\lt}0.3\\) and \\(a_3\\geq 0.32\\).","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/case_2_subcase_1","latex_link":"sect0004.html#lem:Subcase2.1","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/case_2_subcase_1","lean_decl_name":"case_2_subcase_1","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section4.lean","lean4_code":"lemma case_2_subcase_1\n    (hd : 5 ≤ d)\n    (hν : 0.66 < ν)\n    (hs₂ : s 2 < 0.3)\n    (hδ : δ ≤ 0.005)\n    (hε₀ : 0 < ε)\n    (hε : ε ≤ 1 / 10000)\n    (h : 0.32 ≤ a 3) :\n    False := by\n  by_cases hmax : ∃ i ∈ Set.Icc 4 d, 3 / 4 * 0.09 < max (b i) (c i)\n  · exact case_2_subcase_1_large_sum ha hdab hdac (by omega) hν (by linear_combination hδ) h hmax\n  simp only [not_exists, not_and, not_lt] at hmax\n  have h₁ : ∀ i ∈ Set.Icc 4 d, b i + c i ≤ 0.135 := by\n    intro i hi\n    have : b i + c i ≤ 2 * max (b i) (c i) := by\n      linear_combination le_max_left (b i) (c i) + le_max_right (b i) (c i)\n    specialize hmax i hi\n    linear_combination this + 2 * hmax\n  have h₂ : ∑ i ≤ d, (i - 1) * (b i + c i) ≤ 4 / 3 + (δ_ d b + δ_ d c) := calc\n    _ = ∑ x ≤ d, x * b x - ∑ x ≤ d, b x + (∑ x ≤ d, x * c x - ∑ x ≤ d, c x) := by\n      simp only [mul_add, sub_one_mul, Finset.sum_add_distrib, Finset.sum_sub_distrib]\n    _ ≤ _ := by linear_combination hb.sum_bound + hc.sum_bound - sum_eq_δ_ d b - sum_eq_δ_ d c\n  have := hg.s21_application_basic ha hb hc (by omega)\n  rw [← max_add_add_left, lt_max_iff, add_zero] at this\n  obtain (h' | h') := this\n  · have := case_2_subcase_1_subsubcase_1 ha hb hc h43bc htab hd hν (h₁ _ (by simp [hd])) hε₀.le hε\n      (by linear_combination hδ) h₂\n    have : (0.66 : ℝ) < 0.6451 := by linear_combination hν + h' + this + hε\n    norm_num1 at this\n  · have := case_2_subcase_1_subsubcase_2 ha hb hc h45b h45c htac (by omega) hν hs₂\n      (hmax _ (by simp; omega)) hε₀ hδ h\n    have : (0.66 : ℝ) < 0.6101 := by linear_combination hν + h' + this + hε\n    norm_num1 at this\n","line":1259,"end_line":1291,"proof_status":"formalized"}
+{"id":"prop:SpecialThue","label":"SpecialThue","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["prop:ThueBound","def:s"],"nlp_text":"To show \\(\\nu \\leq 0.66\\), it suffices to assume \\begin{equation} \\label{eq:Thue} a_j+b_j, \\, a_j+c_j,\\, b_j+c_j {\\lt} 0.34+\\delta , \\end{equation}","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/bound_4_point_12","latex_link":"sect0004.html#prop:SpecialThue","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/bound_4_point_12","lean_decl_name":"bound_4_point_12","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section4.lean","lean4_code":"lemma bound_4_point_12 (j : ℕ) (hj : j ∈ Set.Ioc 1 d) (hν : 0.66 < ν) :\n    a j + b j < 0.34 + δ := by\n  linear_combination htab.very_special ha hb j hj + hν\n","line":451,"end_line":453,"proof_status":"formalized"}
+{"id":"lem:SpecialGeometry","label":"SpecialGeometry","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["prop:GeometryBound"],"nlp_text":"We may assume \\begin{equation} \\label{eq:SpecialGeometry} s_1+s_2\\leq 0.34 +\\delta . \\end{equation}","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/bound_4_point_15","latex_link":"sect0004.html#lem:SpecialGeometry","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/bound_4_point_15","lean_decl_name":"bound_4_point_15","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section4.lean","lean4_code":"lemma bound_4_point_15\n    (hδ : δ ≤ 0.06)\n    (hν : 0.66 < ν) (hd : 2 ≤ d) :\n    s 1 + s 2 ≤ 0.34 + δ := by\n  have h₁ : ν < δ + (max 1 (s 1 + 2 * s 2) - (s 1 + s 2)) := by\n    have := hg.special_s h44 {1, 2} (by simp [Finset.insert_subset_iff]; omega)\n    simpa using this\n  replace h₁ : ν < max (δ + (1 - (s 1 + s 2))) (δ + s 2) := calc\n    _ < δ + (max 1 (s 1 + 2 * s 2) - (s 1 + s 2)) := h₁\n    _ = δ + (1 - (s 1 + s 2)) ⊔ (s 1 + 2 * s 2 - (s 1 + s 2)) := by rw [max_sub_sub_right]\n    _ = δ + max (1 - (s 1 + s 2)) (s 2) := by ring_nf\n    _ = max (δ + (1 - (s 1 + s 2))) (δ + s 2) := by rw [max_add_add_left]\n  by_contra! h₃ : 0.34 + δ < s 1 + s 2\n  replace h₃ : δ + (1 - (s 1 + s 2)) < 0.66 := by linear_combination h₃\n  have h₂ : s 2 < 0.51 + 1.5 * δ :=\n    bound_4_point_14_general ha hb hc htab htac htbc 2 (by simp [hd]) hν\n  replace h₂ : δ + s 2 < 0.66 := calc\n    _ < δ + (0.51 + 1.5 * δ) := by gcongr\n    _ = 0.51 + 2.5 * δ := by ring\n    _ ≤ 0.66 := by linear_combination 2.5 * hδ\n  have h₄ : max (δ + (1 - (s 1 + s 2))) (δ + s 2) < 0.66 := by simp [h₂, h₃]\n  exact (h₁.trans h₄).not_lt hν\n","line":498,"end_line":519,"proof_status":"formalized"}
+{"id":"lem:SubSumGeometry","label":"SubSumGeometry","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["prop:GeometryBound"],"nlp_text":"For any \\(j\\ge 3\\), allow \\(\\tau _j\\) to be an element \\begin{equation} \\label{eq:taujdef} \\tau _j\\in \\{ a_j,b_j,c_j, s_j, a_j+b_j,a_j+c_j,b_j+c_j\\} . \\end{equation}","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/bound_4_point_18","latex_link":"sect0004.html#lem:SubSumGeometry","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/bound_4_point_18","lean_decl_name":"bound_4_point_18","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section4.lean","lean4_code":"lemma bound_4_point_18 (τ : ℝ) (hτ : τ ∈ SubSums 3 a b c) (hd : 3 ≤ d) (hν : 0.66 < ν) :\n    τ ∉ Set.Icc (0.34 - s 1 - s 2 + δ) (0.33 - 1 / 2 * s 2 - 1 / 2 * δ) ∪\n        Set.Icc (0.34 - s 1 + δ) (0.33 - 1 / 2 * δ) := by\n  simp only [Set.mem_union, not_or]\n  exact ⟨bound_4_point_17_3 h44 hg τ hτ hν hd, bound_4_point_18_aux h44 hg τ hτ hν hd⟩\n","line":627,"end_line":631,"proof_status":"formalized"}
+{"id":"lem:Subcase2.2","label":"Subcase2.2","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["lem:Subcase2.1","lem:SubSumGeometry","lem:ThreeLowerBounds"],"nlp_text":"\\(\\nu \\leq 0.66\\) in the case \\(s.2 {\\lt} 0.3\\) and \\(b_3+c_3{\\lt}0.33-\\frac{s_2}{2}-\\frac{\\delta }{2}\\).","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/case_2_subcase_2","latex_link":"sect0004.html#lem:Subcase2.2","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/case_2_subcase_2","lean_decl_name":"case_2_subcase_2","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section4.lean","lean4_code":"lemma case_2_subcase_2\n    (hd : 5 ≤ d)\n    (hν : 0.66 < ν)\n    (hs₂ : s 2 < 0.3)\n    (hδ : δ ≤ 0.005)\n    (hε₀ : 0 < ε)\n    (hε : ε ≤ 1 / 10000)\n    (h : b 3 + c 3 < 0.33 - 1 / 2 * s 2 - 1 / 2 * δ) :\n    False := by\n  have h₁ : b 3 + c 3 < 0.34 - s 1 - s 2 + δ := by\n    have := bound_4_point_17_3 h44 hg (b 3 + c 3) (by simp [SubSums]) hν (by omega)\n    contrapose! this\n    exact ⟨this, h.le⟩\n  suffices h₂ : 0.34 - s 1 + δ < a 3 by\n    refine case_2_subcase_1 ha hb hc h43bc h45b h45c hdab hdac htab htac hg hd hν hs₂\n      (by linear_combination hδ) hε₀ hε ?_\n    have h₃ := bound_4_point_18_aux h44 hg (a 3) (by simp [SubSums]) hν (by omega)\n    contrapose! h₃\n    exact ⟨h₂.le, by linear_combination h₃ + 1 / 2 * hδ⟩\n  have h421 := bound_4_point_21 ha hb hc (by omega)\n  have h410 := bound_4_point_10_upper hε₀ (by linear_combination hε) h43ab h43ac h43bc\n  have h414 := bound_4_point_14_two_four ha hb hc htab htac htbc (by omega) hν\n  linear_combination s_apply a b c 3 + h₁ +\n    h421 + 5 / 2 * h410 + 1 / 2 * h414 + 11 / 4 * hδ + 5 / 2 * hε\n","line":1294,"end_line":1317,"proof_status":"formalized"}
+{"id":"lem:Subcase2.3","label":"Subcase2.3","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["lem:Subcase2.2","lem:Case2Basic2"],"nlp_text":"\\(\\nu \\leq 0.66\\) in the case \\(s.2 {\\lt} 0.3\\) and \\(4s_1+3s_2{\\gt}0.71\\).","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/case_2_subcase_3","latex_link":"sect0004.html#lem:Subcase2.3","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/case_2_subcase_3","lean_decl_name":"case_2_subcase_3","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section4.lean","lean4_code":"lemma case_2_subcase_3\n    (hd : 5 ≤ d) (hν : 0.66 < ν) (hs₂ : s 2 < 0.3)\n    (hδ : δ ≤ 0.003) (hε₀ : 0 < ε) (hε : ε ≤ 1 / 10000)\n    (hba : b 3 ≤ a 3) (hcb : c 3 ≤ b 3)\n    (h : 0.73 < 4 * s 1 + 3 * s 2) :\n    False := by\n  suffices b 3 + c 3 < 0.33 - 1 / 2 * s 2 - 1 / 2 * δ from\n    case_2_subcase_2 ha hb hc h43ab h43ac h43bc h44 h45b h45c hdab hdac htab htac htbc hg (by omega)\n      hν hs₂ (by linear_combination hδ) hε₀ hε this\n  obtain ⟨h₁, h₂⟩ := bound_4_point_26_aux ha hb h44 htab hg (by omega) hν hs₂\n    (by linear_combination hδ) hba hcb\n  linear_combination h₁ + h₂ + 1 / 2 * h + 5 / 2 * hδ\n","line":1320,"end_line":1331,"proof_status":"formalized"}
+{"id":"lem:Subcase2.5","label":"Subcase2.5","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["lem:Subcase2.2","lem:Case2Basic2"],"nlp_text":"\\(\\nu \\leq 0.66\\) in the case \\(s.2 {\\lt} 0.3\\) and \\(0.066\\leq s_2\\leq 0.204\\).","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/case_2_subcase_5","latex_link":"sect0004.html#lem:Subcase2.5","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/case_2_subcase_5","lean_decl_name":"case_2_subcase_5","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section4.lean","lean4_code":"lemma case_2_subcase_5\n    (hd : 5 ≤ d)\n    (hν : 0.66 < ν)\n    (hs₂ : s 2 < 0.3)\n    (hδ : δ ≤ 0.003)\n    (hε₀ : 0 < ε)\n    (hε : ε ≤ 1 / 100000)\n    (hba : b 3 ≤ a 3)\n    (hcb : c 3 ≤ b 3)\n    (h : s 2 ∈ Set.Icc 0.07 0.199) :\n    False := by\n\n  suffices b 3 + c 3 < 0.33 - 1 / 2 * s 2 - 1 / 2 * δ from\n    case_2_subcase_2 ha hb hc h43ab h43ac h43bc h44 h45b h45c hdab hdac htab htac htbc hg (by omega)\n      hν hs₂ (by linear_combination hδ) hε₀ (by linear_combination hε) this\n\n  simp only [Set.mem_Icc] at h\n  have h426 :=\n    bound_4_point_26 ha hb hc h44 htab hg (by omega) hν hs₂ (by linear_combination hδ) hba hcb\n  have h410 :=\n    bound_4_point_10_upper hε₀ (by linear_combination hε) h43ab h43ac h43bc\n  have h₁ : a 3 > 0.34 - s 1 - s 2 + δ := by\n    linear_combination h426 + 4 * h410 + h.1 + 3 * hδ + 4 * hε\n  replace h₁ : a 3 > 0.33 - 1 / 2 * s 2 - 1 / 2 * δ := by\n    have := bound_4_point_17_3 h44 hg (a 3) (by simp [SubSums]) hν (by omega)\n    contrapose! this\n    exact ⟨h₁.le, this⟩\n\n  have h₂ := bound_4_point_12 ha hb htab 3 (by simp; omega) hν\n  have h₃ := bound_4_point_12 ha hc htac 3 (by simp; omega) hν\n\n  linear_combination h₂ + h₃ + 2 * h₁ + (3 / 2) * h.2 + (7 / 2) * hδ\n","line":1359,"end_line":1390,"proof_status":"formalized"}
+{"id":"lem:Subcase2.4","label":"Subcase2.4","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["lem:Subcase2.2","lem:Case2Basic2"],"nlp_text":"\\(\\nu \\leq 0.66\\) in the case \\(s.2 {\\lt} 0.3\\) and \\(4s_1+s_2{\\lt}0.4\\).","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/case_2_subcase_4","latex_link":"sect0004.html#lem:Subcase2.4","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/case_2_subcase_4","lean_decl_name":"case_2_subcase_4","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section4.lean","lean4_code":"lemma case_2_subcase_4\n    (hd : 5 ≤ d)\n    (hν : 0.66 < ν)\n    (hs₂ : s 2 < 0.3)\n    (hδ : δ ≤ 0.004)\n    (hε₀ : 0 < ε)\n    (hε : ε ≤ 1 / 100000)\n    (hba : b 3 ≤ a 3)\n    (hcb : c 3 ≤ b 3)\n    (h : 4 * s 1 + s 2 < 0.35) :\n    False := by\n  suffices b 3 + c 3 < 0.33 - 1 / 2 * s 2 - 1 / 2 * δ from\n    case_2_subcase_2 ha hb hc h43ab h43ac h43bc h44 h45b h45c hdab hdac htab htac htbc hg hd hν hs₂\n      (by linear_combination hδ) hε₀ (by linear_combination hε) this\n  have h₁ : b 3 ≤ 0.34 - a 3 + δ := by\n    linear_combination bound_4_point_12 ha hb htab 3 (by simp; omega) hν\n  have h₂ : c 3 ≤ 0.34 - a 3 + δ := by\n    linear_combination bound_4_point_12 ha hc htac 3 (by simp; omega) hν\n  have : 0.34 - a 3 + δ < 0.15 - 1 / 4 * s 2 + 3 * δ + 4 * ε := by\n    linear_combination\n      bound_4_point_26 ha hb hc h44 htab hg (by omega) hν hs₂ (by linear_combination hδ) hba hcb +\n      1 / 4 * h + 4 * bound_4_point_10_upper hε₀ (by linear_combination hε) h43ab h43ac h43bc\n  linear_combination h₁ + h₂ + 2 * this + 8 * hε + 13 / 2 * hδ\n","line":1334,"end_line":1356,"proof_status":"formalized"}
+{"id":"lem:Case2","label":"Case2","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["lem:Subcase2.6"],"nlp_text":"Assuming 4.27, 4.28, 4.29, 4.30, \\(\\nu \\leq 0.66\\) in the case \\(s_2 {\\lt} 0.3\\).","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/case_2","latex_link":"sect0004.html#lem:Case2","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/case_2","lean_decl_name":"case_2","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section4.lean","lean4_code":"lemma case_2_subcase_1_subsubcase_2\n    (hd : 4 ≤ d)\n    (hν : 0.66 < ν)\n    (hs₂ : s 2 < 0.3)\n    (h₄ : max (b 4) (c 4) ≤ 3 / 4 * 0.09)\n    (hε₀ : 0 < ε)\n    (hδ : δ ≤ 0.005)\n    (h : 0.32 ≤ a 3) :\n    a 3 + b 3 + min (b 4) (c 4) + ∑ i ∈ Finset.Icc 5 d, (b i + c i) +\n      (∑ i ∈ Finset.Icc 1 d, i * a i + (b 1 + c 1) + 2 * (b 2 + c 2) -\n        3 * (a 3 - c 3) + 4 * max (b 4) (c 4) - 1) < 0.61 := by calc\n    _ ≤ a 3 + b 3 + min (b 4) (c 4) + ∑ i ∈ Finset.Icc 5 d, (b i + c i) +\n        ((b 1 + c 1) + 2 * (b 2 + c 2) - 3 * (a 3 - c 3) + 4 * max (b 4) (c 4)) := by\n      linear_combination ha.sum_restrict_bound\n    _ = a 3 + b 3 + min (b 4) (c 4) + ((2 / 3 - δ_ d b - δ_ d c) - ∑ i ∈ range 5, (b i + c i)) +\n        ((b 1 + c 1) + 2 * (b 2 + c 2) - 3 * (a 3 - c 3) + 4 * max (b 4) (c 4)) := by\n      congr 2\n      rw [eq_sub_iff_add_eq', ← sum_Icc_eq_sum_range_add_sum_Icc (by omega), sum_add_distrib,\n        sum_eq_δ_, sum_eq_δ_]\n      ring_nf\n    _ = 2 / 3 - δ_ d b - δ_ d c - 2 * (a 3 - c 3) + b 2 + c 2 + 3 * (b 4 ⊔ c 4) := by\n      rw [Finset.sum_range, Fin.sum_univ_five]\n      norm_num [hb.zero, hc.zero]\n      linear_combination min_add_max (b 4) (c 4)\n    _ ≤ 11 / 30 - δ_ d b - δ_ d c + 3 * (b 4 ⊔ c 4) + 2 * δ := by\n      have : a 3 + c 3 < 0.34 + δ := bound_4_point_12 ha hc htac 3 (by simp; omega) hν\n      linear_combination hs₂ - s_apply a b c 2 + ha.nonneg 2 + 2 * this + 4 * h\n    _ < 0.61 := by\n      have h := bound_4_point_9_lower hε₀ b h45b\n      have h' := bound_4_point_9_lower hε₀ _ h45c\n      linear_combination h + h' + 3 * h₄ + 4 * hδ\n","line":1226,"end_line":1256,"proof_status":"formalized"}
+{"id":"lem:Case1","label":"Case1","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["lem:Subcase1.2","lem:Subcase1.1"],"nlp_text":"\\(\\nu \\leq 0.66\\) in the case \\(s_2 \\geq 0.3\\).","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/case_1","latex_link":"sect0004.html#lem:Case1","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/case_1","lean_decl_name":"case_1","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section4.lean","lean4_code":"lemma case_1\n    (hd : 6 ≤ d) (hν : 0.66 < ν) (hs₂ : 0.3 ≤ s 2) (hδ : δ ≤ 0.003)\n    (hε₀ : 0 < ε) (hε : ε ≤ 1 / 10000)\n    (hcb : c 3 ≤ b 3) (hba : b 3 ≤ a 3) :\n    False := by\n  obtain hb₃ | hb₃ := le_or_lt (b 3) (0.34 - s 1 - s 2 + δ)\n  · exact subcase_1_point_1 ha hb hc h43ab h43ac h43bc h44 htab htac htbc hg (by omega) hν\n      (by linear_combination hδ) hcb hs₂ hε hε₀ hb₃\n  · exact subcase_1_point_2 ha hb hc h43ab h44 h45a h45b hfab htab htac htbc hg hd hν hba hs₂ hb₃\n      hε₀ hδ hε\n","line":1074,"end_line":1083,"proof_status":"formalized"}
+{"id":"lem:Subcase1.1","label":"Subcase1.1","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["prop:SpecialThue","lem:SubSumGeometry","lem:deltaBounds","lem:Case1Basic","lem:ThreeLowerBounds"],"nlp_text":"\\(\\nu \\leq 0.66\\) in the case \\(s_2 \\geq 0.3\\) and \\(b_3\\le 0.34-s_1-s_2+\\delta \\).","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/subcase_1_point_1","latex_link":"sect0004.html#lem:Subcase1.1","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/subcase_1_point_1","lean_decl_name":"subcase_1_point_1","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section4.lean","lean4_code":"lemma subcase_1_point_1\n    (hd : 4 ≤ d)\n    (hν : 0.66 < ν)\n    (hδ : δ ≤ 0.003)\n    (hcb : c 3 ≤ b 3)\n    (hs₂ : 0.3 ≤ s 2)\n    (hε : ε ≤ 1 / 10000)\n    (hε₀ : 0 < ε)\n    (hb₃ : b 3 ≤ 0.34 - s 1 - s 2 + δ) :\n    False := by\n  -- We have an upper bound on b₃ + c₃\n  have h₁ : b 3 + c 3 ≤ 0.33 - 1 / 2 * s 2 - 1 / 2 * δ := by\n    linear_combination hcb + 2 * hb₃ + 2 * s_nonneg ha hb hc 1 + 3 / 2 * hs₂ + 5 / 2 * hδ\n  have hbcs : b 3 + c 3 ∈ SubSums 3 a b c := by simp [SubSums]\n  -- From 4.17, the upper bound strengthens\n  have h₂ : b 3 + c 3 < 0.34 - s 1 - s 2 + δ := by\n    simpa [h₁.not_lt, -one_div] using\n      bound_4_point_17_3 h44 hg (b 3 + c 3) hbcs hν (by omega)\n  -- Collect applications of earlier inequalities\n  have h_4_20 : 1 - 5 / 2 * δₛ - 2 * s 1 - 3 / 2 * s 2 - 1 / 2 * s 4 ≤ s 3 :=\n    bound_4_point_21 ha hb hc hd\n  have h_4_21 : s 1 ≤ 0.04 + δ :=\n    bound_4_point_22 ha hb hc h44 htab htac htbc hg hν hs₂ (by linear_combination hδ) (by omega)\n  have h_4_14 : s 2 + s 4 < 0.51 + 1.5 * δ :=\n    bound_4_point_14_two_four ha hb hc htab htac htbc hd hν\n\n  -- Combine the above facts to deduce a lower bound on a₃\n  have h₃ : 0.365 - 5 / 2 * δₛ - 11 / 4 * δ ≤ a 3 := calc\n    _ ≤ s 3 - (b 3 + c 3) := by\n      linear_combination h_4_20 + h_4_21 + (1 / 2) * h_4_14 + h₂\n    _ = a 3 := by simp [s_apply]\n\n  -- We have a simple upper bound on a₃ in terms of δₐ\n  have h₄ : a 3 ≤ 1 / 3 - δ_ d a := calc\n    a 3 ≤ ∑ i ≤ d, a i := Finset.single_le_sum (fun i hi ↦ ha.nonneg i) (by simp; omega)\n    _ = 1 / 3 - δ_ d a := sum_eq_δ_ _ _\n\n  have i : δₛ = δ_ d a + δ_ d b + δ_ d c := by simp [δₛ_eq]\n  -- Combining the bounds on a₃, we derive the following inequality\n  replace h₄ : 0.365 - 1 / 3 ≤ 3 / 2 * δₛ + δ_ d b + δ_ d c + 11 / 4 * δ := by\n    linear_combination h₃ + h₄ + i\n\n  -- But this inequality is easily contradicted by 4.8 and 4.10\n  have h_4_8 := bound_4_point_8 h43bc\n  have h_4_10 := bound_4_point_10_upper hε₀ (by linear_combination hε) h43ab h43ac h43bc\n  have : ε ^ 2 ≤ ε := by linear_combination hε * ε + 9999/10000 * hε₀\n\n  have : 3 / 2 * δₛ + δ_ d b + δ_ d c + 11 / 4 * δ < 0.365 - 1 / 3 := by\n    linear_combination (3 / 2) * h_4_10 + h_4_8 + this + (11 / 4) * hδ + (5 / 2) * hε\n\n  exact h₄.not_lt this\n","line":696,"end_line":746,"proof_status":"formalized"}
+{"id":"thm:ABCExponentNu","label":"ABCExponentNu","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["lem:Case1","lem:Case2"],"nlp_text":"\\(\\nu \\le 0.66\\).","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/thm_4_point_3","latex_link":"sect0004.html#thm:ABCExponentNu","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/thm_4_point_3","lean_decl_name":"thm_4_point_3","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section4.lean","lean4_code":"theorem thm_4_point_3\n    (hd : 6 ≤ d) (hδ : δ ≤ 0.001)\n    (hε₀ : 0 < ε) (hε : ε ≤ 1 / 100000) :\n    ν ≤ 0.66 := by\n  wlog hba : b 3 ≤ a 3 generalizing a b\n  · exact this hb ha h43ab.symm h43bc h43ac h44.left_comm h45b h45a hfab.symm hfbc hfac hdab.symm\n      hdbc hdac htab.symm htbc htac hg.left_comm (le_of_not_le hba)\n  wlog hcb : c 3 ≤ b 3 generalizing a b c\n  · obtain hca | hac := le_total (c 3) (a 3)\n    · exact this hb h45b ha hc h43ac h43ab h43bc.symm h44.right_comm h45a h45c hfac hfab hfbc.symm\n        hdac hdab hdbc.symm htac htab htbc.symm hg.right_comm hca (le_of_not_le hcb)\n    · exact this hb h45b hc ha h43ac.symm h43bc.symm h43ab h44.right_comm.left_comm h45c h45a\n        hfac.symm hfbc.symm hfab hdac.symm hdbc.symm hdab htac.symm htbc.symm htab\n        hg.right_comm.left_comm hac hba\n  exact thm_4_point_3_asymm ha hb hc h43ab h43ac h43bc h44 h45a h45b h45c hfab hfac hfbc hdab hdac\n    htab htac htbc hg hd hδ hε₀ hε hba hcb\n","line":1590,"end_line":1605,"proof_status":"formalized"}
+{"id":"lem:Case2Basic1","label":"Case2Basic1","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["prop:SpecialThue"],"nlp_text":"\\begin{equation} \\label{eq:2b3} b_3 {\\lt} 0.17+\\frac{\\delta }{2}. \\end{equation}","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/bound_4_point_24","latex_link":"sect0004.html#lem:Case2Basic1","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/bound_4_point_24","lean_decl_name":"bound_4_point_24","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section4.lean","lean4_code":"lemma bound_4_point_24 (hd : 6 ≤ d) (hba : b 3 ≤ a 3) :\n    2 * ν - 1 - δ < 2 / 15 + 2 / 5 * (δ_ d a + δ_ d b) + 1 / 5 *\n      (6 * max (a 1) (b 1) + min (a 1) (b 1) + 4 * a 3 - b 3 +\n        3 * max (a 4) (b 4) + 2 * max (a 5) (b 5) + max (a 6) (b 6)) :=\n  calc _ < 2 / 3 - (δ_ d a + δ_ d b) - ∑ i ≤ 6 with i ≠ 2, min (a i) (b i) - (a 2 + b 2) :=\n      apply_fourier ha hb hfab hd\n    _ ≤ 2 / 15 + 2 / 5 * (δ_ d a + δ_ d b) - ∑ i ≤ 6 with i ≠ 2, min (a i) (b i) +\n        (1 / 5 * ∑ i ≤ 6 with i ≠ 2, (7 - i) * a i +\n         1 / 5 * ∑ i ≤ 6 with i ≠ 2, (7 - i) * b i) := by\n      linear_combination two_bound ha hd + two_bound hb hd\n    _ = 2 / 15 + 2 / 5 * (δ_ d a + δ_ d b) - ∑ i ≤ 6 with i ≠ 2, min (a i) (b i) +\n        1 / 5 * ∑ i ≤ 6 with i ≠ 2, (7 - i) * (a i + b i) := by\n      simp only [mul_add, Finset.sum_add_distrib]\n    _ = 2 / 15 + 2 / 5 * (δ_ d a + δ_ d b) - ∑ i ≤ 6 with i ≠ 2, min (a i) (b i) +\n        1 / 5 * ∑ i ≤ 6 with i ≠ 2, (7 - i) * (min (a i) (b i) + max (a i) (b i)) := by\n      simp only [min_add_max]\n    _ = 2 / 15 + 2 / 5 * (δ_ d a + δ_ d b) + 1 / 5 * ∑ i ≤ 6 with i ≠ 2,\n        ((7 - i) * (min (a i) (b i) + max (a i) (b i)) - 5 * min (a i) (b i)) := by\n      simp only [Finset.sum_sub_distrib, mul_sub, ← Finset.mul_sum]\n      ring\n    _ ≤ _ := by\n      simp only [summation_range_eq]\n      have h4 : 0 ≤ min (a 4) (b 4) := le_min (ha.nonneg _) (hb.nonneg _)\n      have h5 : 0 ≤ min (a 5) (b 5) := le_min (ha.nonneg _) (hb.nonneg _)\n      have h6 : 0 ≤ min (a 6) (b 6) := le_min (ha.nonneg _) (hb.nonneg _)\n      norm_num [- min_add_max, ha.zero, hb.zero, hba]\n      linear_combination 2 * h4 + 3 * h5 + 4 * h6\n","line":1016,"end_line":1042,"proof_status":"formalized"}
+{"id":"lem:Case1Basic","label":"Case1Basic","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["lem:SpecialGeometry"],"nlp_text":"If \\(s_2 \\geq 0.3\\), \\begin{equation} \\label{eq:s1} s_1 \\leq 0.04+\\delta \\end{equation}","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/bound_4_point_22","latex_link":"sect0004.html#lem:Case1Basic","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/bound_4_point_22","lean_decl_name":"bound_4_point_22","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section4.lean","lean4_code":"lemma bound_4_point_22 (hν : 0.66 < ν) (hs₂ : 0.3 ≤ s 2) (hδ : δ ≤ 0.06) (hd : 2 ≤ d) :\n    s 1 ≤ 0.04 + δ := by\n  linear_combination bound_4_point_15 ha hb hc h44 htab htac htbc hg hδ hν hd + hs₂\n","line":686,"end_line":688,"proof_status":"formalized"}
+{"id":"lem:Case2Basic2","label":"Case2Basic2","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["lem:SubSumGeometry","lem:ThreeLowerBounds"],"nlp_text":"\\begin{equation} \\label{eq:Case2Basic2} a_3 \\geq 0.32 - 4 \\delta _s - s_1 - 2 \\delta . \\end{equation}","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/bound_4_point_25","latex_link":"sect0004.html#lem:Case2Basic2","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/bound_4_point_25","lean_decl_name":"bound_4_point_25","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section4.lean","lean4_code":"lemma bound_4_point_25\n    (hd : 3 ≤ d)\n    (hν : 0.66 < ν)\n    (h : b 3 ≤ a 3) :\n    b 3 < 0.17 + 1 / 2 * δ := by\n  linear_combination 1 / 2 * h + 1 / 2 * bound_4_point_12 ha hb htab 3 (by simp; omega) hν\n","line":1086,"end_line":1091,"proof_status":"formalized"}
+{"id":"def:delta","label":"delta","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"It will be convenient to define constants \\(\\delta _a,\\delta _b,\\delta _c\\) via \\begin{equation} \\label{eq:ai1/3} \\sum _{i\\leq d} a_i \\, = \\frac{1}{3}-\\delta _a, \\quad \\sum _{i\\leq d} b_i \\, = \\frac{1}{3}-\\delta _b, \\quad \\sum _{i\\leq d} c_i = \\frac{1}{3}-\\delta _c, \\end{equation}","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/δ_","latex_link":"sect0004.html#def:delta","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/δ_","lean_decl_name":"δ_","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section4.lean","lean4_code":"/-- Define `δ_ f` to be `1 / 3 - ∑ i ≤ d, f i`. -/\ndef δ_ (d : ℕ) (f : ℕ → ℝ) : ℝ := 1 / 3 - ∑ i ≤ d, f i\n","line":365,"end_line":366,"proof_status":"formalized"}
+{"id":"lem:deltaBounds","label":"deltaBounds","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["def:delta","prop:Eqn4Point5"],"nlp_text":"We have \\begin{equation} \\label{eq:robin2} \\delta _{ab}, \\delta _{ac}, \\delta _{bc} \\leq 0.00\\overline{6} + \\varepsilon ^2, \\end{equation}","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/bound_4_point_8","latex_link":"sect0004.html#lem:deltaBounds","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/bound_4_point_8","lean_decl_name":"bound_4_point_8","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section4.lean","lean4_code":"lemma bound_4_point_8 (h43 : Bound4Point3 d ε a b) : δ_ d a + δ_ d b ≤ 2 / 300 + ε ^ 2 := by\n  simp only [δ_, sub_add_sub_comm, ← Finset.sum_add_distrib, Bound4Point3] at h43 ⊢\n  linear_combination h43\n","line":371,"end_line":373,"proof_status":"formalized"}
+{"id":"lem:DiophantineFactor","label":"DiophantineFactor","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(\\varepsilon \\in (0,1/2)\\), and let \\(2\\leq n\\leq X\\) be an integer. Then there exists a factorisation \\begin{align*} n=c\\prod _{j\\leq \\frac{5}{2}\\varepsilon ^{-2}}x_j^{j}, \\end{align*}","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/exists_nice_factorization","latex_link":"sect0002.html#lem:DiophantineFactor","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/exists_nice_factorization","lean_decl_name":"exists_nice_factorization","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section2.lean","lean4_code":"theorem exists_nice_factorization :\n  ∃ (x : (Fin d) → ℕ), ∃ c : ℕ,\n    n = c * ∏ j, x j ^ (j.val + 1:ℕ) ∧\n    c ≤ (X:ℝ)^(ε) ∧\n    (∀ i j, i ≠ j → Nat.gcd (x i) (x j) = 1) ∧\n    (X:ℝ)^(- ε) * ∏ j, x j ≤ (radical n : ℕ) ∧ (radical n : ℕ) ≤ (X:ℝ)^(ε) * ∏ j, x j := by\n  refine ⟨x, c, c_mul_prod_x_eq_n.symm, c_le_X_pow, x_pairwise_coprime, X_pow_mul_prod_le_radical,\n    radical_le_X_pow_mul_prod⟩\n","line":1098,"end_line":1105,"proof_status":"formalized"}
+{"id":"prop:DiophantineReduction","label":"DiophantineReduction","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["lem:DiophantineFactor","def:BdDiophantineCount","def:Sstar"],"nlp_text":"Let \\(\\alpha ,\\beta ,\\gamma \\in (0,1]\\) be fixed and let \\(X\\geq 2\\). For any \\(\\varepsilon {\\gt}0\\) there exists an integer \\(d=d(\\varepsilon )\\geq 1\\) such that the following holds. There exist \\(X_1,\\ldots , X_d,Y_1,\\ldots , Y_d, Z_1,\\ldots , Z_d\\geq 1\\) satisfying \"> \\begin{align} \\label{eq:xiyizi_1} X^{\\alpha -\\varepsilon }\\ll _{\\varepsilon } \\prod _{j=1}^{d}X_j\\leq 2 X^{\\alpha +\\varepsilon },\\quad X^{\\beta -\\varepsilon }\\ll _{\\varepsilon }\\prod _{j=1}^{d}Y_j\\leq 2 X^{\\beta +\\varepsilon },\\quad X^{\\gamma -\\varepsilon }\\ll _{\\varepsilon }\\prod _{j=1}^{d}Z_j\\leq 2 X^{\\gamma +\\varepsilon } \\end{align}","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/refinedCountTriplesStar_isBigO_B","latex_link":"sect0002.html#prop:DiophantineReduction","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/refinedCountTriplesStar_isBigO_B","lean_decl_name":"refinedCountTriplesStar_isBigO_B","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section2.lean","lean4_code":"theorem refinedCountTriplesStar_isBigO_B\n  {α β γ : ℝ}\n  /- I'm surprised these assumptions are not necessary.\n    Shoud think about if I've done something wrong - Arend -/\n  -- (hα_pos : 0 < α) (hβ_pos : 0 < β) (hγ_pos : 0 < γ)\n  -- (hα1 : α ≤ 1) (hβ1 : β ≤ 1) (hγ1 : γ ≤ 1)\n  {x : ℕ} (h2X : 2 ≤ x) {ε : ℝ} (hε_pos : 0 < ε) (hε : ε < 1/2) :\n  ∃ s : Finset ((Fin (d ε) → ℕ) × (Fin (d ε) → ℕ) × (Fin (d ε) → ℕ) × (Fin 3 → ℕ)),\n    refinedCountTriplesStar α β γ x ≤\n      const ε * (x : ℝ) ^ ε * ↑(s.sup (fun ⟨X, Y, Z, c⟩ ↦ B (d ε) c X Y Z): ℕ) ∧\n    ∀ X Y Z : Fin (d ε) → ℕ,\n    ∀ c : Fin 3 → ℕ,\n    ⟨X, Y, Z, c⟩ ∈ s →\n    (x:ℝ)^(α - ε) ≤ 2 ^ d ε * ∏ j, X j ∧ ∏ j, X j ≤ 2 * (x : ℝ) ^ (α + ε) ∧\n    (x:ℝ)^(β - ε) ≤ 2 ^ d ε * ∏ j, Y j ∧ ∏ j, Y j ≤ 2 * (x : ℝ) ^ (β + ε) ∧\n    (x:ℝ)^(γ - ε) ≤ 2 ^ d ε * ∏ j, Z j ∧ ∏ j, Z j ≤ 2 * (x : ℝ) ^ (γ + ε) ∧\n    ∏ i, X i ^ (i.val + 1) ≤ x ∧\n    ∏ i, Y i ^ (i.val + 1) ≤ x ∧\n    ∏ i, Z i ^ (i.val + 1) ≤ x ∧\n    (x : ℝ) ^ (1 - ε^2) ≤ 2^(Nat.choose (d ε + 1) 2 + 1) * ∏ i, Z i ^ (i.val + 1) ∧\n    (Nat.Coprime (c 0) (c 1)) ∧ (Nat.Coprime (c 1) (c 2)) ∧ (Nat.Coprime (c 0) (c 2)) ∧\n    (∀ i, 1 ≤ c i) ∧\n    (∀ i, (c i : ℝ) ≤ (x : ℝ) ^ ε)\n    := by\n  have h₁ := refinedCountTriplesStar_le_card_BUnion α β γ (d := d ε) x ε hε_pos hε rfl\n  simp_rw [BUnion, Finset.sup_eq_biUnion] at h₁\n  have h₂ := h₁.trans Finset.card_biUnion_le |>.trans (sum_le_card_mul_sup ..)\n  use (indexSet' α β γ (d ε) x ε).image fun ⟨u, v, w, c⟩ ↦\n    ⟨fun i ↦ 2 ^ u i, fun i ↦ 2 ^ v i, fun i ↦ 2 ^ w i, c⟩\n  simp only [Finset.sup_image, Finset.mem_image, Prod.mk.injEq, Prod.exists, Nat.cast_prod,\n    Nat.cast_pow, forall_exists_index, and_imp]\n  refine ⟨?_, ?_⟩\n  · calc\n      _ ≤ ((_ : ℕ) : ℝ) := Nat.cast_le.2 h₂\n      _ ≤ _ := by\n        push_cast\n        gcongr\n        · have := const_nonneg (ε := ε)\n          positivity\n        · exact card_indexSet'_le_pow ε α β γ (d ε) x rfl h2X hε_pos hε\n        rfl\n  rintro X Y Z _ u v w c huvwc rfl rfl rfl rfl\n  simp only [Nat.cast_pow, Nat.cast_ofNat]\n  revert huvwc\n  simp only [indexSet', Finset.mem_filter, Finset.mem_product, Fintype.mem_piFinset, Finset.mem_Icc,\n    zero_le, true_and, Finset.mem_Ioc, and_imp]\n  rintro _ _ _ hc _ _ _ _ _ _ _ _ _ _ _ _ _\n  refine ⟨by assumption, by assumption, by assumption, by assumption, by assumption, by assumption,\n    by assumption, by assumption, by assumption, by assumption, by assumption, by assumption,\n    by assumption, ?_, ?_⟩\n  · intro i\n    apply Nat.succ_le_of_lt\n    apply hc i |>.1\n  · intro i\n    calc\n      (c i : ℝ) ≤ (⌊(x:ℝ) ^ (ε / 4)⌋₊ : ℝ) := by\n        norm_cast\n        apply (hc i).2\n      _ ≤ (x : ℝ)^(ε/4) := by\n        apply Nat.floor_le\n        positivity\n      _ ≤  _ := by\n        gcongr\n        · norm_cast; omega\n        · linarith\n","line":1561,"end_line":1625,"proof_status":"formalized"}
+{"id":"lem:aibiciconstraints","label":"aibiciconstraints","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["prop:DiophantineReduction","def:aibici"],"nlp_text":"\\begin{equation} \\label{eq:oat1} \\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 d} ic_i \\leq 1. \\end{equation}","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/baseAssumptions","latex_link":"sect0004.html#lem:aibiciconstraints","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/baseAssumptions","lean_decl_name":"baseAssumptions","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section4.lean","lean4_code":"/-- A structure containing the basic assumptions for each vector. -/\nstructure baseAssumptions (d : ℕ) (a : ℕ → ℝ) : Prop where\n(nonneg : ∀ i, 0 ≤ a i)\n(zero : a 0 = 0)\n(sum_bound : ∑ i ≤ d, i * a i ≤ 1)\n","line":27,"end_line":31,"proof_status":"formalized"}
+{"id":"lem:ThreeLowerBounds","label":"ThreeLowerBounds","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["lem:aibiciconstraints"],"nlp_text":"We have \\begin{equation} \\label{eq:black1} a_3 \\ge \\frac{1}{3}-4\\delta _a - 3a_1 - 2a_2, \\end{equation}","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/bound_4_point_19_first","latex_link":"sect0004.html#lem:ThreeLowerBounds","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/bound_4_point_19_first","lean_decl_name":"bound_4_point_19_first","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section4.lean","lean4_code":"lemma bound_4_point_19_first (hd : 3 ≤ d) :\n    1 / 3 - 4 * δ_ d a - 3 * a 1 - 2 * a 2 ≤ a 3 := by\n  have hd' : 4 ≤ d + 1 := by omega\n  have h₁ : ∑ i ∈ Finset.Icc 4 d, a i ≤ ∑ i ∈ Finset.Icc 4 d, (i - 3) * a i := by\n    gcongr ∑ _ ∈ _, ?_ with i hi\n    simp only [Finset.mem_Icc] at hi\n    have : (4 : ℝ) ≤ i := by simp_all\n    have ha' := ha.nonneg i\n    linear_combination a i * this\n  have h₂ : ∑ i ∈ Finset.Icc 4 d, (i - 3) * a i ≤ 2 * a 1 + a 2 + 3 * δ_ d a :=\n    bound_4_point_11_4 ha hd\n  have h₃ : a 1 + a 2 + a 3 + ∑ i ∈ Finset.Icc 4 d, a i = 1 / 3 - δ_ d a := by\n    rw [← sum_eq_δ_, sum_Icc_eq_sum_range_add_sum_Icc hd']\n    simp [Finset.sum_range, Fin.sum_univ_four, ha.zero]\n  linear_combination h₁ + h₂ - h₃\n","line":634,"end_line":648,"proof_status":"formalized"}
+{"id":"thm:BdExceptionalBound","label":"BdExceptionalBound","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["thm:ABCExponentNu"],"nlp_text":"There exists \\(\\varepsilon {\\gt}0\\) such that for all \\(\\mathbf{c}\\in \\mathbb {Z}^3\\) and \\(\\mathbf{X},\\mathbf{Y},\\mathbf{Z}\\in \\mathbb {R}_{{\\gt}0}^{d}\\). we have \\begin{align} B_d({\\bf c, X,Y,Z}) \\ll X^{0.66-\\varepsilon }. \\end{align}","doc_url":null,"latex_link":"sect0002.html#thm:BdExceptionalBound","lean_link":null,"lean_decl_name":"BdExceptionalBound","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"thm:ABCExceptionalBound","label":"ABCExceptionalBound","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["thm:BdExceptionalBound","lem:NlambdatoSstar"],"nlp_text":"Let \\(\\lambda \\in (0, 1.001)\\) be fixed. Then \\(N_\\lambda (X)=O( X^{33/50})\\).","doc_url":null,"latex_link":"sect0001.html#thm:ABCExceptionalBound","lean_link":null,"lean_decl_name":"ABCExceptionalBound","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"thm:HeathBrownGeometry","label":"HeathBrownGeometry","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(\\gcd (a_1,a_2,a_3)=1\\). Then for \\(X_1,X_2,X_3{\\gt}1\\), we have \\begin{align*} \\# \\left((x_1,x_2,x_3)\\in \\mathbb {Z}^3 \\; :\\; \\gcd (x_1,x_2,x_3)=1,\\, |x_i| \\le X_i ,\\, a_1x_1+a_2x_2+a_3x_3=0\\right) \\ll \\ 1 \\ + \\ & \\frac{X_1X_2X_3}{\\max _i\\{ |a_i|X_i\\} }. \\end{align*}","doc_url":null,"latex_link":"sect0003.html#thm:HeathBrownGeometry","lean_link":null,"lean_decl_name":"HeathBrownGeometry","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"prop:GeometryofNumbers","label":"GeometryofNumbers","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["thm:HeathBrownGeometry"],"nlp_text":"Let \\(d\\geq 1\\) and \\(\\varepsilon {\\gt}0\\) be fixed, and let \\[ X_1,\\ldots , X_d, Y_1,\\ldots , Y_d,Z_1,\\ldots , Z_d\\geq 1. \\]","doc_url":null,"latex_link":"sect0003.html#prop:GeometryofNumbers","lean_link":null,"lean_decl_name":"GeometryofNumbers","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:deBruijnRadical","label":"deBruijnRadical","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For any \\(\\varepsilon {\\gt}0\\), we have \\begin{equation} \\label{eq:bruijn} \\# \\left\\{ n\\leq x : \\operatorname{rad}(n)\\leq x^\\lambda \\right\\} \\ll _\\varepsilon x^{\\lambda +\\varepsilon }. \\end{equation}","doc_url":null,"latex_link":"sect0001.html#lem:deBruijnRadical","lean_link":null,"lean_decl_name":"deBruijnRadical","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:TrivialBdforS","label":"TrivialBdforS","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["lem:deBruijnRadical"],"nlp_text":"\\begin{equation} \\label{eq:trivial} S_{\\alpha ,\\beta ,\\gamma }(X)\\ll _\\varepsilon X^{\\min \\{ \\alpha +\\beta , \\alpha +\\gamma , \\beta +\\gamma \\} +\\varepsilon }, \\end{equation}","doc_url":null,"latex_link":"sect0001.html#lem:TrivialBdforS","lean_link":null,"lean_decl_name":"TrivialBdforS","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"prop:ABCTrivialBound","label":"ABCTrivialBound","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["lem:deBruijnRadical","def:Nlambda"],"nlp_text":"Let \\(\\lambda {\\gt}0\\). Then \\(N_\\lambda (X)=O_\\varepsilon ( X^{2\\lambda /3+\\varepsilon })\\), for any \\(\\varepsilon {\\gt}0\\).","doc_url":null,"latex_link":"sect0001.html#prop:ABCTrivialBound","lean_link":null,"lean_decl_name":"ABCTrivialBound","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"prop:TrivialBoundforSstar","label":"TrivialBoundforSstar","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["lem:TrivialBdforS","def:Sstar"],"nlp_text":"Let \\(\\alpha ,\\beta ,\\gamma {\\gt}0\\), and let \\(\\varepsilon {\\gt}0\\) be fixed. Then \\[ S^*_{\\alpha ,\\beta ,\\gamma }(X)\\ll X^{0.66-\\varepsilon ^2/2}, \\]","doc_url":null,"latex_link":"sect0004.html#prop:TrivialBoundforSstar","lean_link":null,"lean_decl_name":"TrivialBoundforSstar","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:Trivialaibici","label":"Trivialaibici","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["prop:TrivialBoundforSstar","def:aibici"],"nlp_text":"We have \\begin{equation} \\label{eq:oat4} \\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-\\varepsilon ^2,\\quad \\sum _{i\\leq d} (b_i+c_i) \\geq 0.66-\\varepsilon ^2 \\end{equation}","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/Bound4Point3","latex_link":"sect0004.html#lem:Trivialaibici","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/Bound4Point3","lean_decl_name":"Bound4Point3","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section4.lean","lean4_code":"/--\nA statement of equation 4.3 from the paper, formulated for `a, b`. We explicitly take `a, b` as\ninputs to simplify symmetry arguments in the conclusion.\n-/\ndef Bound4Point3 (d : ℕ) (ε : ℝ) (a b : ℕ → ℝ) : Prop :=\n  0.66 - ε ^ 2 ≤ ∑ i ≤ d, (a i + b i)\n","line":39,"end_line":44,"proof_status":"formalized"}
+{"id":"lem:MixedDetMethods","label":"MixedDetMethods","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["thm:BombieriPilaforDet","thm:HeathBrownpadicDet","lem:EisensteinCrit"],"nlp_text":"Let \\(\\varepsilon {\\gt}0\\) and \\(D\\geq 1\\) and assume that \\(p,q,r\\in [1,D]\\) are integers. Then \\[ N(X,Y,Z)\\ll _{\\varepsilon ,D} Z \\min \\left(X^{\\frac{1}{q}}, Y^{\\frac{1}{p}} \\right) (XY)^\\varepsilon , \\]","doc_url":null,"latex_link":"sect0003.html#lem:MixedDetMethods","lean_link":null,"lean_decl_name":"MixedDetMethods","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"prop:DeterminantMethod","label":"DeterminantMethod","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["lem:MixedDetMethods"],"nlp_text":"Let \\(d\\geq 1\\), and let \\[ X_1,\\ldots , X_d, Y_1,\\ldots , Y_d,Z_1,\\ldots , Z_d\\geq 1. \\]","doc_url":null,"latex_link":"sect0003.html#prop:DeterminantMethod","lean_link":null,"lean_decl_name":"DeterminantMethod","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"prop:Eqn4Point5","label":"Eqn4Point5","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["lem:Trivialaibici"],"nlp_text":"\\begin{equation} \\label{eq:oat5} 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 - \\frac{1}{2}\\varepsilon , \\end{equation}","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/Bound4Point5","latex_link":"sect0004.html#prop:Eqn4Point5","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/Bound4Point5","lean_decl_name":"Bound4Point5","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section4.lean","lean4_code":"/--\nA statement of equation 4.5 from the paper, formulated for `a`.\n\nWe will at some point show that 4.5 can be safely assumed in context, after we've assumed\n1.2 and 4.4\n-/\nstructure Bound4Point5 (d : ℕ) (δ ε : ℝ) (a : ℕ → ℝ) : Prop where\n(lower : 0.32 - δ ≤ ∑ i ≤ d, a i)\n(upper : ∑ i ≤ d, a i ≤ 0.34 + δ - ε / 2)\n","line":80,"end_line":88,"proof_status":"statement_formalized"}
+{"id":"thm:BombieriPilaforDet","label":"BombieriPilaforDet","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(f(x)\\) be a \\(C^\\infty \\) function on a closed subinterval of \\([0,N]\\), and suppose that\\(F (x, f ) = 0\\), where \\(F(x, y) \\in \\mathbb {R}[x, y]\\) is absolutely irreducible of degree \\(d \\ge 2\\). Suppose that \\(|f'(x)| \\le 1\\). Then \\begin{align*} \\# \\{ (x,f(x))\\in \\{ 1,\\ldots ,N\\} ^2 \\} \\ll _d (\\log N)^{O(N)} N^{1/d} \\end{align*}","doc_url":null,"latex_link":"sect0003.html#thm:BombieriPilaforDet","lean_link":null,"lean_decl_name":"BombieriPilaforDet","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"thm:HeathBrownpadicDet","label":"HeathBrownpadicDet","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(F\\in \\mathbb {Z}[x_1 ,\\ldots , x_n]\\) be an absolutely irreducible polynomial of degree \\(d\\), and let \\(\\varepsilon {\\gt} 0\\) and \\(B_1,\\ldots ,B_n \\ge 1\\) be given. Define \\begin{align*} B = \\max _{(e_1,\\ldots ,e_n)}\\Big( \\prod _{i\\le n} B_i^{e_i} \\Big) \\end{align*}","doc_url":null,"latex_link":"sect0003.html#thm:HeathBrownpadicDet","lean_link":null,"lean_decl_name":"HeathBrownpadicDet","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:aibici","label":"aibici","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Define \\(a_i,b_i,c_i\\in \\mathbb {R}_{\\geq 0}\\) via \\[ X_i=X^{a_i}, \\quad Y_i=X^{b_i}, \\quad Z_i=X^{c_i}, \\]","doc_url":null,"latex_link":"sect0004.html#def:aibici","lean_link":null,"lean_decl_name":"aibici","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:s","label":"s","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["def:aibici"],"nlp_text":"Define \\(s_i := a_i + b_i + c_i\\).","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/S","latex_link":"sect0004.html#def:s","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/S","lean_decl_name":"S","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section4.lean","lean4_code":"/-- We define `S i` to be `a i + b i + c i`. -/\ndef S (a b c : ℕ → ℝ) (i : ℕ) := a i + b i + c i\n","line":250,"end_line":251,"proof_status":"formalized"}
+{"id":"thm:QuadraticFormsEstimate","label":"QuadraticFormsEstimate","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["lem:DivisorBoundQuadField"],"nlp_text":"Given integers \\(n,a_1,a_2\\neq 0\\), there are at most \\(O_{\\varepsilon ,D}(|na_1a_2 X_1 X_2|^{\\varepsilon })\\) many solutions \\((x_1,x_2)\\in \\mathbb {Z}^2\\) such that \\(|x_i|\\le X_i\\) and \\begin{align*} n = a_1 x_1^2 + a_2 x_2^2. \\end{align*}","doc_url":null,"latex_link":"sect0003.html#thm:QuadraticFormsEstimate","lean_link":null,"lean_decl_name":"QuadraticFormsEstimate","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:BdDiophantineCount","label":"BdDiophantineCount","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For \\(\\mathbf{c}\\in \\mathbb {Z}^3\\) and \\(\\mathbf{X},\\mathbf{Y},\\mathbf{Z}\\in \\mathbb {R}_{{\\gt}0}^{d}\\). we have \\begin{equation} \\label{eq:Bk} B_d(\\mathbf{c},\\mathbf{X},\\mathbf{Y},\\mathbf{Z}) := \\# \\left\\{ (\\mathbf{x},\\mathbf{y},\\mathbf{z})\\in \\mathbb {N}^{3d}\\; :\\; \\begin{array}{l} x_i\\sim X_i,\\, y_i\\sim Y_i,\\, z_i\\sim Z_i \\\\ c_1\\prod _{j\\leq d}x_j^j+c_2\\prod _{j\\leq d}y_j^j=c_3\\prod _{j\\leq d}z_j^j\\\\ \\gcd (c_1\\prod _{j\\leq d}x_j,c_2\\prod _{j\\leq d}y_j , c_3\\prod _{j\\leq d}z_j)=1 \\end{array} \\right\\} . \\end{equation}","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/B","latex_link":"sect0002.html#def:BdDiophantineCount","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/B","lean_decl_name":"B","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section2.lean","lean4_code":"/-- Definition 2.4 -/\nnoncomputable def B (d : ℕ) (c : Fin 3 → ℕ) (X Y Z : Fin d → ℕ) : ℕ := (B_finset d c X Y Z).card\n","line":534,"end_line":535,"proof_status":"formalized"}
+{"id":"prop:FourierAnalysis","label":"FourierAnalysis","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["def:BdDiophantineCount"],"nlp_text":"Let \\(d\\geq 1\\), \\(\\varepsilon {\\gt}0\\) and \\(A\\geq 1\\) be fixed. Let \\[ X_1,\\ldots , X_d, Y_1,\\ldots , Y_d,Z_1,\\ldots , Z_d\\geq 1 \\]","doc_url":null,"latex_link":"sect0003.html#prop:FourierAnalysis","lean_link":null,"lean_decl_name":"FourierAnalysis","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:NlambdatoSstar","label":"NlambdatoSstar","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":["def:Sstar","def:Nlambda"],"nlp_text":"We have \\begin{equation} \\label{eq:step1} N_\\lambda (X)\\ll (\\log X)^4 \\max _{\\substack { \\alpha ,\\beta ,\\gamma {\\gt}0\\\\ \\alpha +\\beta +\\gamma \\leq \\lambda }} S^*_{\\alpha ,\\beta ,\\gamma }(X). \\end{equation}","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/countTriples_isBigO_dyadicSup","latex_link":"sect0002.html#lem:NlambdatoSstar","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/countTriples_isBigO_dyadicSup","lean_decl_name":"countTriples_isBigO_dyadicSup","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section2.lean","lean4_code":"theorem countTriples_isBigO_dyadicSup (ε : ℝ) :\n    (fun X ↦ (countTriples ε X : ℝ)) =O[atTop] (fun X ↦ (Real.log X) ^ 4 * dyadicSupBound ε X) := by\n  trans fun X ↦ (Nat.log 2 X+1:ℝ)^4 * dyadicSupBound ε X\n  · apply IsBigO.of_norm_le\n    simp only [Real.norm_natCast, Prod.forall]\n    exact_mod_cast fun b ↦ countTriples_le_log_pow_mul_sup _ _\n  · apply IsBigO.mul _ (isBigO_refl ..)\n    apply IsBigO.pow\n    apply IsBigO.add\n    · exact Nat.log_isBigO_log 2\n    apply IsLittleO.isBigO\n    apply Real.isLittleO_const_log_atTop.natCast_atTop\n","line":486,"end_line":497,"proof_status":"formalized"}
+{"id":"lem:EisensteinCrit","label":"EisensteinCrit","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(r\\geq 1\\) and let \\(g\\in \\mathbb {C}[x]\\) be a polynomial which has at least one root of multiplicity \\(1\\). Then the polynomial \\(g(x)-y^r\\) is absolutely irreducible.","doc_url":null,"latex_link":"sect0003.html#lem:EisensteinCrit","lean_link":null,"lean_decl_name":"EisensteinCrit","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:DivisorBoundQuadField","label":"DivisorBoundQuadField","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(\\varepsilon {\\gt}0\\). Let \\(D\\ge 1\\) be a squarefree integer, and set \\(K=\\mathbb {Q}(\\sqrt{-D})\\). Then for all \\(\\alpha \\in K\\), the number of ideals dividing \\((\\alpha )\\) in \\(K\\) is \\(O(N_\\alpha ^\\varepsilon )\\).","doc_url":null,"latex_link":"sect0003.html#lem:DivisorBoundQuadField","lean_link":null,"lean_decl_name":"DivisorBoundQuadField","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:Sstar","label":"Sstar","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(S^*_{\\alpha ,\\beta ,\\gamma }(X)\\) to be the number of \\((a,b,c)\\in \\mathbb {N}^3\\) with \\(\\gcd (a,b,c)=1\\) and \\begin{align*} c\\in [X/2,X],\\quad a+b=c,\\quad \\operatorname{rad}(a)\\sim X^{\\alpha },\\quad \\operatorname{rad}(b)\\sim X^{\\beta },\\quad \\operatorname{rad}(c)\\sim X^{\\gamma }. \\end{align*}","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/refinedCountTriplesStar","latex_link":"sect0002.html#def:Sstar","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/refinedCountTriplesStar","lean_decl_name":"refinedCountTriplesStar","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section2.lean","lean4_code":"/--\nThis is $$S^*_{α,β,γ}(X)$$ in the paper and blueprint.\n-/\nnoncomputable def refinedCountTriplesStar (α β γ : ℝ) (X : ℕ) : ℕ := #(dyadicPoints α β γ X)\n","line":333,"end_line":336,"proof_status":"formalized"}
+{"id":"def:Nlambda","label":"Nlambda","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For \\(\\lambda {\\gt}0\\) define \\(N_\\lambda (X)\\) as the number of triples \\((a,b,c)\\in \\mathbb {N}^3\\) with \\(a+b=c\\), \\(\\gcd (a,b,c)=1\\) and \\(\\operatorname{rad}(abc) {\\lt} c^{\\lambda }\\).","doc_url":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/countTriples","latex_link":"sect0001.html#def:Nlambda","lean_link":"https://b-mehta.github.io/ABC-Exceptions/docs/find/#doc/countTriples","lean_decl_name":"countTriples","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ABC-Exceptions","source_path":"ABCExceptions/Section2.lean","lean4_code":"/--\nThe number of exceptions to the abc conjecture for a given `ε` which are bounded by `X`.\n-/\nnoncomputable def countTriples (ε : ℝ) (X : ℕ) : ℕ := #(abcExceptionsBelow ε X)\n","line":69,"end_line":72,"proof_status":"formalized"}
+{"id":"def:Sabc","label":"Sabc","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://b-mehta.github.io/ABC-Exceptions/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For \\(\\alpha ,\\beta ,\\gamma {\\gt}0\\), define \\(S_{\\alpha ,\\beta ,\\gamma }(X)\\) as the number of \\((a,b,c)\\in \\mathbb {N}^3\\) with \\(\\gcd (a,b,c)=1\\) such that \\[ a,b,c\\in [1,X], \\quad a+b=c, \\quad \\operatorname{rad}(a)\\leq X^\\alpha , \\quad \\operatorname{rad}(b)\\leq X^\\beta , \\quad \\operatorname{rad}(c)\\leq X^\\gamma . \\]","doc_url":null,"latex_link":"sect0001.html#def:Sabc","lean_link":null,"lean_decl_name":"Sabc","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}

crawled_repos/BlueprintRep.jsonl

@@ -0,0 +1,33 @@
+{"id":"sectPuQu","label":"sectPuQu","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["Pu","Qu"],"nlp_text":"Pour un même YoungTableau, l’intersection de \\(P_{\\mu }\\) et \\(Q_{\\mu }\\) est \\({1}\\)","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/sectPuQu","latex_link":"sect0002.html#sectPuQu","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/sectPuQu","lean_decl_name":"sectPuQu","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"lemma sectPuQu {μ : YoungDiagram} (Yᵤ : YoungTableau μ):\n  (Pu Yᵤ).carrier ∩ (Qu Yᵤ).carrier = {↑1} := by\n  rw[Set.eq_singleton_iff_unique_mem, Set.mem_inter_iff]\n  constructor\n  exact ⟨Subgroup.one_mem (Pu Yᵤ), Subgroup.one_mem (Qu Yᵤ)⟩\n  simp\n  intro f p q\n  rw[Equiv.Perm.ext_iff]\n  intro x\n  obtain ⟨j, hj, a⟩ := preImYu Yᵤ x\n  obtain ⟨k, hk, b⟩ := preImYu Yᵤ (f (Yᵤ.entry j))\n  rw[Eq.comm] at hk\n  rw[Pu] at p\n  rw[Qu] at q\n  have h : (j: μ) = (k : μ) := by\n    ext\n    exact q hk\n    exact p hk\n  rw[← h] at hk\n  rw[Equiv.Perm.coe_one, id_eq, ← hj, hk]\n","line":94,"end_line":113,"proof_status":"formalized"}
+{"id":"LinearTransformation","label":"LinearTransformation","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["sectPuQu","YoungSymmetriser","gInPuQu"],"nlp_text":"\\(\\exists l_{\\mu }\\) une fonction linéaire tq\\(\\forall x \\in \\mathbb {C}[S_{n}]\\), \\(a_{\\mu } x b_{\\mu } = l_{\\mu }(x) c_{\\mu }\\)","doc_url":null,"latex_link":"sect0003.html#LinearTransformation","lean_link":null,"lean_decl_name":"LinearTransformation","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"./BlueprintRep/SpechtModules.lean","lean4_code":"lemma LinearTransformation {μ : YoungDiagram} (Yᵤ : YoungTableau μ) :\n  ∀x : (PermComplex μ), ∃ y : (PermComplex μ), (au Yᵤ) * x * (bu Yᵤ) = y * (cu Yᵤ) := by\n  sorry --TODO\n-/\n\n--SmallerImpZero TODO\n\n/-! Meilleure solution trouvé au problème suivant.\nLean ne me laisse pas multiplier un MonoidAlgebra.single avec un MonoidAlgebra -/\n","line":45,"end_line":53,"proof_status":"statement_formalized"}
+{"id":"CuPropIdempotent","label":"CuPropIdempotent","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["LinearTransformation"],"nlp_text":"\\(c_{\\mu }\\) est proportionel à un idempotent. De façon mathématique \\[ \\exists c \\in \\mathbb {C}, c_{\\mu }^{2} = c \\cdot c_{\\mu } \\]","doc_url":null,"latex_link":"sect0003.html#CuPropIdempotent","lean_link":null,"lean_decl_name":"CuPropIdempotent","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"./BlueprintRep/SpechtModules.lean","lean4_code":"lemma CuPropIdempotent {μ : YoungDiagram} (Yᵤ : YoungTableau μ) :\n  ∃ c : ℂ, (cu Yᵤ) * (cu Yᵤ) = (ComplexToComplexPerm μ c) * (cu Yᵤ) := by\n  sorry --TODO\n-/\n\n","line":58,"end_line":62,"proof_status":"statement_formalized"}
+{"id":"IrreductibleRepresentationSn","label":"IrreductibleRepresentationSn","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["CuPropIdempotent","SpechtModules","SmallerImpZero","HomAeMisoeM"],"nlp_text":"\\(\\forall \\mu \\) partition de n, \\(V_{\\mu }\\) est toute les représentations irréductibles de \\(S_n\\)","doc_url":null,"latex_link":"sect0003.html#IrreductibleRepresentationSn","lean_link":null,"lean_decl_name":"IrreductibleRepresentationSn","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"au","label":"au","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["PuCard"],"nlp_text":"\\[ a_{\\mu } := \\frac{1}{|P_{\\mu }|}\\sum _{g \\in P_{\\mu }}g \\]","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/au","latex_link":"sect0003.html#au","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/au","lean_decl_name":"au","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/SpechtModules.lean","lean4_code":"noncomputable def au {μ : YoungDiagram} (Yᵤ : YoungTableau μ) : PermComplex μ :=\n  Finset.sum ((Pu Yᵤ).carrier.toFinset) (fun σ => MonoidAlgebra.single σ ((↑1 / ↑(PuCard Yᵤ)) : ℂ))\n","line":31,"end_line":32,"proof_status":"formalized"}
+{"id":"YoungSymmetriser","label":"YoungSymmetriser","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["au","bu"],"nlp_text":"Un Young symmetriser est défini par un YoungDiagram \\(\\mu \\)\\(c_{\\mu } := a_{\\mu } b_{\\mu } \\)","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/cu","latex_link":"sect0003.html#YoungSymmetriser","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/cu","lean_decl_name":"cu","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/SpechtModules.lean","lean4_code":"noncomputable def cu {μ : YoungDiagram} (Yᵤ :  YoungTableau μ) : PermComplex μ :=\n  (au Yᵤ) * (bu Yᵤ)\n","line":39,"end_line":40,"proof_status":"formalized"}
+{"id":"SpechtModules","label":"SpechtModules","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["YoungSymmetriser"],"nlp_text":"Soit \\(\\mu \\) un YoungDiagram. \\[ V_{\\mu } := \\mathbb {C}[S_{n}]c_{\\mu } \\]","doc_url":null,"latex_link":"sect0003.html#SpechtModules","lean_link":null,"lean_decl_name":"SpechtModules","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"SmallerImpZero","label":"SmallerImpZero","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["YoungSymmetriser","IneqYoungDiagram"],"nlp_text":"Si \\(\\mu {\\gt} \\lambda \\), alors \\[ a_{\\mu } \\mathbb {C}[S_{n}] b_{\\lambda } = 0 \\]","doc_url":null,"latex_link":"sect0003.html#SmallerImpZero","lean_link":null,"lean_decl_name":"SmallerImpZero","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"QuCard","label":"QuCard","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["Qu"],"nlp_text":"Le nombre d’élément de \\(Q_{\\mu }\\) est fini.","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/QuCard","latex_link":"sect0002.html#QuCard","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/QuCard","lean_decl_name":"QuCard","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"noncomputable def QuCard {μ : YoungDiagram} (Yᵤ : YoungTableau μ) :=\n  Fintype.card ↥(Qu Yᵤ)\n","line":90,"end_line":91,"proof_status":"formalized"}
+{"id":"bu","label":"bu","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["QuCard"],"nlp_text":"\\[ b_{\\mu } := \\frac{1}{|Q_{\\mu }|}\\sum _{g \\in Q_{\\mu }}(-1)^{g}g \\]","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/bu","latex_link":"sect0003.html#bu","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/bu","lean_decl_name":"bu","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/SpechtModules.lean","lean4_code":"noncomputable def bu {μ : YoungDiagram} (Yᵤ : YoungTableau μ) : PermComplex μ :=\n  Finset.sum ((Qu Yᵤ).carrier.toFinset) (fun σ => MonoidAlgebra.single σ ((↑(Equiv.Perm.sign σ)  / ↑(PuCard Yᵤ)) : ℂ))\n","line":35,"end_line":36,"proof_status":"formalized"}
+{"id":"qu","label":"qu","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["staysInY"],"nlp_text":"Soit g une permutation rowToCol,\\(q_{\\mu }\\) est une permutation de [0, n-1] défini comme \\[ q_{\\mu }(m)= Y_{\\mu }((Y_{\\mu }^{-1}(m)).x,(Y_{\\mu }^{-1} \\circ g(m)).y) \\]","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/qu","latex_link":"sect0002.html#qu","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/qu","lean_decl_name":"qu","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"def qu {μ : YoungDiagram} {Yᵤ : YoungTableau μ} {g : (Equiv.Perm (Fin μ.card))} (rtc : rowToCol Yᵤ g) (n : Fin μ.card) :=\n  Yᵤ.entry ⟨((YuInv Yᵤ n).val.fst, (YuInv Yᵤ (g n)).val.snd), staysInY rtc n⟩\n","line":140,"end_line":141,"proof_status":"formalized"}
+{"id":"quInQu","label":"quInQu","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["qu"],"nlp_text":"\\[ q_{\\mu } \\in Q_{\\mu } \\]","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/quPermInQu","latex_link":"sect0002.html#quInQu","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/quPermInQu","lean_decl_name":"quPermInQu","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"lemma quPermInQu {μ : YoungDiagram} {Yᵤ : YoungTableau μ} {g : (Equiv.Perm (Fin μ.card))} (rtc : rowToCol Yᵤ g) :\n  (quPerm rtc) ∈ (Qu Yᵤ) := by\n  rw[quPerm, Qu]\n  simp only [Subtype.forall, Prod.forall, Subgroup.mem_mk, Set.mem_setOf_eq, Equiv.ofBijective_apply]\n  intros ix iy _ jx jy _ h\n  rw[qu, Function.Injective.eq_iff (injYu Yᵤ)] at h\n  simp only [YuInvYu, Subtype.mk.injEq, Prod.mk.injEq] at h\n  obtain ⟨h, _⟩ := h\n  exact h\n","line":175,"end_line":183,"proof_status":"formalized"}
+{"id":"quInv","label":"quInv","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["qu"],"nlp_text":"\\(q_{\\mu }^{-1}\\) est la fonction inverse de \\(q_{\\mu }\\)","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/quInv","latex_link":"sect0002.html#quInv","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/quInv","lean_decl_name":"quInv","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"def quInv {μ : YoungDiagram} {Yᵤ : YoungTableau μ} {g : (Equiv.Perm (Fin μ.card))} (rtc : rowToCol Yᵤ g) :=\n  Fintype.bijInv (bijqu rtc)\n","line":186,"end_line":187,"proof_status":"formalized"}
+{"id":"gInPuQu","label":"gInPuQu","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["quInQu","puInPu","PuQu"],"nlp_text":"Soit g une permutation rowToCol \\[ g \\in P_{\\mu }Q_{\\mu } \\]","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/gInPuQu","latex_link":"sect0002.html#gInPuQu","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/gInPuQu","lean_decl_name":"gInPuQu","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"lemma gInPuQu {μ : YoungDiagram} {Yᵤ : YoungTableau μ} {g : (Equiv.Perm (Fin μ.card))} (rtc : rowToCol Yᵤ g) :\n  g ∈ (PuQu Yᵤ) := by\n  rw[PuQu, Set.mem_setOf_eq]\n  exists (puPerm rtc)\n  constructor\n  exact puPermInPu rtc\n  exists (quPerm rtc)\n  constructor\n  exact quPermInQu rtc\n  rw[puPerm,quPerm, ← puqug]\n  rfl\n  exact rtc\n","line":261,"end_line":272,"proof_status":"formalized"}
+{"id":"staysInX","label":"staysInX","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["quInv"],"nlp_text":"Soit g une permutation rowToCol, \\[ \\forall m \\in [0,n-1], ((Y_{\\mu }^{-1} \\circ g \\circ q_{\\mu }^{-1}(m)).x, (Y_{\\mu }^{-1}(m)).y) \\in \\mu \\]","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/staysInX","latex_link":"sect0002.html#staysInX","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/staysInX","lean_decl_name":"staysInX","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"lemma staysInX {μ : YoungDiagram} {Yᵤ : YoungTableau μ} {g : (Equiv.Perm (Fin μ.card))} (rtc : rowToCol Yᵤ g) (n : Fin μ.card) :\n  ((YuInv Yᵤ (g (quInv rtc n))).val.fst,(YuInv Yᵤ n).val.snd ) ∈ μ := by\n  have h : ∀ (n' : Fin μ.card), ∃! m', (qu rtc) m' = n' := by\n    rw[← Function.bijective_iff_existsUnique _]\n    exact (bijqu rtc)\n  obtain ⟨m, hm, _⟩ := h n\n  obtain ⟨i, hi, _⟩ := preImYu Yᵤ m\n  obtain ⟨j, hj, _⟩ := preImYu Yᵤ (g (Yᵤ.entry i))\n  rw[← hm, ← hi]\n  simp only [quInvqu]\n  rw[← hj, qu]\n  simp only [YuInvYu, ← hj, Prod.mk.eta, SetLike.coe_mem]\n","line":195,"end_line":206,"proof_status":"formalized"}
+{"id":"pu","label":"pu","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["staysInX"],"nlp_text":"Soit g une permutation rowToCol,\\(p_{\\mu }\\) est une permutation de [0, n-1] défini comme \\[ p_{\\mu }(m)= Y_{\\mu }((Y_{\\mu }^{-1} \\circ g \\circ q^{-1}(m)).x,(Y_{\\mu }^{-1}(m)).y) \\]","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/pu","latex_link":"sect0002.html#pu","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/pu","lean_decl_name":"pu","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"def pu {μ : YoungDiagram} {Yᵤ : YoungTableau μ} {g : (Equiv.Perm (Fin μ.card))} (rtc : rowToCol Yᵤ g) (n : Fin μ.card) :=\n  Yᵤ.entry ⟨((YuInv Yᵤ (g (quInv rtc n))).val.fst,(YuInv Yᵤ n).val.snd ), staysInX rtc n⟩\n","line":209,"end_line":210,"proof_status":"formalized"}
+{"id":"HomAeM","label":"HomAeM","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["Ae"],"nlp_text":"Soit A un anneau unitaire, M un A-module et e un idempotant de A. \\[ \\forall \\phi \\in \\text{Hom}_{A}(Ae,M), \\exists m \\in e M, \\phi (e)=m \\]","doc_url":null,"latex_link":"sect0001.html#HomAeM","lean_link":null,"lean_decl_name":"HomAeM","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"HomAeMisoeM","label":"HomAeMisoeM","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["HomAeM"],"nlp_text":"Soit A un anneau unitaire, M un A-module et e un idempotant de A. \\[ \\text{Hom}_{A}(A e,M) \\cong e M \\]","doc_url":null,"latex_link":"sect0001.html#HomAeMisoeM","lean_link":null,"lean_decl_name":"HomAeMisoeM","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"staysInY","label":"staysInY","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["YuInv","rowToCol"],"nlp_text":"Soit g une permutation rowToCol, \\[ \\forall m \\in [0,n-1], (Y_{\\mu }^{-1}(m).x,Y_{\\mu }^{-1}(g(m)).y) \\in \\mu \\]","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/staysInY","latex_link":"sect0002.html#staysInY","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/staysInY","lean_decl_name":"staysInY","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"lemma staysInY {μ : YoungDiagram} {Yᵤ : YoungTableau μ} {g : (Equiv.Perm (Fin μ.card))} (rtc : rowToCol Yᵤ g) (n : Fin μ.card) :\n  ((YuInv Yᵤ n).val.fst, ((YuInv Yᵤ) (g n)).val.snd) ∈ μ := by\n  sorry --TODO\n","line":135,"end_line":137,"proof_status":"statement_formalized"}
+{"id":"puqug","label":"puqug","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["pu"],"nlp_text":"Soit g une permutation rowToCol, \\[ g = p_{\\mu } \\circ q_{\\mu } \\]","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/puqug","latex_link":"sect0002.html#puqug","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/puqug","lean_decl_name":"puqug","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"lemma puqug {μ : YoungDiagram} {Yᵤ : YoungTableau μ} {g : (Equiv.Perm (Fin μ.card))} (rtc : rowToCol Yᵤ g) :\n  (pu rtc) ∘ (qu rtc) = g := by\n  ext n\n  obtain ⟨i, hi, _⟩ := preImYu Yᵤ n\n  obtain ⟨j, hj, _⟩ := preImYu Yᵤ (g (Yᵤ.entry i))\n  rw[← hi, ← hj, Function.comp_apply]\n  have hq : (qu rtc (Yᵤ.entry i)) = (qu rtc (Yᵤ.entry i)) := by rfl\n  nth_rewrite 2 [qu] at hq\n  simp only [YuInvYu, ← hj] at hq\n  simp only [pu, quInvqu, hq, YuInvYu]\n  have hq' : (quInv rtc) (qu rtc (Yᵤ.entry i)) = (quInv rtc) (qu rtc (Yᵤ.entry i)) := by rfl\n  nth_rewrite 1 [quInvqu, hq] at hq'\n  simp only[← hq', ← hj, YuInvYu]\n","line":213,"end_line":225,"proof_status":"formalized"}
+{"id":"bijpu","label":"bijpu","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["puqug"],"nlp_text":"Soit g une permutation rowToCol, \\(p_{\\mu }\\) est une bijection","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/bijpu","latex_link":"sect0002.html#bijpu","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/bijpu","lean_decl_name":"bijpu","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"lemma bijpu {μ : YoungDiagram} {Yᵤ : YoungTableau μ} {g : (Equiv.Perm (Fin μ.card))} (rtc : rowToCol Yᵤ g) :\n  Function.Bijective (pu rtc) := by\n  have h : g ∘ (quInv rtc) = g ∘ (quInv rtc) := by rfl\n  nth_rewrite 1 [← puqug rtc, Function.comp_assoc] at h\n  have h' : (qu rtc) ∘ (quInv rtc) = id := by\n    ext n\n    have j : ((quInv rtc) ((qu rtc) ((quInv rtc) n))) = ((quInv rtc) ((qu rtc) ((quInv rtc) n))) := by rfl\n    nth_rewrite 2 [quInvqu] at j\n    rw[Function.Injective.eq_iff] at j\n    rw[Function.comp_apply, id_eq, j]\n    rw[quInv]\n    exact (Fintype.bijective_bijInv (bijqu rtc)).injective\n  rw[h', Function.comp_id] at h\n  rw[h, quInv]\n  exact Function.Bijective.comp (Equiv.bijective g) (Fintype.bijective_bijInv (bijqu rtc))\n","line":228,"end_line":242,"proof_status":"formalized"}
+{"id":"puInPu","label":"puInPu","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["bijpu"],"nlp_text":"\\[ p_{\\mu } \\in P_{\\mu } \\]","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/puPermInPu","latex_link":"sect0002.html#puInPu","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/puPermInPu","lean_decl_name":"puPermInPu","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"lemma puPermInPu {μ : YoungDiagram} {Yᵤ : YoungTableau μ} {g : (Equiv.Perm (Fin μ.card))} (rtc : rowToCol Yᵤ g) :\n  (puPerm rtc) ∈ (Pu Yᵤ) := by\n  rw[puPerm, Pu]\n  simp only [Subtype.forall, Prod.forall, Subgroup.mem_mk, Set.mem_setOf_eq, Equiv.ofBijective_apply]\n  intros ix iy _ jx jy _ h\n  rw[pu, Function.Injective.eq_iff (injYu Yᵤ)] at h\n  simp only [YuInvYu, Subtype.mk.injEq, Prod.mk.injEq] at h\n  obtain ⟨_, h⟩ := h\n  exact h\n","line":250,"end_line":258,"proof_status":"formalized"}
+{"id":"YuInv","label":"YuInv","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["bijYu"],"nlp_text":"\\(Y_{\\mu }^{-1}\\) est une l’inverse de \\(Y_{\\mu }\\)","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/YuInv","latex_link":"sect0002.html#YuInv","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/YuInv","lean_decl_name":"YuInv","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"def YuInv {μ : YoungDiagram} (Yᵤ : YoungTableau μ) :=\n  Fintype.bijInv (bijYu Yᵤ)\n","line":126,"end_line":127,"proof_status":"formalized"}
+{"id":"IneqYoungDiagram","label":"IneqYoungDiagram","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soit \\(\\mu \\) et \\(\\lambda \\) deux YoungDiagram de même cardinalité.On dit que \\(\\mu {\\gt} \\lambda \\) si \\(\\exists i \\in \\mathbb {N}\\) tq \\(\\mu _{i}{\\gt}\\lambda _{i}\\) et \\(\\forall j \\in \\mathbb {N}_{{\\lt}i}\\), \\(\\mu _{j} = \\lambda _{j}\\).","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/IneqYoungDiagram","latex_link":"sect0002.html#IneqYoungDiagram","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/IneqYoungDiagram","lean_decl_name":"IneqYoungDiagram","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/SpechtModules.lean","lean4_code":"def IneqYoungDiagram (μ1 : YoungDiagram) (μ2 : YoungDiagram) :=\n  ∃ i : ℕ, (μ1.colLen i) > (μ2.colLen i) ∧ (∀ j : ℕ, (μ1.colLen j) = (μ2.colLen j))\n","line":8,"end_line":9,"proof_status":"formalized"}
+{"id":"Ae","label":"Ae","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soit \\(A\\) un anneau et \\(e \\in A\\) \\[ Ae := \\{ a' \\in A | \\exists a \\in A, a' = a \\cdot e \\} \\]","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/Ae","latex_link":"sect0001.html#Ae","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/Ae","lean_decl_name":"Ae","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Module.lean","lean4_code":"instance Ae' (A : Type) [Ring A] (e : A) : AddCommMonoid {a' : A | ∃ a : A, a'= a * e} where\n  add := fun a b => ⟨a.1 + b.1, by\n    obtain ⟨a', ha⟩ := a.2\n    obtain ⟨b', hb⟩ := b.2\n    use (a' + b')\n    rw [ha, hb, add_mul]⟩\n  zero := ⟨0, by use 0; rw[zero_mul]⟩\n  add_assoc := by\n    intros _ _ _\n    apply Subtype.ext\n    apply add_assoc\n  zero_add := by\n    intro _\n    apply Subtype.ext\n    apply zero_add\n  add_zero := by\n    intro _\n    apply Subtype.ext\n    apply add_zero\n  add_comm := by\n    intro _ _\n    apply Subtype.ext\n    apply add_comm\n  --Why do I need to do this?\n  nsmul := fun n a => ⟨n • a.1, by\n    obtain ⟨a', ha⟩ := a.2\n    use n • a'\n    rw [ha, smul_mul_assoc]⟩\n  nsmul_zero := by\n    intro a\n    apply Subtype.ext\n    simp [nsmul_zero]\n    rfl\n  nsmul_succ := by\n    intro n a\n    apply Subtype.ext\n    simp\n    rw[add_mul, one_mul]\n    rfl\n","line":4,"end_line":42,"proof_status":"formalized"}
+{"id":"Pu","label":"Pu","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["bijYu"],"nlp_text":"\\(P_{\\mu }\\) est un sous groupe de \\(S_{n}\\), défini de la façon suivante:Un élément de \\(P_{\\mu }\\) permute les entré du YoungDiagram s’ils sont sur la même rangé.","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/Pu","latex_link":"sect0002.html#Pu","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/Pu","lean_decl_name":"Pu","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"def Pu {μ : YoungDiagram} (Yᵤ : YoungTableau μ) : Subgroup (Equiv.Perm (Fin μ.card)) where\n  carrier := {x : (Equiv.Perm (Fin μ.card)) | ∀ {i j : μ}, (x (Yᵤ.entry i) = Yᵤ.entry j) → i.val.snd = j.val.snd}\n  mul_mem' := by\n    intros α β a b i j αβ\n    rw[Equiv.Perm.coe_mul, Function.comp_apply] at αβ\n    obtain ⟨k, hk, _⟩ := preImYu Yᵤ (β (Yᵤ.entry i))\n    rw[Eq.comm] at hk\n    rw[hk] at αβ\n    rw[b hk, a αβ]\n  one_mem' := by\n    intros i j h\n    rw[Equiv.Perm.coe_one, id_eq, Function.Injective.eq_iff] at h\n    rw[h]\n    exact injYu Yᵤ\n  inv_mem' := by\n    intros f h1 i j h2\n    rw[Eq.comm, Equiv.Perm.eq_inv_iff_eq] at h2\n    rw[Eq.comm]\n    exact h1 h2\n","line":34,"end_line":52,"proof_status":"formalized"}
+{"id":"PuCard","label":"PuCard","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["Pu"],"nlp_text":"Le nombre d’élément de \\(P_{\\mu }\\) est fini.","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/PuCard","latex_link":"sect0002.html#PuCard","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/PuCard","lean_decl_name":"PuCard","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"noncomputable def PuCard {μ : YoungDiagram} (Yᵤ : YoungTableau μ) :=\n  Fintype.card ↥(Pu Yᵤ)\n","line":60,"end_line":61,"proof_status":"formalized"}
+{"id":"PuQu","label":"PuQu","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["Pu","Qu"],"nlp_text":"\\(P_{\\mu }Q_{\\mu } := \\{ g : [0,n-1] \\to [0,n-1] | \\exists p \\in P_{\\mu } \\land \\exists q \\in Q_{\\mu }, g = p q \\} \\)","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/PuQu","latex_link":"sect0002.html#PuQu","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/PuQu","lean_decl_name":"PuQu","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"lemma sectPuQu {μ : YoungDiagram} (Yᵤ : YoungTableau μ):\n  (Pu Yᵤ).carrier ∩ (Qu Yᵤ).carrier = {↑1} := by\n  rw[Set.eq_singleton_iff_unique_mem, Set.mem_inter_iff]\n  constructor\n  exact ⟨Subgroup.one_mem (Pu Yᵤ), Subgroup.one_mem (Qu Yᵤ)⟩\n  simp\n  intro f p q\n  rw[Equiv.Perm.ext_iff]\n  intro x\n  obtain ⟨j, hj, a⟩ := preImYu Yᵤ x\n  obtain ⟨k, hk, b⟩ := preImYu Yᵤ (f (Yᵤ.entry j))\n  rw[Eq.comm] at hk\n  rw[Pu] at p\n  rw[Qu] at q\n  have h : (j: μ) = (k : μ) := by\n    ext\n    exact q hk\n    exact p hk\n  rw[← h] at hk\n  rw[Equiv.Perm.coe_one, id_eq, ← hj, hk]\n","line":94,"end_line":113,"proof_status":"formalized"}
+{"id":"Qu","label":"Qu","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["bijYu"],"nlp_text":"\\(Q_{\\mu }\\) est un sous groupe de \\(S_{n}\\), défini de la façon suivante:Un élément de \\(Q_{\\mu }\\) permute les entré du YoungDiagram s’ils sont sur la même colonne.","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/Qu","latex_link":"sect0002.html#Qu","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/Qu","lean_decl_name":"Qu","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"def Qu {μ : YoungDiagram} (Yᵤ : YoungTableau μ) : Subgroup (Equiv.Perm (Fin μ.card)) where\n  carrier := {x : (Equiv.Perm (Fin μ.card)) | ∀ {i j : μ}, (x (Yᵤ.entry i) = Yᵤ.entry j) → i.val.fst = j.val.fst}\n  mul_mem' := by\n    intros α β a b i j αβ\n    rw[Equiv.Perm.coe_mul, Function.comp_apply] at αβ\n    obtain ⟨k, hk, _⟩ := preImYu Yᵤ (β (Yᵤ.entry i))\n    rw[Eq.comm] at hk\n    rw[hk] at αβ\n    rw[b hk, a αβ]\n  one_mem' := by\n    intros i j h\n    rw[Equiv.Perm.coe_one, id_eq, Function.Injective.eq_iff] at h\n    rw[h]\n    exact injYu Yᵤ\n  inv_mem' := by\n    intros f h1 i j h2\n    rw[Eq.comm, Equiv.Perm.eq_inv_iff_eq] at h2\n    rw[Eq.comm]\n    exact h1 h2\n","line":64,"end_line":82,"proof_status":"formalized"}
+{"id":"YoungTableau","label":"YoungTableau","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Un YoungTableau est une fonction des cellules d’un YoungDiagram de taille n et retourne un naturel de 0 à n-1","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/YoungTableau","latex_link":"sect0002.html#YoungTableau","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/YoungTableau","lean_decl_name":"YoungTableau","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"./BlueprintRep/SpechtModules.lean","lean4_code":"noncomputable instance PuElementFinite {μ : YoungDiagram} (Yᵤ : YoungTableau μ) :\n  Fintype (Pu Yᵤ).carrier := Fintype.ofFinite ↑(Pu Yᵤ).carrier\n\n/-! Nécessaire pour indiqué que l'ensemble des éléments de Qᵤ est fini -/\nnoncomputable instance QuElementFinite {μ : YoungDiagram} (Yᵤ : YoungTableau μ) :\n  Fintype (Qu Yᵤ).carrier := Fintype.ofFinite ↑(Qu Yᵤ).carrier\n\n/-! Nécessaire pour utilisé Finset.sum -/\nnoncomputable instance Addition (μ : YoungDiagram) : AddCommMonoid (PermComplex μ) :=\n  MonoidAlgebra.addCommMonoid ℂ (Equiv.Perm (Fin μ.card))\n\n/-! Nécessaire pour multiplier deux éléments -/\nnoncomputable instance Multiplication (μ : YoungDiagram) : Mul (PermComplex μ) :=\n  ⟨fun a b => MonoidAlgebra.mul' a b⟩\n\n/-! Définition de aᵤ -/\n","line":15,"end_line":30,"proof_status":"formalized"}
+{"id":"injYu","label":"injYu","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["YoungTableau"],"nlp_text":"Un YoungTableau est injectif sur les entrés qui sont dans le YoungDiagram","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/injYu","latex_link":"sect0002.html#injYu","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/injYu","lean_decl_name":"injYu","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"lemma injYu {μ : YoungDiagram} (Yᵤ : YoungTableau μ) :\n  Function.Injective Yᵤ.entry := by\n  rw [Function.Injective]\n  intros i j h\n  exact Yᵤ.inj h\n","line":12,"end_line":16,"proof_status":"formalized"}
+{"id":"rowToCol","label":"rowToCol","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["YoungTableau"],"nlp_text":"Soit \\(g\\) est une permutation de [0, n-1]. Elle est rowToCol si \\[ \\forall i,j,k,l \\in \\mu , ((i \\neq j) \\land (g \\circ Y_{\\mu } (i) = Y_{\\mu } (k)) \\land (g \\circ Y_{\\mu } (j) = Y_{\\mu } (l))) → ((i.x \\neq j.x) \\lor (k.y \\neq l.y)) \\]","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/rowToCol","latex_link":"sect0002.html#rowToCol","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/rowToCol","lean_decl_name":"rowToCol","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"def rowToCol {μ : YoungDiagram} (Yᵤ : YoungTableau μ) (g : (Equiv.Perm (Fin μ.card))) :=\n  ∀ {i j k l : μ}, ((i ≠ j) ∧ (g (Yᵤ.entry i) = (Yᵤ.entry k)) ∧ (g (Yᵤ.entry j) = (Yᵤ.entry l))) → ((i.val.fst ≠ j.val.fst) ∨ (k.val.snd ≠ l.val.snd))\n","line":122,"end_line":123,"proof_status":"formalized"}
+{"id":"bijYu","label":"bijYu","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["injYu"],"nlp_text":"Un YoungTableau est une bijection entre les case de son YoungDiagram et les naturels de 0 à n-1","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/bijYu","latex_link":"sect0002.html#bijYu","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/bijYu","lean_decl_name":"bijYu","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"lemma bijYu {μ : YoungDiagram} (Yᵤ : YoungTableau μ) :\n  Function.Bijective Yᵤ.entry := by\n  rw[Fintype.bijective_iff_injective_and_card]\n  simp\n  exact injYu Yᵤ\n","line":19,"end_line":23,"proof_status":"formalized"}

crawled_repos/BonnAnalysis.jsonl

@@ -0,0 +1,49 @@
+{"id":"lem:snormEssSup_eq_sSup_snorm","label":"snormEssSup_eq_sSup_snorm","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(f\\) be measurable and the measure \\(\\mu \\) be \\(\\sigma \\)-finite. Then \\[ \\| f\\| _{L^\\infty } = \\sup _{\\| g\\| _{L^1} \\leq 1, \\ g \\text{ simple}} \\| fg \\| _{L^1}. \\]","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/MeasureTheory.snormEssSup_eq_sSup_snorm","latex_link":"chap-interpolation.html#lem:snormEssSup_eq_sSup_snorm","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/MeasureTheory.snormEssSup_eq_sSup_snorm","lean_decl_name":"MeasureTheory.snormEssSup_eq_sSup_snorm","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/fpvandoorn/BonnAnalysis/f3acd7b6e64e9949e697d595d495e01dc4c3dbf9/BonnAnalysis/ComplexInterpolation.lean","lean4_code":"lemma snormEssSup_eq_sSup_snorm [SigmaFinite μ] (f : α → ℂ) (hf : Measurable f)\n(hf' : sSup {x |∃ g : SimpleFunc α ℂ, snorm g 1 μ ≤ 1 ∧\nx = snorm (f * (g : α → ℂ)) 1 μ} < ⊤):\nsnormEssSup f μ = sSup {x |∃ g : SimpleFunc α ℂ, snorm g 1 μ ≤ 1\n∧ x = snorm (f * (g : α → ℂ)) 1 μ} := by\n   apply le_antisymm ?_\n   .  apply sSup_le\n      intro s ⟨g, hg₁, hg₂⟩\n      rw [hg₂, mul_comm]\n      simp [snorm, snorm']\n      apply le_trans (lintegral_norm_mul_le_one_top _ (by measurability))\n      apply mul_le_of_le_one_of_le hg₁ (by simp [snorm])\n   .  set M := sSup {x |∃ g : SimpleFunc α ℂ, snorm g 1 μ ≤ 1 ∧\n       x = snorm (f * (g : α → ℂ)) 1 μ}\n      apply essSup_le_of_ae_le\n      simp only [EventuallyLE, eventually_iff, mem_ae_iff]\n      by_contra h\n\n      have aux1 : ∃ B : Set α, MeasurableSet B ∧ B ⊆ {x | ‖f x‖₊ ≤ M}ᶜ ∧ 0 < μ B ∧ μ B < ⊤ := by\n        apply Measure.exists_subset_measure_lt_top (MeasurableSet.compl _) ((ne_eq _ _) ▸ h).bot_lt\n        convert hf.ennnorm measurableSet_Iic\n      rcases aux1 with ⟨B, hB⟩\n\n      let g : SimpleFunc α ℂ := {\n        toFun := B.indicator (fun _ ↦ ((μ B)⁻¹.toReal : ℂ))\n        measurableSet_fiber' := by\n          intro x\n          simp [indicator_preimage]\n          rcases eq_or_ne x ((μ B)⁻¹.toReal : ℂ) with (H | H)\n          simp [H]\n          apply MeasurableSet.ite hB.1 (by simp only [MeasurableSet.univ, M])\n          apply measurableSet_preimage measurable_zero (MeasurableSet.singleton _)\n          apply MeasurableSet.ite hB.1 _ (measurableSet_preimage measurable_zero\n          (MeasurableSet.singleton _))\n          rw [preimage_const_of_not_mem]\n          apply MeasurableSet.empty\n          rw [mem_singleton_iff]\n          exact H.symm\n        finite_range' := by\n          have : range (B.indicator fun _ ↦ ((μ B)⁻¹.toReal : ℂ))\n          ⊆ {0, ((μ B)⁻¹.toReal : ℂ)}  := by\n            intro x\n            rw [Set.mem_range_indicator]\n            simp\n            rintro (h | h)\n            exact Or.intro_left _ h.1\n            exact Or.intro_right _ h.2.symm\n          apply Set.Finite.subset _ this\n          simp only [finite_singleton, Finite.insert]\n      }\n\n      have aux_μB : (μ B)⁻¹ =  ‖(μ B)⁻¹.toReal‖₊  := by\n        have : (μ B)⁻¹ < ⊤ := by simp [hB.2.2.1]\n        generalize (μ B)⁻¹ = y at this -- better solution???\n        cases y\n        simp only [lt_self_iff_false] at this\n        simp only [coe_toReal, nnnorm_eq]\n\n      have : ¬ (M < snorm (f * (g : α → ℂ)) 1 μ) := by\n        simp\n        apply le_sSup\n        use g\n        constructor\n        simp [snorm, snorm', lintegral_indicator_const]\n        have : ∫⁻ (a : α), (B.indicator (fun _ ↦ (μ B)⁻¹) a) ∂ μ ≤ 1 := by\n          rw [lintegral_indicator_const hB.1,\n              ENNReal.inv_mul_cancel hB.2.2.1.ne.symm hB.2.2.2.ne]\n        convert this\n        rename α => x\n        by_cases hx : x ∈ B\n        simp [indicator_of_mem hx, aux_μB.symm]\n        simp [indicator_of_not_mem hx]\n        rfl\n\n      apply this\n      calc M = ∫⁻ _ in B, (μ B)⁻¹ * M ∂ μ := by {\n        rw [setLIntegral_const, mul_assoc, mul_comm M, ← mul_assoc,\n        ENNReal.inv_mul_cancel hB.2.2.1.ne.symm hB.2.2.2.ne, one_mul]\n      }\n      _ < ∫⁻ x in B, (μ B)⁻¹ * ‖f x‖₊ ∂ μ := by\n          apply setLIntegral_strict_mono hB.1 hB.2.2.1.ne.symm (hf.ennnorm.const_mul _)\n          rw [setLIntegral_const]\n          apply mul_ne_top (mul_ne_top (ENNReal.inv_ne_top.mpr\n          hB.2.2.1.ne.symm) hf'.ne) hB.2.2.2.ne\n          apply eventually_of_forall\n          intro x hx\n          rw [ENNReal.mul_lt_mul_left (ENNReal.inv_ne_zero.mpr hB.2.2.2.ne)\n          (ENNReal.inv_ne_top.mpr hB.2.2.1.ne.symm)]\n          convert hB.2.1 hx\n          simp only [mem_compl_iff, mem_setOf_eq, not_le]\n      _ = ∫⁻ x : α, B.indicator (fun a ↦ ‖f a‖₊ * ‖(fun _ ↦ ((μ B)⁻¹.toReal : ℂ)) a‖₊) x ∂μ := by\n          rw [lintegral_indicator _ hB.1]\n          simp [mul_comm]\n          congr with x; congr\n      _ = snorm (f * B.indicator fun _ ↦ ((μ B)⁻¹.toReal : ℂ)) 1 μ := by\n          simp [snorm, snorm']\n          congr with x\n          by_cases hx : x ∈ B\n          simp only [indicator_of_mem hx, nnnorm_real]\n          simp only [indicator_of_not_mem hx, nnnorm_zero, ENNReal.coe_zero, mul_zero]","line":1025,"end_line":1124,"proof_status":"formalized"}
+{"id":"a0000000211","label":"a0000000211","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For \\(\\lambda {\\gt} -d\\), Let \\(H_\\lambda \\) be the tempered distribution associated to \\(|x|^\\lambda \\in L^1_{loc}\\). If \\(-d {\\lt} \\lambda {\\lt} 0\\) then \\[ \\widehat{H_\\lambda } = c_\\lambda H_{-d-\\lambda } \\]","doc_url":null,"latex_link":"sect0001.html#a0000000211","lean_link":null,"lean_decl_name":"a0000000211","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000168","label":"a0000000168","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Then \\(C^\\infty (\\mathbb {R}^d)\\) is dense in \\(\\mathcal{D}'(\\mathbb {R}^d)\\).","doc_url":null,"latex_link":"sect0001.html#a0000000168","lean_link":null,"lean_decl_name":"a0000000168","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:hausdorff_young","label":"hausdorff_young","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["thm:riesz_interpolation"],"nlp_text":"Let \\(X=[0,2\\pi ]\\) with normalized Lebesgue measure \\(\\frac{d\\theta }{2\\pi }\\) and let \\(Y=\\mathbb {Z}\\) with counting measure.Consider the operator \\(T: f \\mapsto \\{ a_n\\} _{n\\in \\mathbb {Z}}\\) where \\[ a_n = \\frac{1}{2\\pi } \\int _0^{2\\pi } f(\\theta ) e^{-in\\theta } d\\theta \\]","doc_url":null,"latex_link":"chap-interpolation.html#lem:hausdorff_young","lean_link":null,"lean_decl_name":"hausdorff_young","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"thm:fourier-inversion-L1","label":"fourier-inversion-L1","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["lem:fourier-multiplication","thm:kernel-approximation"],"nlp_text":"Let \\(f\\in L^1(V,E)\\). If \\(\\widehat f\\in L^1(V,E)\\), then \\(\\mathcal F^{-1}\\mathcal Ff=f\\).","doc_url":null,"latex_link":"chap-plancherel.html#thm:fourier-inversion-L1","lean_link":null,"lean_decl_name":"fourier-inversion-L1","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:fourier-gaussian","label":"fourier-gaussian","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["lem:fourier-prop"],"nlp_text":"Let \\(x\\in V\\) and \\(\\delta {\\gt}0\\). Define the modulated Gaussian \\[ u_{x,\\delta }(y):V\\to \\mathbb {C},\\quad y\\mapsto e^{-\\delta \\pi |y|^2}e^{2\\pi i x y}. \\]","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/fourierIntegral_gaussian_innerProductSpace'","latex_link":"chap-plancherel.html#lem:fourier-gaussian","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/fourierIntegral_gaussian_innerProductSpace'","lean_decl_name":"fourierIntegral_gaussian_innerProductSpace'","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/leanprover-community/mathlib4/b387c548d1dc9cea5c5fcb71e5b3370cca1a04cc/Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean","lean4_code":"theorem _root_.fourierIntegral_gaussian_innerProductSpace' (hb : 0 < b.re) (x w : V) :\n    𝓕 (fun v ↦ cexp (- b * ‖v‖^2 + 2 * π * Complex.I * ⟪x, v⟫)) w =\n      (π / b) ^ (FiniteDimensional.finrank ℝ V / 2 : ℂ) * cexp (-π ^ 2 * ‖x - w‖ ^ 2 / b) := by\n  simp only [neg_mul, fourierIntegral_eq', ofReal_neg, ofReal_mul, ofReal_ofNat,\n    smul_eq_mul, ← Complex.exp_add, real_inner_comm w]\n  convert integral_cexp_neg_mul_sq_norm_add hb (2 * π * Complex.I) (x - w) using 3 with v\n  · congr 1\n    simp [inner_sub_left]\n    ring\n  · have : b ≠ 0 := by contrapose! hb; rw [hb, zero_re]\n    field_simp [mul_pow]\n    ring","line":369,"end_line":380,"proof_status":"formalized"}
+{"id":"thm:plancherel","label":"plancherel","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["lem:fourier-gaussian","lem:fourier-multiplication","thm:kernel-approximation"],"nlp_text":"Suppose that \\(f : V \\to E\\) is in \\(L^1(V,E)\\cap L^2(V,E)\\) and let \\(\\widehat{f}\\) be the Fourier transform of \\(f\\). Then \\(\\widehat{f},\\check{f}\\in L^2(V,E)\\) and \\[ \\| \\widehat{f}\\| _{L^2} = \\| f\\| _{L^2}=\\| \\check f\\| _{L^2}. \\]","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/MeasureTheory.memℒp_fourierIntegral","latex_link":"chap-plancherel.html#thm:plancherel","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/MeasureTheory.memℒp_fourierIntegral","lean_decl_name":"MeasureTheory.memℒp_fourierIntegral","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/fpvandoorn/BonnAnalysis/f3acd7b6e64e9949e697d595d495e01dc4c3dbf9/BonnAnalysis/Plancherel.lean","lean4_code":"theorem memℒp_fourierIntegral {f : V → ℂ} (hf : Integrable f) (h2f : Memℒp f 2) :\n    Memℒp (𝓕 f) 2 := by\n  unfold Memℒp; constructor\n  · apply Continuous.aestronglyMeasurable\n    exact VectorFourier.fourierIntegral_continuous continuous_fourierChar\n        (by simp only [innerₗ_apply]; exact continuous_inner) hf\n  · rw [snorm_fourierIntegral hf h2f]\n    exact h2f.2","line":473,"end_line":480,"proof_status":"statement_formalized"}
+{"id":"lem:fourier12-cauchy","label":"fourier12-cauchy","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["thm:plancherel","lem:L12-dense"],"nlp_text":"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]{}f\\). Then \\((\\widehat f_n)_n\\) is a Cauchy sequence, hence converges in \\(L^2(V,E)\\).","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/","latex_link":"chap-plancherel.html#lem:fourier12-cauchy","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/","lean_decl_name":"","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:invFourier-L2","label":"invFourier-L2","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["lem:fourier12-cauchy"],"nlp_text":"Define analogously \\(\\mathcal F^{-1}f:=\\check f:=\\lim _{n\\to \\infty }\\check f_n\\), if \\(f_n\\xrightarrow [L^2]{}f\\in L^2(V,E)\\) with \\((f_n)_n\\subset L^1(V,E)\\cap L^2(V,E)\\). By the same arguments as above, this is well-defined.","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/","latex_link":"chap-plancherel.html#def:invFourier-L2","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/","lean_decl_name":"","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:fourier-L2","label":"fourier-L2","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["lem:fourier12-cauchy"],"nlp_text":"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 [L^2]{}f\\). Set \\[ \\mathcal Ff:=\\widehat f:=\\lim _{n\\to \\infty }\\widehat{f_n}, \\]","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/","latex_link":"chap-plancherel.html#def:fourier-L2","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/","lean_decl_name":"","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:fourier-bounded","label":"fourier-bounded","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["def:fourier-transform"],"nlp_text":"Let \\(f\\in L^1(V,E)\\). Then its Fourier transform \\(\\widehat f\\) is well-defined and bounded. In particular, the Fourier transform defines a map \\(\\mathcal F:L^1(V,E)\\to L^\\infty (V,E)\\).","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/VectorFourier.fourierIntegral_convergent_iff","latex_link":"chap-plancherel.html#lem:fourier-bounded","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/VectorFourier.fourierIntegral_convergent_iff","lean_decl_name":"VectorFourier.fourierIntegral_convergent_iff","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Analysis/Fourier/FourierTransform.lean","lean4_code":"theorem fourierIntegral_convergent_iff (he : Continuous e)\n    (hL : Continuous fun p : V × W ↦ L p.1 p.2) {f : V → E} (w : W) :\n    Integrable (fun v : V ↦ e (-L v w) • f v) μ ↔ Integrable f μ := by\n  -- first prove one-way implication\n  have aux {g : V → E} (hg : Integrable g μ) (x : W) :\n      Integrable (fun v : V ↦ e (-L v x) • g v) μ := by\n    have c : Continuous fun v ↦ e (-L v x) := he.comp (hL.comp (.prodMk_left _)).neg\n    simp_rw [← integrable_norm_iff (c.aestronglyMeasurable.smul hg.1), Circle.norm_smul]\n    exact hg.norm\n  -- then use it for both directions\n  refine ⟨fun hf ↦ ?_, fun hf ↦ aux hf w⟩\n  have := aux hf (-w)\n  simp_rw [← mul_smul (e _) (e _) (f _), ← e.map_add_eq_mul, LinearMap.map_neg, neg_add_cancel,\n    e.map_zero_eq_one, one_smul] at this -- the `(e _)` speeds up elaboration considerably\n  exact this\n\n","line":124,"end_line":139,"proof_status":"formalized"}
+{"id":"a0000000173","label":"a0000000173","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For a multiindex \\(\\alpha \\) and a distribution \\(F\\) define the distribution \\[ \\partial ^\\alpha F (\\phi ) = (-1)^{|\\alpha |}(F \\partial ^\\alpha \\phi ) \\]","doc_url":null,"latex_link":"sect0001.html#a0000000173","lean_link":null,"lean_decl_name":"a0000000173","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000191","label":"a0000000191","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The fourier transformation is a continuous bijection \\begin{align*} \\mathcal{S}& \\to \\mathcal{S}\\\\ \\phi & \\mapsto \\hat\\phi = (\\xi \\mapsto \\int _{\\mathbb {R}^d} \\phi (x) e^{-2\\pi i x \\xi } \\ \\mathrm{d} x) \\end{align*}","doc_url":null,"latex_link":"sect0001.html#a0000000191","lean_link":null,"lean_decl_name":"a0000000191","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000195","label":"a0000000195","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Define \\[ \\hat F (\\phi ) = F (\\hat\\phi ) \\]","doc_url":null,"latex_link":"sect0001.html#a0000000195","lean_link":null,"lean_decl_name":"a0000000195","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:fourier-cont","label":"fourier-cont","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["def:fourier-transform"],"nlp_text":"Let \\(f\\in L^1(V,E)\\). Then \\(\\widehat f\\) is continuous.","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/VectorFourier.fourierIntegral_continuous","latex_link":"chap-plancherel.html#lem:fourier-cont","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/VectorFourier.fourierIntegral_continuous","lean_decl_name":"VectorFourier.fourierIntegral_continuous","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Analysis/Fourier/FourierTransform.lean","lean4_code":"theorem fourierIntegral_continuous [FirstCountableTopology W] (he : Continuous e)\n    (hL : Continuous fun p : V × W ↦ L p.1 p.2) {f : V → E} (hf : Integrable f μ) :\n    Continuous (fourierIntegral e μ L f) := by\n  apply continuous_of_dominated\n  · exact fun w ↦ ((fourierIntegral_convergent_iff he hL w).2 hf).1\n  · exact fun w ↦ ae_of_all _ fun v ↦ le_of_eq (Circle.norm_smul _ _)\n  · exact hf.norm\n  · refine ae_of_all _ fun v ↦ (he.comp ?_).smul continuous_const\n    exact (hL.comp (.prodMk_right _)).neg\n\nend Continuous\n\nsection Fubini\n\nvariable [TopologicalSpace 𝕜] [IsTopologicalRing 𝕜] [TopologicalSpace V] [BorelSpace V]\n  [TopologicalSpace W] [MeasurableSpace W] [BorelSpace W]\n  {e : AddChar 𝕜 𝕊} {μ : Measure V} {L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜}\n  {ν : Measure W} [SigmaFinite μ] [SigmaFinite ν] [SecondCountableTopology V]\n\nvariable [CompleteSpace E] [CompleteSpace F]\n\n/-- The Fourier transform satisfies `∫ 𝓕 f * g = ∫ f * 𝓕 g`, i.e., it is self-adjoint.\nVersion where the multiplication is replaced by a general bilinear form `M`. -/\n","line":151,"end_line":173,"proof_status":"formalized"}
+{"id":"lem:fourier-multiplication","label":"fourier-multiplication","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["lem:fourier-cont"],"nlp_text":"Let \\(f,g\\in L^1(V,E)\\). Then \\[ \\int _WM(\\widehat f(w),g(w))\\, d\\nu (w)=\\int _VM(f(v),\\widehat g(v))\\, d\\mu (v). \\]","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/VectorFourier.integral_bilin_fourierIntegral_eq_flip","latex_link":"chap-plancherel.html#lem:fourier-multiplication","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/VectorFourier.integral_bilin_fourierIntegral_eq_flip","lean_decl_name":"VectorFourier.integral_bilin_fourierIntegral_eq_flip","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Analysis/Fourier/FourierTransform.lean","lean4_code":"theorem integral_bilin_fourierIntegral_eq_flip\n    {f : V → E} {g : W → F} (M : E →L[ℂ] F →L[ℂ] G) (he : Continuous e)\n    (hL : Continuous fun p : V × W ↦ L p.1 p.2) (hf : Integrable f μ) (hg : Integrable g ν) :\n    ∫ ξ, M (fourierIntegral e μ L f ξ) (g ξ) ∂ν =\n      ∫ x, M (f x) (fourierIntegral e ν L.flip g x) ∂μ := by\n  by_cases hG : CompleteSpace G; swap; · simp [integral, hG]\n  calc\n  _ = ∫ ξ, M.flip (g ξ) (∫ x, e (-L x ξ) • f x ∂μ) ∂ν := rfl\n  _ = ∫ ξ, (∫ x, M.flip (g ξ) (e (-L x ξ) • f x) ∂μ) ∂ν := by\n    congr with ξ\n    apply (ContinuousLinearMap.integral_comp_comm _ _).symm\n    exact (fourierIntegral_convergent_iff he hL _).2 hf\n  _ = ∫ x, (∫ ξ, M.flip (g ξ) (e (-L x ξ) • f x) ∂ν) ∂μ := by\n    rw [integral_integral_swap]\n    have : Integrable (fun (p : W × V) ↦ ‖M‖ * (‖g p.1‖ * ‖f p.2‖)) (ν.prod μ) :=\n      (hg.norm.mul_prod hf.norm).const_mul _\n    apply this.mono\n    · -- This proof can be golfed but becomes very slow; breaking it up into steps\n      -- speeds up compilation.\n      change AEStronglyMeasurable (fun p : W × V ↦ (M (e (-(L p.2) p.1) • f p.2) (g p.1))) _\n      have A : AEStronglyMeasurable (fun (p : W × V) ↦ e (-L p.2 p.1) • f p.2) (ν.prod μ) := by\n        refine (Continuous.aestronglyMeasurable ?_).smul hf.1.comp_snd\n        exact he.comp (hL.comp continuous_swap).neg\n      have A' : AEStronglyMeasurable (fun p ↦ (g p.1, e (-(L p.2) p.1) • f p.2) : W × V → F × E)\n        (Measure.prod ν μ) := hg.1.comp_fst.prodMk A\n      have B : Continuous (fun q ↦ M q.2 q.1 : F × E → G) := M.flip.continuous₂\n      apply B.comp_aestronglyMeasurable A' -- `exact` works, but `apply` is 10x faster!\n    · filter_upwards with ⟨ξ, x⟩\n      rw [Function.uncurry_apply_pair, Submonoid.smul_def, (M.flip (g ξ)).map_smul,\n        ← Submonoid.smul_def, Circle.norm_smul, ContinuousLinearMap.flip_apply,\n        norm_mul, norm_norm M, norm_mul, norm_norm, norm_norm, mul_comm (‖g _‖), ← mul_assoc]\n      exact M.le_opNorm₂ (f x) (g ξ)\n  _ = ∫ x, (∫ ξ, M (f x) (e (-L.flip ξ x) • g ξ) ∂ν) ∂μ := by\n      simp only [ContinuousLinearMap.flip_apply, ContinuousLinearMap.map_smul_of_tower,\n      ContinuousLinearMap.coe_smul', Pi.smul_apply, LinearMap.flip_apply]\n  _ = ∫ x, M (f x) (∫ ξ, e (-L.flip ξ x) • g ξ ∂ν) ∂μ := by\n    congr with x\n    apply ContinuousLinearMap.integral_comp_comm\n    apply (fourierIntegral_convergent_iff he _ _).2 hg\n    exact hL.comp continuous_swap\n\n/-- The Fourier transform satisfies `∫ 𝓕 f * g = ∫ f * 𝓕 g`, i.e., it is self-adjoint. -/\n","line":174,"end_line":215,"proof_status":"formalized"}
+{"id":"thm:fourier-inversion","label":"fourier-inversion","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["lem:fourier-multiplication","thm:kernel-approximation"],"nlp_text":"Let \\(f:V\\to E\\) be integrable and continuous. Assume \\(\\widehat f\\) is integrable as well. Then \\[ \\mathcal F^{-1}\\mathcal F f=f. \\]","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/MeasureTheory.Integrable.fourier_inversion","latex_link":"chap-plancherel.html#thm:fourier-inversion","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/MeasureTheory.Integrable.fourier_inversion","lean_decl_name":"MeasureTheory.Integrable.fourier_inversion","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/leanprover-community/mathlib4/b387c548d1dc9cea5c5fcb71e5b3370cca1a04cc/Mathlib/Analysis/Fourier/Inversion.lean","lean4_code":"theorem MeasureTheory.Integrable.fourier_inversion\n    (hf : Integrable f) (h'f : Integrable (𝓕 f)) {v : V}\n    (hv : ContinuousAt f v) : 𝓕⁻ (𝓕 f) v = f v :=\n  tendsto_nhds_unique (Real.tendsto_integral_gaussian_smul hf h'f v)\n    (Real.tendsto_integral_gaussian_smul' hf hv)","line":159,"end_line":163,"proof_status":"formalized"}
+{"id":"thm:fourier2-properties","label":"fourier2-properties","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["thm:fourier-inversion","def:invFourier-L2","lem:fourier2-welldef"],"nlp_text":"Plancherel’s Theorem, the inversion formula, and the properties of Lemma 2.5 hold for the Fourier transform on \\(L^2(V,E)\\) as well.","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/","latex_link":"chap-plancherel.html#thm:fourier2-properties","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/","lean_decl_name":"","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:fourier-prop","label":"fourier-prop","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["def:fourier-transform"],"nlp_text":"Lef \\(f,g\\in L^1(V,E)\\), \\(t\\in \\mathbb {R}\\) and \\(a,b\\in \\mathbb {C}\\). The Fourier transform satisfies the following elementary properties: \\(\\mathcal F(af+bg)=a\\mathcal Ff+b\\mathcal Fg\\)(Linearity) \\(\\mathcal F(f(x-t)) = e^{-2\\pi i ty}\\mathcal F f(y)\\)(Shifting) \\(\\mathcal F(f(tx)) = \\frac1{|t|}\\mathcal F f(\\frac yt)\\)(Scaling) If \\(E\\) admits a conjugation, then \\(\\mathcal F(\\overline{f(x)}) = \\overline{\\mathcal Ff(-y)}\\)(Conjugation) Define the convolution of \\(f\\) and \\(g\\) w.r.t. a bilinear map \\(M:E\\times E\\to F\\) as \\[ (f\\ast _Mg)(w):=\\int _VM(f(v),g(w-v))\\, d\\mu (v). \\]","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/VectorFourier.fourierIntegral_add","latex_link":"chap-plancherel.html#lem:fourier-prop","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/VectorFourier.fourierIntegral_add","lean_decl_name":"VectorFourier.fourierIntegral_add","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Analysis/Fourier/FourierTransform.lean","lean4_code":"theorem fourierIntegral_add (he : Continuous e) (hL : Continuous fun p : V × W ↦ L p.1 p.2)\n    {f g : V → E} (hf : Integrable f μ) (hg : Integrable g μ) :\n    fourierIntegral e μ L (f + g) = fourierIntegral e μ L f + fourierIntegral e μ L g := by\n  ext1 w\n  dsimp only [Pi.add_apply, fourierIntegral]\n  simp_rw [smul_add]\n  rw [integral_add]\n  · exact (fourierIntegral_convergent_iff he hL w).2 hf\n  · exact (fourierIntegral_convergent_iff he hL w).2 hg\n\n/-- The Fourier integral of an `L^1` function is a continuous function. -/\n","line":140,"end_line":150,"proof_status":"formalized"}
+{"id":"lem:fourier2-welldef","label":"fourier2-welldef","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["def:fourier-L2"],"nlp_text":"This is well-defined: By Lemma 2.15, the limit exists. Further it does not depend on the choice of sequence \\((f_n)_n\\). If \\(f\\in L^1(V,E)\\cap L^2(V,E)\\), this definition agrees with the Fourier transform on \\(L^1(V,E)\\).","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/","latex_link":"chap-plancherel.html#lem:fourier2-welldef","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/","lean_decl_name":"","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"thm:fourier-is-l2-linear","label":"fourier-is-l2-linear","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["thm:fourier2-properties"],"nlp_text":"The Fourier transform induces a continuous linear map \\(L^2(V,E) \\to L^2(V,E)\\).","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/MeasureTheory.fourierIntegralL2","latex_link":"chap-plancherel.html#thm:fourier-is-l2-linear","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/MeasureTheory.fourierIntegralL2","lean_decl_name":"MeasureTheory.fourierIntegralL2","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/fpvandoorn/BonnAnalysis/f3acd7b6e64e9949e697d595d495e01dc4c3dbf9/BonnAnalysis/Plancherel.lean","lean4_code":"def fourierIntegralL2 : (V →₂[volume] ℂ) →L[ℝ] (V →₂[volume] ℂ) :=\n  sorry","line":619,"end_line":620,"proof_status":"formalized"}
+{"id":"lem:weierstrass-kernel","label":"weierstrass-kernel","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(K_\\delta (v)=\\delta ^{-n/2}e^{-\\pi |v|^2/\\delta }\\) as in Lemma 2.6. This is a good kernel, called the Weierstrass kernel, satisfying \\[ \\int _{V}K_\\delta (x)\\, dx= 1\\quad \\text{and}\\quad \\int _{|x|{\\gt}\\eta }K_\\delta (x)\\, dx\\xrightarrow {\\delta \\to 0}0\\quad \\text{for all }\\eta {\\gt}0. \\]","doc_url":null,"latex_link":"chap-plancherel.html#lem:weierstrass-kernel","lean_link":null,"lean_decl_name":"weierstrass-kernel","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"thm:kernel-approximation","label":"kernel-approximation","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["lem:weierstrass-kernel"],"nlp_text":"Let \\(f:V\\to E\\) be integrable. Let \\(K_\\delta \\) be the Weierstrass kernel from Lemma 2.7, or indeed any family of functions satisfying the conditions of Lemma 2.7. Then \\[ (K_\\delta \\ast f)(x):=\\int _V K_\\delta (y)f(x-y)\\, d\\mu (y)\\xrightarrow {\\delta \\to 0}f(x) \\]","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/Real.tendsto_integral_gaussian_smul'","latex_link":"chap-plancherel.html#thm:kernel-approximation","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/Real.tendsto_integral_gaussian_smul'","lean_decl_name":"Real.tendsto_integral_gaussian_smul'","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/leanprover-community/mathlib4/b387c548d1dc9cea5c5fcb71e5b3370cca1a04cc/Mathlib/Analysis/Fourier/Inversion.lean","lean4_code":"lemma tendsto_integral_gaussian_smul' (hf : Integrable f) {v : V} (h'f : ContinuousAt f v) :\n    Tendsto (fun (c : ℝ) ↦\n      ∫ w : V, ((π * c : ℂ) ^ (finrank ℝ V / 2 : ℂ) * cexp (-π ^ 2 * c * ‖v - w‖ ^ 2)) • f w)\n    atTop (𝓝 (f v)) := by\n  let φ : V → ℝ := fun w ↦ π ^ (finrank ℝ V / 2 : ℝ) * Real.exp (-π^2 * ‖w‖^2)\n  have A : Tendsto (fun (c : ℝ) ↦ ∫ w : V, (c ^ finrank ℝ V * φ (c • (v - w))) • f w)\n      atTop (𝓝 (f v)) := by\n    apply tendsto_integral_comp_smul_smul_of_integrable'\n    · exact fun x ↦ by positivity\n    · rw [integral_mul_left, GaussianFourier.integral_rexp_neg_mul_sq_norm (by positivity)]\n      nth_rewrite 2 [← pow_one π]\n      rw [← rpow_natCast, ← rpow_natCast, ← rpow_sub pi_pos, ← rpow_mul pi_nonneg,\n        ← rpow_add pi_pos]\n      ring_nf\n      exact rpow_zero _\n    · have A : Tendsto (fun (w : V) ↦ π^2 * ‖w‖^2) (cobounded V) atTop := by\n        rw [tendsto_const_mul_atTop_of_pos (by positivity)]\n        apply (tendsto_pow_atTop two_ne_zero).comp tendsto_norm_cobounded_atTop\n      have B := tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero (finrank ℝ V / 2) 1\n        zero_lt_one |>.comp A |>.const_mul (π ^ (-finrank ℝ V / 2 : ℝ))\n      rw [mul_zero] at B\n      convert B using 2 with x\n      simp only [neg_mul, one_mul, Function.comp_apply, ← mul_assoc, ← rpow_natCast, φ]\n      congr 1\n      rw [mul_rpow (by positivity) (by positivity), ← rpow_mul pi_nonneg,\n        ← rpow_mul (norm_nonneg _), ← mul_assoc, ← rpow_add pi_pos, mul_comm]\n      congr <;> ring\n    · exact hf\n    · exact h'f\n  have B : Tendsto\n      (fun (c : ℝ) ↦ ∫ w : V, ((c^(1/2:ℝ)) ^ finrank ℝ V * φ ((c^(1/2:ℝ)) • (v - w))) • f w)\n      atTop (𝓝 (f v)) :=\n    A.comp (tendsto_rpow_atTop (by norm_num))\n  apply B.congr'\n  filter_upwards [Ioi_mem_atTop 0] with c (hc : 0 < c)\n  congr with w\n  rw [← coe_smul]\n  congr\n  rw [ofReal_mul, ofReal_mul, ofReal_exp, ← mul_assoc]\n  congr\n  · rw [mul_cpow_ofReal_nonneg pi_nonneg hc.le, ← rpow_natCast, ← rpow_mul hc.le, mul_comm,\n      ofReal_cpow pi_nonneg, ofReal_cpow hc.le]\n    simp [div_eq_inv_mul]\n  · norm_cast\n    simp only [one_div, norm_smul, Real.norm_eq_abs, mul_pow, _root_.sq_abs, neg_mul, neg_inj,\n      ← rpow_natCast, ← rpow_mul hc.le, mul_assoc]\n    norm_num","line":104,"end_line":150,"proof_status":"formalized"}
+{"id":"lem:threelines","label":"threelines","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(S\\) be the strip \\(S:=\\{ z \\in \\mathbb {C}\\ | \\ 0 {\\lt} \\mathrm{Re} \\, z {\\lt} 1 \\} \\). Let \\(f: \\overline{S} \\to \\mathbb {C}\\) be a function that is holomorphic on \\(S\\) and continuous and bounded on \\(\\overline{S}\\).Assume \\(M_0, M_1\\) are positive real numbers such that for all values of \\(y\\) in \\(\\mathbb {R}\\), we have \\[ | \\phi (iy) | \\leq M_0 \\ \\qquad | \\phi (1+iy) | \\leq M_1 \\]","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/DiffContOnCl.norm_le_pow_mul_pow","latex_link":"chap-interpolation.html#lem:threelines","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/DiffContOnCl.norm_le_pow_mul_pow","lean_decl_name":"DiffContOnCl.norm_le_pow_mul_pow","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/fpvandoorn/BonnAnalysis/f3acd7b6e64e9949e697d595d495e01dc4c3dbf9/BonnAnalysis/ComplexInterpolation.lean","lean4_code":"theorem DiffContOnCl.norm_le_pow_mul_pow {a b : ℝ} {f : ℂ → ℂ} (hab: a<b)\n    (hf : DiffContOnCl ℂ f { z | z.re ∈ Ioo a b})\n    (h2f : IsBounded (f '' { z | z.re ∈ Icc a b}))\n    {M₀ M₁ : ℝ} (hM₀ : 0 < M₀) (hM₁ : 0 < M₁)\n    (h₀f : ∀ y : ℝ, ‖f (a + I * y)‖ ≤ M₀) (h₁f : ∀ y : ℝ, ‖f (b + I * y)‖ ≤ M₁)\n    {x y t s : ℝ} (ht : 0 ≤ t) (hs : 0 ≤ s) (hx : x = t * a + s * b) (hts : t + s = 1) :\n    ‖f (x + I * y)‖ ≤ M₀ ^ (1-((t-1)*a+s*b)/(b-a)) * M₁ ^ (((t-1)*a+s*b)/(b-a)) := by{\n\n      have hb_sub_a: b - a ≠ 0 := ne_of_gt (by simp [hab])\n\n      have hts'' : s = 1-t := eq_sub_of_add_eq (add_comm t s ▸ hts)\n\n      have hax: x ≥ a := by{\n        simp [hx]\n        rw [eq_sub_of_add_eq hts]\n        ring_nf\n        have : -(s * a) + s * b = s * (b-a) := by ring\n        rw [this]\n        simp only [le_add_iff_nonneg_left, ge_iff_le]\n        rw [mul_nonneg_iff]\n        exact Or.inl ⟨hs, by simp; exact le_of_lt hab⟩\n      }\n\n      -- Essentially same as above with minor tweaks\n      have hxb: b ≥ x := by{\n        simp [hx]\n        rw [hts'']\n        ring_nf\n        have : t * a - t * b = t * (a-b) := by ring\n        rw [this]\n        simp\n        rw [mul_nonpos_iff]\n        exact Or.inl ⟨ht, by simp; exact le_of_lt hab⟩\n      }\n\n      let g : ℂ → ℂ := fun z ↦ f (a + z * (b-a))\n      have hg: DiffContOnCl ℂ g { z | z.re ∈ Ioo 0 1} := by{\n        let h : ℂ → ℂ := fun z ↦ a + z *(b-a)\n        have hcomp: g = f ∘ h := rfl\n        rw [hcomp]\n        apply DiffContOnCl.comp (s:={ z | z.re ∈ Ioo a b}) hf ?_ ?_\n        · simp [h]\n          have : (fun (x:ℂ) ↦ x * (↑b - ↑a) ) = (fun (x:ℂ) ↦ x • ((b:ℂ) - (a:ℂ))) := rfl\n          exact DiffContOnCl.const_add (this ▸ DiffContOnCl.smul_const DiffContOnCl.id _) _\n        · simp [h, MapsTo]\n          refine fun z hz₀ hz₁ ↦ ⟨?_, ?_⟩\n          · apply Real.mul_pos hz₀\n            simp [hab]\n          · calc\n            a + z.re * (b-a) < a + 1 *(b - a) := by gcongr; simp [hab]\n            _ = b := by simp\n      }\n\n      have h2g: IsBounded (g '' { z | z.re ∈ Icc 0 1}) := by{\n        simp only [g]\n        apply IsBounded.subset h2f\n        intro z hz\n        obtain ⟨w, hw₁, hw₂⟩ := hz\n        simp\n        use ↑a + w * (↑b - ↑a)\n        simp\n        simp at hw₁\n        constructor\n        · constructor\n          · rw [mul_nonneg_iff]; left\n            constructor\n            · exact hw₁.1\n            · simp; exact le_of_lt hab\n          · calc\n            a + w.re * (b-a) ≤ a + 1 * (b-a) := by gcongr; simp; exact le_of_lt hab; exact hw₁.2\n            _ = b := by ring\n        · exact hw₂\n      }\n\n      have h₀g : ∀ y : ℝ, ‖g (I * y)‖ ≤ M₀ := by{\n        simp only [g]\n        simp\n        intro y\n        specialize h₀f (y* (b-a))\n        simp at h₀f\n        rw [mul_assoc]\n        exact h₀f\n      }\n\n      have h₁g : ∀ y : ℝ, ‖g (1 + I * y)‖ ≤ M₁ := by{\n        simp only [g]\n        simp\n        intro y\n        specialize h₁f (y * (b-a))\n        simp at h₁f\n        have : ↑a + (1 + I * ↑y) * (↑b - ↑a) = ↑b + I * (↑y * (↑b - ↑a)) := by ring\n        rw [this]\n        exact h₁f\n      }\n\n      let t':= (x-a)/(b-a)\n      let s':= 1 - t'\n      have ht' : 0 ≤ t' := by\n        simp only [t']\n        rw [div_nonneg_iff]\n        refine Or.inl ⟨by simp [hax], by simp; exact le_of_lt hab⟩\n\n      have hs' : 0 ≤ s' := by {\n        simp only [s', t']\n        simp\n        rw [div_le_one]\n        · simp [hxb]\n        · simp [hab]\n      }\n      have hts' : t' + s' = 1 := by simp [s']\n\n      have hgoal := DiffContOnCl.norm_le_pow_mul_pow₀₁ hg h2g hM₀ hM₁ h��g h₁g ht' hs' hts' (y:=y/(b-a))\n\n      simp [g] at hgoal\n      simp only [t'] at hgoal\n      simp at hgoal\n      rw [add_mul, mul_comm_div, div_self (by norm_cast), mul_assoc, mul_comm_div, div_self (by norm_cast), ← add_assoc] at hgoal\n      simp at hgoal\n\n      have ht'₁: t'=((t - 1) * a + s * b)/(b-a) := by{\n        simp only [t', hx]\n        ring\n      }\n      simp only [s'] at hgoal\n      rw [← ht'₁]\n      exact hgoal\n    }","line":704,"end_line":830,"proof_status":"formalized"}
+{"id":"thm:riesz_interpolation","label":"riesz_interpolation","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["lem:threelines"],"nlp_text":"Let \\((X, \\mu )\\) and \\((Y, \\nu )\\) be measure spaces and consider all \\(L^p\\) spaces to be complex valued.Suppose \\(T\\) is a linear map \\(L^{p_0}(X) + L^{p_1}(X) \\to L^{q_0}(Y) + L^{q_1}(Y)\\) that restricts to bounded operators \\(L^{p_0} \\to L^{q_0}\\) and \\(L^{p_1} \\to L^{q_1}\\). Let \\(M_0, M_1\\) be the respective bounds, i.e., \\[ \\begin{cases} ||Tf||_{L^{q_0}} \\leq M_0 ||f||_{L^{p_0}} \\\\ ||Tf||_{L^{q_1}} \\leq M_1 ||f||_{L^{p_1}} \\end{cases} \\]","doc_url":null,"latex_link":"chap-interpolation.html#thm:riesz_interpolation","lean_link":null,"lean_decl_name":"riesz_interpolation","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:hausdorff_young_dual","label":"hausdorff_young_dual","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["thm:riesz_interpolation"],"nlp_text":"For \\(1\\leq p \\leq 2\\) and \\(q\\) conjugate exponent to \\(p\\), we have \\[ ||T' \\{ a_n\\} ||_{L^q} \\leq ||\\{ a_n\\} ||_{L^p} \\]","doc_url":null,"latex_link":"chap-interpolation.html#lem:hausdorff_young_dual","lean_link":null,"lean_decl_name":"hausdorff_young_dual","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000225","label":"a0000000225","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\(F\\) is a fundamental solution of \\(L = \\frac{\\partial }{\\partial t} - \\Delta _x\\).","doc_url":null,"latex_link":"sect0001.html#a0000000225","lean_link":null,"lean_decl_name":"a0000000225","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000164","label":"a0000000164","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(f,g \\in L^1_{loc}(\\Omega )\\) such that \\(\\Lambda _f = \\Lambda _g\\). Then \\(f = g\\) almost everywhere.","doc_url":null,"latex_link":"sect0001.html#a0000000164","lean_link":null,"lean_decl_name":"a0000000164","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000189","label":"a0000000189","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"All \\(\\partial ^\\alpha F\\) are tempered. Let \\(\\psi \\in C^\\infty \\) be slowly increasing, i.e. for each \\(\\alpha \\) exists \\(N_\\alpha \\) such that \\(\\partial _x^\\alpha \\psi (x) = O(|x|^{N_\\alpha })\\). Then \\(\\psi F\\), defined by \\((\\psi F)(\\phi )(F(\\psi \\phi )\\) is tempered.","doc_url":null,"latex_link":"sect0001.html#a0000000189","lean_link":null,"lean_decl_name":"a0000000189","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:snorm_eq_sSup_snorm","label":"snorm_eq_sSup_snorm","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(p\\) and \\(q\\) be real conjugate exponents. Let \\(f\\) be measurable. Then \\[ \\| f\\| _{L^q} = \\sup _{\\| g\\| _{L^p} \\leq 1, \\ g \\text{ simple}} \\| fg \\| _{L^1}. \\]","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/MeasureTheory.snorm_eq_sSup_snorm","latex_link":"chap-interpolation.html#lem:snorm_eq_sSup_snorm","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/MeasureTheory.snorm_eq_sSup_snorm","lean_decl_name":"MeasureTheory.snorm_eq_sSup_snorm","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/fpvandoorn/BonnAnalysis/f3acd7b6e64e9949e697d595d495e01dc4c3dbf9/BonnAnalysis/ComplexInterpolation.lean","lean4_code":"lemma snorm_eq_sSup_snorm (p q : ℝ≥0) (hpq : NNReal.IsConjExponent p q) (f : α → ℂ)\n(hf : Measurable f) (hf' : snorm f p μ ≠ ∞) (hf0 : snorm f p μ ≠ 0):\nsnorm f p μ = sSup {snorm (f * (g.1 : α → ℂ)) 1 μ | g : Lp.simpleLe1 ℂ q μ} := by\n  apply le_antisymm ?_\n  . apply sSup_le\n    rintro b ⟨g, hg⟩\n    rw [← hg]\n    calc snorm (f * (g.1 : α → ℂ)) 1 μ = ∫⁻ a, ‖f a‖₊ * ‖g.1 a‖₊ ∂μ := by simp [snorm, snorm']\n    _ = ∫⁻ a, ((‖f ·‖₊) * (‖(g.1 : α → ℂ) ·‖₊)) a ∂μ := lintegral_congr (by\n      simp only [Pi.mul_apply, ENNReal.coe_mul, implies_true])\n    _ ≤ snorm f p μ * snorm g.1 q μ  := by\n      simp only [snorm, coe_toReal, snorm', ENNReal.coe_eq_zero,\n      hpq.ne_zero, ↓reduceIte, coe_ne_top, hpq.symm.ne_zero]\n      apply ENNReal.lintegral_mul_le_Lp_mul_Lq _ (NNReal.IsConjExponent.coe hpq\n      ) hf.aemeasurable.ennnorm (AEMeasurable.ennnorm (SimpleFunc.aemeasurable _))\n    _ ≤ snorm f p μ := mul_le_of_le_one_right (by positivity) g.2\n  . rcases eq_or_ne (snorm f p μ) 0 with hf0 | hf0\n    . simp [hf0]\n    . set g : α → ℝ≥0∞ :=\n      ENNReal.ofNNReal ∘ fun a ↦ ‖f a‖₊ ^ ((p : ℝ) - 1) * (snorm f p μ).toNNReal ^ (1 - (p : ℝ))\n      have g_meas : Measurable g:=\n        ((hf.nnnorm.pow_const _).mul_const _).coe_nnreal_ennreal\n      have g_norm : (∫⁻ (a : α), g a ^ (q : ℝ) ∂μ) ^ (q : ℝ)⁻¹ = 1 := by\n        simp only [g, Function.comp_apply, ENNReal.coe_mul]\n        calc (∫⁻ (a : α), ↑(‖f a‖₊ ^ ((p : ℝ) - 1) * (snorm f p μ).toNNReal ^\n        (1 - (p : ℝ))) ^ (q : ℝ) ∂μ) ^ (q : ℝ)⁻¹ = (∫⁻ (a : α), (↑‖f a‖₊ ^ (((p : ℝ) - 1) * q)\n        * (snorm f p μ).toNNReal ^ ((1 - (p : ℝ)) * q)) ∂μ) ^ (q : ℝ)⁻¹ := by {\n          congr 1; apply lintegral_congr (by\n          intro a; simp; rw [ENNReal.mul_rpow_of_nonneg (hz := by norm_num),\n        ← ENNReal.coe_rpow_of_nonneg _ (by norm_num; exact hpq.one_le),\n        ← ENNReal.coe_rpow_of_ne_zero (by rw [ENNReal.toNNReal_ne_zero]; exact ⟨hf0, hf'⟩),\n        ← ENNReal.rpow_mul, ← ENNReal.rpow_mul])}\n        _ = 1 := by\n          rw [lintegral_mul_const'' _ ((AEMeasurable.pow_const hf.aemeasurable.ennnorm) _),\n          (isConjExponent_coe.mpr hpq).sub_one_mul_conj, ENNReal.coe_toNNReal hf',\n          ← snorm_eq_lintegral_rpow_nnnorm' _ _ hpq.ne_zero, ← ENNReal.rpow_add _ _ hf0 hf',\n          sub_mul (1 : ℝ), (isConjExponent_coe.mpr hpq).mul_eq_add, one_mul,\n          sub_add_cancel_right, add_right_neg, ENNReal.rpow_zero, ENNReal.one_rpow]\n      have f_norm : ∫⁻ (a : α), ‖f a‖₊ * (g a) ∂ μ = snorm f p μ := by\n        simp only [g, Function.comp_apply]\n        calc ∫⁻ (a : α), ↑‖f a‖₊ * (↑(‖f a‖₊ ^ ((p : ℝ) - 1))\n          * ↑((snorm f p μ).toNNReal ^ (1 - (p : ℝ)))) ∂μ\n          = ∫⁻ (a : α), ↑‖f a‖₊ ^ (p : ℝ) * ↑((snorm f p μ).toNNReal ^ (1 - (p : ℝ))) ∂μ :=\n              lintegral_congr (by\n              intro _; rw [← mul_assoc]; congr\n              rw [← ENNReal.rpow_one ↑‖f _‖₊, ← ENNReal.coe_rpow_of_nonneg _ (by\n                norm_num; exact hpq.one_le),\n                ← ENNReal.rpow_add_of_nonneg _ _ (by norm_num) (by norm_num; exact hpq.one_le),\n                ENNReal.rpow_one, add_sub_cancel]\n              )\n        _ = (∫⁻ (a : α), ↑‖f a‖₊ ^ (p : ℝ) ∂ μ) * ↑((snorm f p μ).toNNReal ^ (1 - (p : ℝ))) := by\n          rw [lintegral_mul_const'' _ ((AEMeasurable.pow_const hf.aemeasurable.ennnorm) _)]\n        _ = snorm f p μ := by\n          rw [← snorm_eq_lintegral_rpow_nnnorm' _ _ hpq.ne_zero,\n              ← ENNReal.coe_rpow_of_ne_zero (ENNReal.toNNReal_ne_zero.mpr ⟨hf0, hf'⟩),\n              ENNReal.coe_toNNReal hf', ← ENNReal.rpow_add _ _ hf0 hf',\n              add_sub_cancel, ENNReal.rpow_one]\n      calc snorm f p μ = ∫⁻ (a : α), ‖f a‖₊ * (g a) ∂ μ := f_norm.symm\n      _ = ⨆ n, ∫⁻ a, ↑‖f a‖₊  * (eapprox g n a) ∂μ := by\n        apply mul_lintegral_eq_iSup_mul_eapprox_lintegral (f := fun a ↦ (‖f a‖₊ : ℝ≥0∞))\n          hf.ennnorm g_meas\n      _ ≤ sSup {∫⁻ (a : α), ‖f a‖₊ * (g.1 a) ∂ μ | g : {f : SimpleFunc α ℝ≥0∞ //\n      (∫⁻ a, (f a) ^ (q : ℝ) ∂ μ) ^ (q : ℝ)⁻¹ ≤ 1} } := by\n        apply iSup_le; intro n; apply le_sSup\n        simp only [Subtype.exists, exists_prop, mem_setOf_eq]\n        use eapprox g n\n        exact ⟨by\n          apply le_trans ?_ g_norm.le\n          gcongr\n          rw [← iSup_eapprox_apply _ g_meas]\n          apply le_iSup _ n, rfl⟩\n      _ ≤ sSup {∫⁻ (a : α), ‖f a‖₊ * (g.1 a) ∂ μ | g : {f : SimpleFunc α ℝ≥0 //\n       (∫⁻ a, (f a) ^ (q : ℝ) ∂ μ) ^ (q : ℝ)⁻¹ ≤ 1} } := by\n        gcongr\n        rintro x ⟨h, hh⟩\n        have ae := ae_lt_top_of_LpNorm_ne_top hpq.symm.ne_zero (SimpleFunc.measurable _)\n          (ne_top_of_le_ne_top one_ne_top h.2)\n        have : (∫⁻ (a : α), ↑(h.1 a).toNNReal ^ (q : ℝ) ∂μ) ^ (q : ℝ)⁻¹\n          = (∫⁻ (a : α), (h.1 a) ^ (q : ℝ) ∂μ) ^ (q : ℝ)⁻¹ := by\n          congr 1\n          apply lintegral_congr_ae\n          rw [Filter.EventuallyEq, Filter.Eventually, ← Filter.exists_mem_subset_iff]\n          use {a | h.1 a < ⊤}\n          exact ⟨ae, by simp; intro a ha; rw [ENNReal.coe_toNNReal ha.ne_top]⟩\n        use ⟨SimpleFunc.map ENNReal.toNNReal h.1, by simp [this, h.2]⟩\n        simp [← hh]\n        apply lintegral_congr_ae\n        rw [Filter.EventuallyEq, Filter.Eventually, ← Filter.exists_mem_subset_iff]\n        use {a | h.1 a < ⊤}\n        exact ⟨ae, by simp; intro a ha; rw [ENNReal.coe_toNNReal ha.ne_top]⟩\n      _ ≤ sSup {snorm (f * (g.1 : α → ℂ)) 1 μ | g : Lp.simpleLe1 ℂ q μ} := by\n        gcongr; rintro x ⟨h, hh⟩\n        use toLpSimpLe1 q hpq.symm.ne_zero _ h.2\n        convert hh\n        simp [snorm, snorm', toLpSimpLe1]","line":913,"end_line":1007,"proof_status":"formalized"}
+{"id":"a0000000151","label":"a0000000151","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(F \\in \\mathcal{D}'(\\Omega ) , \\psi \\in D\\). The following two distributions coincide: The distribution determined by the smooth function \\(x \\mapsto F(\\tau _x (\\psi ^R))\\). The distribution \\(\\phi \\mapsto F(\\psi ^R * \\phi )\\). We write this distribution as \\(F * \\psi \\).","doc_url":null,"latex_link":"sect0001.html#a0000000151","lean_link":null,"lean_decl_name":"a0000000151","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000176","label":"a0000000176","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For \\(F \\in D', \\phi \\in D \\) , We have \\[ \\partial ^\\alpha (F * \\phi ) = (\\partial ^\\alpha F) * \\phi = F * \\partial ^\\alpha \\phi \\]","doc_url":null,"latex_link":"sect0001.html#a0000000176","lean_link":null,"lean_decl_name":"a0000000176","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lemma:cont","label":"cont","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Convolution \\(F * \\psi \\) is continuous in both variables.","doc_url":null,"latex_link":"sect0001.html#lemma:cont","lean_link":null,"lean_decl_name":"cont","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000185","label":"a0000000185","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"We have a continuous inclusions \\(\\mathcal{D}\\subset \\mathcal{S}\\), hence \\(\\mathcal{S}' \\subset \\mathcal{D}'(\\mathbb {R}^d)\\). Moreover, this inclusion is dense. Hence being tempered is a property of distributions.","doc_url":null,"latex_link":"sect0001.html#a0000000185","lean_link":null,"lean_decl_name":"a0000000185","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"thm:maximum_modulus","label":"maximum_modulus","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(U\\) be a connected open set in a complex normed space \\(E\\). Let \\(f:E\\to F\\) be a function that complex differentiable on \\(U\\) and continuous on \\(\\bar{U}\\).If \\(|f(z)|\\) takes its maximum on a point \\(u\\in U\\), then it must be constant on \\(\\bar{U}\\).","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/Analysis.Complex.norm_eqOn_closure_of_isPreconnected_of_isMaxOn","latex_link":"chap-interpolation.html#thm:maximum_modulus","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/Analysis.Complex.norm_eqOn_closure_of_isPreconnected_of_isMaxOn","lean_decl_name":"Analysis.Complex.norm_eqOn_closure_of_isPreconnected_of_isMaxOn","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Analysis/Complex/AbsMax.lean","lean4_code":"theorem norm_eqOn_closure_of_isPreconnected_of_isMaxOn {f : E → F} {U : Set E} {c : E}\n    (hc : IsPreconnected U) (ho : IsOpen U) (hd : DiffContOnCl ℂ f U) (hcU : c ∈ U)\n    (hm : IsMaxOn (norm ∘ f) U c) : EqOn (norm ∘ f) (const E ‖f c‖) (closure U) :=\n  (norm_eqOn_of_isPreconnected_of_isMaxOn hc ho hd.differentiableOn hcU hm).of_subset_closure\n    hd.continuousOn.norm continuousOn_const subset_closure Subset.rfl\n\nsection StrictConvex\n\n/-!\n### The case of a strictly convex codomain\n\nIf the codomain `F` is a strictly convex space, then we can claim equalities like `f w = f z`\ninstead of `‖f w‖ = ‖f z‖`.\n\nInstead of repeating the proof starting with lemmas about integrals, we apply a corresponding lemma\nabove twice: for `f` and for `(f · + f c)`.  Then we have `‖f w‖ = ‖f z‖` and\n`‖f w + f z‖ = ‖f z + f z‖`, thus `‖f w + f z‖ = ‖f w‖ + ‖f z‖`. This is only possible if\n`f w = f z`, see `eq_of_norm_eq_of_norm_add_eq`.\n-/\n\nvariable [StrictConvexSpace ℝ F]\n\n/-- **Maximum modulus principle** on a connected set. Let `U` be a (pre)connected open set in a\ncomplex normed space.  Let `f : E → F` be a function that is complex differentiable on `U`. Suppose\nthat `‖f x‖` takes its maximum value on `U` at `c ∈ U`. Then `f x = f c` for all `x ∈ U`.\n\nTODO: change assumption from `IsMaxOn` to `IsLocalMax`. -/\n","line":246,"end_line":272,"proof_status":"formalized"}
+{"id":"lem:L12-dense","label":"L12-dense","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\(L^1(V,E)\\cap L^2(V,E)\\) is dense in \\(L^2(V,E)\\).","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/","latex_link":"chap-plancherel.html#lem:L12-dense","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/","lean_decl_name":"","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:fourier-transform","label":"fourier-transform","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(f\\in L^1(V,E)\\). Its Fourier transform (w.r.t. \\(L\\)) is the function \\(\\mathcal Ff=\\widehat f:W\\to E\\) given by \\[ \\mathcal Ff(w):=\\widehat f(w):=\\int _V e^{-2\\pi i L(v,w)}f(v)\\, d\\mu . \\]","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/VectorFourier.fourierIntegral","latex_link":"chap-plancherel.html#def:fourier-transform","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/VectorFourier.fourierIntegral","lean_decl_name":"VectorFourier.fourierIntegral","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Analysis/Fourier/FourierTransform.lean","lean4_code":"def fourierIntegral (e : AddChar 𝕜 𝕊) (μ : Measure V) (L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜) (f : V → E)\n    (w : W) : E :=\n  ∫ v, e (-L v w) • f v ∂μ\n\n","line":77,"end_line":80,"proof_status":"formalized"}
+{"id":"a0000000139","label":"a0000000139","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\(\\mathcal{D}(\\Omega ) = C_c^\\infty (\\Omega )\\) is the set of test functions together with a topology determined by its converging sequences : \\(\\phi _n \\to \\phi \\) in \\(\\mathcal{D}(\\Omega )\\) if there exists a compact subset \\(K \\subset \\Omega \\) such that \\(\\operatorname{Supp}(\\phi _n) \\subset K\\). For all multiindices \\(\\alpha \\) we have \\(\\partial _\\alpha \\phi _n \\to \\partial _\\alpha \\phi \\) in uniformly. Convention: if \\(x \\in \\mathbb {R}^d \\setminus \\Omega \\), then \\(\\phi (x) := 0\\).\\(\\mathcal{D}'(\\Omega )\\) is the topological dual space, i.e. the space of continuous linear functionals \\(D(\\Omega ) \\to \\mathbb {C}\\) with the weak-*-convergence, i.e. pointwise convergence.","doc_url":null,"latex_link":"sect0001.html#a0000000139","lean_link":null,"lean_decl_name":"a0000000139","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000182","label":"a0000000182","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Consider the increasing sequence of norms on \\(C^\\infty (\\Omega )\\) defined by \\[ \\| \\phi \\| _N = \\sup \\{ | x^\\beta (\\partial _x^\\alpha \\phi (x)) | \\mid x \\in \\mathbb {R}^d , |\\alpha |,\\| \\beta \\| \\le N\\} \\]","doc_url":null,"latex_link":"sect0001.html#a0000000182","lean_link":null,"lean_decl_name":"a0000000182","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000201","label":"a0000000201","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A fundamental solution of \\(L\\) is a distribution \\(F\\) such that \\(L(F) = \\delta \\)","doc_url":null,"latex_link":"sect0001.html#a0000000201","lean_link":null,"lean_decl_name":"a0000000201","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000207","label":"a0000000207","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The characteristic polynomial of \\(L\\) is \\[ P(\\xi ) = \\sum _{|\\alpha |\\le m} a_\\alpha (2 \\pi i \\xi )^\\alpha \\]","doc_url":null,"latex_link":"sect0001.html#a0000000207","lean_link":null,"lean_decl_name":"a0000000207","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lemma:slowlyInc","label":"slowlyInc","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(f \\in L^1_{loc}(\\mathbb {R}^d)\\) such that there exists \\(N \\ge 0\\) with \\[ \\int _{|x| {\\lt} R} |f(x)| \\ \\mathrm{d} x = O(R^N) , \\text{ as } R \\to \\infty \\]","doc_url":null,"latex_link":"sect0001.html#lemma:slowlyInc","lean_link":null,"lean_decl_name":"slowlyInc","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000193","label":"a0000000193","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"We have \\[ \\Lambda _{\\hat\\psi }(\\phi ) = \\int _{\\mathbb {R}^d} \\hat\\psi (x) \\phi (x) \\ \\mathrm{d} x = \\int _{\\mathbb {R}^d} \\psi (x) \\hat\\phi (x) \\ \\mathrm{d} x = \\Lambda _{\\psi }(\\hat\\phi ) \\]","doc_url":null,"latex_link":"sect0001.html#a0000000193","lean_link":null,"lean_decl_name":"a0000000193","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lemma:distribCommInt","label":"distribCommInt","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(\\phi \\in C_c^{\\infty }(\\Omega \\times \\Omega )\\). Then for any \\(F \\in D'(\\Omega )\\) we have \\[ F \\left(\\int _\\Omega \\phi (x,\\_ ) \\ \\mathrm{d} x \\right) = \\int _\\Omega F(\\phi (x ,\\_ )) \\ \\mathrm{d} x \\]","doc_url":null,"latex_link":"sect0001.html#lemma:distribCommInt","lean_link":null,"lean_decl_name":"distribCommInt","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000188","label":"a0000000188","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(F \\in \\mathcal{D}'\\) has compact support then it is tempered: Choose \\(\\eta \\in \\mathcal{D}\\) such that \\(\\eta |_U = 1\\) on some neighborhood \\(U \\supset \\operatorname{Supp}(F)\\). Then \\(F(\\eta \\phi ) = F(\\phi )\\) for all \\(\\phi \\in \\mathcal{D}\\) and \\(\\phi \\mapsto F (\\eta \\phi )\\) defines a continuous functional on \\(\\mathcal{S}\\). It does not depend on the choice of \\(\\eta \\), because given another such \\(\\eta '\\) and any \\(\\phi \\in \\mathcal{S}\\) we have \\(\\operatorname{Supp}((\\eta - \\eta ') \\cdot \\phi ) \\cap \\operatorname{Supp}(F) = \\varnothing \\).","doc_url":null,"latex_link":"sect0001.html#a0000000188","lean_link":null,"lean_decl_name":"a0000000188","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"proposition:deltaInClosure","label":"deltaInClosure","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"There exists a sequence \\(\\psi _n \\in C_c^{\\infty }(\\Omega )\\) such that \\(\\Lambda \\psi _n \\to \\delta _0\\) in \\(D'(\\Omega )\\).","doc_url":null,"latex_link":"sect0001.html#proposition:deltaInClosure","lean_link":null,"lean_decl_name":"deltaInClosure","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000214","label":"a0000000214","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(d = 2\\), the function \\(F := 1/(2\\pi ) \\log |x| \\in L^1_{loc}\\) is a fundamental solution of \\(\\Delta \\)","doc_url":null,"latex_link":"sect0001.html#a0000000214","lean_link":null,"lean_decl_name":"a0000000214","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000202","label":"a0000000202","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The operator \\[ f \\mapsto T(f) := F * f \\]","doc_url":null,"latex_link":"sect0001.html#a0000000202","lean_link":null,"lean_decl_name":"a0000000202","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:hoelder'","label":"hoelder'","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\((X, \\mu )\\) be a measure space and \\(0{\\lt} p_0 {\\lt}p_1\\leq \\infty \\), \\(0 \\le p \\le \\infty \\). Assume we have \\(t \\ge 0\\) such that \\[ \\frac{1}{p} = \\frac{1-t}{p_0} + \\frac{t}{p_1} \\]","doc_url":null,"latex_link":"chap-interpolation.html#lem:hoelder'","lean_link":null,"lean_decl_name":"hoelder'","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}

crawled_repos/ChandraFurstLipton.jsonl

@@ -0,0 +1,31 @@
+{"id":"def:forget","label":"forget","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For an index \\(i : [d]\\), we define \\begin{align} \\operatorname{forget}_i : G^d & \\to G^{\\{ j : [d] \\mid j \\ne i\\} } \\\\ x & \\mapsto j \\mapsto x_j \\end{align}","doc_url":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.forget","latex_link":"sect0001.html#def:forget","lean_link":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.forget","lean_decl_name":"NOF.forget","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChandraFurstLipton","source_path":"ChandraFurstLipton/MultidimCorners.lean","lean4_code":"def forget (i : ι) (x : ι → G) (j : {j : ι // j ≠ i}) : G := x j\n","line":6,"end_line":6,"proof_status":"formalized"}
+{"id":"def:broadcast","label":"broadcast","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:forget","def:protocol"],"nlp_text":"Given a NOF protocol \\(P\\), the NOF broadcast on input \\(x : G^d\\) is inductively defined by \\begin{align} \\operatorname{broad}(x) : \\mathbb {N}& \\to \\operatorname{List}\\operatorname{Bool}\\\\ 0 & \\mapsto [] \\\\ t + 1 & \\mapsto \\operatorname{strat}_{t \\% d}(\\operatorname{forget}_{t \\% d}(x), \\operatorname{broad}(x, t)) :: \\operatorname{broad}(x, t) \\end{align}","doc_url":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.Protocol.broadcast","latex_link":"sect0002.html#def:broadcast","lean_link":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.Protocol.broadcast","lean_decl_name":"NOF.Protocol.broadcast","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChandraFurstLipton","source_path":"ChandraFurstLipton/NOFModel.lean","lean4_code":"def broadcast (P : Protocol G d) (x : ZMod d → G) : ℕ → List Bool\n  | 0 => []\n  | t + 1 => P.nextBit t (forget (t : ZMod d) x) (broadcast P x t) :: broadcast P x t\n","line":19,"end_line":21,"proof_status":"formalized"}
+{"id":"lem:forbidden-pattern-project-multicorner","label":"forbidden-pattern-project-multicorner","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:forget","def:multicorner","def:eval"],"nlp_text":"If \\((a_1, \\dots , a_d)\\) is a forbidden pattern such that \\(\\operatorname{eval}(a_i) = 1\\) for all \\(i\\), then \\[ (\\operatorname{forget}_i(a_1), \\dots , \\operatorname{forget}_i(a_d)) \\]","doc_url":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.isMultidimCorner_forget_of_isForbiddenPattern","latex_link":"sect0003.html#lem:forbidden-pattern-project-multicorner","lean_link":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.isMultidimCorner_forget_of_isForbiddenPattern","lean_decl_name":"NOF.isMultidimCorner_forget_of_isForbiddenPattern","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChandraFurstLipton","source_path":"ChandraFurstLipton/MultidimCorners.lean","lean4_code":"lemma isMultidimCorner_forget_of_isForbiddenPattern (a : ι → ι → G) (h : IsForbiddenPattern a)\n    (hS : ∀ i, ∑ j, a i j = 0) (i : ι) :\n    IsMultidimCorner (fun j ↦ forget i (a j)) (forget i (a i)) := by\n    rw [IsForbiddenPattern] at h\n    obtain ⟨v, hv⟩ := h\n    refine ⟨fun k l ↦ ?_, fun k l hneq ↦ ?_⟩\n    · rw [← sub_eq_zero]\n      calc\n        ∑ j : {j // j ≠ i}, a k j - ∑ j : { j // j ≠ i }, a l j\n          = ∑ j : {j // j ≠ i}, (a k j - a l j) := by rw [Finset.sum_sub_distrib]\n        _ = ∑ j ∈ {i}ᶜ, (a k j - a l j) := (Finset.sum_subtype _ (by simp) (a k - a l)).symm\n        _ = ∑ j ∈ {i}ᶜ, (a k j - a l j) + (a k i - a l i) := by simp [hv k.2, hv l.2]\n        _ = ∑ j : ι, (a k j - a l j) := by rw [← Fintype.sum_eq_sum_compl_add]\n        _ = 0 := by rw [Finset.sum_sub_distrib, sub_eq_zero, hS k, hS l]\n    · rw [forget, forget, hv, hv l.2.symm]\n      aesop\n","line":27,"end_line":42,"proof_status":"formalized"}
+{"id":"lem:length-broadcast","label":"length-broadcast","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:broadcast"],"nlp_text":"For every NOF protocol \\(P\\), every input \\(x : G^d\\) and every time \\(t\\), \\(\\operatorname{broad}(x, t)\\) has length \\(t\\).","doc_url":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.Protocol.length_broadcast","latex_link":"sect0002.html#lem:length-broadcast","lean_link":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.Protocol.length_broadcast","lean_decl_name":"NOF.Protocol.length_broadcast","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChandraFurstLipton","source_path":"ChandraFurstLipton/NOFModel.lean","lean4_code":"@[simp]\nlemma length_broadcast (P : Protocol G d) (x : ZMod d → G) : ∀ t, (broadcast P x t).length = t\n  | 0 => rfl\n  | t + 1 => by simpa [broadcast_succ] using length_broadcast _ _ _\n","line":32,"end_line":35,"proof_status":"formalized"}
+{"id":"def:valid-protocol","label":"valid-protocol","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:broadcast"],"nlp_text":"Given a function \\(F : G^d \\to \\operatorname{Bool}\\), the NOF protocol \\(P\\) is valid in \\(F\\) at time \\(t\\) on input \\(x\\) if all participants correctly guess \\(F(x)\\), namely if \\[ \\operatorname{guess}_i(\\operatorname{forget}_i(x), \\operatorname{broad}(x, t)) = F(x) \\]","doc_url":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.Protocol.IsValid","latex_link":"sect0002.html#def:valid-protocol","lean_link":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.Protocol.IsValid","lean_decl_name":"NOF.Protocol.IsValid","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChandraFurstLipton","source_path":"ChandraFurstLipton/NOFModel.lean","lean4_code":"def IsValid (P : Protocol G d) (F : (ZMod d → G) → Bool) (t : ℕ) : Prop :=\n  ∀ x i, P.guess i (forget i x) (broadcast P x t) = F x\n","line":29,"end_line":30,"proof_status":"formalized"}
+{"id":"thm:det-fun-complex-eval-ge-corner-color-num","label":"det-fun-complex-eval-ge-corner-color-num","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["lem:forbidden-pattern-project-multicorner","lem:mono-forbidden-pattern-trivial","def:det-fun-complex","def:corner-color-num"],"nlp_text":"\\[ D(\\operatorname{eval}) \\ge \\lceil \\log _2\\chi _d(G)\\rceil \\]","doc_url":null,"latex_link":"sect0003.html#thm:det-fun-complex-eval-ge-corner-color-num","lean_link":null,"lean_decl_name":"det-fun-complex-eval-ge-corner-color-num","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"thm:det-fun-complex-eval-le-corner-color-num","label":"det-fun-complex-eval-le-corner-color-num","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["lem:forbidden-pattern-project-multicorner","lem:non-monochromatic-protocol-valid","def:det-fun-complex","def:corner-color-num"],"nlp_text":"\\[ D(\\operatorname{eval}) \\le \\lceil \\log _2\\chi _d(G)\\rceil + d \\]","doc_url":null,"latex_link":"sect0004.html#thm:det-fun-complex-eval-le-corner-color-num","lean_link":null,"lean_decl_name":"det-fun-complex-eval-le-corner-color-num","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:mono-forbidden-pattern-tip","label":"mono-forbidden-pattern-tip","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["lem:length-broadcast","def:valid-protocol","def:forbidden-pattern"],"nlp_text":"Given \\(P\\) a NOF protocol and a time \\(t\\), if \\((a_1, \\dots , a_d)\\) is a forbidden pattern with tip \\(v\\) such that \\(\\operatorname{broad}(a_i, t)\\) equals some fixed broadcast history \\(b\\) for all \\(i\\), then \\(\\operatorname{broad}(v, t) = b\\) as well.","doc_url":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.IsForbiddenPatternWithTip.broadcast_eq","latex_link":"sect0002.html#lem:mono-forbidden-pattern-tip","lean_link":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.IsForbiddenPatternWithTip.broadcast_eq","lean_decl_name":"NOF.IsForbiddenPatternWithTip.broadcast_eq","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/YaelDillies/ChandraFurstLipton/368d5e4e6edadf50a32df879276214e6fba80af2/ChandraFurstLipton/NOFModel.lean","lean4_code":"lemma IsForbiddenPatternWithTip.broadcast_eq (hF : IsForbiddenPatternWithTip a v)\n    (hB : ∀ i, P.broadcast (a i) t = B) : P.broadcast v t = B := by\n  induction t generalizing B with\n  | zero => simpa using hB 0\n  | succ t ih =>\n  simp_rw [Protocol.broadcast_succ] at *\n  obtain _ | ⟨b, B⟩ := B\n  · cases hB 0\n  simp only [List.cons.injEq, forall_and] at hB\n  specialize ih hB.2\n  subst ih\n  have h₁ : forget (t : ZMod d) v = forget (t : ZMod d) (a t) := hF.forget t\n  have h₂ : P.broadcast v t = P.broadcast (a t) t := (hB.2 t).symm\n  rw [h₁, h₂, hB.1]","line":51,"end_line":64,"proof_status":"formalized"}
+{"id":"def:rand-protocol-complex","label":"rand-protocol-complex","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:valid-protocol"],"nlp_text":"The communication complexity of a randomised NOF protocol \\(P\\) for \\(F\\) with error \\(\\epsilon \\) is the smallest time \\(t\\) such that \\[ \\mathbb P(x \\mid P \\text{ is not valid at time } t) \\le \\epsilon \\]","doc_url":null,"latex_link":"sect0005.html#def:rand-protocol-complex","lean_link":null,"lean_decl_name":"rand-protocol-complex","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:trivial-protocol-valid","label":"trivial-protocol-valid","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:valid-protocol","def:trivial-protocol"],"nlp_text":"For all \\(F\\), the trivial protocol for \\(F\\) is valid in time \\(d\\left\\lceil \\log _2 n\\right\\rceil \\).","doc_url":null,"latex_link":"sect0002.html#lem:trivial-protocol-valid","lean_link":null,"lean_decl_name":"trivial-protocol-valid","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:det-protocol-complex","label":"det-protocol-complex","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:valid-protocol"],"nlp_text":"The communication complexity of a NOF protocol \\(P\\) for \\(F\\) is the smallest time \\(t\\) such that \\(P\\) is valid in \\(F\\) at time \\(t\\) on all inputs \\(x\\), or \\(\\infty \\) if no such \\(t\\) exists.","doc_url":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.Protocol.complexity","latex_link":"sect0002.html#def:det-protocol-complex","lean_link":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.Protocol.complexity","lean_decl_name":"NOF.Protocol.complexity","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChandraFurstLipton","source_path":"ChandraFurstLipton/NOFModel.lean","lean4_code":"noncomputable\ndef complexity (P : Protocol G d) (F : (ZMod d → G) → Bool) : ENat :=\n  ⨅ (t : ℕ) (_ : IsValid P F t), t\n","line":37,"end_line":39,"proof_status":"formalized"}
+{"id":"lem:non-monochromatic-protocol-valid","label":"non-monochromatic-protocol-valid","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:valid-protocol","def:non-monochromatic-protocol"],"nlp_text":"If \\(c\\) is a coloring such that all monochromatic forbidden patterns are trivial, then the non-monochromatic protocol for \\(c\\) is valid in time \\(\\lceil \\log _2\\chi _d(G)\\rceil + d\\).","doc_url":null,"latex_link":"sect0004.html#lem:non-monochromatic-protocol-valid","lean_link":null,"lean_decl_name":"non-monochromatic-protocol-valid","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:rand-eq-test-protocol-eval-valid","label":"rand-eq-test-protocol-eval-valid","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:valid-protocol","def:rand-eq-test-protocol-eval"],"nlp_text":"The randomised equality testing protocol is valid for \\(\\operatorname{eval}\\) at time \\(2d\\).","doc_url":null,"latex_link":"sect0005.html#lem:rand-eq-test-protocol-eval-valid","lean_link":null,"lean_decl_name":"rand-eq-test-protocol-eval-valid","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"cor:det-fun-complex-eval-ge-corner-free-num","label":"det-fun-complex-eval-ge-corner-free-num","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["thm:det-fun-complex-eval-ge-corner-color-num","lem:corner-num-lower"],"nlp_text":"\\[ D(\\operatorname{eval}) \\ge d\\log _2\\frac N{r_d(G)} \\]","doc_url":null,"latex_link":"sect0003.html#cor:det-fun-complex-eval-ge-corner-free-num","lean_link":null,"lean_decl_name":"det-fun-complex-eval-ge-corner-free-num","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"cor:det-fun-complex-eval-le-corner-free-num","label":"det-fun-complex-eval-le-corner-free-num","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["thm:det-fun-complex-eval-le-corner-color-num","lem:corner-num-upper"],"nlp_text":"\\[ D(\\operatorname{eval}) \\le 2d\\log _2\\frac N{r_d(G)} \\]","doc_url":null,"latex_link":"sect0004.html#cor:det-fun-complex-eval-le-corner-free-num","lean_link":null,"lean_decl_name":"det-fun-complex-eval-le-corner-free-num","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:mono-forbidden-pattern-trivial","label":"mono-forbidden-pattern-trivial","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["lem:mono-forbidden-pattern-tip","def:eval"],"nlp_text":"Given \\(P\\) a NOF protocol valid in time \\(t\\) for \\(\\operatorname{eval}\\), all monochromatic forbidden patterns are trivial.","doc_url":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.trivial_of_isForbiddenPattern_of_isValid_eval","latex_link":"sect0003.html#lem:mono-forbidden-pattern-trivial","lean_link":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.trivial_of_isForbiddenPattern_of_isValid_eval","lean_decl_name":"NOF.trivial_of_isForbiddenPattern_of_isValid_eval","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChandraFurstLipton","source_path":"ChandraFurstLipton/LowerBoundEval.lean","lean4_code":"lemma trivial_of_isForbiddenPattern_of_isValid_eval (ha : IsForbiddenPattern a)\n    (hP : P.IsValid eval t) (hE : ∀ i, eval (a i) = true) (hB : ∀ i, P.broadcast (a i) t = B) :\n    ∀ i j, a i = a j := by\n  obtain ⟨v, ha⟩ := ha\n  have : P.broadcast v t = B := ha.broadcast_eq hB\n  have h i : eval (a i) = eval v :=\n    calc\n      _ = P.guess i (forget i (a i)) (P.broadcast (a i) t) := by rw [hP]\n      _ = P.guess i (forget i v) (P.broadcast v t) := by rw [hB, ha.broadcast_eq hB, ha.forget]\n      _ = eval v := by rw [hP]\n  suffices h : ∀ i, a i = v by simp [h]\n  intro i\n  ext j\n  obtain hij | rfl := ne_or_eq i j\n  · exact ha hij\n  simp only [hE, Bool.true_eq, eval_eq_true, forall_const] at h\n  rw [← sub_eq_zero]\n  calc\n    a i i - v i = ∑ j, (a i j - v j) := by\n      rw [Fintype.sum_eq_single]; simpa [eq_comm, sub_eq_zero] using @ha i\n    _ = ∑ j, a i j - ∑ j, v j := by rw [Finset.sum_sub_distrib]\n    _ = 0 := by simpa [h] using hE i\n","line":18,"end_line":39,"proof_status":"formalized"}
+{"id":"def:rand-complex","label":"rand-complex","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:rand-protocol-complex"],"nlp_text":"The randomised communication complexity of a function \\(F\\) with error \\(\\epsilon \\), denoted \\(R_\\epsilon (F)\\), is the minimum of the randomised communication complexity of \\(P\\) when \\(P\\) ranges over all randomised NOF protocols.","doc_url":null,"latex_link":"sect0005.html#def:rand-complex","lean_link":null,"lean_decl_name":"rand-complex","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:trivial-bound-det-fun-complex","label":"trivial-bound-det-fun-complex","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["lem:trivial-protocol-valid","def:det-fun-complex"],"nlp_text":"The communication complexity of any function \\(F\\) is at most \\(d\\left\\lceil \\log _2 n\\right\\rceil \\).","doc_url":null,"latex_link":"sect0002.html#lem:trivial-bound-det-fun-complex","lean_link":null,"lean_decl_name":"trivial-bound-det-fun-complex","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:det-fun-complex","label":"det-fun-complex","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:det-protocol-complex"],"nlp_text":"The deterministic communication complexity of a function \\(F\\), denoted \\(D(F)\\), is the minimum of the communication complexity of \\(P\\) when \\(P\\) ranges over all NOF protocols.","doc_url":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.funComplexity","latex_link":"sect0002.html#def:det-fun-complex","lean_link":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.funComplexity","lean_decl_name":"NOF.funComplexity","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChandraFurstLipton","source_path":"ChandraFurstLipton/NOFModel.lean","lean4_code":"noncomputable def funComplexity (F : (ZMod d → G) → Bool) := ⨅ P : Protocol G d, P.complexity F\n","line":46,"end_line":46,"proof_status":"formalized"}
+{"id":"thm:eval-rand-complexity","label":"eval-rand-complexity","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["lem:rand-eq-test-protocol-eval-valid","def:rand-complex","def:eval"],"nlp_text":"\\[ R_\\epsilon (\\operatorname{eval}) \\le 2d\\lceil \\log _2\\epsilon ^{-1}\\rceil \\]","doc_url":null,"latex_link":"sect0005.html#thm:eval-rand-complexity","lean_link":null,"lean_decl_name":"eval-rand-complexity","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:corner-num-upper","label":"corner-num-upper","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:corner-free-num","def:corner-color-num"],"nlp_text":"\\[ r_d(G) \\chi _d(G) \\le 2d N^d \\log N \\]","doc_url":null,"latex_link":"sect0001.html#lem:corner-num-upper","lean_link":null,"lean_decl_name":"corner-num-upper","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:corner-num-lower","label":"corner-num-lower","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:corner-free-num","def:corner-color-num"],"nlp_text":"\\[ r_d(G) \\chi _d(G) \\ge N^d \\]","doc_url":null,"latex_link":"sect0001.html#lem:corner-num-lower","lean_link":null,"lean_decl_name":"corner-num-lower","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:corner-free-num","label":"corner-free-num","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:multicorner"],"nlp_text":"The \\(d\\)-dimensional corner-free number of \\(G\\), denoted \\(r_d(G)\\) is the size of the largest set \\(A\\) in \\(G^d\\) such that \\(A\\) doesn’t contain a non-trivial corner.","doc_url":null,"latex_link":"sect0001.html#def:corner-free-num","lean_link":null,"lean_decl_name":"corner-free-num","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:protocol","label":"protocol","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A NOF protocol \\(P\\) consists of maps \\begin{align} \\operatorname{strat}: & [d] \\to G^{d - 1} \\to \\operatorname{List}\\operatorname{Bool}\\to \\operatorname{Bool}\\\\ \\operatorname{guess}: & [d] \\to G^{d - 1} \\to \\operatorname{List}\\operatorname{Bool}\\to \\operatorname{Bool}\\end{align}","doc_url":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.Protocol","latex_link":"sect0002.html#def:protocol","lean_link":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.Protocol","lean_decl_name":"NOF.Protocol","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChandraFurstLipton","source_path":"./ChandraFurstLipton/LowerBoundEval.lean","lean4_code":"def Protocol.trivial (hd : 3 ≤ d) (F : (ZMod d → G) → Bool) : Protocol G d where\n  nextBit i x B := by\n    refine (Nat.bits (Fintype.equivFin G (x ⟨i + 1, ?_ ⟩))).getI (B.length / d)\n    rw [Ne, add_eq_left, ← Nat.cast_one, ZMod.natCast_eq_zero_iff, Nat.dvd_one]\n    omega\n  guess i x B := F fun j ↦\n    if h : j = i then\n      (Fintype.equivFin G).symm <| (ZMod.finEquiv _).symm\n        (BitVec.toNat (BitVec.ofBoolListLE (getBits B i.val d)))\n    else\n      x ⟨j, h⟩\n\n","line":50,"end_line":61,"proof_status":"formalized"}
+{"id":"def:rand-eq-test-protocol-eval","label":"rand-eq-test-protocol-eval","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:protocol"],"nlp_text":"The randomised equality testing protocol for \\(\\operatorname{eval}\\) has domain \\(\\Omega := (\\operatorname{Bool}^d)^{\\lceil \\log _2\\epsilon ^{-1}\\rceil }\\) with the uniform measure and is defined by making participant \\(i\\) do “Compute \\[ a_{i, k} = \\sum _{j \\ne i} \\omega _{j, k}x_j \\]","doc_url":null,"latex_link":"sect0005.html#def:rand-eq-test-protocol-eval","lean_link":null,"lean_decl_name":"rand-eq-test-protocol-eval","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:trivial-protocol","label":"trivial-protocol","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:protocol"],"nlp_text":"For all \\(F\\), we define the trivial protocol by making participant \\(i\\) do \"Send the \\(t / d\\)-th bit of the number of participant \\(i + 1\\)\" and \"Compute \\(x_i\\) from the binary representation given by participant \\(i - 1\\), then compute \\(F(x)\\)\".","doc_url":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.Protocol.trivial","latex_link":"sect0002.html#def:trivial-protocol","lean_link":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.Protocol.trivial","lean_decl_name":"NOF.Protocol.trivial","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChandraFurstLipton","source_path":"ChandraFurstLipton/LowerBoundEval.lean","lean4_code":"noncomputable\ndef Protocol.trivial (hd : 3 ≤ d) (F : (ZMod d → G) → Bool) : Protocol G d where\n  nextBit i x B := by\n    refine (Nat.bits (Fintype.equivFin G (x ⟨i + 1, ?_ ⟩))).getI (B.length / d)\n    rw [Ne, add_eq_left, ← Nat.cast_one, ZMod.natCast_eq_zero_iff, Nat.dvd_one]\n    omega\n  guess i x B := F fun j ↦\n    if h : j = i then\n      (Fintype.equivFin G).symm <| (ZMod.finEquiv _).symm\n        (BitVec.toNat (BitVec.ofBoolListLE (getBits B i.val d)))\n    else\n      x ⟨j, h⟩\n","line":49,"end_line":60,"proof_status":"statement_formalized"}
+{"id":"def:non-monochromatic-protocol","label":"non-monochromatic-protocol","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:protocol"],"nlp_text":"Given a coloring \\(c : \\{ x \\mid \\operatorname{eval}x = 1\\} \\to [C]\\), writing \\(a_i\\) the vector whose \\(j\\)-th coordinate is \\(x_j\\) except when \\(j = i\\) in which case it is \\(-\\sum _{j \\ne i} x_j\\), we define the non-monochromatic protocol for \\(c\\) by making participant \\(i\\) do “Send the \\(t / d\\)-th bit of \\(c(a_i)\\) until time \\(\\lceil \\log _2\\chi _d(G)\\rceil \\), then send \\(1\\) iff \\(c(a_i)\\) agrees with the broadcast from time \\(1\\) to time \\(\\lceil \\log _2\\chi _d(G)\\rceil \\) read as a color” and “Send \\(1\\) iff the broadcasts from time \\(\\lceil \\log _2\\chi _d(G)\\rceil \\) to time \\(\\lceil \\log _2\\chi _d(G)\\rceil + d\\) were all \\(1\\)”.","doc_url":null,"latex_link":"sect0004.html#def:non-monochromatic-protocol","lean_link":null,"lean_decl_name":"non-monochromatic-protocol","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:forbidden-pattern","label":"forbidden-pattern","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"We say a tuple \\((a_1, \\dots , a_d) : (G^d)^d\\) is a forbidden pattern with tip \\(v : G^d\\) if \\[ a_{i, j} = v_j \\]","doc_url":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.IsForbiddenPatternWithTip","latex_link":"sect0001.html#def:forbidden-pattern","lean_link":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.IsForbiddenPatternWithTip","lean_decl_name":"NOF.IsForbiddenPatternWithTip","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChandraFurstLipton","source_path":"./ChandraFurstLipton/NOFModel.lean","lean4_code":"lemma IsForbiddenPatternWithTip.broadcast_eq (hF : IsForbiddenPatternWithTip a v)\n    (hB : ∀ i, P.broadcast (a i) t = B) : P.broadcast v t = B := by\n  induction t generalizing B with\n  | zero => simpa using hB 0\n  | succ t ih =>\n  simp_rw [Protocol.broadcast_succ] at *\n  obtain _ | ⟨b, B⟩ := B\n  · cases hB 0\n  simp only [List.cons.injEq, forall_and] at hB\n  specialize ih hB.2\n  subst ih\n  have h₁ : forget (t : ZMod d) v = forget (t : ZMod d) (a t) := hF.forget t\n  have h₂ : P.broadcast v t = P.broadcast (a t) t := (hB.2 t).symm\n  rw [h₁, h₂, hB.1]\n\n","line":51,"end_line":65,"proof_status":"formalized"}
+{"id":"def:multicorner","label":"multicorner","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:forbidden-pattern"],"nlp_text":"A multidimensional corner in \\(d\\) dimensions is a tuple of the form \\((x, x + ce_1, \\dots , x + ce_d)\\) for some \\(x : G^d\\) and \\(c : G\\), where \\(ce_i\\) is the vector of all zeroes except in position \\(i\\) where it is \\(c\\). Such a corner is said to be trivial if \\(c = 0\\).","doc_url":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.IsMultidimCorner","latex_link":"sect0001.html#def:multicorner","lean_link":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.IsMultidimCorner","lean_decl_name":"NOF.IsMultidimCorner","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChandraFurstLipton","source_path":"ChandraFurstLipton/MultidimCorners.lean","lean4_code":"structure IsMultidimCorner (a : ι → ι → G) (b : ι → G) : Prop where\n  sum_eq_sum : ∀ i j, ∑ k, a i k = ∑ k, a j k\n  isForbiddenPatternWithTip : IsForbiddenPatternWithTip a b\n","line":21,"end_line":23,"proof_status":"formalized"}
+{"id":"def:corner-color-num","label":"corner-color-num","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:multicorner"],"nlp_text":"The \\(d\\)-dimensional corner-coloring number of \\(G\\), denoted \\(\\chi _d(G)\\), is the smallest number of colors one needs to color \\(G^d\\) such that no non-trivial \\(d\\)-dimensional corner is monochromatic.","doc_url":null,"latex_link":"sect0001.html#def:corner-color-num","lean_link":null,"lean_decl_name":"corner-color-num","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:eval","label":"eval","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The \\(\\operatorname{eval}\\) function is defined by \\begin{align} \\operatorname{eval}: G^d & \\to \\operatorname{Bool}\\\\ x & \\mapsto \\begin{cases} 1 & \\text{ if } \\sum _i x_i = 0 \\\\ 0 & \\text{ else} \\end{cases}\\end{align}","doc_url":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.eval","latex_link":"sect0003.html#def:eval","lean_link":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.eval","lean_decl_name":"NOF.eval","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChandraFurstLipton","source_path":"ChandraFurstLipton/LowerBoundEval.lean","lean4_code":"def eval (x : ι → G) : Bool :=\n  ∑ i, x i == 0\n","line":11,"end_line":12,"proof_status":"formalized"}

crawled_repos/ChebyshevCircles.jsonl

@@ -0,0 +1,18 @@
+{"id":"def:rotated_roots","label":"rotated_roots","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"When the \\(N\\)-th roots of unity are rotated by an angle \\(\\theta \\) (equivalently, multiplied by \\(\\mathrm{e}^{i\\theta }\\)) and projected onto the real axis, we obtain the \\(N\\) real numbers \\begin{equation} r_k(\\theta ) = \\cos \\left(\\theta + \\frac{2\\pi k}{N}\\right), \\quad k = 0, 1, \\ldots , N-1. \\end{equation}","doc_url":"../../docs/find/#doc/realProjection_eq_cos","latex_link":"sect0001.html#def:rotated_roots","lean_link":"../../docs/find/#doc/realProjection_eq_cos","lean_decl_name":"realProjection_eq_cos","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":"./ChebyshevCircles/Proofs/RootsOfUnity.lean","lean4_code":"theorem realProjection_eq_cos (N : ℕ) (θ : ℝ) (k : ℕ) (_hk : k < N) :\n    (exp (I * (θ + 2 * π * k / N))).re = Real.cos (θ + 2 * π * k / N) := by\n  rw [mul_comm]\n  convert exp_ofReal_mul_I_re (θ + 2 * π * k / N)\n  norm_cast\n\n/-- All real projections lie in the closed interval [-1, 1]. -/\n","line":40,"end_line":46,"proof_status":"formalized"}
+{"id":"def:unscaled_poly","label":"unscaled_poly","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":["def:rotated_roots"],"nlp_text":"Let \\(P_N(x; \\theta )\\) denote the monic polynomial having the rotated roots as roots: \\begin{equation} P_N(x; \\theta ) = \\prod _{k=0}^{N-1} \\left(x - r_k(\\theta )\\right). \\end{equation}","doc_url":"../../docs/find/#doc/polynomialFromRealRoots_degree","latex_link":"sect0001.html#def:unscaled_poly","lean_link":"../../docs/find/#doc/polynomialFromRealRoots_degree","lean_decl_name":"polynomialFromRealRoots_degree","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":"./ChebyshevCircles/Proofs/PolynomialConstruction.lean","lean4_code":"theorem polynomialFromRealRoots_degree (roots : List ℝ) :\n    (polynomialFromRealRoots roots).degree = roots.length := by\n  induction roots with\n  | nil => simp [polynomialFromRealRoots]\n  | cons r rs ih =>\n    unfold polynomialFromRealRoots at ih ⊢\n    simp only [List.foldr, List.length_cons]\n    rw [degree_mul, degree_X_sub_C, ih]\n    norm_cast\n    ring\n\n/-- The unscaled polynomial has degree N. -/\n","line":58,"end_line":69,"proof_status":"formalized"}
+{"id":"def:scaled_poly","label":"scaled_poly","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":["def:unscaled_poly"],"nlp_text":"To match the normalization of Chebyshev polynomials, which have leading coefficient \\(2^{N-1}\\) for \\(N \\geq 1\\), we define the scaled polynomial: \\begin{equation} S_N(x; \\theta ) = 2^{N-1} \\cdot P_N(x; \\theta ). \\end{equation}","doc_url":"../../docs/find/#doc/scaledPolynomial_degree","latex_link":"sect0001.html#def:scaled_poly","lean_link":"../../docs/find/#doc/scaledPolynomial_degree","lean_decl_name":"scaledPolynomial_degree","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":"./ChebyshevCircles/Proofs/PolynomialConstruction.lean","lean4_code":"theorem scaledPolynomial_degree (N : ℕ) (θ : ℝ) (_hN : 0 < N) :\n    (scaledPolynomial N θ).degree = N := by\n  unfold scaledPolynomial\n  rw [degree_C_mul, unscaledPolynomial_degree N θ]\n  exact pow_ne_zero (N - 1) two_ne_zero\n\n/-- The unscaled polynomial is monic (leading coefficient is 1). -/\n","line":76,"end_line":82,"proof_status":"formalized"}
+{"id":"thm:main","label":"main","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":["def:scaled_poly","lem:chebyshev_roots","thm:coefficients_independent"],"nlp_text":"For any positive integer \\(N \\geq 1\\) and any angle \\(\\theta \\in \\mathbb {R}\\), \"> \\begin{equation} \\label{eq:main_result} S_N(x; \\theta ) = T_N(x) + c(\\theta ), \\end{equation}","doc_url":"../../docs/find/#doc/mainTheorem","latex_link":"sect0001.html#thm:main","lean_link":"../../docs/find/#doc/mainTheorem","lean_decl_name":"mainTheorem","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":"./index.html","lean4_code":"theorem mainTheorem (N : ℕ) (θ : ℝ) (hN : 0 < N) :<br>\n&nbsp;&nbsp;scaledPolynomial N θ = Polynomial.Chebyshev.T ℝ N + C (explicitConstant N θ)\n                </div>\n                <p style=\"margin-top: 0.75rem; font-size: 0.9rem; color: #555;\">\n                    where <code>explicitConstant N θ = 2^(N-1) * (-1)^N * ∏_{k=0}^{N-1} cos(θ + 2πk/N) - T_N(0)</code>\n                </p>\n                <p style=\"margin-top: 1rem; font-size: 0.95rem;\">\n                    <a href=\"./docs/ChebyshevCircles/ProofOfMainTheorem.html#mainTheorem\" style=\"color: #3498db; text-decoration: none; font-weight: 500;\">View theorem statement in API docs →</a>\n                </p>\n                <div class=\"construction\">\n                    <strong>Construction:</strong>\n                    <ul>\n                        <li>Projected roots: <code>cos(θ + 2πk/N)</code> for k = 0, ..., N-1 <a href=\"./blueprint/sect0002.html\" style=\"color: #3498db; text-decoration: none; font-size: 0.85rem;\">(blueprint §2)</a></li>\n                        <li>Unscaled polynomial: <code>P(x) = ∏(x - cos(θ + 2πk/N))</code> <a href=\"./blueprint/sect0003.html\" style=\"color: #3498db; text-decoration: none; font-size: 0.85rem;\">(blueprint §3)</a></li>\n                        <li>Scaled polynomial: <code>S(x) = 2<sup>N-1</sup> · P(x)</code> <a href=\"./blueprint/sect0003.html\" style=\"color: #3498db; text-decoration: none; font-size: 0.85rem;\">(blueprint §3)</a></li>\n                        <li>Result: <code>S(x) = T<sub>N</sub>(x) + c(θ)</code> <a href=\"./blueprint/sect0001.html\" style=\"color: #3498db; text-decoration: none; font-size: 0.85rem;\">(blueprint §1)</a></li>\n                    </ul>\n                </div>\n            </div>\n\n            <div class=\"visualization\">\n                <figure>\n                    <video autoplay loop muted playsinline style=\"max-width: 100%; height: auto; border-radius: 8px; box-shadow: 0 4px 12px rgba(0,0,0,0.15);\">\n                        <source src=\"chebyshev_videos/Chebyshev_N5.mp4\" type=\"video/mp4\">\n                        Your browser does not support the video tag. Please upgrade to a modern browser.\n                    </video>\n                    <figcaption>\n                        Video animation showing N=5 roots of unity being rotated. The regular pentagon (coral edges) rotates with\n                        the roots (red points). Vertical projection lines show how roots project to the real axis (pink points),\n                        which serve as roots for the blue polynomial curve. The displayed polynomial equation updates in real-time,\n                        showing that the curve's shape remains a 5th-degree Chebyshev polynomial (T₅(x)) while only the constant\n                        term <em>c</em> changes with rotation angle θ. <strong>Video specifications:</strong> 1080x1920 @ 60fps portrait.\n                        Rendered with Manim Community Edition.\n                    </figcaption>\n                </figure>\n            </div>\n\n            <h2>Project Components</h2>\n\n            <div class=\"components\">\n                <div class=\"component-card\">\n                    <h3>Formal Verification</h3>\n                    <p>Complete Lean 4 formalization across 12 modules with full proofs.</p>\n                    <div class=\"stats\">\n                        <div class=\"stat\">\n                            <div class=\"stat-number\">3,548</div>\n                            <div class=\"stat-label\">Lines of Proof</div>\n                        </div>\n                        <div class=\"stat\">\n                            <div class=\"stat-number\">0</div>\n","line":359,"end_line":408,"proof_status":"formalized"}
+{"id":"lem:discrete_orthogonality","label":"discrete_orthogonality","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For any integer \\(m\\), \\begin{equation} \\sum _{k=0}^{N-1} \\mathrm{e}^{2\\pi i m k/N} = \\begin{cases} N & \\text{if } N \\mid m, \\\\ 0 & \\text{otherwise.} \\end{cases} \\end{equation}","doc_url":"../../docs/find/#doc/sum_primitive_root_pow_mul","latex_link":"sect0007.html#lem:discrete_orthogonality","lean_link":"../../docs/find/#doc/sum_primitive_root_pow_mul","lean_decl_name":"sum_primitive_root_pow_mul","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":"./ChebyshevCircles/Proofs/PowerSums.lean","lean4_code":"lemma sum_primitive_root_pow_mul (N m : ℕ) (hN : 0 < N) (hm : 0 < m) (hm' : m < N) :\n    let ω := Complex.exp (2 * π * Complex.I / N)\n    ∑ k ∈ Finset.range N, ω ^ (k * m) = 0 := by\n  intro ω\n  have hN' : N ≠ 0 := Nat.pos_iff_ne_zero.mp hN\n  have hω : IsPrimitiveRoot ω N := Complex.isPrimitiveRoot_exp N hN'\n  conv_lhs => arg 2; ext k; rw [mul_comm k m, pow_mul]\n  have h_ne_one : ω ^ m ≠ 1 := by\n    intro h_eq\n    have h_div : N ∣ m := (hω.pow_eq_one_iff_dvd m).mp h_eq\n    have : N ��� m := Nat.le_of_dvd hm h_div\n    omega\n  have h_pow_N : (ω ^ m) ^ N = 1 := by\n    calc (ω ^ m) ^ N = ω ^ (m * N) := by rw [← pow_mul]\n      _ = ω ^ (N * m) := by rw [mul_comm]\n      _ = (ω ^ N) ^ m := by rw [← pow_mul]\n      _ = 1 ^ m := by rw [hω.pow_eq_one]\n      _ = 1 := one_pow m\n  have h_geom : (ω ^ m - 1) * ∑ k ∈ Finset.range N, (ω ^ m) ^ k = (ω ^ m) ^ N - 1 :=\n    mul_geom_sum (ω ^ m) N\n  rw [h_pow_N] at h_geom\n  have h_eq_zero : (ω ^ m - 1) * ∑ k ∈ Finset.range N, (ω ^ m) ^ k = 0 := by\n    rw [h_geom]; ring\n  exact eq_zero_of_ne_zero_of_mul_left_eq_zero (sub_ne_zero_of_ne h_ne_one) h_eq_zero\n\n/-- Sum of cosines at integer multiples of angles vanishes for non-zero integers with |m| < N. -/\n","line":193,"end_line":218,"proof_status":"formalized"}
+{"id":"lem:sum_cos_m_phi_vanishes","label":"sum_cos_m_phi_vanishes","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":["lem:discrete_orthogonality"],"nlp_text":"For any integers \\(N \\geq 1\\), \\(m\\) with \\(0 {\\lt} m {\\lt} N\\), and any \\(\\theta \\in \\mathbb {R}\\), \\begin{equation} \\sum _{k=0}^{N-1} \\cos \\left(m\\left(\\theta + \\frac{2\\pi k}{N}\\right)\\right) = 0. \\end{equation}","doc_url":"../../docs/find/#doc/sum_cos_multiple_rotated_roots","latex_link":"sect0010.html#lem:sum_cos_m_phi_vanishes","lean_link":"../../docs/find/#doc/sum_cos_multiple_rotated_roots","lean_decl_name":"sum_cos_multiple_rotated_roots","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":"./ChebyshevCircles/Proofs/TrigonometricIdentities.lean","lean4_code":"lemma sum_cos_multiple_rotated_roots (N : ℕ) (m : ℕ) (θ : ℝ)\n    (hN : 0 < N) (hm : 0 < m) (hm' : m < N) :\n    ∑ k ∈ Finset.range N, Real.cos (m * (θ + 2 * π * k / N)) = 0 := by\n  conv_lhs =>\n    arg 2\n    ext k\n    rw [← Complex.exp_ofReal_mul_I_re (m * (θ + 2 * π * k / N))]\n  rw [← Complex.re_sum]\n  simp only [Complex.ofReal_mul, Complex.ofReal_add, Complex.ofReal_div, Complex.ofReal_natCast]\n  have hN' : N ≠ 0 := Nat.pos_iff_ne_zero.mp hN\n  let ζ := Complex.exp (2 * π * I / N)\n  have hζ : IsPrimitiveRoot ζ N := Complex.isPrimitiveRoot_exp N hN'\n  suffices h_sum_zero : ∑ k ∈ Finset.range N, cexp (↑m * (↑2 * ↑π * ↑k / ↑N) * I) = 0 by\n    simp only [mul_add, add_mul]\n    conv_lhs =>\n      arg 1\n      arg 2\n      ext x\n      rw [Complex.exp_add]\n    rw [← Finset.mul_sum]\n    have : ∑ i ∈ Finset.range N, cexp (↑m * (↑2 * ↑π * ↑i / ↑N) * I) = 0 := h_sum_zero\n    simp [this]\n  calc ∑ k ∈ Finset.range N, cexp (↑m * (↑2 * ↑π * ↑k / ↑N) * I)\n      = ∑ k ∈ Finset.range N, ζ ^ (m * k) := by\n        congr 1 with k\n        simp only [ζ]\n        rw [← Complex.exp_nat_mul]\n        congr 1\n        field_simp\n        norm_cast\n    _ = ∑ k ∈ Finset.range N, (ζ ^ m) ^ k := by\n        congr 1 with k\n        rw [← pow_mul]\n    _ = 0 := by\n        have h_ne_one : ζ ^ m ≠ 1 := by\n          intro h_eq\n          have h_div : N ∣ m := by\n            have := hζ.pow_eq_one_iff_dvd m\n            exact this.mp h_eq\n          have : N ≤ m := Nat.le_of_dvd (by omega) h_div\n          omega\n        have h_pow_N : (ζ ^ m) ^ N = 1 := by\n          rw [← pow_mul, mul_comm m N, pow_mul]\n          simp [hζ.pow_eq_one]\n        have h_geom : (ζ ^ m - 1) * ∑ k ∈ Finset.range N, (ζ ^ m) ^ k = (ζ ^ m) ^ N - 1 :=\n          mul_geom_sum (ζ ^ m) N\n        rw [h_pow_N] at h_geom\n        have : (ζ ^ m - 1) * ∑ k ∈ Finset.range N, (ζ ^ m) ^ k = 0 := by\n          rw [h_geom]; ring\n        exact eq_zero_of_ne_zero_of_mul_left_eq_zero (sub_ne_zero_of_ne h_ne_one) this\n","line":81,"end_line":130,"proof_status":"formalized"}
+{"id":"lem:sum_cos_phi_vanishes","label":"sum_cos_phi_vanishes","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":["lem:discrete_orthogonality"],"nlp_text":"For any \\(N \\geq 2\\) and any \\(\\theta \\in \\mathbb {R}\\), \\begin{equation} \\sum _{k=0}^{N-1} \\cos \\left(\\theta + \\frac{2\\pi k}{N}\\right) = 0. \\end{equation}","doc_url":"../../docs/find/#doc/realProjectionsList_sum","latex_link":"sect0010.html#lem:sum_cos_phi_vanishes","lean_link":"../../docs/find/#doc/realProjectionsList_sum","lean_decl_name":"realProjectionsList_sum","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":"./ChebyshevCircles/Proofs/RootsOfUnity.lean","lean4_code":"theorem realProjectionsList_sum (N : ℕ) (θ : ℝ) :\n    (realProjectionsList N θ).sum = ∑ k ∈ Finset.range N, Real.cos (θ + 2 * π * k / N) := by\n  rw [realProjectionsList]\n  convert list_range_map_sum_eq_finset_sum N (fun k => Real.cos (θ + 2 * π * ↑k / ↑N)) using 2\n\n/-- The power sum of realProjectionsList equals the Finset power sum. -/\n","line":71,"end_line":76,"proof_status":"formalized"}
+{"id":"lem:chebyshev_angle_sum_odd","label":"chebyshev_angle_sum_odd","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":["lem:discrete_orthogonality"],"nlp_text":"For odd \\(m\\) with \\(0 {\\lt} m {\\lt} N\\), \\begin{equation} \\sum _{k=0}^{N-1} \\cos \\left(m \\cdot \\frac{(2k + 1)\\pi }{2N}\\right) = 0. \\end{equation}","doc_url":"../../docs/find/#doc/ChebyshevCircles.sum_cos_chebyshev_angles_vanishes","latex_link":"sect0012.html#lem:chebyshev_angle_sum_odd","lean_link":"../../docs/find/#doc/ChebyshevCircles.sum_cos_chebyshev_angles_vanishes","lean_decl_name":"ChebyshevCircles.sum_cos_chebyshev_angles_vanishes","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":"./ChebyshevCircles/Proofs/ChebyshevOrthogonality.lean","lean4_code":"lemma sum_cos_chebyshev_angles_vanishes (N : ℕ) (m : ℤ) (hN : 0 < N)\n    (hm : m ≠ 0) (hm_odd : Odd m) (hm_bound : |m| < 2 * N) :\n    ∑ k ∈ range N, Real.cos (m * (2 * k + 1 : ℝ) * π / (2 * N)) = 0 := by\n  -- Define the term we're summing\n  let f : ℕ → ℝ := fun k ↦ Real.cos (m * (2 * k + 1 : ℝ) * π / (2 * N))\n\n  -- We'll show the sum vanishes by pairing terms using symmetry\n  -- Key observation: θ_k + θ_{N-1-k} = (2k+1)π/(2N) + (2(N-1-k)+1)π/(2N) = π\n\n  -- First, prove the pairing lemma: f(k) + f(N-1-k) = 0\n  have pair_sum_zero : ∀ k ∈ range N, f k + f (N - 1 - k) = 0 := by\n    intro k hk\n    simp only [f]\n    rw [mem_range] at hk\n    have hk_le : k ≤ N - 1 := Nat.le_pred_of_lt hk\n\n    -- First establish the cast identity for (N - 1 - k)\n    have cast_eq : ((N - 1 - k) : ℝ) = N - 1 - k := by\n      rfl\n\n    -- The key insight: (2*(N-1-k)+1)*π/(2N) + (2*k+1)*π/(2N) = π\n    -- So cos(m * first_angle) = cos(m*(π - second_angle)) = (-1)^m * cos(m*second_angle)\n\n    -- Simplify the angle for N-1-k\n    have angle_identity : (2 * (N - 1 - k : ℝ) + 1) * π /\n      (2 * N) + (2 * k + 1) * π / (2 * N) = π := by\n      have h2N_ne : (2 * N : ℝ) ≠ 0 := by\n        have : (0 : ℝ) < N := Nat.cast_pos.mpr hN\n        linarith\n      rw [← add_div, ← add_mul]\n      -- Show the sum of coefficients equals 2N\n      have coeff_sum : 2 * (N - 1 - k : ℝ) + 1 + (2 * k + 1) = 2 * N := by\n        have : (2 * (N - 1 - k) + 1 : ℕ) + (2 * k + 1) = 2 * N := by omega\n        linarith\n      rw [coeff_sum]\n      field_simp\n\n    -- Rewrite second angle as π minus first angle\n    have second_angle : (2 * (N - 1 - k : ℝ) + 1) * π / (2 * N) =\n      π - (2 * k + 1) * π / (2 * N) := by\n      linarith [angle_identity]\n\n    -- Substitute and use cosine identity\n    -- Need to show the second cosine argument can be rewritten\n    have key : m * (2 * ((N - 1 - k) : ℝ) + 1) * π / (2 * N) =\n      m * (π - (2 * k + 1) * π / (2 * N)) := by\n      calc m * (2 * (N - 1 - k : ℝ) + 1) * π / (2 * N)\n          = m * ((2 * (N - 1 - k : ℝ) + 1) * π / (2 * N)) := by ring\n        _ = m * (π - (2 * k + 1) * π / (2 * N)) := by rw [second_angle]\n\n","line":57,"end_line":106,"proof_status":"formalized"}
+{"id":"lem:chebyshev_angle_sum_even","label":"chebyshev_angle_sum_even","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":["lem:discrete_orthogonality"],"nlp_text":"For even \\(m\\) with \\(0 {\\lt} m {\\lt} N\\), \\begin{equation} \\sum _{k=0}^{N-1} \\cos \\left(m \\cdot \\frac{(2k + 1)\\pi }{2N}\\right) = 0. \\end{equation}","doc_url":"../../docs/find/#doc/ChebyshevCircles.binomial_terms_vanish_chebyshev","latex_link":"sect0012.html#lem:chebyshev_angle_sum_even","lean_link":"../../docs/find/#doc/ChebyshevCircles.binomial_terms_vanish_chebyshev","lean_decl_name":"ChebyshevCircles.binomial_terms_vanish_chebyshev","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":"./ChebyshevCircles/Proofs/ChebyshevOrthogonality.lean","lean4_code":"lemma binomial_terms_vanish_chebyshev (N j r : ℕ) (hN : 0 < N) (hj' : j < N)\n    (hr : r ∈ range (j + 1)) (hr_neq : (2 * r - j : ℤ) ≠ 0) :\n    ∑ k ∈ range N, Real.cos ((2 * r - j : ℤ) * (2 * k + 1 : ℝ) * π / (2 * N)) = 0 := by\n  -- We need to show:\n  -- 1. (2r - j) is odd\n  -- 2. |2r - j| < 2N\n  -- Then apply sum_cos_chebyshev_angles_vanishes\n\n  -- First establish the bound: r ≤ j\n  have hr_le : r ≤ j := by\n    simp only [mem_range] at hr\n    omega\n\n  -- Split by parity of j\n  by_cases h_parity : Odd j\n  · -- Case 1: j is odd, so (2r - j) is odd\n    have h_odd : Odd (2 * r - j : ℤ) := by\n      obtain ⟨m, hm⟩ := h_parity\n      use (r : ℤ) - m - 1\n      have : (j : ℤ) = 2 * m + 1 := by exact_mod_cast hm\n      calc (2 * r - j : ℤ)\n        _ = 2 * (r : ℤ) - (2 * m + 1) := by rw [this]\n        _ = 2 * ((r : ℤ) - m - 1) + 1 := by ring\n\n    -- Prove |2r - j| < 2N\n    have h_bound : |(2 * r - j : ℤ)| < 2 * N := by\n      by_cases hr_case : 2 * r ≥ j\n      · rw [abs_of_nonneg (Int.sub_nonneg_of_le (by omega : (j : ℤ) ≤ (2 * r : ℤ)))]\n        calc (2 * r - j : ℤ)\n          _ ≤ (2 * j - j : ℤ) := by omega\n          _ = (j : ℤ) := by ring\n          _ < (N : ℤ) := by omega\n          _ < (2 * N : ℤ) := by omega\n      · push_neg at hr_case\n        rw [abs_of_neg (Int.sub_neg_of_lt (by omega : (2 * r : ℤ) < (j : ℤ)))]\n        calc -(2 * r - j : ℤ)\n          _ = (j - 2 * r : ℤ) := by ring\n          _ ≤ (j : ℤ) := by omega\n          _ < (N : ℤ) := by omega\n          _ < (2 * N : ℤ) := by omega\n\n    -- Apply odd vanishing lemma\n    convert sum_cos_chebyshev_angles_vanishes N (2 * r - j) hN hr_neq h_odd h_bound using 2\n\n  · -- Case 2: j is even, so (2r - j) is even\n    have j_even : Even j := Nat.not_odd_iff_even.mp h_parity\n    have h_even : Even (2 * r - j : ℤ) := by\n      obtain ⟨m, hm⟩ := j_even\n      use (r : ℤ) - m\n      have : (j : ℤ) = (m + m : ℕ) := by simp [hm]\n","line":410,"end_line":459,"proof_status":"formalized"}
+{"id":"thm:newton_identities","label":"newton_identities","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":["lem:discrete_orthogonality"],"nlp_text":"The power sums and elementary symmetric polynomials are related by: \\begin{equation} k \\cdot e_k = \\sum _{i=1}^k (-1)^{i-1} e_{k-i} p_i, \\quad k = 1, 2, \\ldots , N, \\end{equation}","doc_url":"../../docs/find/#doc/esymm_eq_of_psum_eq","latex_link":"sect0008.html#thm:newton_identities","lean_link":"../../docs/find/#doc/esymm_eq_of_psum_eq","lean_decl_name":"esymm_eq_of_psum_eq","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":"./ChebyshevCircles/Proofs/NewtonIdentities.lean","lean4_code":"lemma esymm_eq_of_psum_eq (s t : Multiset ℝ) (h_card : s.card = t.card)\n    (h_psum : ∀ j, 0 < j → j < s.card → (s.map (· ^ j)).sum = (t.map (· ^ j)).sum) :\n    ∀ m, m < s.card → s.esymm m = t.esymm m := by\n  intro m hm\n  -- We prove by strong induction on m\n  induction m using Nat.strong_induction_on with\n  | h m IH =>\n    by_cases hm_zero : m = 0\n    · -- Base case: esymm 0 = 1 for any multiset\n      rw [hm_zero]\n      simp [Multiset.esymm]\n    · -- Inductive case: m > 0, use Newton's identity\n      have hm_pos : 0 < m := Nat.pos_of_ne_zero hm_zero\n      -- Apply Newton's identity to both s and t\n      have h_newton_s := multiset_newton_identity s m hm_pos\n      have h_newton_t := multiset_newton_identity t m hm_pos\n      -- The RHS of Newton's identity should be equal for s and t\n      have h_rhs_eq : (Finset.antidiagonal m).sum (fun a =>\n            if a.1 < m then (-1)^a.1 * s.esymm a.1 * (s.map (fun x => x ^ a.2)).sum\n            else 0) =\n          (Finset.antidiagonal m).sum (fun a =>\n            if a.1 < m then (-1)^a.1 * t.esymm a.1 * (t.map (fun x => x ^ a.2)).sum\n            else 0) := by\n        apply Finset.sum_congr rfl\n        intro a ha\n        -- a ∈ antidiagonal m means a.1 + a.2 = m\n        have h_sum : a.1 + a.2 = m := Finset.mem_antidiagonal.mp ha\n        split_ifs with ha_lt\n        · -- When a.1 < m, show the terms are equal\n          congr 1\n          · congr 1  -- esymm a.1 equal by IH\n            have ha1_bound : a.1 < s.card := by omega\n            exact IH a.1 ha_lt ha1_bound\n          · -- Power sums equal by hypothesis\n            by_cases ha2_zero : a.2 = 0\n            · -- When a.2 = 0, both power sums are cardinalities\n              simp only [ha2_zero, pow_zero]\n              -- Simplify x^0 = 1, then sum of 1s equals card\n              convert_to (s.card : ℝ) = (t.card : ℝ) using 1 <;>\n                simp\n              exact_mod_cast h_card\n            · have ha2_pos : 0 < a.2 := Nat.pos_of_ne_zero ha2_zero\n              have ha2_bound : a.2 < s.card := by omega\n              exact h_psum a.2 ha2_pos ha2_bound\n        · rfl\n      -- Now use Newton's identity to extract esymm m\n      have h_eq : (m : ℝ) * s.esymm m = (m : ℝ) * t.esymm m := by\n        rw [h_newton_s, h_newton_t, h_rhs_eq]\n      -- Cancel the factor of m\n      have hm_ne_zero : (m : ℝ) ≠ 0 := by\n","line":132,"end_line":181,"proof_status":"formalized"}
+{"id":"lem:power_sum_invariance","label":"power_sum_invariance","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":["lem:sum_cos_m_phi_vanishes"],"nlp_text":"For any integers \\(N \\geq 1\\), \\(j\\) with \\(1 \\leq j {\\lt} N\\), and any \\(\\theta \\in \\mathbb {R}\\), \\begin{equation} \\sum _{k=0}^{N-1} \\cos ^j\\left(\\theta + \\frac{2\\pi k}{N}\\right) \\end{equation}","doc_url":"../../docs/find/#doc/cos_cube_formula","latex_link":"sect0020.html#lem:power_sum_invariance","lean_link":"../../docs/find/#doc/cos_cube_formula","lean_decl_name":"cos_cube_formula","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":"./ChebyshevCircles/Proofs/TrigonometricIdentities.lean","lean4_code":"lemma cos_cube_formula (x : ℝ) :\n    Real.cos x ^ 3 = (Real.cos (3 * x) + 3 * Real.cos x) / 4 := by\n  linarith [Real.cos_three_mul x]\n\n","line":133,"end_line":136,"proof_status":"formalized"}
+{"id":"lem:chebyshev_roots","label":"chebyshev_roots","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":["lem:chebyshev_angle_sum_odd","lem:chebyshev_angle_sum_even","lem:chebyshev_roots_characterization"],"nlp_text":"Let \\(\\xi _k = \\cos ((2k + 1)\\pi /(2N))\\) for \\(k = 0, \\ldots , N-1\\) denote the roots of \\(T_N(x)\\). Then for any \\(1 \\leq j {\\lt} N\\), \\begin{equation} \\sum _{k=0}^{N-1} \\xi _k^j = \\begin{cases} \\frac{N}{2^j} \\binom {j}{j/2} & \\text{if } j \\text{ is even}, \\\\ 0 & \\text{if } j \\text{ is odd.} \\end{cases} \\end{equation}","doc_url":"../../docs/find/#doc/ChebyshevCircles.sum_cos_pow_chebyshev_binomial","latex_link":"sect0027.html#lem:chebyshev_roots","lean_link":"../../docs/find/#doc/ChebyshevCircles.sum_cos_pow_chebyshev_binomial","lean_decl_name":"ChebyshevCircles.sum_cos_pow_chebyshev_binomial","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":"./ChebyshevCircles/Proofs/ChebyshevOrthogonality.lean","lean4_code":"lemma sum_cos_pow_chebyshev_binomial (N j : ℕ) (_hN : 0 < N) (_hj : 0 < j) (_hj' : j < 2 * N) :\n    ∑ k ∈ range N, (Real.cos ((2 * k + 1 : ℝ) * π / (2 * N))) ^ j =\n    (2 : ℝ) ^ (-(j : ℤ)) * ∑ r ∈ range (j + 1), (j.choose r : ℝ) *\n    ∑ k ∈ range N, Real.cos ((2 * r - j : ℤ) * (2 * k + 1 : ℝ) * π / (2 * N)) := by\n  -- Follows sum_cos_pow_eq_sum_binomial from PowerSums.lean\n  simp only [cos_as_exp_chebyshev]\n\n  -- Pull power inside re: (z.re)^j = (z^j).re when z is real\n  have h_re_pow : ∀ k, ((Complex.exp (↑((2 * k + 1 : ℝ) * π / (2 * N)) * Complex.I) +\n      Complex.exp (-(↑((2 * k + 1 : ℝ) * π / (2 * N)) * Complex.I))) / 2).re ^ j =\n    (((Complex.exp (↑((2 * k + 1 : ℝ) * π / (2 * N)) * Complex.I) +\n      Complex.exp (-(↑((2 * k + 1 : ℝ) * π / (2 * N)) * Complex.I))) / 2) ^ j).re := by\n    intro k\n    set z := (Complex.exp (↑((2 * k + 1 : ℝ) * π / (2 * N)) * Complex.I) +\n        Complex.exp (-(↑((2 * k + 1 : ℝ) * π / (2 * N)) * Complex.I))) / 2 with z_def\n    have h_real : z.im = 0 := by\n      rw [z_def]\n      have h1 : (Complex.exp (↑((2 * k + 1 : ℝ) * π / (2 * N)) * Complex.I) +\n          Complex.exp (-(↑((2 * k + 1 : ℝ) * π / (2 * N)) * Complex.I))).im = 0 := by\n        rw [Complex.add_im, Complex.exp_ofReal_mul_I_im]\n        have : (Complex.exp (-(↑((2 * k + 1 : ℝ) * π / (2 * N)) * Complex.I))).im =\n          Real.sin (-((2 * k + 1 : ℝ) * π / (2 * N))) := by\n          rw [show -(↑((2 * k + 1 : ℝ) * π / (2 * N)) * Complex.I) =\n            (↑(-((2 * k + 1 : ℝ) * π / (2 * N))) : ℂ) * Complex.I by\n            apply Complex.ext <;> simp]\n          exact Complex.exp_ofReal_mul_I_im _\n        rw [this, Real.sin_neg]\n        ring\n      rw [Complex.div_im, h1]\n      simp\n    have h_eq : z = (z.re : ℂ) := Complex.ext rfl h_real\n    calc z.re ^ j\n      _ = (z.re : ℂ).re ^ j := by simp\n      _ = ((z.re : ℂ) ^ j).re := by simp [← Complex.ofReal_pow]\n      _ = (z ^ j).re := by rw [← h_eq]\n\n  conv_lhs =>\n    arg 2\n    ext k\n    rw [h_re_pow k]\n  rw [← Complex.re_sum]\n\n  simp only [div_pow]\n\n  rw [show ∑ x ∈ range N, (Complex.exp (↑((2 * ↑x + 1) * π / (2 * ↑N)) * Complex.I) +\n      Complex.exp (-(↑((2 * ↑x + 1) * π / (2 * ↑N)) * Complex.I))) ^ j / (2 : ℂ) ^ j =\n    ∑ x ∈ range N, (∑ r ∈ range (j + 1), (j.choose r : ℂ) *\n      Complex.exp ((2 * r - j : ℤ) * ↑((2 * ↑x + 1) * π / (2 * ↑N)) * Complex.I)) / (2 : ℂ) ^ j by\n    apply Finset.sum_congr rfl\n    intro i _\n","line":194,"end_line":243,"proof_status":"formalized"}
+{"id":"lem:esymm_invariant","label":"esymm_invariant","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":["thm:newton_identities","lem:power_sum_invariance"],"nlp_text":"Let \\(r_k(\\theta ) = \\cos (\\theta + 2\\pi k/N)\\) for \\(k = 0, \\ldots , N-1\\). Then for any \\(m\\) with \\(1 \\leq m \\leq N\\), the elementary symmetric polynomial \\begin{equation} e_m(r_0(\\theta ), \\ldots , r_{N-1}(\\theta )) \\end{equation}","doc_url":"../../docs/find/#doc/esymm_rotated_roots_invariant","latex_link":"sect0022.html#lem:esymm_invariant","lean_link":"../../docs/find/#doc/esymm_rotated_roots_invariant","lean_decl_name":"esymm_rotated_roots_invariant","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":"./ChebyshevCircles/Proofs/NewtonIdentities.lean","lean4_code":"lemma esymm_rotated_roots_invariant (N : ℕ) (m : ℕ) (θ₁ θ₂ : ℝ)\n    (hN : 0 < N) (hm : 0 < m) (hm' : m < N) :\n    let l1 := (realProjectionsList N θ₁ : Multiset ℝ)\n    let l2 := (realProjectionsList N θ₂ : Multiset ℝ)\n    l1.esymm m = l2.esymm m := by\n  intro l1 l2\n  induction m using Nat.strong_induction_on with\n  | h k IH =>\n    cases k with\n    | zero => omega\n    | succ k' =>\n      cases k' with\n      | zero =>\n        -- N ≥ 2, so sum of cosines at N equally spaced angles = 0\n        -- esymm 1 is the sum, and both sums equal 0\n        rw [multiset_esymm_one_eq_sum, multiset_esymm_one_eq_sum]\n        simp only [l1, l2]\n        have hN2 : 2 ≤ N := by omega\n        rw [multiset_coe_realProjectionsList_sum, multiset_coe_realProjectionsList_sum]\n        rw [sum_cos_roots_of_unity N θ₁ hN2, sum_cos_roots_of_unity N θ₂ hN2]\n      | succ k'' =>\n        -- Use Newton's identity to express esymm (k''+2) in terms of smaller esymm and power sums\n        have h_newton_l1 := multiset_newton_identity l1 (k'' + 2) (by omega)\n        have h_newton_l2 := multiset_newton_identity l2 (k'' + 2) (by omega)\n\n        -- The RHS of Newton's identity is the same for l1 and l2\n        have h_rhs_eq : (Finset.antidiagonal (k'' + 2)).sum (fun a =>\n              if a.1 < k'' + 2 then (-1)^a.1 * l1.esymm a.1 * (l1.map (fun x => x ^ a.2)).sum\n              else 0) =\n            (Finset.antidiagonal (k'' + 2)).sum (fun a =>\n              if a.1 < k'' + 2 then (-1)^a.1 * l2.esymm a.1 * (l2.map (fun x => x ^ a.2)).sum\n              else 0) := by\n          -- Each term in the sum is equal\n          apply Finset.sum_congr rfl\n          intro a ha_mem\n          -- a ∈ antidiagonal (k'' + 2), so a.1 + a.2 = k'' + 2\n          have h_sum : a.1 + a.2 = k'' + 2 := Finset.mem_antidiagonal.mp ha_mem\n          split_ifs with ha\n          · -- When a.1 < k'' + 2, we need to show the terms are equal\n            congr 1\n            · -- esymm a.1 is invariant by IH\n              by_cases ha0 : a.1 = 0\n              · simp only [ha0, pow_zero, one_mul]\n                -- esymm 0 = 1 for any multiset\n                simp only [Multiset.esymm, Multiset.powersetCard_zero_left]\n              · have ha_pos : 0 < a.1 := Nat.pos_of_ne_zero ha0\n                -- Need to show a.1 < N\n                have ha_N : a.1 < N := by\n                  calc a.1\n                    _ < k'' + 2 := ha\n","line":238,"end_line":287,"proof_status":"formalized"}
+{"id":"cor:power_sums_determine_coefficients","label":"power_sums_determine_coefficients","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":["thm:newton_identities"],"nlp_text":"Let \\(\\alpha _1, \\ldots , \\alpha _N\\) and \\(\\beta _1, \\ldots , \\beta _N\\) be two collections of numbers. If \\begin{equation} \\sum _{i=1}^N \\alpha _i^j = \\sum _{i=1}^N \\beta _i^j \\quad \\text{for all } j = 1, 2, \\ldots , N, \\end{equation}","doc_url":"../../docs/find/#doc/Multiset.prod_X_sub_C_coeff","latex_link":"sect0008.html#cor:power_sums_determine_coefficients","lean_link":"../../docs/find/#doc/Multiset.prod_X_sub_C_coeff","lean_decl_name":"Multiset.prod_X_sub_C_coeff","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":null,"lean4_code":null,"line":30,"end_line":30,"proof_status":"formalized"}
+{"id":"thm:power_sum_equality","label":"power_sum_equality","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":["lem:power_sum_invariance","lem:chebyshev_roots"],"nlp_text":"For \\(1 \\leq j {\\lt} N\\), \\begin{equation} \\sum _{k=0}^{N-1} \\cos ^j\\left(\\theta + \\frac{2\\pi k}{N}\\right) = \\sum _{k=0}^{N-1} \\cos ^j\\left(\\frac{(2k + 1)\\pi }{2N}\\right). \\end{equation}","doc_url":"../../docs/find/#doc/general_powersum_equality","latex_link":"sect0027.html#thm:power_sum_equality","lean_link":"../../docs/find/#doc/general_powersum_equality","lean_decl_name":"general_powersum_equality","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":"./ChebyshevCircles/Proofs/PowerSumEquality.lean","lean4_code":"theorem general_powersum_equality (N j : ℕ) (hN : 0 < N) (hj : 0 < j) (hj' : j < N) :\n    ∑ k ∈ Finset.range N, (Real.cos (2 * π * k / N)) ^ j =\n    ∑ k ∈ Finset.range N, (Real.cos ((2 * k + 1 : ℝ) * π / (2 * N))) ^ j := by\n  -- Both sums equal the same closed-form expression\n  rw [rotated_roots_powersum_value N j hN hj hj']\n  rw [chebyshev_roots_powersum_value N j hN hj hj']\n\nend PowerSumEquality\n\n","line":264,"end_line":272,"proof_status":"formalized"}
+{"id":"thm:coefficients_independent","label":"coefficients_independent","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":["lem:esymm_invariant","cor:power_sums_determine_coefficients"],"nlp_text":"For any \\(N \\geq 1\\) and any \\(k\\) with \\(1 \\leq k \\leq N\\), the coefficient of \\(x^k\\) in \\(S_N(x; \\theta )\\) is independent of \\(\\theta \\). (See Theorem 5.1 in ../paper/chebyshev_circles.pdfthe paper)","doc_url":"../../docs/find/#doc/constant_term_only_varies","latex_link":"sect0022.html#thm:coefficients_independent","lean_link":"../../docs/find/#doc/constant_term_only_varies","lean_decl_name":"constant_term_only_varies","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":"./ChebyshevCircles/Proofs/PolynomialProperties.lean","lean4_code":"theorem constant_term_only_varies (N : ℕ) (θ₁ θ₂ : ℝ) (k : ℕ) (hN : 0 < N)\n    (hk : 0 < k) : (scaledPolynomial N θ₁).coeff k = (scaledPolynomial N θ₂).coeff k := by\n  unfold scaledPolynomial unscaledPolynomial polynomialFromRealRoots\n  simp only [coeff_C_mul]\n  congr 1\n  -- Convert to multiset form\n  rw [list_foldr_eq_multiset_prod, list_foldr_eq_multiset_prod]\n  -- Factor out the card lemma\n  have h_card (θ : ℝ) : ((realProjectionsList N θ : Multiset ℝ)).card = N := by\n    rw [Multiset.coe_card, card_realProjectionsList]\n  -- Split cases\n  by_cases hk_le : k ≤ N\n  · -- k ≤ N: use Vieta's formula\n    rw [Multiset.prod_X_sub_C_coeff (realProjectionsList N θ₁ : Multiset ℝ) (by rwa [h_card]),\n        Multiset.prod_X_sub_C_coeff (realProjectionsList N θ₂ : Multiset ℝ) (by rwa [h_card])]\n    congr 1\n    · rw [h_card, h_card]\n    · by_cases hk_eq : k = N\n      · rw [hk_eq, h_card, h_card]; norm_num [Multiset.esymm, Multiset.powersetCard_zero_left]\n      · rw [h_card, h_card]\n        exact esymm_rotated_roots_invariant N (N - k) θ₁ θ₂ hN (by omega) (by omega)\n  · -- k > N: both coefficients are 0\n    have deg (θ : ℝ) : (Multiset.map (fun r => X - C r)\n        (realProjectionsList N θ : Multiset ℝ)).prod.natDegree = N := by\n      rw [Polynomial.natDegree_multiset_prod_X_sub_C_eq_card, h_card]\n    rw [Polynomial.coeff_eq_zero_of_natDegree_lt, Polynomial.coeff_eq_zero_of_natDegree_lt]\n    · rw [deg]; omega\n    · rw [deg]; omega\n\n/-- The Chebyshev polynomial T_N has degree N for N ≥ 1. -/\n","line":39,"end_line":68,"proof_status":"formalized"}
+{"id":"prop:chebyshev_properties","label":"chebyshev_properties","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The Chebyshev polynomials satisfy the following properties: Recurrence: \\(T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)\\) for \\(n \\geq 1\\). Leading coefficient: The leading coefficient of \\(T_N(x)\\) is \\(2^{N-1}\\) for \\(N \\geq 1\\). Roots: The roots of \\(T_N(x)\\) are \"> \\begin{equation} \\label{eq:chebyshev_roots} x_k = \\cos \\left(\\frac{(2k + 1)\\pi }{2N}\\right), \\quad k = 0, 1, \\ldots , N-1. \\end{equation}","doc_url":"../../docs/find/#doc/Polynomial.Chebyshev.T","latex_link":"sect0006.html#prop:chebyshev_properties","lean_link":"../../docs/find/#doc/Polynomial.Chebyshev.T","lean_decl_name":"Polynomial.Chebyshev.T","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":null,"lean4_code":null,"line":31,"end_line":31,"proof_status":"formalized"}
+{"id":"lem:chebyshev_roots_characterization","label":"chebyshev_roots_characterization","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://e-vergo.github.io/ChebyshevCircles/blueprint/dep_graph_document.html","dependencies":["prop:chebyshev_properties"],"nlp_text":"The roots of the Chebyshev polynomial \\(T_N(x)\\) are \\begin{equation} \\xi _k = \\cos \\left(\\frac{(2k + 1)\\pi }{2N}\\right), \\quad k = 0, 1, \\ldots , N-1. \\end{equation}","doc_url":"../../docs/find/#doc/chebyshevRoot_in_Icc","latex_link":"sect0025.html#lem:chebyshev_roots_characterization","lean_link":"../../docs/find/#doc/chebyshevRoot_in_Icc","lean_decl_name":"chebyshevRoot_in_Icc","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChebyshevCircles","source_path":"./ChebyshevCircles/Proofs/ChebyshevRoots.lean","lean4_code":"lemma chebyshevRoot_in_Icc (N k : ℕ) : chebyshevRoot N k ∈ Set.Icc (-1) 1 := by\n  unfold chebyshevRoot\n  exact Real.cos_mem_Icc _\n\n/-- Backward direction: T_N(cos((2k+1)π/(2N))) = 0 -/\n","line":37,"end_line":41,"proof_status":"formalized"}

crawled_repos/Cours_ED.jsonl

@@ -0,0 +1,35 @@
+{"id":"trans et nb rotation ratio","label":"trans et nb rotation ratio","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Si \\(\\rho (T) \\in \\mathbb {Q}/\\mathbb {Z}\\), alors aucune orbite de \\(T\\) n’est dense dans \\({\\mathbb {T}}\\). De façon équivalente: si \\(T\\) est transitive, alors nécessairement \\(\\rho (T) \\notin \\mathbb {Q}/\\mathbb {Z}\\).","doc_url":null,"latex_link":"sect0002.html#trans et nb rotation ratio","lean_link":null,"lean_decl_name":"trans et nb rotation ratio","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"delta orbite","label":"delta orbite","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soient \\(n,m \\in \\mathbb {Z}\\) avec \\(n\\neq m\\). Soit \\(x_0 \\in {\\mathbb {T}}\\). Notons \\(\\Delta := [T^n(x_0),T^m(x_0)]\\) et \\(\\Delta ':=[T^m(x_0),T^n(x_0)]\\) les deux arcs de cercle joignant \\(T^n(x_0)\\) et \\(T^m(x_0)\\).Alors, pour tout \\(k_0 \\in {\\mathbb {N}}\\), il existe des entiers \\(k_1\\geq k_0\\) et \\(k_1'\\geq k_0\\) tels que \\[ {\\mathbb {T}}=\\overset {k_1}{\\underset {j=k_0}{\\bigcup }}T^{j(m-n)}\\Delta =\\overset {k_1’}{\\underset {j=k_0}{\\bigcup }}T^{j(m-n)}\\Delta '. \\]","doc_url":null,"latex_link":"sect0002.html#delta orbite","lean_link":null,"lean_decl_name":"delta orbite","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"E+","label":"E+","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Propriétés autour de \\({\\mathcal{E}}^+\\) (\\({\\mathcal{E}}\\),\\(\\circ \\)) et (\\(\\mathcal{E}^+\\),\\(\\circ \\)) sont des sous-groupes de \\(\\mathrm{Hom}(\\mathbb {R})\\). Pour tout \\(f \\in \\mathcal{E}^+\\), \\(f - \\mathrm{Id_{\\mathbb {R}}}\\) est continue et \\(1-\\)périodique sur \\(\\mathbb {R}\\), donc bornée et uniformément continue. Pour tout \\(f \\in \\mathcal{E}^+\\), \\(f\\) est uniformément continue sur \\(\\mathbb {R}\\). Par récurrence, on démontre facilement que pour tous \\(f \\in \\mathcal{E}^+\\), \\(k \\in \\mathbb {Z}\\), et \\(x \\in \\mathbb {R}\\), \\(f(x+k)=f(x)+k\\) et \\(f^k(x+1)=f^k(x)+1\\) (\\(f^k\\) désigne la composée et non le produit).","doc_url":null,"latex_link":"sect0002.html#E+","lean_link":null,"lean_decl_name":"E+","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Tactic/Ring/Basic.lean","lean4_code":"instance {α : Q(Type u)} {E : Q($α) → Type} {e : Q($α)} [Inhabited (Σ e, E e)] :\n    Inhabited (Result E e) :=\n  let ⟨e', v⟩ : Σ e, E e := default; ⟨e', v, default⟩\n\nvariable {α : Q(Type u)} (sα : Q(CommSemiring $α)) {R : Type*} [CommSemiring R]\n\n/--\nConstructs the expression corresponding to `.const n`.\n(The `.const` constructor does not check that the expression is correct.)\n-/\n","line":245,"end_line":254,"proof_status":"statement_formalized"}
+{"id":"a0000000047","label":"a0000000047","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Notons \\(d_q'=\\min (d_q,1/2)\\). L’application \\(\\iota \\) obtenue dans le corollaire 6 est une isométrie de (\\({\\mathcal{E}}^+/\\mathcal{T}_{\\mathrm{ent}}, d_q')\\) dans \\((\\mathrm{Hom}^+({\\mathbb {T}}),\\overline{d}_\\infty )\\).Autrement dit: \\[ \\forall \\ \\overline{f}, \\overline{g} \\in {\\mathcal{E}}^+/\\mathcal{T}_{\\mathrm{ent}},\\ \\overline{d}_\\infty (T_f,T_g)=\\min (d_q(\\overline{f},\\overline{g}), 1/2). \\]","doc_url":null,"latex_link":"sect0002.html#a0000000047","lean_link":null,"lean_decl_name":"a0000000047","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"conjugaison","label":"conjugaison","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Le nombre de rotation est invariant par conjugaison topologique: pour tous \\(T_1\\), \\(T_2\\) dans \\(\\mathrm{Hom}^+({\\mathbb {T}})\\), \\[ \\rho (T_{1}\\circ T_{2}\\circ T_{1}^{-1})=\\rho (T_{2}). \\]","doc_url":null,"latex_link":"sect0002.html#conjugaison","lean_link":null,"lean_decl_name":"conjugaison","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000141","label":"a0000000141","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"L’ensemble fermé \\(L_T\\) des valeurs d’adhérence de la suite \\((T^n(x))_{n\\geq 0}\\), qui ne dépend pas de \\(x \\in {\\mathbb {T}}\\), est appelé l’ensemble dérivé de \\(T\\).","doc_url":null,"latex_link":"sect0002.html#a0000000141","lean_link":null,"lean_decl_name":"a0000000141","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000095","label":"a0000000095","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Si \\(f:\\mathbb {R} \\to \\mathbb {R}\\) est une fonction uniformément continue. On définit l’application module de continuité \\(\\omega _f: \\mathbb {R} \\to \\mathbb {R}^+\\) par \\(\\omega _f(\\delta ) := \\sup E_{f}(\\delta )\\), où \\[ E_{f}(\\delta ):=\\lbrace |f(x_1) - f(x_2)|: (x_1;x_2) \\in \\mathbb {R}^2,\\ |x_1 - x_2| \\leq \\delta \\rbrace \\]","doc_url":null,"latex_link":"sect0002.html#a0000000095","lean_link":null,"lean_decl_name":"a0000000095","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000106","label":"a0000000106","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soit \\(f: X \\to X\\) un homéomorphisme. On dit que \\(f\\) est transitif si et seulement si il existe un point \\(x \\in X\\) tel que l’orbite \\(O_x\\) est dense dans \\(X\\). On dit que \\(f\\) est minimal si et seulement si pour tout \\(x \\in X\\), l’orbite \\(O_x\\) est dense dans \\(X\\). Autrement dit, \\(f\\) est minimal si toutes les orbites sont denses dans \\(X\\).","doc_url":null,"latex_link":"sect0002.html#a0000000106","lean_link":null,"lean_decl_name":"a0000000106","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"bornesnbrot","label":"bornesnbrot","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Lemme fondamentalSoit f un relèvement de T. Notons \\[ a=\\underset {x \\in \\mathbb {R}}{\\inf } f(x) - x \\text{ et } b=\\underset {x \\in \\mathbb {R}}{\\sup } f(x) - x. \\]","doc_url":null,"latex_link":"sect0002.html#bornesnbrot","lean_link":null,"lean_decl_name":"bornesnbrot","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"psi","label":"psi","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Notons \\(\\mathcal{C}_U(\\mathbb {R},\\mathbb {R})\\) l’ensemble des fonctions uniformément continues définies de \\(\\mathbb {R}\\) dans \\({\\mathbb {R}}\\). L’application \\(\\psi : \\mathcal{C}_U(\\mathbb {R},\\mathbb {R}) \\times \\mathcal{C}_U(\\mathbb {R},\\mathbb {R}) \\to \\mathcal{C}_U(\\mathbb {R},\\mathbb {R})\\) définie par \\(\\psi (f,g)=f\\circ g\\) est continue pour la topologie associée à la norme de la convergence uniforme.","doc_url":null,"latex_link":"sect0002.html#psi","lean_link":null,"lean_decl_name":"psi","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"equivrationnel","label":"equivrationnel","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"On a l’équivalence \\[ \\rho (T) \\in \\mathbb {Q}/\\mathbb {Z} \\Longleftrightarrow \\exists k \\in \\mathbb {N}^*,\\ T^k \\text{a un point fixe.} \\]","doc_url":null,"latex_link":"sect0002.html#equivrationnel","lean_link":null,"lean_decl_name":"equivrationnel","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000145","label":"a0000000145","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soit \\(T\\in \\mathrm{Hom}^-({\\mathbb {T}})\\) un hom��omorphisme renversant l’orientation. Alors \\(T\\) possède exactement deux points fixes.","doc_url":null,"latex_link":"sect0002.html#a0000000145","lean_link":null,"lean_decl_name":"a0000000145","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000146","label":"a0000000146","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soit \\(T \\in \\mathrm{Hom}^-({\\mathbb {T}})\\). Alors: \\(T^2 \\in \\mathrm{Hom}^+({\\mathbb {T}})\\) et \\(\\rho (T^2)=\\overline{0}\\). Pour tout \\(x \\in {\\mathbb {T}}\\), les suites \\((T^{2n}(x))_{n\\geq }0\\), \\((T^{2n+1}(x))_{n\\geq }0\\), \\((T^{-2n}(x))_{n\\geq }0\\) et \\((T^{-2n-1}(x))_{n\\geq }0\\) convergent vers des points fixes de \\(T^2\\). Par conséquent, l’adhérence de l’orbite de \\(x\\) est dénombrable.","doc_url":null,"latex_link":"sect0002.html#a0000000146","lean_link":null,"lean_decl_name":"a0000000146","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"Denjoy","label":"Denjoy","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Exemple de DenjoySoit \\(\\alpha \\) un irrationnel fixé. Tout ensemble \\(L\\) fermé d’intérieur non vide sans point isolé peut être vu comme ensemble dérivé d’un homéomorphisme de nombre de rotation \\(\\alpha \\).","doc_url":null,"latex_link":"sect0002.html#Denjoy","lean_link":null,"lean_decl_name":"Denjoy","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"minimal équivaut transitif","label":"minimal équivaut transitif","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Il y a équivalence entre les 3 propriétés suivantes: Il existe \\(x\\in {\\mathbb {T}}\\) tel que \\(O_x\\) soit dense dans \\({\\mathbb {T}}\\) (autrement dit, \\(T\\) est transitif). \\(T\\) est conjugué à une rotation d’angle irrationnel. Pour tout \\(x\\in {\\mathbb {T}}\\), l’orbite \\(O_x\\) de \\(x\\) est dense dans \\({\\mathbb {T}}\\) (autrement dit T est minimal).","doc_url":null,"latex_link":"sect0002.html#minimal équivaut transitif","lean_link":null,"lean_decl_name":"minimal équivaut transitif","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000101","label":"a0000000101","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Le nombre de rotation \\(\\rho (T)\\) dépend continûment de T.","doc_url":null,"latex_link":"sect0002.html#a0000000101","lean_link":null,"lean_decl_name":"a0000000101","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"poincaré","label":"poincaré","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Supposons que \\(\\rho (T) \\notin \\mathbb {Q}/\\mathbb {Z}\\), ie supposons que pour tout relèvement \\( f\\) de \\(T\\), \\(\\alpha (f) \\in {\\mathbb {R}}\\setminus {\\mathbb {Q}}\\). Notons \\(\\alpha _T\\) l’unique représentant de \\(\\rho (T)\\) dans [0,1[. La classification de Poincaré distingue deux cas: Si \\(T\\) est transitive, alors \\(T\\) est conjuguée à la rotation \\(R_{\\alpha _T}\\). Si \\(T\\) n’est pas transitive, \\(T\\) est seulement semi-conjuguée à la rotation \\(R_{\\alpha _T}\\). Plus particulièrement, il existe une application continue surjective \\(h\\) telle que \\[ h\\circ T = R_{\\alpha _T} \\circ h. \\]","doc_url":null,"latex_link":"sect0002.html#poincaré","lean_link":null,"lean_decl_name":"poincaré","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"ensemble dérivé","label":"ensemble dérivé","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soit \\(x \\in {\\mathbb {T}}\\). Notons \\(L_x\\) l’ensemble des valeurs d’adhérence de la suite \\((T^n(x))_{n\\geq 0}\\). L’ensemble \\(L_x\\) possède les trois propriétés suivantes: Il est invariant par \\(T\\) et \\(T^{-1}\\). Il ne dépend pas de \\(x\\). Il vaut soit \\({\\mathbb {T}}\\) tout entier, soit un fermé de \\({\\mathbb {T}}\\) d’intérieur vide sans point isolé.","doc_url":null,"latex_link":"sect0002.html#ensemble dérivé","lean_link":null,"lean_decl_name":"ensemble dérivé","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"distance sur HomT","label":"distance sur HomT","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"La métrique \\(d_q\\) définit la topologie quotient sur \\({\\mathcal{E}}^+/\\mathcal{T}_{\\mathrm{ent}}\\).","doc_url":null,"latex_link":"sect0002.html#distance sur HomT","lean_link":null,"lean_decl_name":"distance sur HomT","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"isom relation d'équiv sur les homéo","label":"isom relation d'équiv sur les homéo","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Isomorphisme entre \\(\\mathcal{E}^+/\\mathcal{T}_{\\mathrm{ent}}\\) et \\(\\mathrm{Hom}^+({\\mathbb {T}})\\)L’application \\(f\\mapsto T_f\\) de \\({\\mathcal{E}}^+\\) dans \\(\\mathrm{Hom}^+({\\mathbb {T}})\\) passe au quotient et fournit un isomorphisme \\(\\iota \\) de groupes de \\({\\mathcal{E}}^+/\\thicksim ={\\mathcal{E}}^+/\\mathcal{T}_{\\mathrm{ent}}\\) vers \\(\\mathrm{Hom}^+({\\mathbb {T}})\\).L’application \\(\\iota \\) est définie par: \\[ \\forall \\ \\overline{f} \\in {\\mathcal{E}}^+/\\mathcal{T}_{\\mathrm{ent}}, \\ \\iota (\\overline{f})= T_f. \\]","doc_url":null,"latex_link":"sect0002.html#isom relation d'équiv sur les homéo","lean_link":null,"lean_decl_name":"isom relation d'équiv sur les homéo","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"T[x;y]","label":"T[x;y]","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soient \\(x,y \\in {\\mathbb {R}}/{\\mathbb {Z}}\\), on a l’égalité \\(T([x,y])=[T(x),T(y)].\\)","doc_url":null,"latex_link":"sect0002.html#T[x;y]","lean_link":null,"lean_decl_name":"T[x;y]","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"k rho","label":"k rho","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Pour tout \\(k \\in \\mathbb {Z}\\), \\(\\rho (T^k)= k \\times \\rho (T)\\). En particulier, on a \\(\\rho (\\mathrm{Id_{{\\mathbb {T}}}})=0\\) et \\(\\rho (T^{-1})=-\\rho (T)\\).","doc_url":null,"latex_link":"sect0002.html#k rho","lean_link":null,"lean_decl_name":"k rho","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000051","label":"a0000000051","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soit \\((u_n)_{n \\in \\mathbb {N}}\\) une suite réelle.On dit que \\((u_n)_{n \\in \\mathbb {N}}\\) est sous-additive si pour tout couple d’entier naturels \\((p,q)\\), on a: \\(u_{p+q} \\leq u_p + u_q\\)On dit que \\((u_n)_{n \\in \\mathbb {N}}\\) est sur-additive si pour tout couple d’entiers naturels \\((p, q)\\), on a: \\(u_{p+q} \\geq u_p + u_q\\)","doc_url":null,"latex_link":"sect0002.html#a0000000051","lean_link":null,"lean_decl_name":"a0000000051","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000110","label":"a0000000110","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soient \\(X\\) et \\(Y\\) deux espaces topologiques et \\(f:X\\to X\\), \\(g:Y \\to Y\\) deux homéomorphismes. On dit que \\(f\\) et \\(g\\) sont semi-conjuguées (resp. conjuguées) s’il existe une application continue surjective (resp. bijective) \\(h : Y \\to X\\) telle que \\(f\\circ h = h \\circ g\\).","doc_url":null,"latex_link":"sect0002.html#a0000000110","lean_link":null,"lean_decl_name":"a0000000110","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"préli relèvement","label":"préli relèvement","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soit \\(y_0 \\in {\\mathbb {R}}\\), et \\(\\varepsilon \\in \\ ]0,1/2[\\). La restriction de \\(\\pi \\) à \\([y_0-\\varepsilon , y_0 +\\varepsilon ]\\) est un homéomorphisme de \\([y_0-\\varepsilon , y_0 +\\varepsilon ]\\) vers son image.","doc_url":null,"latex_link":"sect0002.html#préli relèvement","lean_link":null,"lean_decl_name":"préli relèvement","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"ouvertUr","label":"ouvertUr","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soit \\(r \\in \\mathbb {Z}\\).Posons \\(U_r:=\\lbrace f \\in \\mathcal{E}^+: \\mathrm{Im}(f-Id_{\\mathbb {R}}) \\subset \\ ]r,r+1[\\rbrace \\). \\(U_r\\) est un ouvert de \\(\\mathcal{E}^+\\) muni de la topologie issue de la norme de la convergence uniforme.","doc_url":null,"latex_link":"sect0002.html#ouvertUr","lean_link":null,"lean_decl_name":"ouvertUr","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000111","label":"a0000000111","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soit f un relèvement de T. Supposons \\(\\alpha (f) \\notin \\mathbb {Q}\\).Soient \\(n_1,n_2, m_1, m_2 \\in \\mathbb {Z}\\), il y a équivalence entre: \\(\\forall \\ x \\in \\mathbb {R}, \\ f^{n_1}(x)+m_1 {\\lt} f^{n_2}(x)+m_2\\). \\(n_1 \\alpha (f) + m_1 {\\lt} n_2 \\alpha (f) + m_2\\). \\(\\exists \\ x \\in \\mathbb {R}, \\ f^{n_1}(x)+m_1 {\\lt} f^{n_2}(x)+m_2\\).","doc_url":null,"latex_link":"sect0002.html#a0000000111","lean_link":null,"lean_decl_name":"a0000000111","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000033","label":"a0000000033","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"On définit sur \\({\\mathcal{E}}^+\\) la relation d’équivalence \\(\\thicksim \\) par: \\[ f\\ \\thicksim \\ g \\Longleftrightarrow f-g \\text{ est constante et à valeurs dans\\ } \\mathbb {Z}. \\]","doc_url":null,"latex_link":"sect0002.html#a0000000033","lean_link":null,"lean_decl_name":"a0000000033","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"passage au quotient","label":"passage au quotient","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Passage au quotientSoit \\(f\\in {\\mathcal{E}}\\), i.e. \\(f: \\mathbb {R} \\to \\mathbb {R}\\) une fonction continue et strictement monotone telle que \\(x\\mapsto f(x+1) -f(x)\\) est constante égale à \\(\\pm 1\\).Alors, il existe une unique application \\(T_f \\in \\mathrm{Hom}({\\mathbb {T}})\\), telle que \\(T_f\\circ \\pi = \\pi \\circ f\\).","doc_url":null,"latex_link":"sect0002.html#passage au quotient","lean_link":null,"lean_decl_name":"passage au quotient","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"rho commute","label":"rho commute","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soient \\(T_1\\), \\(T_2 \\in \\mathrm{Hom}^+({\\mathbb {T}})\\) tels que \\(T_1\\circ T_2=T_1 \\circ T_2\\). Alors, \\[ \\rho (T_1\\circ T_2)=\\rho (T_1)+\\rho (T_2). \\]","doc_url":null,"latex_link":"sect0002.html#rho commute","lean_link":null,"lean_decl_name":"rho commute","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"phin","label":"phin","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soit \\(n \\in \\mathbb {N}\\), l’application \\(\\varphi _n : \\mathcal{C}_U(\\mathbb {R},\\mathbb {R}) \\to \\mathcal{C}_U(\\mathbb {R},\\mathbb {R})\\) définie par \\(\\varphi _n(f)=f^n\\) est continue pour la topologie associée à la norme de la convergence uniforme.","doc_url":null,"latex_link":"sect0002.html#phin","lean_link":null,"lean_decl_name":"phin","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"suiteadd","label":"suiteadd","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soit \\((u_n)_{n \\in \\mathbb {N}}\\) une suite réelle. Si \\((u_n)_{n \\in \\mathbb {N}}\\) est sous-additive, alors \\[ \\frac{u_n}{n} \\underset {n \\to + \\infty }{\\longrightarrow } \\underset {p \\geq 1}{\\inf }\\ \\frac{u_p}{p} \\in \\mathbb {R} \\cup \\lbrace - \\infty \\rbrace . \\]","doc_url":null,"latex_link":"sect0002.html#suiteadd","lean_link":null,"lean_decl_name":"suiteadd","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000142","label":"a0000000142","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"On a les équivalences suivantes: \\(L_T= {\\mathbb {T}}\\). \\(T\\) est minimal. \\(T\\) est transitif.","doc_url":null,"latex_link":"sect0002.html#a0000000142","lean_link":null,"lean_decl_name":"a0000000142","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"equiv amel","label":"equiv amel","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soient f \\(\\in \\mathcal{E}^+\\) et \\(k,r \\in \\mathbb {Z}\\). Alors \\[ \\mathrm{Im}(f^k - \\mathrm{Id})\\subset \\ ]r,r+1[ \\Longleftrightarrow \\alpha (f^k) \\in ]r,r+1[. \\]","doc_url":null,"latex_link":"sect0002.html#equiv amel","lean_link":null,"lean_decl_name":"equiv amel","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"modulecont","label":"modulecont","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Pour toute fonction \\(f\\) uniformément continue, le module de continuité \\(w_f\\) possède les 3 propriétés suivantes: \\(\\omega _f\\) est croissant. \\(\\omega _f\\) est sous-additif: \\(\\forall \\delta _1, \\delta _2 \\in \\mathbb {R}^+\\), \\(\\omega _f(\\delta _1 + \\delta _2) \\leq \\omega _f(\\delta _1) +\\omega _f(\\delta _2)\\). \\(\\omega _f(\\delta ) \\underset {\\delta \\to 0}{\\longrightarrow }0\\).","doc_url":null,"latex_link":"sect0002.html#modulecont","lean_link":null,"lean_decl_name":"modulecont","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}

crawled_repos/FLT_ch10.jsonl

@@ -0,0 +1,39 @@
+{"id":"MeasureTheory.ringHaarChar_mul_integral","label":"MeasureTheory.ringHaarChar_mul_integral","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":["MeasureTheory.ringHaarChar","MeasureTheory.addEquivAddHaarChar_smul_integral_map"],"nlp_text":"If \\(f:R\\to \\mathbb {R}\\) is a Borel measurable function and \\(u\\in R^\\times \\) then \\(\\delta _R(u)\\int f(ux)d\\mu (x)=\\int f(x)d\\mu (x)\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.ringHaarChar_mul_integral","latex_link":"Haar_char_project.html#MeasureTheory.ringHaarChar_mul_integral","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.ringHaarChar_mul_integral","lean_decl_name":"MeasureTheory.ringHaarChar_mul_integral","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch10","source_path":"FLT/HaarMeasure/HaarChar/Ring.lean","lean4_code":"lemma ringHaarChar_mul_integral\n    (μ : Measure R) [IsAddHaarMeasure μ] [μ.Regular]\n    {f : R → ℝ} (hf : Measurable f) (u : Rˣ) :\n    (ringHaarChar u) * ∫ (r : R), f (u * r) ∂μ = ∫ a, f a ∂μ := by\n  symm\n  convert (addEquivAddHaarChar_smul_integral_map μ (ContinuousAddEquiv.mulLeft u) (f := f)).symm\n    using 1\n  simp only [ringHaarChar_toFun, NNReal.smul_def, smul_eq_mul, mul_eq_mul_left_iff,\n    NNReal.coe_eq_zero]\n  rw [MeasureTheory.integral_map (by fun_prop) (by fun_prop)]\n  simp\n","line":115,"end_line":125,"proof_status":"formalized"}
+{"id":"MeasureTheory.ringHaarChar_continuous","label":"MeasureTheory.ringHaarChar_continuous","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":["MeasureTheory.ringHaarChar_mul_integral"],"nlp_text":"The function \\(\\delta _R:R^\\times \\to \\mathbb {R}_{{\\gt}0}\\) is continuous.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.ringHaarChar_continuous","latex_link":"Haar_char_project.html#MeasureTheory.ringHaarChar_continuous","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.ringHaarChar_continuous","lean_decl_name":"MeasureTheory.ringHaarChar_continuous","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch10","source_path":"FLT/HaarMeasure/HaarChar/Ring.lean","lean4_code":"lemma ringHaarChar_continuous :\n    Continuous (fun (u : Rˣ) ↦ addEquivAddHaarChar (ContinuousAddEquiv.mulLeft u)) := by\n  suffices\n    hf : Continuous (fun (u : Rˣ) ↦ (addEquivAddHaarChar (ContinuousAddEquiv.mulLeft u) : ℝ)) from\n    continuous_induced_rng.mpr hf\n  obtain ⟨⟨f, f_cont⟩, f_comp, f_nonneg, f_one⟩ :\n    ∃ f : C(R, ℝ), HasCompactSupport f ∧ 0 ≤ f ∧ f 1 ≠ 0 := exists_continuous_nonneg_pos 1\n  have int_f_ne_zero : ∫ x, f x ∂addHaar ≠ 0 :=\n    ne_of_gt (f_cont.integral_pos_of_hasCompactSupport_nonneg_nonzero f_comp f_nonneg f_one)\n  have h (u : Rˣ) :=\n      addEquivAddHaarChar_smul_integral_map addHaar (ContinuousAddEquiv.mulLeft u) (f := f)\n  conv at h => ext; rw [integral_map (by fun_prop) (by fun_prop)]\n  simp only [ContinuousAddEquiv.mulLeft_apply, NNReal.smul_def, smul_eq_mul] at h\n  let g (u : Rˣ) (x : R) := f (u * x)\n  have int_g_ne_zero (u : Rˣ) : ∫ (x : R), g u x ∂addHaar ≠ 0 := by\n    have hu := h u\n    contrapose! hu\n    simp [g, hu, int_f_ne_zero.symm]\n  rw [← funext (fun u ↦ div_eq_of_eq_mul (int_g_ne_zero u) (h u).symm)]\n  refine Continuous.div continuous_const ?_ (fun u ↦ int_g_ne_zero u)\n  rw [continuous_iff_continuousAt]\n  intro u₀\n  obtain ⟨K, hK, hu₀⟩ := exists_compact_mem_nhds (↑u₀⁻¹ : R)\n  let s := (fun (u : Rˣ) ↦ (↑u⁻¹ : R)) ⁻¹' K\n  refine ContinuousOn.continuousAt ?_\n    (ContinuousAt.preimage_mem_nhds (by fun_prop) (by exact hu₀) : s ∈ nhds u₀)\n  apply continuousOn_integral_of_compact_support (hK.mul f_comp) (by fun_prop)\n  intro p x hps hx\n  unfold g\n  apply image_eq_zero_of_notMem_tsupport\n  contrapose! hx\n  exact ⟨(↑p⁻¹ : R) , hps, p * x, hx, by simp⟩\n","line":57,"end_line":88,"proof_status":"formalized"}
+{"id":"MeasureTheory.addEquivAddHaarChar_restrictedProductCongrRight","label":"MeasureTheory.addEquivAddHaarChar_restrictedProductCongrRight","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":[],"nlp_text":"With \\(A\\), \\(A_i\\), \\(C_i\\), \\(\\phi _i\\), \\(\\phi \\) defined as above, we have \\(\\delta _A(\\phi )=\\prod _i\\delta _{A_i}(\\phi _i)\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.addEquivAddHaarChar_restrictedProductCongrRight","latex_link":"Haar_char_project.html#MeasureTheory.addEquivAddHaarChar_restrictedProductCongrRight","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.addEquivAddHaarChar_restrictedProductCongrRight","lean_decl_name":"MeasureTheory.addEquivAddHaarChar_restrictedProductCongrRight","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/HaarMeasure/HaarChar/AddEquiv.lean","lean4_code":"@[to_additive]","line":399,"end_line":399,"proof_status":"formalized"}
+{"id":"NumberField.AdeleRing.isCentralSimple_addHaarScalarFactor_left_mul_eq_right_mul","label":"NumberField.AdeleRing.isCentralSimple_addHaarScalarFactor_left_mul_eq_right_mul","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":["MeasureTheory.addEquivAddHaarChar_restrictedProductCongrRight","IsSimpleRing.ringHaarChar_eq_addEquivAddHaarChar_mulRight"],"nlp_text":"Let \\(B\\) be a finite-dimensional central simple \\(K\\)-algebra. Say \\(u\\in B_{\\mathbb {A}}^\\times \\), and define \\(\\ell _u\\) and \\(r_u:B_{\\mathbb {A}}\\to B_{\\mathbb {A}}\\) by \\(\\ell _u(x)=ux\\) and \\(r_u(x)=xu\\). Then \\(d_{B_{\\mathbb {A}}}(\\ell _u)=d_{B_{\\mathbb {A}}}(r_u)\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.isCentralSimple_addHaarScalarFactor_left_mul_eq_right_mul","latex_link":"Haar_char_project.html#NumberField.AdeleRing.isCentralSimple_addHaarScalarFactor_left_mul_eq_right_mul","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.isCentralSimple_addHaarScalarFactor_left_mul_eq_right_mul","lean_decl_name":"NumberField.AdeleRing.isCentralSimple_addHaarScalarFactor_left_mul_eq_right_mul","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch10","source_path":"FLT/DivisionAlgebra/Finiteness.lean","lean4_code":"lemma isCentralSimple_addHaarScalarFactor_left_mul_eq_right_mul\n    [Algebra.IsCentral K D] (u : D_𝔸ˣ) :\n    addEquivAddHaarChar (ContinuousAddEquiv.mulLeft u) =\n    addEquivAddHaarChar (ContinuousAddEquiv.mulRight u) := by\n  open IsDedekindDomain MeasureTheory in\n  let u' := D𝔸_prodRight_units K D u\n  have hl :\n      addEquivAddHaarChar (ContinuousAddEquiv.mulLeft u)\n      = addEquivAddHaarChar (ContinuousAddEquiv.mulLeft u'.1)\n      * addEquivAddHaarChar (ContinuousAddEquiv.mulLeft u'.2) := by\n    rw [← addEquivAddHaarChar_prodCongr\n      (ContinuousAddEquiv.mulLeft u'.1) (ContinuousAddEquiv.mulLeft u'.2)]\n    apply addEquivAddHaarChar_eq_addEquivAddHaarChar_of_continuousAddEquiv (D𝔸_prodRight'' K D)\n    intro x; simp; rfl\n  have hr :\n      addEquivAddHaarChar (ContinuousAddEquiv.mulRight u)\n      = addEquivAddHaarChar (ContinuousAddEquiv.mulRight u'.1)\n      * addEquivAddHaarChar (ContinuousAddEquiv.mulRight u'.2) := by\n    rw [← addEquivAddHaarChar_prodCongr\n      (ContinuousAddEquiv.mulRight u'.1) (ContinuousAddEquiv.mulRight u'.2)]\n    apply addEquivAddHaarChar_eq_addEquivAddHaarChar_of_continuousAddEquiv (D𝔸_prodRight'' K D)\n    intro x; simp; rfl\n  simp [hl, hr, Dinfx, Dfx, Df,\n    InfiniteAdeleRing.isCentralSimple_infinite_addHaarScalarFactor_left_mul_eq_right_mul _,\n    FiniteAdeleRing.isCentralSimple_addHaarScalarFactor_left_mul_eq_right_mul K D _]\n","line":622,"end_line":646,"proof_status":"statement_formalized"}
+{"id":"MeasureTheory.ringHaarChar_adeles_rat","label":"MeasureTheory.ringHaarChar_adeles_rat","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":["MeasureTheory.addEquivAddHaarChar_restrictedProductCongrRight","MeasureTheory.ringHaarChar_real","MeasureTheory.ringHaarChar_padic","MeasureTheory.addEquivAddHaarChar_prodCongr"],"nlp_text":"If \\(x\\in \\mathbb {A}_{\\mathbb {Q}}^\\times \\) then \\(\\delta _{\\mathbb {A}_{\\mathbb {Q}}}(x)=\\prod _v|x_v|_v.\\)","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.ringHaarChar_adeles_rat","latex_link":"Haar_char_project.html#MeasureTheory.ringHaarChar_adeles_rat","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.ringHaarChar_adeles_rat","lean_decl_name":"MeasureTheory.ringHaarChar_adeles_rat","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/HaarMeasure/HaarChar/AdeleRing.lean","lean4_code":"lemma MeasureTheory.ringHaarChar_adeles_rat (x : (𝔸 ℚ)ˣ) :\n  ringHaarChar x = ringHaarChar (MulEquiv.prodUnits x).1 *\n    (∏ᶠ p, ringHaarChar (MulEquiv.restrictedProductUnits (MulEquiv.prodUnits x).2 p)) := by\n  unfold AdeleRing\n  rw [ringHaarChar_prod' x]\n  congr\n  have := Fact.mk <| NumberField.isOpenAdicCompletionIntegers ℚ\n  have := NumberField.instCompactSpaceAdicCompletionIntegers ℚ\n  convert addEquivAddHaarChar_restrictedProductCongrRight\n    (C := fun p ↦ (p.adicCompletionIntegers ℚ).toAddSubgroup)\n    (fun p ↦\n      (ContinuousAddEquiv.mulLeft (MulEquiv.restrictedProductUnits (MulEquiv.prodUnits x).2 p))) _\n  exact (MulEquiv.restrictedProductUnits (MulEquiv.prodUnits x).2).2.mono\n    (fun p hp ↦ Equiv.bijOn' _\n      (fun x hx ↦ Subring.mul_mem _ ((Submonoid.mem_units_iff _ _).mp hp).1 hx)\n      (fun x hx ↦ Subring.mul_mem _ ((Submonoid.mem_units_iff _ _).mp hp).2 hx))","line":43,"end_line":58,"proof_status":"formalized"}
+{"id":"MeasureTheory.addHaarScalarFactor_tensor_adeles_eq_one","label":"MeasureTheory.addHaarScalarFactor_tensor_adeles_eq_one","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":["MeasureTheory.addEquivAddHaarChar_restrictedProductCongrRight","NumberField.AdeleRing.ModuleBaseChangeContinuousAddEquiv"],"nlp_text":"In the above situation (\\(V\\) a finite-dimensional \\(\\mathbb {Q}\\)-vector space, \\(\\phi :V\\cong V\\) is \\(\\mathbb {Q}\\)-linear, \\(\\phi _{\\mathbb {A}}\\) the base extension to \\(V_{\\mathbb {A}}:=V\\otimes _{\\mathbb {Q}}{\\mathbb {A}_{\\mathbb {Q}}}\\), a continuous linear endomorphism of \\(V_{\\mathbb {A}}\\) with the \\(\\mathbb {A}_{\\mathbb {Q}}\\)-module topology), we have \\(d_{V_{\\mathbb {A}}}(\\phi _{\\mathbb {A}})=1.\\)","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.addHaarScalarFactor_tensor_adeles_eq_one","latex_link":"Haar_char_project.html#MeasureTheory.addHaarScalarFactor_tensor_adeles_eq_one","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.addHaarScalarFactor_tensor_adeles_eq_one","lean_decl_name":"MeasureTheory.addHaarScalarFactor_tensor_adeles_eq_one","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/HaarMeasure/HaarChar/AdeleRing.lean","lean4_code":"lemma MeasureTheory.addHaarScalarFactor_tensor_adeles_eq_one (φ : V ≃ₗ[K] V)\n    [MeasurableSpace (V ⊗[K] 𝔸 K)] [BorelSpace (V ⊗[K] 𝔸 K)] :\n    addEquivAddHaarChar\n      (ContinuousLinearEquiv.baseChange K (𝔸 K) V V φ).toContinuousAddEquiv = 1 := by\n  -- we deduce this from the corresponding statement for `K = ℚ`.\n  -- A K-module is a ℚ-module\n  let : Module ℚ V := Module.compHom V (algebraMap ℚ K)\n  have : Module.Finite ℚ V := FiniteDimensional.trans ℚ K V\n  let : Module (AdeleRing (𝓞 ℚ) ℚ) (V ⊗[K] AdeleRing (𝓞 K) K) :=\n    Module.compHom _ (algebraMap (AdeleRing (𝓞 ℚ) ℚ) (AdeleRing (𝓞 K) K))\n  have : IsScalarTower (AdeleRing (𝓞 ℚ) ℚ) (AdeleRing (𝓞 K) K) (V ⊗[K] AdeleRing (𝓞 K) K) :=\n    IsScalarTower.of_algebraMap_smul fun r ↦ congrFun rfl\n  -- and V ⊗[K] 𝔸_K ≃ V ⊗[ℚ] 𝔸_ℚ\n  let f := NumberField.AdeleRing.ModuleBaseChangeContinuousAddEquiv ℚ K V\n  borelize (V ⊗[ℚ] AdeleRing (𝓞 ℚ) ℚ)\n  -- and the obvious diagram commutes\n  have := MeasureTheory.addEquivAddHaarChar_eq_addEquivAddHaarChar_of_continuousAddEquiv f\n    (ContinuousLinearEquiv.baseChange ℚ (𝔸 ℚ) V V (φ.restrictScalars ℚ)).toContinuousAddEquiv\n    (ContinuousLinearEquiv.baseChange K (𝔸 K) V V φ).toContinuousAddEquiv\n  rw [← this]\n  -- so the result follows from the case K=ℚ\n  · apply MeasureTheory.addHaarScalarFactor_tensor_adeles_rat_eq_one\n  · intro x\n    induction x with\n    | zero => simp\n    | tmul x y => rfl\n    | add x y hx hy => simp [hx, hy]","line":165,"end_line":191,"proof_status":"formalized"}
+{"id":"MeasureTheory.ringHaarChar_restrictedProduct","label":"MeasureTheory.ringHaarChar_restrictedProduct","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":["MeasureTheory.addEquivAddHaarChar_restrictedProductCongrRight"],"nlp_text":"If \\(u=(u_i)_i\\in R^\\times \\) then \\(\\delta _R(u)=\\prod _i\\delta _{R_i}(u_i)\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.ringHaarChar_restrictedProduct","latex_link":"Haar_char_project.html#MeasureTheory.ringHaarChar_restrictedProduct","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.ringHaarChar_restrictedProduct","lean_decl_name":"MeasureTheory.ringHaarChar_restrictedProduct","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch10","source_path":"FLT/HaarMeasure/HaarChar/Ring.lean","lean4_code":"lemma ringHaarChar_restrictedProduct (u : (Πʳ i, [A i, C i])ˣ) :\n    ringHaarChar u = ∏ᶠ i, ringHaarChar (MulEquiv.restrictedProductUnits u i) := by\n  set u := MulEquiv.restrictedProductUnits u\n  apply addEquivAddHaarChar_restrictedProductCongrRight (C := (C · |>.toAddSubgroup))\n    (ContinuousAddEquiv.mulLeft <| u ·)\n  refine Filter.Eventually.and u.coe_prop u⁻¹.coe_prop |>.mono fun i ⟨hu, hv⟩ ↦ ⟨?_, ?_, ?_⟩\n  · exact fun _ ↦ (C i).mul_mem ((C i).mem_units_iff _ |>.mp hu).1\n  · exact Set.injOn_of_injective (ContinuousAddEquiv.injective _)\n  · exact fun c hc ↦ ⟨(u i)⁻¹ * c, (C i).mul_mem ((C i).mem_units_iff _ |>.mp hv).1 hc, by simp⟩\n","line":200,"end_line":208,"proof_status":"formalized"}
+{"id":"MeasureTheory.ringHaarChar_adeles_units_rat_eq_one","label":"MeasureTheory.ringHaarChar_adeles_units_rat_eq_one","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":["MeasureTheory.ringHaarChar_adeles_rat"],"nlp_text":"If \\(x\\in \\mathbb {Q}^\\times \\subseteq \\mathbb {A}_{\\mathbb {Q}}^\\times \\) then \\(\\delta _{\\mathbb {A}_{\\mathbb {Q}}}(x)=1.\\)","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.ringHaarChar_adeles_units_rat_eq_one","latex_link":"Haar_char_project.html#MeasureTheory.ringHaarChar_adeles_units_rat_eq_one","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.ringHaarChar_adeles_units_rat_eq_one","lean_decl_name":"MeasureTheory.ringHaarChar_adeles_units_rat_eq_one","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/HaarMeasure/HaarChar/AdeleRing.lean","lean4_code":"lemma MeasureTheory.ringHaarChar_adeles_units_rat_eq_one (x : ℚˣ) :\n  ringHaarChar (Units.map (algebraMap ℚ (𝔸 ℚ)) x : (𝔸 ℚ)ˣ) = 1 := by\n  rw [ringHaarChar_adeles_rat (Units.map (algebraMap ℚ (𝔸 ℚ)) x : (𝔸 ℚ)ˣ)]\n  ext; simp only [NNReal.coe_mul, NNReal.coe_one]\n  rw [← NumberField.prod_abs_eq_one (K := ℚ) (x := x) (Units.ne_zero x)]; congr\n  · -- infinite place\n    simp only [InfiniteAdeleRing, ringHaarChar_pi', NNReal.coe_prod, Rat.infinitePlace_apply,\n      Rat.cast_abs]\n    congr; ext v; rw [Subsingleton.elim v Rat.infinitePlace]\n    let : Algebra ℤ Rat.infinitePlace.Completion := Ring.toIntAlgebra _\n    simp [InfinitePlace.mult, Rat.isReal_infinitePlace,\n      ringHaarChar_eq_ringHaarChar_of_continuousAlgEquiv {\n        __ := Rat.infinitePlace_completion_continuousAlgEquiv\n        commutes' := by simp},\n      ringHaarChar_real, ← Rat.infinitePlace_completion_continuousAlgEquiv_apply_algebraMap,\n      -eq_ratCast]\n    rfl\n  · -- finite places\n    rw [← finprod_comp_equiv FinitePlace.equivHeightOneSpectrum.symm]\n    conv_lhs =>\n      apply NNReal.toRealHom.map_finprod_of_injective (.of_eq_imp_le fun {_ _} a ↦ a.le)\n    apply finprod_congr; intro p\n    let : Algebra ℤ (p.adicCompletion ℚ) := Ring.toIntAlgebra _\n    simp [FinitePlace.equivHeightOneSpectrum,\n      ringHaarChar_eq_ringHaarChar_of_continuousAlgEquiv {\n        __ := (Rat.HeightOneSpectrum.adicCompletion.padicEquiv p)\n        commutes' := by simp},\n      Rat.HeightOneSpectrum.adicCompletion.padicEquiv_norm_eq]\n    rfl","line":60,"end_line":88,"proof_status":"formalized"}
+{"id":"NumberField.AdeleRing.units_mem_ringHaarCharacter_ker","label":"NumberField.AdeleRing.units_mem_ringHaarCharacter_ker","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":["MeasureTheory.addHaarScalarFactor_tensor_adeles_eq_one"],"nlp_text":"If \\(B\\) is a finite-dimensional \\(\\mathbb {Q}\\)-algebra (for example a number field, or a quaternion algebra over a number field), if \\(B_{\\mathbb {A}}\\) denotes the ring \\(B\\otimes _{\\mathbb {Q}}\\mathbb {A}_{\\mathbb {Q}}\\), and if \\(b\\in B^\\times \\), then \\(\\delta _{B_{\\mathbb {A}}}(b)=1\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.units_mem_ringHaarCharacter_ker","latex_link":"Haar_char_project.html#NumberField.AdeleRing.units_mem_ringHaarCharacter_ker","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.units_mem_ringHaarCharacter_ker","lean_decl_name":"NumberField.AdeleRing.units_mem_ringHaarCharacter_ker","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/HaarMeasure/HaarChar/AdeleRing.lean","lean4_code":"/-- Left multiplication by an element of Bˣ on B ⊗ 𝔸_K does not scale additive\nHaar measure. In other words, Bˣ is in the kernel of the `ringHaarChar` of `B ⊗ 𝔸_K`.\n-/\nlemma NumberField.AdeleRing.units_mem_ringHaarCharacter_ker\n    [MeasurableSpace (B ⊗[K] AdeleRing (𝓞 K) K)] [BorelSpace (B ⊗[K] AdeleRing (𝓞 K) K)]\n    (b : Bˣ) :\n    (Units.map Algebra.TensorProduct.includeLeftRingHom.toMonoidHom b :\n      (B ⊗[K] AdeleRing (𝓞 K) K)ˣ) ∈\n    ringHaarChar_ker (B ⊗[K] AdeleRing (𝓞 K) K) := by\n  rw [mem_ringHaarChar_ker, ringHaarChar_apply]\n  convert MeasureTheory.addHaarScalarFactor_tensor_adeles_eq_one K B (LinearEquiv.mulLeft K b)\n  ext c\n  change _ = (ContinuousLinearEquiv.baseChange K _ _ _ _) c\n  induction c with\n  | zero => simp\n  | tmul x y => simp [LinearEquiv.mulLeft]\n  | add x y hx hy => simp_all [mul_add]","line":194,"end_line":210,"proof_status":"formalized"}
+{"id":"NumberField.AdeleRing.addEquivAddHaarChar_mulRight_unit_eq_one","label":"NumberField.AdeleRing.addEquivAddHaarChar_mulRight_unit_eq_one","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":["MeasureTheory.addHaarScalarFactor_tensor_adeles_eq_one"],"nlp_text":"If \\(B\\) is a finite-dimensional \\(\\mathbb {Q}\\)-algebra and if \\(b\\in B^\\times \\) then right multiplication by \\(b\\) does not change Haar measure on \\(B_{\\mathbb {A}}\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.addEquivAddHaarChar_mulRight_unit_eq_one","latex_link":"Haar_char_project.html#NumberField.AdeleRing.addEquivAddHaarChar_mulRight_unit_eq_one","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.addEquivAddHaarChar_mulRight_unit_eq_one","lean_decl_name":"NumberField.AdeleRing.addEquivAddHaarChar_mulRight_unit_eq_one","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/HaarMeasure/HaarChar/AdeleRing.lean","lean4_code":"/-- Right multiplication by an element of Bˣ on B ⊗ 𝔸_K does not scale additive\nHaar measure.\n-/\nlemma NumberField.AdeleRing.addEquivAddHaarChar_mulRight_unit_eq_one\n    [MeasurableSpace (B ⊗[K] AdeleRing (𝓞 K) K)] [BorelSpace (B ⊗[K] AdeleRing (𝓞 K) K)]\n    (b : Bˣ) :\n    addEquivAddHaarChar\n      (ContinuousAddEquiv.mulRight\n        (Units.map Algebra.TensorProduct.includeLeftRingHom.toMonoidHom b :\n      (B ⊗[K] AdeleRing (𝓞 K) K)ˣ)) = 1 := by\n  convert addHaarScalarFactor_tensor_adeles_eq_one K B (LinearEquiv.mulRight K b)\n  ext c\n  change _ = (ContinuousLinearEquiv.baseChange K _ _ _ _) c\n  induction c with\n  | zero => simp\n  | tmul x y => simp [LinearEquiv.mulRight]\n  | add x y hx hy => simp_all [add_mul]","line":213,"end_line":229,"proof_status":"formalized"}
+{"id":"MeasureTheory.addEquivAddHaarChar","label":"MeasureTheory.addEquivAddHaarChar","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":[],"nlp_text":"If \\(A\\) is a locally compact topological additive abelian group, if \\(\\mu \\) is a regular Haar measure on \\(A\\), and if \\(\\phi :A\\to A\\) is an additive homeomorphism, then we let \\(d_A(\\phi )\\) denote the unique positive real number such that \\(\\mu (X)=d_A(\\phi )(\\phi _*\\mu )(X)\\) for any Borel set&#160;\\(X\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.addEquivAddHaarChar","latex_link":"Haar_char_project.html#MeasureTheory.addEquivAddHaarChar","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.addEquivAddHaarChar","lean_decl_name":"MeasureTheory.addEquivAddHaarChar","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/leanprover-community/mathlib4/98379c78c5dd2e3614cd24ab7ad22f5713e0eb63/Mathlib/MeasureTheory/Measure/Haar/MulEquivHaarChar.lean","lean4_code":"@[to_additive /-- If `φ : A ≃ₜ+ A` then `addEquivAddHaarChar φ` is the positive\nreal factor by which `φ` scales Haar measures on `A`. -/]","line":38,"end_line":39,"proof_status":"formalized"}
+{"id":"MeasureTheory.addEquivAddHaarChar_eq","label":"MeasureTheory.addEquivAddHaarChar_eq","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":["MeasureTheory.addEquivAddHaarChar"],"nlp_text":"\\(d_A(\\phi )\\) is independent of choice of regular Haar measure.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.addEquivAddHaarChar_eq","latex_link":"Haar_char_project.html#MeasureTheory.addEquivAddHaarChar_eq","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.addEquivAddHaarChar_eq","lean_decl_name":"MeasureTheory.addEquivAddHaarChar_eq","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/leanprover-community/mathlib4/98379c78c5dd2e3614cd24ab7ad22f5713e0eb63/Mathlib/MeasureTheory/Measure/Haar/MulEquivHaarChar.lean","lean4_code":"@[to_additive]","line":47,"end_line":47,"proof_status":"formalized"}
+{"id":"MeasureTheory.addEquivAddHaarChar_refl","label":"MeasureTheory.addEquivAddHaarChar_refl","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":["MeasureTheory.addEquivAddHaarChar_eq"],"nlp_text":"\\(d_A(id)=1.\\)","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.addEquivAddHaarChar_refl","latex_link":"Haar_char_project.html#MeasureTheory.addEquivAddHaarChar_refl","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.addEquivAddHaarChar_refl","lean_decl_name":"MeasureTheory.addEquivAddHaarChar_refl","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/leanprover-community/mathlib4/98379c78c5dd2e3614cd24ab7ad22f5713e0eb63/Mathlib/MeasureTheory/Measure/Haar/MulEquivHaarChar.lean","lean4_code":"@[to_additive (attr := simp)]","line":111,"end_line":111,"proof_status":"formalized"}
+{"id":"MeasureTheory.addEquivAddHaarChar_smul_map","label":"MeasureTheory.addEquivAddHaarChar_smul_map","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":["MeasureTheory.addEquivAddHaarChar_eq"],"nlp_text":"If \\(\\mu \\) is any regular Haar measure on \\(A\\) then \\(d_A(\\phi )(\\phi _*\\mu ) = \\mu .\\)","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.addEquivAddHaarChar_smul_map","latex_link":"Haar_char_project.html#MeasureTheory.addEquivAddHaarChar_smul_map","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.addEquivAddHaarChar_smul_map","lean_decl_name":"MeasureTheory.addEquivAddHaarChar_smul_map","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/leanprover-community/mathlib4/98379c78c5dd2e3614cd24ab7ad22f5713e0eb63/Mathlib/MeasureTheory/Measure/Haar/MulEquivHaarChar.lean","lean4_code":"@[to_additive addEquivAddHaarChar_smul_map]","line":62,"end_line":62,"proof_status":"formalized"}
+{"id":"MeasureTheory.ringHaarChar","label":"MeasureTheory.ringHaarChar","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":["MeasureTheory.addEquivAddHaarChar_trans","MeasureTheory.addEquivAddHaarChar_refl"],"nlp_text":"We define \\(\\delta _R(u)\\) (or just \\(\\delta (u)\\) when the ring \\(R\\) is clear) to be \\(d_R(\\ell _u)\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.ringHaarChar","latex_link":"Haar_char_project.html#MeasureTheory.ringHaarChar","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.ringHaarChar","lean_decl_name":"MeasureTheory.ringHaarChar","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch10","source_path":"FLT/HaarMeasure/HaarChar/Ring.lean","lean4_code":"/-- `ringHaarChar : Rˣ →ₜ* ℝ≥0` is the function sending a unit of\na locally compact topological ring `R` to the positive real factor\nwhich left multiplication by the unit scales additive Haar measure by.\n-/\n@[simps (isSimp := false)]\nnoncomputable def ringHaarChar : Rˣ →ₜ* ℝ≥0 where\n  toFun r := addEquivAddHaarChar (ContinuousAddEquiv.mulLeft r)\n  map_one' := by convert addEquivAddHaarChar_refl (G := R); ext; simp\n  map_mul' φ ψ := by\n    rw [mul_comm]\n    convert addEquivAddHaarChar_trans (G := R); ext; simp [mul_assoc]\n  continuous_toFun := ringHaarChar_continuous\n","line":90,"end_line":101,"proof_status":"formalized"}
+{"id":"MeasureTheory.addEquivAddHaarChar_eq_ringHaarChar_det, MeasureTheory.addEquivAddHaarChar_eq_ringHaarChar_det_of_existsListTransvecEtc","label":"MeasureTheory.addEquivAddHaarChar_eq_ringHaarChar_det, MeasureTheory.addEquivAddHaarChar_eq_ringHaarChar_det_of_existsListTransvecEtc","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":["MeasureTheory.ringHaarChar"],"nlp_text":"Assume that there’s an \\(F\\)-basis for \\(V\\) such that \\(\\phi \\) is a product of elementary and diagonal matrices (note that this is automatic if \\(F\\) is a field and mathlib has this). Then \\(d_V(\\phi )=\\delta _F(\\det (\\phi ))\\), where \\(\\det (\\phi )\\in F\\) is the determinant of \\(\\phi \\) as an \\(F\\)-linear map.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.addEquivAddHaarChar_eq_ringHaarChar_det","latex_link":"Haar_char_project.html#MeasureTheory.addEquivAddHaarChar_eq_ringHaarChar_det, MeasureTheory.addEquivAddHaarChar_eq_ringHaarChar_det_of_existsListTransvecEtc","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.addEquivAddHaarChar_eq_ringHaarChar_det","lean_decl_name":"MeasureTheory.addEquivAddHaarChar_eq_ringHaarChar_det","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch10","source_path":"FLT/HaarMeasure/HaarChar/FiniteDimensional.lean","lean4_code":"/-- Haar measure scaling for invertible linear maps on a finite-dimensional vector space\nover a field F assuming `[SecondCountableTopology F]`. -/\ntheorem addEquivAddHaarChar_eq_ringHaarChar_det [SecondCountableTopology F] (ρ : V ≃L[F] V) :\n    addEquivAddHaarChar ρ.toContinuousAddEquiv = ringHaarChar ρ.toLinearEquiv.det :=\n  addEquivAddHaarChar_eq_ringHaarChar_det_of_existsListTransvecEtc ρ _ (finBasis F V)\n  (Pivot.exists_list_transvec_mul_diagonal_mul_list_transvec'\n    ((LinearMap.toMatrix (finBasis F V) (finBasis F V)) ↑ρ.toLinearEquiv))\n","line":261,"end_line":267,"proof_status":"formalized"}
+{"id":"MeasureTheory.ringHaarChar_mul_volume","label":"MeasureTheory.ringHaarChar_mul_volume","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":["MeasureTheory.ringHaarChar"],"nlp_text":"If \\(X\\) is a Borel subset of \\(R\\) and \\(r\\in R^\\times \\) then \\(\\mu (rX)=\\delta _R(r)\\mu (X)\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.ringHaarChar_mul_volume","latex_link":"Haar_char_project.html#MeasureTheory.ringHaarChar_mul_volume","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.ringHaarChar_mul_volume","lean_decl_name":"MeasureTheory.ringHaarChar_mul_volume","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch10","source_path":"FLT/HaarMeasure/HaarChar/Ring.lean","lean4_code":"lemma ringHaarChar_mul_volume (μ : Measure R) [IsAddHaarMeasure μ] [μ.Regular]\n    {X : Set R} (u : Rˣ) :\n    μ (u • X) = ringHaarChar u * μ X := by\n  rw [ringHaarChar_toFun, (addEquivAddHaarChar_smul_preimage _ (ContinuousAddEquiv.mulLeft u)).symm]\n  simp\n","line":128,"end_line":132,"proof_status":"formalized"}
+{"id":"MeasureTheory.ringHaarChar_complex","label":"MeasureTheory.ringHaarChar_complex","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":["MeasureTheory.ringHaarChar_mul_volume"],"nlp_text":"If \\(R=\\mathbb {C}\\) then \\(\\delta _R(u)=|u|^2\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.ringHaarChar_complex","latex_link":"Haar_char_project.html#MeasureTheory.ringHaarChar_complex","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.ringHaarChar_complex","lean_decl_name":"MeasureTheory.ringHaarChar_complex","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/HaarMeasure/HaarChar/RealComplex.lean","lean4_code":"/-- The distributive Haar character of the action of `ℂˣ` on `ℂ` is the usual norm squared.\n\nThis means that `volume (z • s) = ‖z‖ ^ 2 * volume s` for all `z : ℂ` and `s : Set ℂ`.\nSee `Complex.volume_complex_smul`. -/\nlemma MeasureTheory.ringHaarChar_complex (z : ℂˣ) : ringHaarChar z = ‖(z : ℂ)‖₊ ^ 2 := by\n  -- We compute that `volume (x • ([0, 1] × [0, 1])) = ‖x‖₊ ^ 2 * volume ([0, 1] × [0, 1])`.\n  refine ringHaarChar_eq_of_measure_smul_eq_mul (s := Icc 0 1 ×ℂ Icc 0 1) (μ := volume)\n    (measure_pos_of_nonempty_interior _ <| by simp [interior_reProdIm]).ne'\n    (isCompact_Icc.reProdIm isCompact_Icc).measure_ne_top ?_\n  -- The determinant of left multiplication by `z⁻¹` as a `ℝ`-linear map is `‖z‖₊ ^ (-2)`.\n  have key : ((LinearMap.mul ℂ ℂ z⁻¹).restrictScalars ℝ).det = ‖z.val‖₊ ^ (-2 : ℤ) := by\n    refine Complex.ofReal_injective ?_\n    rw [LinearMap.det_restrictScalars]\n    simp [Algebra.norm_complex_apply, normSq_eq_norm_sq, zpow_ofNat]\n  -- Massaging, we find the result.\n  convert addHaar_preimage_linearMap (E := ℂ) volume\n    (f := (LinearMap.mul ℂ ℂ z⁻¹).restrictScalars ℝ) _ _ using 2\n  · simpa [LinearMap.mul, LinearMap.mk₂, LinearMap.mk₂', LinearMap.mk₂'ₛₗ, Units.smul_def, eq_comm]\n      using preimage_smul_inv z (Icc 0 1 ×ℂ Icc 0 1)\n  · simp [key, ENNReal.ofReal_pow, zpow_ofNat]; rfl\n  · simp [key, zpow_ofNat]","line":45,"end_line":65,"proof_status":"formalized"}
+{"id":"MeasureTheory.ringHaarChar_real","label":"MeasureTheory.ringHaarChar_real","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":["MeasureTheory.ringHaarChar_mul_volume"],"nlp_text":"If \\(R=\\mathbb {R}\\) then \\(\\delta _R(u)=|u|\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.ringHaarChar_real","latex_link":"Haar_char_project.html#MeasureTheory.ringHaarChar_real","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.ringHaarChar_real","lean_decl_name":"MeasureTheory.ringHaarChar_real","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/HaarMeasure/HaarChar/RealComplex.lean","lean4_code":"/-- The distributive Haar character of the action of `ℝˣ` on `ℝ` is the usual norm.\n\nThis means that `volume (x • s) = ‖x‖ * volume s` for all `x : ℝ` and `s : Set ℝ`.\nSee `Real.volume_real_smul`. -/\nlemma MeasureTheory.ringHaarChar_real (x : ℝˣ) : ringHaarChar x = ‖(x : ℝ)‖₊ :=\n  -- We compute that `volume (x • [0, 1]) = ‖x‖₊ * volume [0, 1]`.\n  ringHaarChar_eq_of_measure_smul_eq_mul (s := Icc 0 1) (μ := volume)\n    (measure_pos_of_nonempty_interior _ <| by simp).ne' isCompact_Icc.measure_ne_top\n      (Real.volume_real_smul ..)","line":35,"end_line":43,"proof_status":"formalized"}
+{"id":"MeasureTheory.ringHaarChar_padic","label":"MeasureTheory.ringHaarChar_padic","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":["MeasureTheory.ringHaarChar_mul_volume"],"nlp_text":"If \\(R=\\mathbb {Q}_p\\) then \\(\\delta _R(u)=|u|_p\\), the usual \\(p\\)-adic norm.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.ringHaarChar_padic","latex_link":"Haar_char_project.html#MeasureTheory.ringHaarChar_padic","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.ringHaarChar_padic","lean_decl_name":"MeasureTheory.ringHaarChar_padic","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/HaarMeasure/HaarChar/Padic.lean","lean4_code":"/-- The distributive Haar character of the action of `ℚ_[p]ˣ` on `ℚ_[p]` is the usual p-adic norm.\n\nThis means that `volume (x • s) = ‖x‖ * volume s` for all `x : ℚ_[p]` and `s : Set ℚ_[p]`.\nSee `Padic.volume_padic_smul` -/\n@[simp]\nlemma MeasureTheory.ringHaarChar_padic (x : ℚ_[p]ˣ) : ringHaarChar x = ‖(x : ℚ_[p])‖₊ := by\n  -- Write the RHS as the application of a monoid hom `g`.\n  let g : ℚ_[p]ˣ →* ℝ≥0 := {\n    toFun := fun x => ‖(x : ℚ_[p])‖₊\n    map_one' := by simp\n    map_mul' := by simp\n  }\n  change ringHaarChar.toMulHom x = _\n  revert x\n  suffices ringHaarChar (R := ℚ_[p]) = g by simp [this, g]\n  -- By density of `ℤ_[p]⁰` inside `ℚ_[p]ˣ`, it's enough to check that `ringHaarChar ℚ_[p]` and\n  -- `g` agree on `ℤ_[p]⁰`.\n  refine MonoidHom.eq_of_eqOn_dense (PadicInt.closure_nonZeroDivisors_padicInt (p := p)) ?_\n  -- But this is what we proved in `ringHaarChar_padic_padicInt`.\n  simp only [eqOn_range, g]\n  ext x\n  simp only [MonoidHom.coe_coe, Function.comp_apply, MonoidHom.coe_mk,\n    OneHom.coe_mk, Units.val_mk0, coe_nnnorm, PadicInt.padic_norm_e_of_padicInt,\n    ringHaarChar_padic_padicInt]","line":60,"end_line":83,"proof_status":"formalized"}
+{"id":"MeasureTheory.addEquivAddHaarChar_smul_preimage","label":"MeasureTheory.addEquivAddHaarChar_smul_preimage","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":["MeasureTheory.addEquivAddHaarChar_smul_map"],"nlp_text":"If \\(X\\) is a Borel set then \\(\\mu (X)=d_A(\\phi )\\mu (\\phi ^{-1}X)\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.addEquivAddHaarChar_smul_preimage","latex_link":"Haar_char_project.html#MeasureTheory.addEquivAddHaarChar_smul_preimage","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.addEquivAddHaarChar_smul_preimage","lean_decl_name":"MeasureTheory.addEquivAddHaarChar_smul_preimage","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/leanprover-community/mathlib4/98379c78c5dd2e3614cd24ab7ad22f5713e0eb63/Mathlib/MeasureTheory/Measure/Haar/MulEquivHaarChar.lean","lean4_code":"@[to_additive addEquivAddHaarChar_smul_preimage]","line":103,"end_line":103,"proof_status":"formalized"}
+{"id":"MeasureTheory.addEquivAddHaarChar_trans","label":"MeasureTheory.addEquivAddHaarChar_trans","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":["MeasureTheory.addEquivAddHaarChar_smul_preimage"],"nlp_text":"\\(d_A(\\phi \\circ \\psi )=d_A(\\phi )d_A(\\psi ).\\)","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.addEquivAddHaarChar_trans","latex_link":"Haar_char_project.html#MeasureTheory.addEquivAddHaarChar_trans","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.addEquivAddHaarChar_trans","lean_decl_name":"MeasureTheory.addEquivAddHaarChar_trans","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/leanprover-community/mathlib4/98379c78c5dd2e3614cd24ab7ad22f5713e0eb63/Mathlib/MeasureTheory/Measure/Haar/MulEquivHaarChar.lean","lean4_code":"@[to_additive]","line":116,"end_line":116,"proof_status":"formalized"}
+{"id":"MeasureTheory.addEquivAddHaarChar_smul_integral_map","label":"MeasureTheory.addEquivAddHaarChar_smul_integral_map","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":["MeasureTheory.addEquivAddHaarChar_smul_map"],"nlp_text":"If \\(f:A\\to \\mathbb {R}\\) is a Borel measurable function then \\(d_A(\\phi )\\int f(x)d\\phi _*\\mu (x)=\\int f(x)d\\mu (x)\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.addEquivAddHaarChar_smul_integral_map","latex_link":"Haar_char_project.html#MeasureTheory.addEquivAddHaarChar_smul_integral_map","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.addEquivAddHaarChar_smul_integral_map","lean_decl_name":"MeasureTheory.addEquivAddHaarChar_smul_integral_map","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/leanprover-community/mathlib4/98379c78c5dd2e3614cd24ab7ad22f5713e0eb63/Mathlib/MeasureTheory/Measure/Haar/MulEquivHaarChar.lean","lean4_code":"@[to_additive addEquivAddHaarChar_smul_integral_map]","line":82,"end_line":82,"proof_status":"formalized"}
+{"id":"MeasureTheory.addEquivAddHaarChar_comap","label":"MeasureTheory.addEquivAddHaarChar_comap","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":["MeasureTheory.addEquivAddHaarChar_smul_map"],"nlp_text":"If \\(\\mu \\) is any regular Haar measure on \\(A\\) then \\(d_A(\\phi )\\mu = \\phi ^*\\mu .\\)","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.addEquivAddHaarChar_smul_eq_comap","latex_link":"Haar_char_project.html#MeasureTheory.addEquivAddHaarChar_comap","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.addEquivAddHaarChar_smul_eq_comap","lean_decl_name":"MeasureTheory.addEquivAddHaarChar_smul_eq_comap","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/leanprover-community/mathlib4/98379c78c5dd2e3614cd24ab7ad22f5713e0eb63/Mathlib/MeasureTheory/Measure/Haar/MulEquivHaarChar.lean","lean4_code":"@[to_additive addEquivAddHaarChar_smul_eq_comap]","line":70,"end_line":70,"proof_status":"formalized"}
+{"id":"MeasureTheory.addEquivAddHaarChar_smul_integral_comap","label":"MeasureTheory.addEquivAddHaarChar_smul_integral_comap","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":["MeasureTheory.addEquivAddHaarChar_comap"],"nlp_text":"If \\(f:A\\to \\mathbb {R}\\) is a Borel measurable function then \\(d_A(\\phi )\\int f(x)d\\mu (x)=\\int f(x)d\\phi ^*\\mu (x)\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.integral_comap_eq_addEquivAddHaarChar_smul","latex_link":"Haar_char_project.html#MeasureTheory.addEquivAddHaarChar_smul_integral_comap","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.integral_comap_eq_addEquivAddHaarChar_smul","lean_decl_name":"MeasureTheory.integral_comap_eq_addEquivAddHaarChar_smul","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/leanprover-community/mathlib4/98379c78c5dd2e3614cd24ab7ad22f5713e0eb63/Mathlib/MeasureTheory/Measure/Haar/MulEquivHaarChar.lean","lean4_code":"@[to_additive integral_comap_eq_addEquivAddHaarChar_smul]","line":89,"end_line":89,"proof_status":"formalized"}
+{"id":"MeasureTheory.addEquivAddHaarChar_eq_addEquivAddHaarChar_of_isOpenEmbedding","label":"MeasureTheory.addEquivAddHaarChar_eq_addEquivAddHaarChar_of_isOpenEmbedding","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":["MeasureTheory.addEquivAddHaarChar_smul_integral_comap","Topology.IsOpenEmbedding.isHaarMeasure_comap","Topology.IsOpenEmbedding.regular_comap"],"nlp_text":"If \\(f:A\\to B\\) is a group homomorphism and open embedding between locally compact topological additive groups and if \\(\\alpha :A\\to A\\) and \\(\\beta :B\\to B\\) are additive homeomorphisms such that the square commutes (i.e., \\(f\\circ \\alpha =\\beta \\circ f\\)) then \\(d_A(\\alpha )=d_B(\\beta )\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.addEquivAddHaarChar_eq_addEquivAddHaarChar_of_isOpenEmbedding","latex_link":"Haar_char_project.html#MeasureTheory.addEquivAddHaarChar_eq_addEquivAddHaarChar_of_isOpenEmbedding","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.addEquivAddHaarChar_eq_addEquivAddHaarChar_of_isOpenEmbedding","lean_decl_name":"MeasureTheory.addEquivAddHaarChar_eq_addEquivAddHaarChar_of_isOpenEmbedding","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/HaarMeasure/HaarChar/AddEquiv.lean","lean4_code":"@[to_additive]","line":88,"end_line":88,"proof_status":"formalized"}
+{"id":"MeasureTheory.addEquivAddHaarChar_piCongrRight","label":"MeasureTheory.addEquivAddHaarChar_piCongrRight","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":["MeasureTheory.addEquivAddHaarChar_prodCongr"],"nlp_text":"If \\(A_i\\) are a finite collection of locally compact topological abelian groups, with \\(\\phi _i:A_i\\to A_i\\) additive homeomorphisms, then \\(d_{\\prod _i A_i}(\\prod _i\\phi _i)=\\prod _i d_{A_i}(\\phi _i)\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.addEquivAddHaarChar_piCongrRight","latex_link":"Haar_char_project.html#MeasureTheory.addEquivAddHaarChar_piCongrRight","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.addEquivAddHaarChar_piCongrRight","lean_decl_name":"MeasureTheory.addEquivAddHaarChar_piCongrRight","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/HaarMeasure/HaarChar/AddEquiv.lean","lean4_code":"@[to_additive]","line":287,"end_line":287,"proof_status":"formalized"}
+{"id":"MeasureTheory.ringHaarChar_pi","label":"MeasureTheory.ringHaarChar_pi","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":["MeasureTheory.addEquivAddHaarChar_piCongrRight"],"nlp_text":"If \\(R_i\\) are a finite collection of locally compact topological rings, and \\(u_i\\in R_i^\\times \\) then \\(\\delta _{\\prod _i R_i}((u_i)_i)=\\prod _i\\delta _{R_i}(u_i)\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.ringHaarChar_pi","latex_link":"Haar_char_project.html#MeasureTheory.ringHaarChar_pi","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.ringHaarChar_pi","lean_decl_name":"MeasureTheory.ringHaarChar_pi","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch10","source_path":"FLT/HaarMeasure/HaarChar/Ring.lean","lean4_code":"lemma ringHaarChar_pi [Fintype ι] [∀ i, SecondCountableTopology (A i)] (u : Π i, (A i)ˣ) :\n    ringHaarChar (MulEquiv.piUnits.symm u) = ∏ i, ringHaarChar (u i) :=\n  addEquivAddHaarChar_piCongrRight (fun i ↦ ContinuousAddEquiv.mulLeft (u i))\n","line":177,"end_line":179,"proof_status":"formalized"}
+{"id":"IsModuleTopology.continuous_bilinear_of_finite_left","label":"IsModuleTopology.continuous_bilinear_of_finite_left","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":[],"nlp_text":"Say \\(R\\) and \\(S\\) are topological rings, and \\(S\\) is an \\(R\\)-algebra, finite as an \\(R\\)-module. Assume that the topology on \\(S\\) is the \\(R\\)-module topology. Now say \\(M\\) is an \\(S\\)-module, and give it the induced \\(R\\)-module structure. Then the \\(R\\)-module topology and \\(S\\)-module topology on&#160;\\(M\\) coincide.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsModuleTopology.continuous_bilinear_of_finite_left","latex_link":"Haar_char_project.html#IsModuleTopology.continuous_bilinear_of_finite_left","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsModuleTopology.continuous_bilinear_of_finite_left","lean_decl_name":"IsModuleTopology.continuous_bilinear_of_finite_left","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Topology/Algebra/Module/ModuleTopology.lean","lean4_code":"theorem continuous_bilinear_of_finite_left [Module.Finite R A]\n    (bil : A →ₗ[R] B →ₗ[R] C) : Continuous (fun ab ↦ bil ab.1 ab.2 : (A × B → C)) := by\n  -- `A` is finite and hence admits a surjection from `Rⁿ` for some finite `n`.\n  obtain ⟨m, f, hf⟩ := Module.Finite.exists_fin' R A\n  -- The induced linear map `φ : Rⁿ × B → A × B` is surjective\n  let bil' : (Fin m → R) →ₗ[R] B →ₗ[R] C := bil.comp f\n  let φ := f.prodMap (LinearMap.id : B →ₗ[R] B)\n  have hφ : Function.Surjective φ := Function.Surjective.prodMap hf fun b ↦ ⟨b, rfl⟩\n  -- ... and thus a quotient map, so it suffices to prove that the composite `Rⁿ × B → C` is\n  -- continuous.\n  rw [Topology.IsQuotientMap.continuous_iff (isQuotientMap_of_surjective hφ)]\n  -- But this follows from an earlier result.\n  exact continuous_bilinear_of_pi_fintype (Fin m) bil'\n\n/--\nIf `A`, `B` and `C` have the module topology, and if furthermore `B` is a finite `R`-module,\nthen any bilinear map `A × B → C` is automatically continuous\n-/\n","line":556,"end_line":573,"proof_status":"formalized"}
+{"id":"NumberField.AdeleRing.ModuleBaseChangeContinuousAddEquiv","label":"NumberField.AdeleRing.ModuleBaseChangeContinuousAddEquiv","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":["IsModuleTopology.continuous_bilinear_of_finite_left"],"nlp_text":"If \\(K\\) is a number field and \\(V\\) is an \\(K\\)-module, then the natural isomorphism \\(V\\otimes _K\\mathbb {A}_K=V\\otimes _{\\mathbb {Q}}\\mathbb {A}_{\\mathbb {Q}}\\) induced by the natural isomorphism \\(\\mathbb {A}_K=K\\otimes _K\\mathbb {A}_{\\mathbb {Q}}\\) is a homeomorphism if the left hand side has the \\(\\mathbb {A}_K\\)-module topology and the right hand side has the \\(\\mathbb {A}_{\\mathbb {Q}}\\)-module topology.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.ModuleBaseChangeContinuousAddEquiv","latex_link":"Haar_char_project.html#NumberField.AdeleRing.ModuleBaseChangeContinuousAddEquiv","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.ModuleBaseChangeContinuousAddEquiv","lean_decl_name":"NumberField.AdeleRing.ModuleBaseChangeContinuousAddEquiv","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch10","source_path":"FLT/NumberField/AdeleRing.lean","lean4_code":"/-- 𝔸_K ⊗[K] V = 𝔸_L ⊗[L] V as topological additive groups\nfor V an L-module and K ⊆ L number fields. -/\nnoncomputable def ModuleBaseChangeContinuousAddEquiv\n    (V : Type*) [AddCommGroup V] [Module L V] [Module K V]\n    [IsScalarTower K L V] [FiniteDimensional L V] [FiniteDimensional K V]\n    [Module (AdeleRing (𝓞 K) K) (V ⊗[L] AdeleRing (𝓞 L) L)]\n    [IsScalarTower (AdeleRing (𝓞 K) K) (AdeleRing (𝓞 L) L) (V ⊗[L] AdeleRing (𝓞 L) L)] :\n    V ⊗[K] (𝔸 K) ≃ₜ+ (V ⊗[L] (𝔸 L)) := by\n  haveI : ContinuousSMul (AdeleRing (𝓞 K) K) (V ⊗[L] AdeleRing (𝓞 L) L) :=\n    IsScalarTower.continuousSMul (AdeleRing (𝓞 L) L)\n  haveI : IsModuleTopology (AdeleRing (𝓞 K) K) (V ⊗[L] AdeleRing (𝓞 L) L) := {\n    eq_moduleTopology' := by\n      obtain ⟨h2⟩ : IsModuleTopology (AdeleRing (𝓞 L) L) (V ⊗[L] AdeleRing (𝓞 L) L) :=\n        inferInstance\n      rwa [moduleTopology.trans (𝔸 K) (𝔸 L) (V ⊗[L] (𝔸 L))] }\n  exact {\n  __ := (NumberField.AdeleRing.ModuleBaseChangeAddEquiv K L V).toAddEquiv\n  continuous_toFun := IsModuleTopology.continuous_of_linearMap\n      (ModuleBaseChangeAddEquiv' K L V : V ⊗[K] (𝔸 K) ≃ₗ[𝔸 K] (V ⊗[L] (𝔸 L))).toLinearMap\n  continuous_invFun := IsModuleTopology.continuous_of_linearMap\n      (ModuleBaseChangeAddEquiv' K L V : V ⊗[K] (𝔸 K) ≃ₗ[𝔸 K] (V ⊗[L] (𝔸 L))).symm.toLinearMap\n  }\n","line":246,"end_line":267,"proof_status":"formalized"}
+{"id":"MeasureTheory.addEquivAddHaarChar_prodCongr","label":"MeasureTheory.addEquivAddHaarChar_prodCongr","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":[],"nlp_text":"If \\((A,+)\\) and \\((B,+)\\) are locally compact topological abelian groups, and if \\(\\phi :A\\to A\\) and \\(\\psi :B\\to B\\) are additive homeomorphisms, then \\(\\phi \\times \\psi :A\\times B\\to A\\times B\\) is an additive homeomorphism (this is obvious), and \\(d_{A\\times B}(\\phi \\times \\psi )=d_A(\\phi )d_B(\\psi )\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.addEquivAddHaarChar_prodCongr","latex_link":"Haar_char_project.html#MeasureTheory.addEquivAddHaarChar_prodCongr","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.addEquivAddHaarChar_prodCongr","lean_decl_name":"MeasureTheory.addEquivAddHaarChar_prodCongr","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/HaarMeasure/HaarChar/AddEquiv.lean","lean4_code":"@[to_additive MeasureTheory.addEquivAddHaarChar_prodCongr]","line":158,"end_line":158,"proof_status":"formalized"}
+{"id":"MeasureTheory.ringHaarChar_prod","label":"MeasureTheory.ringHaarChar_prod","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":["MeasureTheory.addEquivAddHaarChar_prodCongr"],"nlp_text":"If \\(R\\) and \\(S\\) are locally compact topological rings, then \\(\\delta _{R\\times S}(r,s)=\\delta _R(r)\\times \\delta _S(s)\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.ringHaarChar_prod","latex_link":"Haar_char_project.html#MeasureTheory.ringHaarChar_prod","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.ringHaarChar_prod","lean_decl_name":"MeasureTheory.ringHaarChar_prod","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch10","source_path":"FLT/HaarMeasure/HaarChar/Ring.lean","lean4_code":"lemma ringHaarChar_prod (u : Rˣ) (v : Sˣ) [SecondCountableTopologyEither R S] :\n    ringHaarChar (MulEquiv.prodUnits.symm (u, v)) = ringHaarChar u * ringHaarChar v :=\n  addEquivAddHaarChar_prodCongr (ContinuousAddEquiv.mulLeft u) (ContinuousAddEquiv.mulLeft v)\n","line":160,"end_line":162,"proof_status":"formalized"}
+{"id":"IsSimpleRing.ringHaarChar_eq_addEquivAddHaarChar_mulRight","label":"IsSimpleRing.ringHaarChar_eq_addEquivAddHaarChar_mulRight","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":["IsSimpleRing.mulLeft_det_eq_mulRight_det"],"nlp_text":"If \\(B\\) is a central simple algebra over a locally compact field \\(F\\), and if \\(u\\in B^\\times \\), then \\(d_B(r_u)=\\delta _B(u)\\) (recall that the latter is defined to mean \\(d_B(\\ell _u)\\)).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsSimpleRing.ringHaarChar_eq_addEquivAddHaarChar_mulRight","latex_link":"Haar_char_project.html#IsSimpleRing.ringHaarChar_eq_addEquivAddHaarChar_mulRight","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsSimpleRing.ringHaarChar_eq_addEquivAddHaarChar_mulRight","lean_decl_name":"IsSimpleRing.ringHaarChar_eq_addEquivAddHaarChar_mulRight","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/HaarMeasure/HaarChar/FiniteDimensional.lean","lean4_code":"lemma _root_.IsSimpleRing.ringHaarChar_eq_addEquivAddHaarChar_mulRight (u : Dˣ) :\n    ringHaarChar u = addEquivAddHaarChar (ContinuousAddEquiv.mulRight u) := by\n  rw [algebra_ringHaarChar_eq_ringHaarChar_det F u]\n  rw [IsSimpleRing.mulLeft_det_eq_mulRight_det']\n  symm\n  exact addEquivAddHaarChar_eq_ringHaarChar_det (ContinuousLinearEquiv.mulRight F u)","line":336,"end_line":341,"proof_status":"formalized"}
+{"id":"Continuous.restrictedProduct_congrRight","label":"Continuous.restrictedProduct_congrRight","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":[],"nlp_text":"If the \\(A_i\\) and \\(B_i\\) are topological spaces and the \\(\\phi _i\\) are continuous functions, then the restricted product \\(\\phi = \\prod '_i\\phi _i\\) is a continuous function.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Continuous.restrictedProduct_congrRight","latex_link":"Haar_char_project.html#Continuous.restrictedProduct_congrRight","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Continuous.restrictedProduct_congrRight","lean_decl_name":"Continuous.restrictedProduct_congrRight","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean","lean4_code":"@[fun_prop]\ntheorem Continuous.restrictedProduct_congrRight {φ : (i : ι) → G i → H i}\n    (hφ : ∀ᶠ i in ℱ, Set.MapsTo (φ i) (C i) (D i))\n    (hφcont : ∀ i, Continuous (φ i)) :\n    Continuous (map φ hφ) :=\n  mapAlong_continuous G H id Filter.tendsto_id φ hφ hφcont","line":18,"end_line":23,"proof_status":"formalized"}
+{"id":"MeasureTheory.algebra_ringHaarChar_eq_ringHaarChar_det","label":"MeasureTheory.algebra_ringHaarChar_eq_ringHaarChar_det","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":[],"nlp_text":"If \\(u\\in R^\\times \\) then \\(\\delta _R(u)=\\delta _F(\\det (\\ell _u))\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.algebra_ringHaarChar_eq_ringHaarChar_det","latex_link":"Haar_char_project.html#MeasureTheory.algebra_ringHaarChar_eq_ringHaarChar_det","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.algebra_ringHaarChar_eq_ringHaarChar_det","lean_decl_name":"MeasureTheory.algebra_ringHaarChar_eq_ringHaarChar_det","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch10","source_path":"FLT/HaarMeasure/HaarChar/FiniteDimensional.lean","lean4_code":"lemma algebra_ringHaarChar_eq_ringHaarChar_det (u : Aˣ) :\n    ringHaarChar u = ringHaarChar (LinearEquiv.mulLeft F u).det :=\n  addEquivAddHaarChar_eq_ringHaarChar_det (ContinuousLinearEquiv.mulLeft F u)\n","line":316,"end_line":318,"proof_status":"formalized"}
+{"id":"Topology.IsOpenEmbedding.isHaarMeasure_comap","label":"Topology.IsOpenEmbedding.isHaarMeasure_comap","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":[],"nlp_text":"Let \\(A\\) and \\(B\\) be locally compact topological groups and let \\(f:A\\to B\\) be both a group homomorphism and open embedding. The pullback along \\(f\\) of a Haar measure on \\(B\\) is a Haar measure on \\(A\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Topology.IsOpenEmbedding.isHaarMeasure_comap","latex_link":"Haar_char_project.html#Topology.IsOpenEmbedding.isHaarMeasure_comap","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Topology.IsOpenEmbedding.isHaarMeasure_comap","lean_decl_name":"Topology.IsOpenEmbedding.isHaarMeasure_comap","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/Mathlib/MeasureTheory/Group/Measure.lean","lean4_code":"@[to_additive]\nlemma Topology.IsOpenEmbedding.isHaarMeasure_comap {G H : Type*}\n    [Group G] [TopologicalSpace G] [MeasurableSpace G] [MeasurableMul G] [BorelSpace G]\n    [Group H] [TopologicalSpace H] [MeasurableSpace H] [MeasurableMul H] [BorelSpace H]\n    {φ : G →* H} (hφ : IsOpenEmbedding φ) (μ : Measure H) [IsHaarMeasure μ] :\n    IsHaarMeasure (comap φ μ) where\n  map_mul_left_eq_self := (hφ.measurableEmbedding.isMulLeftInvariant_comap μ).map_mul_left_eq_self\n  lt_top_of_isCompact := (hφ.isFiniteMeasureOnCompacts_comap μ).lt_top_of_isCompact\n  open_pos := (IsOpenPosMeasure.comap μ hφ).open_pos","line":8,"end_line":16,"proof_status":"formalized"}
+{"id":"IsSimpleRing.mulLeft_det_eq_mulRight_det","label":"IsSimpleRing.mulLeft_det_eq_mulRight_det","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":[],"nlp_text":"Say \\(B\\) is a finite-dimensional central simple algebra over a field&#160;\\(k\\), and \\(u\\in B^\\times \\). Let \\(\\ell _u:B\\to B\\) be the \\(k\\)-linear mapping \\(x\\) to \\(ux\\) and let \\(r_u:B\\to B\\) be the \\(k\\)-linear map sending \\(x\\) to \\(xu\\). Then \\(\\det (\\ell _u)=\\det (r_u)\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsSimpleRing.mulLeft_det_eq_mulRight_det","latex_link":"Haar_char_project.html#IsSimpleRing.mulLeft_det_eq_mulRight_det","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsSimpleRing.mulLeft_det_eq_mulRight_det","lean_decl_name":"IsSimpleRing.mulLeft_det_eq_mulRight_det","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/Mathlib/LinearAlgebra/Determinant.lean","lean4_code":"lemma IsSimpleRing.mulLeft_det_eq_mulRight_det (d : D) :\n    (LinearMap.mulLeft k d).det = (LinearMap.mulRight k d).det := by\n  let K' := AlgebraicClosure k\n  obtain ⟨n, hn, ⟨e⟩⟩ := IsSimpleRing.exists_algEquiv_matrix_of_isAlgClosed K' (K' ⊗[k] D)\n  have h1 : (LinearMap.mulLeft k d).baseChange K' = LinearMap.mulLeft K' ((1 : K') ⊗ₜ[k] d) := by\n    ext; simp\n  have h2 : (LinearMap.mulRight k d).baseChange K' = LinearMap.mulRight K' ((1 : K') ⊗ₜ[k] d) := by\n    ext; simp\n  apply FaithfulSMul.algebraMap_injective k K'\n  rw [LinearMap.det_baseChange (LinearMap.mulLeft k d) |>.symm, LinearMap.det_baseChange\n    (LinearMap.mulRight k d) |>.symm, h1, h2]\n  have h5 : LinearMap.det (LinearMap.mulLeft K' ((1 : K') ⊗ₜ[k] d)) =\n    LinearMap.det (LinearMap.mulLeft K' (e ((1 : K') ⊗ₜ d))) := by\n    rw [← LinearMap.det_conj (LinearMap.mulLeft _ _) e.toLinearEquiv, mulLeft_conj]\n    rfl\n  have h6: LinearMap.det (LinearMap.mulRight K' ((1 : K') ⊗ₜ[k] d)) =\n    LinearMap.det (LinearMap.mulRight K' (e ((1 : K') ⊗ₜ d))) := by\n    rw [← LinearMap.det_conj (LinearMap.mulRight _ _) e.toLinearEquiv, mulRight_conj]\n    rfl\n  rw [h5, h6, ← LinearMap.det_conj (LinearMap.mulRight K' (e (1 ⊗ₜ[k] d))) <|\n    (Matrix.ofLinearEquiv K').symm, LinearEquiv.symm_symm, mulRight_conj_ofLinear,\n    LinearMap.det_pi, ← LinearMap.det_conj (LinearMap.mulLeft K' (e (1 ⊗ₜ[k] d))) <|\n    (((Matrix.ofLinearEquiv K') ≪≫ₗ Matrix.transposeLinearEquiv (Fin n) (Fin n) K' K')).symm,\n    LinearEquiv.symm_symm, mulLeft_conj_ofLinear, LinearMap.det_pi]\n  simp [LinearMap.det_toLin', Finset.prod_const, Finset.card_univ, Fintype.card_fin]","line":61,"end_line":85,"proof_status":"formalized"}
+{"id":"Topology.IsOpenEmbedding.regular_comap","label":"Topology.IsOpenEmbedding.regular_comap","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":[],"nlp_text":"The pullback of a regular Borel measure along an open embedding is a regular Borel measure.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Topology.IsOpenEmbedding.regular_comap","latex_link":"Haar_char_project.html#Topology.IsOpenEmbedding.regular_comap","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Topology.IsOpenEmbedding.regular_comap","lean_decl_name":"Topology.IsOpenEmbedding.regular_comap","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch10","source_path":"FLT/Mathlib/MeasureTheory/Measure/Regular.lean","lean4_code":"lemma regular_comap\n    (φ : X → Y) (hφ : IsOpenEmbedding φ) (μ : Measure Y) [Regular μ] :\n    Regular (comap φ μ) where\n  lt_top_of_isCompact := (hφ.isFiniteMeasureOnCompacts_comap μ).lt_top_of_isCompact\n  outerRegular := (hφ.outerRegular_comap μ).outerRegular\n  innerRegular := hφ.innerRegularWRT_comap Regular.innerRegular\n","line":41,"end_line":46,"proof_status":"formalized"}
+{"id":"MeasureTheory.mulEquivHaarChar_eq_one_of_compactSpace","label":"MeasureTheory.mulEquivHaarChar_eq_one_of_compactSpace","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_10.html","dependencies":[],"nlp_text":"Say \\(A\\) is a compact topological additive group and \\(\\phi :A\\to A\\) is an additive isomorphism. Then \\(d_A(\\phi )=1.\\)","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.mulEquivHaarChar_eq_one_of_compactSpace","latex_link":"Haar_char_project.html#MeasureTheory.mulEquivHaarChar_eq_one_of_compactSpace","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MeasureTheory.mulEquivHaarChar_eq_one_of_compactSpace","lean_decl_name":"MeasureTheory.mulEquivHaarChar_eq_one_of_compactSpace","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch10","source_path":"FLT/HaarMeasure/HaarChar/AddEquiv.lean","lean4_code":"@[to_additive addEquivAddHaarChar_eq_one_of_compactSpace]\nlemma mulEquivHaarChar_eq_one_of_compactSpace [CompactSpace G] (φ : G ≃ₜ* G) :\n    mulEquivHaarChar φ = 1 := by\n  set μ := haarMeasure (⟨⟨univ, isCompact_univ⟩, by simp⟩ : PositiveCompacts G)\n  have hμ : μ univ = 1 := haarMeasure_self\n  rw [mulEquivHaarChar_eq μ]\n  suffices (μ.haarScalarFactor (map φ μ) : ℝ≥0∞) = 1 by exact_mod_cast this\n  calc\n    _ = μ.haarScalarFactor (map φ μ) • (1 : ℝ≥0∞) := by rw [ENNReal.smul_def, smul_eq_mul, mul_one]\n    _ = μ.haarScalarFactor (map φ μ) • (map φ μ univ) := by\n          rw [map_apply (map_continuous φ).measurable .univ, Set.preimage_univ, hμ]\n    _ = μ univ := by\n          conv_rhs => rw [isMulInvariant_eq_smul_of_compactSpace μ (map φ μ), Measure.smul_apply]\n    _ = 1 := hμ\n","line":72,"end_line":85,"proof_status":"formalized"}

crawled_repos/FLT_ch11.jsonl

@@ -0,0 +1,16 @@
+{"id":"NumberField.AdeleRing.DivisionAlgebra.Aux.C","label":"NumberField.AdeleRing.DivisionAlgebra.Aux.C","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_11.html","dependencies":["NumberField.AdeleRing.DivisionAlgebra.Aux.T"],"nlp_text":"Define \\(C:= (T^{-1}.X) \\times X\\subset D_{\\mathbb {A}}\\times D_{\\mathbb {A}}\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.DivisionAlgebra.Aux.C","latex_link":"Fujisaki_project.html#NumberField.AdeleRing.DivisionAlgebra.Aux.C","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.DivisionAlgebra.Aux.C","lean_decl_name":"NumberField.AdeleRing.DivisionAlgebra.Aux.C","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch11","source_path":"FLT/DivisionAlgebra/Finiteness.lean","lean4_code":"/-- An auxiliary set C used in the proof of Fukisaki's lemma. Defined as T⁻¹X × X. -/\ndef C : Set (D_𝔸 × D_𝔸) := ((((↑) : D_𝔸ˣ → D_𝔸) '' (T K D)⁻¹) * X K D) ×ˢ X K D\n","line":868,"end_line":869,"proof_status":"formalized"}
+{"id":"NumberField.AdeleRing.DivisionAlgebra.Aux.C_compact","label":"NumberField.AdeleRing.DivisionAlgebra.Aux.C_compact","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_11.html","dependencies":["NumberField.AdeleRing.DivisionAlgebra.Aux.C","NumberField.AdeleRing.DivisionAlgebra.Aux.T_finite","NumberField.AdeleRing.DivisionAlgebra.Aux.X_compact"],"nlp_text":"\\(C\\) is compact.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.DivisionAlgebra.Aux.C_compact","latex_link":"Fujisaki_project.html#NumberField.AdeleRing.DivisionAlgebra.Aux.C_compact","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.DivisionAlgebra.Aux.C_compact","lean_decl_name":"NumberField.AdeleRing.DivisionAlgebra.Aux.C_compact","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch11","source_path":"FLT/DivisionAlgebra/Finiteness.lean","lean4_code":"lemma C_compact : IsCompact (C K D) := by\n  refine IsCompact.prod ?_ (X_compact K D)\n  simpa only [Set.image_prod, Set.image2_mul] using\n    (IsCompact.image_of_continuousOn (IsCompact.prod (IsCompact.image_of_continuousOn\n    (IsCompact.inv (Set.Finite.isCompact (T_finite K D))) (Continuous.comp_continuousOn'\n    (Units.continuous_val) (continuousOn_id' (T K D)⁻¹)))\n    (X_compact K D)) ((continuous_fst.mul continuous_snd).continuousOn))\n","line":871,"end_line":877,"proof_status":"formalized"}
+{"id":"NumberField.AdeleRing.DivisionAlgebra.Aux.antidiag_mem_C","label":"NumberField.AdeleRing.DivisionAlgebra.Aux.antidiag_mem_C","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_11.html","dependencies":["NumberField.AdeleRing.DivisionAlgebra.Aux.C","NumberField.AdeleRing.DivisionAlgebra.Aux.X_meets_kernel'","NumberField.AdeleRing.DivisionAlgebra.Aux.X_meets_kernel"],"nlp_text":"For every \\(\\beta \\in D_{\\mathbb {A}}^{(1)}\\), there exists \\(b\\in D^\\times \\) and \\(\\nu \\in D_{\\mathbb {A}}^{(1)}\\) such that \\(\\beta =b\\nu \\) and \\((\\nu ,\\nu ^{-1})\\in C.\\)","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.DivisionAlgebra.Aux.antidiag_mem_C","latex_link":"Fujisaki_project.html#NumberField.AdeleRing.DivisionAlgebra.Aux.antidiag_mem_C","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.DivisionAlgebra.Aux.antidiag_mem_C","lean_decl_name":"NumberField.AdeleRing.DivisionAlgebra.Aux.antidiag_mem_C","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch11","source_path":"FLT/DivisionAlgebra/Finiteness.lean","lean4_code":"lemma antidiag_mem_C [Algebra.IsCentral K D] {β : D_𝔸ˣ} (hβ : β ∈ ringHaarChar_ker D_𝔸) :\n    ∃ b ∈ Set.range (incl K D : Dˣ → D_𝔸ˣ),\n    ∃ ν ∈ ringHaarChar_ker D_𝔸,\n    β = b * ν ∧ ((ν : D_𝔸), ((ν⁻¹ : D_𝔸ˣ) : D_𝔸)) ∈ C K D := by\n  obtain ⟨x1, hx1, b1, ⟨b1, rfl⟩, eq1⟩ := X_meets_kernel K D hβ\n  obtain ⟨x2, hx2, b2, ⟨b2, rfl⟩, eq2⟩ := X_meets_kernel' K D hβ\n  obtain ⟨x1, rfl⟩ : IsUnit x1 := ⟨↑β⁻¹ * incl K D b1,\n    ((Units.eq_inv_mul_iff_mul_eq β).mpr eq1).symm⟩\n  obtain ⟨x2, rfl⟩ : IsUnit x2 := ⟨incl K D b2 * β, ((Units.mul_inv_eq_iff_eq_mul β).mp eq2).symm⟩\n  have h : x2 * x1 ∈ T K D := ⟨by simpa only [Y] using Set.mul_mem_mul hx2 hx1,\n    b2 * b1, by norm_cast at eq1 eq2; rw [map_mul, ← eq2, ← eq1]; group⟩\n  refine ⟨incl K D b1, by simp only [Set.mem_range, exists_apply_eq_apply],  x1⁻¹, ?_,\n    eq_mul_inv_of_mul_eq (Units.val_inj.mp eq1), ?_, hx1⟩\n  · rw [(Eq.symm (inv_mul_eq_of_eq_mul (eq_mul_inv_of_mul_eq (Units.val_inj.mp eq1))))]\n    exact (Subgroup.mul_mem_cancel_right (ringHaarChar_ker (D ⊗[K] AdeleRing (𝓞 K) K)) hβ).mpr\n      ((Subgroup.inv_mem_iff (ringHaarChar_ker (D ⊗[K] AdeleRing (𝓞 K) K))).mpr\n      (NumberField.AdeleRing.units_mem_ringHaarCharacter_ker K D b1))\n  · obtain ⟨t, ht, ht1⟩ := exists_eq_right'.mpr h\n    simp_rw [(Eq.symm (inv_mul_eq_of_eq_mul (eq_mul_inv_of_mul_eq ht1)))]\n    exact Set.mem_mul.mpr ⟨↑t⁻¹, Set.mem_image_of_mem Units.val ht, x2, hx2, rfl⟩\n","line":879,"end_line":898,"proof_status":"formalized"}
+{"id":"NumberField.AdeleRing.DivisionAlgebra.compact_quotient","label":"NumberField.AdeleRing.DivisionAlgebra.compact_quotient","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_11.html","dependencies":["NumberField.AdeleRing.DivisionAlgebra.Aux.C_compact","NumberField.AdeleRing.DivisionAlgebra.Aux.antidiag_mem_C"],"nlp_text":"If \\(D\\) is a division algebra then the quotient \\(D^\\times \\backslash D_{\\mathbb {A}}^{(1)}\\) with its quotient topology coming from \\(D_{\\mathbb {A}}^{(1)}\\), is compact.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.DivisionAlgebra.compact_quotient","latex_link":"Fujisaki_project.html#NumberField.AdeleRing.DivisionAlgebra.compact_quotient","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.DivisionAlgebra.compact_quotient","lean_decl_name":"NumberField.AdeleRing.DivisionAlgebra.compact_quotient","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch11","source_path":"FLT/DivisionAlgebra/Finiteness.lean","lean4_code":"/-- Dˣ \\ D_𝔸^{(1)} is compact. -/\nlemma compact_quotient [Algebra.IsCentral K D] :\n    CompactSpace (_root_.Quotient (QuotientGroup.rightRel\n    ((MonoidHom.range (incl K D)).comap (ringHaarChar_ker D_𝔸).subtype))) :=\n  isCompact_univ_iff.mp (by simpa only [toQuot_surjective, Set.image_univ] using\n    (((IsCompact.image (M_compact K D) (toQuot_cont K D)))))\n","line":1044,"end_line":1049,"proof_status":"formalized"}
+{"id":"NumberField.FiniteAdeleRing.DivisionAlgebra.units_cocompact","label":"NumberField.FiniteAdeleRing.DivisionAlgebra.units_cocompact","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_11.html","dependencies":["NumberField.AdeleRing.DivisionAlgebra.compact_quotient"],"nlp_text":"\\(D^\\times \\backslash (D\\otimes _K\\mathbb {A}_K^\\infty )^\\times \\) is compact.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.FiniteAdeleRing.DivisionAlgebra.units_cocompact","latex_link":"Fujisaki_project.html#NumberField.FiniteAdeleRing.DivisionAlgebra.units_cocompact","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.FiniteAdeleRing.DivisionAlgebra.units_cocompact","lean_decl_name":"NumberField.FiniteAdeleRing.DivisionAlgebra.units_cocompact","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/DivisionAlgebra/Finiteness.lean","lean4_code":"/-- Dˣ \\ D_𝔸^fˣ is compact. -/\ntheorem _root_.NumberField.FiniteAdeleRing.DivisionAlgebra.units_cocompact :\n    CompactSpace (_root_.Quotient (QuotientGroup.rightRel (incl₁ K D).range)) := by\n  have := isCompact_univ_iff.mpr (NumberField.AdeleRing.DivisionAlgebra.compact_quotient K D)\n  apply isCompact_univ_iff.mp\n  have := IsCompact.image (this) (incl_D𝔸quot_continuous K D)\n  rw [Set.image_univ_of_surjective (incl_D𝔸quot_surjective K D)] at this\n  exact this","line":1053,"end_line":1060,"proof_status":"formalized"}
+{"id":"NumberField.AdeleRing.DivisionAlgebra.Aux.Y","label":"NumberField.AdeleRing.DivisionAlgebra.Aux.Y","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_11.html","dependencies":["NumberField.AdeleRing.DivisionAlgebra.Aux.X"],"nlp_text":"Define \\(Y:=X.X:=\\{ xy:x,y\\in X\\} \\subseteq D_{\\mathbb {A}}\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.DivisionAlgebra.Aux.Y","latex_link":"Fujisaki_project.html#NumberField.AdeleRing.DivisionAlgebra.Aux.Y","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.DivisionAlgebra.Aux.Y","lean_decl_name":"NumberField.AdeleRing.DivisionAlgebra.Aux.Y","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch11","source_path":"FLT/DivisionAlgebra/Finiteness.lean","lean4_code":"/-- An auxiliary set Y used in the proof of Fukisaki's lemma. Defined as X * X. -/\ndef Y : Set D_𝔸 := X K D * X K D\n","line":809,"end_line":810,"proof_status":"formalized"}
+{"id":"NumberField.AdeleRing.DivisionAlgebra.Aux.T","label":"NumberField.AdeleRing.DivisionAlgebra.Aux.T","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_11.html","dependencies":["NumberField.AdeleRing.DivisionAlgebra.Aux.Y"],"nlp_text":"Let \\(T:=Y\\cap D^\\times \\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.DivisionAlgebra.Aux.T","latex_link":"Fujisaki_project.html#NumberField.AdeleRing.DivisionAlgebra.Aux.T","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.DivisionAlgebra.Aux.T","lean_decl_name":"NumberField.AdeleRing.DivisionAlgebra.Aux.T","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch11","source_path":"FLT/DivisionAlgebra/Finiteness.lean","lean4_code":"/-- We put the Borel measurable space structure on D_𝔸 in this file. -/\ninstance : MeasurableSpace D_𝔸 := borel _\n","line":119,"end_line":120,"proof_status":"formalized"}
+{"id":"NumberField.AdeleRing.DivisionAlgebra.Aux.Y_compact","label":"NumberField.AdeleRing.DivisionAlgebra.Aux.Y_compact","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_11.html","dependencies":["NumberField.AdeleRing.DivisionAlgebra.Aux.Y"],"nlp_text":"\\(Y\\) is a compact subset of \\(D_{\\mathbb {A}}\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.DivisionAlgebra.Aux.Y_compact","latex_link":"Fujisaki_project.html#NumberField.AdeleRing.DivisionAlgebra.Aux.Y_compact","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.DivisionAlgebra.Aux.Y_compact","lean_decl_name":"NumberField.AdeleRing.DivisionAlgebra.Aux.Y_compact","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch11","source_path":"FLT/DivisionAlgebra/Finiteness.lean","lean4_code":"lemma Y_compact : IsCompact (Y K D) := by\n  simpa only [Set.image_prod, Set.image2_mul] using (IsCompact.image_of_continuousOn\n    ((X_compact K D).prod (X_compact K D)) ((continuous_fst.mul continuous_snd).continuousOn))\n","line":816,"end_line":818,"proof_status":"formalized"}
+{"id":"NumberField.AdeleRing.DivisionAlgebra.Aux.T_finite","label":"NumberField.AdeleRing.DivisionAlgebra.Aux.T_finite","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_11.html","dependencies":["NumberField.AdeleRing.DivisionAlgebra.Aux.T","NumberField.AdeleRing.DivisionAlgebra.Aux.Y_compact"],"nlp_text":"\\(T\\) is finite.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.DivisionAlgebra.Aux.T_finite","latex_link":"Fujisaki_project.html#NumberField.AdeleRing.DivisionAlgebra.Aux.T_finite","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.DivisionAlgebra.Aux.T_finite","lean_decl_name":"NumberField.AdeleRing.DivisionAlgebra.Aux.T_finite","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch11","source_path":"FLT/DivisionAlgebra/Finiteness.lean","lean4_code":"lemma T_finite : Set.Finite (T K D) := by\n  have h := IsCompact.finite (T_finite_extracted1 K D)\n    ⟨(inter_Discrete (includeLeft_subgroup K D).carrier (Y K D))⟩\n  have h1 : Units.val '' T K D ⊆ (Y K D) ∩\n      (Set.range (Algebra.TensorProduct.includeLeft : D →ₐ[K] D_𝔸)) := by\n    rintro _ ⟨t, ⟨ht1, d, rfl⟩, rfl⟩\n    exact ⟨ht1, d, rfl⟩\n  exact Set.Finite.of_finite_image (Set.Finite.subset h h1)\n    (Function.Injective.injOn Units.val_injective)\n","line":857,"end_line":865,"proof_status":"formalized"}
+{"id":"NumberField.AdeleRing.DivisionAlgebra.Aux.X_meets_kernel'","label":"NumberField.AdeleRing.DivisionAlgebra.Aux.X_meets_kernel'","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_11.html","dependencies":["NumberField.AdeleRing.DivisionAlgebra.Aux.X"],"nlp_text":"Similarly, if \\(\\beta \\in D_{\\mathbb {A}}^{(1)}\\) then \\(X\\beta ^{-1}\\cap D^\\times \\not=\\emptyset \\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.DivisionAlgebra.Aux.X_meets_kernel'","latex_link":"Fujisaki_project.html#NumberField.AdeleRing.DivisionAlgebra.Aux.X_meets_kernel'","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.DivisionAlgebra.Aux.X_meets_kernel'","lean_decl_name":"NumberField.AdeleRing.DivisionAlgebra.Aux.X_meets_kernel'","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/DivisionAlgebra/Finiteness.lean","lean4_code":"lemma X_meets_kernel' [Algebra.IsCentral K D] {β : D_𝔸ˣ} (hβ : β ∈ ringHaarChar_ker D_𝔸) :\n    ∃ x ∈ X K D, ∃ d ∈ Set.range (incl K D : Dˣ → D_𝔸ˣ), x * β⁻¹ = d := by\n  obtain ⟨e1, he1, e2, he2, noteq, b, hb⟩ := E_noninjective_right K D hβ\n  refine ⟨e1 - e2, by simpa only using (Set.sub_mem_sub he1 he2), ?_⟩\n  obtain ⟨b1, rfl⟩ : IsUnit b := by\n    simp_rw [← mul_sub_right_distrib, Algebra.TensorProduct.includeLeft_apply] at hb\n    have h1 : (e1 - e2) * ↑β⁻¹ ≠ 0 := by\n      simpa only [ne_eq, Units.mul_left_eq_zero] using (sub_ne_zero_of_ne noteq)\n    simp only [isUnit_iff_ne_zero, ne_eq]\n    rintro rfl\n    simp only [← hb, TensorProduct.zero_tmul, ne_eq, not_true_eq_false] at h1\n  exact ⟨incl K D b1, ⟨b1, rfl⟩, by simpa [sub_mul] using hb.symm⟩","line":833,"end_line":844,"proof_status":"formalized"}
+{"id":"NumberField.FiniteAdeleRing.DivisionAlgebra.finiteDoubleCoset","label":"NumberField.FiniteAdeleRing.DivisionAlgebra.finiteDoubleCoset","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_11.html","dependencies":["NumberField.FiniteAdeleRing.DivisionAlgebra.units_cocompact"],"nlp_text":"If \\(U\\) is an open subgroup of \\((D\\otimes _K \\mathbb {A}_K^\\infty )^\\times \\) then the double coset space \\(D^\\times \\backslash (D\\otimes _K \\mathbb {A}_K^\\infty )^\\times /U\\) is finite.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.FiniteAdeleRing.DivisionAlgebra.finiteDoubleCoset","latex_link":"Fujisaki_project.html#NumberField.FiniteAdeleRing.DivisionAlgebra.finiteDoubleCoset","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.FiniteAdeleRing.DivisionAlgebra.finiteDoubleCoset","lean_decl_name":"NumberField.FiniteAdeleRing.DivisionAlgebra.finiteDoubleCoset","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/DivisionAlgebra/Finiteness.lean","lean4_code":"theorem _root_.NumberField.FiniteAdeleRing.DivisionAlgebra.finiteDoubleCoset\n    {U : Subgroup (Dfx K D)} (hU : IsOpen (U : Set (Dfx K D))) :\n    Finite (DoubleCoset.Quotient (Set.range (incl₁ K D)) U) := by\n  have ToFinCover := IsCompact.elim_finite_subcover\n    (ι := (DoubleCoset.Quotient (Set.range (incl₁ K D)) U))\n    (U := fun q ↦ Quot.mk ⇑(QuotientGroup.rightRel (incl₁ K D).range) ''\n    DoubleCoset.doubleCoset (Quotient.out q) (Set.range ⇑(incl₁ K D)) U) (isCompact_univ_iff.mpr\n    (NumberField.FiniteAdeleRing.DivisionAlgebra.units_cocompact K D))\n  have ⟨t, FinCover_descended⟩ := ToFinCover (DoubleCoset.isOpen_doubleCoset_rightrel_mk\n    ((incl₁ K D).range) U hU) (DoubleCoset.union_image_mk_rightRel (incl₁ K D).range U\n    ▸ Set.Subset.rfl)\n  apply (DoubleCoset.iUnion_finset_quotTodoubleCoset ((incl₁ K D).range) U).mp\n  exact ⟨t, DoubleCoset.union_finset_rightrel_cover ((incl₁ K D).range) U t FinCover_descended⟩","line":1069,"end_line":1081,"proof_status":"formalized"}
+{"id":"NumberField.AdeleRing.DivisionAlgebra.Aux.E","label":"NumberField.AdeleRing.DivisionAlgebra.Aux.E","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_11.html","dependencies":["NumberField.AdeleRing.DivisionAlgebra.Aux.existsE"],"nlp_text":"We let \\(E\\) denote any compact set satisfying the hypothesis of the previous lemma.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.DivisionAlgebra.Aux.E","latex_link":"Fujisaki_project.html#NumberField.AdeleRing.DivisionAlgebra.Aux.E","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.DivisionAlgebra.Aux.E","lean_decl_name":"NumberField.AdeleRing.DivisionAlgebra.Aux.E","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch11","source_path":"FLT/DivisionAlgebra/Finiteness.lean","lean4_code":"lemma existsE : ∃ E : Set (D_𝔸), IsCompact E ∧\n    ∀ φ : D_𝔸 ≃ₜ+ D_𝔸, addEquivAddHaarChar φ = 1 → ∃ e₁ ∈ E, ∃ e₂ ∈ E,\n    e₁ ≠ e₂ ∧ φ e₁ - φ e₂ ∈ Set.range (Algebra.TensorProduct.includeLeft : D →ₐ[K] D_𝔸) := by\n  obtain ⟨B, hB⟩ := not_injective_of_large_measure K D\n  obtain ⟨r, hr⟩ := E_family_unbounded K D B\n  let E := Efamily K D r\n  obtain ⟨U, hU, hKU, hU'⟩ := exists_isOpen_superset_and_isCompact_closure (E_family_compact K D r)\n  use closure U, hU'\n  intro φ hφ\n  specialize hB (φ.symm ⁻¹' U) (hU.preimage φ.symm.continuous)\n  replace hr : B < Measure.addHaar U := hr.trans_le (measure_mono hKU)\n  replace hφ : addEquivAddHaarChar φ.symm = 1 := by\n    simpa [hφ] using (addEquivAddHaarChar_trans (φ := φ) (ψ := φ.symm)).symm\n  have foo : Measure.addHaar U = Measure.addHaar (⇑φ.symm ⁻¹' U) := by\n    rw [← one_smul NNReal (Measure.addHaar (φ.symm ⁻¹' U)), ← hφ,\n      addEquivAddHaarChar_smul_preimage]\n  rw [foo] at hr\n  specialize hB hr\n  simp only [Set.InjOn, not_forall] at hB\n  obtain ⟨x, hx, y, hy, h, hne⟩ := hB\n  rw [QuotientAddGroup.eq_iff_sub_mem] at h\n  exact ⟨φ.symm x, subset_closure hx, φ.symm y, subset_closure hy, by simpa, by simpa⟩\n","line":762,"end_line":783,"proof_status":"formalized"}
+{"id":"NumberField.AdeleRing.DivisionAlgebra.Aux.X","label":"NumberField.AdeleRing.DivisionAlgebra.Aux.X","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_11.html","dependencies":["NumberField.AdeleRing.DivisionAlgebra.Aux.E"],"nlp_text":"Define \\(X:=E-E:=\\{ e-f:e,f\\in E\\} \\subseteq D_{\\mathbb {A}}\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.DivisionAlgebra.Aux.X","latex_link":"Fujisaki_project.html#NumberField.AdeleRing.DivisionAlgebra.Aux.X","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.DivisionAlgebra.Aux.X","lean_decl_name":"NumberField.AdeleRing.DivisionAlgebra.Aux.X","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch11","source_path":"FLT/DivisionAlgebra/Finiteness.lean","lean4_code":"/-- An auxiliary set X used in the proof of Fukisaki's lemma. Defined as E - E. -/\ndef X : Set D_𝔸 := E K D - E K D\n","line":805,"end_line":806,"proof_status":"formalized"}
+{"id":"NumberField.AdeleRing.DivisionAlgebra.Aux.X_meets_kernel","label":"NumberField.AdeleRing.DivisionAlgebra.Aux.X_meets_kernel","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_11.html","dependencies":["NumberField.AdeleRing.DivisionAlgebra.Aux.X"],"nlp_text":"If \\(\\beta \\in D_{\\mathbb {A}}^{(1)}\\) then \\(\\beta X\\cap D^\\times \\not=\\emptyset \\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.DivisionAlgebra.Aux.X_meets_kernel","latex_link":"Fujisaki_project.html#NumberField.AdeleRing.DivisionAlgebra.Aux.X_meets_kernel","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.DivisionAlgebra.Aux.X_meets_kernel","lean_decl_name":"NumberField.AdeleRing.DivisionAlgebra.Aux.X_meets_kernel","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch11","source_path":"FLT/DivisionAlgebra/Finiteness.lean","lean4_code":"lemma X_meets_kernel {β : D_𝔸ˣ} (hβ : β ∈ ringHaarChar_ker D_𝔸) :\n    ∃ x ∈ X K D, ∃ d ∈ Set.range (incl K D : Dˣ → D_𝔸ˣ), β * x = d := by\n  obtain ⟨e1, he1, e2, he2, noteq, b, hb⟩ := E_noninjective_left K D hβ\n  refine ⟨e1 - e2, by simpa only using (Set.sub_mem_sub he1 he2), ?_⟩\n  obtain ⟨b1, rfl⟩ : IsUnit b := by\n    simp_rw [← mul_sub_left_distrib, Algebra.TensorProduct.includeLeft_apply] at hb\n    have h1 : ↑β * (e1 - e2) ≠ 0 := by\n      simpa only [ne_eq, not_not, Units.mul_right_eq_zero] using (sub_ne_zero_of_ne noteq)\n    simp only [isUnit_iff_ne_zero, ne_eq]\n    rintro rfl\n    simp only [← hb, TensorProduct.zero_tmul, ne_eq, not_true_eq_false] at h1\n  exact ⟨incl K D b1, ⟨b1, rfl⟩, by simpa [mul_sub] using hb.symm⟩\n","line":820,"end_line":831,"proof_status":"formalized"}
+{"id":"NumberField.AdeleRing.DivisionAlgebra.Aux.X_compact","label":"NumberField.AdeleRing.DivisionAlgebra.Aux.X_compact","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_11.html","dependencies":["NumberField.AdeleRing.DivisionAlgebra.Aux.X"],"nlp_text":"\\(X\\) is a compact subset of \\(D_{\\mathbb {A}}\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.DivisionAlgebra.Aux.X_compact","latex_link":"Fujisaki_project.html#NumberField.AdeleRing.DivisionAlgebra.Aux.X_compact","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.DivisionAlgebra.Aux.X_compact","lean_decl_name":"NumberField.AdeleRing.DivisionAlgebra.Aux.X_compact","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch11","source_path":"FLT/DivisionAlgebra/Finiteness.lean","lean4_code":"lemma X_compact : IsCompact (X K D) := by\n  simpa only [Set.image_prod, Set.image2_sub] using (IsCompact.image_of_continuousOn\n    ((E_compact K D).prod (E_compact K D)) ((continuous_fst.sub continuous_snd).continuousOn))\n","line":812,"end_line":814,"proof_status":"formalized"}
+{"id":"NumberField.AdeleRing.DivisionAlgebra.Aux.existsE","label":"NumberField.AdeleRing.DivisionAlgebra.Aux.existsE","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_11.html","dependencies":[],"nlp_text":"There’s a compact subset \\(E\\) of \\(D_{\\mathbb {A}}\\) with the property that for all \\(x\\in D_{\\mathbb {A}}^{(1)}\\), the obvious map \\(xE\\to D\\backslash D_{\\mathbb {A}}\\) is not injective.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.DivisionAlgebra.Aux.existsE","latex_link":"Fujisaki_project.html#NumberField.AdeleRing.DivisionAlgebra.Aux.existsE","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.DivisionAlgebra.Aux.existsE","lean_decl_name":"NumberField.AdeleRing.DivisionAlgebra.Aux.existsE","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch11","source_path":"FLT/DivisionAlgebra/Finiteness.lean","lean4_code":"lemma existsE : ∃ E : Set (D_𝔸), IsCompact E ∧\n    ∀ φ : D_𝔸 ≃ₜ+ D_𝔸, addEquivAddHaarChar φ = 1 → ∃ e₁ ∈ E, ∃ e₂ ∈ E,\n    e₁ ≠ e₂ ∧ φ e₁ - φ e₂ ∈ Set.range (Algebra.TensorProduct.includeLeft : D →ₐ[K] D_𝔸) := by\n  obtain ⟨B, hB⟩ := not_injective_of_large_measure K D\n  obtain ⟨r, hr⟩ := E_family_unbounded K D B\n  let E := Efamily K D r\n  obtain ⟨U, hU, hKU, hU'⟩ := exists_isOpen_superset_and_isCompact_closure (E_family_compact K D r)\n  use closure U, hU'\n  intro φ hφ\n  specialize hB (φ.symm ⁻¹' U) (hU.preimage φ.symm.continuous)\n  replace hr : B < Measure.addHaar U := hr.trans_le (measure_mono hKU)\n  replace hφ : addEquivAddHaarChar φ.symm = 1 := by\n    simpa [hφ] using (addEquivAddHaarChar_trans (φ := φ) (ψ := φ.symm)).symm\n  have foo : Measure.addHaar U = Measure.addHaar (⇑φ.symm ⁻¹' U) := by\n    rw [← one_smul NNReal (Measure.addHaar (φ.symm ⁻¹' U)), ← hφ,\n      addEquivAddHaarChar_smul_preimage]\n  rw [foo] at hr\n  specialize hB hr\n  simp only [Set.InjOn, not_forall] at hB\n  obtain ⟨x, hx, y, hy, h, hne⟩ := hB\n  rw [QuotientAddGroup.eq_iff_sub_mem] at h\n  exact ⟨φ.symm x, subset_closure hx, φ.symm y, subset_closure hy, by simpa, by simpa⟩\n","line":762,"end_line":783,"proof_status":"formalized"}

crawled_repos/FLT_ch12.jsonl

@@ -0,0 +1,6 @@
+{"id":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.module","label":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.module","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_12.html","dependencies":["TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.addCommGroup"],"nlp_text":"If \\(R\\) is a commutative ring then pointwise scalar multiplication \\((r\\cdot f)(g):= r\\cdot (f(g))\\) makes \\(S^D(R)\\) into an \\(R\\)-module.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.module","latex_link":"Quat_alg_project.html#TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.module","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.module","lean_decl_name":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.module","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch12","source_path":"FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Abstract.lean","lean4_code":"instance module [Ring R] [Module R A] [SMulCommClass G R A] : Module R (fixedPoints G A) where\n  one_smul a := one_smul _ _\n  mul_smul r s a := mul_smul _ _ _\n  smul_zero a := by\n    ext\n    exact smul_zero _\n  smul_add r s a := by\n    ext\n    exact smul_add _ _ _\n  add_smul r s a := by\n    ext\n    exact add_smul _ _ _\n  zero_smul a := by\n    ext\n    exact zero_smul _ _\n","line":93,"end_line":107,"proof_status":"formalized"}
+{"id":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.distribMulAction","label":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.distribMulAction","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_12.html","dependencies":["TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.module"],"nlp_text":"The group \\(D_{\\mathbb {A}^f}^\\times \\) acts on the additive abelian group \\(S^D(R)\\) by \\((g\\cdot f)(x)=f(xg)\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.distribMulAction","latex_link":"Quat_alg_project.html#TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.distribMulAction","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.distribMulAction","lean_decl_name":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.distribMulAction","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Algebra/GroupWithZero/Action/Pointwise/Finset.lean","lean4_code":"protected noncomputable def distribMulAction [Monoid α] [AddMonoid β] [DistribMulAction α β] :\n    DistribMulAction α (Finset β) :=\n  coe_injective.distribMulAction coeAddMonoidHom coe_smul_finset\n\n/-- A multiplicative action of a monoid on a monoid `β` gives a multiplicative action on `Set β`. -/\nprotected noncomputable def mulDistribMulAction [Monoid α] [Monoid β] [MulDistribMulAction α β] :\n    MulDistribMulAction α (Finset β) :=\n  coe_injective.mulDistribMulAction coeMonoidHom coe_smul_finset\n\nscoped[Pointwise] attribute [instance] Finset.smulZeroClass Finset.distribSMul\n  Finset.distribMulAction Finset.mulDistribMulAction\n\n","line":38,"end_line":49,"proof_status":"formalized"}
+{"id":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.finiteDimensional","label":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.finiteDimensional","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_12.html","dependencies":["TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.module"],"nlp_text":"Let \\(k\\) be a field. Then the space \\(S^D(U;k)\\) is a finite-dimensional \\(k\\)-vector space.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.finiteDimensional","latex_link":"Quat_alg_project.html#TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.finiteDimensional","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.finiteDimensional","lean_decl_name":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.finiteDimensional","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/NumberTheory/Cyclotomic/Basic.lean","lean4_code":"theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]\n    [IsCyclotomicExtension S K C] : FiniteDimensional K C :=\n  IsCyclotomicExtension.finite S K C\n\nend Fintype\n\nsection\n\nvariable {A B}\n\n","line":397,"end_line":406,"proof_status":"statement_formalized"}
+{"id":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicFormOfLevel","label":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicFormOfLevel","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_12.html","dependencies":["TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.distribMulAction"],"nlp_text":"The quaternionic modular forms of level \\(U\\), with notation \\(S^D(U;R)\\), are the \\(U\\)-invariants for the \\(D_{\\mathbb {A}^\\infty }^\\times \\)-action on \\(S^D(R)\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicFormOfLevel","latex_link":"Quat_alg_project.html#TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicFormOfLevel","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicFormOfLevel","lean_decl_name":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicFormOfLevel","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch12","source_path":"FLT/AutomorphicForm/QuaternionAlgebra/Defs.lean","lean4_code":"/--\n`WeightTwoAutomorphicFormOfLevel U R` is the `R`-valued weight 2 automorphic forms of a fixed\nlevel `U` for a totally definite quaternion algebra over a totally real field.\n-/\ndef WeightTwoAutomorphicFormOfLevel (U : Subgroup (Dfx F D))\n    (R : Type*) [CommRing R] : Type _ := MulAction.FixedPoints U (WeightTwoAutomorphicForm F D R)\n","line":285,"end_line":290,"proof_status":"formalized"}
+{"id":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.addCommGroup","label":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.addCommGroup","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_12.html","dependencies":["TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm"],"nlp_text":"Pointwise addition \\((f_1+f_2)(g):=f_1(g)+f_2(g)\\) makes \\(S^D(R)\\) into an additive abelian group.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.addCommGroup","latex_link":"Quat_alg_project.html#TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.addCommGroup","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.addCommGroup","lean_decl_name":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.addCommGroup","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/CategoryTheory/Localization/CalculusOfFractions/Preadditive.lean","lean4_code":"noncomputable def addCommGroup' : AddCommGroup (L.obj X ⟶ L.obj Y) := by\n  letI : Zero (L.obj X ⟶ L.obj Y) := ⟨L.map 0⟩\n  letI : Add (L.obj X ⟶ L.obj Y) := ⟨add' W⟩\n  letI : Neg (L.obj X ⟶ L.obj Y) := ⟨neg' W⟩\n  exact\n    { add_assoc := add'_assoc _\n      add_zero := add'_zero _\n      add_comm := add'_comm _\n      zero_add := zero_add' _\n      neg_add_cancel := neg'_add'_self _\n      nsmul := nsmulRec\n      zsmul := zsmulRec }\n\nvariable {X Y}\n\nvariable {L}\nvariable {X' Y' Z' : D} (eX : L.obj X ≅ X') (eY : L.obj Y ≅ Y') (eZ : L.obj Z ≅ Z')\n\n/-- The bijection `(X' ⟶ Y') ≃ (L.obj X ⟶ L.obj Y)` induced by isomorphisms\n`eX : L.obj X ≅ X'` and `eY : L.obj Y ≅ Y'`. -/\n","line":223,"end_line":242,"proof_status":"formalized"}
+{"id":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm","label":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_12.html","dependencies":[],"nlp_text":"The space of \\(R\\)-valued automorphic forms for \\(D^\\times \\) is the set of functions \\(f:D_{\\mathbb {A}^\\infty }^\\times \\to R\\) satisfying the following axioms: \\(f(dg)=f(g)\\) for all \\(d\\in D^\\times \\) and \\(g\\in D_{\\mathbb {A}^\\infty }^\\times \\). \\(f(gz)=f(g)\\) for all \\(g\\in D_{\\mathbb {A}^\\infty }^\\times \\). There exists a compact open subgroup \\(U\\subseteq (D_{\\mathbb {A}^f}^\\times )\\) such that \\(f(gu)=f(g)\\) for all \\(g\\in D_{\\mathbb {A}^\\infty }^\\times \\) and \\(u\\in U\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm","latex_link":"Quat_alg_project.html#TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm","lean_decl_name":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch12","source_path":"FLT/AutomorphicForm/QuaternionAlgebra/Defs.lean","lean4_code":"/--\nThis definition is made in mathlib-generality but is *not* the definition of a\nweight 2 automorphic form unless `Dˣ` is compact mod centre at infinity.\nThis hypothesis will be true if `D` is a totally definite quaternion algebra\nover a totally real field.\n-/\nstructure WeightTwoAutomorphicForm\n  -- defined over R\n  (R : Type*) [AddCommMonoid R] where\n  /-- The function underlying an automorphic form. -/\n  toFun : Dfx F D → R\n  left_invt : ∀ (δ : Dˣ) (g : Dfx F D),\n    toFun (incl₁ F D δ * g) = (toFun g)\n  right_invt : ∃ (U : Subgroup (Dfx F D)),\n    IsOpen (U : Set (Dfx F D)) ∧\n    ∀ (g : Dfx F D),\n    ∀ u ∈ U, toFun (g * u) = toFun g\n  trivial_central_char (z : (FiniteAdeleRing (𝓞 F) F)ˣ)\n      (g : Dfx F D) :\n      toFun (g * incl₂ F D z) = toFun g\n","line":82,"end_line":101,"proof_status":"formalized"}

crawled_repos/FLT_ch13.jsonl

@@ -0,0 +1,21 @@
+{"id":"AbstractHeckeOperator.HeckeOperator","label":"AbstractHeckeOperator.HeckeOperator","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_13.html","dependencies":["AbstractHeckeOperator.HeckeOperator_toFun"],"nlp_text":"This function is well-defined (that is, independent of the choice of \\(g_i\\)), has image in \\(A^U\\) and is \\(R\\)-linear.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/AbstractHeckeOperator.HeckeOperator","latex_link":"HeckeOperator_project.html#AbstractHeckeOperator.HeckeOperator","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/AbstractHeckeOperator.HeckeOperator","lean_decl_name":"AbstractHeckeOperator.HeckeOperator","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch13","source_path":"FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Abstract.lean","lean4_code":"noncomputable def HeckeOperator : fixedPoints V A →ₗ[R] fixedPoints U A where\n  toFun := HeckeOperator_toFun h\n  map_add' a b := by\n    ext\n    simp only [HeckeOperator_toFun, FixedPoints.coe_add, smul_add,\n      finsum_mem_add_distrib (h.image Quotient.out)]\n  map_smul' r a := by\n    ext\n    simp only [HeckeOperator_toFun, FixedPoints.coe_smul, smul_comm,\n      smul_finsum_mem (h.image Quotient.out), RingHom.id_apply]\n","line":186,"end_line":195,"proof_status":"formalized"}
+{"id":"AbstractHeckeOperator.comm","label":"AbstractHeckeOperator.comm","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_13.html","dependencies":["AbstractHeckeOperator.HeckeOperator"],"nlp_text":"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 all \\(i,j\\). Then \\([UgU][UhU]=[UhU][UgU]\\), that is, the Hecke operators acting on \\(A^U\\) commute.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/AbstractHeckeOperator.comm","latex_link":"HeckeOperator_project.html#AbstractHeckeOperator.comm","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/AbstractHeckeOperator.comm","lean_decl_name":"AbstractHeckeOperator.comm","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch13","source_path":"FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Abstract.lean","lean4_code":"theorem comm {g₁ g₂ : G} (h₁ : (QuotientGroup.mk '' (U * {g₁}) : Set (G ⧸ U)).Finite)\n    (h₂ : (QuotientGroup.mk '' (U * {g₂}) : Set (G ⧸ U)).Finite)\n    (hcomm : ∃ s₁ s₂ : Set G,\n      Set.BijOn QuotientGroup.mk s₁ (QuotientGroup.mk '' (U * {g₁}) : Set (G ⧸ U)) ∧\n      Set.BijOn QuotientGroup.mk s₂ (QuotientGroup.mk '' (U * {g₂}) : Set (G ⧸ U)) ∧\n      ∀ a ∈ s₁, ∀ b ∈ s₂, a * b = b * a) :\n    (HeckeOperator g₁ U U h₁ ∘ₗ HeckeOperator g₂ U U h₂ : fixedPoints U A →ₗ[R] fixedPoints U A) =\n    HeckeOperator g₂ U U h₂ ∘ₗ HeckeOperator g₁ U U h₁ := by\n  ext a\n  rcases hcomm with ⟨s₁, s₂, hs₁, hs₂, hcomm⟩\n  simp only [LinearMap.coe_comp, Function.comp_apply]\n  nth_rw 1 [HeckeOperator_apply]\n  rw [← eq_finsum_quotient_out_of_bijOn' _ hs₁]\n  nth_rw 1 [HeckeOperator_apply]\n  rw [← eq_finsum_quotient_out_of_bijOn' _ hs₂]\n  nth_rw 1 [HeckeOperator_apply]\n  rw [← eq_finsum_quotient_out_of_bijOn' _ hs₂]\n  nth_rw 1 [HeckeOperator_apply]\n  rw [← eq_finsum_quotient_out_of_bijOn' _ hs₁]\n  have hfs₁ : s₁.Finite := by rwa [hs₁.finite_iff_finite]\n  have hfs₂ : s₂.Finite := by rwa [hs₂.finite_iff_finite]\n  simp_rw [smul_finsum_mem hfs₁, smul_finsum_mem hfs₂, finsum_mem_comm _ hfs₁ hfs₂]\n  -- I'm sure there's a better way to do this!\n  congr; ext g₂; congr; ext hg₂; congr; ext g₁; congr; ext hg₁;\n  rw [smul_smul, smul_smul, hcomm _ hg₁ _ hg₂]\n","line":202,"end_line":226,"proof_status":"formalized"}
+{"id":"Submonoid.units_isCompact","label":"Submonoid.units_isCompact","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_13.html","dependencies":[],"nlp_text":"If \\(M\\) is a Hausdorff topological monoid and \\(U\\) is a compact submonoid, then the units \\(U^\\times \\) of \\(U\\) are naturally a compact subgroup of \\(M^\\times \\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Submonoid.units_isCompact","latex_link":"HeckeOperator_project.html#Submonoid.units_isCompact","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Submonoid.units_isCompact","lean_decl_name":"Submonoid.units_isCompact","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/leanprover-community/mathlib4/98379c78c5dd2e3614cd24ab7ad22f5713e0eb63/Mathlib/Topology/Algebra/Group/Basic.lean","lean4_code":"lemma _root_.Submonoid.units_isCompact [T1Space α] [ContinuousMul α] {S : Submonoid α}\n    (hS : IsCompact (S : Set α)) : IsCompact (S.units : Set αˣ) := by\n  have : IsCompact (S ×ˢ S.op) := hS.prod (opHomeomorph.isCompact_preimage.mp hS)\n  exact isClosedEmbedding_embedProduct.isCompact_preimage this","line":1225,"end_line":1228,"proof_status":"statement_formalized"}
+{"id":"nolean-compactopen-GL2","label":"nolean-compactopen-GL2","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_13.html","dependencies":["Submonoid.units_isCompact","Submonoid.units_isOpen"],"nlp_text":"\\(GL_2(\\mathcal{O}_v)\\) is a compact open subgroup of \\(GL_2(K_v)\\).","doc_url":null,"latex_link":"HeckeOperator_project.html#nolean-compactopen-GL2","lean_link":null,"lean_decl_name":"nolean-compactopen-GL2","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"nolean-compactopen-U1p","label":"nolean-compactopen-U1p","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_13.html","dependencies":["Submonoid.units_isCompact","Submonoid.units_isOpen"],"nlp_text":"\\(U_v\\) is a compact open subgroup of \\(GL_2(K_v)\\).","doc_url":null,"latex_link":"HeckeOperator_project.html#nolean-compactopen-U1p","lean_link":null,"lean_decl_name":"nolean-compactopen-U1p","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"M2.localFullLevel.isCompact","label":"M2.localFullLevel.isCompact","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_13.html","dependencies":["NumberField.instCompactSpaceAdicCompletionIntegers"],"nlp_text":"\\(M_2(\\mathcal{O}_v)\\) is a compact subring of \\(M_2(K_v)\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.M2.localFullLevel.isCompact","latex_link":"HeckeOperator_project.html#M2.localFullLevel.isCompact","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.M2.localFullLevel.isCompact","lean_decl_name":"IsDedekindDomain.M2.localFullLevel.isCompact","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Topology/Sets/Compacts.lean","lean4_code":"protected theorem isCompact (s : Compacts α) : IsCompact (s : Set α) :=\n  s.isCompact'\n\n","line":50,"end_line":52,"proof_status":"formalized"}
+{"id":"nolean-product-of-U-alpha","label":"nolean-product-of-U-alpha","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_13.html","dependencies":[],"nlp_text":"If \\(v\\in S\\) and \\(\\begin{pmatrix} 1 & * \\\\ 0 & 1 \\end{pmatrix}\\subseteq \\Gamma _v\\subseteq \\begin{pmatrix} * & * \\\\ 0 & * \\end{pmatrix}\\) then \\(U_{\\alpha ,v}U_{\\beta ,v}=U_{\\alpha \\beta ,v}\\).","doc_url":null,"latex_link":"HeckeOperator_project.html#nolean-product-of-U-alpha","lean_link":null,"lean_decl_name":"nolean-product-of-U-alpha","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"QuotientGroup.mk_image_finite_of_compact_of_open","label":"QuotientGroup.mk_image_finite_of_compact_of_open","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_13.html","dependencies":[],"nlp_text":"If \\(U\\) and \\(V\\) are compact subgroups of a topological group&#160;\\(G\\), if \\(V\\) is also open, and if \\(g\\in G\\), then the double coset space \\(UgV\\) is a finite union of left cosets \\(g_iV\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/QuotientGroup.mk_image_finite_of_compact_of_open","latex_link":"HeckeOperator_project.html#QuotientGroup.mk_image_finite_of_compact_of_open","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/QuotientGroup.mk_image_finite_of_compact_of_open","lean_decl_name":"QuotientGroup.mk_image_finite_of_compact_of_open","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Concrete.lean","lean4_code":"lemma QuotientGroup.mk_image_finite_of_compact_of_open\n    (hU : IsCompact (U : Set G)) (hVopen : IsOpen (V : Set G)) :\n    (QuotientGroup.mk '' (U * {g}) : Set (G ⧸ V)).Finite := by\n  have : DiscreteTopology (G ⧸ V) := by\n    rw [discreteTopology_iff_forall_isOpen]\n    intro s\n    rw [← (isQuotientMap_mk V).isOpen_preimage, ← (QuotientGroup.mk_surjective).image_preimage s,\n      preimage_image_mk_eq_iUnion_image, iUnion_subtype]\n    conv in ⋃ x ∈ _, _ => change ⋃ x ∈ (V : Set G), _\n    rw [iUnion_mul_right_image]\n    exact IsOpen.mul_left hVopen\n  exact ((hU.mul <| isCompact_singleton).image continuous_mk).finite_of_discrete","line":63,"end_line":74,"proof_status":"formalized"}
+{"id":"ContinuousMulEquiv.piUnits","label":"ContinuousMulEquiv.piUnits","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_13.html","dependencies":[],"nlp_text":"If \\(U_i\\) are topological monoids then the canonical group isomorphism \\((\\prod _i U_i)^\\times =\\prod _i(U_i^\\times )\\) is a homeomorphism.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/ContinuousMulEquiv.piUnits","latex_link":"HeckeOperator_project.html#ContinuousMulEquiv.piUnits","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/ContinuousMulEquiv.piUnits","lean_decl_name":"ContinuousMulEquiv.piUnits","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Algebra/Group/Pi/Units.lean","lean4_code":"def MulEquiv.piUnits : (Π i, M i)ˣ ≃* Π i, (M i)ˣ where\n  toFun f i := ⟨f.val i, f.inv i, congr_fun f.val_inv i, congr_fun f.inv_val i⟩\n  invFun f := ⟨(val <| f ·), (inv <| f ·), funext (val_inv <| f ·), funext (inv_val <| f ·)⟩\n  map_mul' _ _ := rfl\n\n","line":21,"end_line":25,"proof_status":"formalized"}
+{"id":"ContinuousMulEquiv.restrictedProductUnits","label":"ContinuousMulEquiv.restrictedProductUnits","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_13.html","dependencies":["ContinuousMulEquiv.piUnits","Submonoid.units_isOpen"],"nlp_text":"If \\(M_i\\) are a family of topological monoids equipped with open submonoids \\(U_i\\), then the canonical map \\((\\prod '_iM_i)^\\times \\to \\prod '_i(M_i^\\times )\\) is a homeomorphism.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/ContinuousMulEquiv.restrictedProductUnits","latex_link":"HeckeOperator_project.html#ContinuousMulEquiv.restrictedProductUnits","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/ContinuousMulEquiv.restrictedProductUnits","lean_decl_name":"ContinuousMulEquiv.restrictedProductUnits","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch13","source_path":"FLT/Mathlib/Topology/Algebra/RestrictedProduct/Equiv.lean","lean4_code":"/-- The isomorphism between the units of a restricted product of monoids,\nand the restricted product of the units of the monoids. -/\ndef MulEquiv.restrictedProductUnits {ι : Type*} {ℱ : Filter ι}\n    {M : ι → Type*} [(i : ι) → Monoid (M i)]\n    {S : ι → Type*} [∀ i, SetLike (S i) (M i)] [∀ i, SubmonoidClass (S i) (M i)]\n    {A : Π i, S i} :\n    (Πʳ i, [M i, A i]_[ℱ])ˣ ≃*\n      Πʳ i, [(M i)ˣ, (Submonoid.ofClass (A i)).units]_[ℱ] where\n        toFun u := ⟨fun i ↦ ⟨u.1 i, u⁻¹.1 i, congr($u.mul_inv i), congr($u.inv_mul i)⟩,\n          by filter_upwards [u.val.2, u⁻¹.val.2] using fun i hi hi' ↦ ⟨hi, hi'⟩⟩\n        invFun ui := ⟨⟨fun i ↦ ui i, by filter_upwards [ui.2] using fun i hi ↦ hi.1⟩,\n          ⟨fun i ↦ ui⁻¹ i, by filter_upwards [ui⁻¹.2] using fun i hi ↦ hi.1⟩,\n          by ext i; exact (ui i).mul_inv,\n          by ext i; exact (ui i).inv_mul⟩\n        left_inv u := by ext; rfl\n        right_inv ui := by ext; rfl\n        map_mul' u v := by ext; rfl\n","line":65,"end_line":81,"proof_status":"formalized"}
+{"id":"GL2.restrictedProduct","label":"GL2.restrictedProduct","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_13.html","dependencies":["ContinuousMulEquiv.restrictedProductUnits","Homeomorph.restrictedProductMatrix"],"nlp_text":"\\(G\\) is isomorphic and homeomorphic to the restricted product of \\(GL_2(K_v)\\) with respect to the compact open subgroups \\(GL_2(\\mathcal{O}_v)\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.FiniteAdeleRing.GL2.restrictedProduct","latex_link":"HeckeOperator_project.html#GL2.restrictedProduct","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.FiniteAdeleRing.GL2.restrictedProduct","lean_decl_name":"IsDedekindDomain.FiniteAdeleRing.GL2.restrictedProduct","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch13","source_path":"FLT/QuaternionAlgebra/NumberField.lean","lean4_code":"/-- `GL_2(𝔸_F^∞)` is isomorphic and homeomorphic to the\nrestricted product of the local components `GL_2(F_v)`.\n-/\nnoncomputable def GL2.restrictedProduct :\n    GL (Fin 2) (FiniteAdeleRing (𝓞 F) F) ≃ₜ*\n    Πʳ (v : HeightOneSpectrum (𝓞 F)),\n      [(GL (Fin 2) (v.adicCompletion F)), (M2.localFullLevel v).units] :=\n  ContinuousMulEquiv.restrictedProductMatrixUnits (NumberField.isOpenAdicCompletionIntegers F)\n","line":226,"end_line":233,"proof_status":"formalized"}
+{"id":"NumberField.instCompactSpaceAdicCompletionIntegers","label":"NumberField.instCompactSpaceAdicCompletionIntegers","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_13.html","dependencies":[],"nlp_text":"\\(\\mathcal{O}_v\\) is a compact subring of \\(K_v\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.instCompactSpaceAdicCompletionIntegers","latex_link":"HeckeOperator_project.html#NumberField.instCompactSpaceAdicCompletionIntegers","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.instCompactSpaceAdicCompletionIntegers","lean_decl_name":"NumberField.instCompactSpaceAdicCompletionIntegers","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/NumberField/Completion/Finite.lean","lean4_code":"instance NumberField.instCompactSpaceAdicCompletionIntegers :\n    CompactSpace (v.adicCompletionIntegers K) :=\n  Valued.WithZeroMulInt.integer_compactSpace (v.adicCompletion K) inferInstance","line":44,"end_line":46,"proof_status":"formalized"}
+{"id":"M2.localFullLevel.isOpen","label":"M2.localFullLevel.isOpen","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_13.html","dependencies":["NumberField.isOpenAdicCompletionIntegers"],"nlp_text":"\\(M_2(\\mathcal{O}_v)\\) is an open subring of \\(M_2(K_v)\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.M2.localFullLevel.isOpen","latex_link":"HeckeOperator_project.html#M2.localFullLevel.isOpen","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.M2.localFullLevel.isOpen","lean_decl_name":"IsDedekindDomain.M2.localFullLevel.isOpen","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Topology/Sets/Opens.lean","lean4_code":"protected theorem isOpen (U : Opens α) : IsOpen (U : Set α) :=\n  U.is_open'\n\n","line":107,"end_line":109,"proof_status":"formalized"}
+{"id":"bijOn_unipotent_mul_diagU1_U1diagU1","label":"bijOn_unipotent_mul_diagU1_U1diagU1","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_13.html","dependencies":[],"nlp_text":"The double coset space \\(UgU\\) is the disjoint union of \\(g_tU\\) as \\(t\\) ranges through \\(\\mathcal{O}_v/\\alpha \\mathcal{O}_v\\) and \\(g_t:=\\begin{pmatrix} \\alpha & \\tilde{t} \\\\ 0 & 1 \\end{pmatrix}\\), where \\(\\tilde{t}\\) is any lift of \\(t\\) to \\(\\mathcal{O}_v\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.Local.bijOn_unipotent_mul_diagU1_U1diagU1","latex_link":"HeckeOperator_project.html#bijOn_unipotent_mul_diagU1_U1diagU1","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.Local.bijOn_unipotent_mul_diagU1_U1diagU1","lean_decl_name":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.Local.bijOn_unipotent_mul_diagU1_U1diagU1","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch13","source_path":"FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Local.lean","lean4_code":"/-- The double coset space `U1diagU1` is the disjoint union of\n`unipotent_mul_diagU1` as t ranges over `O_v / αO_v`. -/\ntheorem bijOn_unipotent_mul_diagU1_U1diagU1 :\n    Set.BijOn (unipotent_mul_diagU1 v α hα) ⊤ (U1diagU1 v α hα) :=\n  ⟨mapsTo_unipotent_mul_diagU1_U1diagU1 α hα,\n    injOn_unipotent_mul_diagU1 α hα,\n    surjOn_unipotent_mul_diagU1_U1diagU1 α hα⟩\n","line":296,"end_line":302,"proof_status":"formalized"}
+{"id":"nolean-hecke-algebra-commutative-noetherian","label":"nolean-hecke-algebra-commutative-noetherian","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_13.html","dependencies":[],"nlp_text":"Say&#160;\\(R\\) is a Noetherian ring. Then the subalgebra of the \\(R\\)-linear endomorphisms of \\(A^U\\) generated by the Hecke operators \\(T_v\\) for \\(v\\notin S\\) and \\(U_{v,\\alpha }\\) for \\(v\\in S\\) is a Noetherian commutative ring.","doc_url":null,"latex_link":"HeckeOperator_project.html#nolean-hecke-algebra-commutative-noetherian","lean_link":null,"lean_decl_name":"nolean-hecke-algebra-commutative-noetherian","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"Homeomorph.restrictedProductPi","label":"Homeomorph.restrictedProductPi","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_13.html","dependencies":["Homeomorph.restrictedProductProd"],"nlp_text":"Restricted products (with respect to open subspaces) commute with finite products. In other words, if \\(j\\) runs through a finite set \\(J\\) and \\(i\\) runs through an arbitrary set \\(I\\), and if \\(X_{ji}\\) are topological spaces equipped with open subspaces \\(Y_{ji}\\), then the obvious bijection \\(\\prod '_i(\\prod _j X_{ji})=\\prod _j\\left(\\prod '_i X_{ji}\\right)\\) is a homeomorphism.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Homeomorph.restrictedProductPi","latex_link":"HeckeOperator_project.html#Homeomorph.restrictedProductPi","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Homeomorph.restrictedProductPi","lean_decl_name":"Homeomorph.restrictedProductPi","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch13","source_path":"FLT/Mathlib/Topology/Algebra/RestrictedProduct/Module.lean","lean4_code":"/-- Canonical linear equivalence between `Π' R^n` and `(Π' R)^n` -/\ndef _root_.LinearEquiv.restrictedProductPi\n    : Πʳ i, [n → R i, piSubringSubmodule B n i]_[ℱ] ≃ₗ[Πʳ i, [R i, B i]_[ℱ]]\n      n → Πʳ i, [R i, B i]_[ℱ] where\n  toFun x j := map (fun i y ↦ y j)\n    (by\n      filter_upwards with i r hr\n      rw [piSubringSubmodule, Submodule.coe_pi, Set.mem_univ_pi] at hr\n      exact hr j)\n    x\n  invFun x := ⟨fun i j ↦ x j i, by simpa [piSubringSubmodule] using fun j ↦ (x j).eventually⟩\n  map_add' x y := rfl\n  map_smul' x y := rfl\n","line":176,"end_line":188,"proof_status":"formalized"}
+{"id":"Homeomorph.restrictedProductMatrix","label":"Homeomorph.restrictedProductMatrix","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_13.html","dependencies":["Homeomorph.restrictedProductPi"],"nlp_text":"If \\(X_i\\) are topological spaces and the \\(Y_i\\) are open subspaces, then the obvious map \\(M_n(\\prod '_iX_i)=\\prod '_iM_n(X_i)\\) is a homeomorphism.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Homeomorph.restrictedProductMatrix","latex_link":"HeckeOperator_project.html#Homeomorph.restrictedProductMatrix","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Homeomorph.restrictedProductMatrix","lean_decl_name":"Homeomorph.restrictedProductMatrix","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch13","source_path":"FLT/Mathlib/Topology/Algebra/RestrictedProduct/Equiv.lean","lean4_code":"/-- The bijection between a restricted product of m x n matrices, and m x n matrices\nof restricted products.\n-/\ndef Equiv.restrictedProductMatrix {ι : Type*} {m n : Type*} [Fintype m] [Fintype n]\n    {A : ι → Type*}\n    {C : (i : ι) → Set (A i)} :\n    Πʳ i, [Matrix m n (A i), {f | ∀ a b, f a b ∈ C i}] ≃ Matrix m n (Πʳ i, [A i, C i]) :=\n  Equiv.restrictedProductPi.trans (Equiv.piCongrRight fun _ ↦ Equiv.restrictedProductPi)\n","line":454,"end_line":461,"proof_status":"formalized"}
+{"id":"Submonoid.units_isOpen","label":"Submonoid.units_isOpen","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_13.html","dependencies":[],"nlp_text":"If \\(M\\) is a topological monoid and \\(U\\) is an open submonoid, then the units \\(U^\\times \\) of \\(U\\) are naturally an open subgroup of \\(M^\\times \\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Submonoid.units_isOpen","latex_link":"HeckeOperator_project.html#Submonoid.units_isOpen","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Submonoid.units_isOpen","lean_decl_name":"Submonoid.units_isOpen","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/Mathlib/Topology/Algebra/Group/Units.lean","lean4_code":"lemma Submonoid.units_isOpen {M : Type*} [TopologicalSpace M] [Monoid M]\n  {U : Submonoid M} (hU : IsOpen (U : Set M)) : IsOpen (U.units : Set Mˣ) :=\n  (hU.preimage Units.continuous_val).inter (hU.preimage Units.continuous_coe_inv)","line":7,"end_line":9,"proof_status":"formalized"}
+{"id":"NumberField.isOpenAdicCompletionIntegers","label":"NumberField.isOpenAdicCompletionIntegers","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_13.html","dependencies":[],"nlp_text":"\\(\\mathcal{O}_v\\) is an open subring of \\(K_v\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.isOpenAdicCompletionIntegers","latex_link":"HeckeOperator_project.html#NumberField.isOpenAdicCompletionIntegers","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.isOpenAdicCompletionIntegers","lean_decl_name":"NumberField.isOpenAdicCompletionIntegers","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/NumberField/Completion/Finite.lean","lean4_code":"lemma NumberField.isOpenAdicCompletionIntegers :\n    IsOpen (v.adicCompletionIntegers K : Set (v.adicCompletion K)) :=\n  Valued.isOpen_valuationSubring _","line":53,"end_line":55,"proof_status":"formalized"}
+{"id":"Homeomorph.restrictedProductProd","label":"Homeomorph.restrictedProductProd","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_13.html","dependencies":[],"nlp_text":"If \\(A_i\\) is a family of topological spaces equipped with open subsets \\(B_i\\), and if \\(C_i\\) is a family of topological spaces equipped with open subsets \\(D_i\\), and if we equip \\(A_i\\times C_i\\) with the open subset \\(B_i\\times D_i\\), then the natural bijection \\(\\prod '_i(A_i\\times C_i)=\\left(\\prod '_iA_i\\right)\\times \\left(\\prod '_iB_i\\right)\\) is a homeomorphism.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Homeomorph.restrictedProductProd","latex_link":"HeckeOperator_project.html#Homeomorph.restrictedProductProd","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Homeomorph.restrictedProductProd","lean_decl_name":"Homeomorph.restrictedProductProd","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch13","source_path":"FLT/Mathlib/Topology/Algebra/RestrictedProduct/Equiv.lean","lean4_code":"/-- The bijection between a restricted product of binary products, and the binary product\nof the restricted products. -/\n@[simps]\ndef Equiv.restrictedProductProd :\n    Πʳ i, [A i × B i, C i ×ˢ D i]_[ℱ] ≃ (Πʳ i, [A i, C i]_[ℱ]) × (Πʳ i, [B i, D i]_[ℱ]) where\n  toFun x := (map (fun i (t : A i × B i) ↦ t.1) (by simp +contextual [Set.MapsTo]) x,\n              map (fun i (t : A i × B i) ↦ t.2) (by simp +contextual [Set.MapsTo]) x)\n  invFun yz :=\n    ⟨fun i ↦ (yz.1 i, yz.2 i), by\n    filter_upwards [yz.1.2, yz.2.2] with i using Set.mk_mem_prod⟩\n  left_inv x := by ext <;> rfl\n  right_inv y := by ext <;> rfl\n","line":410,"end_line":421,"proof_status":"formalized"}
+{"id":"AbstractHeckeOperator.HeckeOperator_toFun","label":"AbstractHeckeOperator.HeckeOperator_toFun","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_13.html","dependencies":[],"nlp_text":"Assuming \\(UgV\\) is a finite union of cosets \\(g_iV\\), we define \\([UgV]:A^V\\to A^U\\) to be the map sending \\(a\\in A^V\\) to \\(\\sum _i g_ia.\\)","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/AbstractHeckeOperator.HeckeOperator_toFun","latex_link":"HeckeOperator_project.html#AbstractHeckeOperator.HeckeOperator_toFun","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/AbstractHeckeOperator.HeckeOperator_toFun","lean_decl_name":"AbstractHeckeOperator.HeckeOperator_toFun","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch13","source_path":"FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Abstract.lean","lean4_code":"/-- The Hecke operator T_g = [UgV] : A^V → A^U associated to the double coset UgV. -/\nnoncomputable def HeckeOperator_toFun (a : fixedPoints V A) : fixedPoints U A :=\n  ⟨∑ᶠ gᵢ ∈ Quotient.out '' (QuotientGroup.mk '' (U * {g}) : Set (G ⧸ V)), gᵢ • a.1, by\n  rintro ⟨u, huU⟩\n  rw [smul_finsum_mem (h.image Quotient.out), ← eq_finsum_quotient_out_of_bijOn' a]\n  · rw [finsum_mem_eq_of_bijOn (fun g ↦ u • g)]\n    · exact Set.InjOn.bijOn_image <| Set.injOn_of_injective (MulAction.injective u)\n    · simp [mul_smul]\n  · apply (Set.bijOn_comp_iff (Set.injOn_of_injective (MulAction.injective u))).1\n    change Set.BijOn ((fun xbar ↦ u • xbar) ∘ (QuotientGroup.mk : G → G ⧸ V)) _ _\n    rw [Set.bijOn_comp_iff]\n    · rw [← Set.image_comp]\n      simp only [Function.comp_apply, Quotient.out_eq, Set.image_id']\n      refine Set.bijOn_image_image (f := fun (x : G) ↦ u • x) (p₁ := (QuotientGroup.mk : G → G ⧸ V))\n        (fun a ↦ rfl) ?_ (Set.injOn_of_injective (MulAction.injective u))\n      apply Set.bijOn_smul _ _ _ huU\n    · refine Set.InjOn.image_of_comp ?_\n      simp only [Function.comp_def, Quotient.out_eq]\n      exact Function.Injective.injOn Function.injective_id\n    ⟩\n","line":151,"end_line":170,"proof_status":"formalized"}

crawled_repos/FLT_ch14.jsonl

@@ -0,0 +1,30 @@
+{"id":"local_galois_coh_top_degree","label":"local_galois_coh_top_degree","background_color":"white","border_color":"#FFAA33","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_14.html","dependencies":[],"nlp_text":"\\(H^2(G_K,\\mu _n)\\) is “canonically” isomorphic to \\(\\mathbb {Z}/n\\mathbb {Z}\\).","doc_url":null,"latex_link":"ch_bestiary.html#local_galois_coh_top_degree","lean_link":null,"lean_decl_name":"local_galois_coh_top_degree","kind_description":"theorems and lemmas","status_from_color":"unknown","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000300","label":"a0000000300","background_color":"white","border_color":"#FFAA33","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_14.html","dependencies":[],"nlp_text":"There is a “canonical” isomorphism \\(H^2(K,\\mu _\\infty )=\\mathbb {Q}/\\mathbb {Z}\\).","doc_url":null,"latex_link":"ch_bestiary.html#a0000000300","lean_link":null,"lean_decl_name":"a0000000300","kind_description":"theorems and lemmas","status_from_color":"unknown","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"Skinner_Wiles_CFT_trick","label":"Skinner_Wiles_CFT_trick","background_color":"white","border_color":"#FFAA33","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_14.html","dependencies":["global_class_field_theory"],"nlp_text":"Let \\(S\\) be a finite set of places of a number field \\(K\\) . For each \\(v \\in S\\) let \\(L_v/K_v\\) be a finite Galois extension. Then there is a finite solvable Galois extension \\(L/K\\) such that if \\(w\\) is a place of \\(L\\) dividing \\(v \\in S\\), then \\(L_w/K_v\\) is isomorphic to \\(L_v/K_v\\) as \\(K_v\\)-algebra. Moreover, if \\(K^{\\operatorname{avoid}} /K\\) is any finite extension then we can choose \\(L\\) to be linearly disjoint from \\(K^{\\operatorname{avoid}}\\).","doc_url":null,"latex_link":"ch_bestiary.html#Skinner_Wiles_CFT_trick","lean_link":null,"lean_decl_name":"Skinner_Wiles_CFT_trick","kind_description":"theorems and lemmas","status_from_color":"unknown","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"slowly_increasing_well_defined","label":"slowly_increasing_well_defined","background_color":"white","border_color":"#FFAA33","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_14.html","dependencies":["slowly_increasing"],"nlp_text":"This is independent of the choice of \\(\\rho \\) as above.","doc_url":null,"latex_link":"ch_bestiary.html#slowly_increasing_well_defined","lean_link":null,"lean_decl_name":"slowly_increasing_well_defined","kind_description":"theorems and lemmas","status_from_color":"unknown","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"automorphic_form","label":"automorphic_form","background_color":"white","border_color":"#FFAA33","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_14.html","dependencies":["slowly_increasing_well_defined","topology_on_affine_variety_computation","lie_group_from_algebraic_group","connected_reductive_group"],"nlp_text":"An automorphic form is a function \\(\\phi :G(\\mathbb {A}_N)\\to \\mathbb {C}\\) satisfying the following conditions: \\(\\phi \\) is locally constant on \\(G(\\mathbb {A}_N^f)\\) and \\(C^\\infty \\) on \\(G(N_\\infty )\\). In other words, for every \\(g_\\infty \\), \\(\\phi (-,g_\\infty )\\) is locally constant, and for every \\(g_f\\), \\(\\phi (g_f,-)\\) is smooth. \\(\\phi \\) is left-invariant under \\(G(N)\\); \\(\\phi \\) is right-\\(U_\\infty \\)-finite (that is, the space spanned by \\(x\\mapsto \\phi (xu)\\) as \\(u\\) varies over \\(U_\\infty \\) is finite-dimensional); \\(\\phi \\) is right \\(K_f\\)-finite, where \\(K_f\\) is one (or equivalently all) compact open subgroups of \\(G(\\mathbb {A}_N^f)\\); \\(\\phi \\) is \\(\\mathcal{z}\\)-finite, where \\(\\mathcal{z}\\) is the centre of the universal enveloping algebra of the Lie algebra of \\(G(N_\\infty )\\), acting via differential operators. Equivalently \\(\\phi \\) is annihiliated by a finite index ideal of this centre, so morally \\(\\phi \\) satisfies lots of differential equations of a certain type; For all \\(g_f\\), the function \\(g_\\infty \\mapsto \\phi (g_f g\\infty )\\) is slowly-increasing in the sense above.","doc_url":null,"latex_link":"ch_bestiary.html#automorphic_form","lean_link":null,"lean_decl_name":"automorphic_form","kind_description":"definitions","status_from_color":"unknown","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"automorphic_form_actions","label":"automorphic_form_actions","background_color":"white","border_color":"#FFAA33","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_14.html","dependencies":["automorphic_form"],"nlp_text":"The group \\(G(\\mathbb {A}_N)\\) acts on itself on the right, and this induces a left action of its subgroup \\(G(\\mathbb {A}_N^f)\\times U_\\infty \\) on the spaces of automorphic forms and cusp forms. The Lie algebra \\(\\mathfrak {g}\\) of \\(G(N_\\infty )\\) also acts, via differential operators. Furthermore the actions of \\(\\mathfrak {g}\\) and \\(U_\\infty \\) are compatible in the sense that the differential of the \\(U_\\infty \\) action is the action of its Lie algebra considered as a subalgebra of \\(\\mathfrak {g}\\). We say that the spaces are \\((G(\\mathbb {A}_N^f)\\times U_\\infty ,\\mathfrak {g})\\)-modules.","doc_url":null,"latex_link":"ch_bestiary.html#automorphic_form_actions","lean_link":null,"lean_decl_name":"automorphic_form_actions","kind_description":"definitions","status_from_color":"unknown","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"cuspidal_automorphic_form","label":"cuspidal_automorphic_form","background_color":"white","border_color":"#FFAA33","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_14.html","dependencies":["automorphic_form"],"nlp_text":"An automorphic form is cuspidal (or “a cusp form”) if it furthermore satisfies \\(\\int _{U(N)\\backslash U(\\mathbb {A}_N)}\\phi (ux)du=0\\), where \\(P\\) runs through all the proper parabolic subgroups of \\(G\\) defined over \\(N\\) and \\(U\\) is the unipotent radical of \\(P\\), and the integral is with respect to the measure coming from Haar measure.","doc_url":null,"latex_link":"ch_bestiary.html#cuspidal_automorphic_form","lean_link":null,"lean_decl_name":"cuspidal_automorphic_form","kind_description":"definitions","status_from_color":"unknown","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"cuspidal_automorphic_form_decomposition","label":"cuspidal_automorphic_form_decomposition","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_14.html","dependencies":["automorphic_form_actions","cuspidal_automorphic_form"],"nlp_text":"The cusp forms decompose as a (typically infinite) direct sum of irreducible \\((G(\\mathbb {A}_N^f)\\times U_\\infty ,\\mathfrak {g})\\)-modules.","doc_url":null,"latex_link":"ch_bestiary.html#cuspidal_automorphic_form_decomposition","lean_link":null,"lean_decl_name":"cuspidal_automorphic_form_decomposition","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"automorphic_representation","label":"automorphic_representation","background_color":"white","border_color":"#FFAA33","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_14.html","dependencies":["automorphic_form_actions"],"nlp_text":"An automorphic representation is an irreducible \\((G(\\mathbb {A}_N^f)\\times U_\\infty ,\\mathfrak {g})\\)-module isomorphic to an irreducible subquotient of the space of automorphic forms.","doc_url":null,"latex_link":"ch_bestiary.html#automorphic_representation","lean_link":null,"lean_decl_name":"automorphic_representation","kind_description":"definitions","status_from_color":"unknown","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"automorphic_representation_local_decomposition","label":"automorphic_representation_local_decomposition","background_color":"white","border_color":"#FFAA33","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_14.html","dependencies":["automorphic_representation"],"nlp_text":"An irreducible admissible \\((G(\\mathbb {A}_N^f)\\times U_\\infty ,\\mathfrak {g})\\)-module is a restricted tensor product of irreducible representations \\(\\pi _v\\) of \\(G(N_v)\\) as \\(v\\) runs through the finite places of \\(N\\), tensored with a tensor product of irreducible \\((\\mathfrak {g}_v,U_{\\infty ,v})\\)-modules as \\(v\\) runs through the infinite places of \\(N\\). The representations \\(\\pi _v\\) are unramified for all but finitely many \\(v\\).","doc_url":null,"latex_link":"ch_bestiary.html#automorphic_representation_local_decomposition","lean_link":null,"lean_decl_name":"automorphic_representation_local_decomposition","kind_description":"theorems and lemmas","status_from_color":"unknown","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"local_Weil_group","label":"local_Weil_group","background_color":"white","border_color":"#FFAA33","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_14.html","dependencies":["maximal_unramified_extension_of_p-adic_field"],"nlp_text":"The topological group described above is called the Weil group of \\(K\\).","doc_url":null,"latex_link":"ch_bestiary.html#local_Weil_group","lean_link":null,"lean_decl_name":"local_Weil_group","kind_description":"definitions","status_from_color":"unknown","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"local_class_field_theory","label":"local_class_field_theory","background_color":"white","border_color":"#FFAA33","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_14.html","dependencies":["local_Weil_group"],"nlp_text":"If \\(K\\) is a finite extension of \\(\\mathbb {Q}_p\\) then there are two “canonical” isomorphisms of topological abelian groups, between \\(K^\\times \\) and the abelianisation of the Weil group of \\(K\\).","doc_url":null,"latex_link":"ch_bestiary.html#local_class_field_theory","lean_link":null,"lean_decl_name":"local_class_field_theory","kind_description":"theorems and lemmas","status_from_color":"unknown","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"global_class_field_theory","label":"global_class_field_theory","background_color":"white","border_color":"#FFAA33","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_14.html","dependencies":["local_class_field_theory"],"nlp_text":"If \\(N\\) is a finite extension of \\(\\mathbb {Q}\\) then there are two “canonical” isomorphisms of topological groups between the profinite abelian groups \\(\\pi _0(\\mathbb {A}_N^\\times /N^\\times )\\) and \\(\\operatorname{Gal}(\\overline{N}/N)^{\\operatorname{ab}}\\); one sends local uniformisers to arithmetic Frobenii and the other to geometric Frobenii; each of the global isomorphisms is compatible with the local isomorphisms above.","doc_url":null,"latex_link":"ch_bestiary.html#global_class_field_theory","lean_link":null,"lean_decl_name":"global_class_field_theory","kind_description":"theorems and lemmas","status_from_color":"unknown","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"topology_on_affine_variety_points","label":"topology_on_affine_variety_points","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_14.html","dependencies":[],"nlp_text":"If \\(X\\) is an affine scheme of finite type over \\(K\\), and if \\(R\\) is a \\(K\\)-algebra which is also a topological ring, then we define a topology on the \\(R\\)-points \\(X(R)\\) of \\(K\\) by embedding the \\(K\\)-algebra homomorphisms from \\(A\\) to \\(R\\) into the set-theoretic maps from \\(A\\) to \\(R\\) with its product topology, and giving it the subspace topology.","doc_url":null,"latex_link":"ch_bestiary.html#topology_on_affine_variety_points","lean_link":null,"lean_decl_name":"topology_on_affine_variety_points","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"topology_on_affine_variety_computation","label":"topology_on_affine_variety_computation","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_14.html","dependencies":["topology_on_affine_variety_points"],"nlp_text":"If \\(X\\) is as above and \\(X\\to \\mathbb {A}^n_K\\) is a closed immersion, then the induced map from \\(X(R)\\) with its topology as above to \\(R^n\\) is an embedding of topological spaces (that is, a homeomorphism onto its image).","doc_url":null,"latex_link":"ch_bestiary.html#topology_on_affine_variety_computation","lean_link":null,"lean_decl_name":"topology_on_affine_variety_computation","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"cuspidal_automorphic_representation","label":"cuspidal_automorphic_representation","background_color":"white","border_color":"#FFAA33","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_14.html","dependencies":["cuspidal_automorphic_form_decomposition"],"nlp_text":"A cuspidal automorphic representation is an irreducible \\((G(\\mathbb {A}_N^f)\\times U_\\infty ,\\mathfrak {g})\\)-module isomorphic to an irreducible summand of the space of cusp forms.","doc_url":null,"latex_link":"ch_bestiary.html#cuspidal_automorphic_representation","lean_link":null,"lean_decl_name":"cuspidal_automorphic_representation","kind_description":"definitions","status_from_color":"unknown","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"Galois_representation_from_automorphic_representation_on_GL_2_form","label":"Galois_representation_from_automorphic_representation_on_GL_2_form","background_color":"white","border_color":"#FFAA33","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_14.html","dependencies":["automorphic_representation","compatible_family","Shimura_varieties"],"nlp_text":"Given an automorphic representation \\(\\pi \\) for an inner form of \\(\\operatorname{GL}_2\\) over a totally real field and with reflex field \\(E\\), such that \\(\\pi \\) is weight 2 discrete series at every infinite place, there exists a compatible family of 2-dimensional Galois representations associated to \\(\\pi \\), with \\(S\\) being the places at which \\(\\pi \\) is ramified, and \\(F_{{\\mathfrak {p}}}(X)\\) being the monic polynomial with roots the two Satake parameters for \\(\\pi \\) at \\({\\mathfrak {p}}\\).","doc_url":null,"latex_link":"ch_bestiary.html#Galois_representation_from_automorphic_representation_on_GL_2_form","lean_link":null,"lean_decl_name":"Galois_representation_from_automorphic_representation_on_GL_2_form","kind_description":"theorems and lemmas","status_from_color":"unknown","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"maximal_unramified_extension_of_p-adic_field","label":"maximal_unramified_extension_of_p-adic_field","background_color":"white","border_color":"#FFAA33","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_14.html","dependencies":[],"nlp_text":"The maximal unramified extension \\(K^{un}\\) in a given algebraic closure of \\(K\\) is Galois over \\(K\\) with Galois group “canonically” isomorphic to \\(\\widehat{\\mathbb {Z}}\\) in two ways; one of these two isomorphisms identifies \\(1\\in \\widehat{\\mathbb {Z}}\\) with an arithmetic Frobenius (the endomorphism inducing \\(x\\mapsto x^q\\) on the residue field of \\(K^{un}\\), where \\(q\\) is the size of the residue field of \\(K\\)). The other identifies 1 with geometric Frobenius (defined to be the inverse of arithmetic Frobenius).","doc_url":null,"latex_link":"ch_bestiary.html#maximal_unramified_extension_of_p-adic_field","lean_link":null,"lean_decl_name":"maximal_unramified_extension_of_p-adic_field","kind_description":"theorems and lemmas","status_from_color":"unknown","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"local_galois_coh_euler_poincare","label":"local_galois_coh_euler_poincare","background_color":"white","border_color":"#FFAA33","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_14.html","dependencies":[],"nlp_text":"If \\(h^i(M)\\) denotes the order of \\(H^i(G_K,M)\\) then \\(h^0(M)-h^1(M)+h^2(M)=0\\).","doc_url":null,"latex_link":"ch_bestiary.html#local_galois_coh_euler_poincare","lean_link":null,"lean_decl_name":"local_galois_coh_euler_poincare","kind_description":"theorems and lemmas","status_from_color":"unknown","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"local_galois_coh_poincare","label":"local_galois_coh_poincare","background_color":"white","border_color":"#FFAA33","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_14.html","dependencies":[],"nlp_text":"If \\(\\mu =\\bigcup _{n\\geq 1}\\mu _n\\) and \\(M':=\\operatorname{Hom}(M,\\mu )\\) is the dual of \\(M\\) then for \\(0\\leq i\\leq 2\\) the cup product pairing \\(H^i(G_K,M)\\times H^{2-i}(G_K,M')\\to H^2(G_K,\\mu )=\\mathbb {Q}/\\mathbb {Z}\\) is perfect.","doc_url":null,"latex_link":"ch_bestiary.html#local_galois_coh_poincare","lean_link":null,"lean_decl_name":"local_galois_coh_poincare","kind_description":"theorems and lemmas","status_from_color":"unknown","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"compatible_family","label":"compatible_family","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_14.html","dependencies":[],"nlp_text":"Let \\(N\\) be a number field. A compatible family of \\(d\\)-dimensional Galois representations over \\(N\\) is a finite set of finite places \\(S\\) of \\(N\\), a number field \\(E\\), a monic degree \\(d\\) polynomial \\(F_{{\\mathfrak {p}}}(X)\\in E[X]\\) for each finite place \\({\\mathfrak {p}}\\) of \\(K\\) not in \\(S\\) and, for each prime number \\(\\ell \\) and field embedding \\(\\phi : E\\to \\overline{\\mathbb {Q}}_\\ell \\) (or essentially equivalently for each finite place of \\(E\\)), a continuous homomorphism \\(\\rho :\\operatorname{Gal}(K^{\\operatorname{sep}}/K)\\to \\operatorname{GL}_2(\\overline{\\mathbb {Q}}_\\ell )\\) unramified outside \\(S\\) and the primes of \\(K\\) above \\(\\ell \\), such that \\(\\rho (\\operatorname{Frob}_{\\mathfrak {p}})\\) has characteristic polynomial \\(P_\\pi (X)\\) if \\(\\pi \\) lies above a prime number \\(p\\not=\\ell \\) with \\(p\\not\\in S\\).","doc_url":null,"latex_link":"ch_bestiary.html#compatible_family","lean_link":null,"lean_decl_name":"compatible_family","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"local_galois_coh_dim_two","label":"local_galois_coh_dim_two","background_color":"white","border_color":"#FFAA33","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_14.html","dependencies":[],"nlp_text":"If \\(M\\) is torsion then \\(H^i(G_K,M)=0\\) if \\(i{\\gt}2\\).","doc_url":null,"latex_link":"ch_bestiary.html#local_galois_coh_dim_two","lean_link":null,"lean_decl_name":"local_galois_coh_dim_two","kind_description":"theorems and lemmas","status_from_color":"unknown","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"manifold_on_algebraic_variety_computation","label":"manifold_on_algebraic_variety_computation","background_color":"white","border_color":"#FFAA33","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_14.html","dependencies":["manifold_on_algebraic_variety_points"],"nlp_text":"If \\(X\\) is as in the previous definition and \\(X\\to \\mathbb {A}^n_K\\) is a closed immersion, then the induced map from \\(X(K)\\) with its manifold structure to \\(K^n\\) is an embedding of manifolds.","doc_url":null,"latex_link":"ch_bestiary.html#manifold_on_algebraic_variety_computation","lean_link":null,"lean_decl_name":"manifold_on_algebraic_variety_computation","kind_description":"theorems and lemmas","status_from_color":"unknown","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lie_group_from_algebraic_group","label":"lie_group_from_algebraic_group","background_color":"white","border_color":"#FFAA33","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_14.html","dependencies":["manifold_on_algebraic_variety_computation"],"nlp_text":"If \\(G\\) is an affine algebraic group of finite type over \\(K=\\mathbb {R}\\) or \\(\\mathbb {C}\\) then \\(G(K)\\) is naturally a real or complex Lie group.","doc_url":null,"latex_link":"ch_bestiary.html#lie_group_from_algebraic_group","lean_link":null,"lean_decl_name":"lie_group_from_algebraic_group","kind_description":"theorems and lemmas","status_from_color":"unknown","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"moret-bailly","label":"moret-bailly","background_color":"white","border_color":"#FFAA33","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_14.html","dependencies":[],"nlp_text":"Let \\(K^{\\operatorname{avoid}}/K\\) be a Galois extension of number fields. Suppose also that \\(S\\) is a finite set of places of \\(K\\). For \\(v\\in S\\) let \\(L_v/K_v\\) be a finite Galois extension. Suppose also that \\(T /K\\) is a smooth, geometrically connected curve and that for each \\(v\\in S\\) we are given a nonempty, \\(\\operatorname{Gal}(L_v/K_v)\\)-invariant, open subset \\(\\Omega _v\\subseteq (L_v)\\). Then there is a finite Galois extension \\(L/K\\) and a point \\(P ∈ T (L)\\) such that \\(L/K\\) is Galois and linearly disjoint from \\(K^{\\operatorname{avoid}}\\) over \\(K\\); if \\(v\\in S\\) and \\(w\\) is a prime of \\(L\\) above \\(v\\) then \\(L_w /K_v\\) is isomorphic to \\(L_v/K_v\\); and \\(P \\in \\Omega _v\\subseteq T (L_v) \\cong (L_w)\\) via one such \\(K_v\\)-algebra morphism (this makes sense as \\(\\Omega _v\\) is \\(\\operatorname{Gal}(L_v/K v)\\)-invariant).","doc_url":null,"latex_link":"ch_bestiary.html#moret-bailly","lean_link":null,"lean_decl_name":"moret-bailly","kind_description":"theorems and lemmas","status_from_color":"unknown","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"connected_reductive_group","label":"connected_reductive_group","background_color":"white","border_color":"#FFAA33","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_14.html","dependencies":[],"nlp_text":"An affine algebraic group \\(G\\) of finite type over a field \\(k\\) is said to be connected if it is connected as a scheme, and reductive if \\(G_{\\overline{k}}\\) has no nontrivial smooth connected unipotent normal \\(k\\)-subgroup.","doc_url":null,"latex_link":"ch_bestiary.html#connected_reductive_group","lean_link":null,"lean_decl_name":"connected_reductive_group","kind_description":"definitions","status_from_color":"unknown","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"manifold_on_algebraic_variety_points","label":"manifold_on_algebraic_variety_points","background_color":"white","border_color":"#FFAA33","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_14.html","dependencies":[],"nlp_text":"Let \\(K\\) be a field equipped with an isomorphism to the reals, complexes, or a finite extension of the \\(p\\)-adic numbers. Let \\(X\\) be a smooth affine algebraic variety over \\(K\\). Then the points \\(X(K)\\) naturally inherit the structure of a manifold over \\(K\\).","doc_url":null,"latex_link":"ch_bestiary.html#manifold_on_algebraic_variety_points","lean_link":null,"lean_decl_name":"manifold_on_algebraic_variety_points","kind_description":"definitions","status_from_color":"unknown","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"slowly_increasing","label":"slowly_increasing","background_color":"white","border_color":"#FFAA33","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_14.html","dependencies":[],"nlp_text":"A function \\(f : G(N_\\infty )\\to \\mathbb {C}\\) is slowly-increasing if there exists some \\(C{\\gt}0\\) and \\(n\\geq 1\\) such that \\(|f(x)\\leq C||x||_\\rho ^n\\).","doc_url":null,"latex_link":"ch_bestiary.html#slowly_increasing","lean_link":null,"lean_decl_name":"slowly_increasing","kind_description":"definitions","status_from_color":"unknown","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"Shimura_varieties","label":"Shimura_varieties","background_color":"white","border_color":"#FFAA33","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_14.html","dependencies":[],"nlp_text":"We need the definition of (the canonical model over \\(F\\) of) the Shimura curve attached to an inner form of \\(\\operatorname{GL}_2\\) with precisely one split infinite place, and the same for the Shimura surface associated to an inner form split at two infinite places (and ramified elsewhere, so it’s compact).","doc_url":null,"latex_link":"ch_bestiary.html#Shimura_varieties","lean_link":null,"lean_decl_name":"Shimura_varieties","kind_description":"definitions","status_from_color":"unknown","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"local_galois_coh_finite","label":"local_galois_coh_finite","background_color":"white","border_color":"#FFAA33","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_14.html","dependencies":[],"nlp_text":"If \\(M\\) is finite then the cohomology groups \\(H^i(G_K,M)\\) all finite.","doc_url":null,"latex_link":"ch_bestiary.html#local_galois_coh_finite","lean_link":null,"lean_decl_name":"local_galois_coh_finite","kind_description":"theorems and lemmas","status_from_color":"unknown","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}

crawled_repos/FLT_ch2.jsonl

@@ -0,0 +1,10 @@
+{"id":"FreyPackage","label":"FreyPackage","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_2.html","dependencies":[],"nlp_text":"A Frey package \\((a,b,c,p)\\) is three nonzero pairwise-coprime integers \\(a\\), \\(b\\), \\(c\\), with \\(a\\equiv 3\\pmod4\\) and \\(b\\equiv 0\\pmod2\\), and a prime \\(p\\geq 5\\), such that \\(a^p+b^p=c^p\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FreyPackage","latex_link":"ch_reductions.html#FreyPackage","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FreyPackage","lean_decl_name":"FreyPackage","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch2","source_path":"./FLT/Basic/FreyPackage.lean","lean4_code":"structure FreyPackage where\n  /-- The integer `a` in the Frey package. -/\n  a : ℤ\n  /-- The integer `b` in the Frey package. -/\n  b : ℤ\n  /-- The integer `c` in the Frey package. -/\n  c : ℤ\n  ha0 : a ≠ 0\n  hb0 : b ≠ 0\n  hc0 : c ≠ 0\n  /-- The prime number `p` in the Frey package. -/\n  p : ℕ\n  pp : Nat.Prime p\n  hp5 : 5 ≤ p\n  hFLT : a ^ p + b ^ p = c ^ p\n  hgcdab : gcd a b = 1 -- same as saying a,b,c pairwise coprime\n  ha4 : (a : ZMod 4) = 3\n  hb2 : (b : ZMod 2) = 0\n\nnamespace FreyPackage\n\n","line":80,"end_line":100,"proof_status":"formalized"}
+{"id":"FreyPackage.of_not_FermatLastTheorem_p_ge_5","label":"FreyPackage.of_not_FermatLastTheorem_p_ge_5","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_2.html","dependencies":["FreyPackage"],"nlp_text":"If Fermat’s Last Theorem is false for \\(p \\ge 5\\) and prime, then there exists a Frey package.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FreyPackage.of_not_FermatLastTheorem_p_ge_5","latex_link":"ch_reductions.html#FreyPackage.of_not_FermatLastTheorem_p_ge_5","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FreyPackage.of_not_FermatLastTheorem_p_ge_5","lean_decl_name":"FreyPackage.of_not_FermatLastTheorem_p_ge_5","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch2","source_path":"FLT/Basic/FreyPackage.lean","lean4_code":"/-- Given a counterexample a^p+b^p=c^p to Fermat's Last Theorem with p>=5,\nthere exists a Frey package. -/\nlemma of_not_FermatLastTheorem_p_ge_5 {a b c : ℤ} (ha : a ≠ 0) (hb : b ≠ 0) (hc : c ≠ 0)\n    {p : ℕ} (pp : p.Prime) (hp5 : 5 ≤ p) (H : a ^ p + b ^ p = c ^ p) : Nonempty FreyPackage := by\n  have p_odd := pp.odd_of_ne_two (by omega)\n  -- First, show that we can make a,b coprime by dividing through by gcd a b\n  have ⟨a, b, c, a0, b0, c0, ab, H⟩ :\n      ∃ (a b c : ℤ), a ≠ 0 ∧ b ≠ 0 ∧ c ≠ 0 ∧ Int.gcd a b = 1 ∧ a^p + b^p = c^p := by\n    obtain ⟨d, a', b', d0, cop, a_eq, b_eq⟩ :=\n      Int.exists_gcd_one' (Int.gcd_pos_of_ne_zero_left b ha)\n    simp only [a_eq, mul_pow, b_eq] at H\n    rw [← add_mul, mul_comm] at H\n    obtain ⟨c', rfl⟩ := (Int.pow_dvd_pow_iff pp.ne_zero).1 ⟨_, H.symm⟩\n    rw [mul_pow] at H\n    have a0' := left_ne_zero_of_mul (a_eq ▸ ha)\n    have b0' := left_ne_zero_of_mul (b_eq ▸ hb)\n    have c0' := right_ne_zero_of_mul hc\n    exact ⟨a', b', c', a0', b0', c0', cop, mul_left_cancel₀ (pow_ne_zero _ (mod_cast d0.ne')) H⟩\n  -- Then show that WLOG we can take b to be even,\n  -- because at least one of a,b,c is even and we can permute if needed\n  have ⟨a, b, c, a0, b0, c0, ab, eb, H⟩ :\n      ∃ (a b c : ℤ), a ≠ 0 ∧ b ≠ 0 ∧ c ≠ 0 ∧ Int.gcd a b = 1 ∧ Even b ∧ a^p + b^p = c^p := by\n    if eb : Even b then\n      exact ⟨a, b, c, a0, b0, c0, ab, eb, H⟩\n    else if ea : Even a then\n      exact ⟨b, a, c, b0, a0, c0, Int.gcd_comm a b ▸ ab, ea, by rwa [add_comm]⟩\n    else\n      refine ⟨a, -c, -b, a0, neg_ne_zero.2 c0, neg_ne_zero.2 b0, ?_, even_neg.2 ?_, ?_⟩\n      · refine Int.gcd_neg.trans (.trans (.symm ?_) ab)\n        exact Nat.cast_inj.1 (gcdab_eq_gcdac pp.pos H)\n      · refine ((Int.even_pow (n := p)).1 (H.symm ▸ Int.even_add.2 (iff_of_false ?_ ?_))).1\n        · exact fun h => ea (Int.even_pow.1 h).1\n        · exact fun h => eb (Int.even_pow.1 h).1\n      · simp [p_odd.neg_pow, ← H]\n  -- We can ensure additionally that a ≡ 3 [ZMOD 4] by negating everything if necessary\n  have ⟨a, b, c, ha0, hb0, hc0, ab, ha3, eb, hFLT⟩ :\n      ∃ (a b c : ℤ), a ≠ 0 ∧ b ≠ 0 ∧ c ≠ 0 ∧ Int.gcd a b = 1 ∧\n        a ≡ 3 [ZMOD 4] ∧ Even b ∧ a^p + b^p = c^p := by\n    -- Since b is even, a cannot also be even\n    have a_odd' : ∀ {i}, a ≡ i [ZMOD 4] → ¬2 ∣ i := fun ai ei => by\n      have ea := (dvd_sub_right ei).1 (.trans (by decide) (Int.modEq_iff_dvd.1 ai))\n      simpa (config := {decide := true}) [gcd, ab] using dvd_gcd ea (even_iff_two_dvd.1 eb)\n    mod_cases a_mod : a % 4\n    · cases a_odd' a_mod (by decide)\n    · exact ⟨-a, -b, -c, neg_ne_zero.2 a0, neg_ne_zero.2 b0, neg_ne_zero.2 c0,\n        by rwa [Int.neg_gcd, Int.gcd_neg], a_mod.neg, eb.neg,\n        by simp [p_odd.neg_pow, ← H, add_comm]⟩\n    · cases a_odd' a_mod (by decide)\n    · exact ⟨a, b, c, a0, b0, c0, ab, a_mod, eb, H⟩\n  -- Build the Frey package from the assumptions\n  exact ⟨{\n    a, b, c, ha0, hb0, hc0, p, pp, hp5, hFLT\n    hgcdab := by simp [gcd, ab]\n    ha4 := (ZMod.intCast_eq_intCast_iff ..).2 ha3\n    hb2 := (ZMod.intCast_zmod_eq_zero_iff_dvd ..).2 (even_iff_two_dvd.1 eb)\n  }⟩\n","line":154,"end_line":209,"proof_status":"formalized"}
+{"id":"FreyCurve","label":"FreyCurve","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_2.html","dependencies":["FreyPackage"],"nlp_text":"Given a Frey package \\((a,b,c,p)\\), the corresponding Frey curve (considered by Frey and, before him, Hellegouarch) is the elliptic curve over \\(\\mathbb {Q}\\) defined by the equation \\(Y^2=X(X-a^p)(X+b^p).\\)","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FreyPackage.freyCurve","latex_link":"ch_reductions.html#FreyCurve","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FreyPackage.freyCurve","lean_decl_name":"FreyPackage.freyCurve","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch2","source_path":"FLT/Basic/FreyPackage.lean","lean4_code":"/-- The elliptic curve over `ℚ` associated to a Frey package. The conditions imposed\nupon a Frey package guarantee that the running hypotheses in\nSection 4.1 of [Serre] all hold. We put the curve into the form where the\nequation is semistable at 2, rather than the usual `Y^2=X(X-a^p)(X+b^p)` form.\nThe change of variables is `X=4x` and `Y=8y+4x`, and then divide through by 64. -/\ndef freyCurve (P : FreyPackage) : WeierstrassCurve ℚ where\n  a₁ := 1\n  -- a₂ is an integer because of the congruences assumed e.g. P.ha4\n  a₂ := (P.b ^ P.p - 1 - P.a ^ P.p) / 4\n  a₃ := 0\n  a₄ := -(P.a ^ P.p) * (P.b ^ P.p) / 16 -- this is also an integer\n  a₆ := 0\n","line":232,"end_line":243,"proof_status":"formalized"}
+{"id":"FLT","label":"FLT","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_2.html","dependencies":["FreyPackage.of_not_FermatLastTheorem_p_ge_5","FreyPackage.false","FermatLastTheorem.of_p_ge_5"],"nlp_text":"Fermat’s Last Theorem is true. In other words, there are no positive integers \\(a,b,c\\) and natural \\(n\\geq 3\\) such that \\(a^n+b^n=c^n\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Wiles_Taylor_Wiles","latex_link":"ch_reductions.html#FLT","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Wiles_Taylor_Wiles","lean_decl_name":"Wiles_Taylor_Wiles","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch2","source_path":"FLT/Basic/Reductions.lean","lean4_code":"/-- Fermat's Last Theorem is true -/\ntheorem Wiles_Taylor_Wiles : FermatLastTheorem := by\n  -- Suppose Fermat's Last Theorem is false\n  by_contra h\n  -- then there exists a Frey package\n  obtain ⟨P : FreyPackage⟩ := FreyPackage.of_not_FermatLastTheorem h\n  -- but there is no Frey package\n  exact P.false\n","line":68,"end_line":75,"proof_status":"formalized"}
+{"id":"Wiles_Frey","label":"Wiles_Frey","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_2.html","dependencies":["FreyCurve"],"nlp_text":"If \\(\\rho \\) is the mod \\(p\\) Galois representation associated to a Frey package \\((a,b,c,p)\\) then \\(\\rho \\) is reducible.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Wiles_Frey","latex_link":"ch_reductions.html#Wiles_Frey","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Wiles_Frey","lean_decl_name":"Wiles_Frey","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch2","source_path":"FLT/Basic/Reductions.lean","lean4_code":"theorem Wiles_Frey (P : FreyPackage) :\n    haveI : Fact P.p.Prime := ⟨P.pp⟩\n    ¬ GaloisRep.IsIrreducible (P.freyCurve.galoisRep P.p P.hppos) :=\n  FreyCurve.torsion_not_isIrreducible P\n","line":47,"end_line":50,"proof_status":"statement_formalized"}
+{"id":"Mazur_Frey","label":"Mazur_Frey","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_2.html","dependencies":["FreyCurve"],"nlp_text":"If \\(\\rho \\) is the mod \\(p\\) Galois representation associated to a Frey package \\((a,b,c,p)\\) then \\(\\rho \\) is irreducible.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Mazur_Frey","latex_link":"ch_reductions.html#Mazur_Frey","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Mazur_Frey","lean_decl_name":"Mazur_Frey","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch2","source_path":"FLT/Basic/Reductions.lean","lean4_code":"theorem Mazur_Frey (P : FreyPackage) :\n    haveI : Fact P.p.Prime := ⟨P.pp⟩\n    GaloisRep.IsIrreducible (P.freyCurve.galoisRep P.p P.hppos) :=\n  sorry\n","line":35,"end_line":38,"proof_status":"statement_formalized"}
+{"id":"FreyPackage.false","label":"FreyPackage.false","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_2.html","dependencies":["Wiles_Frey","Mazur_Frey"],"nlp_text":"There is no Frey package.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FreyPackage.false","latex_link":"ch_reductions.html#FreyPackage.false","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FreyPackage.false","lean_decl_name":"FreyPackage.false","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Tactic/ITauto.lean","lean4_code":"@[match_pattern] def IProp.not (a : IProp) : IProp := a.imp .false\n\n/-- Constructor for `xor p q`. -/\n","line":115,"end_line":117,"proof_status":"formalized"}
+{"id":"FermatLastTheorem.of_odd_primes","label":"FermatLastTheorem.of_odd_primes","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_2.html","dependencies":[],"nlp_text":"If there is a counterexample to Fermat’s Last Theorem, then there is a counterexample \\(a^p+b^p=c^p\\) with \\(p\\) an odd prime.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FermatLastTheorem.of_odd_primes","latex_link":"ch_reductions.html#FermatLastTheorem.of_odd_primes","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FermatLastTheorem.of_odd_primes","lean_decl_name":"FermatLastTheorem.of_odd_primes","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/leanprover-community/mathlib4/98379c78c5dd2e3614cd24ab7ad22f5713e0eb63/Mathlib/NumberTheory/FLT/Four.lean","lean4_code":"/--\nTo prove Fermat's Last Theorem, it suffices to prove it for odd prime exponents.\n-/\ntheorem FermatLastTheorem.of_odd_primes\n    (hprimes : ∀ p : ℕ, Nat.Prime p → Odd p → FermatLastTheoremFor p) : FermatLastTheorem := by\n  intro n h\n  obtain hdvd | ⟨p, hpprime, hdvd, hpodd⟩ := Nat.four_dvd_or_exists_odd_prime_and_dvd_of_two_lt h\n    <;> apply FermatLastTheoremWith.mono hdvd\n  · exact fermatLastTheoremFour\n  · exact hprimes p hpprime hpodd","line":278,"end_line":287,"proof_status":"formalized"}
+{"id":"FermatLastTheorem.of_p_ge_5","label":"FermatLastTheorem.of_p_ge_5","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_2.html","dependencies":["FermatLastTheorem.of_odd_primes","fermatLastTheoremThree"],"nlp_text":"If there is a counterexample to Fermat’s Last Theorem, then there is a counterexample \\(a^p+b^p=c^p\\) with \\(p\\) prime and \\(p\\geq 5\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FermatLastTheorem.of_p_ge_5","latex_link":"ch_reductions.html#FermatLastTheorem.of_p_ge_5","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FermatLastTheorem.of_p_ge_5","lean_decl_name":"FermatLastTheorem.of_p_ge_5","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/Basic/FreyPackage.lean","lean4_code":"/-- If Fermat's Last Theorem is true for primes `p ≥ 5`, then FLT is true. -/\nlemma FermatLastTheorem.of_p_ge_5 (H : ∀ p ≥ 5, p.Prime → FermatLastTheoremFor p) :\n    FermatLastTheorem := by\n  apply FermatLastTheorem.of_odd_primes -- this is Fermat's proof for n=4, plus reduction to\n                                        -- the case n prime.\n  intro p pp p_odd\n  if hp5 : 5 ≤ p then\n    exact H _ hp5 pp\n  else\n    have hp2 := pp.two_le\n    interval_cases p\n    · contradiction\n    · exact fermatLastTheoremThree -- this is Euler's proof for n=3\n    · contradiction","line":58,"end_line":71,"proof_status":"formalized"}
+{"id":"fermatLastTheoremThree","label":"fermatLastTheoremThree","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_2.html","dependencies":[],"nlp_text":"There are no solutions in positive integers to \\(a^3+b^3=c^3\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/fermatLastTheoremThree","latex_link":"ch_reductions.html#fermatLastTheoremThree","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/fermatLastTheoremThree","lean_decl_name":"fermatLastTheoremThree","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/NumberTheory/FLT/Three.lean","lean4_code":"theorem fermatLastTheoremThree : FermatLastTheoremFor 3 := by\n  classical\n  let K := CyclotomicField 3 ℚ\n  let hζ := IsCyclotomicExtension.zeta_spec 3 ℚ K\n  have : NumberField K := IsCyclotomicExtension.numberField {3} ℚ _\n  apply FermatLastTheoremForThree_of_FermatLastTheoremThreeGen hζ\n  intro a b c u hc ha hb hcdvd coprime H\n  let S' : FermatLastTheoremForThreeGen.Solution' hζ :=\n  { a := a\n    b := b\n    c := c\n    u := u\n    ha := ha\n    hb := hb\n    hc := hc\n    coprime := coprime\n    hcdvd := hcdvd\n    H := H }\n  obtain ⟨S, -⟩ := FermatLastTheoremForThreeGen.exists_Solution_of_Solution' S'\n  obtain ⟨Smin, hSmin⟩ := S.exists_minimal\n  obtain ⟨Sfin, hSfin⟩ := Smin.exists_Solution_multiplicity_lt\n  linarith [hSmin Sfin]\n","line":743,"end_line":764,"proof_status":"formalized"}

crawled_repos/FLT_ch3.jsonl

@@ -0,0 +1,8 @@
+{"id":"hardly_ramified_reducible","label":"hardly_ramified_reducible","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_3.html","dependencies":["hardly_ramified_lifts","hardly_ramified_spreads_out","hardly_ramified_3adic_reducible"],"nlp_text":"If \\(\\ell \\geq 3\\) is a prime and \\(\\rho :\\operatorname{Gal}(\\overline{\\mathbb {Q}}/\\mathbb {Q})\\to \\operatorname{GL}_2(\\mathbb {Z}/\\ell \\mathbb {Z})\\) is hardly ramified, then \\(\\rho \\) is reducible.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FreyCurve.torsion_not_isIrreducible","latex_link":"ch_freyreduction.html#hardly_ramified_reducible","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FreyCurve.torsion_not_isIrreducible","lean_decl_name":"FreyCurve.torsion_not_isIrreducible","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/GaloisRepresentation/HardlyRamified/Frey.lean","lean4_code":"theorem FreyCurve.torsion_not_isIrreducible :\n    haveI : Fact (P.p.Prime) := ⟨P.pp⟩\n    ¬ GaloisRep.IsIrreducible (P.freyCurve.galoisRep P.p P.hppos) :=\n  sorry -- TODO prove this","line":22,"end_line":25,"proof_status":"statement_formalized"}
+{"id":"Wiles_Frey_again","label":"Wiles_Frey_again","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_3.html","dependencies":["hardly_ramified_reducible","Frey_curve_hardly_ramified"],"nlp_text":"If \\(\\overline{\\rho }\\) is the mod \\(p\\) Galois representation associated to a Frey package \\((a,b,c,p)\\) then \\(\\overline{\\rho }\\) is reducible.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Wiles_Frey","latex_link":"ch_freyreduction.html#Wiles_Frey_again","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Wiles_Frey","lean_decl_name":"Wiles_Frey","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch3","source_path":"FLT/Basic/Reductions.lean","lean4_code":"theorem Wiles_Frey (P : FreyPackage) :\n    haveI : Fact P.p.Prime := ⟨P.pp⟩\n    ¬ GaloisRep.IsIrreducible (P.freyCurve.galoisRep P.p P.hppos) :=\n  FreyCurve.torsion_not_isIrreducible P\n","line":47,"end_line":50,"proof_status":"statement_formalized"}
+{"id":"hardly_ramified_lifts","label":"hardly_ramified_lifts","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_3.html","dependencies":["hardly_ramified"],"nlp_text":"If \\(\\ell \\geq 3\\) is prime and \\(\\overline{\\rho }:\\operatorname{Gal}(\\overline{\\mathbb {Q}}/\\mathbb {Q})\\to \\operatorname{GL}_2(\\mathbb {Z}/\\ell \\mathbb {Z})\\) is hardly ramified and irreducible, then there exists a finite extension \\(K\\) of \\(\\mathbb {Q}_\\ell \\) with integer ring \\(\\mathcal{O}\\) and maximal ideal \\(\\mathfrak {m}\\) and a hardly ramified representation \\(\\rho :\\operatorname{Gal}(\\overline{\\mathbb {Q}}/\\mathbb {Q})\\to \\operatorname{GL}_2(\\mathcal{O})\\) whose reduction modulo \\(\\mathfrak {m}\\) is isomorphic to \\(\\rho \\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/GaloisRepresentation.IsHardlyRamified.lifts","latex_link":"ch_freyreduction.html#hardly_ramified_lifts","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/GaloisRepresentation.IsHardlyRamified.lifts","lean_decl_name":"GaloisRepresentation.IsHardlyRamified.lifts","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch3","source_path":"FLT/GaloisRepresentation/HardlyRamified/Lift.lean","lean4_code":"/--\nAn irreducible mod p hardly ramified representation lifts to a p-adic one.\n-/\ntheorem lifts (ρ : GaloisRep ℚ k V) (hρirred : ρ.IsIrreducible)\n    (hρ : IsHardlyRamified hpodd hV ρ) :\n    ∃ (R : Type u) (_ : CommRing R) (_ : IsLocalRing R)\n      (_ : TopologicalSpace R) (_ : IsTopologicalRing R)\n      (_ : Algebra ℤ_[p] R) (_ : IsLocalHom (algebraMap ℤ_[p] R))\n      (_ : Module.Finite ℤ_[p] R) (_ : Module.Free ℤ_[p] R)\n      (_ : IsModuleTopology ℤ_[p] R)\n      (_ : Algebra R k) (_ : IsScalarTower ℤ_[p] R k) (_ : ContinuousSMul R k)\n      (W : Type v) (_ : AddCommGroup W) (_ : Module R W) (_ : Module.Finite R W)\n      (_ : Module.Free R W) (hW : Module.rank R W = 2)\n      (σ : GaloisRep ℚ R W) (r : k ⊗[R] W ≃ₗ[k] V),\n    IsHardlyRamified hpodd hW σ ∧ (σ.baseChange k).conj r = ρ := sorry\n","line":17,"end_line":31,"proof_status":"statement_formalized"}
+{"id":"hardly_ramified","label":"hardly_ramified","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_3.html","dependencies":[],"nlp_text":"Let \\(R\\) be a coefficient ring with finite residue field of characteristic \\(\\ell \\geq 3\\). Let \\(V\\) be a finite free \\(R\\)-module of rank 2, equipped with the product topology. A continuous representation \\(\\rho : \\operatorname{Gal}(\\overline{\\mathbb {Q}}/\\mathbb {Q})\\to \\operatorname{GL}_R(V)\\) is said to be hardly ramified if it satisfies the following four conditions: \\(\\det (\\rho ):\\operatorname{Gal}(\\overline{\\mathbb {Q}}/\\mathbb {Q})\\to R^\\times \\) is the cyclotomic character; \\(\\rho \\) is unramified outside \\(2\\ell \\); The restriction of \\(\\rho \\) to \\(\\operatorname{Gal}(\\overline{\\mathbb {Q}}_2/\\mathbb {Q}_2)\\) is reducible (more precisely, there is a short exact sequence \\(0\\to R\\to V\\to R\\to 0\\) which is stable under the action of \\(\\operatorname{Gal}(\\overline{\\mathbb {Q}}_2/\\mathbb {Q}_2)\\)) and the Galois action on the 1-dimensional quotient is an unramified representation of \\(\\operatorname{Gal}(\\overline{\\mathbb {Q}}_2/\\mathbb {Q}_2)\\) whose square is trivial; The restriction of \\(\\rho \\) to \\(\\operatorname{Gal}(\\overline{\\mathbb {Q}}_\\ell /\\mathbb {Q}_\\ell )\\) is flat, by which we mean that for all open ideals \\(I\\) of \\(R\\), the (finite image) representation \\(\\rho \\) mod \\(I:\\operatorname{Gal}(\\overline{\\mathbb {Q}}_\\ell /\\mathbb {Q}_\\ell )\\to \\operatorname{GL}_{R/I}(V/I)\\) comes from a finite flat group scheme.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/GaloisRepresentation.IsHardlyRamified","latex_link":"ch_freyreduction.html#hardly_ramified","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/GaloisRepresentation.IsHardlyRamified","lean_decl_name":"GaloisRepresentation.IsHardlyRamified","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch3","source_path":"FLT/GaloisRepresentation/HardlyRamified/Defs.lean","lean4_code":"/-- Let `R` be a compact Hausdorff local topological ring (for example any complete Noetherian\nlocal ring with the maximal ideal-adic topology) having finite residue field of\ncharacteristic `ℓ > 2`, and let `ρ : Gal(Qbar/Q) → GL_2(R)` be a continuous 2-dimensional\nrepresentation. We say that `ρ` is *hardly ramified* if it has cyclotomic determinant, is\nunramified outside `2ℓ`, flat at `ℓ` and upper-triangular at 2 with a 1-dimensional quotient which\nis unramified and whose square is trivial. -/\nstructure IsHardlyRamified {ℓ : ℕ} [Fact ℓ.Prime] (hℓOdd : Odd ℓ)\n    {R : Type u} [CommRing R] [TopologicalSpace R] [IsTopologicalRing R] [IsLocalRing R]\n    [Algebra ℤ_[ℓ] R] --[IsLocalHom (algebraMap ℤ_[ℓ] R)] -- a convenient way of saying \"residue\n    -- field has char ell\"\n    -- Rather than GL_2(R) we use the automorphisms of a finite free rank 2 `R`-module `V`.\n    {V : Type*} [AddCommGroup V] [Module R V]\n    [Module.Finite R V] [Module.Free R V] (hdim : Module.rank R V = 2)\n  -- Let `ρ` be a continuous action of the absolute Galois group of `ℚ` on `V`.\n    (ρ : GaloisRep ℚ R V) : Prop where\n  -- We say `ρ` is *hardly ramified* if\n  -- `det(ρ)` is the ell-adic cyclotomic character;\n  det : ∀ g, ρ.det g = algebraMap ℤ_[ℓ] R (cyclotomicCharacter (ℚ ᵃˡᵍ) ℓ g.toRingEquiv)\n  -- `ρ` is unramified outside `2` and `ℓ`;\n  isUnramified : ∀ p (hp : p.Prime), p ≠ 2 ∧ p ≠ ℓ →\n    ρ.IsUnramifiedAt hp.toHeightOneSpectrumRingOfIntegersRat\n  -- ρ is flat at ℓ;\n  isFlat : ρ.IsFlatAt (Nat.Prime.toHeightOneSpectrumRingOfIntegersRat (Fact.out : ℓ.Prime))\n  -- and ρ has a 1-dimensional quotient π : ρ → δ such that\n  isTameAtTwo : ∃ (π : V →ₗ[R] R) (_ : Function.Surjective π) (δ : GaloisRep ℚ_[2] R R),\n    ∀ g : Γ ℚ_[2], ∀ v : V, π (ρ.map (algebraMap ℚ ℚ_[2]) g v) = δ g (π v) ∧\n    -- δ is unramified and\n    (AddSubgroup.inertia ((𝔪 Z2bar).toAddSubgroup : AddSubgroup Z2bar) (Γ ℚ_[2]) ≤ δ.ker) ∧\n    -- δ² = 1.\n    (∀ g : Γ ℚ_[2], δ g * δ g = 1)\n","line":80,"end_line":109,"proof_status":"formalized"}
+{"id":"Frey_curve_hardly_ramified","label":"Frey_curve_hardly_ramified","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_3.html","dependencies":["hardly_ramified"],"nlp_text":"The \\(\\ell \\)-torsion \\(\\rho :\\operatorname{Gal}(\\overline{\\mathbb {Q}}/\\mathbb {Q})\\to \\operatorname{GL}_2(\\mathbb {Z}/\\ell \\mathbb {Z})\\) in the Frey curve associated to a Frey package \\((a,b,c,\\ell )\\) is hardly ramified.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FreyCurve.torsion_isHardlyRamified","latex_link":"ch_freyreduction.html#Frey_curve_hardly_ramified","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FreyCurve.torsion_isHardlyRamified","lean_decl_name":"FreyCurve.torsion_isHardlyRamified","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/GaloisRepresentation/HardlyRamified/Frey.lean","lean4_code":"theorem FreyCurve.torsion_isHardlyRamified :\n    haveI : Fact (P.p.Prime) := ⟨P.pp⟩\n    IsHardlyRamified P.hp_odd sorry\n      (P.freyCurve.galoisRep P.p (show 0 < P.p from P.hppos)) :=\n  sorry","line":16,"end_line":20,"proof_status":"statement_formalized"}
+{"id":"hardly_ramified_spreads_out","label":"hardly_ramified_spreads_out","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_3.html","dependencies":["hardly_ramified"],"nlp_text":"If \\(\\ell \\geq 3\\) is prime, \\(K\\) is a finite extension of \\(\\mathbb {Q}_\\ell \\) with integers \\(\\mathcal{O}\\) and if \\(\\rho :\\operatorname{Gal}(\\overline{\\mathbb {Q}}/\\mathbb {Q})\\to \\operatorname{GL}_2(\\mathcal{O})\\) is a hardly ramified representation whose reduction is irreducible, then there exists a number field \\(M\\) and, for each finite place \\(\\mu \\) of \\(M\\) of characteristic prime to 2, with completion \\(M_\\mu \\) having integer ring \\(R_\\mu \\), a hardly ramified semisimple representation \\(\\rho _\\mu :\\operatorname{Gal}(\\overline{\\mathbb {Q}}/\\mathbb {Q})\\to \\operatorname{GL}_2(R_\\mu )\\) (by which we mean the generic fibre is semisimple), with the following properties: There is some \\(\\lambda \\mid \\ell \\) of \\(M\\) such that \\(\\rho _\\lambda \\cong \\rho \\), the isomorphism happening over some appropriate local field containing a copy of \\(M_\\lambda \\) and a copy of&#160;\\(K\\); If \\(\\mu _1\\) and \\(\\mu _2\\) are two finite places of \\(M\\) with odd residue characteristics \\(m_1\\) and \\(m_2\\), and if \\(p\\nmid 2m_1m_2\\) is prime, then \\(\\rho _{\\mu _1}\\) and \\(\\rho _{\\mu _2}\\) are both unramified at&#160;\\(p\\) and the characteristic polynomials \\(\\rho _{\\mu _1}(\\operatorname{Frob}_p)\\) and \\(\\rho _{\\mu _2}(\\operatorname{Frob}_p)\\) lie in \\(M[X]\\) and are equal.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/GaloisRepresentation.IsHardlyRamified.mem_isCompatible","latex_link":"ch_freyreduction.html#hardly_ramified_spreads_out","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/GaloisRepresentation.IsHardlyRamified.mem_isCompatible","lean_decl_name":"GaloisRepresentation.IsHardlyRamified.mem_isCompatible","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch3","source_path":"FLT/GaloisRepresentation/HardlyRamified/Family.lean","lean4_code":"theorem mem_isCompatible (hρ : IsHardlyRamified hpodd hv ρ) :\n    -- Then `ρ` lives in a compatible family of Galois representations\n    -- i.e., there's a family σ of 2-dimensional representations of Γ_ℚ\n    -- parametrised by maps from a number field M → ℚ_p-bar\n    ∃ (E : Type v) (_ : Field E) (_ : NumberField E) (σ : GaloisRepFamily ℚ E 2),\n    -- which are compatible, and\n    σ.isCompatible ∧\n    -- are \"hardly ramified\" for ℓ>2,\n    (∀ {ℓ : ℕ} (hℓ : Fact ℓ.Prime) (hℓodd : Odd ℓ) (φ : E →+* AlgebraicClosure ℚ_[ℓ]),\n      -- by which we mean that for a representation σ_φ in the family,\n      -- there's a hardly-ramified representation `τ` to GL_2(A)\n      -- for A a module-finite free ℤ_ℓ-algebra\n      ∃ (A : Type u) (_ : CommRing A) (_ : TopologicalSpace A) (_ : IsTopologicalRing A)\n        (_ : IsLocalRing A) (_ : Algebra ℤ_[ℓ] A) (_ : Module.Finite ℤ_[ℓ] A)\n        (_ : Module.Free ℤ_[ℓ] A) (_ : IsDomain A) (_ : Algebra A (AlgebraicClosure ℚ_[ℓ]))\n        (_ : IsScalarTower ℤ_[ℓ] A (AlgebraicClosure ℚ_[ℓ])) (_ : IsModuleTopology ℤ_[ℓ] A)\n        (_ : ContinuousSMul A (AlgebraicClosure ℚ_[ℓ]))\n        (W : Type v) (_ : AddCommGroup W) (_ : Module A W) (_ : Module.Finite A W)\n        (_ : Module.Free A W) (hW : Module.rank A W = 2)\n        (τ : GaloisRep ℚ A W)\n        (r : AlgebraicClosure ℚ_[ℓ] ⊗[A] W ≃ₗ[AlgebraicClosure ℚ_[ℓ]]\n          Fin 2 → AlgebraicClosure ℚ_[ℓ]),\n        IsHardlyRamified hℓodd hW τ ∧\n        -- whose base extension to GL_2(ℚ_p-bar) is φ_σ\n        (τ.baseChange (AlgebraicClosure ℚ_[ℓ])).conj r = σ hℓ φ) ∧\n    -- and `ρ` is part of the family.\n    (∃ (_ : Algebra R (AlgebraicClosure ℚ_[p])) (_ : ContinuousSMul R (AlgebraicClosure ℚ_[p]))\n      (ψ : E →+* AlgebraicClosure ℚ_[p])\n      (r' : AlgebraicClosure ℚ_[p] ⊗[R] V ≃ₗ[AlgebraicClosure ℚ_[p]]\n        Fin 2 → AlgebraicClosure ℚ_[p]),\n      (ρ.baseChange (AlgebraicClosure ℚ_[p])).conj r' = σ hp ψ) :=\n  sorry\n","line":21,"end_line":52,"proof_status":"statement_formalized"}
+{"id":"hardly_ramified_mod3_reducible","label":"hardly_ramified_mod3_reducible","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_3.html","dependencies":["hardly_ramified"],"nlp_text":"Suppose \\(k\\) is a finite field of characteristic 3, and suppose \\(\\overline{rho}:\\operatorname{Gal}(\\overline{\\mathbb {Q}}/\\mathbb {Q})\\to \\operatorname{GL}_2(k)\\) is hardly ramified. Then \\(\\overline{\\rho }\\) is an extension of the cyclotomic character by the trivial representation.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/GaloisRepresentation.IsHardlyRamified.mod_three","latex_link":"ch_freyreduction.html#hardly_ramified_mod3_reducible","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/GaloisRepresentation.IsHardlyRamified.mod_three","lean_decl_name":"GaloisRepresentation.IsHardlyRamified.mod_three","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch3","source_path":"FLT/GaloisRepresentation/HardlyRamified/ModThree.lean","lean4_code":"/-- A mod 3 hardly ramified representation is an extension of trivial by cyclo -/\n-- Probably `Field k` can be replaced with `(3 : k) = 0`\ntheorem mod_three {k : Type u} [Fintype k] [Field k] [Algebra ℤ_[3] k] --\n    [TopologicalSpace k] [DiscreteTopology k]\n    (V : Type*) [AddCommGroup V] [Module k V] [Module.Finite k V] [Module.Free k V]\n    (hV : Module.rank k V = 2) {ρ : GaloisRep ℚ k V}\n    (hρ : IsHardlyRamified (show Odd 3 by decide) hV ρ) :\n    ∃ (π : V →ₗ[k] k) (_ : Function.Surjective π),\n    ∀ g : Γ ℚ, ∀ v : V, π (ρ g v) = π v := by\n  sorry\n","line":9,"end_line":18,"proof_status":"statement_formalized"}
+{"id":"hardly_ramified_3adic_reducible","label":"hardly_ramified_3adic_reducible","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_3.html","dependencies":["hardly_ramified_mod3_reducible"],"nlp_text":"Suppose \\(L/\\mathbb {Q}_3\\) is a finite extension, with integer ring \\(\\mathcal{O}_L\\), and suppose \\(\\rho _3:\\operatorname{Gal}(\\overline{\\mathbb {Q}}/\\mathbb {Q})\\to \\operatorname{GL}_2(\\mathcal{O}_L)\\) is hardly ramified. Then (considered as a representation to \\(\\operatorname{GL}_2(L)\\)) \\(\\rho _3^{ss}=1\\oplus \\chi _3\\) where \\(1\\) is the trivial character and \\(\\chi _3\\) is the 3-adic cyclotomic character.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/GaloisRepresentation.IsHardlyRamified.three_adic","latex_link":"ch_freyreduction.html#hardly_ramified_3adic_reducible","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/GaloisRepresentation.IsHardlyRamified.three_adic","lean_decl_name":"GaloisRepresentation.IsHardlyRamified.three_adic","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch3","source_path":"FLT/GaloisRepresentation/HardlyRamified/Threeadic.lean","lean4_code":"/--\nA 3-adic hardly ramified representation has trace(Frob_p) = 1 + p for all p ≠ 2,3\n-/\ntheorem three_adic {R : Type*} [CommRing R] [Algebra ℤ_[3] R] [Module.Finite ℤ_[3] R]\n    [Module.Free ℤ_[3] R] [TopologicalSpace R] [IsTopologicalRing R] [IsLocalRing R]\n    [IsModuleTopology ℤ_[3] R]\n    (V : Type*) [AddCommGroup V] [Module R V] [Module.Finite R V] [Module.Free R V]\n    (hV : Module.rank R V = 2) {ρ : GaloisRep ℚ R V}\n    (hρ : IsHardlyRamified (show Odd 3 by decide) hV ρ) :\n    ∀ p (hp : Nat.Prime p) (hp5 : 5 ≤ p),\n      letI v := hp.toHeightOneSpectrumRingOfIntegersRat -- p as a finite place of ℚ\n      (ρ.toLocal v (Frob v)).trace _ _ = 1 + p := sorry\n","line":11,"end_line":22,"proof_status":"statement_formalized"}

crawled_repos/FLT_ch5.jsonl

@@ -0,0 +1,35 @@
+{"id":"ZHat.multiples","label":"ZHat.multiples","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":["ZHat.commRing"],"nlp_text":"The multiples of \\(N\\) in \\(\\widehat{\\mathbb {Z}}\\) are precisely the compatible collections \\((z_i)_i\\in \\widehat{\\mathbb {Z}}\\) with \\(z_N=0\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/ZHat.multiples","latex_link":"sect0002.html#ZHat.multiples","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/ZHat.multiples","lean_decl_name":"ZHat.multiples","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch5","source_path":"FLT/Data/QHat.lean","lean4_code":"lemma multiples (N : ℕ+) (z : ZHat) : (∃ (y : ZHat), N * y = z) ↔ z N = 0 := by\n  constructor\n  · intro ⟨y, hy⟩\n    rw [← hy]\n    change N * (y N) = 0\n    simp\n  · intro h\n    let y : ZHat := {\n      val := fun j ↦ (z (N * j)).val / (N : ℕ)\n      property := by\n        intro j k hjk\n        have hj := z.prop N (N * j) (by simp only [PNat.mul_coe, dvd_mul_right])\n        have hk := z.prop N (N * k) (by simp only [PNat.mul_coe, dvd_mul_right])\n        rw [h, ZMod.castHom_apply, ZMod.cast_eq_val, ZMod.natCast_eq_zero_iff] at hj\n        rw [h, ZMod.castHom_apply, ZMod.cast_eq_val, ZMod.natCast_eq_zero_iff] at hk\n        have hNjk := z.prop (N * j) (N * k) (mul_dvd_mul (dvd_refl _) hjk)\n        rw [ZMod.castHom_apply, ZMod.cast_eq_val] at hNjk\n        simp only [PNat.mul_coe, map_natCast, ZMod.natCast_eq_natCast_iff]\n        apply Nat.ModEq.mul_right_cancel' (c := N) (by simp)\n        rw [Nat.div_mul_cancel hj, Nat.div_mul_cancel hk,\n          mul_comm (j : ℕ) (N : ℕ), ← ZMod.natCast_eq_natCast_iff, hNjk]\n        simp\n    }\n    refine ⟨y, ?_⟩\n    ext j\n    exact y_mul_N_eq_z N z h j\n","line":238,"end_line":263,"proof_status":"formalized"}
+{"id":"QHat.unitsrat_join_unitszHat","label":"QHat.unitsrat_join_unitszHat","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":["ZHat.multiples","QHat.canonicalForm"],"nlp_text":"The product of \\(\\mathbb {Q}^\\times \\) and \\(\\widehat{\\mathbb {Z}}^\\times \\) in \\(\\widehat{\\mathbb {Q}}^\\times \\) is all of \\(\\widehat{\\mathbb {Q}}^\\times \\). More precisely, every element of \\(\\widehat{\\mathbb {Q}}^\\times \\) can be written as \\(qz\\) with \\(q\\in \\mathbb {Q}^\\times \\) and \\(z\\in \\widehat{\\mathbb {Z}}^\\times \\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/QHat.unitsrat_join_unitszHat","latex_link":"sect0002.html#QHat.unitsrat_join_unitszHat","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/QHat.unitsrat_join_unitszHat","lean_decl_name":"QHat.unitsrat_join_unitszHat","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch5","source_path":"FLT/Data/QHat.lean","lean4_code":"lemma unitsrat_join_unitszHat : unitsratsub ⊔ unitszHatsub = ⊤ := by\n  rw [eq_top_iff]\n  rintro y -\n  rcases canonicalForm y.val with ⟨N, x, hy⟩\n  rcases canonicalForm (y⁻¹.val) with ⟨N2, x2, hy2⟩\n  set xinv := (1 / (N * N2) : ℚ) ⊗ₜ[ℤ] x2 with xinv_def\n  have : (i₂ x) * xinv = 1 := by\n    rw [xinv_def, Algebra.TensorProduct.includeRight_apply, one_div, mul_inv_rev,\n      Algebra.TensorProduct.tmul_mul_tmul,one_mul,mul_comm,← Algebra.TensorProduct.tmul_mul_tmul,\n      ← one_div, ← one_div, ← hy, ← hy2, ← Units.val_mul, mul_inv_cancel, Units.val_one]\n  let xunit : QHatˣ := ⟨i₂ x, xinv, this, by rw [mul_comm]; exact this⟩\n  suffices h : ∀ (u : QHatˣ), (u : QHat) ∈ zHatsub → u ∈ unitsratsub ⊔ unitszHatsub by\n    specialize h xunit\n    simp only [Algebra.TensorProduct.includeRight_apply, AddMonoidHom.mem_range,\n      AddMonoidHom.coe_coe, exists_apply_eq_apply, forall_const, Subgroup.mem_sup, xunit] at h\n    rcases h with ⟨w, ⟨v, rfl⟩, z, ⟨t, rfl⟩, wzx⟩\n    rw [Subgroup.mem_sup]\n    let q : ℚˣ := ⟨v / N, N / v, by field_simp, by field_simp⟩\n    use ((Units.map ↑i₁) q)\n    simp only [MonoidHom.mem_range, exists_exists_eq_and]\n    refine ⟨⟨q, rfl⟩, t, ?_⟩\n    simp only [← Units.val_inj, hy, Units.map_mk, MonoidHom.coe_coe,\n      Algebra.TensorProduct.includeLeft_apply, Units.val_mul, one_div, q]\n    rw [← mul_one (N⁻¹ : ℚ), ← one_mul x, ← Algebra.TensorProduct.tmul_mul_tmul, div_eq_mul_inv,\n      mul_comm (v : ℚ), ← mul_one 1, ← Algebra.TensorProduct.tmul_mul_tmul, mul_assoc, mul_one]\n    congr\n    simpa only [← Units.val_inj, Units.val_mul, Units.coe_map, MonoidHom.coe_coe,\n      Algebra.TensorProduct.includeLeft_apply, xunit, q] using wzx\n  clear * -\n  intro x hx\n  rcases canonicalForm (x⁻¹.val) with ⟨M, y, hxinv⟩\n  have : x * (i₂ y) = M := by\n    rw [← one_mul (M : QHat), ← Units.val_one, ← mul_inv_cancel x, Units.val_mul, mul_assoc]\n    congr!\n    have h : (M : QHat) = (M : ℚ) ⊗ₜ[ℤ] 1 := by norm_cast\n    rw [Algebra.TensorProduct.includeRight_apply, hxinv, h, Algebra.TensorProduct.tmul_mul_tmul,\n      mul_one, one_div, inv_mul_cancel₀]\n    simp only [ne_eq, Rat.natCast_eq_zero_iff, PNat.ne_zero, not_false_eq_true]\n  rcases hx with ⟨X, hX⟩\n  let I := Ideal.span {X}\n  let J := I.comap (Int.castRingHom ZHat)\n  have Jnonzero : (M : ℤ) ∈ J := by\n    simp only [J, I, Ideal.mem_comap]\n    rw [Ideal.mem_span_singleton']\n    use y\n    apply injective_zHat\n    simp only [mul_comm, ← hX, AddMonoidHom.coe_coe, Algebra.TensorProduct.includeRight_apply,\n      Algebra.TensorProduct.tmul_mul_tmul, mul_one] at this\n    rw [Algebra.TensorProduct.includeRight_apply, this, map_natCast,\n      Algebra.TensorProduct.includeRight_apply, Algebra.TensorProduct.one_tmul_natCast]\n  obtain ⟨g, hg⟩ := IsPrincipalIdealRing.principal (R := ℤ) J\n  wlog gpos : 0 < g with H\n  · specialize H x M y hxinv this X hX Jnonzero (-g)\n    apply H (by rw [← Set.neg_singleton, Submodule.span_neg, ← hg])\n    rw [Int.neg_pos, Int.lt_iff_le_and_ne, ← Int.not_gt_eq]\n    refine ⟨gpos, ?_⟩\n    rintro rfl\n    rw [Submodule.span_zero_singleton, Submodule.eq_bot_iff] at hg\n    specialize hg (M : ℤ) Jnonzero\n    simp only [Int.natCast_eq_zero, PNat.ne_zero] at hg\n  clear this hxinv y Jnonzero M\n  let N : ℕ+ := ⟨g.toNat, Int.pos_iff_toNat_pos.1 gpos⟩\n  suffices h : Ideal.span {X} = Ideal.span {(g : ZHat)} by\n    obtain ⟨y, hy⟩ : ∃ y, y * X = g := by\n      rw [← Ideal.mem_span_singleton', h, Ideal.mem_span_singleton]\n    obtain ⟨z, hz⟩ : ∃ z, z * g = X := by\n      rw [← Ideal.mem_span_singleton', ← h, Ideal.mem_span_singleton]\n    have : y * z = 1 := by\n      rw [mul_comm, ← sub_right_inj (a := (1 : ZHat)), sub_self]\n      apply ZHat.eq_zero_of_mul_eq_zero N\n      rw [PNat.mk_coe, ← Int.cast_natCast, Int.natCast_toNat_eq_self.2 (le_of_lt gpos), mul_sub,\n        mul_one, sub_eq_zero, ← mul_assoc, mul_comm _ z, hz, mul_comm, hy]\n    simp only [Subgroup.mem_sup, MonoidHom.mem_range, exists_exists_eq_and]\n    set G : ℚ := 1 / g with G_def\n    have gG : g * G = 1 := by\n      rw [G_def, one_div, mul_inv_cancel₀]\n      simp only [ne_eq, Rat.intCast_eq_zero_iff, Int.ne_of_gt gpos, not_false_eq_true]\n    use ⟨g, G, gG, mul_comm _ G ▸ gG⟩\n    use ⟨z, y, by rw[mul_comm]; exact this, this⟩\n    simp only [← Units.val_inj, ← hX, Units.map_mk, MonoidHom.coe_coe, map_intCast,\n      Algebra.TensorProduct.includeLeft_apply, Algebra.TensorProduct.includeRight_apply,\n      Units.val_mul, AddMonoidHom.coe_coe]\n    rw [← hz, ← mul_one 1, ← Algebra.TensorProduct.tmul_mul_tmul, mul_one, mul_comm,\n      Algebra.TensorProduct.one_tmul_intCast]\n  have hgx : Ideal.span {(g : ZHat)} ≤ Ideal.span {X} := by\n    have : g ∈ J := by\n      rw [hg, Ideal.submodule_span_eq]\n      apply Ideal.mem_span_singleton_self\n    simp only [J, I] at this\n    exact (Ideal.span_singleton_le_iff_mem _).2 this\n  refine le_antisymm ?_ hgx\n  suffices h : X N = 0 by\n    rcases (ZHat.multiples N X).2 h with ⟨y, hy⟩\n    rw [Ideal.span_singleton_le_span_singleton, ← hy, PNat.mk_coe]\n    exact ⟨y, by rw [← Int.cast_natCast, Int.natCast_toNat_eq_self.2 (le_of_lt gpos)]⟩\n  let xg := (X N).val\n  have : (xg - X) N = 0 := by\n    simp only [ZHat, ZMod.castHom_apply, ZHat.instDFunLikePNatZModVal,\n      AddSubgroupClass.coe_sub, SubringClass.coe_natCast, Pi.sub_apply, Pi.natCast_apply, xg]\n    simp only [ZMod.natCast_val, ZMod.cast_id', id_eq, sub_self]\n  rcases (ZHat.multiples N _).2 this with ⟨y, hy⟩\n  have : (xg : ZHat) ∈ Ideal.span {X} := by\n    rw [← sub_add_cancel (xg : ZHat) X]\n    apply Ideal.add_mem\n    · apply hgx\n      rw [Ideal.mem_span_singleton', ← hy, mul_comm, PNat.mk_coe]\n      exact ⟨y, by rw [← Int.cast_natCast, Int.natCast_toNat_eq_self.2 (le_of_lt gpos)]⟩\n    apply Ideal.mem_span_singleton_self\n  have hxg : (xg : ℤ) ∈ J := by\n    rw [Ideal.mem_comap]\n    simp only [Int.coe_castRingHom, Int.cast_natCast, I, this]\n  rw [hg, Submodule.mem_span_singleton] at hxg\n  rcases hxg with ⟨a, ha⟩\n  rw [← ZMod.val_eq_zero, ← Int.natCast_eq_zero]\n  apply Int.eq_zero_of_dvd_of_nonneg_of_lt (n := g) (Int.natCast_nonneg _)\n  · exact Int.lt_of_toNat_lt (ZMod.val_lt (X N))\n  exact ⟨a, by rw [mul_comm, ← smul_eq_mul, ha]⟩\n","line":654,"end_line":770,"proof_status":"formalized"}
+{"id":"QHat.rat_join_zHat","label":"QHat.rat_join_zHat","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":["QHat.canonicalForm"],"nlp_text":"The sum of \\(\\mathbb {Q}\\) and \\(\\widehat{\\mathbb {Z}}\\) in \\(\\widehat{\\mathbb {Q}}\\) is \\(\\widehat{\\mathbb {Q}}\\). More precisely, every element of \\(\\widehat{\\mathbb {Q}}\\) can be written as \\(q+z\\) with \\(q\\in \\mathbb {Q}\\) and \\(z\\in \\widehat{\\mathbb {Z}}\\), or more precisely as \\(q\\otimes _t 1+1\\otimes _t z\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/QHat.rat_join_zHat","latex_link":"sect0002.html#QHat.rat_join_zHat","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/QHat.rat_join_zHat","lean_decl_name":"QHat.rat_join_zHat","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch5","source_path":"FLT/Data/QHat.lean","lean4_code":"lemma rat_join_zHat : ratsub ⊔ zHatsub = ⊤ := by\n  rw [eq_top_iff]\n  intro x _\n  rcases x.canonicalForm with ⟨N, z, hNz⟩\n  rcases ZHat.nat_dense N z with ⟨q, r, hz, _⟩\n  have h : z - r = N * q := sub_eq_of_eq_add hz\n  rw [AddSubgroup.mem_sup]\n  use ((r : ℤ) / N : ℚ) ⊗ₜ[ℤ] 1\n  constructor\n  · simp\n  use 1 ⊗ₜ[ℤ] q\n  constructor\n  · simp\n  nth_rw 1 [← mul_one ((r : ℤ) / N : ℚ), div_mul_comm,\n    mul_comm, ← zsmul_eq_mul, TensorProduct.smul_tmul, zsmul_eq_mul, mul_one]\n  have : 1 = 1 / (N : ℚ) * (N : ℤ) := by simp\n  nth_rw 2 [this]\n  rw [mul_comm, ← zsmul_eq_mul, TensorProduct.smul_tmul, zsmul_eq_mul]\n  norm_cast; rw [← h, ← TensorProduct.tmul_add]\n  simp [hNz]\n","line":543,"end_line":562,"proof_status":"formalized"}
+{"id":"Hurwitz.norm_eq_mul_conj","label":"Hurwitz.norm_eq_mul_conj","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":["Hurwitz.norm","Hurwitz.starRing"],"nlp_text":"We have \\(N(x)=x\\overline{x}\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Hurwitz.norm_eq_mul_conj","latex_link":"sect0002.html#Hurwitz.norm_eq_mul_conj","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Hurwitz.norm_eq_mul_conj","lean_decl_name":"Hurwitz.norm_eq_mul_conj","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch5","source_path":"FLT/Data/Hurwitz.lean","lean4_code":"lemma norm_eq_mul_conj (z : 𝓞) : (norm z : 𝓞) = z * star z := by\n  ext <;> simp only [norm, intCast_re, intCast_im_o, intCast_im_i, intCast_im_oi,\n    mul_re, mul_im_o, mul_im_i, mul_im_oi, star_re, star_im_o, star_im_i, star_im_oi] <;> ring\n","line":371,"end_line":373,"proof_status":"formalized"}
+{"id":"Hurwitz.norm_zero","label":"Hurwitz.norm_zero","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":["Hurwitz.norm"],"nlp_text":"The norm of \\(0\\) is \\(0\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Hurwitz.norm_zero","latex_link":"sect0002.html#Hurwitz.norm_zero","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Hurwitz.norm_zero","lean_decl_name":"Hurwitz.norm_zero","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch5","source_path":"FLT/Data/Hurwitz.lean","lean4_code":"lemma norm_zero : norm 0 = 0 := by simp [norm]\n","line":386,"end_line":386,"proof_status":"formalized"}
+{"id":"Hurwitz.left_ideal_princ","label":"Hurwitz.left_ideal_princ","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":["Hurwitz.quot_rem"],"nlp_text":"All left ideals of \\(\\mathcal{O}\\) are principal.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Hurwitz.left_ideal_princ","latex_link":"sect0002.html#Hurwitz.left_ideal_princ","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Hurwitz.left_ideal_princ","lean_decl_name":"Hurwitz.left_ideal_princ","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch5","source_path":"FLT/Data/Hurwitz.lean","lean4_code":"lemma left_ideal_princ (I : Submodule 𝓞 𝓞) : ∃ a : 𝓞, I = Submodule.span 𝓞 {a} := by\n  by_cases h_bot : I = ⊥\n  · use 0\n    rw [Eq.comm]\n    simp only [h_bot, Submodule.span_singleton_eq_bot]\n  let S := {a : 𝓞 // a ∈ I ∧ a ≠ 0}\n  have : Nonempty S := by\n    simp only [ne_eq, nonempty_subtype, S]\n    exact Submodule.exists_mem_ne_zero_of_ne_bot h_bot\n  have hbdd : BddBelow <| Set.range (fun i : S ↦ norm i) := by\n    use 0\n    simp only [ne_eq, mem_lowerBounds, Set.mem_range]\n    rintro _ ⟨_, rfl⟩\n    exact norm_nonneg _\n  obtain ⟨a, ha⟩ : ∃ a : S, norm a = ⨅ i : S, norm i :=\n    exists_eq_ciInf_of_not_isPredPrelimit hbdd (Order.not_isPredPrelimit)\n  use a\n  apply le_antisymm\n  · intro i hi\n    rw [Submodule.mem_span_singleton]\n    simp only [ne_eq]\n    obtain ⟨q, r, hqr⟩ := quot_rem i a a.2.2\n    rw [ha] at hqr\n    have hrI : r ∈ I := by\n      rw [show r = i - q • a by apply eq_sub_of_add_eq; rw [add_comm]; exact hqr.1.symm ]\n      apply I.sub_mem hi (I.smul_mem _ a.2.1)\n    have hr : r = 0 := by\n      by_contra hr\n      lift r to S using ⟨hrI, hr⟩\n      apply (ciInf_le hbdd r).not_gt hqr.2\n    rw [hr, add_zero] at hqr\n    refine ⟨q, hqr.1.symm⟩\n  · rw [Submodule.span_singleton_le_iff_mem]\n    exact a.2.1\n","line":525,"end_line":558,"proof_status":"formalized"}
+{"id":"QHat.injective_zHat","label":"QHat.injective_zHat","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":["QHat","ZHat.torsionfree"],"nlp_text":"The ring homomorphism \\(\\widehat{\\mathbb {Z}}\\to \\widehat{\\mathbb {Q}}\\) sending \\(z\\) to \\(1\\otimes _t z\\) is injective.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/QHat.injective_zHat","latex_link":"sect0002.html#QHat.injective_zHat","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/QHat.injective_zHat","lean_decl_name":"QHat.injective_zHat","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch5","source_path":"FLT/Data/QHat.lean","lean4_code":"lemma injective_zHat :\n    Function.Injective i₂ := by\n      intro a b h\n      have h₁ := LinearMap.rTensor_tmul ZHat (f := Algebra.linearMap ℤ ℚ) a 1\n      have h₂ := LinearMap.rTensor_tmul ZHat (f := Algebra.linearMap ℤ ℚ) b 1\n      simp only [Algebra.linearMap_apply, map_one] at h₁ h₂\n      dsimp at h\n      rw [← h₁, ← h₂] at h\n      replace h := Module.Flat.rTensor_preserves_injective_linearMap\n        (M := ZHat) (Algebra.linearMap ℤ ℚ) (fun _ _ ↦ by simp) h\n      simp only at h\n      have := congrArg (TensorProduct.lid ℤ ZHat) h\n      simpa using this\n","line":329,"end_line":341,"proof_status":"formalized"}
+{"id":"Qhat.unitsrat_meet_unitszHat","label":"Qhat.unitsrat_meet_unitszHat","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":["QHat.rat_meet_zHat"],"nlp_text":"The intersection of \\(\\mathbb {Q}^\\times \\) and \\(\\widehat{\\mathbb {Z}}^\\times \\) in \\(\\widehat{\\mathbb {Q}}^\\times \\) is \\(\\mathbb {Z}^\\times \\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/QHat.unitsrat_meet_unitszHat","latex_link":"sect0002.html#Qhat.unitsrat_meet_unitszHat","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/QHat.unitsrat_meet_unitszHat","lean_decl_name":"QHat.unitsrat_meet_unitszHat","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch5","source_path":"FLT/Data/QHat.lean","lean4_code":"lemma unitsrat_meet_unitszHat : unitsratsub ⊓ unitszHatsub = unitszsub := by\n  apply le_antisymm\n  · intro x ⟨⟨q, hxq⟩, ⟨zHat, hxzHat⟩⟩\n    obtain ⟨z, (hz : (z : QHat) = x)⟩ : (x : QHat) ∈ zsub := by\n      rw [← rat_meet_zHat]\n      exact ⟨⟨q, by simp [← hxq]⟩, zHat, by simp [← hxzHat]⟩\n    have znez : z ≠ 0 := by\n      rintro rfl\n      simp [Eq.comm] at hz\n    let a := Int.sign z\n    let b := Int.natAbs z\n    set zinvRat : ℚ := a / b with zinvRat_def\n    have hzinvRat : z * zinvRat = 1 := by\n      rw [mul_div, div_eq_one_iff_eq]\n      · rw_mod_cast [Int.mul_sign_self z]\n      · exact_mod_cast Int.natAbs_ne_zero.mpr znez\n    let zinvZHat : ZHatˣ := zHat⁻¹\n    have hzinvZHat : ↑zHat * ↑zinvZHat = (1 : ZHat) := Units.mul_inv zHat\n    let xinv : QHatˣ := x⁻¹\n    have h1 : zinvRat ⊗ₜ[ℤ] (1 : ZHat) = xinv := by\n      apply Units.eq_inv_of_mul_eq_one_left\n      rw [← hz, ← zsmul_eq_mul, TensorProduct.smul_tmul', zsmul_eq_mul,\n        hzinvRat, Algebra.TensorProduct.one_def]\n    have h2 : (1 : ℚ) ⊗ₜ[ℤ] (Units.val zinvZHat) = xinv := by\n      apply Units.eq_inv_of_mul_eq_one_left\n      have hzHat : (1 : ℚ) ⊗ₜ[ℤ] (zHat : ZHat) = (x : QHat) := by simp [← hxzHat]\n      rw [← hzHat, Algebra.TensorProduct.tmul_mul_tmul, mul_one, hzinvZHat,\n        Algebra.TensorProduct.one_def]\n    have h3 : zinvRat ⊗ₜ[ℤ] (1 : ZHat) = (1 / b : ℚ) ⊗ₜ[ℤ] (a : ZHat) := by\n      rw [zinvRat_def, ← mul_one (a : ℚ), ← mul_div,\n      ← zsmul_eq_mul, TensorProduct.smul_tmul, zsmul_eq_mul, mul_one]\n    have bpos : 0 < b := Int.natAbs_pos.2 znez\n    have heq : (1 / (((Nat.toPNat b bpos) : ℕ) : ℚ)) ⊗ₜ[ℤ] (a : ZHat) =\n        (1 / (((1 : ℕ+) : ℕ) : ℚ)) ⊗ₜ[ℤ] ↑zinvZHat := by\n      have : ↑(Nat.toPNat b bpos) = b := by\n        unfold Nat.toPNat\n        rw [PNat.mk_coe]\n      rw [PNat.val_ofNat, Nat.cast_one, div_self one_ne_zero, this, ← h3, h1, h2]\n    have cop1 : IsCoprime (b.toPNat bpos) ↑a := by\n      rw [IsCoprime, ZHat.intCast_val, ← ZMod.isUnit_natAbs,\n        ZMod.isUnit_iff_coprime, Int.natAbs_sign_of_ne_zero znez]\n      exact Nat.coprime_one_left _\n    have cop2 : IsCoprime 1 ↑zinvZHat := by\n      simp only [IsCoprime, PNat.val_ofNat, isUnit_of_subsingleton]\n    obtain ⟨hb, ha⟩ := (lowestTerms ↑x).2 (Nat.toPNat b bpos) 1 ↑a ↑zinvZHat ⟨cop1, cop2, heq⟩\n    have b1 : b = 1 := PNat.coe_eq_one_iff.2 hb\n    obtain ⟨u, rfl⟩ := Int.isUnit_iff_natAbs_eq.2 b1\n    use u\n    ext\n    norm_cast at hz\n  · intro x ⟨xz, hxz⟩\n    constructor\n    · use (Units.map ↑(Int.castRingHom ℚ)) xz\n      norm_cast\n    · use (Units.map ↑(Int.castRingHom ZHat)) xz\n      rw [← hxz, ← MonoidHom.comp_apply, ← Units.map_comp]\n      congr\n      ext x\n      · simp only [MonoidHom.coe_comp, MonoidHom.coe_coe, Function.comp_apply, Int.coe_castRingHom,\n        Algebra.TensorProduct.includeRight_apply, Algebra.TensorProduct.one_tmul_intCast]\n      simp\n","line":577,"end_line":637,"proof_status":"formalized"}
+{"id":"Hurwitz.norm_mul","label":"Hurwitz.norm_mul","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":["Hurwitz.norm"],"nlp_text":"The norm of a product is the product of the norms.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Hurwitz.norm_mul","latex_link":"sect0002.html#Hurwitz.norm_mul","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Hurwitz.norm_mul","lean_decl_name":"Hurwitz.norm_mul","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch5","source_path":"FLT/Data/Hurwitz.lean","lean4_code":"lemma norm_mul (x y : 𝓞) : norm (x * y) = norm x * norm y := by\n  rw [← Int.cast_inj (α := 𝓞)]\n  simp_rw [norm_eq_mul_conj, star_mul]\n  rw [mul_assoc, ← mul_assoc y, ← norm_eq_mul_conj]\n  rw [Int.cast_comm, ← mul_assoc, ← norm_eq_mul_conj, Int.cast_mul]\n","line":390,"end_line":394,"proof_status":"formalized"}
+{"id":"Hurwitz.quot_rem","label":"Hurwitz.quot_rem","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":["Hurwitz.exists_near"],"nlp_text":"Given two Hurwitz quaternions \\(a\\) and \\(b\\) with \\(b\\) nonzero, there exists \\(q\\) and \\(r\\) such that \\(a=qb+r\\) and \\(N(r){\\lt}N(b)\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Hurwitz.quot_rem","latex_link":"sect0002.html#Hurwitz.quot_rem","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Hurwitz.quot_rem","lean_decl_name":"Hurwitz.quot_rem","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch5","source_path":"FLT/Data/Hurwitz.lean","lean4_code":"lemma quot_rem (a b : 𝓞) (hb : b ≠ 0) : ∃ q r : 𝓞, a = q * b + r ∧ norm r < norm b := by\n  let a' := toQuaternion a\n  let b' := toQuaternion b\n  have hb' : b' ≠ 0 := toQuaternion_ne_zero_iff.mpr hb\n  let q' := a' / b'\n  obtain ⟨q : 𝓞, hq : dist q' (toQuaternion q) < 1⟩ : ∃ _, _ := exists_near q'\n  refine ⟨q, a - q * b, (add_sub_cancel _ _).symm, ?_⟩\n  rw [← Int.cast_lt (R := ℝ), ← normSq_toQuaternion, ← normSq_toQuaternion]\n  rw [normSq_eq_norm_mul_self, normSq_eq_norm_mul_self]\n  refine mul_self_lt_mul_self ?_ ?_\n  · exact _root_.norm_nonneg (a - q * b).toQuaternion\n  rw [toQuaternion_sub, ← dist_eq_norm]\n  calc\n    _ = dist (q' * b') (q.toQuaternion * b') := ?_\n    _ = dist q' (q.toQuaternion) * ‖b'‖ := ?_\n    _ < _ := ?_\n  · rw [toQuaternion_mul]\n    dsimp only [b', q']\n    rw [div_mul_cancel₀ a' hb']\n  · -- Surprised that this doesn't seem to exist in mathlib.\n    rw [dist_eq_norm_sub', ← sub_mul, _root_.norm_mul, ← dist_eq_norm_sub']\n  · rw [← norm_pos_iff] at hb'\n    exact mul_lt_of_lt_one_left hb' hq\n","line":501,"end_line":523,"proof_status":"formalized"}
+{"id":"QHat","label":"QHat","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":["ZHat"],"nlp_text":"The profinite completion \\(\\widehat{\\mathbb {Q}}\\) of \\(\\mathbb {Q}\\) is the tensor product \\(\\mathbb {Q}\\otimes _{\\mathbb {Z}}\\widehat{\\mathbb {Z}}\\), or \\(\\widehat{\\mathbb {Q}}=\\mathbb {Q}\\otimes \\widehat{\\mathbb {Z}}\\) for short.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/QHat","latex_link":"sect0002.html#QHat","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/QHat","lean_decl_name":"QHat","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch5","source_path":"./FLT/Data/QHat.lean","lean4_code":"instance nontrivial_QHat : Nontrivial QHat where\n  exists_pair_ne := ⟨1 ⊗ₜ 0, 1 ⊗ₜ 1, injective_zHat.ne ZHat.zeroNeOne⟩\n\nnoncomputable abbrev i₁ : ℚ →ₐ[ℤ] QHat := Algebra.TensorProduct.includeLeft\n","line":343,"end_line":346,"proof_status":"formalized"}
+{"id":"QHat.injective_rat","label":"QHat.injective_rat","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":["QHat","ZHat.commRing"],"nlp_text":"The ring homomorphism \\(\\mathbb {Q}\\to \\widehat{\\mathbb {Q}}\\) sending \\(q\\) to \\(q\\otimes _t 1\\) is injective.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/QHat.injective_rat","latex_link":"sect0002.html#QHat.injective_rat","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/QHat.injective_rat","lean_decl_name":"QHat.injective_rat","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch5","source_path":"FLT/Data/QHat.lean","lean4_code":"lemma injective_rat :\n    Function.Injective i₁ := RingHom.injective i₁.toRingHom\n","line":347,"end_line":348,"proof_status":"formalized"}
+{"id":"HurwitzRatHat.canonicalForm","label":"HurwitzRatHat.canonicalForm","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":["QHat","Hurwitz"],"nlp_text":"Every element of \\(\\widehat{D}\\) can be written as \\(z/N\\) with \\(z\\in \\widehat{\\mathcal{O}}\\) and \\(N\\in \\mathbb {N}^+\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/HurwitzRatHat.canonicalForm","latex_link":"sect0002.html#HurwitzRatHat.canonicalForm","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/HurwitzRatHat.canonicalForm","lean_decl_name":"HurwitzRatHat.canonicalForm","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch5","source_path":"FLT/Data/QHat.lean","lean4_code":"lemma canonicalForm (z : QHat) : ∃ (N : ℕ+) (z' : ZHat), z = (1 / N : ℚ) ⊗ₜ z' := by\n  induction z using TensorProduct.induction_on with\n  | zero =>\n    refine ⟨1, 0, ?_⟩\n    simp\n  | tmul q z =>\n    refine ⟨⟨q.den, q.den_pos ⟩, q.num * z, ?_⟩\n    simp_rw [← zsmul_eq_mul, TensorProduct.tmul_smul, TensorProduct.smul_tmul']\n    simp only [PNat.mk_coe, zsmul_eq_mul]\n    simp only [← q.mul_den_eq_num, mul_assoc,\n      one_div, ne_eq, Nat.cast_eq_zero, Rat.den_ne_zero, not_false_eq_true,\n        mul_one, mul_inv_cancel₀]\n  | add x y hx hy =>\n    obtain ⟨N₁, z₁, rfl⟩ := hx\n    obtain ⟨N₂, z₂, rfl⟩ := hy\n    refine ⟨N₁ * N₂, (N₁ : ℤ) * z₂ + (N₂ : ℤ) * z₁, ?_⟩\n    simp only [TensorProduct.tmul_add, ← zsmul_eq_mul,\n      TensorProduct.tmul_smul, TensorProduct.smul_tmul']\n    simp only [one_div, PNat.mul_coe, Nat.cast_mul, mul_inv_rev, zsmul_eq_mul, Int.cast_natCast,\n      ne_eq, Nat.cast_eq_zero, PNat.ne_zero, not_false_eq_true, mul_inv_cancel_left₀]\n    rw [add_comm]\n    congr\n    simp [mul_comm]\n","line":285,"end_line":307,"proof_status":"statement_formalized"}
+{"id":"QHat.canonicalForm","label":"QHat.canonicalForm","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":["QHat","ZHat.commRing"],"nlp_text":"Every element of \\(\\widehat{\\mathbb {Q}}:=\\mathbb {Q}\\otimes \\widehat{\\mathbb {Z}}\\) can be written as \\(q\\otimes _t z\\) with \\(q\\in \\mathbb {Q}\\) and \\(z\\in \\widehat{\\mathbb {Z}}\\). Furthermore one can even assume that \\(q=\\frac{1}{N}\\) for some positive integer \\(N\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/QHat.canonicalForm","latex_link":"sect0002.html#QHat.canonicalForm","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/QHat.canonicalForm","lean_decl_name":"QHat.canonicalForm","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch5","source_path":"FLT/Data/QHat.lean","lean4_code":"lemma canonicalForm (z : QHat) : ∃ (N : ℕ+) (z' : ZHat), z = (1 / N : ℚ) ⊗ₜ z' := by\n  induction z using TensorProduct.induction_on with\n  | zero =>\n    refine ⟨1, 0, ?_⟩\n    simp\n  | tmul q z =>\n    refine ⟨⟨q.den, q.den_pos ⟩, q.num * z, ?_⟩\n    simp_rw [← zsmul_eq_mul, TensorProduct.tmul_smul, TensorProduct.smul_tmul']\n    simp only [PNat.mk_coe, zsmul_eq_mul]\n    simp only [← q.mul_den_eq_num, mul_assoc,\n      one_div, ne_eq, Nat.cast_eq_zero, Rat.den_ne_zero, not_false_eq_true,\n        mul_one, mul_inv_cancel₀]\n  | add x y hx hy =>\n    obtain ⟨N₁, z₁, rfl⟩ := hx\n    obtain ⟨N₂, z₂, rfl⟩ := hy\n    refine ⟨N₁ * N₂, (N₁ : ℤ) * z₂ + (N₂ : ℤ) * z₁, ?_⟩\n    simp only [TensorProduct.tmul_add, ← zsmul_eq_mul,\n      TensorProduct.tmul_smul, TensorProduct.smul_tmul']\n    simp only [one_div, PNat.mul_coe, Nat.cast_mul, mul_inv_rev, zsmul_eq_mul, Int.cast_natCast,\n      ne_eq, Nat.cast_eq_zero, PNat.ne_zero, not_false_eq_true, mul_inv_cancel_left₀]\n    rw [add_comm]\n    congr\n    simp [mul_comm]\n","line":285,"end_line":307,"proof_status":"formalized"}
+{"id":"QHat.rat_meet_zHat","label":"QHat.rat_meet_zHat","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":["QHat","QHat.lowestTerms"],"nlp_text":"The intersection of \\(\\mathbb {Q}\\) and \\(\\widehat{\\mathbb {Z}}\\) in \\(\\widehat{\\mathbb {Q}}\\) is \\(\\mathbb {Z}\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/QHat.rat_meet_zHat","latex_link":"sect0002.html#QHat.rat_meet_zHat","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/QHat.rat_meet_zHat","lean_decl_name":"QHat.rat_meet_zHat","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch5","source_path":"FLT/Data/QHat.lean","lean4_code":"lemma rat_meet_zHat : ratsub ⊓ zHatsub = zsub := by\n  apply le_antisymm\n  · intro x ⟨⟨l, hl⟩, ⟨r, hr⟩⟩\n    simp only [AddMonoidHom.coe_coe, Algebra.TensorProduct.includeLeft_apply,\n      Algebra.TensorProduct.includeRight_apply] at hl hr\n    rcases lowestTerms x with ⟨⟨N, z, hNz, hx⟩, unique⟩\n    have cop1 : IsCoprime l.den.toPNat' l.num := by\n      simp_rw [IsCoprime, ZHat.intCast_val, ← ZMod.isUnit_natAbs, ZMod.isUnit_iff_coprime,\n        PNat.toPNat'_coe l.den_pos]\n      exact l.reduced\n    have cop2 : IsCoprime 1 r := by\n      simp only [IsCoprime, PNat.val_ofNat]\n      exact isUnit_of_subsingleton _\n    have hcanon : x = (1/(l.den : ℚ)) ⊗ₜ[ℤ] (l.num : ZHat) := by\n      nth_rw 1 [← hl, ← Rat.num_div_den l, ← mul_one ((l.num : ℚ) / l.den), div_mul_comm,\n      mul_comm, ← zsmul_eq_mul, TensorProduct.smul_tmul, zsmul_eq_mul, mul_one]\n    rw [← PNat.toPNat'_coe l.den_pos, hx] at hcanon\n    obtain ⟨rfl, rfl⟩ := unique _ _ _ _ ⟨hNz, cop1, hcanon⟩\n    have : 1 = 1 / (((1 : ℕ+) : ℕ) : ℚ) := by simp\n    nth_rw 1 [← hx, ← hr, this] at hcanon\n    use l.num; rw [hx, (unique _ 1 _ r ⟨hNz, cop2, hcanon.symm⟩).1]\n    simp\n  · exact fun x ⟨k, hk⟩ ↦ by constructor <;>\n      (use k; simp only [AddMonoidHom.coe_coe,\n        map_intCast]; exact hk)\n","line":517,"end_line":541,"proof_status":"formalized"}
+{"id":"HurwitzRatHat.completed_units","label":"HurwitzRatHat.completed_units","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":["HurwitzRatHat.canonicalForm"],"nlp_text":"The group of units of \\(\\widehat{D}\\) is \\(D^\\times \\widehat{\\mathcal{O}}^\\times \\). More precisely, every element of \\(\\widehat{D}^\\times \\) can be written as a product \\(\\delta u\\) with \\(\\delta \\in D^\\times \\) and \\(u\\in \\widehat{\\mathcal{O}}^\\times \\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/HurwitzRatHat.completed_units","latex_link":"sect0002.html#HurwitzRatHat.completed_units","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/HurwitzRatHat.completed_units","lean_decl_name":"HurwitzRatHat.completed_units","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch5","source_path":"FLT/Data/HurwitzRatHat.lean","lean4_code":"lemma completed_units (z : D^ˣ) : ∃ (u : Dˣ) (v : 𝓞^ˣ), (z : D^) = j₁ u * j₂ v := sorry\n","line":64,"end_line":64,"proof_status":"statement_formalized"}
+{"id":"ZHat.charZero","label":"ZHat.charZero","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":[],"nlp_text":"The map from the naturals into \\(\\widehat{\\mathbb {Z}}\\) sending \\(n\\) to \\(n\\) is injective.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/ZHat.charZero","latex_link":"sect0002.html#ZHat.charZero","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/ZHat.charZero","lean_decl_name":"ZHat.charZero","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Algebra/Ring/CharZero.lean","lean4_code":"lemma charZero (ϕ : R →+* S) [CharZero S] : CharZero R where\n  cast_injective a b h := CharZero.cast_injective (R := S) <| by\n    rw [← map_natCast ϕ, ← map_natCast ϕ, h]\n\n","line":48,"end_line":51,"proof_status":"formalized"}
+{"id":"ZHat.torsionfree","label":"ZHat.torsionfree","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":["ZHat.charZero","ZHat.commRing"],"nlp_text":"If \\(0{\\lt}N\\) is an integer then multiplication by \\(N\\) is injective on \\(\\widehat{\\mathbb {Z}}\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/ZHat.torsionfree","latex_link":"sect0002.html#ZHat.torsionfree","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/ZHat.torsionfree","lean_decl_name":"ZHat.torsionfree","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch5","source_path":"FLT/Data/QHat.lean","lean4_code":"lemma torsionfree (N : ℕ+) : Function.Injective (fun z : ZHat ↦ N * z) := by\n  rw [← AddMonoidHom.coe_mulLeft, injective_iff_map_eq_zero]\n  intro a ha\n  rw [AddMonoidHom.coe_mulLeft] at ha\n  exact eq_zero_of_mul_eq_zero N a ha\n","line":205,"end_line":209,"proof_status":"formalized"}
+{"id":"ZHat.e_not_in_Int","label":"ZHat.e_not_in_Int","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":["ZHat.e"],"nlp_text":"The element \\((e_N)_N\\) of \\(\\widehat{\\mathbb {Z}}\\) is not in \\(\\mathbb {Z}\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/ZHat.e_not_in_Int","latex_link":"sect0002.html#ZHat.e_not_in_Int","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/ZHat.e_not_in_Int","lean_decl_name":"ZHat.e_not_in_Int","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch5","source_path":"FLT/Data/QHat.lean","lean4_code":"/-- Nonarchimedean $e$ is not an integer. -/\nlemma e_not_in_Int : ∀ a : ℤ, e ≠ a := by\n  rintro (a|a) ha\n  · obtain ⟨j, honelt, hj⟩ : ∃ j : ℕ, 1 < j ∧ a < ∑ i ∈ range (j + 1), i ! := by\n      refine ⟨a + 2, ?_, ?_⟩\n      · simp only [lt_add_iff_pos_left, add_pos_iff, zero_lt_one, or_true]\n      rw [sum_range_add]\n      apply lt_add_of_nonneg_of_lt\n      · positivity\n      rw [range_one, sum_singleton, add_zero]\n      exact (Nat.lt_add_of_pos_right two_pos).trans_le (self_le_factorial _)\n    let N : ℕ+ := ⟨(j + 1)!, by positivity⟩\n    apply lt_irrefl (e N).val\n    have h₀ : ∑ i ∈ range (j + 1), i ! < (j + 1) ! := sum_factorial_lt_factorial_succ honelt\n    calc\n      _ = _ := by simp [ha, N, mod_eq_of_lt (hj.trans h₀)]\n      _ < _ := hj\n      _ = _ := by simp only [PNat.mk_coe, e_factorial_succ, ZMod.val_natCast, mod_eq_of_lt h₀, N]\n  · obtain ⟨j, honelt, hj⟩ : ∃ j, 1 < j ∧ (a + 1) + ∑ i ∈ range (j + 1), i ! < (j + 1)! := by\n      refine ⟨a + 3, ?_, ?_⟩\n      · omega\n      calc\n        _ < (a + 1) * 1 + 2 * (a + 3)! := ?_\n        _ ≤ (a + 1) * (a + 3)! + 2 * (a + 3)! + 0 := ?_\n        _ < (a + 1) * (a + 3)! + 2 * (a + 3)! + (a + 3)! := ?_\n        _ = (a + 4)! := ?_\n      · rw [mul_one]\n        have : 3 ≤ a + 3 := by omega\n        have := sum_factorial_lt_two_mul_factorial this\n        gcongr\n      · rw [add_zero]\n        have : 1 ≤ (a + 3)! := Nat.one_le_of_lt (factorial_pos _)\n        gcongr\n      · gcongr\n        exact factorial_pos _\n      · rw [factorial_succ (a + 3)]\n        ring\n    let N : ℕ+ := ⟨(j + 1)!, by positivity⟩\n    apply lt_irrefl (e N).val\n    calc\n      _ < N - (a + 1) := ?_\n      _ = (e N).val := ?_\n    · dsimp [N]\n      apply lt_sub_of_add_lt\n      rwa [add_comm, e_factorial_succ, ZMod.val_natCast,\n        mod_eq_of_lt (sum_factorial_lt_factorial_succ honelt)]\n    · have : a + 1 < N := lt_of_le_of_lt (Nat.le_add_right _ _) hj\n      rw [ha, intCast_val, Int.cast_negSucc, ZMod.neg_val, ZMod.val_natCast, if_neg,\n        mod_eq_of_lt this]\n      rw [ZMod.natCast_eq_zero_iff]\n      contrapose! this\n      apply le_of_dvd (zero_lt_succ a) this\n","line":124,"end_line":175,"proof_status":"formalized"}
+{"id":"QHat.lowestTerms","label":"QHat.lowestTerms","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":["QHat.IsCoprime"],"nlp_text":"Every element of \\(\\widehat{\\mathbb {Q}}\\) can be uniquely written as \\(z/N\\) with \\(z\\in \\widehat{\\mathbb {Z}}\\), \\(N\\in \\mathbb {N}^+\\), and with \\(N\\) and \\(z\\) coprime.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/QHat.lowestTerms","latex_link":"sect0002.html#QHat.lowestTerms","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/QHat.lowestTerms","lean_decl_name":"QHat.lowestTerms","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch5","source_path":"FLT/Data/QHat.lean","lean4_code":"lemma lowestTerms (x : QHat) : (∃ N z, IsCoprime N z ∧ x = (1 / N : ℚ) ⊗ₜ z) ∧\n    (∀ N₁ N₂ z₁ z₂,\n    IsCoprime N₁ z₁ ∧ IsCoprime N₂ z₂ ∧ (1 / N₁ : ℚ) ⊗ₜ z₁ = (1 / N₂ : ℚ) ⊗ₜ[ℤ] z₂ →\n      N₁ = N₂ ∧ z₁ = z₂) := by\n  constructor\n  · -- Existence: by the previous lemma, an arbitrary element [x] can be written as z/N;\n    obtain ⟨N, z, h⟩ := canonicalForm x\n    -- let D be the greatest common divisor of N and z_N (lifted to a natural).\n    let D : PNat := ⟨Nat.gcd N (z N).val, Nat.gcd_pos_of_pos_left _ N.pos⟩\n    cases D.one_le.eq_or_lt with\n    | inl hD =>\n      -- If D = 1 then the fraction is by definition in lowest terms.\n      use N, z, ?_, h\n      symm at hD\n      simp_rw [D, ← PNat.coe_eq_one_iff, PNat.mk_coe] at hD\n      rwa [isCoprime_iff_coprime, Nat.coprime_iff_gcd_eq_one]\n    | inr hD =>\n      -- However if 1 < D ∣ N then z_D is the reduction of z_N and is hence 0.\n      have hDN : D ∣ N := PNat.dvd_iff.mpr (Nat.gcd_dvd_left N (z N).val)\n      have := z.prop D N (PNat.dvd_iff.mp hDN)\n      have : z D = 0 := by\n        rw [← this, ZMod.castHom_apply, ZMod.cast_eq_val, ZMod.natCast_eq_zero_iff]\n        exact Nat.gcd_dvd_right N (z N).val\n      -- By lemma 5.9 (ZHat.multiples) we deduce that z = Dy is a multiple of D,\n      obtain ⟨y, hy⟩ : ∃ y, D * y = z := by rwa [ZHat.multiples]\n      obtain ⟨E, hE⟩ := hDN\n      use E, y, ?_, ?_\n      swap\n      · -- and hence [x = ] z / N = 1/N ⨂ₜ Dy = 1/E ⨂ y, where E = N / D.\n        rw [h, hE, ← hy]\n        have : (D : ZHat) • y = (D : ℤ) • y := by simp\n        simp_rw [PNat.mul_coe, Nat.cast_mul, one_div, mul_inv, ← smul_eq_mul, this,\n          ← TensorProduct.smul_tmul]\n        simp\n      · -- Now if a natural divided both y_E and E\n        rw [isCoprime_iff_coprime]\n        apply Nat.coprime_of_dvd fun k hk hk1 hk2 => ?_\n        -- then this natural would divide both z_N/D [ = z_ED/D = y_N = y_ED] and N/D [ = E],\n        -- contradicting the fact that D is the greatest common divisors\n        suffices k ∣ (z N).val / D ∧ k ∣ N / D by\n          have := Nat.dvd_gcd this.2 this.1\n          simp [D, Nat.gcd_div_gcd_div_gcd_of_pos_left, hk.ne_one] at this\n        constructor\n        swap\n        · simp [hE, hk1]\n        simp only [← hy, ZHat.nat_mul_apply, ZMod.val_mul, ZMod.val_natCast, Nat.mod_mul_mod]\n        nth_rw 3 [hE]\n        have := y.prop E N (by simp [hE])\n        simp only [ZMod.castHom_apply, ZMod.cast_eq_val] at this\n        rwa [PNat.mul_coe, Nat.mul_mod_mul_left, Nat.mul_div_cancel_left _ (by simp),\n          ← ZMod.val_natCast, this]\n  · -- Uniqueness:\n    rintro N M z w ⟨hcpz, hcpw, h⟩\n    -- if z/N = w/M, we deduce 1 ⨂ₜ Mz = 1 ⨂ₜ Nw\n    have : i₂ (M * z) = i₂ (N * w) := by\n      apply_fun ((M * N : ℤ) • ·) at h\n      conv_lhs at h => rw [mul_comm]\n      simpa [← TensorProduct.smul_tmul_smul] using h\n    let y := M * z\n    -- and by injectivity of ZHat → QHat\n    have hNz := injective_zHat this\n    -- we deduce that Mz = Nw = y.\n    have hy₁ : y = M * z := rfl\n    have hy₂ : y = N * w := by rw [← hNz]\n    -- In particular, if L is the lowest common multiple of M and N\n    let L : ℕ+ := PNat.lcm N M\n    -- then y_L is a multiple of both M and N and is hence zero,\n    have : y L = 0 := by\n      suffices (L : ℕ) ∣ (y L).val by\n        simpa [← ZMod.natCast_eq_zero_iff]\n      apply lcm_dvd <;> [rw [hy₂]; rw [hy₁]] <;>\n      · simp only [ZHat.pnat_mul_apply, ZMod.val_mul, ZMod.val_natCast, Nat.mod_mul_mod]\n        refine (Nat.dvd_mod_iff ?_).mpr (Nat.dvd_mul_right _ _)\n        simp only [PNat.lcm_coe, Nat.dvd_lcm_left, Nat.dvd_lcm_right, L]\n    -- so y = Lx is a multiple of L by 5.9 (ZHat.multiples),\n    obtain ⟨x, hx⟩ := (ZHat.multiples _ _).mpr this\n    -- and we deduce from torsionfreeness that z = (L/M)x [ = M'x] and w = (L/N)x [ = N'x].\n    obtain ⟨N', hN'⟩ : N ∣ L := PNat.dvd_lcm_left N M\n    have hN'' : (N' : ℕ) = L / N := by simp [hN']\n    obtain ⟨M', hM'⟩ : M ∣ L := PNat.dvd_lcm_right N M\n    have hM'' : (M' : ℕ) = L / M := by simp [hM']\n    have hz : z = M' * x := by\n      apply ZHat.torsionfree M\n      dsimp\n      rw [← hy₁, ← hx, ← mul_assoc, ← Nat.cast_mul, ← PNat.mul_coe, ← hM']\n    have hw : w = N' * x :=  by\n      apply ZHat.torsionfree N\n      dsimp\n      rw [← hy₂, ← hx, ← mul_assoc, ← Nat.cast_mul, ← PNat.mul_coe, ← hN']\n    -- If some prime divided L/M [ = M'] then it would have to divide N\n    -- which means that z is not in lowest terms;\n    -- similarly if some prime divided L/N [ = N'] then w/M would not be in lowest terms.\n    have dvd (n m p : Nat) (hm : 0 < m) : p ∣ (n.lcm m / m) → p ∣ n := by\n      intro h\n      rw [Nat.lcm_eq_mul_div] at h\n      rw [Nat.div_div_eq_div_mul] at h\n      rw [Nat.mul_div_mul_right _ _ hm] at h\n      apply h.trans\n      refine Nat.div_dvd_of_dvd ?_\n      exact Nat.gcd_dvd_left n m\n    -- We deduce that L = M = N and hence z = w by torsionfreeness.\n    have {n m : ℕ+} {Z : ZHat} (hcp : IsCoprime m Z) (hZ : ((n.lcm m / n : ℕ) : ZHat) ∣ Z) :\n        n.lcm m = n := by\n      rw [isCoprime_iff_coprime] at *\n      apply PNat.eq\n      symm\n      apply Nat.eq_of_dvd_of_div_eq_one\n      · refine PNat.dvd_iff.mp ?_\n        exact PNat.dvd_lcm_left n m\n      contrapose! hcp\n      let f := Nat.minFac (n.lcm m / n : ℕ)\n      have hf : f ∣ _ := Nat.minFac_dvd (n.lcm m / n : ℕ)\n      have hfprime : Nat.Prime f := Nat.minFac_prime <| by simpa\n      have := dvd m n f (by simp) (by simpa [← PNat.lcm_coe, Nat.lcm_comm] using hf)\n      apply Nat.not_coprime_of_dvd_of_dvd hfprime.one_lt this\n      obtain ⟨g, hg⟩ : (f : ZHat) ∣ Z := by\n        apply dvd_trans ?_ hZ\n        obtain ⟨g, hg⟩ := hf\n        simp only [PNat.lcm_coe] at hg\n        simp [hg]\n      rw [hg]\n      simp only [ZHat.nat_mul_apply, ZMod.val_mul, Nat.dvd_mod_iff this]\n      apply dvd_mul_of_dvd_left\n      simp only [ZMod.val_natCast]\n      rw [Nat.dvd_mod_iff this]\n    have hw' : ((L / N : ℕ) : ZHat) ∣ w := by\n      rw [hw, hN'']\n      exact dvd_mul_right _ _\n    have hz' : ((M.lcm N / M : ℕ) : ZHat) ∣ z := by\n      rw [hz, hM'', PNat.lcm_comm]\n      exact dvd_mul_right _ _\n    have hN : L = N := this hcpw hw'\n    have hM : L = M := PNat.lcm_comm _ _ |>.trans <| this hcpz hz'\n    have hNM' : N' = M' := by\n      apply mul_left_cancel (a := L)\n      conv_lhs =>\n        rw [hN, ← hN']\n      conv_rhs =>\n        rw [hM, ← hM']\n    rw [hz, hw, ← hN, ← hM, hNM']\n    exact ⟨rfl, rfl⟩\n","line":353,"end_line":493,"proof_status":"statement_formalized"}
+{"id":"Hurwitz.norm_one","label":"Hurwitz.norm_one","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":["Hurwitz.norm"],"nlp_text":"The norm of \\(1\\) is \\(1\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Hurwitz.norm_one","latex_link":"sect0002.html#Hurwitz.norm_one","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Hurwitz.norm_one","lean_decl_name":"Hurwitz.norm_one","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch5","source_path":"FLT/Data/Hurwitz.lean","lean4_code":"lemma norm_one : norm 1 = 1 := by simp [norm]\n","line":388,"end_line":388,"proof_status":"formalized"}
+{"id":"ZHat.e","label":"ZHat.e","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":["ZHat.commRing"],"nlp_text":"The infinite sum \\(0!+1!+2!+3!+4!+5!+\\cdots \\) looks like it makes no sense at all; it is the sum of an infinite series of larger and larger positive numbers. However, the sum is finite modulo \\(N\\) for every positive integer \\(N\\), because all the terms from \\(N!\\) onwards are multiples of \\(N\\) and thus are zero in \\(\\mathbb {Z}/N\\mathbb {Z}\\). Thus it makes sense to define \\(e_N\\) to be the value of the finite sum modulo \\(N\\). Explicitly, \\(e_N=0!+1!+\\cdots +(N-1)!\\) modulo \\(N\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/ZHat.e","latex_link":"sect0002.html#ZHat.e","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/ZHat.e","lean_decl_name":"ZHat.e","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch5","source_path":"./FLT/Deformations/Lemmas.lean","lean4_code":"instance {R S} [Semiring R] [Semiring S] (e : R ≃+* S) : IsLocalHom e.toRingHom :=\n  ⟨fun x hx ↦ by convert hx.map e.symm; simp⟩\n\n","line":154,"end_line":156,"proof_status":"formalized"}
+{"id":"ZHat.e_def","label":"ZHat.e_def","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":["ZHat.e"],"nlp_text":"The collection \\((e_N)_N\\) is an element of \\(\\widehat{\\mathbb {Z}}\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/ZHat.e_def","latex_link":"sect0002.html#ZHat.e_def","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/ZHat.e_def","lean_decl_name":"ZHat.e_def","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch5","source_path":"FLT/Data/QHat.lean","lean4_code":"lemma e_def (n : ℕ+) : e n = ∑ i ∈ range (n : ℕ), (i ! : ZMod n) := rfl\n","line":90,"end_line":90,"proof_status":"formalized"}
+{"id":"Hurwitz.norm","label":"Hurwitz.norm","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":["Hurwitz"],"nlp_text":"The Hurwitz quaternions come equipped with an integer-valued norm, which is \\(a^2+b^2+c^2+d^2\\) on \\(a+bi+cj+dk\\) but needs to be modified a bit to deal with \\(\\omega \\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Hurwitz.norm","latex_link":"sect0002.html#Hurwitz.norm","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Hurwitz.norm","lean_decl_name":"Hurwitz.norm","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch5","source_path":"FLT/Data/Hurwitz.lean","lean4_code":"def norm (z : 𝓞) : ℤ :=\n  z.re * z.re + z.im_o * z.im_o + z.im_i * z.im_i + z.im_oi * z.im_oi\n  - z.re * (z.im_o + z.im_oi) + z.im_i * (z.im_o - z.im_oi)\n","line":367,"end_line":369,"proof_status":"formalized"}
+{"id":"Hurwitz.norm_nonneg","label":"Hurwitz.norm_nonneg","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":["Hurwitz.norm"],"nlp_text":"The norm of an element is nonnegative.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Hurwitz.norm_nonneg","latex_link":"sect0002.html#Hurwitz.norm_nonneg","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Hurwitz.norm_nonneg","lean_decl_name":"Hurwitz.norm_nonneg","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch5","source_path":"FLT/Data/Hurwitz.lean","lean4_code":"lemma norm_nonneg (x : 𝓞) : 0 ≤ norm x := by\n  rw [← Int.cast_nonneg_iff (R := ℝ), coe_norm]\n  positivity\n","line":396,"end_line":398,"proof_status":"formalized"}
+{"id":"Hurwitz.norm_eq_zero","label":"Hurwitz.norm_eq_zero","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":["Hurwitz.norm"],"nlp_text":"The norm of an element is zero if and only if the element is zero.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Hurwitz.norm_eq_zero","latex_link":"sect0002.html#Hurwitz.norm_eq_zero","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Hurwitz.norm_eq_zero","lean_decl_name":"Hurwitz.norm_eq_zero","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch5","source_path":"FLT/Data/Hurwitz.lean","lean4_code":"lemma norm_eq_zero (x : 𝓞) : norm x = 0 ↔ x = 0 := by\n  constructor\n  swap\n  · rintro rfl; exact norm_zero\n  intro h\n  rw [← Int.cast_eq_zero (α := ℝ), coe_norm] at h\n  field_simp at h\n  norm_cast at h\n  have h4 := eq_zero_of_add_nonpos_right (by positivity) (by positivity) h.le\n  rw [sq_eq_zero_iff, sub_eq_zero] at h4\n  have h1 := eq_zero_of_add_nonpos_left (by positivity) (by positivity) h.le\n  have h3 := eq_zero_of_add_nonpos_right (by positivity) (by positivity) h1.le\n  rw [h4] at h3\n  simp only [ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, pow_eq_zero_iff, add_self_eq_zero] at h3\n  rw [h3] at h4\n  simp only [h4, sub_zero, h3, add_zero, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, zero_pow]\n    at h1\n  have h2 := eq_zero_of_add_nonpos_right (by positivity) (by positivity) h1.le\n  simp only [ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, pow_eq_zero_iff, mul_eq_zero,\n    false_or] at h2\n  simp only [h2, mul_zero, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, zero_pow, add_zero,\n    pow_eq_zero_iff, mul_eq_zero, or_false] at h1\n  ext <;> assumption\n","line":400,"end_line":422,"proof_status":"formalized"}
+{"id":"Hurwitz.exists_near","label":"Hurwitz.exists_near","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":["Hurwitz.norm"],"nlp_text":"Given a “usual” quaternion \\(a=x+yi+zj+wk\\) with \\(x,y,z,w\\in \\mathbb {R}\\), there exists a Hurwitz quaternion \\(q\\) such that \\(N(a-q){\\lt}1\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Hurwitz.exists_near","latex_link":"sect0002.html#Hurwitz.exists_near","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Hurwitz.exists_near","lean_decl_name":"Hurwitz.exists_near","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch5","source_path":"FLT/Data/Hurwitz.lean","lean4_code":"lemma exists_near (a : ℍ) : ∃ q : 𝓞, dist a (toQuaternion q) < 1 := by\n  have four_inv : (4⁻¹ : ℝ) = 2⁻¹ ^ 2 := by norm_num\n  have (r : ℝ) : (r - round r) ^ 2 ≤ 4⁻¹ := by\n    rw [four_inv, sq_le_sq]\n    apply (abs_sub_round _).trans_eq\n    rw [abs_of_nonneg]\n    all_goals norm_num\n  let x := round a.re\n  let y := round a.imI\n  let z := round a.imJ\n  let w := round a.imK\n  by_cases H : |a.re - x| = 2⁻¹ ∧ |a.imI - y| = 2⁻¹ ∧ |a.imJ - z| = 2⁻¹ ∧ |a.imK - w| = 2⁻¹\n  · use fromQuaternion a\n    convert zero_lt_one' ℝ\n    rw [NormedRing.dist_eq, ← sq_eq_zero_iff, sq, ← Quaternion.normSq_eq_norm_mul_self, normSq_def']\n    rw [add_eq_zero_iff_of_nonneg (by positivity) (by positivity)]\n    rw [add_eq_zero_iff_of_nonneg (by positivity) (by positivity)]\n    rw [add_eq_zero_iff_of_nonneg (by positivity) (by positivity)]\n    simp_rw [and_assoc, sq_eq_zero_iff, re_sub, imI_sub, imJ_sub, imK_sub, sub_eq_zero,\n      ← Quaternion.ext_iff]\n    symm\n    apply leftInvOn_toQuaternion_fromQuaternion\n    · simp only [Set.mem_setOf]\n      have {r : ℝ} {z : ℤ} (h : |r - z| = 2⁻¹) : ∃ z' : ℤ, r = z' + 2⁻¹  := by\n        cases (abs_eq (by positivity)).mp h with (rw [sub_eq_iff_eq_add'] at h)\n        | inl h => use z\n        | inr h => use z - 1; rw [h, Int.cast_sub, Int.cast_one, add_comm_sub]; norm_num\n      obtain ⟨x', hx'⟩ := this H.1\n      obtain ⟨y', hy'⟩ := this H.2.1\n      obtain ⟨z', hz'⟩ := this H.2.2.1\n      obtain ⟨w', hw'⟩ := this H.2.2.2\n      use x', y', z', w', Or.inr ?_\n      ext <;> simp [*]\n  use fromQuaternion ⟨x,y,z,w⟩\n  rw [aux]\n  rw [NormedRing.dist_eq, ← sq_lt_one_iff₀ (_root_.norm_nonneg _), sq,\n    ← Quaternion.normSq_eq_norm_mul_self, normSq_def']\n  simp only [re_sub, imI_sub, imJ_sub, imK_sub]\n  apply aux2 <;> try apply this\n  contrapose! H\n  suffices ∀ r : ℝ, |r| = 2⁻¹ ↔ r ^ 2 = 4⁻¹ by\n    simpa [this]\n  intro r\n  rw [four_inv, sq_eq_sq_iff_abs_eq_abs, abs_of_nonneg (a := (2⁻¹ : ℝ))]\n  norm_num\n","line":454,"end_line":498,"proof_status":"formalized"}
+{"id":"ZHat.commRing","label":"ZHat.commRing","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":["ZHat"],"nlp_text":"\\(\\widehat{\\mathbb {Z}}\\) is a subring of \\(\\prod _{N\\geq 1}(Z/N\\mathbb {Z})\\) and in particular is a ring.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/ZHat.commRing","latex_link":"sect0002.html#ZHat.commRing","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/ZHat.commRing","lean_decl_name":"ZHat.commRing","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/Data/QHat.lean","lean4_code":"instance commRing : CommRing ZHat := inferInstance","line":56,"end_line":56,"proof_status":"formalized"}
+{"id":"ZHat.nontrivial","label":"ZHat.nontrivial","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":["ZHat.commRing"],"nlp_text":"\\(0\\not=1\\) in \\(\\widehat{\\mathbb {Z}}\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/ZHat.nontrivial","latex_link":"sect0002.html#ZHat.nontrivial","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/ZHat.nontrivial","lean_decl_name":"ZHat.nontrivial","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Logic/Nontrivial/Basic.lean","lean4_code":"/-- As a convenience, provide an instance automatically if `(f default)` is nontrivial.\n\nIf a different index has the non-trivial type, then use `haveI := nontrivial_at that_index`.\n-/\n","line":84,"end_line":87,"proof_status":"formalized"}
+{"id":"QHat.IsCoprime","label":"QHat.IsCoprime","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":["ZHat.commRing"],"nlp_text":"If \\(N\\in \\mathbb {N}^+\\) and \\(z\\in \\widehat{\\mathbb {Z}}\\) then we say that \\(N\\) and \\(z\\) are coprime if \\(z_N\\in (\\mathbb {Z}/N\\mathbb {Z})^\\times \\). We write \\(z/N\\) as notation for the element \\(\\frac{1}{N}\\otimes _tz\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/QHat.IsCoprime","latex_link":"sect0002.html#QHat.IsCoprime","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/QHat.IsCoprime","lean_decl_name":"QHat.IsCoprime","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch5","source_path":"FLT/Data/QHat.lean","lean4_code":"def IsCoprime (N : ℕ+) (z : ZHat) : Prop := IsUnit (z N)\n","line":309,"end_line":309,"proof_status":"formalized"}
+{"id":"ZHat","label":"ZHat","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":[],"nlp_text":"The profinite completion \\(\\widehat{\\mathbb {Z}}\\) of \\(\\mathbb {Z}\\) is the set of all compatible collections \\(c=(c_N)_N\\) of elements of \\(\\mathbb {Z}/N\\mathbb {Z}\\) indexed by \\(\\mathbb {N}^+:=\\{ 1,2,3,\\ldots \\} \\). A collection is said to be compatible if for all positive integers \\(D\\mid N\\), we have \\(c_N\\) mod \\(D\\) equals \\(c_D\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/ZHat","latex_link":"sect0002.html#ZHat","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/ZHat","lean_decl_name":"ZHat","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch5","source_path":"FLT/Data/QHat.lean","lean4_code":"/-- We define the profinite completion of ℤ explicitly as compatible elements of ℤ/Nℤ for\nall positive integers `N`. We declare it as a subring of `∏_{N ≥ 1} (ℤ/Nℤ)`, and then promote it\nto a type. -/\ndef ZHat : Type := {\n  carrier := { f : Π M : ℕ+, ZMod M | ∀ (D N : ℕ+) (h : (D : ℕ) ∣ N),\n    ZMod.castHom h (ZMod D) (f N) = f D },\n  zero_mem' := by simp\n  neg_mem' := fun {x} hx => by\n    simp only [ZMod.castHom_apply, Set.mem_setOf_eq, Pi.neg_apply] at *\n    peel hx with D N hD hx\n    rw [ZMod.cast_neg hD, hx]\n  add_mem' := fun {a b} ha hb => by\n    simp only [ZMod.castHom_apply, Set.mem_setOf_eq, Pi.add_apply] at *\n    intro D N hD\n    rw [ZMod.cast_add hD, ha _ _ hD, hb _ _ hD]\n  one_mem' := by\n    simp only [ZMod.castHom_apply, Set.mem_setOf_eq, Pi.one_apply]\n    intro D N hD\n    rw [ZMod.cast_one hD]\n  mul_mem' := fun {a b} ha hb => by\n    simp only [ZMod.castHom_apply, Set.mem_setOf_eq, Pi.mul_apply] at *\n    intro D N hD\n    rw [ZMod.cast_mul hD, ha _ _ hD, hb _ _ hD]\n  : Subring (Π n : ℕ+, ZMod n)}\nderiving CommRing\n","line":11,"end_line":35,"proof_status":"formalized"}
+{"id":"Hurwitz.starRing","label":"Hurwitz.starRing","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":["Hurwitz.ring"],"nlp_text":"There’s a conjugation map (which we’ll call \"star\") from the Hurwitz quaternions to themselves, sending integers to themselves and purely imaginary elements like \\(2\\omega +1\\) to minus themselves. It satisfies \\((x^*)^*=x\\), \\((xy)^*=y^*x^*\\) and \\((x+y)^*=x^*+y^*\\). In particular, the Hurwitz quaternions are a \"star ring\" in the sense of mathlib.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Hurwitz.starRing","latex_link":"sect0002.html#Hurwitz.starRing","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Hurwitz.starRing","lean_decl_name":"Hurwitz.starRing","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Algebra/Star/RingQuot.lean","lean4_code":"def starRing {R : Type u} [Semiring R] [StarRing R] (r : R → R → Prop)\n    (hr : ∀ a b, r a b → r (star a) (star b)) : StarRing (RingQuot r) where\n  star := star' r hr\n  star_involutive := by\n    rintro ⟨⟨⟩⟩\n    simp [star'_quot]\n  star_mul := by\n    rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩\n    simp [star'_quot, mul_quot, star_mul]\n  star_add := by\n    rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩\n    simp [star'_quot, add_quot, star_add]\n\nend StarRing\n\n","line":41,"end_line":55,"proof_status":"formalized"}
+{"id":"Hurwitz.ring","label":"Hurwitz.ring","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":["Hurwitz"],"nlp_text":"The Hurwitz quaternions form a ring.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Hurwitz.ring","latex_link":"sect0002.html#Hurwitz.ring","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Hurwitz.ring","lean_decl_name":"Hurwitz.ring","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch5","source_path":"./FLT/Hacks/RightActionInstances.lean","lean4_code":"noncomputable section ring\n\nvariable (R A B M : Type*) [CommRing R]\n    [CommRing A] [Algebra R A]\n    [Ring B] [Algebra R B]\n    [AddCommGroup M] [Module R M]\n\nscoped instance [TopologicalSpace A] [Module.Finite R M] :\n    IsTopologicalAddGroup (M ⊗[R] A) := IsModuleTopology.topologicalAddGroup A (M ⊗[R] A)\n\nscoped instance [TopologicalSpace A] [IsTopologicalRing A] [Module.Finite R B] :\n    IsTopologicalRing (B ���[R] A) :=\n  IsModuleTopology.Module.topologicalRing A _\n\nscoped instance [TopologicalSpace A] [IsTopologicalRing A] [Module.Finite R B] :\n    IsTopologicalRing (B ⊗[R] A) :=\n  IsModuleTopology.Module.topologicalRing A _\n\nscoped instance [TopologicalSpace A] [IsTopologicalRing A]\n    [LocallyCompactSpace A] [Module.Finite R M] :\n    LocallyCompactSpace (M ⊗[R] A) := IsModuleTopology.locallyCompactSpaceOfFinite A\n\nend ring -- section\n\n","line":201,"end_line":224,"proof_status":"formalized"}
+{"id":"Hurwitz.surjective_pnat_quotient","label":"Hurwitz.surjective_pnat_quotient","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":["Hurwitz"],"nlp_text":"If \\(N\\) is a positive natural then the obvious map \\(\\mathcal{O}\\to \\widehat{\\mathcal{O}}/N\\widehat{\\mathcal{O}}\\) is surjective.","doc_url":null,"latex_link":"sect0002.html#Hurwitz.surjective_pnat_quotient","lean_link":null,"lean_decl_name":"Hurwitz.surjective_pnat_quotient","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"Hurwitz","label":"Hurwitz","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_5.html","dependencies":[],"nlp_text":"The Hurwitz quaternions are the set \\(\\mathcal{O}:= \\mathbb {Z}\\oplus \\mathbb {Z}\\omega \\oplus \\mathbb {Z}i\\oplus \\mathbb {Z}i\\omega \\) (as an abstract abelian group or as a subgroup of the usual quaternions). Here \\(\\omega =\\frac{-1+(i+j+k)}{2}\\) and note that \\((i+j+k)^2=-3\\). We have \\(\\overline{\\omega }=\\omega ^2=-(\\omega +1)\\). A general quaternion \\(a+bi+cj+dk\\) is a Hurwitz quaternion if either \\(a,b,c,d\\in \\mathbb {Z}\\) or \\(a,b,c,d\\in \\mathbb {Z}+\\frac{1}{2}\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Hurwitz","latex_link":"sect0002.html#Hurwitz","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Hurwitz","lean_decl_name":"Hurwitz","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch5","source_path":"./FLT/Data/Hurwitz.lean","lean4_code":"structure Hurwitz : Type where\n  re : ℤ -- 1\n  im_o : ℤ -- ω\n  im_i : ℤ -- i\n  im_oi : ℤ -- ωi -- note iω + ωi + 1 + i = 0\n\nnotation \"𝓞\" => Hurwitz -- 𝓞 = \\MCO\nnamespace Hurwitz\n\nopen Quaternion in\n","line":4,"end_line":13,"proof_status":"formalized"}

crawled_repos/FLT_ch6.jsonl

@@ -0,0 +1,2 @@
+{"id":"MatrixRing.isCentralSimple","label":"MatrixRing.isCentralSimple","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_6.html","dependencies":[],"nlp_text":"If \\(n\\geq 1\\) then the \\(n\\times n\\) matrices \\(M_n(K)\\) are a central simple algebra over \\(K\\).","doc_url":null,"latex_link":"sect0003.html#MatrixRing.isCentralSimple","lean_link":null,"lean_decl_name":"MatrixRing.isCentralSimple","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"IsCentralSimple.baseChange","label":"IsCentralSimple.baseChange","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_6.html","dependencies":[],"nlp_text":"If \\(D\\) is a central simple algebra over \\(K\\) and \\(L/K\\) is a field extension, then \\(L\\otimes _KD\\) is a central simple algebra over \\(L\\).","doc_url":null,"latex_link":"sect0003.html#IsCentralSimple.baseChange","lean_link":null,"lean_decl_name":"IsCentralSimple.baseChange","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch6","source_path":"./FLT/Deformations/RepresentationTheory/GaloisRep.lean","lean4_code":"def GaloisRep.baseChange [IsTopologicalRing B] [Algebra A B] [ContinuousSMul A B]\n    [Module.Finite A M] [Module.Free A M]\n    (ρ : GaloisRep K A M) : GaloisRep K B (B ⊗[A] M) :=\n  letI := moduleTopology A (Module.End A M)\n  letI := moduleTopology B (Module.End B (B ⊗[A] M))\n  letI : ContinuousMul _ := ⟨IsModuleTopology.continuous_mul_of_finite B (Module.End B (B ⊗[A] M))⟩\n  letI := IsModuleTopology.toContinuousAdd B (Module.End B (B ⊗[A] M))\n  let F : Module.End A M →+* Module.End B (B ⊗[A] M) := Module.End.baseChangeHom A B M\n  have : Continuous F := by\n    have : IsTopologicalSemiring (Module.End B (B ⊗[A] M)) := ⟨⟩\n    have : Continuous (algebraMap A (Module.End B (B ⊗[A] M))) := by\n      rw [IsScalarTower.algebraMap_eq A B, RingHom.coe_comp]\n      exact (continuous_algebraMap _ _).comp (continuous_algebraMap _ _)\n    exact IsModuleTopology.continuous_of_ringHom (R := A) F (by simpa [F])\n  .comp ⟨F, this⟩ ρ\n\nomit [IsTopologicalRing A] [NumberField K] in\nopen TensorProduct in\n","line":191,"end_line":208,"proof_status":"statement_formalized"}

crawled_repos/FLT_ch7.jsonl

@@ -0,0 +1,10 @@
+{"id":"AutomorphicForm.GLn.IsSmooth","label":"AutomorphicForm.GLn.IsSmooth","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_7.html","dependencies":[],"nlp_text":"A function \\(f:\\operatorname{GL}_n(\\mathbb {A}_{\\mathbb {Q}}^f)\\times \\operatorname{GL}_n(\\mathbb {R})\\to \\mathbb {C}\\) is smooth if it has the following three properties. \\(f\\) is continuous. For all \\(x\\in \\operatorname{GL}_n(\\mathbb {A}_{\\mathbb {Q}}^f)\\), the function \\(y\\mapsto f(x,y)\\) is smooth. For all \\(y\\in \\operatorname{GL}_n(\\mathbb {R})\\), the function \\(x\\mapsto f(x,y)\\) is locally constant.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/AutomorphicForm.GLn.IsSmooth","latex_link":"sect0004.html#AutomorphicForm.GLn.IsSmooth","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/AutomorphicForm.GLn.IsSmooth","lean_decl_name":"AutomorphicForm.GLn.IsSmooth","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch7","source_path":"FLT/GlobalLanglandsConjectures/GLnDefs.lean","lean4_code":"structure IsSmooth (f : GL (Fin n) (FiniteAdeleRing ℤ ℚ) × GL (Fin n) ℝ → ℂ) : Prop where\n  continuous : Continuous f\n  loc_cst (y : GL (Fin n) ℝ) :\n    IsLocallyConstant (fun x ↦ f (x, y))\n  smooth (x : GL (Fin n) (FiniteAdeleRing ℤ ℚ)) :\n    ContMDiff 𝓘(ℝ, Matrix (Fin n) (Fin n) ℝ) 𝓘(ℝ, ℂ) ∞ (fun y ↦ f (x, y))\n","line":199,"end_line":204,"proof_status":"statement_formalized"}
+{"id":"AutomorphicForm.GLn.AutomorphicFormForGLnOverQ","label":"AutomorphicForm.GLn.AutomorphicFormForGLnOverQ","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_7.html","dependencies":["AutomorphicForm.GLn.IsSmooth","instCentreAction","AutomorphicForm.GLn.Weight","AutomorphicForm.GLn.IsSlowlyIncreasing"],"nlp_text":"A smooth function \\(f:\\operatorname{GL}_n(\\mathbb {A}_{\\mathbb {Q}}^f)\\times \\operatorname{GL}_n(\\mathbb {R})\\to \\mathbb {C}\\) is an \\(O_n(\\mathbb {R})\\)-automorphic form on \\(\\operatorname{GL}_n(\\mathbb {A}_{\\mathbb {Q}})\\) if it satisfies the following five conditions. (periodicity) For all \\(g\\in \\operatorname{GL}_n(\\mathbb {Q})\\), we have \\(f(gx,gy)=f(x,y)\\). (has a finite level) There exists a compact open subgroup \\(U\\subseteq \\operatorname{GL}_n(\\mathbb {A}_{\\mathbb {Q}}^f)\\) such that \\(f(xu,y)=f(x,y)\\) for all \\(u\\in U\\), \\(x\\in \\operatorname{GL}_n(\\mathbb {A}_{\\mathbb {Q}}^f)\\) and \\(y\\in \\operatorname{GL}_n(\\mathbb {R})\\). (weight \\(\\rho \\)) There exists a continuous finite-dimensional irreducible complex representation \\(\\rho \\) of \\(O_n(\\mathbb {R})\\) such that for every \\((x,y)\\in \\operatorname{GL}_n(\\mathbb {A}_{\\mathbb {Q}})\\), the set of functions \\(k\\mapsto f(x,yk)\\) span a finite-dimensional complex vector space isomorphic as \\(O_n(\\mathbb {R})\\)-representation to a direct sum of copies of \\(\\rho \\). (has an infinite level) There is an ideal \\(I\\) of the centre \\(Z_n\\) described in the previous section, which has finite complex codimension, and which annihiliates the function \\(y \\mapsto f(x,y)\\) for all \\(x\\in \\operatorname{GL}_n(\\mathbb {A}_{\\mathbb {Q}}^f)\\). Note that this is a very fancy way of saying “the function satisfies some natural differential equations”. In the case of modular forms, the differential equations are the Cauchy-Riemann equations, which is why modular forms are holomorphic. (growth condition) For every \\(x\\in \\operatorname{GL}_n(\\mathbb {A}_{\\mathbb {Q}}^f)\\), the function \\(y\\mapsto f(x,y)\\) on \\(\\operatorname{GL}_n(\\mathbb {R})\\) is slowly-increasing.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/AutomorphicForm.GLn.AutomorphicFormForGLnOverQ","latex_link":"sect0004.html#AutomorphicForm.GLn.AutomorphicFormForGLnOverQ","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/AutomorphicForm.GLn.AutomorphicFormForGLnOverQ","lean_decl_name":"AutomorphicForm.GLn.AutomorphicFormForGLnOverQ","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch7","source_path":"FLT/GlobalLanglandsConjectures/GLnDefs.lean","lean4_code":"/-- Automorphic forms for GL_n/Q with weight ρ. -/\nstructure AutomorphicFormForGLnOverQ (n : ℕ) (ρ : Weight n) where\n  toFun : GL (Fin n) (FiniteAdeleRing ℤ ℚ) × GL (Fin n) ℝ → ℂ\n  is_smooth : IsSmooth toFun\n  is_periodic : ∀ (g : GL (Fin n) ℚ) (x : GL (Fin n) (FiniteAdeleRing ℤ ℚ)) (y : GL (Fin n) ℝ),\n    toFun (RingHom.GL (algebraMap _ _) _ g * x, RingHom.GL (algebraMap _ _) _ g * y) = toFun (x, y)\n  is_slowly_increasing (x : GL (Fin n) (FiniteAdeleRing ℤ ℚ)) :\n    IsSlowlyIncreasing (fun y ↦ toFun (x, y))\n  has_finite_level : ∃ U, IsConstantOn U toFun\n  is_finite_cod (x : GL (Fin n) (FiniteAdeleRing ℤ ℚ)) :\n    haveI f : C^∞⟮𝓘(ℝ, _), _; 𝓘(ℝ, ℂ), ℂ⟯ := ⟨fun y ↦ toFun (x, y), is_smooth.smooth x⟩\n    letI m := (actionTensorCAlg'3 (GL (Fin n) ℝ) (Matrix (Fin n) (Fin n) ℝ)).toLinearMap\n    FiniteDimensional ℂ (Z (GL (Fin n) ℝ) (Matrix (Fin n) (Fin n) ℝ) ⧸ (annihilator f).comap m)\n","line":261,"end_line":273,"proof_status":"statement_formalized"}
+{"id":"instComplexLieAlgebraAction","label":"instComplexLieAlgebraAction","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_7.html","dependencies":["instLieAlgebraAction"],"nlp_text":"This extends to is a natural complex Lie algebra action of the complexification of the real Lie algebra, on the smooth complex functions on \\(\\operatorname{GL}_n(\\mathbb {R})\\).","doc_url":null,"latex_link":"sect0004.html#instComplexLieAlgebraAction","lean_link":null,"lean_decl_name":"instComplexLieAlgebraAction","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"instUniversalEnvelopingAlgebraAction","label":"instUniversalEnvelopingAlgebraAction","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_7.html","dependencies":["instComplexLieAlgebraAction"],"nlp_text":"By functoriality, we get an action of the universal enveloping algebra of this complexified Lie algebra on the smooth complex functions.","doc_url":null,"latex_link":"sect0004.html#instUniversalEnvelopingAlgebraAction","lean_link":null,"lean_decl_name":"instUniversalEnvelopingAlgebraAction","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"instCentreAction","label":"instCentreAction","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_7.html","dependencies":["instUniversalEnvelopingAlgebraAction"],"nlp_text":"Thus the centre \\(\\mathbb {Z}_n\\) of this universal enveloping algebra also acts on the smooth complex functions.","doc_url":null,"latex_link":"sect0004.html#instCentreAction","lean_link":null,"lean_decl_name":"instCentreAction","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"instLieAlgebraAction","label":"instLieAlgebraAction","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_7.html","dependencies":[],"nlp_text":"There is a natural action of the real Lie algebra of \\(\\operatorname{GL}_n(\\mathbb {R})\\) on the complex vector space of smooth complex-valued functions on \\(\\operatorname{GL}_n(\\mathbb {R})\\).","doc_url":null,"latex_link":"sect0004.html#instLieAlgebraAction","lean_link":null,"lean_decl_name":"instLieAlgebraAction","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"AutomorphicForm.GLn.Weight","label":"AutomorphicForm.GLn.Weight","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_7.html","dependencies":[],"nlp_text":"The weight of an automorphic form for \\(\\operatorname{GL}_n/\\mathbb {Q}\\) can be thought of as a finite-dimensional continuous complex representation \\(\\rho \\) of a maximal compact subgroup of \\(\\operatorname{GL}_n(\\mathbb {R})\\), and it’s convenient to choose one (they’re all conjugate) so we choose \\(O_n(\\mathbb {R})\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/AutomorphicForm.GLn.Weight","latex_link":"sect0004.html#AutomorphicForm.GLn.Weight","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/AutomorphicForm.GLn.Weight","lean_decl_name":"AutomorphicForm.GLn.Weight","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch7","source_path":"FLT/GlobalLanglandsConjectures/GLnDefs.lean","lean4_code":"structure Weight (n : ℕ) where\n  w : preweight n\n  isSimple : Simple w.fdRep\n","line":239,"end_line":241,"proof_status":"statement_formalized"}
+{"id":"a0000000139","label":"a0000000139","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_7.html","dependencies":[],"nlp_text":"The group \\(\\operatorname{GL}_n(\\mathbb {A}_{\\mathbb {Q}}^f)\\) acts (on the left) on the space of automorphic forms for \\(\\operatorname{GL}_n(\\mathbb {A}_{\\mathbb {Q}})\\) by the formula \\((g\\cdot f)(x,y)=f(xg,y)\\).","doc_url":null,"latex_link":"sect0004.html#a0000000139","lean_link":null,"lean_decl_name":"a0000000139","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"AutomorphicForm.GLn.IsSlowlyIncreasing","label":"AutomorphicForm.GLn.IsSlowlyIncreasing","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_7.html","dependencies":[],"nlp_text":"We say that a function \\(f:\\operatorname{GL}_n(\\mathbb {R})\\to \\mathbb {C}\\) is slowly-increasing if there’s some real constant \\(C\\) and positive integer \\(n\\) such that \\(|f(M)|\\leq Cs(M)^n\\) for all \\(M\\in \\operatorname{GL}_n(\\mathbb {R})\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/AutomorphicForm.GLn.IsSlowlyIncreasing","latex_link":"sect0004.html#AutomorphicForm.GLn.IsSlowlyIncreasing","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/AutomorphicForm.GLn.IsSlowlyIncreasing","lean_decl_name":"AutomorphicForm.GLn.IsSlowlyIncreasing","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch7","source_path":"FLT/GlobalLanglandsConjectures/GLnDefs.lean","lean4_code":"structure IsSlowlyIncreasing (f : GeneralLinearGroup (Fin n) ℝ → ℂ) : Prop where\n  bounded_by : ∃ (C : ℝ) (N : ℕ),\n    ∀ (M : GeneralLinearGroup (Fin n) ℝ),\n    ‖f M‖ ≤ C * (s (M : Matrix (Fin n) (Fin n) ℝ)) ^ N\n","line":211,"end_line":214,"proof_status":"formalized"}
+{"id":"a0000000141","label":"a0000000141","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_7.html","dependencies":[],"nlp_text":"This function is well-defined, i.e., it sends a \\(U\\)-invariant form to a \\(U\\)-invariant form which is independent of the choice of \\(g_i\\).","doc_url":null,"latex_link":"sect0004.html#a0000000141","lean_link":null,"lean_decl_name":"a0000000141","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}

crawled_repos/FLT_ch8.jsonl

@@ -0,0 +1,12 @@
+{"id":"MulSemiringAction.monic_charpoly","label":"MulSemiringAction.monic_charpoly","background_color":"#1CAC78","border_color":"darkgreen","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_8.html","dependencies":["MulSemiringAction.charpoly"],"nlp_text":"\\(F_b\\) is monic.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MulSemiringAction.monic_charpoly","latex_link":"Frobenius_project.html#MulSemiringAction.monic_charpoly","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MulSemiringAction.monic_charpoly","lean_decl_name":"MulSemiringAction.monic_charpoly","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Invariant/Basic.lean","lean4_code":"theorem monic_charpoly (b : B) : (charpoly G b).Monic :=\n  monic_prod_of_monic _ _ (fun _ _ ↦ monic_X_sub_C _)\n\n","line":134,"end_line":136,"proof_status":"formalized"}
+{"id":"Algebra.IsInvariant.isIntegral","label":"Algebra.IsInvariant.isIntegral","background_color":"#1CAC78","border_color":"darkgreen","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_8.html","dependencies":["MulSemiringAction.monic_charpoly","Algebra.IsInvariant.charpoly_mem_lifts"],"nlp_text":"\\(B/A\\) is integral.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Algebra.IsInvariant.isIntegral","latex_link":"Frobenius_project.html#Algebra.IsInvariant.isIntegral","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Algebra.IsInvariant.isIntegral","lean_decl_name":"Algebra.IsInvariant.isIntegral","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/NumberTheory/NumberField/Basic.lean","lean4_code":"theorem isIntegral (x : 𝓞 K) : IsIntegral ℤ x := by\n  obtain ⟨P, hPm, hP⟩ := x.isIntegral_coe\n  refine ⟨P, hPm, ?_⟩\n  rwa [IsScalarTower.algebraMap_eq (S := 𝓞 K), ← Polynomial.hom_eval₂, coe_eq_zero_iff] at hP\n\n","line":238,"end_line":242,"proof_status":"formalized"}
+{"id":"fixed_of_fixed2","label":"fixed_of_fixed2","background_color":"#1CAC78","border_color":"darkgreen","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_8.html","dependencies":["Algebra.IsInvariant.isIntegral","IsAlgebraic.exists_smul_eq_mul","fixed_of_fixed1"],"nlp_text":"Let \\(x \\in L\\). Suppose that \\(x\\) is fixed by the stabilizer subgroup \\(D_Q\\). Then \\(x\\) is fixed by the automorphism group \\(\\operatorname{Aut}(L/K)\\).","doc_url":null,"latex_link":"Frobenius_project.html#fixed_of_fixed2","lean_link":null,"lean_decl_name":"fixed_of_fixed2","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Invariant/Basic.lean","lean4_code":"private theorem fixed_of_fixed2 (f : L ≃ₐ[K] L) (x : L)\n    (hx : ∀ g : MulAction.stabilizer G Q, IsFractionRing.stabilizerHom G P Q K L g x = x) :\n    f x = x := by\n  obtain ⟨_⟩ := nonempty_fintype G\n  have : P.IsPrime := Ideal.over_def Q P ▸ Ideal.IsPrime.under A Q\n  have : Algebra.IsIntegral A B := Algebra.IsInvariant.isIntegral A B G\n  obtain ⟨x, y, hy, rfl⟩ := IsFractionRing.div_surjective (A := B ⧸ Q) x\n  obtain ⟨b, a, ha, h⟩ := (Algebra.IsAlgebraic.isAlgebraic (R := A ⧸ P) y).exists_smul_eq_mul x hy\n  replace ha : algebraMap (A ⧸ P) L a ≠ 0 := by\n    rwa [Ne, algebraMap_apply (A ⧸ P) K L, algebraMap_eq_zero_iff, algebraMap_eq_zero_iff]\n  replace hy : algebraMap (B ⧸ Q) L y ≠ 0 :=\n    mt (algebraMap_eq_zero_iff (B ⧸ Q) L).mp (nonZeroDivisors.ne_zero hy)\n  replace h : algebraMap (B ⧸ Q) L x / algebraMap (B ⧸ Q) L y =\n      algebraMap (B ⧸ Q) L b / algebraMap (A ⧸ P) L a := by\n    rw [mul_comm, Algebra.smul_def, mul_comm] at h\n    rw [div_eq_div_iff hy ha, ← map_mul, ← h, map_mul, ← algebraMap_apply]\n  simp only [h, map_div₀, algebraMap_apply (A ⧸ P) K L, AlgEquiv.commutes] at hx ⊢\n  simp only [← algebraMap_apply, div_left_inj' ha] at hx ⊢\n  exact fixed_of_fixed1 G P Q K L f b (fun g ↦ IsFractionRing.injective (B ⧸ Q) L\n    ((IsFractionRing.fieldEquivOfAlgEquiv_algebraMap K L L\n      (Ideal.Quotient.stabilizerHom Q P G g) b).symm.trans (hx g)))\n\n/-- The stabilizer subgroup of `Q` surjects onto `Aut(Frac(B/Q)/Frac(A/P))`. -/\n","line":342,"end_line":364,"proof_status":"formalized"}
+{"id":"IsFractionRing.stabilizerHom_surjective","label":"IsFractionRing.stabilizerHom_surjective","background_color":"#1CAC78","border_color":"darkgreen","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_8.html","dependencies":["fixed_of_fixed2","FixedPoints.toAlgAut_surjective"],"nlp_text":"The map \\(g\\mapsto \\phi _g\\) from \\(D_Q\\) to \\(\\operatorname{Aut}(L/K)\\) defined above is surjective.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsFractionRing.stabilizerHom_surjective","latex_link":"Frobenius_project.html#IsFractionRing.stabilizerHom_surjective","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsFractionRing.stabilizerHom_surjective","lean_decl_name":"IsFractionRing.stabilizerHom_surjective","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/leanprover-community/mathlib4/98379c78c5dd2e3614cd24ab7ad22f5713e0eb63/Mathlib/RingTheory/Invariant/Basic.lean","lean4_code":"/-- The stabilizer subgroup of `Q` surjects onto `Aut(Frac(B/Q)/Frac(A/P))`. -/\ntheorem IsFractionRing.stabilizerHom_surjective :\n    Function.Surjective (stabilizerHom G P Q K L) := by\n  let _ := MulSemiringAction.compHom L (stabilizerHom G P Q K L)\n  intro f\n  obtain ⟨g, hg⟩ := FixedPoints.toAlgAut_surjective (MulAction.stabilizer G Q) L\n    (AlgEquiv.ofRingEquiv (f := f) (fun x ↦ fixed_of_fixed2 G P Q K L f x x.2))\n  exact ⟨g, by rwa [AlgEquiv.ext_iff] at hg ⊢⟩","line":375,"end_line":382,"proof_status":"formalized"}
+{"id":"Algebra.IsInvariant.charpoly_mem_lifts","label":"Algebra.IsInvariant.charpoly_mem_lifts","background_color":"#1CAC78","border_color":"darkgreen","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_8.html","dependencies":["MulSemiringAction.charpoly"],"nlp_text":"\\(F_b\\) is the lift of some monic polynomial \\(M_b\\) in \\(A[X]\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Algebra.IsInvariant.charpoly_mem_lifts","latex_link":"Frobenius_project.html#Algebra.IsInvariant.charpoly_mem_lifts","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Algebra.IsInvariant.charpoly_mem_lifts","lean_decl_name":"Algebra.IsInvariant.charpoly_mem_lifts","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Invariant/Basic.lean","lean4_code":"theorem charpoly_mem_lifts [Fintype G] (b : B) :\n    charpoly G b ∈ Polynomial.lifts (algebraMap A B) :=\n  (charpoly G b).lifts_iff_coeff_lifts.mpr fun n ↦ isInvariant _ (smul_coeff_charpoly b n)\n\n","line":159,"end_line":162,"proof_status":"formalized"}
+{"id":"fixed_of_fixed1_aux1","label":"fixed_of_fixed1_aux1","background_color":"#1CAC78","border_color":"darkgreen","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_8.html","dependencies":["Algebra.IsInvariant.charpoly_mem_lifts"],"nlp_text":"There exist elements \\(a,b \\in B\\), with \\(a \\notin Q\\) and \\(a\\) in the image of \\(A\\) such that for all \\(g\\in G\\), If \\(g \\cdot Q = Q\\), then \\(g \\cdot b \\equiv a \\pmod{Q}\\). If \\(g \\cdot Q \\neq Q\\), then \\(g \\cdot b \\equiv 0 \\pmod{Q}\\).","doc_url":null,"latex_link":"Frobenius_project.html#fixed_of_fixed1_aux1","lean_link":null,"lean_decl_name":"fixed_of_fixed1_aux1","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Invariant/Basic.lean","lean4_code":"private theorem fixed_of_fixed1_aux1 [DecidableEq (Ideal B)] :\n    ∃ a b : B, (∀ g : G, g • a = a) ∧ a ∉ Q ∧\n    ∀ g : G, algebraMap B (B ⧸ Q) (g • b) = algebraMap B (B ⧸ Q) (if g • Q = Q then a else 0) := by\n  obtain ⟨_⟩ := nonempty_fintype G\n  let P := Finset.inf {g : G | g • Q ≠ Q} (fun g ↦ g • Q)\n  have h1 : ¬ P ≤ Q := by\n    rw [Ideal.IsPrime.inf_le' inferInstance]\n    rintro ⟨g, hg1, hg2⟩\n    exact (Finset.mem_filter.mp hg1).2 (smul_eq_of_smul_le hg2)\n  obtain ⟨b, hbP, hbQ⟩ := SetLike.not_le_iff_exists.mp h1\n  replace hbP : ∀ g : G, g • Q ≠ Q → b ∈ g • Q :=\n    fun g hg ↦ (Finset.inf_le (Finset.mem_filter.mpr ⟨Finset.mem_univ g, hg⟩) : P ≤ g • Q) hbP\n  let f := MulSemiringAction.charpoly G b\n  obtain ⟨q, hq, hq0⟩ :=\n    (f.map (algebraMap B (B ⧸ Q))).exists_eq_pow_rootMultiplicity_mul_and_not_dvd\n      (Polynomial.map_monic_ne_zero (MulSemiringAction.monic_charpoly G b)) 0\n  rw [map_zero, sub_zero] at hq hq0\n  let j := (f.map (algebraMap B (B ⧸ Q))).rootMultiplicity 0\n  let k := q.natDegree\n  let r := ∑ i ∈ Finset.range (k + 1), Polynomial.monomial i (f.coeff (i + j))\n  have hr : r.map (algebraMap B (B ⧸ Q)) = q := by\n    ext n\n    rw [Polynomial.coeff_map, Polynomial.finset_sum_coeff]\n    simp only [Polynomial.coeff_monomial, Finset.sum_ite_eq', Finset.mem_range_succ_iff]\n    split_ifs with hn\n    · rw [← Polynomial.coeff_map, hq, Polynomial.coeff_X_pow_mul]\n    · rw [map_zero, eq_comm, Polynomial.coeff_eq_zero_of_natDegree_lt (lt_of_not_ge hn)]\n  have hf : f.eval b = 0 := MulSemiringAction.eval_charpoly G b\n  have hr : r.eval b ∈ Q := by\n    rw [← Ideal.Quotient.eq_zero_iff_mem, ← Ideal.Quotient.algebraMap_eq] at hbQ ⊢\n    replace hf := congrArg (algebraMap B (B ⧸ Q)) hf\n    rw [← Polynomial.eval₂_at_apply, ← Polynomial.eval_map] at hf ⊢\n    rwa [map_zero, hq, ← hr, Polynomial.eval_mul, Polynomial.eval_pow, Polynomial.eval_X,\n      mul_eq_zero, or_iff_right (pow_ne_zero _ hbQ)] at hf\n  let a := f.coeff j\n  have ha : ∀ g : G, g • a = a := MulSemiringAction.smul_coeff_charpoly b j\n  have hr' : ∀ g : G, g • Q ≠ Q → a - r.eval b ∈ g • Q := by\n    intro g hg\n    have hr : r = ∑ i ∈ Finset.range (k + 1), Polynomial.monomial i (f.coeff (i + j)) := rfl\n    rw [← Ideal.neg_mem_iff, neg_sub, hr, Finset.sum_range_succ', Polynomial.eval_add,\n        Polynomial.eval_monomial, zero_add, pow_zero, mul_one, add_sub_cancel_right]\n    simp only [ ← Polynomial.monomial_mul_X]\n    rw [← Finset.sum_mul, Polynomial.eval_mul_X]\n    exact Ideal.mul_mem_left (g • Q) _ (hbP g hg)\n  refine ⟨a, a - r.eval b, ha, ?_, fun h ↦ ?_⟩\n  · rwa [← Ideal.Quotient.eq_zero_iff_mem, ← Ideal.Quotient.algebraMap_eq, ← Polynomial.coeff_map,\n      ← zero_add j, hq, Polynomial.coeff_X_pow_mul, ← Polynomial.X_dvd_iff]\n  · rw [← sub_eq_zero, ← map_sub, Ideal.Quotient.algebraMap_eq, Ideal.Quotient.eq_zero_iff_mem,\n      ← Ideal.smul_mem_pointwise_smul_iff (a := h⁻¹), smul_sub, inv_smul_smul]\n    simp only [← eq_inv_smul_iff (g := h), eq_comm (a := Q)]\n","line":218,"end_line":267,"proof_status":"formalized"}
+{"id":"fixed_of_fixed1_aux2","label":"fixed_of_fixed1_aux2","background_color":"#1CAC78","border_color":"darkgreen","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_8.html","dependencies":["fixed_of_fixed1_aux1"],"nlp_text":"Let \\(b_0 \\in B\\). Suppose that the image of \\(b_0\\) in the quotient \\(B/Q\\) is fixed by the stabilizer subgroup \\(D_Q\\). Then there exist elements \\(a,b \\in B\\), with \\(a\\notin Q\\) and \\(a\\) in the image of \\(A\\) such that for all \\(g\\in G\\), If \\(g \\cdot Q = Q\\), then \\(g \\cdot b \\equiv ab_0 \\pmod{Q}\\). If \\(g \\cdot Q \\neq Q\\), then \\(g \\cdot b \\equiv 0 \\pmod{Q}\\).","doc_url":null,"latex_link":"Frobenius_project.html#fixed_of_fixed1_aux2","lean_link":null,"lean_decl_name":"fixed_of_fixed1_aux2","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Invariant/Basic.lean","lean4_code":"private theorem fixed_of_fixed1_aux2 [DecidableEq (Ideal B)] (b₀ : B)\n    (hx : ∀ g : G, g • Q = Q → algebraMap B (B ⧸ Q) (g • b₀) = algebraMap B (B ⧸ Q) b₀) :\n    ∃ a b : B, (∀ g : G, g • a = a) ∧ a ∉ Q ∧\n    (∀ g : G, algebraMap B (B ⧸ Q) (g • b) =\n      algebraMap B (B ⧸ Q) (if g • Q = Q then a * b₀ else 0)) := by\n  obtain ⟨a, b, ha1, ha2, hb⟩ := fixed_of_fixed1_aux1 G Q\n  refine ⟨a, b * b₀, ha1, ha2, fun g ↦ ?_⟩\n  rw [smul_mul', map_mul, hb]\n  specialize hb g\n  split_ifs with hg\n  · rw [map_mul, hx g hg]\n  · rw [map_zero, zero_mul]\n\n/-- A technical lemma for `fixed_of_fixed1`. -/\nprivate theorem fixed_of_fixed1_aux3 [NoZeroDivisors B] {b : B} {i j : ℕ} {p : Polynomial A}\n    (h : p.map (algebraMap A B) = (X - C b) ^ i * X ^ j) (f : B ≃ₐ[A] B) (hi : i ≠ 0) :\n    f b = b := by\n  by_cases ha : b = 0\n  · rw [ha, map_zero]\n  have hf := congrArg (eval b) (congrArg (Polynomial.mapAlgHom f.toAlgHom) h)\n  rw [coe_mapAlgHom, map_map, f.toAlgHom.comp_algebraMap, h] at hf\n  simp_rw [Polynomial.map_mul, Polynomial.map_pow, Polynomial.map_sub, map_X, map_C,\n    eval_mul, eval_pow, eval_sub, eval_X, eval_C, sub_self, zero_pow hi, zero_mul,\n    zero_eq_mul, or_iff_left (pow_ne_zero j ha), pow_eq_zero_iff hi, sub_eq_zero] at hf\n  exact hf.symm\n\n/-- This theorem will be made redundant by `IsFractionRing.stabilizerHom_surjective`. -/\nprivate theorem fixed_of_fixed1 [NoZeroSMulDivisors (B ⧸ Q) L] (f : L ≃ₐ[K] L) (b : B ⧸ Q)\n    (hx : ∀ g : MulAction.stabilizer G Q, Ideal.Quotient.stabilizerHom Q P G g b = b) :\n    f (algebraMap (B ⧸ Q) L b) = (algebraMap (B ⧸ Q) L b) := by\n  classical\n  cases nonempty_fintype G\n  obtain ⟨b₀, rfl⟩ := Ideal.Quotient.mk_surjective b\n  rw [← Ideal.Quotient.algebraMap_eq]\n  obtain ⟨a, b, ha1, ha2, hb⟩ := fixed_of_fixed1_aux2 G Q b₀ (fun g hg ↦ hx ⟨g, hg⟩)\n  obtain ⟨M, key⟩ := (mem_lifts _).mp (Algebra.IsInvariant.charpoly_mem_lifts A B G b)\n  replace key := congrArg (map (algebraMap B (B ⧸ Q))) key\n  rw [map_map, ← algebraMap_eq, algebraMap_eq A (A ⧸ P) (B ⧸ Q),\n      ← map_map, MulSemiringAction.charpoly, Polynomial.map_prod] at key\n  have key₀ : ∀ g : G, (X - C (g • b)).map (algebraMap B (B ⧸ Q)) =\n      if g • Q = Q then X - C (algebraMap B (B ⧸ Q) (a * b₀)) else X := by\n    intro g\n    rw [Polynomial.map_sub, map_X, map_C, hb]\n    split_ifs\n    · rfl\n    · rw [map_zero, map_zero, sub_zero]\n  simp only [key₀, Finset.prod_ite, Finset.prod_const] at key\n  replace key := congrArg (map (algebraMap (B ⧸ Q) L)) key\n  rw [map_map, ← algebraMap_eq, algebraMap_eq (A ⧸ P) K L,\n      ← map_map, Polynomial.map_mul, Polynomial.map_pow, Polynomial.map_pow, Polynomial.map_sub,\n","line":274,"end_line":323,"proof_status":"formalized"}
+{"id":"fixed_of_fixed1","label":"fixed_of_fixed1","background_color":"#1CAC78","border_color":"darkgreen","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_8.html","dependencies":["fixed_of_fixed1_aux2","IsFractionRing.stabilizerHom"],"nlp_text":"Let \\(b_0 \\in B/Q\\). Suppose that \\(b_0\\) is fixed by the stabilizer subgroup \\(D_Q\\). Then \\(b_0\\) is fixed by \\(\\operatorname{Aut}(L/K)\\).","doc_url":null,"latex_link":"Frobenius_project.html#fixed_of_fixed1","lean_link":null,"lean_decl_name":"fixed_of_fixed1","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Invariant/Basic.lean","lean4_code":"private theorem fixed_of_fixed1 [NoZeroSMulDivisors (B ⧸ Q) L] (f : L ≃ₐ[K] L) (b : B ⧸ Q)\n    (hx : ∀ g : MulAction.stabilizer G Q, Ideal.Quotient.stabilizerHom Q P G g b = b) :\n    f (algebraMap (B ⧸ Q) L b) = (algebraMap (B ⧸ Q) L b) := by\n  classical\n  cases nonempty_fintype G\n  obtain ⟨b₀, rfl⟩ := Ideal.Quotient.mk_surjective b\n  rw [← Ideal.Quotient.algebraMap_eq]\n  obtain ⟨a, b, ha1, ha2, hb⟩ := fixed_of_fixed1_aux2 G Q b₀ (fun g hg ↦ hx ⟨g, hg⟩)\n  obtain ⟨M, key⟩ := (mem_lifts _).mp (Algebra.IsInvariant.charpoly_mem_lifts A B G b)\n  replace key := congrArg (map (algebraMap B (B ⧸ Q))) key\n  rw [map_map, ← algebraMap_eq, algebraMap_eq A (A ⧸ P) (B ⧸ Q),\n      ← map_map, MulSemiringAction.charpoly, Polynomial.map_prod] at key\n  have key₀ : ∀ g : G, (X - C (g • b)).map (algebraMap B (B ⧸ Q)) =\n      if g • Q = Q then X - C (algebraMap B (B ⧸ Q) (a * b₀)) else X := by\n    intro g\n    rw [Polynomial.map_sub, map_X, map_C, hb]\n    split_ifs\n    · rfl\n    · rw [map_zero, map_zero, sub_zero]\n  simp only [key₀, Finset.prod_ite, Finset.prod_const] at key\n  replace key := congrArg (map (algebraMap (B ⧸ Q) L)) key\n  rw [map_map, ← algebraMap_eq, algebraMap_eq (A ⧸ P) K L,\n      ← map_map, Polynomial.map_mul, Polynomial.map_pow, Polynomial.map_pow, Polynomial.map_sub,\n      map_X, map_C] at key\n  replace key := fixed_of_fixed1_aux3 key f (Finset.card_ne_zero_of_mem\n    (Finset.mem_filter.mpr ⟨Finset.mem_univ 1, one_smul G Q⟩))\n  simp only [map_mul] at key\n  obtain ⟨a, rfl⟩ := Algebra.IsInvariant.isInvariant (A := A) a ha1\n  rwa [← algebraMap_apply A B (B ⧸ Q), algebraMap_apply A (A ⧸ P) (B ⧸ Q),\n      ← algebraMap_apply, algebraMap_apply (A ⧸ P) K L, f.commutes, mul_right_inj'] at key\n  rwa [← algebraMap_apply, algebraMap_apply (A ⧸ P) (B ⧸ Q) L,\n      ← algebraMap_apply A (A ⧸ P) (B ⧸ Q), algebraMap_apply A B (B ⧸ Q),\n      Ne, algebraMap_eq_zero_iff, Ideal.Quotient.algebraMap_eq, Ideal.Quotient.eq_zero_iff_mem]\n\nvariable [IsFractionRing (A ⧸ P) K] [IsFractionRing (B ⧸ Q) L]\n\n/-- If `Q` lies over `P`, then the stabilizer of `Q` acts on `Frac(B/Q)/Frac(A/P)`. -/\n","line":301,"end_line":337,"proof_status":"formalized"}
+{"id":"MulSemiringAction.charpoly","label":"MulSemiringAction.charpoly","background_color":"#B0ECA3","border_color":"darkgreen","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_8.html","dependencies":[],"nlp_text":"If \\(b\\in B\\) then define the characteristic polynomial \\(F_b(X) \\in B[X]\\) of \\(b\\) to be \\(\\prod _{g\\in G}(X-g\\cdot b)\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MulSemiringAction.charpoly","latex_link":"Frobenius_project.html#MulSemiringAction.charpoly","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MulSemiringAction.charpoly","lean_decl_name":"MulSemiringAction.charpoly","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch8","source_path":"./FLT/Deformations/RepresentationTheory/GaloisRep.lean","lean4_code":"def GaloisRep.charFrob (ρ : GaloisRep K A M) : Polynomial A := (ρ.toLocal v Frobᵥ).charpoly\n\nomit [IsTopologicalRing A] in\n","line":316,"end_line":318,"proof_status":"formalized"}
+{"id":"IsAlgebraic.exists_smul_eq_mul","label":"IsAlgebraic.exists_smul_eq_mul","background_color":"#1CAC78","border_color":"darkgreen","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_8.html","dependencies":[],"nlp_text":"If \\(R/S\\) is an algebraic extension of integral domains, then any fraction \\(a/b\\) with \\(a,b\\in R\\) can be written as \\(c/d\\) with \\(c\\in R\\) and \\(d\\in S\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsAlgebraic.exists_smul_eq_mul","latex_link":"Frobenius_project.html#IsAlgebraic.exists_smul_eq_mul","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsAlgebraic.exists_smul_eq_mul","lean_decl_name":"IsAlgebraic.exists_smul_eq_mul","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/leanprover-community/mathlib4/98379c78c5dd2e3614cd24ab7ad22f5713e0eb63/Mathlib/RingTheory/Algebraic/Basic.lean","lean4_code":"/-- A fraction `(a : S) / (b : S)` can be reduced to `(c : S) / (d : R)`,\nif `b` is algebraic over `R`. -/\ntheorem IsAlgebraic.exists_smul_eq_mul\n    (a : S) {b : S} (hRb : IsAlgebraic R b) (hb : b ∈ S⁰) :\n    ∃ᵉ (c : S) (d ≠ (0 : R)), d • a = b * c :=\n  have ⟨r, hr, s, h⟩ := hRb.exists_nonzero_dvd hb\n  ⟨s * a, r, hr, by rw [Algebra.smul_def, h, mul_assoc]⟩","line":557,"end_line":563,"proof_status":"formalized"}
+{"id":"IsFractionRing.stabilizerHom","label":"IsFractionRing.stabilizerHom","background_color":"#B0ECA3","border_color":"darkgreen","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_8.html","dependencies":[],"nlp_text":"Choose \\(g\\in D_Q\\). Then the action of \\(g\\) on \\(B\\) gives us an induced \\(A/P\\)-algebra automorphism of \\(B/Q\\) which extends to a \\(K\\)-algebra automorphism \\(\\phi (g)\\) of \\(L\\). This construction \\(g\\mapsto \\phi (g)\\) defines a group homomorphism from \\(D_Q\\) to \\(\\operatorname{Aut}(L/K)\\) (all the proofs implicit in the definition here are straightforward).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsFractionRing.stabilizerHom","latex_link":"Frobenius_project.html#IsFractionRing.stabilizerHom","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsFractionRing.stabilizerHom","lean_decl_name":"IsFractionRing.stabilizerHom","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Invariant/Basic.lean","lean4_code":"noncomputable def IsFractionRing.stabilizerHom : MulAction.stabilizer G Q →* (L ≃ₐ[K] L) :=\n  MonoidHom.comp (IsFractionRing.fieldEquivOfAlgEquivHom K L) (Ideal.Quotient.stabilizerHom Q P G)\n\n/-- This theorem will be made redundant by `IsFractionRing.stabilizerHom_surjective`. -/\nprivate theorem fixed_of_fixed2 (f : L ≃ₐ[K] L) (x : L)\n    (hx : ∀ g : MulAction.stabilizer G Q, IsFractionRing.stabilizerHom G P Q K L g x = x) :\n    f x = x := by\n  obtain ⟨_⟩ := nonempty_fintype G\n  have : P.IsPrime := Ideal.over_def Q P ▸ Ideal.IsPrime.under A Q\n  have : Algebra.IsIntegral A B := Algebra.IsInvariant.isIntegral A B G\n  obtain ⟨x, y, hy, rfl⟩ := IsFractionRing.div_surjective (A := B ⧸ Q) x\n  obtain ⟨b, a, ha, h⟩ := (Algebra.IsAlgebraic.isAlgebraic (R := A ⧸ P) y).exists_smul_eq_mul x hy\n  replace ha : algebraMap (A ⧸ P) L a ≠ 0 := by\n    rwa [Ne, algebraMap_apply (A ⧸ P) K L, algebraMap_eq_zero_iff, algebraMap_eq_zero_iff]\n  replace hy : algebraMap (B ⧸ Q) L y ≠ 0 :=\n    mt (algebraMap_eq_zero_iff (B ⧸ Q) L).mp (nonZeroDivisors.ne_zero hy)\n  replace h : algebraMap (B ⧸ Q) L x / algebraMap (B ⧸ Q) L y =\n      algebraMap (B ⧸ Q) L b / algebraMap (A ⧸ P) L a := by\n    rw [mul_comm, Algebra.smul_def, mul_comm] at h\n    rw [div_eq_div_iff hy ha, ← map_mul, ← h, map_mul, ← algebraMap_apply]\n  simp only [h, map_div₀, algebraMap_apply (A ⧸ P) K L, AlgEquiv.commutes] at hx ⊢\n  simp only [← algebraMap_apply, div_left_inj' ha] at hx ⊢\n  exact fixed_of_fixed1 G P Q K L f b (fun g ↦ IsFractionRing.injective (B ⧸ Q) L\n    ((IsFractionRing.fieldEquivOfAlgEquiv_algebraMap K L L\n      (Ideal.Quotient.stabilizerHom Q P G g) b).symm.trans (hx g)))\n\n/-- The stabilizer subgroup of `Q` surjects onto `Aut(Frac(B/Q)/Frac(A/P))`. -/\n","line":338,"end_line":364,"proof_status":"formalized"}
+{"id":"FixedPoints.toAlgAut_surjective","label":"FixedPoints.toAlgAut_surjective","background_color":"#1CAC78","border_color":"darkgreen","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_8.html","dependencies":[],"nlp_text":"Let \\(H\\) be a finite group acting on a field \\(F\\) by field automorphisms. Then the map \\(H \\to \\operatorname{Aut}(F/F^H)\\) is surjective.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FixedPoints.toAlgAut_surjective","latex_link":"Frobenius_project.html#FixedPoints.toAlgAut_surjective","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FixedPoints.toAlgAut_surjective","lean_decl_name":"FixedPoints.toAlgAut_surjective","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/FieldTheory/Fixed.lean","lean4_code":"theorem toAlgAut_surjective [Finite G] :\n    Function.Surjective (MulSemiringAction.toAlgAut G (FixedPoints.subfield G F) F) := by\n  let f : G →* F ≃ₐ[FixedPoints.subfield G F] F :=\n    MulSemiringAction.toAlgAut G (FixedPoints.subfield G F) F\n  let Q := G ⧸ f.ker\n  let _ : MulSemiringAction Q F := MulSemiringAction.compHom _ (QuotientGroup.kerLift f)\n  have : FaithfulSMul Q F := ⟨by\n    intro q₁ q₂\n    refine Quotient.inductionOn₂' q₁ q₂ (fun g₁ g₂ h ↦ QuotientGroup.eq.mpr ?_)\n    rwa [MonoidHom.mem_ker, map_mul, map_inv, inv_mul_eq_one, AlgEquiv.ext_iff]⟩\n  intro f\n  obtain ⟨q, hq⟩ := (toAlgAut_bijective Q F).surjective\n    (AlgEquiv.ofRingEquiv (f := f) (fun ⟨x, hx⟩ ↦ f.commutes' ⟨x, fun g ↦ hx g⟩))\n  revert hq\n  refine QuotientGroup.induction_on q (fun g hg ↦ ⟨g, ?_⟩)\n  rwa [AlgEquiv.ext_iff] at hg ⊢\n\n","line":360,"end_line":376,"proof_status":"formalized"}

crawled_repos/FLT_ch9.jsonl

@@ -0,0 +1,52 @@
+{"id":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom'","label":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom'","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom"],"nlp_text":"The product of the maps \\(K_v\\to L_w\\) for \\(w|v\\) is a natural ring map \\(K_v\\to \\prod _{w|v}L_w\\) lying over \\(K\\to L\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom'","latex_link":"Adele_miniproject.html#IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom'","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom'","lean_decl_name":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom'","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/DedekindDomain/Completion/BaseChange.lean","lean4_code":"/-- The canonical map `K_v → ∏_{w|v} L_w` extending K → L. -/\nnoncomputable def adicCompletionComapSemialgHom' (v : HeightOneSpectrum A) :\n  v.adicCompletion K →ₛₐ[algebraMap K L] ∀ w : v.Extension B, w.1.adicCompletion L :=\n  Pi.semialgHom _ _ fun i ↦ adicCompletionComapSemialgHom A K L B v i.1 i.2","line":205,"end_line":208,"proof_status":"formalized"}
+{"id":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapAlgEquiv","label":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapAlgEquiv","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom'","IsDedekindDomain.HeightOneSpectrum.Extension.finite"],"nlp_text":"The induced \\(L\\)-algebra homomorphism \\(L\\otimes _KK_v\\to \\prod _{w|v}L_w\\) is an isomorphism of rings.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.adicCompletionComapAlgEquiv","latex_link":"Adele_miniproject.html#IsDedekindDomain.HeightOneSpectrum.adicCompletionComapAlgEquiv","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.adicCompletionComapAlgEquiv","lean_decl_name":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapAlgEquiv","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/DedekindDomain/Completion/BaseChange.lean","lean4_code":"/-- The L-algebra isomorphism `L ⊗[K] K_v ≅ ∏_{w|v} L_w`. -/\nnoncomputable def adicCompletionComapAlgEquiv (v : HeightOneSpectrum A) :\n    L ⊗[K] v.adicCompletion K ≃ₐ[L] (∀ w : v.Extension B, w.1.adicCompletion L) :=\n  AlgEquiv.ofBijective (tensorAdicCompletionComapAlgHom A K L B v) <|\n    tensorAdicCompletionComapAlgHom_bijective A K L B v\n","line":889,"end_line":893,"proof_status":"statement_formalized"}
+{"id":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapAlgEquiv_integral","label":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapAlgEquiv_integral","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.HeightOneSpectrum.adicCompletionComapAlgEquiv"],"nlp_text":"The isomorphism \\(L\\otimes _KK_v\\to \\prod _{w|v}L_w\\) induces an isomorphism \\(B\\otimes _AA_v\\to \\prod _{w|v}B_w\\) for all \\(v\\) in the height one spectrum of \\(A\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.adicCompletionComapAlgEquiv_integral","latex_link":"Adele_miniproject.html#IsDedekindDomain.HeightOneSpectrum.adicCompletionComapAlgEquiv_integral","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.adicCompletionComapAlgEquiv_integral","lean_decl_name":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapAlgEquiv_integral","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/DedekindDomain/Completion/BaseChange.lean","lean4_code":"/-- The image of `B ⊗[A] 𝓞_v` in `∏_w L_w` is `∏_w 𝓞_w`. -/\ntheorem adicCompletionComapAlgEquiv_integral :\n    AlgHom.range (((tensorAdicCompletionComapAlgHom A K L B v).restrictScalars B).comp\n      (tensorAdicCompletionIntegersTo A K L B v)) =\n    Subalgebra.pi Set.univ (fun _ ↦ adicCompletionIntegersSubalgebra _ _) := by\n  have hlhs := tensorAdicCompletionIntegersToRange_eq_closureIntegers\n  have hrhs := prodAdicCompletionsIntegers_eq_closureIntegers\n  have hrange :\n    SetLike.coe (algebraMap B ((w : Extension B v) → adicCompletion L w.1)).range =\n      (tensorAdicCompletionComapAlgHom A K L B v) ''\n      SetLike.coe (algebraMap B (L ⊗[K] adicCompletion K v)).range := by\n    ext x\n    simp [Algebra.algebraMap_eq_smul_one]\n  have hrange' := adicCompletionComapAlgHom_map_closure_is_closed A K L B v\n  rw [AlgHom.range_comp, ← SetLike.coe_set_eq, Subalgebra.coe_map, AlgHom.coe_restrictScalars',\n      hlhs, hrhs, hrange, ← IsClosed.closure_eq hrange']\n  exact closure_image_closure\n    (tensorAdicCompletionComapLinearMap_isOpenQuotientMap A K L B v).continuous\n","line":594,"end_line":611,"proof_status":"formalized"}
+{"id":"IsDedekindDomain.FiniteAdeleRing.baseChangeAlgEquiv","label":"IsDedekindDomain.FiniteAdeleRing.baseChangeAlgEquiv","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.HeightOneSpectrum.adicCompletionComapAlgEquiv_integral","IsDedekindDomain.FiniteAdeleRing.baseChangeIntegralAlgEquiv","IsDedekindDomain.AKLB.tensorProduct_module_algEquiv"],"nlp_text":"This natural map \\(L\\otimes _K\\mathbb {A}_{A,K}^\\infty \\to \\mathbb {A}_{B,L}^\\infty \\) is an isomorphism.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.FiniteAdeleRing.baseChangeAlgEquiv","latex_link":"Adele_miniproject.html#IsDedekindDomain.FiniteAdeleRing.baseChangeAlgEquiv","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.FiniteAdeleRing.baseChangeAlgEquiv","lean_decl_name":"IsDedekindDomain.FiniteAdeleRing.baseChangeAlgEquiv","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/DedekindDomain/FiniteAdeleRing/BaseChange.lean","lean4_code":"/-- The `L`-algebra isomorphism `L ⊗_K 𝔸_K^∞ ≅ 𝔸_L^∞`. -/\ndef FiniteAdeleRing.baseChangeAlgEquiv :\n    L ⊗[K] FiniteAdeleRing A K ≃ₐ[L] FiniteAdeleRing B L :=\n  AlgEquiv.ofBijective\n    (SemialgHom.baseChange_of_algebraMap <| FiniteAdeleRing.mapSemialgHom A K L B)\n    (FiniteAdeleRing.baseChange_bijective A K L B)\n","line":282,"end_line":287,"proof_status":"formalized"}
+{"id":"IsDedekindDomain.FiniteAdeleRing.baseChangeContinuousAlgEquiv","label":"IsDedekindDomain.FiniteAdeleRing.baseChangeContinuousAlgEquiv","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.HeightOneSpectrum.adicCompletionComapAlgEquiv_integral"],"nlp_text":"The induced \\(L\\)-algebra morphism \\(L\\otimes _K\\mathbb {A}_{A,K}^\\infty \\to \\mathbb {A}_{B,L}^\\infty \\) is a topological isomorphism.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.FiniteAdeleRing.baseChangeContinuousAlgEquiv","latex_link":"Adele_miniproject.html#IsDedekindDomain.FiniteAdeleRing.baseChangeContinuousAlgEquiv","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.FiniteAdeleRing.baseChangeContinuousAlgEquiv","lean_decl_name":"IsDedekindDomain.FiniteAdeleRing.baseChangeContinuousAlgEquiv","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/DedekindDomain/FiniteAdeleRing/BaseChange.lean","lean4_code":"/-- The continuous `L`-algebra isomorphism `L ⊗_K 𝔸_K^∞ ≅ 𝔸_L^∞` -/\nnoncomputable def FiniteAdeleRing.baseChangeContinuousAlgEquiv :\n    L ⊗[K] FiniteAdeleRing A K ≃A[L] FiniteAdeleRing B L where\n  __ := FiniteAdeleRing.baseChangeAlgEquiv A K L B\n  __ := FiniteAdeleRing.baseChangeAdeleContinuousAlgEquiv A K L B\n","line":407,"end_line":411,"proof_status":"formalized"}
+{"id":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom.mapadicCompletionIntegers","label":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom.mapadicCompletionIntegers","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.HeightOneSpectrum.valued_adicCompletionComap"],"nlp_text":"The map \\(i_v:K_v\\to L_w\\) sends the integer ring \\(A_v\\) into \\(B_w\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom.mapadicCompletionIntegers","latex_link":"Adele_miniproject.html#IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom.mapadicCompletionIntegers","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom.mapadicCompletionIntegers","lean_decl_name":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom.mapadicCompletionIntegers","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/DedekindDomain/Completion/BaseChange.lean","lean4_code":"/-- The canonical map K_v → L_w sends 𝓞_v to 𝓞_w. -/\nlemma adicCompletionComapSemialgHom.mapadicCompletionIntegers (v : HeightOneSpectrum A)\n    (w : HeightOneSpectrum B) (hvw : w.comap A = v) :\n    (adicCompletionComapSemialgHom A K L B v w hvw) '' (v.adicCompletionIntegers K) ≤\n    w.adicCompletionIntegers L := by\n  rintro y ⟨x, hx, rfl⟩\n  rw [SetLike.mem_coe, mem_adicCompletionIntegers] at hx ⊢\n  rw [valued_adicCompletionComap A K L B v w hvw]\n  rwa [pow_le_one_iff]\n  exact ramificationIdx_ne_zero A B (algebraMap_injective_of_field_isFractionRing A B K L) w","line":188,"end_line":197,"proof_status":"formalized"}
+{"id":"IsDedekindDomain.FiniteAdeleRing.mapSemialgHom","label":"IsDedekindDomain.FiniteAdeleRing.mapSemialgHom","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom.mapadicCompletionIntegers"],"nlp_text":"There’s a natural ring homomorphism \\(\\mathbb {A}_{A,K}^\\infty \\to \\mathbb {A}_{B,L}^\\infty \\) lying over \\(K\\to L\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.FiniteAdeleRing.mapSemialgHom","latex_link":"Adele_miniproject.html#IsDedekindDomain.FiniteAdeleRing.mapSemialgHom","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.FiniteAdeleRing.mapSemialgHom","lean_decl_name":"IsDedekindDomain.FiniteAdeleRing.mapSemialgHom","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/Mathlib/Topology/Algebra/UniformRing.lean","lean4_code":"noncomputable def mapSemialgHom {α : Type*} [CommRing α] [UniformSpace α]\n    [IsTopologicalRing α] [IsUniformAddGroup α] {β : Type*} [UniformSpace β] [CommRing β]\n    [IsUniformAddGroup β] [IsTopologicalRing β] (f : α →+* β) (hf : Continuous f) :\n    Completion α →ₛₐ[f] Completion β where\n  __ := UniformSpace.Completion.mapRingHom f hf\n  map_smul' m x := by\n    simp only [RingHom.toMonoidHom_eq_coe, OneHom.toFun_eq_coe, MonoidHom.toOneHom_coe,\n      MonoidHom.coe_coe]\n    rw [Algebra.smul_def, map_mul, Algebra.smul_def]\n    congr\n    exact extensionHom_coe _ _ m\n","line":16,"end_line":26,"proof_status":"formalized"}
+{"id":"NumberField.instCompactSpaceAdicCompletionIntegers","label":"NumberField.instCompactSpaceAdicCompletionIntegers","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"If \\(K\\) is a number field and \\(v\\) is a nonzero prime ideal of the integers of \\(K\\), then the integers of \\(K_v\\) is a compact open subgroup.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.instCompactSpaceAdicCompletionIntegers","latex_link":"Adele_miniproject.html#NumberField.instCompactSpaceAdicCompletionIntegers","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.instCompactSpaceAdicCompletionIntegers","lean_decl_name":"NumberField.instCompactSpaceAdicCompletionIntegers","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/NumberField/Completion/Finite.lean","lean4_code":"instance NumberField.instCompactSpaceAdicCompletionIntegers :\n    CompactSpace (v.adicCompletionIntegers K) :=\n  Valued.WithZeroMulInt.integer_compactSpace (v.adicCompletion K) inferInstance","line":44,"end_line":46,"proof_status":"statement_formalized"}
+{"id":"NumberField.AdeleRing.baseChangeEquiv","label":"NumberField.AdeleRing.baseChangeEquiv","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.FiniteAdeleRing.baseChangeAlgEquiv","NumberField.InfiniteAdeleRing.baseChangeEquiv"],"nlp_text":"If \\(K\\to L\\) is a ring homomorphism between two number fields then there is a natural isomorphism (both topological and algebraic) \\(L\\otimes _K\\mathbb {A}_K\\cong \\mathbb {A}_L\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.baseChangeEquiv","latex_link":"Adele_miniproject.html#NumberField.AdeleRing.baseChangeEquiv","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.baseChangeEquiv","lean_decl_name":"NumberField.AdeleRing.baseChangeEquiv","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/NumberField/InfiniteAdeleRing.lean","lean4_code":"/-- The canonical `L`-algebra homeomorphism from `L ⊗_K K_∞` to `L_∞` induced by the\n`K`-algebra base change map `K_∞ → L_∞`. -/\nnoncomputable\ndef baseChangeEquiv :\n    L ⊗[K] InfiniteAdeleRing K ≃A[L] InfiniteAdeleRing L :=\n  /- Because both `L ⊗[K] K_∞` and `L_∞` have the `K_∞` module topology, we obtain a _continuous_\n  `L`-algebra equivalence, using the following set up of algebras\n  ```\n  K_∞    L\n    \\   /\n    \\  /\n     K\n  ```\n  -/\n  IsModuleTopology.continuousAlgEquivOfIsScalarTower K _ (baseChangeEquivAux K L)\n    (by simp_rw [baseChangeEquivAux_apply]; exact SemialgHom.baseChange_of_algebraMap_tmul_right _)\n","line":111,"end_line":126,"proof_status":"formalized"}
+{"id":"NumberField.AdeleRing.cocompact","label":"NumberField.AdeleRing.cocompact","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["NumberField.AdeleRing.baseChangeEquiv","Rat.AdeleRing.cocompact"],"nlp_text":"The quotient \\(\\mathbb {A}_K/K\\) is compact.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.cocompact","latex_link":"Adele_miniproject.html#NumberField.AdeleRing.cocompact","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.cocompact","lean_decl_name":"NumberField.AdeleRing.cocompact","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Topology/Defs/Filter.lean","lean4_code":"def Filter.cocompact : Filter X :=\n  ⨅ (s : Set X) (_ : IsCompact s), 𝓟 sᶜ\n\nvariable (X) in\n/-- `Filter.coclosedCompact` is the filter generated by complements to closed compact sets.\nIn a Hausdorff space, this is the same as `Filter.cocompact`. -/\n","line":323,"end_line":328,"proof_status":"statement_formalized"}
+{"id":"NumberField.AdeleRing.baseChange_moduleTopology","label":"NumberField.AdeleRing.baseChange_moduleTopology","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["NumberField.AdeleRing.baseChangeEquiv"],"nlp_text":"If \\(K\\to L\\) is a ring homomorphism between two number fields then the topology on \\(\\mathbb {A}_L\\) is the \\(\\mathbb {A}_K\\)-module topology, where the module structure comes from the natural map \\(\\mathbb {A}_K\\to \\mathbb {A}_L\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.instBaseChangeIsModuleTopology","latex_link":"Adele_miniproject.html#NumberField.AdeleRing.baseChange_moduleTopology","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.instBaseChangeIsModuleTopology","lean_decl_name":"NumberField.AdeleRing.instBaseChangeIsModuleTopology","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/NumberField/AdeleRing.lean","lean4_code":"instance instBaseChangeIsModuleTopology : IsModuleTopology (𝔸 K) (𝔸 L) := by\n  exact IsModuleTopology.instProd' (A := InfiniteAdeleRing K)\n    (B := FiniteAdeleRing (𝓞 K) K) (M := InfiniteAdeleRing L) (N := FiniteAdeleRing (𝓞 L) L)","line":67,"end_line":69,"proof_status":"formalized"}
+{"id":"NumberField.AdeleRing.zero_discrete","label":"NumberField.AdeleRing.zero_discrete","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["NumberField.AdeleRing.baseChangeEquiv","Rat.AdeleRing.zero_discrete"],"nlp_text":"There’s an open subset of \\(\\mathbb {A}_{K}\\) whose intersection with \\(K\\) is \\(\\{ 0\\} \\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.zero_discrete","latex_link":"Adele_miniproject.html#NumberField.AdeleRing.zero_discrete","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.zero_discrete","lean_decl_name":"NumberField.AdeleRing.zero_discrete","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/NumberField/AdeleRing.lean","lean4_code":"theorem NumberField.AdeleRing.zero_discrete : ∃ U : Set (AdeleRing (𝓞 K) K),\n    IsOpen U ∧ (algebraMap K (AdeleRing (𝓞 K) K)) ⁻¹' U = {0} := by\n  obtain ⟨V, hV, hV0⟩ := Rat.AdeleRing.zero_discrete\n  use (piEquiv ℚ K) '' {f | ∀i, f i ∈ V }\n  constructor\n  · rw [← (piEquiv ℚ K).coe_toHomeomorph, Homeomorph.isOpen_image, Set.setOf_forall]\n    apply isOpen_iInter_of_finite\n    intro i\n    exact hV.preimage (continuous_apply i)\n  rw [Set.eq_singleton_iff_unique_mem]\n  constructor\n  · rw [Set.eq_singleton_iff_unique_mem, Set.mem_preimage, map_zero] at hV0\n    simp only [Set.mem_preimage, map_zero, Set.mem_image,\n      EmbeddingLike.map_eq_zero_iff, exists_eq_right]\n    exact fun _ => hV0.left\n  intro x ⟨y, hy, hyx⟩\n  apply (Module.Finite.equivPi ℚ K).injective\n  set f := Module.Finite.equivPi ℚ K x\n  let g := fun i => algebraMap ℚ (AdeleRing (𝓞 ℚ) ℚ) (f i)\n  have hfg : ∀ i, (algebraMap _ _) (f i) = g i := fun i => rfl\n  have hg := piEquiv_apply_of_algebraMap hfg\n  simp only [LinearEquiv.symm_apply_apply, f, ← hyx, EquivLike.apply_eq_iff_eq] at hg\n  subst hg\n  ext i\n  rw [map_zero, Pi.zero_apply, ← Set.mem_singleton_iff, ← hV0, Set.mem_preimage]\n  exact hy i","line":411,"end_line":436,"proof_status":"statement_formalized"}
+{"id":"NumberField.AdeleRing.discrete","label":"NumberField.AdeleRing.discrete","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["NumberField.AdeleRing.zero_discrete"],"nlp_text":"The additive subgroup \\(K\\) of \\(\\mathbb {A}_K\\) is discrete.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.discrete","latex_link":"Adele_miniproject.html#NumberField.AdeleRing.discrete","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.discrete","lean_decl_name":"NumberField.AdeleRing.discrete","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Topology/Category/TopCat/Basic.lean","lean4_code":"def discrete : Type u ⥤ TopCat.{u} where\n  obj X := @of X ⊥\n  map f := @ofHom _ _ ⊥ ⊥ <| @ContinuousMap.mk _ _ ⊥ ⊥ f continuous_bot\n\n","line":164,"end_line":167,"proof_status":"statement_formalized"}
+{"id":"a0000000185","label":"a0000000185","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"\\(\\mathbb {\\mathbb {A}_L^\\infty }\\) is homeomorphic to \\(\\prod _v(B\\otimes _AK_v,B\\otimes _AA_v)\\).","doc_url":null,"latex_link":"Adele_miniproject.html#a0000000185","lean_link":null,"lean_decl_name":"a0000000185","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"NumberField.InfinitePlace.Completion.piExtension","label":"NumberField.InfinitePlace.Completion.piExtension","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"Let \\(v\\) be an infinite place of \\(K\\). There is a continuous \\(K\\)-algebra homomorphism \\(K_v \\to \\prod _{w\\mid v}L_w\\), whose restriction to \\(K\\) corresponds to the global embedding of \\(K\\) into \\((L_w)_w\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.piExtension","latex_link":"Adele_miniproject.html#NumberField.InfinitePlace.Completion.piExtension","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.piExtension","lean_decl_name":"NumberField.InfinitePlace.Completion.piExtension","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/NumberField/Completion/Infinite.lean","lean4_code":"/-- The map from `v.Completion` to the product of all completions of `L` lying above `v`. -/\ndef piExtension :\n    v.Completion →ₛₐ[algebraMap K L] (wv : v.Extension L) → wv.1.Completion :=\n  Pi.semialgHom _ _ fun wv => comapHom wv.2\n","line":49,"end_line":52,"proof_status":"formalized"}
+{"id":"NumberField.InfinitePlace.Completion.baseChange","label":"NumberField.InfinitePlace.Completion.baseChange","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["NumberField.InfinitePlace.Completion.piExtension"],"nlp_text":"Let \\(v\\) be an infinite place of \\(K\\). There is a natural \\(L\\)-algebra homomorphism \\(L\\otimes _K K_v \\to \\prod _{w\\mid v}L_w\\), whose restriction to \\(1\\otimes _K K_v\\) corresponds to the map in&#160;9.36.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.baseChange","latex_link":"Adele_miniproject.html#NumberField.InfinitePlace.Completion.baseChange","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.baseChange","lean_decl_name":"NumberField.InfinitePlace.Completion.baseChange","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/Deformations/RepresentationTheory/GaloisRep.lean","lean4_code":"/-- Make a `A`-linear galois rep on `M` into a `B`-linear rep on `B ⊗ M`. -/\nnoncomputable\ndef GaloisRep.baseChange [IsTopologicalRing B] [Algebra A B] [ContinuousSMul A B]\n    [Module.Finite A M] [Module.Free A M]\n    (ρ : GaloisRep K A M) : GaloisRep K B (B ⊗[A] M) :=\n  letI := moduleTopology A (Module.End A M)\n  letI := moduleTopology B (Module.End B (B ⊗[A] M))\n  letI : ContinuousMul _ := ⟨IsModuleTopology.continuous_mul_of_finite B (Module.End B (B ⊗[A] M))⟩\n  letI := IsModuleTopology.toContinuousAdd B (Module.End B (B ⊗[A] M))\n  let F : Module.End A M →+* Module.End B (B ⊗[A] M) := Module.End.baseChangeHom A B M\n  have : Continuous F := by\n    have : IsTopologicalSemiring (Module.End B (B ⊗[A] M)) := ⟨⟩\n    have : Continuous (algebraMap A (Module.End B (B ⊗[A] M))) := by\n      rw [IsScalarTower.algebraMap_eq A B, RingHom.coe_comp]\n      exact (continuous_algebraMap _ _).comp (continuous_algebraMap _ _)\n    exact IsModuleTopology.continuous_of_ringHom (R := A) F (by simpa [F])\n  .comp ⟨F, this⟩ ρ\n","line":189,"end_line":205,"proof_status":"formalized"}
+{"id":"NumberField.InfinitePlace.Completion.baseChangeEquiv","label":"NumberField.InfinitePlace.Completion.baseChangeEquiv","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["NumberField.InfinitePlace.Completion.baseChange","NumberField.InfinitePlace.Completion.baseChange_surjective","NumberField.InfinitePlace.Completion.baseChange_injective","NumberField.InfinitePlace.Completion.instIsModuleTopologyValEqComapAlgebraMap_fLT"],"nlp_text":"Let \\(v\\) be an infinite place of \\(K\\). There is a natural \\(L\\)-algebra homeomorphism \\(L\\otimes _K K_v \\cong _L \\prod _{w\\mid v}L_w\\), whose restriction to \\(1\\otimes _K K_v\\) corresponds to the map in&#160;9.36.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.baseChangeEquiv","latex_link":"Adele_miniproject.html#NumberField.InfinitePlace.Completion.baseChangeEquiv","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.baseChangeEquiv","lean_decl_name":"NumberField.InfinitePlace.Completion.baseChangeEquiv","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/NumberField/InfiniteAdeleRing.lean","lean4_code":"/-- The canonical `L`-algebra homeomorphism from `L ⊗_K K_∞` to `L_∞` induced by the\n`K`-algebra base change map `K_∞ → L_∞`. -/\nnoncomputable\ndef baseChangeEquiv :\n    L ⊗[K] InfiniteAdeleRing K ≃A[L] InfiniteAdeleRing L :=\n  /- Because both `L ⊗[K] K_∞` and `L_∞` have the `K_∞` module topology, we obtain a _continuous_\n  `L`-algebra equivalence, using the following set up of algebras\n  ```\n  K_∞    L\n    \\   /\n    \\  /\n     K\n  ```\n  -/\n  IsModuleTopology.continuousAlgEquivOfIsScalarTower K _ (baseChangeEquivAux K L)\n    (by simp_rw [baseChangeEquivAux_apply]; exact SemialgHom.baseChange_of_algebraMap_tmul_right _)\n","line":111,"end_line":126,"proof_status":"statement_formalized"}
+{"id":"NumberField.InfiniteAdeleRing.baseChangeEquivAux","label":"NumberField.InfiniteAdeleRing.baseChangeEquivAux","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["NumberField.InfinitePlace.Completion.baseChangeEquiv"],"nlp_text":"There is a natural \\(L\\)-algebra isomorphism \\(L \\otimes _K K_{\\infty } \\cong _L L_{\\infty }\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfiniteAdeleRing.baseChangeEquivAux","latex_link":"Adele_miniproject.html#NumberField.InfiniteAdeleRing.baseChangeEquivAux","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfiniteAdeleRing.baseChangeEquivAux","lean_decl_name":"NumberField.InfiniteAdeleRing.baseChangeEquivAux","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/NumberField/InfiniteAdeleRing.lean","lean4_code":"/-- The $L$-algebra isomorphism $L\\otimes_K K_{\\infty} \\cong L_{\\infty}$. -/\nnoncomputable def baseChangeEquivAux :\n    L ⊗[K] InfiniteAdeleRing K ≃ₐ[L] InfiniteAdeleRing L :=\n  -- L ⊗ K_∞ ≃[K_∞] ∏ v, L ⊗ K_v\n  Algebra.TensorProduct.piRight K L L Completion |>.trans\n    -- lift the established equivalence L ⊗ K_v ≃[v.Completion] ∏ w ∣ v, L_w on fibers of comap\n    (AlgEquiv.piCongrFiberwise\n      (fun v : InfinitePlace K => (Completion.baseChangeEquiv L v).toAlgEquiv.symm)).symm\n","line":67,"end_line":74,"proof_status":"statement_formalized"}
+{"id":"NumberField.InfinitePlace.Completion.piEquiv","label":"NumberField.InfinitePlace.Completion.piEquiv","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["NumberField.InfinitePlace.Completion.baseChangeEquiv"],"nlp_text":"Let \\(v\\) be an infinite place of \\(K\\). There is a natural \\(K_v\\)-linear homeomorphism \\(K_v^{[L:K]} \\cong _{K_v} \\prod _{w\\mid v}L_w\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.piEquiv","latex_link":"Adele_miniproject.html#NumberField.InfinitePlace.Completion.piEquiv","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.piEquiv","lean_decl_name":"NumberField.InfinitePlace.Completion.piEquiv","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/NumberField/InfiniteAdeleRing.lean","lean4_code":"/-- The $K_{\\infty}$-linear homeomorphism $K_{\\infty}^{[L:K]} \\cong L_{\\infty}$. -/\nnoncomputable\ndef piEquiv : let d := Module.finrank K L\n    (Fin d → InfiniteAdeleRing K) ≃L[InfiniteAdeleRing K] InfiniteAdeleRing L := by\n  -- I think we could remove convert if we make `InfiniteAdeleRing` an `abbrev`\n  -- (K_∞)^d ≃[K_∞] ∏ v, K_v^d\n  convert (ContinuousLinearEquiv.piScalarPiComm _ _).symm.trans\n    -- lift the equivalence K_v^d ≃[v.Completion] ∏ w ∣ v, L_w on fibers of comap\n    (ContinuousLinearEquiv.piScalarPiCongrFiberwise\n      (fun v : InfinitePlace K => (Completion.piEquiv L v).symm)).symm\n","line":48,"end_line":57,"proof_status":"statement_formalized"}
+{"id":"NumberField.AdeleRing.locallyCompactSpace","label":"NumberField.AdeleRing.locallyCompactSpace","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"The adeles of a number field are locally compact.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.locallyCompactSpace","latex_link":"Adele_miniproject.html#NumberField.AdeleRing.locallyCompactSpace","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.locallyCompactSpace","lean_decl_name":"NumberField.AdeleRing.locallyCompactSpace","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Analysis/Normed/Field/WithAbs.lean","lean4_code":"theorem locallyCompactSpace [LocallyCompactSpace L] (h : ∀ x, ‖f x‖ = v x) :\n    LocallyCompactSpace (v.Completion) :=\n  (isClosedEmbedding_extensionEmbedding_of_comp h).locallyCompactSpace\n\n","line":118,"end_line":121,"proof_status":"statement_formalized"}
+{"id":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComap_isModuleTopology","label":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComap_isModuleTopology","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom"],"nlp_text":"Giving \\(L_w\\) the \\(K_v\\)-algebra structure coming from the natural map \\(K_v\\to L_w\\), the \\(w\\)-adic topology on \\(L_w\\) is the \\(K_v\\)-module topology.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.adicCompletionComap_isModuleTopology","latex_link":"Adele_miniproject.html#IsDedekindDomain.HeightOneSpectrum.adicCompletionComap_isModuleTopology","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.adicCompletionComap_isModuleTopology","lean_decl_name":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComap_isModuleTopology","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/DedekindDomain/Completion/BaseChange.lean","lean4_code":"/-- L_w has the K_v-module topology. -/\nlemma adicCompletionComap_isModuleTopology\n    (v : HeightOneSpectrum A) (w : HeightOneSpectrum B) (hvw : w.comap A = v) :\n    -- temporarily make L_w a K_v-algebra\n    letI := comap_algebra A K L B hvw\n    -- then claim that L_w has the module topology.\n    IsModuleTopology (v.adicCompletion K) (w.adicCompletion L) := by\n  let Kv := adicCompletion K v\n  let Lw := adicCompletion L w\n  let _ : Algebra Kv Lw := comap_algebra A K L B hvw\n  have : ContinuousSMul Kv Lw := comap_algebra_continuousSmul A K L B v w hvw\n  have : Module.Finite Kv Lw := comap_algebra_finite A K L B v w hvw\n  let iso : ((Fin (Module.finrank Kv Lw)) → Kv) ≃L[Kv] Lw :=\n    ContinuousLinearEquiv.ofFinrankEq (Module.finrank_fin_fun Kv)\n  apply IsModuleTopology.iso iso\n","line":334,"end_line":348,"proof_status":"statement_formalized"}
+{"id":"IsDedekindDomain.HeightOneSpectrum.prodAdicCompletionComap_isModuleTopology","label":"IsDedekindDomain.HeightOneSpectrum.prodAdicCompletionComap_isModuleTopology","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.HeightOneSpectrum.adicCompletionComap_isModuleTopology"],"nlp_text":"For \\(v\\) fixed, the product topology on \\(\\prod _{w|v}L_w\\) is the \\(K_v\\)-module topology.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.prodAdicCompletionComap_isModuleTopology","latex_link":"Adele_miniproject.html#IsDedekindDomain.HeightOneSpectrum.prodAdicCompletionComap_isModuleTopology","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.prodAdicCompletionComap_isModuleTopology","lean_decl_name":"IsDedekindDomain.HeightOneSpectrum.prodAdicCompletionComap_isModuleTopology","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/DedekindDomain/Completion/BaseChange.lean","lean4_code":"/-- ∏_{w|v} L_w has the K_v-module topology. -/\nlemma prodAdicCompletionComap_isModuleTopology (v : HeightOneSpectrum A) :\n    -- TODO: the `let _` in the statement below should not be required as it is an instance\n    -- see mathlib PR #22488 for potential fix to this.\n    -- Note that this one does not involve `adicCompletionIntegers` so the\n    -- issue may not be to do with subtype vs. type implementation of\n    -- `adicCompletionIntegers`.\n    let _ := comap_pi_algebra A K L B v |>.toSMul\n    -- the claim that L_w has the module topology.\n    IsModuleTopology (v.adicCompletion K) (Π (w : v.Extension B), w.1.adicCompletion L) := by\n  -- these are defs or lemmas so are required\n  let _ (w : v.Extension B) := comap_algebra A K L B w.2 |>.toModule\n  let _ (w : v.Extension B) := adicCompletionComap_isModuleTopology A K L B v w.1 w.2\n  let _ := Extension.finite A K L B v\n  exact IsModuleTopology.instPi\n","line":351,"end_line":365,"proof_status":"formalized"}
+{"id":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapContinuousAlgEquiv","label":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapContinuousAlgEquiv","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.HeightOneSpectrum.prodAdicCompletionComap_isModuleTopology"],"nlp_text":"If we give \\(L\\otimes _KK_v\\) the \\(K_v\\)-module topology then the \\(L\\)-algebra isomorphism \\(L\\otimes _K K_v\\cong \\prod _{w|v}L_w\\) is also a homeomorphism.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.adicCompletionComapContinuousAlgEquiv","latex_link":"Adele_miniproject.html#IsDedekindDomain.HeightOneSpectrum.adicCompletionComapContinuousAlgEquiv","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.adicCompletionComapContinuousAlgEquiv","lean_decl_name":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapContinuousAlgEquiv","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/DedekindDomain/Completion/BaseChange.lean","lean4_code":"/-- The continuous L-algebra isomorphism `L ⊗[K] K_v ≅ ∏_{w|v} L_w`. -/\nnoncomputable def adicCompletionComapContinuousAlgEquiv (v : HeightOneSpectrum A) :\n    L ⊗[K] v.adicCompletion K ≃A[L] ∀ w : v.Extension B, w.1.adicCompletion L :=\n  {\n    __ := adicCompletionComapAlgEquiv A K L B v\n    __ := adicCompletionComapRightContinuousAlgEquiv A K L B v\n  }\n","line":908,"end_line":914,"proof_status":"statement_formalized"}
+{"id":"NumberField.InfiniteAdeleRing.piEquiv","label":"NumberField.InfiniteAdeleRing.piEquiv","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["NumberField.InfinitePlace.Completion.piEquiv"],"nlp_text":"There is a natural \\(K_{\\infty }\\)-linear homeomorphism \\(K_{\\infty }^{[L:K]} \\cong _{K_{\\infty }} L_{\\infty }\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfiniteAdeleRing.piEquiv","latex_link":"Adele_miniproject.html#NumberField.InfiniteAdeleRing.piEquiv","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfiniteAdeleRing.piEquiv","lean_decl_name":"NumberField.InfiniteAdeleRing.piEquiv","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/NumberField/InfiniteAdeleRing.lean","lean4_code":"/-- The $K_{\\infty}$-linear homeomorphism $K_{\\infty}^{[L:K]} \\cong L_{\\infty}$. -/\nnoncomputable\ndef piEquiv : let d := Module.finrank K L\n    (Fin d → InfiniteAdeleRing K) ≃L[InfiniteAdeleRing K] InfiniteAdeleRing L := by\n  -- I think we could remove convert if we make `InfiniteAdeleRing` an `abbrev`\n  -- (K_∞)^d ≃[K_∞] ∏ v, K_v^d\n  convert (ContinuousLinearEquiv.piScalarPiComm _ _).symm.trans\n    -- lift the equivalence K_v^d ≃[v.Completion] ∏ w ∣ v, L_w on fibers of comap\n    (ContinuousLinearEquiv.piScalarPiCongrFiberwise\n      (fun v : InfinitePlace K => (Completion.piEquiv L v).symm)).symm\n","line":48,"end_line":57,"proof_status":"statement_formalized"}
+{"id":"NumberField.InfiniteAdeleRing.instIsModuleTopology_fLT","label":"NumberField.InfiniteAdeleRing.instIsModuleTopology_fLT","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["NumberField.InfiniteAdeleRing.piEquiv"],"nlp_text":"\\(L_{\\infty }\\) has the \\(K_{\\infty }\\)-module topology.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfiniteAdeleRing.instIsModuleTopology_fLT","latex_link":"Adele_miniproject.html#NumberField.InfiniteAdeleRing.instIsModuleTopology_fLT","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfiniteAdeleRing.instIsModuleTopology_fLT","lean_decl_name":"NumberField.InfiniteAdeleRing.instIsModuleTopology_fLT","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/NumberField/InfiniteAdeleRing.lean","lean4_code":"instance : IsModuleTopology (InfiniteAdeleRing K) (InfiniteAdeleRing L) := by\n  exact IsModuleTopology.iso (piEquiv K L)","line":59,"end_line":60,"proof_status":"statement_formalized"}
+{"id":"NumberField.InfiniteAdeleRing.baseChangeEquiv","label":"NumberField.InfiniteAdeleRing.baseChangeEquiv","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["NumberField.InfiniteAdeleRing.baseChangeEquivAux"],"nlp_text":"If \\(K\\to L\\) is a ring homomorphism between two number fields then there is a natural isomorphism (both topological and algebraic) \\(L\\otimes _KK_\\infty \\cong L_\\infty \\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfiniteAdeleRing.baseChangeEquiv","latex_link":"Adele_miniproject.html#NumberField.InfiniteAdeleRing.baseChangeEquiv","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfiniteAdeleRing.baseChangeEquiv","lean_decl_name":"NumberField.InfiniteAdeleRing.baseChangeEquiv","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/NumberField/InfiniteAdeleRing.lean","lean4_code":"/-- The canonical `L`-algebra homeomorphism from `L ⊗_K K_∞` to `L_∞` induced by the\n`K`-algebra base change map `K_∞ → L_∞`. -/\nnoncomputable\ndef baseChangeEquiv :\n    L ⊗[K] InfiniteAdeleRing K ≃A[L] InfiniteAdeleRing L :=\n  /- Because both `L ⊗[K] K_∞` and `L_∞` have the `K_∞` module topology, we obtain a _continuous_\n  `L`-algebra equivalence, using the following set up of algebras\n  ```\n  K_∞    L\n    \\   /\n    \\  /\n     K\n  ```\n  -/\n  IsModuleTopology.continuousAlgEquivOfIsScalarTower K _ (baseChangeEquivAux K L)\n    (by simp_rw [baseChangeEquivAux_apply]; exact SemialgHom.baseChange_of_algebraMap_tmul_right _)\n","line":111,"end_line":126,"proof_status":"statement_formalized"}
+{"id":"IsDedekindDomain.FiniteAdeleRing.IntegraltensorProductAlgEquiv_aux1","label":"IsDedekindDomain.FiniteAdeleRing.IntegraltensorProductAlgEquiv_aux1","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.pi_tensorProduct"],"nlp_text":"The natural map \\(B\\otimes _A\\mathbb {A}_K^\\infty \\to R\\) is a \\(B\\)-algebra isomorphism.","doc_url":null,"latex_link":"Adele_miniproject.html#IsDedekindDomain.FiniteAdeleRing.IntegraltensorProductAlgEquiv_aux1","lean_link":null,"lean_decl_name":"IsDedekindDomain.FiniteAdeleRing.IntegraltensorProductAlgEquiv_aux1","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"IsDedekindDomain.FiniteAdeleRing.baseChangeIntegralAlgEquiv","label":"IsDedekindDomain.FiniteAdeleRing.baseChangeIntegralAlgEquiv","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.FiniteAdeleRing.IntegraltensorProductAlgEquiv_aux1","IsDedekindDomain.FiniteAdeleRing.IntegraltensorProductAlgEquiv_aux2"],"nlp_text":"The natural map \\(B\\otimes _A\\mathbb {A}_K^\\infty \\to \\mathbb {A}_L^\\infty \\) is a \\(B\\)-algebra isomorphism.","doc_url":null,"latex_link":"Adele_miniproject.html#IsDedekindDomain.FiniteAdeleRing.baseChangeIntegralAlgEquiv","lean_link":null,"lean_decl_name":"IsDedekindDomain.FiniteAdeleRing.baseChangeIntegralAlgEquiv","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"IsDedekindDomain.FiniteAdeleRing.IntegraltensorProductAlgEquiv_aux2","label":"IsDedekindDomain.FiniteAdeleRing.IntegraltensorProductAlgEquiv_aux2","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["RestrictedProduct.relabelIso"],"nlp_text":"The ring \\(R\\) introduced above (the restricted product of the \\(B\\otimes _A K_v\\) with respect to the \\(B\\otimes _A A_v\\)) is isomorphic to \\(\\mathbb {A}_L\\).","doc_url":null,"latex_link":"Adele_miniproject.html#IsDedekindDomain.FiniteAdeleRing.IntegraltensorProductAlgEquiv_aux2","lean_link":null,"lean_decl_name":"IsDedekindDomain.FiniteAdeleRing.IntegraltensorProductAlgEquiv_aux2","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000191","label":"a0000000191","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"If \\(X_v\\) and \\(Y_v\\) are topological spaces with open subspaces \\(C_v\\) and \\(D_v\\), then the obvious bijection \\(\\prod '_v(X_v \\times Y_v,C_v\\times D_v) \\cong \\left(\\prod '_v(X_v,C_v)\\right)\\times \\left(\\prod '_v(Y_v,D_v)\\right)\\) is a homeomorphism, where the restricted products have the restricted product topology and the binary product has the product topology.","doc_url":null,"latex_link":"Adele_miniproject.html#a0000000191","lean_link":null,"lean_decl_name":"a0000000191","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"pi_tensorProduct_of_finitePresentation","label":"pi_tensorProduct_of_finitePresentation","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"If \\(R\\) is a commutative ring, if \\(M\\) is a finitely presented \\(R\\)-module and if \\(N_i\\) are a collection of \\(R\\)-modules, then the canonical map \\(M\\otimes _R\\prod _i N_i\\to \\prod _i(M\\otimes _R N_i)\\) is an isomorphism.","doc_url":null,"latex_link":"Adele_miniproject.html#pi_tensorProduct_of_finitePresentation","lean_link":null,"lean_decl_name":"pi_tensorProduct_of_finitePresentation","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"IsDedekindDomain.pi_tensorProduct","label":"IsDedekindDomain.pi_tensorProduct","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["pi_tensorProduct_of_finitePresentation","IsDedekindDomain.AKLB.finitePresentation"],"nlp_text":"If \\(S\\) is a finite set of nonzero primes of \\(A\\) then the natural map \\(B\\otimes ((\\prod _{v\\in S}K_v)\\times (\\prod _{v\\notin S}A_v))\\to (\\prod _{v\\in S}(B\\otimes _AK_v))\\times (\\prod _{v\\notin S}(B\\otimes _AA_v))\\) is an isomorphism.","doc_url":null,"latex_link":"Adele_miniproject.html#IsDedekindDomain.pi_tensorProduct","lean_link":null,"lean_decl_name":"IsDedekindDomain.pi_tensorProduct","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"NumberField.InfinitePlace.Completion.denseRange_algebraMap_subtype_pi","label":"NumberField.InfinitePlace.Completion.denseRange_algebraMap_subtype_pi","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"Let \\(S\\) be a set of infinite places of \\(K\\). The image of \\(K\\) under the embedding \\(K\\hookrightarrow (K_v)_{v\\in S}; k \\mapsto (k)_v\\) is dense in the product topology.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.denseRange_algebraMap_subtype_pi","latex_link":"Adele_miniproject.html#NumberField.InfinitePlace.Completion.denseRange_algebraMap_subtype_pi","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.denseRange_algebraMap_subtype_pi","lean_decl_name":"NumberField.InfinitePlace.Completion.denseRange_algebraMap_subtype_pi","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/NumberField/InfinitePlace/WeakApproximation.lean","lean4_code":"/--\n*Weak approximation for subcollections*: this is the result that `K` is dense in any subcollection\n`Π v ∈ S, Kᵥ` of completions at infinite places.\n-/\ntheorem denseRange_algebraMap_subtype_pi (p : InfinitePlace K → Prop) [NumberField K] :\n    DenseRange <| algebraMap K ((v : Subtype p) → v.1.Completion) := by\n  apply DenseRange.comp (g := Subtype.restrict p)\n    (f := algebraMap K ((v : InfinitePlace K) → v.1.Completion))\n  · exact Subtype.surjective_restrict (β := fun v => v.1.Completion) p |>.denseRange\n  · exact InfiniteAdeleRing.denseRange_algebraMap K\n  · exact continuous_pi (fun _ => continuous_apply _)\n","line":298,"end_line":308,"proof_status":"statement_formalized"}
+{"id":"NumberField.InfinitePlace.Completion.baseChange_surjective","label":"NumberField.InfinitePlace.Completion.baseChange_surjective","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["NumberField.InfinitePlace.Completion.denseRange_algebraMap_subtype_pi"],"nlp_text":"For a fixed infinite place \\(v\\) of \\(K\\), the map \\(L\\otimes _K K_v \\to \\prod _{w\\mid v}L_w\\) is surjective.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.baseChange_surjective","latex_link":"Adele_miniproject.html#NumberField.InfinitePlace.Completion.baseChange_surjective","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.baseChange_surjective","lean_decl_name":"NumberField.InfinitePlace.Completion.baseChange_surjective","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/NumberField/Completion/Infinite.lean","lean4_code":"theorem baseChange_surjective : Function.Surjective (baseChange L v) := by\n  -- Let `Bw` be a `K_v` basis of `Π v | w, L_w`\n  let Bw := Module.finBasis v.Completion ((w : v.Extension L) → w.1.Completion)\n  -- `L` is dense inside Π v | w, L_w\n  have := denseRange_algebraMap_subtype_pi _ fun w : InfinitePlace L => w.comap (algebraMap K L) = v\n  -- So there exists a vector `α ∈ L^d` whose matrix wrt `Bw` gets close to `1` (has non-zero det)\n  classical\n  let ⟨α, h⟩ := (DenseRange.piMap fun _ => this).exists_matrix_det_ne_zero\n    (Basis.toMatrix_continuous Bw) Bw.toMatrix_self\n  -- Therefore `α` is a basis under the image of `piExtension L v`, hence it's surjective\n  rw [← isUnit_iff_ne_zero, ← Bw.det_apply, ← Module.Basis.is_basis_iff_det Bw] at h\n  rw [← baseChangeRightOfAlgebraMap_coe, ← AlgHom.coe_toLinearMap, ← LinearMap.range_eq_top,\n    ← top_le_iff, ← h.2, Submodule.span_le]\n  rintro x ⟨i, rfl⟩\n  exact ⟨α i ⊗ₜ 1, by simp⟩\n","line":106,"end_line":120,"proof_status":"statement_formalized"}
+{"id":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom","label":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.HeightOneSpectrum.valuation_comap"],"nlp_text":"There’s a natural ring map \\(K_v\\to L_w\\) extending the map \\(K\\to L\\). It is defined by completing the inclusion \\(K\\to L\\) at the finite places \\(v\\) and \\(w\\) (which can be done because the previous lemma shows that the map is uniformly continuous for the \\(v\\)-adic and \\(w\\)-adic topologies).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom","latex_link":"Adele_miniproject.html#IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom","lean_decl_name":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/DedekindDomain/Completion/BaseChange.lean","lean4_code":"/-- If w of L divides v of K, `adicCompletionComapSemialgHom v w pf` is the canonical map\n`Kᵥ → L_w` lying above `K → L`. Here we actually use the type synonyms `WithVal K` and `WithVal L`.\n-/\nnoncomputable def adicCompletionComapSemialgHom (v : HeightOneSpectrum A) (w : HeightOneSpectrum B)\n    (hvw : w.comap A = v) :\n    v.adicCompletion K →ₛₐ[σ v w] w.adicCompletion L :=\n  UniformSpace.Completion.mapSemialgHom _ <|\n  adicValued.continuous_algebraMap A K L B v w hvw\n","line":148,"end_line":155,"proof_status":"formalized"}
+{"id":"IsDedekindDomain.HeightOneSpectrum.valued_adicCompletionComap","label":"IsDedekindDomain.HeightOneSpectrum.valued_adicCompletionComap","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom"],"nlp_text":"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 w(i(k))=v(k)\\), where \\(e\\) is the ramification index of \\(w/v\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.valued_adicCompletionComap","latex_link":"Adele_miniproject.html#IsDedekindDomain.HeightOneSpectrum.valued_adicCompletionComap","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.valued_adicCompletionComap","lean_decl_name":"IsDedekindDomain.HeightOneSpectrum.valued_adicCompletionComap","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/DedekindDomain/Completion/BaseChange.lean","lean4_code":"/--\nThe local ramification index for the extension L_w/K_v is equal to the global ramification\nindex for the extension w/v. In other words, if x in K_v and i:K_v->L_w then w(i(x))=v(x)^e\nwhere e is computed globally.\n-/\nlemma valued_adicCompletionComap\n  (v : HeightOneSpectrum A) (w : HeightOneSpectrum B) (hvw : w.comap A = v) (x) :\n    Valued.v (adicCompletionComapSemialgHom A K L B v w hvw x) = Valued.v x ^\n      Ideal.ramificationIdx (algebraMap A B) (comap A w).asIdeal w.asIdeal := by\n  revert x\n  apply funext_iff.mp\n  symm\n  apply UniformSpace.Completion.ext\n  · exact Valued.continuous_valuation.pow _\n  · exact Valued.continuous_valuation.comp UniformSpace.Completion.continuous_extension\n  intro a\n  simp_rw [adicCompletionComap_coe, adicCompletion, Valued.valuedCompletion_apply,\n    adicValued_apply']\n  subst hvw\n  rw [← valuation_comap A K L B w a]\n","line":166,"end_line":185,"proof_status":"formalized"}
+{"id":"a0000000181","label":"a0000000181","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"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 continuous map sending the subset \\(C_v\\subseteq X_v\\) into \\(D_v\\subseteq Y_v\\) then there’s an induced continuous map \\(\\prod '_v(X_v,C_v)\\to \\prod '_v(Y_v,D_v)\\).","doc_url":null,"latex_link":"Adele_miniproject.html#a0000000181","lean_link":null,"lean_decl_name":"a0000000181","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"NumberField.InfinitePlace.Completion.baseChange_injective","label":"NumberField.InfinitePlace.Completion.baseChange_injective","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["NumberField.InfinitePlace.Completion.finrank_pi_eq_finrank_tensorProduct"],"nlp_text":"For a fixed infinite place \\(v\\) of \\(K\\), the map \\(L\\otimes _K K_v \\to \\prod _{w\\mid v}L_w\\) is injective.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.baseChange_injective","latex_link":"Adele_miniproject.html#NumberField.InfinitePlace.Completion.baseChange_injective","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.baseChange_injective","lean_decl_name":"NumberField.InfinitePlace.Completion.baseChange_injective","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/NumberField/Completion/Infinite.lean","lean4_code":"theorem baseChange_injective :\n    Function.Injective (baseChange L v) := by\n  rw [← baseChangeRightOfAlgebraMap_coe, ← AlgHom.coe_toLinearMap,\n    LinearMap.injective_iff_surjective_of_finrank_eq_finrank\n      (finrank_pi_eq_finrank_tensorProduct L v).symm]\n  simp [baseChange_surjective L v]\n","line":124,"end_line":129,"proof_status":"formalized"}
+{"id":"NumberField.InfinitePlace.Completion.instIsModuleTopologyValEqComapAlgebraMap_fLT","label":"NumberField.InfinitePlace.Completion.instIsModuleTopologyValEqComapAlgebraMap_fLT","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"If \\(w \\mid v\\) is an infinite place of \\(L\\) lying above the infinite place \\(v\\) of \\(K\\), then \\(L_w\\) has the \\(K_v\\)-module topology.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.instIsModuleTopologyValEqComapAlgebraMap_fLT","latex_link":"Adele_miniproject.html#NumberField.InfinitePlace.Completion.instIsModuleTopologyValEqComapAlgebraMap_fLT","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.instIsModuleTopologyValEqComapAlgebraMap_fLT","lean_decl_name":"NumberField.InfinitePlace.Completion.instIsModuleTopologyValEqComapAlgebraMap_fLT","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/NumberField/Completion/Infinite.lean","lean4_code":"instance : IsModuleTopology v.Completion wv.1.Completion :=\n  IsModuleTopology.iso (FiniteDimensional.nonempty_continuousLinearEquiv_of_finrank_eq\n    (Module.finrank_fin_fun v.Completion)).some","line":131,"end_line":133,"proof_status":"statement_formalized"}
+{"id":"RestrictedProduct.relabelIso","label":"RestrictedProduct.relabelIso","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"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 subsets \\(C_v\\), let \\(Y_w\\) be sets with subsets \\(D_w\\), and say for all \\(v\\in V\\) we’re given a bijection \\(X_v\\to \\prod _{w|f(w)=v}Y_w\\), identifying \\(C_v\\) with \\(\\prod _{w:f(w)=v}D_w\\). Then there’s an induced bijection between the restricted products \\(\\prod '_v(X_v,C_v)\\) and \\(\\prod '_w(Y_w,D_w)\\).","doc_url":null,"latex_link":"Adele_miniproject.html#RestrictedProduct.relabelIso","lean_link":null,"lean_decl_name":"RestrictedProduct.relabelIso","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"IsDedekindDomain.AKLB.finitePresentation","label":"IsDedekindDomain.AKLB.finitePresentation","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"\\(B\\) is a finitely-presented \\(A\\)-module.","doc_url":null,"latex_link":"Adele_miniproject.html#IsDedekindDomain.AKLB.finitePresentation","lean_link":null,"lean_decl_name":"IsDedekindDomain.AKLB.finitePresentation","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/AdjoinRoot.lean","lean4_code":"theorem finitePresentation : Algebra.FinitePresentation R (AdjoinRoot f) :=\n  (Algebra.FinitePresentation.polynomial R).quotient (Submodule.fg_span_singleton f)\n\n/-- The adjoined root. -/\n","line":155,"end_line":158,"proof_status":"statement_formalized"}
+{"id":"Rat.AdeleRing.cocompact","label":"Rat.AdeleRing.cocompact","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"The quotient \\(\\mathbb {A}_{\\mathbb {Q}}/\\mathbb {Q}\\) is compact.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Rat.AdeleRing.cocompact","latex_link":"Adele_miniproject.html#Rat.AdeleRing.cocompact","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Rat.AdeleRing.cocompact","lean_decl_name":"Rat.AdeleRing.cocompact","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Topology/Defs/Filter.lean","lean4_code":"def Filter.cocompact : Filter X :=\n  ⨅ (s : Set X) (_ : IsCompact s), 𝓟 sᶜ\n\nvariable (X) in\n/-- `Filter.coclosedCompact` is the filter generated by complements to closed compact sets.\nIn a Hausdorff space, this is the same as `Filter.cocompact`. -/\n","line":323,"end_line":328,"proof_status":"statement_formalized"}
+{"id":"IsDedekindDomain.AKLB.surjective_tensorProduct_map","label":"IsDedekindDomain.AKLB.surjective_tensorProduct_map","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.dvd_norm"],"nlp_text":"The \\(A\\)-bilinear map \\(B\\times K\\to L\\) sending \\((b,k)\\) to \\(bk\\) is surjective.","doc_url":null,"latex_link":"Adele_miniproject.html#IsDedekindDomain.AKLB.surjective_tensorProduct_map","lean_link":null,"lean_decl_name":"IsDedekindDomain.AKLB.surjective_tensorProduct_map","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"IsDedekindDomain.FiniteAdeleRing.tensorProduct_algEquiv","label":"IsDedekindDomain.FiniteAdeleRing.tensorProduct_algEquiv","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.dvd_norm"],"nlp_text":"The natural map \\(B\\otimes _AK\\to L\\) is a \\(B\\)-algebra isomorphism.","doc_url":null,"latex_link":"Adele_miniproject.html#IsDedekindDomain.FiniteAdeleRing.tensorProduct_algEquiv","lean_link":null,"lean_decl_name":"IsDedekindDomain.FiniteAdeleRing.tensorProduct_algEquiv","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000182","label":"a0000000182","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"If all the \\(f_v\\) are homeomorphisms identifying \\(C_v\\) with \\(D_v\\) then the induced map on restricted products is also a homeomorphism (proof: apply the previous construction to \\(f_v\\) and their inverses)","doc_url":null,"latex_link":"Adele_miniproject.html#a0000000182","lean_link":null,"lean_decl_name":"a0000000182","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"IsDedekindDomain.HeightOneSpectrum.Extension.finite","label":"IsDedekindDomain.HeightOneSpectrum.Extension.finite","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"There are only finitely many primes \\(w\\) of \\(B\\) lying above \\(v\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.Extension.finite","latex_link":"Adele_miniproject.html#IsDedekindDomain.HeightOneSpectrum.Extension.finite","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.Extension.finite","lean_decl_name":"IsDedekindDomain.HeightOneSpectrum.Extension.finite","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Order/Interval/Finset/Defs.lean","lean4_code":"Provide the `LocallyFiniteOrder` instance for `α →₀ β` where `β` is locally finite. Provide the\n`LocallyFiniteOrder` instance for `Π₀ i, β i` where all the `β i` are locally finite.\n\nFrom `LinearOrder α`, `NoMaxOrder α`, `LocallyFiniteOrder α`, we can also define an\norder isomorphism `α ≃ ℕ` or `α ≃ ℤ`, depending on whether we have `OrderBot α` or\n`NoMinOrder α` and `Nonempty α`. When `OrderBot α`, we can match `a : α` to `#(Iio a)`.\n\nWe can provide `SuccOrder α` from `LinearOrder α` and `LocallyFiniteOrder α` using\n\n```lean\n","line":69,"end_line":78,"proof_status":"formalized"}
+{"id":"IsDedekindDomain.dvd_norm","label":"IsDedekindDomain.dvd_norm","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"If \\(0\\not=b\\in B\\) then there exists \\(0\\not=a\\in A\\) such that \\(b\\) divides the image of \\(a\\) in \\(B\\).","doc_url":null,"latex_link":"Adele_miniproject.html#IsDedekindDomain.dvd_norm","lean_link":null,"lean_decl_name":"IsDedekindDomain.dvd_norm","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/NumberTheory/NumberField/Norm.lean","lean4_code":"theorem dvd_norm [IsGalois K L] (x : 𝓞 L) : x ∣ algebraMap (𝓞 K) (𝓞 L) (norm K x) := by\n  classical\n  have hint :\n    IsIntegral ℤ (∏ σ ∈ univ.erase (AlgEquiv.refl : L ≃ₐ[K] L), σ x) :=\n    IsIntegral.prod _ (fun σ _ =>\n      ((RingOfIntegers.isIntegral_coe x).map σ))\n  refine ⟨⟨_, hint⟩, ?_⟩\n  ext\n  rw [coe_algebraMap_norm K x, norm_eq_prod_automorphisms]\n  simp [← Finset.mul_prod_erase _ _ (mem_univ AlgEquiv.refl)]\n\nvariable (F : Type*) [Field F] [Algebra K F] [Algebra.IsSeparable K F] [FiniteDimensional K F]\n\n","line":84,"end_line":96,"proof_status":"statement_formalized"}
+{"id":"Rat.AdeleRing.zero_discrete","label":"Rat.AdeleRing.zero_discrete","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"There’s an open subset of \\(\\mathbb {A}_{\\mathbb {Q}}\\) whose intersection with \\(\\mathbb {Q}\\) is \\(\\{ 0\\} \\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Rat.AdeleRing.zero_discrete","latex_link":"Adele_miniproject.html#Rat.AdeleRing.zero_discrete","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Rat.AdeleRing.zero_discrete","lean_decl_name":"Rat.AdeleRing.zero_discrete","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/NumberField/AdeleRing.lean","lean4_code":"theorem Rat.AdeleRing.zero_discrete : ∃ U : Set (AdeleRing (𝓞 ℚ) ℚ),\n    IsOpen U ∧ (algebraMap ℚ (AdeleRing (𝓞 ℚ) ℚ)) ⁻¹' U = {0} := by\n  let integralAdeles := {f : FiniteAdeleRing (𝓞 ℚ) ℚ |\n    ∀ v , f v ∈ IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers ℚ v}\n  use {f | ∀ v, f v ∈ (Metric.ball 0 1)} ×ˢ integralAdeles\n  refine ⟨?_, ?_⟩\n  · apply IsOpen.prod\n    · rw [Set.setOf_forall]\n      apply isOpen_iInter_of_finite\n      intro v\n      exact Metric.isOpen_ball.preimage (continuous_apply v)\n    · exact RestrictedProduct.isOpen_forall_mem fun v ↦ Valued.isOpen_integer _\n  · apply subset_antisymm\n    · intro x hx\n      rw [Set.mem_preimage] at hx\n      simp only [Set.mem_singleton_iff]\n      rw [show (algebraMap ℚ (AdeleRing (𝓞 ℚ) ℚ)) x =\n        (algebraMap ℚ (InfiniteAdeleRing ℚ) x, algebraMap ℚ (FiniteAdeleRing (𝓞 ℚ) ℚ) x)\n        from rfl] at hx\n      rw [Set.mem_prod] at hx\n      obtain ⟨h1, h2⟩ := hx\n      dsimp only at h1 h2\n      simp only [Metric.mem_ball, dist_zero_right, Set.mem_setOf_eq] at h1\n      exact Rat.AdeleRing.integral_and_norm_lt_one x h2 h1\n    · intro x\n      simp only [Set.mem_singleton_iff, Set.mem_preimage]\n      rintro rfl\n      simp only [map_zero]\n      change (0, 0) ∈ _\n      simp only [Prod.mk_zero_zero]\n      constructor\n      · simp only [Metric.mem_ball, dist_zero_right, Set.mem_setOf_eq]\n        intro v\n        have : ‖(0:InfiniteAdeleRing ℚ) v‖ = 0 := by\n          simp only [norm_eq_zero]\n          rfl\n        simp [this, zero_lt_one]\n      · simp only [integralAdeles, Set.mem_setOf_eq]\n        intro v\n        apply zero_mem","line":368,"end_line":407,"proof_status":"statement_formalized"}
+{"id":"IsDedekindDomain.HeightOneSpectrum.valuation_comap","label":"IsDedekindDomain.HeightOneSpectrum.valuation_comap","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"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 ramification index of \\(w/v\\) (recall that valuations here are written additively, unlike in mathlib).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.valuation_comap","latex_link":"Adele_miniproject.html#IsDedekindDomain.HeightOneSpectrum.valuation_comap","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.valuation_comap","lean_decl_name":"IsDedekindDomain.HeightOneSpectrum.valuation_comap","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/DedekindDomain/IntegralClosure.lean","lean4_code":"/-- If w | v then for x ∈ K we have w(x)=v(x)^e where e is the ramification index. -/\nlemma valuation_comap (w : HeightOneSpectrum B) (x : K) :\n    (comap A w).valuation K x ^\n      (Ideal.ramificationIdx (algebraMap A B) (comap A w).asIdeal w.asIdeal) =\n    w.valuation L (algebraMap K L x) := by\n  obtain ⟨x, y, hy, rfl⟩ := IsFractionRing.div_surjective (A := A) x\n  simp [valuation, ← IsScalarTower.algebraMap_apply A K L, IsScalarTower.algebraMap_apply A B L,\n    ← intValuation_comap A B (algebraMap_injective_of_field_isFractionRing A B K L), div_pow]\n","line":116,"end_line":123,"proof_status":"formalized"}
+{"id":"a0000000183","label":"a0000000183","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"In the same setup as definition&#160;9.26 (\\(V,W\\) index sets, \\(f:W\\to V\\), \\(C_v\\subseteq X_v\\) and \\(D_w\\subseteq Y_w\\), bijections \\(b_v:X_v\\to \\prod _{w:f(w)=v}Y_w\\) identifying \\(C_v\\) with \\(\\prod _{w:f(w)=v}D_w\\)), if all the \\(X_v\\) and \\(Y_w\\) are furthermore topological spaces, all the \\(C_v\\) and \\(D_w\\) are open, and all the \\(b_v\\) are homeomorphisms, then the induced map \\(\\prod '_v(X_v,C_v)\\to \\prod '_w(Y_w,D_w)\\) is also a homeomorphism.","doc_url":null,"latex_link":"Adele_miniproject.html#a0000000183","lean_link":null,"lean_decl_name":"a0000000183","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"NumberField.InfinitePlace.Completion.finrank_pi_eq_finrank_tensorProduct","label":"NumberField.InfinitePlace.Completion.finrank_pi_eq_finrank_tensorProduct","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"For a fixed infinite place \\(v\\) of \\(K\\), we have \\(\\text{dim}_{K_v} \\prod _{w\\mid v} L_w = \\text{dim}_{K_v} L\\otimes _K K_v\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.finrank_pi_eq_finrank_tensorProduct","latex_link":"Adele_miniproject.html#NumberField.InfinitePlace.Completion.finrank_pi_eq_finrank_tensorProduct","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.finrank_pi_eq_finrank_tensorProduct","lean_decl_name":"NumberField.InfinitePlace.Completion.finrank_pi_eq_finrank_tensorProduct","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/NumberField/Completion/Infinite.lean","lean4_code":"theorem finrank_pi_eq_finrank_tensorProduct [NumberField K] :\n    Module.finrank v.Completion ((w : v.Extension L) → w.1.Completion) =\n      Module.finrank v.Completion (L ⊗[K] v.Completion) := by\n  rw [(TensorProduct.RightActions.Algebra.TensorProduct.comm\n       K v.Completion L).symm.toLinearEquiv.finrank_eq,\n      Module.finrank_tensorProduct, Module.finrank_self, one_mul,\n    finrank_prod_eq_finrank]\n","line":97,"end_line":103,"proof_status":"statement_formalized"}
+{"id":"IsDedekindDomain.AKLB.tensorProduct_module_algEquiv","label":"IsDedekindDomain.AKLB.tensorProduct_module_algEquiv","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"If \\(M\\) is any \\(K\\)-module then the canonical map \\(B\\otimes _A M\\to L\\otimes _K M\\) is an isomorphism.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.linearEquivTensorProductModule","latex_link":"Adele_miniproject.html#IsDedekindDomain.AKLB.tensorProduct_module_algEquiv","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.linearEquivTensorProductModule","lean_decl_name":"IsDedekindDomain.linearEquivTensorProductModule","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/DedekindDomain/IntegralClosure.lean","lean4_code":"/-- The canonical `A`-linear isomorphism `L ⊗ M ≅ B ⊗ M` for any `K`-module `M`. -/\nnoncomputable def linearEquivTensorProductModule : L ⊗[K] M ≃ₗ[A] B ⊗[A] M :=\n  let f₁ : L ⊗[K] M ≃ₗ[A] L ⊗[A] M := IsLocalization.moduleTensorEquiv (nonZeroDivisors A) K L M\n    |>.restrictScalars A\n  let f₂ : L ≃ₗ[A] B ⊗[A] K := LinearEquivTensorProduct A K L B\n    |>.restrictScalars A\n    |>.trans (TensorProduct.comm A K B)\n  let f₃ : L ⊗[A] M ≃ₗ[A] (B ⊗[A] K) ⊗[A] M := TensorProduct.congr f₂ (LinearEquiv.refl A M)\n  let f₄ : (B ⊗[A] K) ⊗[A] M ≃ₗ[A] B ⊗[A] (K ⊗[A] M) :=\n    TensorProduct.assoc A B K M\n  let f₅ : B ⊗[A] (K ⊗[A] M) ≃ₗ[A] B ⊗[A] M := TensorProduct.congr (LinearEquiv.refl A B)\n    (IsLocalization.moduleLid (nonZeroDivisors A) K M |>.restrictScalars A)\n  f₁.trans f₃ |>.trans f₄ |>.trans f₅\n","line":245,"end_line":257,"proof_status":"formalized"}

crawled_repos/FinDFA.jsonl

@@ -0,0 +1,4 @@
+{"id":"def:dfa","label":"dfa","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AydenLamp.github.io/FinDFA/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Mathlib provides a general definition of deterministic finite automata that does not require the state space or alphabet to be finite or have decidable equality. A DFA \\(\\alpha \\) \\(\\sigma \\) consists of: step : \\(\\sigma \\) \\(\\to \\) \\(\\alpha \\) \\(\\to \\) \\(\\sigma \\) - a transition function that maps a state and input symbol to a new state start : \\(\\sigma \\) - an initial state accept : Set \\(\\sigma \\) - a set of accepting states The DFA structure provides methods such as: eval : List \\(\\alpha \\) \\(\\to \\) \\(\\sigma \\) - evaluates a word from the start state evalFrom : \\(\\sigma \\) \\(\\to \\) List \\(\\alpha \\) \\(\\to \\) \\(\\sigma \\) - evaluates a word from a given state accepts : Set (List \\(\\alpha \\)) - the language accepted by the automaton Decidable equality means that for any two elements of a type, we can computationally determine whether they are equal or not. This is essential for implementing algorithms that need to compare states or symbols.","doc_url":null,"latex_link":"sect0001.html#def:dfa","lean_link":null,"lean_decl_name":"dfa","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:exists-short-access-word","label":"exists-short-access-word","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://AydenLamp.github.io/FinDFA/blueprint/dep_graph_document.html","dependencies":["def:accessible-fin-dfa"],"nlp_text":"A fundamental result for implementing accessibility checking is that if a state is accessible by any word, then it is accessible by some word of length at most the number of states in the automaton. This bound follows from the pigeonhole principle: if a longer word exists, it must revisit some state, creating a loop that can be removed. This theorem enables us to create a decidable procedure for determining state accessibility. Instead of searching the infinite space of all possible words, we only need to check words up to a finite length bound. Using the getWordsLeqLength function, we can enumerate all words of bounded length and test each one. Furthermore, this allows us to implement a language-preserving conversion from any FinDFA to an AccessibleFinDFA by restricting the state space to only the accessible states. The resulting automaton accepts exactly the same language as the original.","doc_url":"https://AydenLamp.github.io/FinDFA/docs/find/#doc/FinDFA.exists_short_access_word","latex_link":"sect0001.html#lem:exists-short-access-word","lean_link":"https://AydenLamp.github.io/FinDFA/docs/find/#doc/FinDFA.exists_short_access_word","lean_decl_name":"FinDFA.exists_short_access_word","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FinDFA","source_path":"./MyProject/DFA/Accessible.lean","lean4_code":"theorem exists_short_access_word (s : σ) {w : List α}\n    (hw : M.eval w = s) :\n    ∃ v : List α, v.length ≤ Fintype.card σ ∧ M.eval v = s := by\n  -- Strong recursion on the length of `w`\n  refine (Nat.strongRecOn\n    (motive := fun n => ∀ w : List α, w.length = n → M.eval w = s →\n      ∃ v : List α, v.length ≤ Fintype.card σ ∧ M.eval v = s)\n    w.length ?_ w rfl hw)\n  simp_all\n  intro n ih u hlen hu\n  by_cases hshort : n ≤ Fintype.card σ\n  · subst hlen\n    use u\n  · have hle : Fintype.card σ ≤ n := by exact Nat.le_of_not_ge hshort\n    -- Use Mathlib's `DFA.evalFrom_split` lemma to split `w` into `a ++ b ++ c` with a loop on `b`.\n    subst hlen\n    have h := (M.evalFrom_split hle hu)\n    rcases h with ⟨q, a, b, c, hsplit, hlen, hbne, hqa, hqb, hqc⟩\n    simp_all\n    have hlen₂ : (a ++ c).length < a.length + (b.length + c.length) := by\n      simp_all\n      exact List.length_pos_iff.mpr hbne\n    specialize ih (a ++ c).length hlen₂ (a ++ c) rfl\n    apply ih\n    simp [eval, evalFrom_of_append, hqa, hqc]\n\nvariable [Fintype α] [DecidableEq α]\n\n/-- A state is accessible iff there exists a word accessing it in the Finset\nof all words of length less than or equal to the cardinality of the state-space. -/\n","line":188,"end_line":217,"proof_status":"formalized"}
+{"id":"def:fin-dfa","label":"fin-dfa","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AydenLamp.github.io/FinDFA/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"We define FinDFA \\(\\alpha \\) \\(\\sigma \\) as a computable version of DFA \\(\\alpha \\) \\(\\sigma \\) that enables algorithmic manipulation. A FinDFA differs from a DFA in several key ways: It requires Fintype instances on both the alphabet \\(\\alpha \\) and state space \\(\\sigma \\). A Fintype is a type that has finitely many elements and provides a way to enumerate all of them. It requires DecidableEq instances on both types, enabling computational equality testing. The accepting states are represented as a Finset \\(\\sigma \\) rather than a Set \\(\\sigma \\). A Finset is a finite set that can be computationally manipulated, unlike the more general Set which may be infinite or non-computable. This structure allows for a decidable procedure to determine if a state is accepting - we can simply check membership in the finite set of accepting states. We provide a coercion from FinDFA to DFA, allowing us to use all the existing DFA definitions for evaluation and language acceptance.","doc_url":"https://AydenLamp.github.io/FinDFA/docs/find/#doc/FinDFA","latex_link":"sect0001.html#def:fin-dfa","lean_link":"https://AydenLamp.github.io/FinDFA/docs/find/#doc/FinDFA","lean_decl_name":"FinDFA","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/AydenLamp/FinDFA/a817bff14f8dd08b3681cda1fa8bb87a89e64805/MyProject/Defs.lean","lean4_code":"/-- A computable DFA. Just like `DFA` but `accept` is a `Finset` -/\nstructure FinDFA (α : Type u) (σ : Type v) where\n  /-- Transition function. -/\n  step : σ → α → σ\n  /-- Start state. -/\n  start : σ\n  /-- Accepting states as a `Finset`. -/\n  accept : Finset σ","line":118,"end_line":125,"proof_status":"formalized"}
+{"id":"def:accessible-fin-dfa","label":"accessible-fin-dfa","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AydenLamp.github.io/FinDFA/blueprint/dep_graph_document.html","dependencies":["def:fin-dfa"],"nlp_text":"A state \\(s\\) in a FinDFA is called accessible if there exists some word \\(w\\) that reaches \\(s\\) from the start state. Formally, FinDFA.IsAccessibleState M s holds when there exists a word \\(w\\) such that evaluating \\(w\\) from the start state of \\(M\\) results in state \\(s\\). An AccessibleFinDFA is a structure that extends FinDFA with the additional requirement that every state in the automaton is accessible from the start state. This ensures that the automaton contains no \"dead\" or unreachable states.","doc_url":"https://AydenLamp.github.io/FinDFA/docs/find/#doc/AccessibleFinDFA","latex_link":"sect0001.html#def:accessible-fin-dfa","lean_link":"https://AydenLamp.github.io/FinDFA/docs/find/#doc/AccessibleFinDFA","lean_decl_name":"AccessibleFinDFA","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/AydenLamp/FinDFA/a817bff14f8dd08b3681cda1fa8bb87a89e64805/MyProject/Defs.lean","lean4_code":"/-- A DFA where every state is accessible from the start state. -/\nstructure AccessibleFinDFA (α : Type u) (σ : Type v) extends M : FinDFA α σ where\n  is_accessible (s : σ) : M.IsAccessibleState s","line":290,"end_line":292,"proof_status":"formalized"}

crawled_repos/ForbiddenMatrix.jsonl

@@ -0,0 +1,2 @@
+{"id":"def:containment","label":"containment","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ForbiddenMatrix/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"An \\(n\\)-by-\\(m\\) binary matrix \\(M\\) contains an \\(n'\\)-by-\\(m'\\) matrix \\(P\\) if there are integers \\(1 \\leq i_1 \\leq \\ldots \\leq i_{n'} \\leq n\\) and \\(1 \\leq j_1 \\leq \\ldots \\leq j_{m'} \\leq m\\) such that for all \\(i' \\in [n'], j' \\in [m']\\), \\(P(i',j') = 1\\) implies \\(M(i_{i'},j_{j'}) = 1\\). If \\(M\\) does not contain \\(P\\), then \\(M\\) avoids \\(P\\).","doc_url":null,"latex_link":"chap-defs.html#def:containment","lean_link":null,"lean_decl_name":"containment","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:extremal function","label":"extremal function","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ForbiddenMatrix/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Given two integers \\(n,m\\), we denote \\(Ex(P,n,m)\\) as the maximum number of 1’s in a matrix avoiding a matrix \\(P\\). We also define \\(Ex(P,n) = Ex(P,n,n)\\).","doc_url":null,"latex_link":"chap-defs.html#def:extremal function","lean_link":null,"lean_decl_name":"extremal function","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}

crawled_repos/GibbsMeasure.jsonl

@@ -0,0 +1,35 @@
+{"id":"def:proper-ker","label":"proper-ker","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:kernel"],"nlp_text":"\\(\\pi \\) is proper iff \\(\\pi (A\\cap B\\mid x)=\\pi (A\\mid x)\\cdot \\mathbf{1}_B(x)\\) for all \\(A\\in \\mathcal X\\), \\(B\\in \\mathcal B\\) and \\(x\\in X\\).","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/ProbabilityTheory.Kernel.IsProper","latex_link":"sect0002.html#def:proper-ker","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/ProbabilityTheory.Kernel.IsProper","lean_decl_name":"ProbabilityTheory.Kernel.IsProper","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"/-- A specification is proper if all its boundary condition kernels are. -/\ndef IsProper (γ : Specification S E) : Prop := ∀ Λ : Finset S, (γ Λ).IsProper\n","line":92,"end_line":93,"proof_status":"formalized"}
+{"id":"lem:cond-exp-proper-ker-lintegral","label":"cond-exp-proper-ker-lintegral","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:proper-ker","def:cond-exp-ker","lem:lebesgue-cond-exp-char"],"nlp_text":"If \\(\\pi \\in \\text{Ker}_{\\mathcal B, \\mathcal X}\\) is a conditional expectation kernel for \\(\\mu \\), then \\[ \\mu [f\\mid \\mathcal B] = \\int f(x)\\ \\pi (\\partial x\\mid \\cdot )\\ \\mu \\text{-a.e.} \\]","doc_url":null,"latex_link":"sect0002.html#lem:cond-exp-proper-ker-lintegral","lean_link":null,"lean_decl_name":"cond-exp-proper-ker-lintegral","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:proper-spec","label":"proper-spec","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:proper-ker","def:spec"],"nlp_text":"A specification \\(\\gamma \\) is proper iff the kernel \\(\\gamma _\\Lambda \\) is proper for every \\(\\Lambda \\in \\operatorname{Finset}(S)\\).","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsProper","latex_link":"sect0002.html#def:proper-spec","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsProper","lean_decl_name":"Specification.IsProper","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"/-- A specification is proper if all its boundary condition kernels are. -/\ndef IsProper (γ : Specification S E) : Prop := ∀ Λ : Finset S, (γ Λ).IsProper\n","line":92,"end_line":93,"proof_status":"formalized"}
+{"id":"lem:cond-exp-proper-ker-integral","label":"cond-exp-proper-ker-integral","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:proper-ker","def:cond-exp-ker","lem:lebesgue-cond-exp-char"],"nlp_text":"If \\(\\pi \\in \\text{Ker}_{\\mathcal B, \\mathcal X}\\) is a conditional expectation kernel for \\(\\mu \\), then \\[ \\mu (f\\mid \\mathcal B) = \\int f(x)\\ \\pi (\\partial x\\mid \\cdot )\\ \\mu \\text{-a.e.} \\]","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/ProbabilityTheory.Kernel.condExp_ae_eq_integral_kernel","latex_link":"sect0002.html#lem:cond-exp-proper-ker-integral","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/ProbabilityTheory.Kernel.condExp_ae_eq_integral_kernel","lean_decl_name":"ProbabilityTheory.Kernel.condExp_ae_eq_integral_kernel","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Prereqs/Kernel/CondExp.lean","lean4_code":"lemma condExp_ae_eq_integral_kernel (f : X → ℝ) :\n    condExp 𝓑 μ f =ᵐ[μ] (fun x₀ ↦ ∫ x, f x ∂(π x₀)) := sorry\n","line":91,"end_line":92,"proof_status":"statement_formalized"}
+{"id":"lem:proper-ker-integral","label":"proper-ker-integral","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:proper-ker"],"nlp_text":"If \\(\\pi \\) is a proper Markov kernel and \\(g\\) is a bounded function, then \\[ \\int f(x) g(x)\\ \\pi (dx\\mid x_0) = g(x_0)\\int f(x)\\ \\pi (dx\\mid x_0) \\]","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/ProbabilityTheory.Kernel.IsProper.integral_bdd_mul","latex_link":"sect0002.html#lem:proper-ker-integral","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/ProbabilityTheory.Kernel.IsProper.integral_bdd_mul","lean_decl_name":"ProbabilityTheory.Kernel.IsProper.integral_bdd_mul","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/james18lpc/GibbsMeasure/4275e8f4c089b3f14f067e521aed77af94624974/GibbsMeasure/Mathlib/Probability/Kernel/Proper.lean","lean4_code":"lemma IsProper.integral_bdd_mul (h𝓑𝓧 : 𝓑 ≤ 𝓧) (hπ : IsProper π) (hf : Integrable[𝓧] f (π x₀))\n    (hg : StronglyMeasurable[𝓑] g) (hgbdd : ∃ C > 0, ∀ x, ‖g x‖ ≤ C) :\n    ∫ x, g x * f x ∂(π x₀) = g x₀ * ∫ x, f x ∂(π x₀) := by\n  obtain ⟨C, hpC, hC⟩ := hgbdd\n  induction f, hf using Integrable.induction' with\n  | indicator c s hs _ =>\n    simp [← smul_indicator_one_apply, mul_left_comm, integral_const_mul,\n      hπ.integral_bdd_mul_indicator h𝓑𝓧 hs hg ⟨C, hC⟩]\n  | add f₁ f₂ hf₁ hf₂ _ hgf₁ hgf₂ =>\n    have : Integrable (fun x ↦ g x * f₁ x) (π x₀) :=\n      hf₁.bdd_mul (hg.mono h𝓑𝓧).aestronglyMeasurable ⟨C, hC⟩\n    have : Integrable (fun x ↦ g x * f₂ x) (π x₀) :=\n      hf₂.bdd_mul (hg.mono h𝓑𝓧).aestronglyMeasurable ⟨C, hC⟩\n    simp [mul_add, integral_add, *]\n  | isClosed =>\n    refine isClosed_eq ?_ <| by fun_prop\n    refine Metric.continuous_iff.mpr fun f2 ε hε ↦ ⟨ε / C, div_pos hε hpC, fun a ha ↦ ?_⟩\n    have hInt1 : Integrable (fun x ↦ g x * a x) (π x₀) :=\n      (L1.integrable_coeFn a).bdd_mul (hg.mono h𝓑𝓧).aestronglyMeasurable ⟨C, hC⟩\n    have hInt2 : Integrable (fun x ↦ g x * f2 x) (π x₀) :=\n      (L1.integrable_coeFn f2).bdd_mul (hg.mono h𝓑𝓧).aestronglyMeasurable ⟨C, hC⟩\n    have hsub :\n        ‖∫ x, g x * a x ∂π x₀ - ∫ x, g x * f2 x ∂π x₀‖ = ‖∫ x, g x * a x - g x * f2 x ∂π x₀‖ := by\n      simp [integral_sub hInt1 hInt2]\n    have hdiff_ae : (fun x ↦ g x * a x - g x * f2 x) =ᵐ[π x₀] (fun x ↦ g x * (a - f2) x) := by\n      filter_upwards [Lp.coeFn_sub a f2] with x hx\n      have hmul : g x * (a - f2) x = g x * (a x - f2 x) := by\n        simpa [hx] using congrArg (fun t ↦ g x * t) hx\n      calc\n        g x * a x - g x * f2 x\n            = g x * (a x - f2 x) := by simp [mul_sub]\n        _ = g x * (a - f2) x := by simp [hmul.symm]\n    have hIntDiff : Integrable (fun x ↦ g x * (a - f2) x) (π x₀) :=\n      (L1.integrable_coeFn (a - f2)).bdd_mul ((hg.mono h𝓑𝓧).aestronglyMeasurable) ⟨C, hC⟩\n    have hdInt : Integrable (fun x ↦ C * ‖(a - f2) x‖) (π x₀) :=\n      ((L1.integrable_coeFn (a - f2)).norm.smul C)\n    have hle_ae : (fun x ↦ ‖g x * (a - f2) x‖) ≤ᵐ[π x₀] (fun x ↦ C * ‖(a - f2) x‖) :=\n      .of_forall fun x ↦ by simpa using mul_le_mul_of_nonneg_right (hC x) (norm_nonneg ((a - f2) x))\n    have hle_int : ‖∫ x, g x * (a - f2) x ∂π x₀‖ ≤ C * ‖a - f2‖ := by\n      calc\n        ‖∫ x, g x * (a - f2) x ∂π x₀‖\n            ≤ ∫ x, ‖g x * (a - f2) x‖ ∂π x₀ := norm_integral_le_integral_norm _\n        _ ≤ ∫ x, C * ‖(a - f2) x‖ ∂π x₀ := integral_mono_ae hIntDiff.norm hdInt hle_ae\n        _ = C * ∫ x, ‖(a - f2) x‖ ∂π x₀ := by simp [integral_const_mul]\n        _ = C * ‖a - f2‖ := by rw [(L1.norm_eq_integral_norm (a - f2) (μ := π x₀)).symm]\n    have hle : ‖∫ x, g x * a x ∂π x₀ - ∫ x, g x * f2 x ∂π x₀‖ ≤ C * ‖a - f2‖ := by\n      have hdiffint : ‖∫ x, g x * a x ∂π x₀ - ∫ x, g x * f2 x ∂π x₀‖\n           = ‖∫ x, g x * (a - f2) x ∂π x₀‖ := by\n        simpa [integral_congr_ae hdiff_ae,integral_sub hInt1 hInt2] using hsub\n      simpa [hdiffint] using hle_int\n    have hlt : C * ‖a - f2‖ < ε := by\n      have hdist : ‖a - f2‖ < ε / C := by simpa [dist_eq_norm] using ha\n      simpa [mul_div,mul_div_cancel_left₀ ε (ne_of_gt hpC)] using (mul_lt_mul_of_pos_left hdist hpC)\n    exact lt_of_le_of_lt hle hlt\n  | ae_congr f₁ f₂ hf₁ hf hgf₁ =>\n    have : (fun x ↦ g x * f₁ x) =ᵐ[π x₀] (fun x ↦ g x * f₂ x) := by\n      filter_upwards [hf] with x hx; simp [hx]\n    simpa [integral_congr_ae this, integral_congr_ae hf] using hgf₁","line":83,"end_line":140,"proof_status":"formalized"}
+{"id":"lem:proper-ker-lintegral","label":"proper-ker-lintegral","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:proper-ker"],"nlp_text":"If \\(\\pi \\) is proper, then \\[ \\int f(x) g(x)\\ \\pi (dx\\mid x_0) = g(x_0)\\int f(x)\\ \\pi (dx\\mid x_0) \\]","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/ProbabilityTheory.Kernel.IsProper.lintegral_mul","latex_link":"sect0002.html#lem:proper-ker-lintegral","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/ProbabilityTheory.Kernel.IsProper.lintegral_mul","lean_decl_name":"ProbabilityTheory.Kernel.IsProper.lintegral_mul","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/leanprover-community/mathlib4/029db123ddaa7f8fd0d18cea3b1b33bf84dacd1e/Mathlib/Probability/Kernel/Proper.lean","lean4_code":"lemma IsProper.lintegral_mul (hπ : IsProper π) (h𝓑𝓧 : 𝓑 ≤ 𝓧) (hf : Measurable[𝓧] f)\n    (hg : Measurable[𝓑] g) (x₀ : X) :\n    ∫⁻ x, g x * f x ∂(π x₀) = g x₀ * ∫⁻ x, f x ∂(π x₀) := by\n  refine hg.ennreal_induction ?_ ?_ ?_\n  · rintro c A hA\n    simp_rw [← smul_indicator_one_apply, smul_mul_assoc, smul_eq_mul]\n    rw [lintegral_const_mul, hπ.lintegral_indicator_mul h𝓑𝓧 hf hA]\n    · measurability\n  · rintro g₁ g₂ - _ hg₂_meas hg₁ hg₂\n    simp only [Pi.add_apply, add_mul]\n    rw [lintegral_add_right, hg₁, hg₂]\n    · exact (hg₂_meas.mono h𝓑𝓧 le_rfl).mul hf\n  · rintro g' hg'_meas hg'_mono hg'\n    simp_rw [ENNReal.iSup_mul]\n    rw [lintegral_iSup (fun n ↦ ((hg'_meas _).mono h𝓑𝓧 le_rfl).mul hf)\n      (hg'_mono.mul_const (zero_le _))]\n    simp_rw [hg']","line":121,"end_line":137,"proof_status":"formalized"}
+{"id":"lem:gibbs-meas-tfae","label":"gibbs-meas-tfae","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["lem:cond-exp-proper-ker-lintegral","def:proper-spec","def:gibbs-meas"],"nlp_text":"Let \\(\\gamma \\) be a proper specification with parameter set \\(S\\) and state space \\((E, \\mathcal{E})\\), and let \\(\\nu \\in \\mathfrak {P}(E^S, \\mathcal{E}^S)\\). TFAE: \\(\\nu \\in \\mathcal{G}(\\gamma )\\). \\(\\gamma _\\Lambda \\circ _m\\nu = \\nu \\) for all \\(\\Lambda \\in \\operatorname{Finset}(S)\\). \\(\\gamma _\\Lambda \\circ _m\\nu = \\nu \\) frequently as \\(\\Lambda \\to S\\).","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.isGibbsMeasure_iff_forall_bind_eq","latex_link":"sect0002.html#lem:gibbs-meas-tfae","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.isGibbsMeasure_iff_forall_bind_eq","lean_decl_name":"Specification.isGibbsMeasure_iff_forall_bind_eq","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"lemma isGibbsMeasure_iff_forall_bind_eq (hγ : γ.IsProper) [IsFiniteMeasure μ] :\n    γ.IsGibbsMeasure μ ↔ ∀ Λ, μ.bind (γ Λ) = μ :=\n  forall_congr' fun _Λ ↦ Kernel.isCondExp_iff_bind_eq_left (hγ _) cylinderEvents_le_pi\n","line":142,"end_line":144,"proof_status":"formalized"}
+{"id":"lem:isssd-proper-spec","label":"isssd-proper-spec","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:proper-spec","def:isssd-spec","def:isssd"],"nlp_text":"\\(\\operatorname{ISSSD}(\\nu )\\) is a proper specification.","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsProper.isssd","latex_link":"sect0002.html#lem:isssd-proper-spec","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsProper.isssd","lean_decl_name":"Specification.IsProper.isssd","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"/-- The **Independent Specification with Single Spin Distribution**.\n\nThis is the specification corresponding to the product measure. -/\n@[simps]\ndef isssd : Specification S E where\n  toFun := isssdFun ν\n  isConsistent' Λ₁ Λ₂ hΛ := by\n    classical\n    rw [isssdFun_comp_isssdFun]\n    ext a s _\n    simp only [Kernel.comap_apply, id_eq, isssdFun_apply, Finset.coe_sort_coe]\n    rw [Finset.union_eq_right.2 hΛ]\n","line":198,"end_line":209,"proof_status":"formalized"}
+{"id":"lem:modif-proper","label":"modif-proper","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:proper-spec","lem:proper-ker-lintegral","def:modif"],"nlp_text":"If \\(\\gamma \\) is a specification and \\(\\rho \\) a modifier of \\(\\gamma \\), then \\(\\rho \\gamma \\) is a proper specification.","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsProper.modification","latex_link":"sect0002.html#lem:modif-proper","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsProper.modification","lean_decl_name":"Specification.IsProper.modification","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"/-- Modification specification.\n\nModifying the specification `γ` by a family indexed by finsets `Λ : Finset S` of densities\n`ρ Λ : (S → E) → ℝ≥0∞` results in a family of kernels `γ.modificationKer ρ _ Λ` whose density is\nthat of `γ Λ` multiplied by `ρ Λ`.\n\nWhen the family of densities `ρ` is a modifier (`Specification.IsModifier`), modifying a\nspecification results in a specification `γ.modification ρ _`. -/\nnoncomputable def modification (γ : Specification S E) (ρ : Finset S → (S → E) → ℝ≥0∞)\n    (hρ : γ.IsModifier ρ) : Specification S E where\n  toFun := modificationKer γ ρ hρ.measurable\n  isConsistent' := hρ.isConsistent\n","line":296,"end_line":307,"proof_status":"formalized"}
+{"id":"lem:isssd-gibbs-meas-uniqueness","label":"isssd-gibbs-meas-uniqueness","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["lem:isssd-proper-spec","def:gibbs-meas"],"nlp_text":"There is at most one Gibbs measure specified by \\(\\operatorname{ISSSD}(\\nu )\\).","doc_url":null,"latex_link":"sect0002.html#lem:isssd-gibbs-meas-uniqueness","lean_link":null,"lean_decl_name":"isssd-gibbs-meas-uniqueness","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:spec","label":"spec","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:cylinder-event","def:kernel"],"nlp_text":"A specification is a family of kernels \\(\\gamma : \\operatorname{Finset}S \\to \\operatorname{Ker}_{\\mathcal{F}_{S\\setminus \\Lambda }, \\mathcal{E}^S}\\) which is consistent, in the sense that \\[ \\forall \\Lambda _1, \\Lambda _2 \\in \\operatorname{Finset}(S), \\Lambda _1 \\subseteq \\Lambda _2 \\implies \\gamma _{\\Lambda _1} \\circ _k \\gamma _{\\Lambda _2} = \\gamma _{\\Lambda _2} \\]","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification","latex_link":"sect0002.html#def:spec","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification","lean_decl_name":"Specification","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"./GibbsMeasure/Specification.lean","lean4_code":"structure Specification [MeasurableSpace E] where\n  /-- The boundary condition kernels of a specification.\n\n  DO NOT USE. Instead use the coercion to function `⇑γ`. Lean should insert it automatically in most\n  cases. -/\n  toFun (Λ : Finset S) : Kernel[cylinderEvents Λᶜ] (S → E) (S → E)\n  /-- The boundary condition kernels of a specification are consistent.\n\n  DO NOT USE. Instead use `Specification.isConsistent`. -/\n  isConsistent' : IsConsistent toFun\n\nnamespace Specification\n\n","line":37,"end_line":49,"proof_status":"formalized"}
+{"id":"def:indep-spec","label":"indep-spec","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:spec"],"nlp_text":"A specification \\(\\gamma \\) is independent iff \\[ \\forall \\Lambda _1, \\Lambda _2 \\in \\operatorname{Finset}(S), \\gamma _{\\Lambda _1} \\circ _k \\gamma _{\\Lambda _2} = \\gamma _{\\Lambda _1\\cup \\Lambda _2} \\]","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsIndep","latex_link":"sect0002.html#def:indep-spec","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsIndep","lean_decl_name":"Specification.IsIndep","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"/-- An independent specification is a specification `γ` where `γ Λ₁ ∘ₖ γ Λ₂ = γ (Λ₁ ∪ Λ₂)` for all\n`Λ₁ Λ₂`. -/\ndef IsIndep (γ : Specification S E) : Prop :=\n  ∀ ⦃Λ₁ Λ₂⦄ [DecidableEq S] , (γ Λ₁).comap id cylinderEvents_le_pi ∘ₖ γ Λ₂ = (γ (Λ₁ ∪ Λ₂)).comap id\n      (measurable_id'' <| by gcongr; exact Finset.subset_union_right)\n","line":70,"end_line":74,"proof_status":"formalized"}
+{"id":"def:markov-spec","label":"markov-spec","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:spec"],"nlp_text":"A specification \\(\\gamma \\) is a Markov specification iff \\(\\gamma _\\Lambda \\) is a probability kernel for every \\(\\Lambda \\in \\operatorname{Finset}(S)\\).","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsMarkov","latex_link":"sect0002.html#def:markov-spec","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsMarkov","lean_decl_name":"Specification.IsMarkov","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"/-- A Markov specification is a specification whose boundary condition kernels are all Markov\nkernels. -/\nclass IsMarkov (γ : Specification S E) : Prop where\n  isMarkovKernel : ∀ Λ, IsMarkovKernel (γ Λ)\n","line":80,"end_line":83,"proof_status":"formalized"}
+{"id":"def:modif","label":"modif","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:spec"],"nlp_text":"A modifier of \\(\\gamma \\) is a family \\[ \\rho : \\operatorname{Finset}(S) \\to \\Omega \\to [0, \\infty [ \\]","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsModifier","latex_link":"sect0002.html#def:modif","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsModifier","lean_decl_name":"Specification.IsModifier","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"/-- A modifier of a specification `γ` is a family indexed by finsets `Λ : Finset S` of densities\n`ρ Λ : (S → E) → ℝ≥0∞` such that:\n* Each `ρ Λ` is measurable.\n* `γ.modificationKer ρ` (informally, `ρ * γ`) is consistent. -/\n@[mk_iff]\nstructure IsModifier (γ : Specification S E) (ρ : Finset S → (S → E) → ℝ≥0∞) : Prop where\n  measurable Λ : Measurable (ρ Λ)\n  isConsistent : IsConsistent (modificationKer γ ρ measurable)\n","line":264,"end_line":271,"proof_status":"formalized"}
+{"id":"def:isssd-spec","label":"isssd-spec","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:spec"],"nlp_text":"\\(\\operatorname{ISSSD}(\\nu )\\) is a specification.","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.isssd","latex_link":"sect0002.html#def:isssd-spec","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.isssd","lean_decl_name":"Specification.isssd","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"/-- The **Independent Specification with Single Spin Distribution**.\n\nThis is the specification corresponding to the product measure. -/\n@[simps]\ndef isssd : Specification S E where\n  toFun := isssdFun ν\n  isConsistent' Λ₁ Λ₂ hΛ := by\n    classical\n    rw [isssdFun_comp_isssdFun]\n    ext a s _\n    simp only [Kernel.comap_apply, id_eq, isssdFun_apply, Finset.coe_sort_coe]\n    rw [Finset.union_eq_right.2 hΛ]\n","line":198,"end_line":209,"proof_status":"formalized"}
+{"id":"def:gibbs-meas","label":"gibbs-meas","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:spec","def:cond-exp-ker"],"nlp_text":"Given a specification \\(\\gamma \\), a Gibbs measures specified by \\(\\gamma \\) is a measure \\(\\nu \\in \\mathfrak {M}(E^S, \\mathcal{E}^S)\\) such that \\(\\gamma _\\Lambda (A|\\cdot )\\) is a conditional expectation kernel for \\(\\nu \\) for all \\(A \\in \\mathcal E^S\\) and \\(\\Lambda \\in \\operatorname{Finset}(S)\\).","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsGibbsMeasure","latex_link":"sect0002.html#def:gibbs-meas","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsGibbsMeasure","lean_decl_name":"Specification.IsGibbsMeasure","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"/-- For a specification `γ`, a Gibbs measure is a measure whose conditional expectation kernels\nconditionally on configurations exterior to finite sets agree with the boundary condition kernels\nof the specification `γ`. -/\ndef IsGibbsMeasure (γ : Specification S E) (μ : Measure (S → E)) : Prop := ∀ Λ, (γ Λ).IsCondExp μ\n","line":136,"end_line":139,"proof_status":"formalized"}
+{"id":"prop:modif-indep-char","label":"modif-indep-char","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:indep-spec","def:modif"],"nlp_text":"If \\(\\rho \\) is a family of measurable densities and \\(\\gamma \\) is a proper independent specification, then TFAE \\(\\rho \\) is a modifier of \\(\\gamma \\) For all \\(\\Lambda _1, \\Lambda _2\\) with \\(\\Lambda _1 \\subseteq \\Lambda _2\\), \\(\\eta : S \\to E\\) and \\(\\gamma _{\\Lambda _2 \\setminus \\Lambda _1}(\\cdot |\\alpha )\\)-almost all \\(\\eta _2 : S \\to E\\), we have \\[ \\rho _{\\Lambda _2}(\\zeta _1)\\rho _{\\Lambda _1}(\\zeta _2) = \\rho _{\\Lambda _2}(\\zeta _2) \\rho _{\\Lambda _1}(\\zeta _1) \\]","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.isModifier_iff_ae_comm","latex_link":"sect0002.html#prop:modif-indep-char","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.isModifier_iff_ae_comm","lean_decl_name":"Specification.isModifier_iff_ae_comm","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"lemma isModifier_iff_ae_comm [DecidableEq S] :\n    γ.IsModifier ρ ↔ (∀ Λ, Measurable (ρ Λ)) ∧\n    ∀ ⦃Λ₁ Λ₂⦄, Λ₁ ⊆ Λ₂ → ∀ η₁, ∀ᵐ η₂ ∂γ (Λ₂ \\ Λ₁) η₁, ∀ᵐ ζ ∂(γ Λ₁ η₂).prod (γ Λ₂ η₂),\n      ρ Λ₂ ζ.1 * ρ Λ₁ ζ.2 = ρ Λ₂ ζ.2 * ρ Λ₁ ζ.1 := by\n  -- simp only [isModifier_iff_ae_eq, and_congr_right_iff]\n  -- refine fun hρ ↦ forall₄_congr fun Λ₁ Λ₂ hΛ η ↦ ?_\n  sorry\n","line":288,"end_line":294,"proof_status":"formalized"}
+{"id":"lem:isssd-isIndep","label":"isssd-isIndep","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:indep-spec","def:isssd"],"nlp_text":"\\(\\operatorname{ISSSD}(\\nu )\\) is independent.","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.isssdFun_comp_isssdFun","latex_link":"sect0002.html#lem:isssd-isIndep","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.isssdFun_comp_isssdFun","lean_decl_name":"Specification.isssdFun_comp_isssdFun","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"/-- The ISSSD of a measure is strongly consistent. -/\nlemma isssdFun_comp_isssdFun [DecidableEq S] (Λ₁ Λ₂ : Finset S) :\n    (isssdFun ν Λ₁).comap id cylinderEvents_le_pi ∘ₖ isssdFun ν Λ₂ =\n      (isssdFun ν (Λ₁ ∪ Λ₂)).comap id\n        (measurable_id'' <| by gcongr; exact Finset.subset_union_right) :=\n  sorry\n","line":191,"end_line":196,"proof_status":"formalized"}
+{"id":"lem:modif-modif","label":"modif-modif","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:modif"],"nlp_text":"Modifying a specification \\(\\gamma \\) by \\(\\rho _1\\) then \\(\\rho _2\\) is the same as modifying it by their product.","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.modification_modification","latex_link":"sect0002.html#lem:modif-modif","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.modification_modification","lean_decl_name":"Specification.modification_modification","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"@[simp] lemma modification_modification (γ : Specification S E) (ρ₁ ρ₂ : Finset S → (S → E) → ℝ≥0∞)\n    (hρ₁ : γ.IsModifier ρ₁) (hρ₂ : (γ.modification ρ₁ hρ₁).IsModifier ρ₂) :\n    (γ.modification ρ₁ hρ₁).modification ρ₂ hρ₂ = γ.modification (ρ₁ * ρ₂) (hρ₁.mul hρ₂) := by\n  ext Λ σ s hs\n  simp only [modification_apply, Pi.mul_apply]\n  rw [withDensity_apply _ hs, withDensity_apply _ hs,\n    setLIntegral_withDensity_eq_setLIntegral_mul _ (hρ₁.measurable Λ) (hρ₂.1 Λ) hs]\n","line":329,"end_line":335,"proof_status":"formalized"}
+{"id":"lem:exists-modif-isssd","label":"exists-modif-isssd","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:modif","def:isssd"],"nlp_text":"If \\(E\\) is countable, \\(\\nu \\) is the counting measure and \\(\\gamma \\) is any specification, then \\[ \\rho _\\Lambda (\\eta ) = \\gamma _\\Lambda (\\{ \\sigma _\\Lambda = \\eta _\\Lambda \\} | \\eta ) \\]","doc_url":null,"latex_link":"sect0002.html#lem:exists-modif-isssd","lean_link":null,"lean_decl_name":"exists-modif-isssd","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"prop:modif-char","label":"modif-char","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:modif"],"nlp_text":"If \\(\\rho \\) is a family of measurable densities and \\(\\gamma \\) is a proper specification, then TFAE \\(\\rho \\) is a modifier of \\(\\gamma \\) For all \\(\\Lambda _1, \\Lambda _2\\) with \\(\\Lambda _1 \\subseteq \\Lambda _2\\) and all \\(\\eta : S \\to E\\), we have \\[ \\rho _{\\Lambda _2} = \\rho _{\\Lambda _1}\\cdot (\\gamma _{\\Lambda _1} \\rho _{\\Lambda _2}) \\quad \\gamma _{\\Lambda _2}(\\cdot |\\eta )\\text{-a.e.} \\]","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.isModifier_iff_ae_eq","latex_link":"sect0002.html#prop:modif-char","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.isModifier_iff_ae_eq","lean_decl_name":"Specification.isModifier_iff_ae_eq","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"lemma isModifier_iff_ae_eq (hγ : γ.IsProper) :\n    γ.IsModifier ρ ↔ (∀ Λ, Measurable (ρ Λ)) ∧ ∀ ⦃Λ₁ Λ₂⦄, Λ₁ ⊆ Λ₂ → ∀ η,\n      ρ Λ₂ =ᵐ[γ Λ₂ η] fun η ↦ ∫⁻ ζ, ρ Λ₂ ζ ∂(γ Λ₁ η).withDensity (ρ Λ₁) := by\n  simp only [isModifier_iff, IsConsistent, modificationKer, Kernel.ext_iff, Kernel.comp_apply,\n    Kernel.coe_mk, Kernel.coe_comap, CompTriple.comp_eq, Measure.ext_iff, exists_prop,\n    and_congr_right_iff]\n  refine fun hρ ↦ forall₄_congr fun Λ₁ Λ₂ hΛ η ↦ ?_\n  sorry\n","line":279,"end_line":286,"proof_status":"statement_formalized"}
+{"id":"lem:isssd-gibbs-meas-existence","label":"isssd-gibbs-meas-existence","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:isssd-spec","def:gibbs-meas","def:product-probability-measure"],"nlp_text":"The product measure \\(\\nu ^S\\) is a Gibbs measure specified by \\(\\operatorname{ISSSD}(\\nu )\\).","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.isGibbsMeasure_isssd_productMeasure","latex_link":"sect0002.html#lem:isssd-gibbs-meas-existence","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.isGibbsMeasure_isssd_productMeasure","lean_decl_name":"Specification.isGibbsMeasure_isssd_productMeasure","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"/-- The product measure `ν ^ S` is a `isssd μ`-Gibbs measure. -/\nlemma isGibbsMeasure_isssd_productMeasure (ν : Measure E) [IsProbabilityMeasure ν] :\n    (isssd ν).IsGibbsMeasure (.infinitePi fun _ : S ↦  ν) := by\n  rintro Λ\n  sorry\n","line":229,"end_line":233,"proof_status":"statement_formalized"}
+{"id":"lem:premodif-modif","label":"premodif-modif","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["prop:modif-indep-char","def:premodif"],"nlp_text":"If \\(h\\) is a premodifier and \\(\\nu \\) is such that \\(0 {\\lt} \\nu _\\Lambda h_\\Lambda {\\lt} \\infty \\) for all \\(\\Lambda \\), then \\[ \\rho _\\Lambda := \\frac{h_\\Lambda }{\\operatorname{ISSSD}(\\nu )_\\Lambda h_\\Lambda } \\]","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsPremodifier.isModifier_div","latex_link":"sect0002.html#lem:premodif-modif","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsPremodifier.isModifier_div","lean_decl_name":"Specification.IsPremodifier.isModifier_div","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/james18lpc/GibbsMeasure/4275e8f4c089b3f14f067e521aed77af94624974/GibbsMeasure/Specification.lean","lean4_code":"lemma IsPremodifier.isModifier_div (hρ : IsPremodifier ρ) (ν : Measure E) [IsProbabilityMeasure ν] :\n    (isssd ν).IsModifier fun Λ σ ↦ ρ Λ σ / ∫⁻ x, ρ Λ x ∂(isssd ν Λ σ) where\n  measurable Λ :=\n    (hρ.measurable Λ).div ((hρ.measurable Λ).lintegral_kernel.mono cylinderEvents_le_pi le_rfl)\n  isConsistent Λ₁ Λ₂ hΛ := by\n    sorry","line":353,"end_line":358,"proof_status":"statement_formalized"}
+{"id":"lem:modif-premodif","label":"modif-premodif","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["prop:modif-char","def:premodif"],"nlp_text":"If \\(\\rho \\) is a modifier of \\(\\operatorname{ISSSD}(\\nu ^S)\\), then it is a premodifier if any of the following conditions hold: \\(E\\) is countable and \\(\\nu \\) is equivalent to the counting measure. \\(E\\) is a second countable Borel space. \\(\\nu \\) is everywhere dense. For all \\(\\Lambda _1 \\subseteq \\Lambda _2\\) and all \\(\\eta : S \\to E\\), \\(\\zeta \\mapsto \\rho _{\\Lambda _1}(\\zeta \\eta _{\\Lambda _1^c})\\) is continuous on \\(E^{\\Lambda _1}\\).","doc_url":null,"latex_link":"sect0002.html#lem:modif-premodif","lean_link":null,"lean_decl_name":"modif-premodif","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:cond-exp-proper-ker-bind","label":"cond-exp-proper-ker-bind","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:cond-exp-ker","lem:lebesgue-cond-exp-char"],"nlp_text":"Let \\(\\mu \\in \\mathfrak {M}(X,\\mathcal X)\\) be a finite measure and let \\(\\pi \\in \\text{Ker}_{\\mathcal B,\\mathcal X}\\) be a proper kernel. Then, \\[ \\pi \\text{ is a conditional expectation kernel for }\\mu \\iff \\mu \\pi = \\mu \\]","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/ProbabilityTheory.Kernel.isCondExp_iff_bind_eq_left","latex_link":"sect0002.html#lem:cond-exp-proper-ker-bind","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/ProbabilityTheory.Kernel.isCondExp_iff_bind_eq_left","lean_decl_name":"ProbabilityTheory.Kernel.isCondExp_iff_bind_eq_left","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Prereqs/Kernel/CondExp.lean","lean4_code":"lemma isCondExp_iff_bind_eq_left (hπ : π.IsProper) (h𝓑𝓧 : 𝓑 ≤ 𝓧) [SigmaFinite (μ.trim h𝓑𝓧)] :\n    IsCondExp π μ ↔ μ.bind π = μ := by\n  simp_rw [isCondExp_iff, Filter.eventuallyEq_comm,\n    toReal_ae_eq_indicator_condExp_iff_forall_meas_inter_eq h𝓑𝓧, Measure.ext_iff]\n  refine ⟨fun h A hA ↦ ?_, fun h A hA B hB ↦ ?_⟩\n  · rw [eq_comm, Measure.bind_apply hA (π.measurable.mono h𝓑𝓧 le_rfl).aemeasurable]\n    simpa using h hA _ .univ\n  · rw [hπ.setLIntegral_eq_comp h𝓑𝓧 hA hB, eq_comm]\n    exact h _ (by measurability)\n","line":15,"end_line":23,"proof_status":"formalized"}
+{"id":"def:cond-exp-ker","label":"cond-exp-ker","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:kernel"],"nlp_text":"Let \\(\\mu \\in \\mathfrak {M}(X,\\mathcal X)\\). Then, \\(\\pi \\in \\text{Ker}_{\\mathcal B,\\mathcal X}\\) is a conditional expectation kernel for \\(\\mu \\) if \\(\\mu (A\\mid \\mathcal B)=\\pi (A\\mid \\cdot )\\) \\(\\mu \\)-a.e.","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/ProbabilityTheory.condExpKernel","latex_link":"sect0002.html#def:cond-exp-ker","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/ProbabilityTheory.condExpKernel","lean_decl_name":"ProbabilityTheory.condExpKernel","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Probability/Kernel/Condexp.lean","lean4_code":"noncomputable irreducible_def condExpKernel (μ : Measure Ω) [IsFiniteMeasure μ]\n    (m : MeasurableSpace Ω) : @Kernel Ω Ω m mΩ :=\n  if _h : Nonempty Ω then\n    Kernel.comap (@condDistrib Ω Ω Ω mΩ _ _ mΩ (m ⊓ mΩ) id id μ _) id\n      (measurable_id'' (inf_le_left : m ⊓ mΩ ≤ m))\n  else 0\n\n","line":71,"end_line":77,"proof_status":"formalized"}
+{"id":"def:premodif","label":"premodif","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A family of measurable functions \\(h_\\Lambda : (S \\to E) \\to [0, \\infty [\\) is a premodifier if \\[ h_{\\Lambda _2}(\\zeta )h_{\\Lambda _1}(\\eta ) = h_{\\Lambda _1}(\\zeta )h_{\\Lambda _2}(\\eta ) \\]","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsPremodifier","latex_link":"sect0002.html#def:premodif","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.IsPremodifier","lean_decl_name":"Specification.IsPremodifier","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"/-- A premodifier is a family indexed by finsets `Λ : Finset S` of densities\n`ρ Λ : (S → E) → ℝ≥0∞` such that:\n* each `ρ Λ` is measurable,\n* `ρ Λ₂ ζ * ρ Λ₁ η = ρ Λ₁ ζ * ρ Λ₂ η` for all `Λ₁ Λ₂ : Finset S` and `ζ η : S → E` such that\n  `Λ₁ ⊆ Λ₂` and `∀ (s : Λ₁ᶜ), ζ s = η s`. -/\nstructure IsPremodifier [MeasurableSpace E] (ρ : Finset S → (S → E) → ℝ≥0∞) : Prop where\n  measurable Λ : Measurable (ρ Λ)\n  comm_of_subset ⦃Λ₁ Λ₂ : Finset S⦄ ⦃ζ η : S → E⦄ (hΛ : Λ₁ ⊆ Λ₂)\n    (hrestrict : ∀ s ∉ Λ₁, ζ s = η s) : ρ Λ₂ ζ * ρ Λ₁ η = ρ Λ₁ ζ * ρ Λ₂ η\n","line":343,"end_line":351,"proof_status":"formalized"}
+{"id":"lem:lebesgue-cond-exp-char","label":"lebesgue-cond-exp-char","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:lebesgue-cond-exp"],"nlp_text":"If \\(f : X \\to [0, \\infty ]\\) is a \\(\\mathcal X\\)-measurable function, then \\(\\mu [f | \\mathcal B]\\) is the \\(\\mu \\)-ae unique \\(\\mathcal B\\)-measurable function \\(X \\to [0, \\infty ]\\) such that \\[ \\int _B \\mu [f | \\mathcal B]\\ \\partial \\mu = \\int _B f\\ \\partial \\mu \\]","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/MeasureTheory.ae_eq_lcondExp_of_forall_setLIntegral_eq","latex_link":"sect0001.html#lem:lebesgue-cond-exp-char","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/MeasureTheory.ae_eq_lcondExp_of_forall_setLIntegral_eq","lean_decl_name":"MeasureTheory.ae_eq_lcondExp_of_forall_setLIntegral_eq","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Prereqs/LebesgueCondExp.lean","lean4_code":"  rw [lintegral_lcondExp, lintegral_indicator hA, setLIntegral_const, one_mul]\n\n/-- **Uniqueness of the conditional expectation**\n\nIf a function is a.e. `m`-measurable, verifies an integrability condition and has same lintegral\nas `f` on all `m`-measurable sets, then it is a.e. equal to `μ⁻[f|hm]`. -/\nlemma ae_eq_lcondExp_of_forall_setLIntegral_eq (hm : m ≤ m₀) [SigmaFinite (μ.trim hm)]\n    {f g : α → ℝ≥0∞}\n    (hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫⁻ x in s, g x ∂μ = ∫⁻ x in s, f x ∂μ)\n    (hgm : AEStronglyMeasurable[m] g μ) : g =ᵐ[μ] μ⁻[f|m] := by\n  refine ae_eq_of_forall_setLIntegral_eq_of_sigmaFinite' hm (fun s hs hμs => ?_) hgm sorry\n","line":113,"end_line":123,"proof_status":"statement_formalized"}
+{"id":"def:lebesgue-cond-exp","label":"lebesgue-cond-exp","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The conditional expectation of a \\(\\mathcal X\\)-measurable function \\(f : X \\to [0, \\infty ]\\) is \\[ \\mu [f | \\mathcal B] = ?? \\]","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/MeasureTheory.lcondExp","latex_link":"sect0001.html#def:lebesgue-cond-exp","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/MeasureTheory.lcondExp","lean_decl_name":"MeasureTheory.lcondExp","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Prereqs/LebesgueCondExp.lean","lean4_code":"/-- Lebesgue conditional expectation of an `ℝ≥0∞`-valued function. It is defined as `0` if any of\nthe following conditions holds:\n* `m` is not a sub-σ-algebra of `m₀`,\n* `μ` is not σ-finite with respect to `m`,\n* `f` is not `μ`-integrable. -/\nnoncomputable def lcondExp : α → ℝ≥0∞ :=\n  open scoped Classical in\n  if hm : m ≤ m₀ then\n    if _h : SigmaFinite (μ.trim hm) then\n      if Measurable[m] f then f\n      else if hf : Measurable[m₀] f then\n        ENNReal.ofReal ∘\n          ⨆ n, μ[ENNReal.toReal ∘ (hf.stronglyMeasurable.finStronglyMeasurable μ).approx n | m]\n      else 0\n    else 0\n","line":12,"end_line":26,"proof_status":"formalized"}
+{"id":"def:cylinder-event","label":"cylinder-event","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\((E,\\mathcal{E})\\) be a measurable space, and let \\(S\\) be a set. Then, \\begin{align} \\mathcal{F}:\\mathcal{P}(S)& \\to \\left\\{ \\text{sigma algebras on } E^S \\right\\} \\\\ \\Delta & \\mapsto \\sigma (\\{ \\text{proj}_\\delta :E^S\\to E\\mid \\delta \\in \\Delta \\} ) \\end{align}","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/MeasureTheory.cylinderEvents","latex_link":"sect0002.html#def:cylinder-event","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/MeasureTheory.cylinderEvents","lean_decl_name":"MeasureTheory.cylinderEvents","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/MeasureTheory/Constructions/Cylinders.lean","lean4_code":"def cylinderEvents (Δ : Set ι) : MeasurableSpace (∀ i, X i) := ⨆ i ∈ Δ, (m i).comap fun σ ↦ σ i\n\n","line":394,"end_line":395,"proof_status":"formalized"}
+{"id":"def:product-probability-measure","label":"product-probability-measure","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(I\\) be a set. Suppose for each \\(i\\in I\\) that \\((\\Omega _i,\\mathcal B_i,P_i)\\) is a probability space. Then, \\(P:=\\bigotimes _{i\\in I}P_i\\) is a well-defined product probability measure on \\(\\prod _{i\\in I}\\Omega _i\\).","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/MeasureTheory.productMeasure","latex_link":"sect0002.html#def:product-probability-measure","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/MeasureTheory.productMeasure","lean_decl_name":"MeasureTheory.productMeasure","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/james18lpc/GibbsMeasure/4275e8f4c089b3f14f067e521aed77af94624974/GibbsMeasure/KolmogorovExtension4/ProductMeasure.lean","lean4_code":"/-- The product measure of an arbitrary family of probability measures. It is defined as the unique\nextension of the function which gives to cylinders the measure given by the associated product\nmeasure. -/\nnoncomputable def productMeasure : Measure ((i : ι) → X i) := (sorry : (∀ i, Measure (X i)) → _) μ","line":106,"end_line":109,"proof_status":"formalized"}
+{"id":"def:isssd","label":"isssd","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:juxtaposition"],"nlp_text":"The Independent Specification with Single Spin Distribution \\(\\nu \\) is \\begin{align} \\operatorname{ISSSD}:\\mathfrak {P}(E, \\mathcal{E})& \\to \\operatorname{Finset}(S)\\to \\mathcal{E}^S\\times E^S\\to \\overline{\\mathbb {R}_{\\geq 0}}\\\\ \\nu & \\mapsto \\Lambda \\mapsto (A\\mid \\omega )\\mapsto \\left(\\nu ^\\Lambda \\left(\\operatorname{Juxt}_{\\omega }^{-1}(A)\\right)\\right) \\end{align}","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.isssdFun","latex_link":"sect0002.html#def:isssd","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/Specification.isssdFun","lean_decl_name":"Specification.isssdFun","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"/-- Auxiliary definition for `Specification.isssd`. -/\n@[simps -fullyApplied]\ndef isssdFun (ν : Measure E) (Λ : Finset S) : Kernel[cylinderEvents Λᶜ] (S → E) (S → E) :=\n  @Kernel.mk _ _ (_) _\n    (fun η ↦ Measure.map (juxt Λ η) (Measure.pi fun _ : Λ ↦ ν))\n    (measurable_isssdFun ν Λ)\n","line":184,"end_line":189,"proof_status":"statement_formalized"}
+{"id":"def:kernel","label":"kernel","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\((X,\\mathcal X)\\) and \\((Y,\\mathcal{Y})\\) be measurable spaces. Then, \\[ \\text{Ker}_{\\mathcal{Y},\\mathcal X}:=\\left\\{ \\pi :\\mathcal X\\times Y\\to [0,\\infty ]~ \\middle \\vert ~ \\forall y\\in Y,\\pi (\\cdot \\mid y)\\in \\mathfrak {M}(X,\\mathcal X);~ \\forall A\\in \\mathcal X,\\pi (A\\mid \\cdot )\\text{ is }\\mathcal{Y}\\text{-measurable}\\right\\} \\]","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/ProbabilityTheory.Kernel","latex_link":"sect0002.html#def:kernel","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/ProbabilityTheory.Kernel","lean_decl_name":"ProbabilityTheory.Kernel","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Probability/Moments/SubGaussian.lean","lean4_code":"structure Kernel.HasSubgaussianMGF (X : Ω → ℝ) (c : ℝ≥0)\n    (κ : Kernel Ω' Ω) (ν : Measure Ω' := by volume_tac) : Prop where\n  integrable_exp_mul : ∀ t, Integrable (fun ω ↦ exp (t * X ω)) (κ ∘ₘ ν)\n  mgf_le : ∀ᵐ ω' ∂ν, ∀ t, mgf X (κ ω') t ≤ exp (c * t ^ 2 / 2)\n```\nAn interesting point is that the integrability condition is not integrability of `exp (t * X)`\nwith respect to `κ ω'` for `ν`-almost all `ω'`, but integrability with respect to `κ ∘ₘ ν`.\nThis is a stronger condition, as the weaker one did not allow to prove interesting results about\nthe sum of two sub-Gaussian random variables.\n\nFor the conditional case, that integrability condition reduces to integrability of `exp (t * X)`\nwith respect to `μ`.\n\n### Definition of `HasCondSubgaussianMGF`\n\nWe define `HasCondSubgaussianMGF` as a special case of `Kernel.HasSubgaussianMGF` with the\nconditional expectation kernel for `m`, `condExpKernel μ m`, and the restriction of `μ` to `m`,\n`μ.trim hm` (where `hm` states that `m` is a sub-sigma-algebra).\nNote that `condExpKernel μ m ∘ₘ μ.trim hm = μ`. The definition is equivalent to the two\nconditions\n* for all `t`, `exp (t * X)` is `μ`-integrable,\n* for `μ.trim hm`-almost all `ω`, for all `t`, the mgf with respect to the the conditional\n  distribution `condExpKernel μ m ω` is bounded by `exp (c * t ^ 2 / 2)`.\n\nFor any `t`, we can write the mgf of `X` with respect to the conditional expectation kernel as\na conditional expectation, `(μ.trim hm)`-almost surely:\n`mgf X (condExpKernel μ m ·) t =ᵐ[μ.trim hm] μ[fun ω' ↦ exp (t * X ω') | m]`.\n\n## References\n\n* [R. Vershynin, *High-dimensional probability: An introduction with applications in data\n  science*][vershynin2018high]\n\n-/\n\nopen MeasureTheory Real\n\nopen scoped ENNReal NNReal Topology\n\nnamespace ProbabilityTheory\n\nsection Kernel\n\nvariable {Ω Ω' : Type*} {mΩ : MeasurableSpace Ω} {mΩ' : MeasurableSpace Ω'}\n  {ν : Measure Ω'} {κ : Kernel Ω' Ω} {X : Ω → ℝ} {c : ℝ≥0}\n\n/-! ### Sub-Gaussian with respect to a kernel and a measure -/\n\n","line":86,"end_line":133,"proof_status":"formalized"}
+{"id":"def:markov-ker","label":"markov-ker","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":["def:kernel"],"nlp_text":"Let \\((X,\\mathcal X)\\) and \\((Y,\\mathcal{Y})\\) be measurable spaces. We say that \\(\\pi \\in \\text{Ker}_{\\mathcal{Y},\\mathcal X}\\) is a Markov kernel iff \\(\\pi (X\\mid \\cdot )=1\\).","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/ProbabilityTheory.IsMarkovKernel","latex_link":"sect0002.html#def:markov-ker","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/ProbabilityTheory.IsMarkovKernel","lean_decl_name":"ProbabilityTheory.IsMarkovKernel","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Specification.lean","lean4_code":"instance IsMarkov.toIsMarkovKernel [γ.IsMarkov] {Λ : Finset S} : IsMarkovKernel (γ Λ) :=\n  isMarkovKernel _\n","line":85,"end_line":86,"proof_status":"formalized"}
+{"id":"def:juxtaposition","label":"juxtaposition","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://james18lpc.github.io/GibbsMeasure/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(E\\) and \\(S\\) be sets. Let \\(\\Delta \\in \\mathcal{P}(S)\\), and let \\(\\omega \\in E^S\\). We define \\begin{align} \\operatorname{Juxt}_{\\omega } : E^\\Delta & \\to E^S\\\\ \\zeta & \\mapsto \\delta \\mapsto \\begin{cases} \\zeta _\\delta & \\delta \\in \\Delta \\\\ \\omega _\\delta & \\delta \\notin \\Delta \\end{cases}\\end{align}","doc_url":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/juxt","latex_link":"sect0002.html#def:juxtaposition","lean_link":"https://james18lpc.github.io/GibbsMeasure/docs/find/#doc/juxt","lean_decl_name":"juxt","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"GibbsMeasure","source_path":"GibbsMeasure/Prereqs/Juxt.lean","lean4_code":"noncomputable def juxt (Λ : Set S) (η : S → E) (ζ : Λ → E) (x : S) : E := by\n  classical exact dite (x ∈ Λ) (fun h ↦ ζ ⟨x, h⟩) (fun _ ↦ η x)\n","line":8,"end_line":9,"proof_status":"formalized"}

crawled_repos/HoTTLean.jsonl

@@ -0,0 +1,54 @@
+{"id":"prop:GroupoidPathObjectFibrant","label":"GroupoidPathObjectFibrant","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(A\\) be a small groupoid. Recall that \\(\\mathbf{grpd}\\) is Cartesian closed, so we can take the image of the above diagram under the functor \\(A^{-}\\).","doc_url":null,"latex_link":"sect0004.html#prop:GroupoidPathObjectFibrant","lean_link":null,"lean_decl_name":"GroupoidPathObjectFibrant","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"prop:GroupoidFibrationComposition","label":"GroupoidFibrationComposition","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The corresponding fact about fibrations is that the composition of two fibrations is a fibration.","doc_url":null,"latex_link":"sect0004.html#prop:GroupoidFibrationComposition","lean_link":null,"lean_decl_name":"GroupoidFibrationComposition","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"prop:GroupoidFibrationPushforward","label":"GroupoidFibrationPushforward","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Stated in terms of fibrations, we have","doc_url":null,"latex_link":"sect0004.html#prop:GroupoidFibrationPushforward","lean_link":null,"lean_decl_name":"GroupoidFibrationPushforward","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"defn:PCat","label":"PCat","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"We will take the category of pointed small categories \\(\\mathbf{cat}_\\bullet \\) to have objects as pairs \\((\\mathbb {C}\\in \\mathbf{cat}, c \\in \\mathbb {C})\\) and morphisms as pairs \\[ (F : \\mathbb {C}_{1} \\to \\mathbb {C}_{0}, \\phi : F c_{1} \\to c_{0}) : (\\mathbb {C}_{1}, c_{1}) \\to (\\mathbb {C}_{0}, c_{0}) \\]","doc_url":null,"latex_link":"sect0004.html#defn:PCat","lean_link":null,"lean_decl_name":"PCat","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./attic/PullbackProofs.lean","lean4_code":"theorem PCat.eqToHom_hom_help {C1 C2 : PCat.{u,v}}(x y: C1)(eq : C1 = C2): (x ⟶ y) = ((eqToHom eq).obj x ⟶ (eqToHom eq).obj y) := by\n  cases eq\n  simp[CategoryStruct.id]\n\n-- /-- This is the turns the hom part of eqToHom pointed functors into a cast-/\n-- theorem PCat.eqToHom_hom {C1 C2 : PCat.{u,v}}{x y: C1}(f : x ⟶ y)(eq : C1 = C2): (eqToHom eq).map f = (cast (PCat.eqToHom_hom_help x y eq) f) := by\n--   cases eq\n--   simp[CategoryStruct.id]\n\n/-- This shows that two objects are equal in Grothendieck A if they have equal bases and fibers that are equal after being cast-/\n","line":55,"end_line":64,"proof_status":"statement_formalized"}
+{"id":"defn:CategoryTheory.Grothendieck","label":"CategoryTheory.Grothendieck","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"From \\(\\mathbb {C}\\) a small category and \\(F : \\mathbb {C}\\to \\mathbf{cat}\\) a functor, we construct a small category \\(\\int F\\). For any \\(c\\) in \\(\\mathbb {C}\\) we refer to \\(F c\\) as the fiber over \\(c\\). The objects of \\(\\int F\\) consist of pairs \\((c \\in \\mathbb {C}, x \\in F c)\\), and morphisms between \\((c, x)\\) and \\((d, y)\\) are pairs \\((f : c \\to d, \\phi : F \\, f \\, x \\to y)\\). This makes the following pullback in \\(\\mathbf{Cat}\\)","doc_url":null,"latex_link":"sect0004.html#defn:CategoryTheory.Grothendieck","lean_link":null,"lean_decl_name":"CategoryTheory.Grothendieck","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./HoTTLean/Grothendieck/Groupoidal/Basic.lean","lean4_code":"@[simp] lemma Grothendieck.Groupoidal.eta {Γ : Type*} [Category Γ]\n  {A : Γ ⥤ Grpd} (x : ∫(A)) : objMk x.base x.fiber = x :=\n  rfl\n\n","line":176,"end_line":179,"proof_status":"statement_formalized"}
+{"id":"defn:GroupoidInterval","label":"GroupoidInterval","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let the interval groupoid \\(\\mathbb {I}\\) be the small groupoid with two objects and a single non-identity isomorphism. There are two distinct morphisms \\(\\delta _{0}, \\delta _{1} : \\bullet \\to \\mathbb {I}\\) and a natural isomorphism \\(i : \\delta _{0} \\Rightarrow \\delta _{1}\\). Note that \\(\\delta _{0}\\) and \\(\\delta _{1}\\) both form adjoint equivalences with the unique map \\(! : \\mathbb {I}\\to \\bullet \\). Denote by \\(\\bullet +\\bullet \\) the small groupoid with two objects and only identity morphisms. Then let \\(\\partial : \\bullet +\\bullet \\to \\mathbb {I}\\) be the unique map factoring \\(\\delta _{0}\\) and \\(\\delta _{1}\\).","doc_url":null,"latex_link":"sect0004.html#defn:GroupoidInterval","lean_link":null,"lean_decl_name":"GroupoidInterval","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"defn:GroupoidGrothendieck","label":"GroupoidGrothendieck","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(\\Gamma \\) be a groupoid and \\(A : \\Gamma \\to \\mathbf{grpd}\\) a functor, we can compose \\(F\\) with the inclusion \\(i : \\mathbf{grpd}\\hookrightarrow \\mathbf{Cat}\\) and form the Grothendieck construction which we denote as \\[ \\Gamma \\cdot A := \\int i \\circ A \\quad \\quad \\mathsf{disp}_{A} : \\Gamma \\cdot A \\to \\Gamma \\]","doc_url":null,"latex_link":"sect0004.html#defn:GroupoidGrothendieck","lean_link":null,"lean_decl_name":"GroupoidGrothendieck","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"defn:GroupoidPi","label":"GroupoidPi","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Given \\(A : \\Gamma \\to \\mathbf{grpd}\\) and \\(B : \\Gamma \\cdot A \\to \\mathbf{grpd}\\) we will define \\(\\Pi _{A} B : \\Gamma \\to \\mathbf{grpd}\\) such that for any \\(C : \\Gamma \\to \\mathbf{grpd}\\) we have an isomorphism \\[ [\\Gamma \\cdot A, \\mathbf{grpd}](\\mathsf{disp}_{A}\\circ C, B) \\cong [\\Gamma , \\mathbf{grpd}](C, \\Pi _{A} B) \\]","doc_url":null,"latex_link":"sect0004.html#defn:GroupoidPi","lean_link":null,"lean_decl_name":"GroupoidPi","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./attic/Tarski/TarskiNaturalModel.lean","lean4_code":"def GroupoidPi (Γ : Grpd) (A : Γ ⥤ Grpd) (B : (Grothendieck.Groupoidal A) ⥤ Grpd) : Γ ⥤ Grpd where\n  obj x := by\n    have t := g (A.obj x ⥤ (GroupoidSigma Γ A B).obj x) (fun(f) => f ⋙ ((GroupoidSigmaBase Γ A B).app x) = 𝟙 (A.obj x))\n    exact Grpd.of {f : A.obj x ⥤ (GroupoidSigma Γ A B).obj x // f ⋙ ((GroupoidSigmaBase Γ A B).app x) = 𝟙 (A.obj x)}\n  map f := by\n    dsimp[Quiver.Hom]\n    rename_i X Y\n    sorry\n\n\nend NaturalModelPi\n\nsection NaturalModelEq\n\nend NaturalModelEq\n\n","line":369,"end_line":384,"proof_status":"statement_formalized"}
+{"id":"defn:GroupoidSigma","label":"GroupoidSigma","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Then given \\(A : \\Gamma \\to \\mathbf{grpd}\\) and \\(B : \\Gamma \\cdot A \\to \\mathbf{grpd}\\) we define \\(\\Sigma _{A}B : \\Gamma \\to \\mathbf{grpd}\\) such that \\(\\Sigma _{A}B\\) acts on objects by forming fiberwise Grothendieck constructions \\[ \\Sigma _{A}B (x) := A(x) \\cdot B \\circ x_{A} \\]","doc_url":null,"latex_link":"sect0004.html#defn:GroupoidSigma","lean_link":null,"lean_decl_name":"GroupoidSigma","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./attic/Tarski/TarskiNaturalModel.lean","lean4_code":"def GroupoidSigma (Γ : Grpd) (A : Γ ⥤ Grpd) (B : (Grothendieck.Groupoidal A) ⥤ Grpd) : Γ ⥤ Grpd where\n  obj x := Grpd.of (Grothendieck.Groupoidal ((PointToFiber A x) ⋙ B))\n  map f := by\n    rename_i X Y\n    have NT' : (PointToFiber A X) ⋙ (B ⋙ Grpd.forgetToCat) ⟶ (A.map f ⋙ PointToFiber A Y) ⋙ (B ⋙ Grpd.forgetToCat) := whiskerRight (PointToFiberNT A X Y f) (B ⋙ Grpd.forgetToCat)\n    exact (Grothendieck.map NT') ⋙ (Grothendieck.Groupoidal.functorial (A.map f) (PointToFiber A Y ⋙ B))\n  map_id := by\n    intro X\n    simp[CategoryStruct.id,whiskerRight,Functor.id]\n    refine CategoryTheory.Functor.ext ?_ ?_\n    . intro X\n      simp[PointToFiber,Grpd.forgetToCat]\n      sorry\n    all_goals sorry\n  map_comp := by\n    intro X Y Z f g\n    simp[Grpd.forgetToCat]\n    sorry\n\n","line":198,"end_line":216,"proof_status":"statement_formalized"}
+{"id":"defn:NaturalModelPi","label":"NaturalModelPi","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":["defn:NaturalModelRussell","defn:UvPoly"],"nlp_text":"We will use \\(P_{\\mathsf{tp}_i}\\) to denote the polynomial endofunctor 5.0.1 associated with a natural model \\(\\mathsf{tp}_i\\). Then additional structure of \\(\\Pi \\) types on our \\(N\\) universes consists of, for each \\(i, j \\le N\\), a pullback square","doc_url":null,"latex_link":"sect0002.html#defn:NaturalModelPi","lean_link":null,"lean_decl_name":"NaturalModelPi","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./attic/Tarski/NaturalModel.lean","lean4_code":"class NaturalModelPi where\n  Pi : (P tp).obj Ty ⟶ M.Ty\n  lam : (P tp).obj Tm ⟶ M.Tm\n  Pi_pullback : IsPullback lam ((P tp).map tp) tp Pi\n\nvariable (Ctx) in\n","line":104,"end_line":109,"proof_status":"statement_formalized"}
+{"id":"defn:ProductsExponentials","label":"ProductsExponentials","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":["defn:NaturalModelPi","defn:NaturalModelSigma","prop:UvPoly.equiv"],"nlp_text":"In the natural model we can construct these by considering first the map \\[ (\\mathsf{fst}, \\mathsf{snd}) : \\mathsf{Ty}_i \\times \\mathsf{Ty}_j \\to P_{\\mathsf{tp}_i}{\\mathsf{Ty}_j} \\]","doc_url":null,"latex_link":"sect0002.html#defn:ProductsExponentials","lean_link":null,"lean_decl_name":"ProductsExponentials","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"defn:IdComp","label":"IdComp","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":["defn:ProductsExponentials"],"nlp_text":"The identity function \\(\\mathsf{id}_{A} : \\Gamma \\to \\mathsf{Tm}\\) of type \\(\\mathsf{Exp}\\circ (A,A) : \\Gamma \\to \\mathsf{Ty}\\) can be defined by the following","doc_url":null,"latex_link":"sect0002.html#defn:IdComp","lean_link":null,"lean_decl_name":"IdComp","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"defn:NaturalModelRussell","label":"NaturalModelRussell","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":["defn:NaturalModel"],"nlp_text":"A collection of \\(N+1\\) natural models with \\(N\\) Russell style universes and lifts consists of For each \\(i \\leq N\\) a natural model \\(\\mathsf{tp}_i : \\mathsf{Tm}_i \\to \\mathsf{Ty}_i\\) For each \\(i {\\lt} N\\) a lift \\(\\mathsf{L}_{i}^{i+1} : \\mathsf{Ty}_i \\to \\mathsf{Ty}_{i+1}\\) For each \\(i {\\lt} N\\) a point \\(\\mathsf{U}_i : 1 \\to \\mathsf{Ty}_{i+1}\\) such that","doc_url":null,"latex_link":"sect0002.html#defn:NaturalModelRussell","lean_link":null,"lean_decl_name":"NaturalModelRussell","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"defn:NaturalModelSigma","label":"NaturalModelSigma","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":["defn:NaturalModelRussell"],"nlp_text":"We will use the polynomial composition of two maps 5.0.6, \\(\\mathsf{tp}_i \\lhd \\mathsf{tp}_j : Q \\to P_{\\mathsf{tp}_i}(\\mathsf{Ty}_j)\\). Then additional structure of \\(\\Sigma \\) types on our \\(N\\) universes consists of, for each \\(i, j \\le N\\), a pullback square","doc_url":null,"latex_link":"sect0002.html#defn:NaturalModelSigma","lean_link":null,"lean_decl_name":"NaturalModelSigma","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./attic/Tarski/NaturalModel.lean","lean4_code":"class NaturalModelSigma where\n  Sig : (P tp).obj Ty ⟶ M.Ty\n  pair : (UvPoly.compDom (uvPoly M.tp) (uvPoly M.tp)) ⟶ M.Tm\n  Sig_pullback : IsPullback pair ((uvPoly M.tp).comp (uvPoly M.tp)).p M.tp Sig\n\n","line":110,"end_line":114,"proof_status":"statement_formalized"}
+{"id":"defn:NaturalModelUnivalentU","label":"NaturalModelUnivalentU","background_color":"white","border_color":"#FFAA33","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":["defn:NaturalModelRussell"],"nlp_text":"Univalence for universe \\(\\mathsf{U}\\) states that IdToEquiv itself is an equivalence \\[ \\mathsf{ua} : \\mathsf{IsBigEquiv} (\\mathsf{IdToEquiv} A \\, B) \\]","doc_url":null,"latex_link":"sect0002.html#defn:NaturalModelUnivalentU","lean_link":null,"lean_decl_name":"NaturalModelUnivalentU","kind_description":"theorems and lemmas","status_from_color":"unknown","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./attic/Tarski/NaturalModel.lean","lean4_code":"def NaturalModelUnivalentU : Type := sorry\n\n/-\nwe will also want to say that the universe U is closed under Sigma, Pi, and Id,\nso that we can say that U is univalent.\n-/\n/-\nit would probably also be useful to have another universe U1 with U : U1,\nand maybe some type formers for U1 as well .\n-/\n\nend NaturalModel\n\nopen NaturalModel in\nvariable (Ctx) in\n","line":198,"end_line":212,"proof_status":"statement_formalized"}
+{"id":"defn:NaturalModelIdUIP","label":"NaturalModelIdUIP","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":["defn:NaturalModelId"],"nlp_text":"Say an identity type in a natural model satisfies UIP if \\(I \\to \\mathsf{tp}\\times _{\\mathsf{Ty}} \\mathsf{tp}\\) is a strict proposition, meaning for any \\((a,b) : \\Gamma \\to \\mathsf{tp}\\times _{\\mathsf{Ty}} \\mathsf{tp}\\) there it at most one lift","doc_url":null,"latex_link":"sect0002.html#defn:NaturalModelIdUIP","lean_link":null,"lean_decl_name":"NaturalModelIdUIP","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"defn:GroupoidNMId.J","label":"GroupoidNMId.J","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"We want to define \\(J : T \\to P_{q}{\\mathsf{Tm}}\\) \\begin{align*} J_{\\Gamma } : \\mathbf{Psh}(\\mathbf{grpd})(\\Gamma , T) \\to \\, & \\mathbf{Psh}(\\mathbf{grpd})(\\Gamma , P_{q}{\\mathsf{Tm}})\\\\ (A, C, \\gamma _{\\mathsf{refl}}) \\mapsto \\, & (A, \\gamma ) \\end{align*}","doc_url":null,"latex_link":"sect0004.html#defn:GroupoidNMId.J","lean_link":null,"lean_decl_name":"GroupoidNMId.J","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./HoTTLean/ForPoly.lean","lean4_code":"instance preservesConnectedLimitsOfShape_of_hasLimitsOfShape {J : Type v₁} [SmallCategory J]\n  [IsConnected J] [HasLimitsOfShape J C] (P : UvPoly E B) :\n    PreservesLimitsOfShape J (P.functor) := by\n  unfold UvPoly.functor\n  infer_instance\n\n","line":620,"end_line":625,"proof_status":"statement_formalized"}
+{"id":"prop:GroupoidSigmaBeckChevalley","label":"GroupoidSigmaBeckChevalley","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(\\sigma : \\Delta \\to \\Gamma \\), \\(A : \\Gamma \\to \\mathbf{grpd}\\) and \\(B : \\Gamma \\cdot A \\to \\mathbf{grpd}\\). Then \\[ (\\Sigma _{A}B) \\circ \\sigma = \\Sigma _{A \\circ \\sigma }(B \\circ \\sigma _{A}) \\]","doc_url":null,"latex_link":"sect0004.html#prop:GroupoidSigmaBeckChevalley","lean_link":null,"lean_decl_name":"GroupoidSigmaBeckChevalley","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./attic/Tarski/TarskiNaturalModel.lean","lean4_code":"theorem GroupoidSigmaBeckChevalley (Δ Γ: Grpd.{v,u}) (σ : Δ ⥤ Γ) (A : Γ ⥤ Grpd.{v,u})\n  (B : (Grothendieck.Groupoidal A) ⥤ Grpd.{v,u}) : σ ⋙ GroupoidSigma Γ A B = GroupoidSigma _ (σ ⋙ A)\n  (Grothendieck.Groupoidal.Map Δ Γ σ A B) := by\n  refine CategoryTheory.Functor.ext ?_ ?_\n  . intros X\n    sorry\n  . intros X Y f\n    sorry\n\n","line":234,"end_line":242,"proof_status":"statement_formalized"}
+{"id":"prop:GroupoidI.epComputation","label":"GroupoidI.epComputation","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"We can compute \\(\\varepsilon : P_{q}{\\mathsf{Tm}} \\to T\\) via \\begin{align*} \\varepsilon _{\\Gamma } : \\, & \\mathbf{Psh}(\\mathbf{grpd})(\\Gamma , P_{q}{\\mathsf{Tm}}) \\to \\mathbf{Psh}(\\mathbf{grpd})(\\Gamma , T) \\\\ & (A, \\gamma ) \\mapsto (A, U \\circ \\gamma , \\gamma \\circ A^{*} \\rho ’) \\end{align*}","doc_url":null,"latex_link":"sect0004.html#prop:GroupoidI.epComputation","lean_link":null,"lean_decl_name":"GroupoidI.epComputation","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"defn:PolynomialComposition","label":"PolynomialComposition","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(f : B \\to A\\) and \\(g : D \\to C\\). Define the polynomial composition \\(f \\lhd g : Q \\to P_{f}{C}\\) as the composition of the two vertical maps in the following","doc_url":null,"latex_link":"sect0005.html#defn:PolynomialComposition","lean_link":null,"lean_decl_name":"PolynomialComposition","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"prop:GroupoidNM.Substitution","label":"GroupoidNM.Substitution","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(\\sigma : \\Delta \\to \\Gamma \\) be a functor between groupoids. Since display maps are pullbacks of the classifier \\(U : \\mathbf{grpd}_\\bullet \\to \\mathbf{grpd}\\) we have the pasting diagram","doc_url":null,"latex_link":"sect0004.html#prop:GroupoidNM.Substitution","lean_link":null,"lean_decl_name":"GroupoidNM.Substitution","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"defn:UvPolySlice","label":"UvPolySlice","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(P_{-} : (\\mathbb {C}/ A)^{\\mathrm{op}} \\to [\\mathbb {C}, \\mathbb {C}]\\) be defined by taking \\(s \\mapsto P_{s}\\) on objects and act on a morphism by","doc_url":null,"latex_link":"sect0005.html#defn:UvPolySlice","lean_link":null,"lean_decl_name":"UvPolySlice","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"defn:NaturalModelId","label":"NaturalModelId","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":["defn:UvPolySlice","defn:NaturalModel"],"nlp_text":"Suppose \\(\\mathsf{tp}: \\mathsf{Tm}\\to \\mathsf{Ty}\\) is a natural model and we have a commutative square (this need not be a pullback)","doc_url":null,"latex_link":"sect0002.html#defn:NaturalModelId","lean_link":null,"lean_decl_name":"NaturalModelId","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./attic/Tarski/NaturalModel.lean","lean4_code":"class NaturalModelId extends NaturalModelIdBase Ctx where\n  data : NaturalModelIdData Ctx\n\n","line":150,"end_line":152,"proof_status":"statement_formalized"}
+{"id":"prop:GroupoidObjectsFibrant","label":"GroupoidObjectsFibrant","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(\\bullet \\) denote the terminal groupoid, namely that with a single object and morphism. Then the unique map \\(\\Gamma \\to \\bullet \\) is a fibration.","doc_url":null,"latex_link":"sect0004.html#prop:GroupoidObjectsFibrant","lean_link":null,"lean_decl_name":"GroupoidObjectsFibrant","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"prop:GroupoidPiBeckChevalley","label":"GroupoidPiBeckChevalley","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(\\sigma : \\Delta \\to \\Gamma \\), \\(A : \\Gamma \\to \\mathbf{grpd}\\) and \\(B : \\Gamma \\cdot A \\to \\mathbf{grpd}\\). Then \\[ (\\Pi _{A}B) \\circ \\sigma = \\Pi _{A \\circ \\sigma }(B \\circ \\sigma _{A}) \\]","doc_url":null,"latex_link":"sect0004.html#prop:GroupoidPiBeckChevalley","lean_link":null,"lean_decl_name":"GroupoidPiBeckChevalley","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"defn:GroupoidQ","label":"GroupoidQ","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Recall the following definition of composition of polynomial endofunctors, specialized to our situation","doc_url":null,"latex_link":"sect0004.html#defn:GroupoidQ","lean_link":null,"lean_decl_name":"GroupoidQ","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"prop:GroupoidNMId.JSection","label":"GroupoidNMId.JSection","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\(J : T \\to P_{q}{\\mathsf{Tm}}\\), as defined above is a section of \\(\\varepsilon \\).","doc_url":null,"latex_link":"sect0004.html#prop:GroupoidNMId.JSection","lean_link":null,"lean_decl_name":"GroupoidNMId.JSection","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:UvPolyEvEq","label":"UvPolyEvEq","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":["defn:UvPolyEv"],"nlp_text":"Suppose \\((\\beta ,y) : \\Gamma \\to R\\), as in 5.0.3 \\[ \\beta : \\Gamma \\to B \\quad \\text{ and } \\quad y : \\Gamma \\cdot t \\circ \\beta \\to Y \\]","doc_url":null,"latex_link":"sect0005.html#lem:UvPolyEvEq","lean_link":null,"lean_decl_name":"UvPolyEvEq","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:GroupoidI.q","label":"GroupoidI.q","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Specialized to \\(q : I \\to \\mathsf{Ty}\\) in \\(\\mathbf{Psh}(\\mathbf{grpd})\\), the characterizing property of polynomial endofunctors 5.0.2 says that a map from a representable \\(\\varepsilon : \\Gamma \\to P_{q}X\\) corresponds to the data of \\[ A : \\Gamma \\to \\mathsf{Ty}\\quad \\quad \\text{ and } \\quad \\quad C : \\Gamma \\cdot A \\cdot A \\cdot \\mathsf{Id}\\to X \\]","doc_url":null,"latex_link":"sect0004.html#lem:GroupoidI.q","lean_link":null,"lean_decl_name":"GroupoidI.q","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./attic/Tarski/NaturalModel.lean","lean4_code":"def q : I ⟶ M.Ty := pullback.fst .. ≫ pullback.fst .. ≫ tp\n","line":133,"end_line":133,"proof_status":"statement_formalized"}
+{"id":"lem:GroupoidT","label":"GroupoidT","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"","doc_url":null,"latex_link":"sect0004.html#lem:GroupoidT","lean_link":null,"lean_decl_name":"GroupoidT","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"prop:GroupoidFibrationPullbackStable","label":"GroupoidFibrationPullbackStable","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"","doc_url":null,"latex_link":"sect0004.html#prop:GroupoidFibrationPullbackStable","lean_link":null,"lean_decl_name":"GroupoidFibrationPullbackStable","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"defn:GroupoidNMDiscreteU","label":"GroupoidNMDiscreteU","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(\\mathsf{U}\\) be the (large) groupoid of small sets, i.e. let \\(\\mathsf{U}\\) have \\(\\mathbf{set}\\) as its objects and morphisms between two small sets as all the bijections between them. This gives us \\(\\ulcorner \\mathsf{U}\\urcorner : \\bullet \\to \\mathsf{Ty}\\). Then we define \\(\\mathsf{El}: \\mathsf{y}\\mathsf{U}\\to \\mathsf{Ty}\\) by defining \\(\\mathsf{El}: \\mathsf{U}\\to \\mathbf{Grpd}\\) as the inclusion - any small set can be regarded as a large discrete groupoid.","doc_url":null,"latex_link":"sect0004.html#defn:GroupoidNMDiscreteU","lean_link":null,"lean_decl_name":"GroupoidNMDiscreteU","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"defn:GroupoidNM","label":"GroupoidNM","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"We would like to define a natural transformation in \\(\\mathbf{Psh}(\\mathbf{grpd})\\) \\[ \\mathsf{tp}: \\mathsf{Tm}\\to \\mathsf{Ty} \\]","doc_url":null,"latex_link":"sect0004.html#defn:GroupoidNM","lean_link":null,"lean_decl_name":"GroupoidNM","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"defn:GroupoidNMPi","label":"GroupoidNMPi","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"We define the natural transformation \\(\\Pi : P_{\\mathsf{tp}} \\mathsf{Ty}\\to \\mathsf{Ty}\\) as that which is induced (4.4.1) by the \\(\\Pi \\)-former operation (4.2.8). Then we define the natural transformation \\(\\lambda : P_{\\mathsf{tp}} \\mathsf{Ty}\\to \\mathsf{Ty}\\) as the natural transformation induced by the following operation: given \\(A : \\Gamma \\to \\mathbf{grpd}\\) and \\(\\beta : \\Gamma \\cdot A \\to \\mathbf{grpd}_\\bullet \\), \\(\\lambda _{A}\\beta : \\Gamma \\to \\mathbf{grpd}_\\bullet \\) will be the functor such that on objects \\(x \\in \\Gamma \\) \\[ \\lambda _{A}\\beta \\, (x) := (\\Pi _{A} B \\, (x) , s_x : a \\mapsto (a, b (x , a))) \\]","doc_url":null,"latex_link":"sect0004.html#defn:GroupoidNMPi","lean_link":null,"lean_decl_name":"GroupoidNMPi","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"defn:GroupoidFibration","label":"GroupoidFibration","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(p : \\mathbb {C}_{1} \\to \\mathbb {C}_{0}\\) be a functor. We say \\(p\\) is a split Grothendieck fibration if we have a dependent function \\(\\mathsf{lift} \\, a \\, f\\) satisfying the following: for any object \\(a\\) in \\(\\mathbb {C}_{1}\\) and morphism \\(f : p \\, a \\to y\\) in the base \\(\\mathbb {C}_{0}\\) we have \\(\\mathsf{lift} \\, a \\, f : a \\to b\\) in \\(\\mathbb {C}_{1}\\) such that \\(p (\\mathsf{lift} \\, a \\, f) = f\\) and moreover \\(\\mathsf{lift} \\, a \\, g \\circ f = \\mathsf{lift} \\, b \\, g \\circ \\mathsf{lift} \\, a \\, f\\)","doc_url":null,"latex_link":"sect0004.html#defn:GroupoidFibration","lean_link":null,"lean_decl_name":"GroupoidFibration","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"defn:NaturalModel","label":"NaturalModel","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Following Awodey [ Awo17 ] , we say that a map \\(\\mathsf{tp}: \\mathsf{Tm}\\to \\mathsf{Ty}\\) in \\(\\mathbf{Psh}(\\mathbb {C})\\) is fiberwise representable or a natural model when every fiber is representable. In other words, given any \\(\\Gamma \\in \\mathbb {C}\\) and any map \\(A : \\mathsf{y}(\\Gamma ) \\to \\mathsf{Ty}\\), there is some representable \\(\\Gamma \\cdot A \\in \\mathbb {C}\\) and maps \\(\\mathsf{disp}_{A} : \\Gamma \\cdot A \\to \\Gamma \\) and \\(\\mathsf{var}_{A} : \\mathsf{y}(\\Gamma \\cdot A) \\to \\mathsf{Tm}\\) forming a pullback","doc_url":null,"latex_link":"sect0002.html#defn:NaturalModel","lean_link":null,"lean_decl_name":"NaturalModel","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./attic/Tarski/TarskiNaturalModel.lean","lean4_code":"instance GroupoidNM : NaturalModel.NaturalModelBase sGrpd.{u} where\n  Ty := Ty\n  Tm := Tm\n  tp := tp\n  ext _ A := sGrpd.of (ext A)\n  disp _ A := disp A\n  var _ A := var A\n  disp_pullback A := disp_pullback A\n\n","line":115,"end_line":123,"proof_status":"statement_formalized"}
+{"id":"defn:NaturalModelEq","label":"NaturalModelEq","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":["defn:NaturalModel"],"nlp_text":"The first option is fully extensional identity types, i.e. those satisfying equality reflection and uniqueness of identity proofs (UIP). Equality reflection says that if one can construct a term satisfying \\(\\mathsf{Id}(a,b)\\) then we have that definitionally \\(a \\equiv b\\), i.e. they are equal morphisms in the natural model. This amounts to just requiring that 3 is a pullback, i.e. \\(\\rho \\) is an isomorphism","doc_url":null,"latex_link":"sect0002.html#defn:NaturalModelEq","lean_link":null,"lean_decl_name":"NaturalModelEq","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./attic/Tarski/NaturalModel.lean","lean4_code":"class NaturalModelEq where\n  Eq : pullback tp tp ⟶ M.Ty\n  refl : Tm ⟶ M.Tm\n  Eq_pullback : IsPullback refl δ tp Eq\n\nvariable (Ctx) in\n","line":117,"end_line":122,"proof_status":"statement_formalized"}
+{"id":"defn:GroupoidEv","label":"GroupoidEv","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Define the operation of evaluation \\(\\mathsf{ev}_{\\alpha } \\, B\\) to take \\(\\alpha : \\Gamma \\to \\mathbf{grpd}_\\bullet \\) and \\(B : \\Gamma \\cdot U \\circ \\alpha \\to \\mathbf{grpd}\\) and return \\(\\mathsf{ev}_{\\alpha } \\, B : \\Gamma \\to \\mathbf{grpd}\\), described below.","doc_url":null,"latex_link":"sect0004.html#defn:GroupoidEv","lean_link":null,"lean_decl_name":"GroupoidEv","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"defn:GroupoidNMSigma","label":"GroupoidNMSigma","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"We define the natural transformation \\[ \\Sigma : P_{\\mathsf{tp}} \\mathsf{Ty}\\to \\mathsf{Ty} \\]","doc_url":null,"latex_link":"sect0004.html#defn:GroupoidNMSigma","lean_link":null,"lean_decl_name":"GroupoidNMSigma","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./attic/Tarski/TarskiNaturalModel.lean","lean4_code":"def GroupoidNMSigma : (P.obj.{u} Ty.{u}) ⟶ Ty.{u} := by\n  fconstructor\n  . dsimp [Quiver.Hom]\n    intros sObj poly\n    let poly' := yonedaEquiv.invFun poly\n    let poly_as_pair := (UvPoly.equiv uv_tp (yoneda.obj (Opposite.unop sObj)) Ty).toFun poly'\n    rcases poly_as_pair with ⟨A, B⟩\n    have B := yonedaEquiv.toFun ((Limits_Sub A).inv ≫ B)\n    exact downFunctor ⋙ (GroupoidSigma (sGrpd.forget.obj (Opposite.unop sObj)) (upFunctor ⋙ (yonedaEquiv.toFun A)) (upFunctor ⋙ B))\n  . intros X Y f\n    funext a\n    refine CategoryTheory.Functor.ext ?_ ?_\n    . intro b\n      sorry\n    . intros b₁ b₂ g\n      sorry\n\n","line":255,"end_line":271,"proof_status":"statement_formalized"}
+{"id":"defn:UvPoly","label":"UvPoly","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(\\mathbb {C}\\) be a locally Cartesian closed category (in our case, presheaves on the category of contexts). This means for each morphism \\(t : B \\to A\\) we have an adjoint triple","doc_url":null,"latex_link":"sect0005.html#defn:UvPoly","lean_link":null,"lean_decl_name":"UvPoly","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./HoTTLean/ForPoly.lean","lean4_code":"instance preservesPullbacks (P : UvPoly E B)\n    {Pb X Y Z : C} (fst : Pb ⟶ X) (snd : Pb ⟶ Y)\n    (f : X ⟶ Z) (g : Y ⟶ Z)\n    (h: IsPullback fst snd f g) :\n    IsPullback (P.functor.map fst) (P.functor.map snd) (P.functor.map f) (P.functor.map g) :=\n    P.functor.map_isPullback h\n\n\n","line":626,"end_line":633,"proof_status":"statement_formalized"}
+{"id":"prop:UvPoly.equiv","label":"UvPoly.equiv","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":["defn:UvPoly"],"nlp_text":"The data of a map into the polynomial applied to an object in \\(\\mathbb {C}\\)","doc_url":null,"latex_link":"sect0005.html#prop:UvPoly.equiv","lean_link":null,"lean_decl_name":"UvPoly.equiv","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./attic/FibrationForMathlib/FibredCats/Basic.lean","lean4_code":"def equiv (c : C) : (P.op ⁻¹ (op c)) ≃ (P⁻¹ c) where\n  toFun := fun x =>  (⟨unop x.1, by rw [obj_over] ⟩)\n  invFun := fun x => ⟨op x.1 , by simp only [Functor.op_obj, unop_op, Fiber.over]⟩\n  left_inv := fun _ ↦ rfl\n  right_inv := fun _ ↦ rfl\n\n/-- The fibres of the opposite functor `P.op` are isomorphic\nto the the fibres of `P`.  -/\n","line":90,"end_line":97,"proof_status":"statement_formalized"}
+{"id":"defn:UvPolyEv","label":"UvPolyEv","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":["defn:UvPoly"],"nlp_text":"Let \\(\\, \\mathsf{counit} \\, : \\rho _B \\to B \\to B^* Y\\) denote the counit of the adjunction \\(f^* \\dashv f_*\\) at the object \\(B^* Y\\), recalling that \\(\\rho _B = t^* t_* B^* Y\\). Then viewing the object \\(B^* Y\\) in the slice as the object \\(Y \\times B\\) in the ambient category, we define \\(\\mathsf{ev}_{} \\, : R \\to Y\\) as the composition","doc_url":null,"latex_link":"sect0005.html#defn:UvPolyEv","lean_link":null,"lean_decl_name":"UvPolyEv","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:R","label":"R","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":["prop:UvPoly.equiv"],"nlp_text":"Use \\(R\\) to denote the fiber product","doc_url":null,"latex_link":"sect0005.html#lem:R","lean_link":null,"lean_decl_name":"R","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"HoTTLean","source_path":"./attic/Groupoids-NaturalModel/Id.lean","lean4_code":"def R : PGrpd ⥤ ∫ id :=\n  (isPullback id).lift refl diag refl_forgetToGrpd\n\n/- This is the composition\n\nPGrpd\n |----> ∫Id\n |  R   |\n |      |\n Diag   | forget\n |      |\n |      V\n ---> BPGrpd\n        |\n        |\n        | BPGrpd.forgetToGrpd\n        |\n        V\n      Grpd\n-/\n-- TODO : consider removal?\n-- def K : Grothendieck.Groupoidal Id ⥤ Grpd := Grothendieck.Groupoidal.forget ⋙  BPGrpd.forgetToGrpd\n\n/- This is the universal lift\n            Refl\nPGrpd ------------>\n |----> ∫Id -----> PGrpd\n |  R   |            |\n |      |            |\n Diag   |            | forget\n |      |            |\n |      V            V\n ---> BPGrpd -----> Grpd\n              Id\n-/\n","line":223,"end_line":257,"proof_status":"statement_formalized"}
+{"id":"prop:GroupoidPtpUniversalProperty","label":"GroupoidPtpUniversalProperty","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Specialized to \\(\\mathsf{tp}: \\mathsf{Tm}\\to \\mathsf{Ty}\\) in \\(\\mathbf{Psh}(\\mathbf{grpd})\\), the characterizing property of polynomial endofunctors 5.0.2 says that a map from a representable \\(\\Gamma \\to P_{\\mathsf{tp}}X\\) corresponds to the data of \\[ A : \\Gamma \\to \\mathsf{Ty}\\quad \\quad \\text{ and } \\quad \\quad B : \\Gamma \\cdot A \\to X \\]","doc_url":null,"latex_link":"sect0004.html#prop:GroupoidPtpUniversalProperty","lean_link":null,"lean_decl_name":"GroupoidPtpUniversalProperty","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"prop:GroupoidEvComputation","label":"GroupoidEvComputation","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The functor \\(\\mathsf{ev}_{\\alpha } \\, B : \\Gamma \\to \\mathbf{grpd}\\) can be computed as \\[ \\mathsf{ev}_{\\alpha } \\, B = B \\circ a \\]","doc_url":null,"latex_link":"sect0004.html#prop:GroupoidEvComputation","lean_link":null,"lean_decl_name":"GroupoidEvComputation","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:GroupoidI","label":"GroupoidI","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"We can then construct the pullback \\(I'\\)","doc_url":null,"latex_link":"sect0004.html#lem:GroupoidI","lean_link":null,"lean_decl_name":"GroupoidI","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:GroupoidBeckChevalleyIffNatural","label":"GroupoidBeckChevalleyIffNatural","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\(X \\in \\mathbf{Psh}(\\mathbf{grpd})\\) be a presheaf. Let \\(F\\) be an operation that takes a groupoid \\(\\Gamma \\), a functor \\(A : \\Gamma \\to \\mathbf{grpd}\\) and \\(B : \\Gamma \\cdot A \\to X\\) and returns a natural transformation \\(F_{A} B : \\Gamma \\to X\\). Then using Yoneda to define \\(\\tilde{F} : P_{\\mathsf{tp}} X \\to X\\) pointwise as \\begin{align*} \\tilde{F}_{\\Gamma } : \\mathbf{Psh}(\\mathbf{grpd})(\\Gamma , P_{\\mathsf{tp}} X) & \\to \\mathbf{Psh}(\\mathbf{grpd})(\\Gamma , X) \\\\ (A, B) & \\mapsto F_{A} B \\end{align*}","doc_url":null,"latex_link":"sect0004.html#lem:GroupoidBeckChevalleyIffNatural","lean_link":null,"lean_decl_name":"GroupoidBeckChevalleyIffNatural","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"prop:ContextExtension","label":"ContextExtension","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For any small groupoid \\(\\Gamma \\) and \\(A : \\mathsf{y}\\Gamma \\to \\mathsf{Ty}\\), the pullback of \\(\\mathsf{tp}\\) along \\(A\\) can be given by the representable map \\(\\mathsf{y}\\mathsf{disp}_{A}\\).","doc_url":null,"latex_link":"sect0004.html#prop:ContextExtension","lean_link":null,"lean_decl_name":"ContextExtension","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"mate","label":"mate","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Suppose","doc_url":null,"latex_link":"sect0005.html#mate","lean_link":null,"lean_decl_name":"mate","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"prop:GroupoidFibrationClassified","label":"GroupoidFibrationClassified","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"There is an adjoint equivalence","doc_url":null,"latex_link":"sect0004.html#prop:GroupoidFibrationClassified","lean_link":null,"lean_decl_name":"GroupoidFibrationClassified","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"defn:UvPolyCartesianAction","label":"UvPolyCartesianAction","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"We can also view taking polynomial endofunctors as a covariant functor on the category of arrows with cartesian squares as morphisms \\[ P_{-} : \\mathsf{CartArr}(\\mathbb {C}) \\to [\\mathbb {C}, \\mathbb {C}] \\]","doc_url":null,"latex_link":"sect0005.html#defn:UvPolyCartesianAction","lean_link":null,"lean_decl_name":"UvPolyCartesianAction","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:GroupoidR","label":"GroupoidR","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"This is a specialization of 5.0.3. Use \\(R\\) to denote the fiber product","doc_url":null,"latex_link":"sect0004.html#lem:GroupoidR","lean_link":null,"lean_decl_name":"GroupoidR","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"prop:UvPolySliceCartesian","label":"UvPolySliceCartesian","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If we have","doc_url":null,"latex_link":"sect0005.html#prop:UvPolySliceCartesian","lean_link":null,"lean_decl_name":"UvPolySliceCartesian","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"defn:GroupoidNMId","label":"GroupoidNMId","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://sinhp.github.io/HoTTLean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"To define the commutative square in \\(\\mathbf{Psh}(\\mathbf{grpd})\\)","doc_url":null,"latex_link":"sect0004.html#defn:GroupoidNMId","lean_link":null,"lean_decl_name":"GroupoidNMId","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}

crawled_repos/Junk-Test.jsonl

@@ -0,0 +1,13 @@
+{"id":"lem: wstar_of_masa","label":"wstar_of_masa","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://JonBannon.github.io/Junk-Test/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A maximal abelian \\(*\\)–subalgebra of a \\(W^{*}\\)–algebra \\(M\\) is also a \\(W^{*}\\)–algebra.","doc_url":null,"latex_link":"sect0001.html#lem: wstar_of_masa","lean_link":null,"lean_decl_name":"wstar_of_masa","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"sigma_cts_of_normal","label":"sigma_cts_of_normal","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://JonBannon.github.io/Junk-Test/blueprint/dep_graph_document.html","dependencies":["lem: wstar_of_masa","lem: exists_uw_ge_normal","lem: msr_th_lemma","lem: fin_lin_approx_of_stonean","lem: cut_down_of_wstar","lem: spec_masa_wstar_stonean","lem: zorn_base","lem: bdd_sigma_cts_iff_bdd_s_cts"],"nlp_text":"Every positive normal linear functional \\(\\varphi \\) on \\(M\\) is \\(\\sigma (M,M_{*})\\)–continuous.","doc_url":null,"latex_link":"sect0001.html#sigma_cts_of_normal","lean_link":null,"lean_decl_name":"sigma_cts_of_normal","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem: selfadjoint_le_norm","label":"selfadjoint_le_norm","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://JonBannon.github.io/Junk-Test/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For all self-adjoint \\(x\\in M\\), \\(\\| x\\| 1-x \\ge 0\\).","doc_url":null,"latex_link":"sect0001.html#lem: selfadjoint_le_norm","lean_link":null,"lean_decl_name":"selfadjoint_le_norm","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem: proj_sub_of_subproj","label":"proj_sub_of_subproj","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://JonBannon.github.io/Junk-Test/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For all \\(p,q\\in \\mathcal{P}(M)\\) such that \\(q\\le p\\), \\(p-q\\in \\mathcal{P}(M)\\).","doc_url":null,"latex_link":"sect0001.html#lem: proj_sub_of_subproj","lean_link":null,"lean_decl_name":"proj_sub_of_subproj","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem: uw_pos_sep_pts","label":"uw_pos_sep_pts","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://JonBannon.github.io/Junk-Test/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For every nonzero element \\(a\\in M\\), there is a \\(\\sigma (M,M_{*})\\)–continuous positive linear functional \\(\\psi \\) on \\(M\\) such that \\(\\psi (a)\\ne 0\\).","doc_url":null,"latex_link":"sect0001.html#lem: uw_pos_sep_pts","lean_link":null,"lean_decl_name":"uw_pos_sep_pts","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem: exists_uw_ge_normal","label":"exists_uw_ge_normal","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://JonBannon.github.io/Junk-Test/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For every positive normal linear functional \\(\\varphi \\) and nonzero \\(p\\in \\mathcal{P}(M)\\) there exists a positive \\(\\sigma (M,M_{*})\\)–continuous linear functional \\(\\psi \\) such that \\(\\varphi (p){\\lt}\\psi (p)\\).","doc_url":null,"latex_link":"sect0001.html#lem: exists_uw_ge_normal","lean_link":null,"lean_decl_name":"exists_uw_ge_normal","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem: msr_th_lemma","label":"msr_th_lemma","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://JonBannon.github.io/Junk-Test/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For all positive linear functionals \\(\\varphi ,\\psi \\) with \\(\\varphi \\) normal and \\(\\psi \\) \\(\\sigma (M,M_{*})\\)–continuous and every nonzero \\(p\\in \\mathcal{P}(M)\\) such that \\(\\varphi (p){\\lt}\\psi (p)\\), there exists a nonzero \\(p_1 \\in \\mathcal{P}(M)\\) such that \\(p_1\\le p\\) and for all nonzero \\(q\\in \\mathcal{P}(M)\\) with \\(q\\le p_1\\), we have \\(\\varphi (q){\\lt}\\psi (q)\\).","doc_url":null,"latex_link":"sect0001.html#lem: msr_th_lemma","lean_link":null,"lean_decl_name":"msr_th_lemma","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem: proj_of_sup_proj","label":"proj_of_sup_proj","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://JonBannon.github.io/Junk-Test/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For every increasing bounded net \\((p_{\\alpha })\\) of projections in \\(M\\), the supremum \\(p\\) is a projection in \\(M\\)","doc_url":null,"latex_link":"sect0001.html#lem: proj_of_sup_proj","lean_link":null,"lean_decl_name":"proj_of_sup_proj","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem: fin_lin_approx_of_stonean","label":"fin_lin_approx_of_stonean","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://JonBannon.github.io/Junk-Test/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(K\\) be a Stonean space. Then every element \\(a\\) in \\(C(K)\\) can be uniformly approximated by finite linear combinations of projections in \\(C(K)\\). If \\(a\\ge 0\\) then the coefficients of the approximating linear combinations may be chosen nonnegative.","doc_url":null,"latex_link":"sect0001.html#lem: fin_lin_approx_of_stonean","lean_link":null,"lean_decl_name":"fin_lin_approx_of_stonean","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem: cut_down_of_wstar","label":"cut_down_of_wstar","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://JonBannon.github.io/Junk-Test/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(p\\in M\\) is a projection, \\(pMp\\) is also a \\(W^{*}\\) algebra with identity \\(p\\).","doc_url":null,"latex_link":"sect0001.html#lem: cut_down_of_wstar","lean_link":null,"lean_decl_name":"cut_down_of_wstar","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem: spec_masa_wstar_stonean","label":"spec_masa_wstar_stonean","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://JonBannon.github.io/Junk-Test/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(C\\) is any maximal commutative \\(C^*\\)–subalgebra of the \\(W^{*}\\)–algebra \\(M\\), its spectrum space (maximal ideal space) is Stonean.","doc_url":null,"latex_link":"sect0001.html#lem: spec_masa_wstar_stonean","lean_link":null,"lean_decl_name":"spec_masa_wstar_stonean","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem: zorn_base","label":"zorn_base","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://JonBannon.github.io/Junk-Test/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(\\varphi \\) be a normal positive linear functional on \\(M\\) and consider the predicate \\(P:\\mathcal{P}(M)\\to \\text{Prop}\\) defined, for \\(p\\in \\mathcal{P}(M)\\), by “\\(M\\ni x \\mapsto \\varphi (xp)\\) is \\(\\sigma (M,M_{*})\\)–continuous”. If \\((p_{\\alpha })\\) is a chain of projections in \\(M\\) such that \\(P(p_{\\alpha })\\) is true for each \\(\\alpha \\), then \\(P(\\sup (p_{\\alpha }))\\) is true. Hence by Zorn’s Lemma there is a maximal \\(p_0\\in \\mathcal{P}(M)\\) such that \\(P(p_0)\\) is true.","doc_url":null,"latex_link":"sect0001.html#lem: zorn_base","lean_link":null,"lean_decl_name":"zorn_base","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem: bdd_sigma_cts_iff_bdd_s_cts","label":"bdd_sigma_cts_iff_bdd_s_cts","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://JonBannon.github.io/Junk-Test/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A linear functional \\(\\rho \\) on \\(M\\) is \\(\\sigma \\) continuous on the unit sphere (hence \\(\\sigma \\)–continuous) if and only if it is \\(s\\)–continuous on the unit sphere (hence \\(s\\)–continuous).","doc_url":null,"latex_link":"sect0001.html#lem: bdd_sigma_cts_iff_bdd_s_cts","lean_link":null,"lean_decl_name":"bdd_sigma_cts_iff_bdd_s_cts","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}

crawled_repos/LBPTester.jsonl

@@ -0,0 +1,7 @@
+{"id":"definition : VectorSpace","label":"VectorSpace","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AdamJasonKern.github.io/LBPTester/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A vector space is a space over a field K with an abelian group V. It has four main properties: smul_add: \\(\\forall (a : K) (x y : V), a \\bullet (x + y) = a \\bullet x + a \\bullet y \\) add_smul: \\(\\forall (a b : K) (x : V), (a + b) \\bullet x = a \\bullet x + b \\bullet x \\) mul_smul: \\(\\forall (a b : K) (x : V), (a * b) \\bullet x = a \\bullet (b \\bullet x) \\) one_smul: \\(\\forall (x : V), (1 : K) \\bullet x = x \\)","doc_url":"https://AdamJasonKern.github.io/LBPTester/docs/find/#doc/LinearAlgebraGame.VectorSpace","latex_link":"sect0001.html#definition : VectorSpace","lean_link":"https://AdamJasonKern.github.io/LBPTester/docs/find/#doc/LinearAlgebraGame.VectorSpace","lean_decl_name":"LinearAlgebraGame.VectorSpace","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"LBPTester","source_path":"Game/Levels/VectorSpaceWorld/Level01.lean","lean4_code":"class VectorSpace (K V : Type) [Field K] [AddCommGroup V] extends Module K V where\n  smul_add_explicit : ∀ (a : K) (x y : V), a • (x + y) = a • x + a • y := smul_add\n  add_smul_explicit : ∀ (a b : K) (x : V), (a + b) • x = a • x + b • x := add_smul\n  mul_smul_explicit : ∀ (a b : K) (x : V), (a * b) • x = a • (b • x) := mul_smul\n  one_smul_explicit : ∀ (x : V), (1 : K) • x = x := one_smul\n","line":100,"end_line":104,"proof_status":"formalized"}
+{"id":"theorem : smul_zero","label":"smul_zero","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://AdamJasonKern.github.io/LBPTester/blueprint/dep_graph_document.html","dependencies":["definition : VectorSpace"],"nlp_text":"In any vector space V over K, any scalar a multiplied by the zero vector gives the zero vector:\\( (a : K) : a \\bullet (0 : V) = (0 : V)\\)","doc_url":"https://AdamJasonKern.github.io/LBPTester/docs/find/#doc/LinearAlgebraGame.smul_zero_v","latex_link":"sect0001.html#theorem : smul_zero","lean_link":"https://AdamJasonKern.github.io/LBPTester/docs/find/#doc/LinearAlgebraGame.smul_zero_v","lean_decl_name":"LinearAlgebraGame.smul_zero_v","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/AdamJasonKern/LBPTester/8eb9f5c46e81d41eb1a4490ee2c761793c3f839d/Game/Levels/VectorSpaceWorld/Level02.lean","lean4_code":"/--\nIn any vector space V over K, any scalar a multiplied by the zero vector gives the zero vector.\n-/\nStatement smul_zero_v (fk : Field K) (acg : AddCommGroup V) (vs : VectorSpace K V) (a : K) : a • (0 : V) = (0 : V) := by\n  Hint (hidden := true) \"Try `apply add_right_cancel (b := a • (0 : V))`\"\n  apply add_right_cancel (b := a • (0 : V))\n  Hint (hidden := true) \"Try `rw[(smul_add a (0 : V) (0 : V)).symm]`\"\n  rw[(smul_add a (0 : V) (0 : V)).symm]\n  Hint (hidden := true) \"Try `rw[zero_add]`\"\n  rw[zero_add]\n  Hint (hidden := true) \"Try `rw[zero_add]`\"\n  rw[zero_add]","line":31,"end_line":42,"proof_status":"formalized"}
+{"id":"theorem : zero_smul","label":"zero_smul","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://AdamJasonKern.github.io/LBPTester/blueprint/dep_graph_document.html","dependencies":["definition : VectorSpace"],"nlp_text":"In any vector space V over K, the scalar 0 multiplied by any vector gives the zero vector: (0 : K) • w = (0 : V)","doc_url":"https://AdamJasonKern.github.io/LBPTester/docs/find/#doc/LinearAlgebraGame.zero_smul_v","latex_link":"sect0001.html#theorem : zero_smul","lean_link":"https://AdamJasonKern.github.io/LBPTester/docs/find/#doc/LinearAlgebraGame.zero_smul_v","lean_decl_name":"LinearAlgebraGame.zero_smul_v","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/AdamJasonKern/LBPTester/8eb9f5c46e81d41eb1a4490ee2c761793c3f839d/Game/Levels/VectorSpaceWorld/Level01.lean","lean4_code":"/--\nIn any vector space V over K, the scalar 0 multiplied by any vector gives the zero vector.\n-/\nStatement zero_smul_v (fk : Field K) (acg : AddCommGroup V) (vs : VectorSpace K V) (w : V) : (0 : K) • w = (0 : V) := by\n  Hint \"Remember, we are trying to prove this backwards. The last step in the normal proof was to\n  cancel out 0 • w from both sides, so what should the first step of the backwards proof be?\"\n  Hint (hidden := true) \"Try `apply add_right_cancel (b := (0 : K) • w)`\"\n  apply add_right_cancel (b := (0 : K) • w)\n  Hint \"Now, there is a theorem we have from the vector space definition that can change the left\n  side of the equation. Think about the second to last step in the normal proof. You may also need to\n  use `.symm` here.\"\n  Hint (hidden := true) \"Try `rw[(add_smul (0 : K) (0 : K) w).symm]`\"\n  rw[(add_smul (0 : K) (0 : K) w).symm]\n  Hint \"Now, we just have to cancel out zeros.\"\n  Hint (hidden := true) \"Try rw[zero_add]\"\n  rw[zero_add]\n  Hint (hidden := true) \"Try rw[zero_add]\"\n  rw[zero_add]","line":184,"end_line":201,"proof_status":"formalized"}
+{"id":"theorem : neg_one_smul","label":"neg_one_smul","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://AdamJasonKern.github.io/LBPTester/blueprint/dep_graph_document.html","dependencies":["definition : VectorSpace"],"nlp_text":"In any vector space V over K, multiplying a vector by -1 gives its additive inverse: \\((v : V) : (-1 : K) \\bullet v = -v\\)","doc_url":"https://AdamJasonKern.github.io/LBPTester/docs/find/#doc/LinearAlgebraGame.neg_one_smul_v","latex_link":"sect0001.html#theorem : neg_one_smul","lean_link":"https://AdamJasonKern.github.io/LBPTester/docs/find/#doc/LinearAlgebraGame.neg_one_smul_v","lean_decl_name":"LinearAlgebraGame.neg_one_smul_v","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/AdamJasonKern/LBPTester/8eb9f5c46e81d41eb1a4490ee2c761793c3f839d/Game/Levels/VectorSpaceWorld/Level03.lean","lean4_code":"/--\nIn any vector space V over K, multiplying a vector by -1 gives its additive inverse.\n-/\nStatement neg_one_smul_v (fk : Field K) (acg : AddCommGroup V) (vs : VectorSpace K V) (v : V) : (-1 : K) • v = -v := by\n  Hint \"A good first step is cancelling out the `-v` term on the right.\"\n  Hint (hidden := true) \"Try `apply add_right_cancel (b := v)`\"\n  apply add_right_cancel (b := v)\n  Hint \"Remember the `nth_rw m [theorem]` tactic to only rewrite the mth instance.\"\n  Hint (hidden := true) \"Try `nth_rw 2 [(one_smul (K := K) v).symm]`\"\n  nth_rw 2 [(one_smul K v).symm]\n  Hint (hidden := true) \"Try `rw[(add_smul (-1 : K) (1 : K) v).symm]`\"\n  rw [(add_smul (-1 : K) (1 : K) v).symm]\n  Hint (hidden := true) \"Try `rw[neg_add_self]`\"\n  rw[neg_add_self]\n  Hint (hidden := true) \"Try `rw[neg_add_self]`\"\n  rw[neg_add_self]\n  Hint \"This looks like something we've done before. Either the `rw` or `exact` tactics should solve the goal\"\n  Hint (hidden := true) \"Try `rw [zero_smul_v]`\"\n  rw [zero_smul_v]","line":52,"end_line":70,"proof_status":"formalized"}
+{"id":"theorem : subspace_contains_zero","label":"subspace_contains_zero","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://AdamJasonKern.github.io/LBPTester/blueprint/dep_graph_document.html","dependencies":["theorem : zero_smul","definition : Subspace"],"nlp_text":"This is a proof that any subspace contains the zero vector:\\((0 : V) \\in W\\)","doc_url":"https://AdamJasonKern.github.io/LBPTester/docs/find/#doc/LinearAlgebraGame.subspace_contains_zero","latex_link":"sect0001.html#theorem : subspace_contains_zero","lean_link":"https://AdamJasonKern.github.io/LBPTester/docs/find/#doc/LinearAlgebraGame.subspace_contains_zero","lean_decl_name":"LinearAlgebraGame.subspace_contains_zero","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/AdamJasonKern/LBPTester/8eb9f5c46e81d41eb1a4490ee2c761793c3f839d/Game/Levels/VectorSpaceWorld/Level04.lean","lean4_code":"/--\nThis is a proof that any subspace contains the zero vector.\n-/\nStatement subspace_contains_zero (fk : Field K) (acg : AddCommGroup V) (vs : VectorSpace K V) {W : Set V} (hW : isSubspace (K := K) (V := V) W) : (0 : V) ∈ W := by\n  Hint \"Try to expand out your hypotheses using `obtain`.\"\n  Hint (hidden := true) \"Try `obtain ⟨h1, h2, h3⟩ := hW`\"\n  obtain ⟨h1, _h2, h3⟩ := hW\n  Hint \"Again, you can use `obtain` to simplify a hypothesis.\"\n  Hint (hidden := true) \"Try `obtain ⟨w, hw⟩ := {h1}`\"\n  obtain ⟨w, hw⟩ := h1\n  Hint \"We know that `0 • {w} ∈ W`. If this was our goal, the level would be easy to solve. Also,\n  remember that if you have to use a theorem you have proven in a previous level, you have to write\n  `theorem_name fk acg vs theorem_args` to show Lean that K V is a vector space.\"\n  Hint (hidden := true) \"Try `rw [(zero_smul_v fk acg vs {w}).symm]`\"\n  rw [(zero_smul_v fk acg vs w).symm]\n  Hint \"Now, apply the fact that subspaces are closed under scalar multiplication.\"\n  Hint (hidden := true) \"Try `apply {h3}`\"\n  apply h3\n  Hint (hidden := true) \"Try `exact {hw}`\"\n  exact hw","line":74,"end_line":93,"proof_status":"formalized"}
+{"id":"theorem : subspace_neg","label":"subspace_neg","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://AdamJasonKern.github.io/LBPTester/blueprint/dep_graph_document.html","dependencies":["theorem : neg_one_smul","definition : Subspace"],"nlp_text":"This is a proof that if a subspace contains a vector ‘x‘, it also contains ‘-x‘:\\( \\forall (x : V), x \\in W \\rightarrow (-x) \\in W\\)","doc_url":"https://AdamJasonKern.github.io/LBPTester/docs/find/#doc/LinearAlgebraGame.subspace_neg","latex_link":"sect0001.html#theorem : subspace_neg","lean_link":"https://AdamJasonKern.github.io/LBPTester/docs/find/#doc/LinearAlgebraGame.subspace_neg","lean_decl_name":"LinearAlgebraGame.subspace_neg","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/AdamJasonKern/LBPTester/8eb9f5c46e81d41eb1a4490ee2c761793c3f839d/Game/Levels/VectorSpaceWorld/Level05.lean","lean4_code":"Statement subspace_neg (fk : Field K) (acg : AddCommGroup V) (vs : VectorSpace K V) {W : Set V} (hW : isSubspace (K := K) (V := V) W) : ∀ (x : V), x ∈ W → (-x) ∈ W := by\n  Hint (hidden := true) \"Try `obtain ⟨h1, h2, h3⟩ := hW`\"\n  Branch\n    intros _x _hx\n    Hint (hidden := true) \"Try `obtain ⟨h1, h2, h3⟩ := hW`\"\n  obtain ⟨_h1, _h2, h3⟩ := hW\n  Hint (hidden := true) \"Try `intros x hx`\"\n  intros x hx\n  Hint (hidden := true) \"Try `rw [(neg_one_smul_v fk acg vs {x}).symm]`\"\n  rw [(neg_one_smul_v fk acg vs x).symm]\n  Hint (hidden := true) \"Try `apply {h3}`\"\n  apply h3\n  Hint (hidden := true) \"Try `exact {hx}`\"\n  exact hx","line":24,"end_line":37,"proof_status":"formalized"}
+{"id":"definition : Subspace","label":"Subspace","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AdamJasonKern.github.io/LBPTester/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A subspace is a subset of a vector space that acts similarly to a vector space itself. It has three main properties: Nonempty: W.Nonempty Closure_Under_Addition: \\(\\forall (x y : V), x ∈ W \\rightarrow y \\in W \\rightarrow x + y \\in W\\) Closure_Under_Scalar_Multiplication: \\(\\forall (a : K) (x : V), x \\in W \\rightarrow a \\bullet x \\in W\\)","doc_url":"https://AdamJasonKern.github.io/LBPTester/docs/find/#doc/LinearAlgebraGame.isSubspace","latex_link":"sect0001.html#definition : Subspace","lean_link":"https://AdamJasonKern.github.io/LBPTester/docs/find/#doc/LinearAlgebraGame.isSubspace","lean_decl_name":"LinearAlgebraGame.isSubspace","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"LBPTester","source_path":"Game/Levels/VectorSpaceWorld/Level04.lean","lean4_code":"def isSubspace {K V : Type} [Field K] [AddCommGroup V] [VectorSpace K V] (W : Set V) : Prop :=\n  W.Nonempty ∧ (∀ (x y : V), x ∈ W → y ∈ W → x + y ∈ W) ∧ (∀ (a : K) (x : V), x ∈ W → a • x ∈ W)\n","line":31,"end_line":32,"proof_status":"formalized"}

crawled_repos/LeanArchitect-example.jsonl

@@ -0,0 +1,8 @@
+{"id":"MyNat","label":"MyNat","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://hanwenzhu.github.io/LeanArchitect-example/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Natural numbers.","doc_url":"https://hanwenzhu.github.io/lean-architect-example/docs/find/#doc/MyNat","latex_link":"index.html#MyNat","lean_link":"https://hanwenzhu.github.io/lean-architect-example/docs/find/#doc/MyNat","lean_decl_name":"MyNat","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:nat-add","label":"nat-add","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://hanwenzhu.github.io/LeanArchitect-example/blueprint/dep_graph_document.html","dependencies":["MyNat"],"nlp_text":"Natural number addition.","doc_url":"https://hanwenzhu.github.io/lean-architect-example/docs/find/#doc/MyNat.add","latex_link":"index.html#def:nat-add","lean_link":"https://hanwenzhu.github.io/lean-architect-example/docs/find/#doc/MyNat.add","lean_decl_name":"MyNat.add","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"LeanArchitect-example","source_path":"./Example/MyNat.lean","lean4_code":"def add (a b : MyNat) : MyNat :=\n  match b with\n  | zero => a\n  | succ b => succ (add a b)\n\n","line":14,"end_line":18,"proof_status":"formalized"}
+{"id":"MyNat.zero_add","label":"MyNat.zero_add","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://hanwenzhu.github.io/LeanArchitect-example/blueprint/dep_graph_document.html","dependencies":["def:nat-add"],"nlp_text":"For any natural number \\(a\\), \\(0 + a = a\\), where \\(+\\) is definition 2.","doc_url":"https://hanwenzhu.github.io/lean-architect-example/docs/find/#doc/MyNat.zero_add","latex_link":"index.html#MyNat.zero_add","lean_link":"https://hanwenzhu.github.io/lean-architect-example/docs/find/#doc/MyNat.zero_add","lean_decl_name":"MyNat.zero_add","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"LeanArchitect-example","source_path":"./Example/MyNat.lean","lean4_code":"theorem zero_add (a : MyNat) : add zero a = a := by\n  /-- The proof follows by induction. -/\n  induction a <;> simp [*, add]\n\n","line":21,"end_line":24,"proof_status":"formalized"}
+{"id":"MyNat.mul","label":"MyNat.mul","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://hanwenzhu.github.io/LeanArchitect-example/blueprint/dep_graph_document.html","dependencies":["def:nat-add"],"nlp_text":"Natural number multiplication.","doc_url":"https://hanwenzhu.github.io/lean-architect-example/docs/find/#doc/MyNat.mul","latex_link":"index.html#MyNat.mul","lean_link":"https://hanwenzhu.github.io/lean-architect-example/docs/find/#doc/MyNat.mul","lean_decl_name":"MyNat.mul","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"LeanArchitect-example","source_path":"./Example/MyNat.lean","lean4_code":"def mul (a b : MyNat) : MyNat := sorry\n\n","line":49,"end_line":50,"proof_status":"statement_formalized"}
+{"id":"MyNat.succ_add","label":"MyNat.succ_add","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://hanwenzhu.github.io/LeanArchitect-example/blueprint/dep_graph_document.html","dependencies":["def:nat-add"],"nlp_text":"For any natural numbers \\(a, b\\), \\((a + 1) + b = (a + b) + 1\\).","doc_url":"https://hanwenzhu.github.io/lean-architect-example/docs/find/#doc/MyNat.succ_add","latex_link":"index.html#MyNat.succ_add","lean_link":"https://hanwenzhu.github.io/lean-architect-example/docs/find/#doc/MyNat.succ_add","lean_decl_name":"MyNat.succ_add","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"LeanArchitect-example","source_path":"./Example/MyNat.lean","lean4_code":"theorem succ_add (a b : MyNat) : add (succ a) b = succ (add a b) := by\n  /-- Proof by induction on $b$. -/\n  -- If the proof contains sorry, the `\\leanok` command will not be added\n  sorry\n\n","line":27,"end_line":31,"proof_status":"statement_formalized"}
+{"id":"MyNat.add_comm","label":"MyNat.add_comm","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://hanwenzhu.github.io/LeanArchitect-example/blueprint/dep_graph_document.html","dependencies":["MyNat.zero_add","MyNat.succ_add"],"nlp_text":"For any natural numbers \\(a, b\\), \\(a + b = b + a\\).","doc_url":"https://hanwenzhu.github.io/lean-architect-example/docs/find/#doc/MyNat.add_comm","latex_link":"index.html#MyNat.add_comm","lean_link":"https://hanwenzhu.github.io/lean-architect-example/docs/find/#doc/MyNat.add_comm","lean_decl_name":"MyNat.add_comm","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"LeanArchitect-example","source_path":"./Example/MyNat.lean","lean4_code":"theorem add_comm (a b : MyNat) : add a b = add b a := by\n  induction b with\n  | zero =>\n    have := trivial\n    /-- The base case follows from \\cref{MyNat.zero_add}. -/\n    simp [add]\n  | succ b ih =>\n    /-- The inductive case follows from \\cref{MyNat.succ_add}. -/\n    sorry_using [succ_add]  -- the `sorry_using` tactic declares that the proof uses succ_add\n\n/-! ## Multiplication -/\n\n","line":34,"end_line":45,"proof_status":"statement_formalized"}
+{"id":"MyNat.mul_comm","label":"MyNat.mul_comm","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://hanwenzhu.github.io/LeanArchitect-example/blueprint/dep_graph_document.html","dependencies":["MyNat.mul"],"nlp_text":"For any natural numbers \\(a, b\\), \\(a * b = b * a\\).","doc_url":"https://hanwenzhu.github.io/lean-architect-example/docs/find/#doc/MyNat.mul_comm","latex_link":"index.html#MyNat.mul_comm","lean_link":"https://hanwenzhu.github.io/lean-architect-example/docs/find/#doc/MyNat.mul_comm","lean_decl_name":"MyNat.mul_comm","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"LeanArchitect-example","source_path":"./Example/MyNat.lean","lean4_code":"theorem mul_comm (a b : MyNat) : mul a b = mul b a := by sorry\n\n/-! ## Fermat's Last Theorem -/\n\n","line":53,"end_line":56,"proof_status":"formalized"}
+{"id":"thm:flt","label":"flt","background_color":"white","border_color":"#FFAA33","kind":"ellipse","dep_graph_url":"https://hanwenzhu.github.io/LeanArchitect-example/blueprint/dep_graph_document.html","dependencies":["MyNat.mul_comm"],"nlp_text":"Fermat’s last theorem.","doc_url":"https://hanwenzhu.github.io/lean-architect-example/docs/find/#doc/MyNat.flt","latex_link":"index.html#thm:flt","lean_link":"https://hanwenzhu.github.io/lean-architect-example/docs/find/#doc/MyNat.flt","lean_decl_name":"MyNat.flt","kind_description":"theorems and lemmas","status_from_color":"unknown","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"LeanArchitect-example","source_path":"./Example/MyNat.lean","lean4_code":"theorem flt : (sorry : Prop) := sorry\n\nend MyNat\n\n-- Finally, these are utility commands for debugging:\n\n","line":66,"end_line":71,"proof_status":"statement_formalized"}

crawled_repos/LeanBlueprintExample.jsonl

@@ -0,0 +1,3 @@
+{"id":"one","label":"one","background_color":"#b0eca3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/LeanBlueprintExample/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"one is 1.","doc_url":"https://utensil.github.io/LeanBlueprintExample/doc//find/#doc/one","latex_link":"sec-subdemo.html#one","lean_link":"https://utensil.github.io/LeanBlueprintExample/doc//find/#doc/one","lean_decl_name":"one","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"LeanBlueprintExample","source_path":"./LeanBlueprintExample/Basic.lean","lean4_code":"def one := 1\n\n","line":3,"end_line":4,"proof_status":"formalized"}
+{"id":"one_plus_one_eq_two","label":"one_plus_one_eq_two","background_color":"#a3d6ff","border_color":"green","kind":"ellipse","dep_graph_url":"https://utensil.github.io/LeanBlueprintExample/blueprint/dep_graph_document.html","dependencies":["one","two"],"nlp_text":"one + one = two","doc_url":"https://utensil.github.io/LeanBlueprintExample/doc//find/#doc/one_plus_one_eq_two","latex_link":"sec-subdemo.html#one_plus_one_eq_two","lean_link":"https://utensil.github.io/LeanBlueprintExample/doc//find/#doc/one_plus_one_eq_two","lean_decl_name":"one_plus_one_eq_two","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"LeanBlueprintExample","source_path":"./LeanBlueprintExample/Basic.lean","lean4_code":"theorem one_plus_one_eq_two : one + one = two := by\n  rfl\n\n","line":7,"end_line":9,"proof_status":"statement_formalized"}
+{"id":"two","label":"two","background_color":"#b0eca3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/LeanBlueprintExample/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"two is 2.","doc_url":"https://utensil.github.io/LeanBlueprintExample/doc//find/#doc/two","latex_link":"sec-subdemo.html#two","lean_link":"https://utensil.github.io/LeanBlueprintExample/doc//find/#doc/two","lean_decl_name":"two","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"LeanBlueprintExample","source_path":"./LeanBlueprintExample/Basic.lean","lean4_code":"def two := 2\n\n","line":5,"end_line":6,"proof_status":"formalized"}

crawled_repos/M2.jsonl

@@ -0,0 +1,49 @@
+{"id":"def:pure_homotopy_k_sheaf","label":"pure_homotopy_k_sheaf","background_color":"blue","border_color":"blue","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A homotopy \\(\\mathcal{K}\\)-sheaf \\(\\mathcal{F}^{\\bullet }\\) is said to be pure on \\(X\\) if: For \\(p\\in X\\) and \\(i\\neq 0\\), \\((H^i\\mathcal{F}^{\\bullet })_p=0\\) \\(H^i\\mathcal{F}^{\\bullet }=0\\) for \\(i{\\lt}{\\lt}0\\) locally on \\(X\\): ie for all \\(p\\in X\\) there is an open neighbourhoud \\(U\\) of \\(p\\) and an integer \\(N\\) such that for \\(i\\leq N\\) and \\(K\\subset U\\): \\(H^i\\mathcal{F}^{\\bullet }(K)=0\\)","doc_url":null,"latex_link":"sect0006.html#def:pure_homotopy_k_sheaf","lean_link":null,"lean_decl_name":"pure_homotopy_k_sheaf","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"prop:cech_of_pure_homotopy_k_sheaf","label":"cech_of_pure_homotopy_k_sheaf","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:pure_homotopy_k_sheaf","lem:first_non_zero_homology_of_homotopy_k_sheaf_is_k_sheaf","lem:csq_of_pure_homotopy_k_sheaf","prop:chech_is_determined_by_stalks_3","lem:homotpy_k_sheaves_stable_by_extension","prop:homotopy_k_sheaf_compute_chech"],"nlp_text":"Let \\(\\mathcal{F}^{\\bullet }\\) be a pure-homotopy \\(\\mathcal{K}\\)-sheaf. Then there is a canonical isomorphism: \\[ H^{\\bullet }\\mathcal{F}^{\\bullet }(X)=\\check{H}^{\\bullet }(X;H^0\\mathcal{F}^{\\bullet }) \\]","doc_url":null,"latex_link":"sect0006.html#prop:cech_of_pure_homotopy_k_sheaf","lean_link":null,"lean_decl_name":"cech_of_pure_homotopy_k_sheaf","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:gluing_pure_homotopy_k_sheaves","label":"gluing_pure_homotopy_k_sheaves","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:pure_homotopy_k_sheaf","lem:hocolim_preserves_homotopy_k_sheaves"],"nlp_text":"","doc_url":null,"latex_link":"sect0008.html#lem:gluing_pure_homotopy_k_sheaves","lean_link":null,"lean_decl_name":"gluing_pure_homotopy_k_sheaves","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:poincare-lefschetz_duality","label":"poincare-lefschetz_duality","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["prop:cech_of_pure_homotopy_k_sheaf","def:pushforward_of_K-sheaf","lem:chech_comm_with_exceptional","def:orientation_sheaf_of_a_manifold","lem:Homotopy_k-sheaf_axioms_for_singular_chain","def:chech_of_compact_sup_of_sh"],"nlp_text":"Let \\(M\\) be a topological manifold of dimension \\(n\\) with boundary, \\(i:X\\to M\\) a closed subset, and \\(N\\subset \\partial M\\) a closed subset that locally looks like \\(\\varnothing \\subset \\mathbb {R}^{n-1}, \\mathbb {R}_{{\\gt}0}\\times R^{n-2}\\subset \\mathbb {R}^{n-1}\\) or \\(\\mathbb {R}^{n-1}\\). Let \\(j:\\overset {\\circ }{M}\\cup \\overset {\\circ }{N}\\to M\\) be the canonical inclusion. Then there is a canonical isomorphism: \\[ H^{\\bullet }[C_{n-1-\\bullet }(N,N\\backslash X)\\to C_{n-1-\\bullet }(M,M\\backslash X)]=\\check{H}_c^{\\bullet }(X;i^*j_!j^*\\mathfrak {o}_M) \\]","doc_url":null,"latex_link":"sect0007.html#lem:poincare-lefschetz_duality","lean_link":null,"lean_decl_name":"poincare-lefschetz_duality","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:first_non_zero_homology_of_homotopy_k_sheaf_is_k_sheaf","label":"first_non_zero_homology_of_homotopy_k_sheaf_is_k_sheaf","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["lem:Mayer_Vietoris","def:homotopy_k_sheaf","def:K_sheaves"],"nlp_text":"If \\(\\mathcal{F}^{\\bullet }\\) is a homotopy-\\(\\mathcal{K}\\)-sheaf, and \\(H^{-1}\\mathcal{F}^{\\bullet }=0\\) then \\(H^{0}\\mathcal{F}^{\\bullet }\\) is a \\(\\mathcal{K}\\)-sheaf","doc_url":null,"latex_link":"sect0003.html#lem:first_non_zero_homology_of_homotopy_k_sheaf_is_k_sheaf","lean_link":null,"lean_decl_name":"first_non_zero_homology_of_homotopy_k_sheaf_is_k_sheaf","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:kshv_equiv_shv","label":"kshv_equiv_shv","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["lem:adj_kshv_and_shv","def:K_stalk","lem:equiv_of_adj","def:stalk"],"nlp_text":"The previous adjoint pair give rise to an equivalence of category between shaeves and \\(\\mathcal{K}\\)-sheaves","doc_url":null,"latex_link":"sect0002.html#lem:kshv_equiv_shv","lean_link":null,"lean_decl_name":"kshv_equiv_shv","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:chech_preserve_quasi_iso","label":"chech_preserve_quasi_iso","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["lem:q-iso_iff_cone_acyclic","def:quasi_iso_of_complex_of_k-prsh","lem:chech_of_acyclic_is_zero","lem:chech_long_exact_sequence"],"nlp_text":"If \\(\\mathcal{F}^{\\bullet }\\to \\mathcal{G}^{\\bullet }\\) is a quasi-isomorphism then the induced maps \\(\\check{H}^i\\mathcal{F}^{\\bullet }\\to \\check{H}^i\\mathcal{G}^{\\bullet }\\) are isomorphims.","doc_url":null,"latex_link":"sect0005.html#lem:chech_preserve_quasi_iso","lean_link":null,"lean_decl_name":"chech_preserve_quasi_iso","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:chech_is_determined_by_stalks_2","label":"chech_is_determined_by_stalks_2","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["lem:chech_preserve_quasi_iso","def:homotopy_k_sheaf","lem:chech_is_determined_by_stalks_1"],"nlp_text":"Let \\(\\mathcal{F}^{\\bullet }\\) be a complex of \\(\\mathcal{K}\\)-presheaves that verify 3 and \\(H^i\\mathcal{F}^{\\bullet }=0\\) for \\(i{\\lt}{\\lt}0\\). Then if the stalks of \\(\\mathcal{F}^{\\bullet }\\) are acyclics, \\(\\check{H}^{\\bullet }(X;\\mathcal{F}^{\\bullet })=0\\)","doc_url":null,"latex_link":"sect0005.html#lem:chech_is_determined_by_stalks_2","lean_link":null,"lean_decl_name":"chech_is_determined_by_stalks_2","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:csq_of_pure_homotopy_k_sheaf","label":"csq_of_pure_homotopy_k_sheaf","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["lem:chech_preserve_quasi_iso"],"nlp_text":"Let \\(\\mathcal{F}^{\\bullet }\\) be a pure-homotopy \\(\\mathcal{K}\\)-sheaf. Then: For \\(i{\\lt}0\\) \\(H^i\\mathcal{F}^{\\bullet }=0\\) \\(H^0\\mathcal{F}^{\\bullet }\\) is a \\(\\mathcal{K}\\)-sheaf.","doc_url":null,"latex_link":"sect0006.html#lem:csq_of_pure_homotopy_k_sheaf","lean_link":null,"lean_decl_name":"csq_of_pure_homotopy_k_sheaf","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"prop:chech_is_determined_by_stalks_3","label":"chech_is_determined_by_stalks_3","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["lem:chech_is_determined_by_stalks_2"],"nlp_text":"Let \\(\\mathcal{F}^{\\bullet }\\) and \\(\\mathcal{G}^{\\bullet }\\) be complexes of \\(\\mathcal{K}\\)-presheaves that verify 3 and \\(H^i\\mathcal{F}^{\\bullet }=H^i\\mathcal{G}^{\\bullet }=0\\) for \\(i\\) small enough. Then if a morphism \\(\\mathcal{F}^{\\bullet }\\to \\mathcal{G}^{\\bullet }\\) induces a quasi-isomorphism on stalks, \\(\\check{H}^{\\bullet }(X;\\mathcal{F}^{\\bullet })=\\check{H}^{\\bullet }(X;\\mathcal{G}^{\\bullet })\\)","doc_url":null,"latex_link":"sect0005.html#prop:chech_is_determined_by_stalks_3","lean_link":null,"lean_decl_name":"chech_is_determined_by_stalks_3","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:q-iso_iff_cone_acyclic","label":"q-iso_iff_cone_acyclic","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A morphism of complex \\(f:A^{\\bullet }\\to B^{\\bullet }\\) is a quasi isomorphism if and only if, its cone is acyclic.","doc_url":null,"latex_link":"sect0001.html#lem:q-iso_iff_cone_acyclic","lean_link":null,"lean_decl_name":"q-iso_iff_cone_acyclic","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:complex_total_of_three_is_acyclic","label":"complex_total_of_three_is_acyclic","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["lem:q-iso_iff_cone_acyclic"],"nlp_text":"If a complex \\([A^{\\bullet }\\to B^{\\bullet -1}\\to C^{\\bullet -2}]\\) is acyclic then there is a long exact sequence \\[ \\ldots \\to H^k A\\to H^k B\\to H^k C\\to \\ldots \\]","doc_url":null,"latex_link":"sect0001.html#lem:complex_total_of_three_is_acyclic","lean_link":null,"lean_decl_name":"complex_total_of_three_is_acyclic","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:Mayer_Vietoris","label":"Mayer_Vietoris","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["lem:complex_total_of_three_is_acyclic"],"nlp_text":"By using 1.4, 3 give rise to a \"Mayer-Vietoris\" long exact sequence: \\[ \\ldots \\to H^k \\mathcal{F}^{\\bullet }(K_1\\cup K_2)\\to H^k \\mathcal{F}^{\\bullet }(K_1)\\bigoplus H^k\\mathcal{F}^{\\bullet }(K_2)\\to H^k \\mathcal{F}^{\\bullet }(K_1\\cap K_2)\\to \\ldots \\]","doc_url":null,"latex_link":"sect0003.html#lem:Mayer_Vietoris","lean_link":null,"lean_decl_name":"Mayer_Vietoris","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:pushforward_of_K-sheaf","label":"pushforward_of_K-sheaf","background_color":"blue","border_color":"blue","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:pushforward_of_sheaf","def:K_sheaves","lem:basis_of_open_of_fm1K"],"nlp_text":"If \\(\\mathcal{F}\\) is a pre-\\(\\mathcal{K}\\)-sheaf over \\(X\\), then the rule \\(K\\mapsto \\mathcal{F}(f^{-1}(K))\\) defines a pre-\\(\\mathcal{K}\\)-sheaf over \\(Y\\). The functor obtained is denoted \\(f_*\\) and named the pushforward by \\(f\\). \\(f_*\\) send \\(\\mathcal{K}\\)-sheaves over \\(X\\) into \\(\\mathcal{K}\\)-sheaves over \\(Y\\).","doc_url":null,"latex_link":"sect0004.html#def:pushforward_of_K-sheaf","lean_link":null,"lean_decl_name":"pushforward_of_K-sheaf","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:total_complex","label":"total_complex","background_color":"blue","border_color":"blue","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(A_0^{\\bullet }\\to \\ldots \\to A_n^{\\bullet }\\) is a sequence of maps (with \\(f_i:A_i\\to A_{i+1}\\) )of complex such that the composition of two consecutive maps is \\(0\\), then let’s denote \\([A_0^{\\bullet }\\to \\ldots \\to A_n^{\\bullet -n}]\\) the total complex of this double complex, defined by the following data: The object in degré \\(k\\) is \\(\\bigoplus \\limits _{i=0}^n A_i[-i]^k\\) The diferential is given by the matrix \\(\\begin{pmatrix} d_{A_0} & 0 & \\ldots & \\ldots & 0 \\\\ f_0 & -d_{A_1} & \\ldots & \\ldots & 0 \\\\ 0 & f_1 & \\ldots & \\ldots & 0 \\\\ \\vdots & \\vdots & \\ddots & & \\vdots \\\\ \\vdots & \\vdots & & (-1)^{n-1}d_{A_{n-1}} & 0 \\\\ 0 & 0 & \\ldots & f_{n-1} & (-1)^nd_{A_n} \\\\ \\end{pmatrix}\\)","doc_url":null,"latex_link":"sect0001.html#def:total_complex","lean_link":null,"lean_decl_name":"total_complex","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:homotopy_k_sheaf","label":"homotopy_k_sheaf","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:total_complex","def:K_pre_sheaves"],"nlp_text":"A complex of \\(\\mathcal{K}\\)-presheaves \\(\\mathcal{F}^{\\bullet }\\) is said to be a Homotopy-\\(\\mathcal{K}\\)-sheave if the folowing conditions are satisfied: \\begin{equation} \\label{axiom:hKsh1} \\mathcal{F}^{\\bullet }(\\varnothing ) \\text{ is an acyclic complex} \\end{equation}","doc_url":null,"latex_link":"sect0003.html#def:homotopy_k_sheaf","lean_link":null,"lean_decl_name":"homotopy_k_sheaf","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:homotpy_k_sheaves_stable_by_extension","label":"homotpy_k_sheaves_stable_by_extension","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:homotopy_k_sheaf"],"nlp_text":"Let \\(\\mathcal{F}^{\\bullet }\\) be a complex of \\(\\mathcal{K}\\)-presheaves. If \\(\\mathcal{F}^{\\bullet }\\) has a finite filtration whose associated graded is a Homotopy-\\(\\mathcal{K}\\)-sheaf, then \\(\\mathcal{F}^{\\bullet }\\) is a Homotopy-\\(\\mathcal{K}\\)-sheaf.","doc_url":null,"latex_link":"sect0003.html#lem:homotpy_k_sheaves_stable_by_extension","lean_link":null,"lean_decl_name":"homotpy_k_sheaves_stable_by_extension","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"prop:homotopy_k_sheaf_compute_chech","label":"homotopy_k_sheaf_compute_chech","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:homotopy_k_sheaf","def:chech_of_compelx_of_k_prsh"],"nlp_text":"Let \\(\\mathcal{F}^{\\bullet }\\) be a complex of \\(\\mathcal{K}\\)-presheaves that verify 1 and 2 then the canonical map \\(H^{\\bullet }\\mathcal{F}^{\\bullet }\\to \\check{H}^{\\bullet }(X;\\mathcal{F}^{\\bullet })\\) is an isomorphism.","doc_url":null,"latex_link":"sect0005.html#prop:homotopy_k_sheaf_compute_chech","lean_link":null,"lean_decl_name":"homotopy_k_sheaf_compute_chech","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:hocolim_preserves_homotopy_k_sheaves","label":"hocolim_preserves_homotopy_k_sheaves","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["lem:homotpy_k_sheaves_stable_by_extension","def:homotopy_colimit"],"nlp_text":"","doc_url":null,"latex_link":"sect0008.html#lem:hocolim_preserves_homotopy_k_sheaves","lean_link":null,"lean_decl_name":"hocolim_preserves_homotopy_k_sheaves","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:sheaves","label":"sheaves","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:pre_sheaves"],"nlp_text":"If \\(\\mathcal{F}\\) is a presheaf on \\(X\\), it is said to be a sheaf if for any \\(U\\subset X\\) open and any covering family of \\(U\\) \\((U_a)_{a\\in A}\\) one has the exact sequence: \\begin{equation} \\label{axiom:Sh} 0\\to \\mathcal{F}(U)\\to \\prod \\limits _{a\\in A}\\mathcal{F}(U_a)\\to \\prod \\limits _{a,b\\in A}\\mathcal{F}(U_a\\cap U_b) \\end{equation}","doc_url":null,"latex_link":"sect0002.html#def:sheaves","lean_link":null,"lean_decl_name":"sheaves","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/AlgebraicGeometry/StructureSheaf.lean","lean4_code":"a morphism between the structure sheaves of `R` and `S`.\n-/\n","line":949,"end_line":950,"proof_status":"formalized"}
+{"id":"def:pullback_of_sheaf","label":"pullback_of_sheaf","background_color":"blue","border_color":"blue","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:sheaves"],"nlp_text":"If \\(\\mathcal{F}\\) is a pre-sheaf over \\(Y\\), then the rule \\(U\\mapsto \\varinjlim \\limits _{f(U)\\subset V}\\mathcal{F}(V)\\) defines a pre-sheaf over \\(Y\\). If \\(\\mathcal{F}\\) is a sheaf, the sheafification of the previous pre-sheaf is a denoted \\(f^*\\mathcal{F}\\) and called the pullback by \\(f\\).","doc_url":null,"latex_link":"sect0004.html#def:pullback_of_sheaf","lean_link":null,"lean_decl_name":"pullback_of_sheaf","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:adj_kshv_and_shv","label":"adj_kshv_and_shv","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:sheaves","def:K_sheaves","def:adj_kprshv_and_prshv","lem:cofinal_syst_of_inter_compact"],"nlp_text":"\\(\\alpha ^*\\) send sheaves to \\(\\mathcal{K}\\)-sheaves \\(\\alpha _*\\) send \\(\\mathcal{K}\\)-sheaves to sheaves The reistrictions obtained still form an adjoint pair between shaeves and \\(\\mathcal{K}\\)-sheaves.","doc_url":null,"latex_link":"sect0002.html#lem:adj_kshv_and_shv","lean_link":null,"lean_decl_name":"adj_kshv_and_shv","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:pushforward_of_sheaf","label":"pushforward_of_sheaf","background_color":"blue","border_color":"blue","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:sheaves"],"nlp_text":"If \\(\\mathcal{F}\\) is a pre-sheaf over \\(X\\), then the rule \\(U\\mapsto \\mathcal{F}(f^{-1}(U))\\) defines a pre-shaef over \\(Y\\). The functor obtained is denoted \\(f_*\\) and named the pushforward by \\(f\\). \\(f_*\\) send sheaves over \\(X\\) into sheaves over \\(Y\\).","doc_url":null,"latex_link":"sect0004.html#def:pushforward_of_sheaf","lean_link":null,"lean_decl_name":"pushforward_of_sheaf","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:exceptional_pushforward_of_sheaf","label":"exceptional_pushforward_of_sheaf","background_color":"blue","border_color":"blue","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:pushforward_of_sheaf"],"nlp_text":"If \\(f:X\\to Y\\) is the inclusion of an open subset, the exceptional pushforward by \\(f\\): \\(f_!\\) is defined by \\(f_!\\mathcal{F}(U)\\) being the subset of \\(f_*\\mathcal{F}(U)\\) of sections that vanish over a neighborhood of \\(Y-X\\). It send the sheaves over \\(X\\) into the sheaves over \\(Y\\)","doc_url":null,"latex_link":"sect0004.html#def:exceptional_pushforward_of_sheaf","lean_link":null,"lean_decl_name":"exceptional_pushforward_of_sheaf","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:chech_comm_with_exceptional","label":"chech_comm_with_exceptional","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:exceptional_pushforward_of_sheaf"],"nlp_text":"Let \\(f:X\\to Y\\) be an inclusion of open subset, then there is a natural isomorphism \\(f_!:\\check{H}_c^{\\bullet }(X;\\mathcal{F})\\to \\check{H}^{\\bullet }(Y;f_!\\mathcal{F})\\)","doc_url":null,"latex_link":"sect0005.html#lem:chech_comm_with_exceptional","lean_link":null,"lean_decl_name":"chech_comm_with_exceptional","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:K_pre_sheaves","label":"K_pre_sheaves","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A \\(\\mathcal{K}\\)-presheave on \\(X\\) is a contravariant functor from the category of compact sets of \\(X\\) to abélian groups.","doc_url":null,"latex_link":"sect0002.html#def:K_pre_sheaves","lean_link":null,"lean_decl_name":"K_pre_sheaves","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:cohomology_of_k-prsh","label":"cohomology_of_k-prsh","background_color":"blue","border_color":"blue","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:K_pre_sheaves"],"nlp_text":"Let \\(\\mathcal{F}^{\\bullet }\\) be complex of \\(\\mathcal{K}\\)-presheaves then taking the cohomology defines a \\(\\mathcal{K}\\)-presheaf denoted \\(H^{\\bullet }\\mathcal{F}^{\\bullet }\\).","doc_url":null,"latex_link":"sect0003.html#def:cohomology_of_k-prsh","lean_link":null,"lean_decl_name":"cohomology_of_k-prsh","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:K_stalk","label":"K_stalk","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:K_pre_sheaves"],"nlp_text":"If \\(\\mathcal{F}\\) is a \\(\\mathcal{K}\\)-presheaf on \\(X\\) and \\(p\\in X\\) then the stalk of \\(\\mathcal{F}\\) at \\(p\\) is the abelian group \\(\\mathcal{F}_p:=\\varinjlim \\limits _{p\\in K\\text{ compact}}\\mathcal{F}(K)=\\mathcal{F}(\\{ p\\} )\\).","doc_url":null,"latex_link":"sect0002.html#def:K_stalk","lean_link":null,"lean_decl_name":"K_stalk","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:K_sheaves","label":"K_sheaves","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:K_pre_sheaves"],"nlp_text":"If \\(\\mathcal{F}\\) is a \\(\\mathcal{K}\\)-presheaf on \\(X\\), it is said to be a \\(\\mathcal{K}\\)-sheaf if the folowing conditions are satisfied: \\begin{equation} \\label{axiom:Ksh1} \\mathcal{F}(\\varnothing )=0 \\end{equation}","doc_url":null,"latex_link":"sect0002.html#def:K_sheaves","lean_link":null,"lean_decl_name":"K_sheaves","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:adj_kprshv_and_prshv","label":"adj_kprshv_and_prshv","background_color":"blue","border_color":"blue","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:K_pre_sheaves","def:pre_sheaves"],"nlp_text":"If \\(\\mathcal{F}\\) is a presheaf then let \\(\\alpha ^*\\mathcal{F}\\) be the \\(\\mathcal{K}\\)-presheaf: \\[ K\\mapsto \\varinjlim \\limits _{K\\subset U \\text{ open}}\\mathcal{F}(U) \\]","doc_url":null,"latex_link":"sect0002.html#def:adj_kprshv_and_prshv","lean_link":null,"lean_decl_name":"adj_kprshv_and_prshv","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:chech_is_determined_by_stalks_1","label":"chech_is_determined_by_stalks_1","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:cohomology_of_k-prsh","def:K_stalk","def:chech_of_compelx_of_k_prsh"],"nlp_text":"Let \\(\\mathcal{F}^{\\bullet }\\) be a complex of \\(\\mathcal{K}\\)-presheaves that verify 4 and such that all the stalks are \\(0\\) then \\(\\check{H}^{\\bullet }(X;\\mathcal{F})=0\\)","doc_url":null,"latex_link":"sect0005.html#lem:chech_is_determined_by_stalks_1","lean_link":null,"lean_decl_name":"chech_is_determined_by_stalks_1","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:quasi_iso_of_complex_of_k-prsh","label":"quasi_iso_of_complex_of_k-prsh","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:cohomology_of_k-prsh"],"nlp_text":"A morphisme of complex of \\(\\mathcal{K}\\)-presheave \\(\\mathcal{F}^{\\bullet }\\to \\mathcal{G}^{\\bullet }\\) is called quasi-isomorphism if it induces isomorphisms \\(H^i\\mathcal{F}^{\\bullet }\\to H^i\\mathcal{G}^{\\bullet }\\) for all \\(i\\).","doc_url":null,"latex_link":"sect0003.html#def:quasi_iso_of_complex_of_k-prsh","lean_link":null,"lean_decl_name":"quasi_iso_of_complex_of_k-prsh","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"pro:adj_kprshv_and_prshv","label":"adj_kprshv_and_prshv","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:adj_kprshv_and_prshv"],"nlp_text":"The pair \\((\\alpha ^*,\\alpha _*)\\) is an adjonit pair.","doc_url":null,"latex_link":"sect0002.html#pro:adj_kprshv_and_prshv","lean_link":null,"lean_decl_name":"adj_kprshv_and_prshv","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:chech_of_sh","label":"chech_of_sh","background_color":"blue","border_color":"blue","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(X\\) is a topological space, and \\(\\mathcal{F}\\) a sheaf over \\(X\\), then let \\(\\check{H}^{\\bullet }(X;\\mathcal{F}))\\) be the Čech cohomology of \\(X\\) with coeficient in \\(\\mathcal{F}\\)","doc_url":null,"latex_link":"sect0005.html#def:chech_of_sh","lean_link":null,"lean_decl_name":"chech_of_sh","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:pullback_for_chech","label":"pullback_for_chech","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:chech_of_sh","def:chech_of_compact_sup_of_sh"],"nlp_text":"Let \\(f:X\\to Y\\) be a continuous map between topological spaces, and \\(\\mathcal{F}\\) a sheaf over \\(X\\). \\(f\\) induces a natural map \\(\\check{H}^{\\bullet }(Y;f_*\\mathcal{F})\\to \\check{H}^{\\bullet }(X;\\mathcal{F})\\). Moreover if \\(f\\) is proper, one gets a natural map \\(\\check{H}_c^{\\bullet }(Y;f_*\\mathcal{F})\\to \\check{H}_c^{\\bullet }(X;\\mathcal{F})\\)","doc_url":null,"latex_link":"sect0005.html#def:pullback_for_chech","lean_link":null,"lean_decl_name":"pullback_for_chech","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:orientation_sheaf_of_a_manifold","label":"orientation_sheaf_of_a_manifold","background_color":"blue","border_color":"blue","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(M\\) be a topological manifold, the rule \\(\\mathfrak {o}_M:K\\mapsto H_{\\dim M}(M,M\\backslash K)\\) defines a \\(\\mathcal{K}-\\)sheaf, called the orientation \\(\\mathcal{K}-\\)sheaf of \\(M\\). If \\(M\\) is a manifold with boundary, let \\(j:M\\backslash \\partial M\\to M\\) denote the canonical inclusion. The the orientation shaeves of \\(M\\) are defined as follows: \\(\\mathfrak {o}_M:= j_*\\mathfrak {o}_{M\\backslash \\partial M}\\) \\(\\mathfrak {o}_{M\\quad rel \\partial }:= j_!\\mathfrak {o}_{M\\backslash \\partial M}\\)","doc_url":null,"latex_link":"sect0007.html#def:orientation_sheaf_of_a_manifold","lean_link":null,"lean_decl_name":"orientation_sheaf_of_a_manifold","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:equiv_of_adj","label":"equiv_of_adj","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(\\mathcal{C}\\) and \\(\\mathcal{D}\\) are two categories, \\(F:\\mathcal{C}\\to \\mathcal{D}\\) and \\(G:\\mathcal{D}\\to \\mathcal{C}\\) two functors such that \\((F,G)\\) is an adjoint pair. Then for \\((F,G)\\) to be an equivalence of category, it’s enough to have that thes canonical naturals transformations \\(\\text{id}_{\\mathcal{D}}\\Rightarrow F\\circ G\\) and \\(G\\circ F\\Rightarrow \\text{id}_{\\mathcal{D}}\\) are isomorphisms.","doc_url":null,"latex_link":"sect0002.html#lem:equiv_of_adj","lean_link":null,"lean_decl_name":"equiv_of_adj","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:basis_of_open_of_fm1K","label":"basis_of_open_of_fm1K","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(K\\) is a compact of \\(Y\\), then \\(\\{ f^{-1}(U)\\} _{K\\subset U}\\) is a basis of open neighborhoods of \\(f^{-1}(K)\\).","doc_url":null,"latex_link":"sect0004.html#lem:basis_of_open_of_fm1K","lean_link":null,"lean_decl_name":"basis_of_open_of_fm1K","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:Homotopy_k-sheaf_axioms_for_singular_chain","label":"Homotopy_k-sheaf_axioms_for_singular_chain","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:singular_chain"],"nlp_text":"Let \\(X\\) be a topological manifold, then we have: \\(C_{\\bullet }(X,X)\\) is an acyclic complex For \\(A\\) and \\(B\\) two closed subsets of \\(X\\) the folowing complex is acyclic: \\[ [C_{\\bullet }(X,X\\backslash (A\\cup B))\\to C_{\\bullet +1}(X,X\\backslash A)\\bigoplus C_{\\bullet +1}(X,X\\backslash B)\\to C_{\\bullet +2}(X,X\\backslash (A\\cap B)) ] \\]","doc_url":null,"latex_link":"sect0007.html#lem:Homotopy_k-sheaf_axioms_for_singular_chain","lean_link":null,"lean_decl_name":"Homotopy_k-sheaf_axioms_for_singular_chain","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:chech_of_acyclic_is_zero","label":"chech_of_acyclic_is_zero","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(mathcal{F}^{\\bullet }\\) be an acyclic complex of \\(\\mathcal{K}\\)-presheaves, then \\(\\check{H}^k(X;mathcal{F}^{\\bullet })=0\\)","doc_url":null,"latex_link":"sect0005.html#lem:chech_of_acyclic_is_zero","lean_link":null,"lean_decl_name":"chech_of_acyclic_is_zero","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:chech_long_exact_sequence","label":"chech_long_exact_sequence","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:chech_of_compelx_of_k_prsh"],"nlp_text":"Let \\(0\\to \\mathcal{F}^{\\bullet }\\to \\mathcal{G}^{\\bullet }\\to \\mathcal{H}^{\\bullet }\\to 0\\) be a short exact sequence of complex of \\(\\mathcal{K}\\)-presheaves. Then there is a long exact seuquence in čech cohomology: \\[ \\ldots \\to \\check{H}^k(X;\\mathcal{F}^{\\bullet })\\to \\check{H}^k(X;\\mathcal{G}^{\\bullet })\\to \\check{H}^k(X;\\mathcal{H}^{\\bullet })\\to \\ldots \\]","doc_url":null,"latex_link":"sect0005.html#lem:chech_long_exact_sequence","lean_link":null,"lean_decl_name":"chech_long_exact_sequence","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:chech_of_compelx_of_k_prsh","label":"chech_of_compelx_of_k_prsh","background_color":"blue","border_color":"blue","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(\\mathcal{F}^{\\bullet }\\) be a complex of \\(\\mathcal{K}\\)-presheaves on a compact space \\(X\\) then we define the Čech cohomology \\(\\check{H}(X;\\mathcal{F}^{\\bullet })\\) by TODO","doc_url":null,"latex_link":"sect0005.html#def:chech_of_compelx_of_k_prsh","lean_link":null,"lean_decl_name":"chech_of_compelx_of_k_prsh","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:chech_of_compact_sup_of_sh","label":"chech_of_compact_sup_of_sh","background_color":"blue","border_color":"blue","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(X\\) is a topological space, \\(K\\) a comapct subset of \\(X\\) and \\(\\mathcal{F}\\) a sheaf over \\(X\\), then let \\(\\check{H}_K^{\\bullet }(X;\\mathcal{F}))\\) be the Čech cohomology of \\(X\\) with support in \\(K\\) with coeficient in \\(\\mathcal{F}\\)","doc_url":null,"latex_link":"sect0005.html#def:chech_of_compact_sup_of_sh","lean_link":null,"lean_decl_name":"chech_of_compact_sup_of_sh","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:pre_sheaves","label":"pre_sheaves","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A presheave on \\(X\\) is a contravariant functor from the category of open sets of \\(X\\) to abélian groups.","doc_url":null,"latex_link":"sect0002.html#def:pre_sheaves","lean_link":null,"lean_decl_name":"pre_sheaves","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:stalk","label":"stalk","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:pre_sheaves"],"nlp_text":"If \\(\\mathcal{F}\\) is a presheaf on \\(X\\) and \\(p\\in X\\) then the stalk of \\(\\mathcal{F}\\) at \\(p\\) is the abelian group \\(\\mathcal{F}_p:=\\varinjlim \\limits _{p\\in U\\text{ open}}\\mathcal{F}(U)\\).","doc_url":null,"latex_link":"sect0002.html#def:stalk","lean_link":null,"lean_decl_name":"stalk","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"M2","source_path":"./M2/alphaup_and_stalk.lean","lean4_code":"def FUtoStalkι : FU (pC p) F (trueCond ) ⟶ (Functor.const _ ).obj (F.stalk p) where\n  app U := germ _ U.unop.obj p (U.unop.property.1 (by rfl)) ≫ F.stalkSpecializes (by rfl)\n\n/-- The cocone induced by FUtoStalkι -/\n","line":29,"end_line":32,"proof_status":"formalized"}
+{"id":"def:homotopy_diagram","label":"homotopy_diagram","background_color":"blue","border_color":"blue","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"","doc_url":null,"latex_link":"sect0008.html#def:homotopy_diagram","lean_link":null,"lean_decl_name":"homotopy_diagram","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:homotopy_colimit","label":"homotopy_colimit","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":["def:homotopy_diagram"],"nlp_text":"","doc_url":null,"latex_link":"sect0008.html#def:homotopy_colimit","lean_link":null,"lean_decl_name":"homotopy_colimit","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:cofinal_syst_of_inter_compact","label":"cofinal_syst_of_inter_compact","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(K_1,\\ldots K_n\\) are comapcts of \\(X\\) then \\(\\{ U_1\\cap \\ldots \\cap U_n\\} _{U_i\\supset K_i\\text{ open in }X}\\) is a cofinal system of neighborhoods of \\(K_1\\cap \\ldots \\cap K_n\\).","doc_url":null,"latex_link":"sect0002.html#lem:cofinal_syst_of_inter_compact","lean_link":null,"lean_decl_name":"cofinal_syst_of_inter_compact","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:singular_chain","label":"singular_chain","background_color":"blue","border_color":"blue","kind":"box","dep_graph_url":"https://ymonbru.github.io/M2/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Singular chains","doc_url":null,"latex_link":"sect0007.html#def:singular_chain","lean_link":null,"lean_decl_name":"singular_chain","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}

crawled_repos/Mills.jsonl

@@ -0,0 +1,9 @@
+{"id":"def:mills","label":"mills","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://unaoya.github.io/Mills/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A positive real number \\(x\\) is Mills if \\(1 {\\lt} x\\) and for all positive integers \\(n\\), the number \\(\\lfloor x^{3^n} \\rfloor \\) is prime.","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Mills","latex_link":"sect0001.html#def:mills","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Mills","lean_decl_name":"Mills","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Mills","source_path":"./Mills/Basic copy.lean","lean4_code":"def Mills (x : ℝ) : Prop := 1 < x ∧ ∀ k : ℕ, 0 < k → Nat.Prime (Nat.floor (x ^ (3 ^ k)))\n","line":121,"end_line":121,"proof_status":"statement_formalized"}
+{"id":"prop:mills","label":"mills","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://unaoya.github.io/Mills/blueprint/dep_graph_document.html","dependencies":["def:mills"],"nlp_text":"There exists a Mills number.","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/exists_Mills","latex_link":"sect0001.html#prop:mills","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/exists_Mills","lean_decl_name":"exists_Mills","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Mills","source_path":"./Mills/Basic.lean","lean4_code":"theorem exists_Mills : ∃ A : ℝ, Mills A := by\n  have : 1 < pp 1 := by rw [pp]; linarith [(pp' 1).property, BHP_const_nat_ge2]\n  use (left_lim pp)\n  constructor\n  · exact (by linarith [(left_gt_1 pp this), (left_le_sup pp (fun n ↦ aux' n) 1)])\n  · intro k h\n    rw [left_floor_eq_seq pp (fun n ↦ (hpp n).left) (fun n ↦ aux' n) k, ← Nat.succ_pred_eq_of_pos h]\n    exact (hpp k.pred).right.right.right\n    exact h\n\n","line":409,"end_line":418,"proof_status":"statement_formalized"}
+{"id":"prop:proposition4","label":"proposition4","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://unaoya.github.io/Mills/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\(W\\) has minimum.","doc_url":null,"latex_link":"sect0001.html#prop:proposition4","lean_link":null,"lean_decl_name":"proposition4","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:lemma6","label":"lemma6","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://unaoya.github.io/Mills/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For all natural numbers \\(k\\), we have \\[ p_k \\leq p_{k+1} {\\lt} (p_k ^3 + 1) ^3 - 1. \\]","doc_url":null,"latex_link":"sect0001.html#lem:lemma6","lean_link":null,"lean_decl_name":"lemma6","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"prop:proposition5","label":"proposition5","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://unaoya.github.io/Mills/blueprint/dep_graph_document.html","dependencies":["lem:lemma6","thm:BHP"],"nlp_text":"\\(\\theta = 21/40\\). There exists a natural number \\(k_0\\) such that for all \\(k \\geq k_0\\), we have \\[ p_k ^3 \\leq p_{k+1} \\leq p_k ^3 + p_k ^{3\\theta }. \\]","doc_url":null,"latex_link":"sect0001.html#prop:proposition5","lean_link":null,"lean_decl_name":"proposition5","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:lemma7","label":"lemma7","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://unaoya.github.io/Mills/blueprint/dep_graph_document.html","dependencies":["prop:proposition5"],"nlp_text":"There exists \\(\\gamma {\\gt} 0\\) and \\(k_1 {\\gt} 1\\) such that for all \\(k \\geq k_1\\), we have \\[ \\lvert A ^{3 ^k} - p_k\\rvert \\leq e^{-3 ^k\\gamma }. \\]","doc_url":null,"latex_link":"sect0001.html#lem:lemma7","lean_link":null,"lean_decl_name":"lemma7","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"thm:mills_irrational","label":"mills_irrational","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://unaoya.github.io/Mills/blueprint/dep_graph_document.html","dependencies":["lem:lemma7"],"nlp_text":"The Mills’ constant is irrational.","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Mills_irrational","latex_link":"sect0001.html#thm:mills_irrational","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Mills_irrational","lean_decl_name":"Mills_irrational","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Mills","source_path":"./Mills/Basic copy.lean","lean4_code":"theorem Mills_irrational : Irrational A := by\n  rw [Irrational]; simp\n  intro r h\n  have h₁ : 1 < r := by\n    have : Rat.cast 1  < (r : ℝ) := by rw [h]; simp; exact Mills_exists.left\n    apply Rat.cast_lt.1 this\n  have h₂ : ∀ n : ℤ, ↑n ≠ r := by\n    intro n h₃\n    rcases (nat_pos r h₁ n h₃) with ⟨m, hm⟩\n    rw [← h₃, ← hm] at h\n    apply A_not_nat m h.symm\n  rcases lem7 with ⟨γ, _, k₁, _, h₃⟩\n  rcases Mahler r γ h₁ h₂ with ⟨K, h₄⟩\n  let k := max K k₁\n  have h₅ : K ≤ 3 ^ k := by\n    calc\n      K ≤ k := le_max_left K k₁\n      _ ≤ 3 ^ k := by apply le_of_lt (Nat.lt_pow_self (by norm_num) k)\n  have h₆ : min_dist (r ^ 3 ^ k) > Real.exp (-γ * 3 ^ k) := by\n    have : Nat.pow 3 k = (3 : ℝ) ^ k := by simp -- 抽象化\n    calc\n      min_dist (r ^ 3 ^ k) = min_dist (r ^ (3 ^ k : ℝ)) := by rw [← Real.rpow_natCast, ← Real.rpow_natCast]; simp\n      _ > Real.exp (-γ * 3 ^ k) := by rw [← this]; apply h₄.right (3 ^ k) h₅\n  have h₇ : min_dist (r ^ 3 ^ k) ≤ Real.exp (-γ * 3 ^ k) := by\n    calc\n      min_dist (r ^ 3 ^ k) ≤ |(r : ℝ) ^ 3 ^ k - Nat.floor ((r : ℝ) ^ 3 ^ k)| := by\n        apply min_dist_floor\n        rw [h, ← Real.rpow_natCast]\n        apply Real.rpow_nonneg (by linarith [Mills_gt_one])\n      _ = |A ^ 3 ^ k - Nat.floor (A ^ 3 ^ k)| := by rw [h]\n      _ = |A ^ (3 : ℝ) ^ k - Nat.floor (A ^ (3 : ℝ) ^ k)| := by\n        rw [← Real.rpow_natCast, ← Real.rpow_natCast]\n        have : (Nat.cast : ℕ → ℝ) (Nat.pow 3 k) = (3 : ℝ) ^ (k : ℝ) := by simp --抽象化\n        rw [← this]; simp\n      _ = |A ^ (3 : ℝ) ^ k - p' k| := by rw [p'eqp''', p''', ← Real.rpow_natCast]\n      _ ≤ Real.exp (-γ * 3 ^ k) := h₃ k (le_max_right K k₁)\n  linarith\n","line":700,"end_line":736,"proof_status":"statement_formalized"}
+{"id":"thm:BHP","label":"BHP","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://unaoya.github.io/Mills/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"BHP","doc_url":null,"latex_link":"sect0001.html#thm:BHP","lean_link":null,"lean_decl_name":"BHP","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"Mills","source_path":"./Mills/Defs.lean","lean4_code":"theorem BHP' (x : ℕ+) (h : BHP_cons_pnat ≤ x) : ∃ p : ℕ+, x ≤ p ∧ p ≤ x + (pn_pow_nnr x θ) ∧ Nat.Prime p := by sorry\n\n","line":98,"end_line":99,"proof_status":"statement_formalized"}
+{"id":"thm:Mahler","label":"Mahler","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://unaoya.github.io/Mills/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Mahler","doc_url":null,"latex_link":"sect0001.html#thm:Mahler","lean_link":null,"lean_decl_name":"Mahler","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}

crawled_repos/algebra-in-lean.jsonl

@@ -0,0 +1,54 @@
+{"id":"theorem : hom_inv_to_inv","label":"hom_inv_to_inv","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Homomorphism"],"nlp_text":"Suppose \\( \\phi : G \\rightarrow H \\) is a homomorphism. If \\(g \\in G \\), then \\( \\phi (\\iota (g)) = \\iota (\\phi (g)) \\).","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.hom_inv_to_inv","latex_link":"sect0002.html#theorem : hom_inv_to_inv","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.hom_inv_to_inv","lean_decl_name":"AlgebraInLean.hom_inv_to_inv","kind_description":"theorems and lemmas","status_from_color":"statement_formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter02/Sheet01.lean","lean4_code":"theorem hom_inv_to_inv (φ : G → H) (hp : Homomorphism φ) (g : G) : φ (ι g) = ι (φ g) := by\n  -- SAMPLE SOLUTION\n  have h1 : μ (φ (ι g)) (φ g) = φ (μ (ι g) g)\n  · rw [Homomorphism_def] at hp\n    rw [hp (ι g) g]\n  have h2 : φ (μ (ι g) g) = φ 𝕖\n  · rw [inv_op]\n  rw [h2] at h1\n  rw [hom_id_to_id φ hp] at h1\n  rw [two_sided_inv (φ (ι g)) (φ g) h1]\n  rw [inv_inv]\n","line":218,"end_line":228,"proof_status":"statement_formalized"}
+{"id":"definition : Center","label":"Center","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Centralizer"],"nlp_text":"The center of \\(G\\), denoted \\(Z(G)\\), is the set of all elements that commute with every element of \\(G\\), or \\(C_G(G)\\).","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Center","latex_link":"sect0003.html#definition : Center","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Center","lean_decl_name":"AlgebraInLean.Center","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter03/Sheet05.lean","lean4_code":"def Center : Subgroup G := Centralizer Set.univ\n","line":171,"end_line":171,"proof_status":"formalized"}
+{"id":"theorem : Pows_eq_Generate_singleton","label":"Pows_eq_Generate_singleton","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Pows","definition : Generate"],"nlp_text":"\\(\\text{Pows}(x) = \\langle \\{ x\\} \\rangle \\).","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Pows_eq_Generate_singleton","latex_link":"sect0003.html#theorem : Pows_eq_Generate_singleton","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Pows_eq_Generate_singleton","lean_decl_name":"AlgebraInLean.Pows_eq_Generate_singleton","kind_description":"theorems and lemmas","status_from_color":"statement_formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter03/Sheet04.lean","lean4_code":"theorem Pows_eq_Generate_singleton (x : G) : Pows x = Generate {x} := by\n  apply le_antisymm\n  · intro g hg\n    intro H hH\n    rw [Set.singleton_subset_iff] at hH\n    obtain ⟨a, ha⟩ := hg\n    rw [←ha]\n    apply gpow_closure\n    exact hH\n  · intro g hg\n    dsimp [Pows]\n    dsimp [Generate] at hg\n    have : {x} ⊆ (Pows x).carrier\n    · rw [Set.singleton_subset_iff]\n      apply Pows_contain_self\n    specialize hg (Pows x) this\n    obtain ⟨n, hn⟩ := hg\n    use n\n","line":247,"end_line":264,"proof_status":"statement_formalized"}
+{"id":"definition : gpow","label":"gpow","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : mpow","definition : Group"],"nlp_text":"For a group \\(G\\) and \\(x \\in G\\), we extend the domain of the power function to include all of \\(\\mathbb {Z}\\) such that if \\(n ~ {\\lt}~ 0\\), then \\(x^n\\) is equal to \\(\\iota (x^{-n})\\). It is defined inductively below: \\begin{equation*} x^n := \\begin{cases} \\mathbb {e}, & n = 0 \\\\ \\mu (x^{n-1}, x), & n ~ {\\gt}~ 0 \\\\ \\iota (\\mu (x^{n+1}, x)), & n ~ {\\lt}~ 0 \\end{cases}\\end{equation*}","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.gpow","latex_link":"sect0003.html#definition : gpow","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.gpow","lean_decl_name":"AlgebraInLean.gpow","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter03/Sheet02.lean","lean4_code":"def gpow {G : Type*} [Group G] (x : G) : ℤ → G\n/- The integer type can be constructed using natural numbers. This is done through an inductive type\nwith two constructors, indicating the two possible states of an integer (those being negative and\nnonnegative). The constructor `Int.ofNat` covers the nonnegative end of the integers (those that are\nequivalent to a `Nat`, hence the name `ofNat`). -/\n| Int.ofNat n => mpow x n\n/-\nSince the integer zero is already covered by `Int.ofNat 0`, it is not helpful for the negative\nconstructor to have its own notion of zero. Instead, the negative constructor offsets the provided\nnatural number by one before negating it. So, (0 : ℕ) maps to (-1 : ℤ), (1 : ℕ) maps to (-2 : ℤ),\nand so on. This warrants the name `negSucc`, as in \"the negation of the successor of the inputted\nnatural number\". Keep this in mind as you work with `gpow`.\n-/\n| Int.negSucc n => mpow (ι x) (n+1)\n","line":115,"end_line":128,"proof_status":"formalized"}
+{"id":"definition : Pows","label":"Pows","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : gpow","definition : Subgroup"],"nlp_text":"For a group \\(G\\), the power subgroup of \\(x \\in G\\) is defined as \\[ \\text{Pows}(x) := \\left\\{ x^n \\middle | n \\in \\mathbb {Z} \\right\\} \\]","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Pows","latex_link":"sect0003.html#definition : Pows","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Pows","lean_decl_name":"AlgebraInLean.Pows","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter03/Sheet04.lean","lean4_code":"def Pows (x : G) : Subgroup G where\n  carrier := {g : G | ∃ a, gpow x a = g}\n  -- EXERCISES\n  has_id := by\n    use 0\n    rw [gpow_zero]\n  mul_closure := by\n    intro g₁ g₂ ⟨a, ha⟩ ⟨b, hb⟩\n    use a + b\n    rw [←ha, ←hb, gpow_add]\n  inv_closure := by\n    intro g ⟨a, ha⟩\n    use -a\n    have : ∀ (g i : G), μ i g = 𝕖 → i = ι g := inv_unique_left\n    apply this\n    rw [←ha, gpow_neg, inv_op]\n","line":220,"end_line":235,"proof_status":"formalized"}
+{"id":"theorem : mod_order_eq_of_gpow_eq","label":"mod_order_eq_of_gpow_eq","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : gpow","definition : order"],"nlp_text":"Let \\(G\\) be a group and \\(x \\in G\\). Suppose that for two integers \\(m\\) and \\(n\\), \\(x^m = x^n\\). Then \\(m \\equiv n\\) (mod \\(|x|\\)). Note that if two integers are equivalent mod 0, then they are equal, implying that if \\(|x| = 0\\), then the power function over \\(\\mathbb {Z}\\) is injective.","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.mod_order_eq_of_gpow_eq","latex_link":"sect0003.html#theorem : mod_order_eq_of_gpow_eq","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.mod_order_eq_of_gpow_eq","lean_decl_name":"AlgebraInLean.mod_order_eq_of_gpow_eq","kind_description":"theorems and lemmas","status_from_color":"statement_formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter03/Sheet03.lean","lean4_code":"lemma mod_order_eq_of_gpow_eq {m n : ℤ}\n  : gpow x m = gpow x n → m % (order x) = n % (order x) := by\n  intro h\n  by_cases h₀ : order x = 0\n  · rw [h₀]\n    simp\n    exact gpow_inj_of_order_zero x h₀ h\n  · obtain ⟨m', hm'⟩ := emod_has_nat m (Nat.zero_lt_of_ne_zero h₀)\n    obtain ⟨n', hn'⟩ := emod_has_nat n (Nat.zero_lt_of_ne_zero h₀)\n    rw [hm', hn']\n    repeat rw [←Int.ofNat_emod]\n    congr 1\n    apply mod_order_eq_of_mpow_eq x m' n' h₀\n    rw [←mpow_mod_order x m', ←mpow_mod_order x n']\n    repeat rw [←gpow_ofNat, Int.ofNat_emod]\n    rw [←hm', ←hn']\n    repeat rw [gpow_mod_order]\n    exact h\n","line":358,"end_line":375,"proof_status":"statement_formalized"}
+{"id":"theorem : Pows_card_eq_order","label":"Pows_card_eq_order","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Pows","definition : order"],"nlp_text":"\\(|\\text{Pows}(x)| = |x|\\).","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Pows_card_eq_order","latex_link":"sect0003.html#theorem : Pows_card_eq_order","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Pows_card_eq_order","lean_decl_name":"AlgebraInLean.Pows_card_eq_order","kind_description":"theorems and lemmas","status_from_color":"statement_formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter03/Sheet04.lean","lean4_code":"theorem Pows_card_eq_order (x : G) : Nat.card (Pows x) = order x := by\n  by_cases h : order x ≠ 0\n  · -- EXERCISE\n    apply Nat.card_eq_of_equiv_fin\n    apply Equiv.symm\n    apply Equiv.ofBijective (finPowMap x (order x))\n    apply finPowMap_order_bijective\n    exact h\n  · -- EXERCISE\n    rw [ne_eq, Decidable.not_not] at h\n    rw [h]\n    apply Set.Infinite.card_eq_zero\n    have e : ℤ ≃ Pows x\n    · apply Equiv.ofBijective (gpowMap x)\n      apply gpowMap_bijective_of_order_zero\n      exact h\n    rw [←Set.infinite_coe_iff, ←Equiv.infinite_iff e]\n    exact Int.infinite\n  done\n","line":327,"end_line":345,"proof_status":"statement_formalized"}
+{"id":"definition : mpow","label":"mpow","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Monoid"],"nlp_text":"For \\(x \\in M\\), the \\(n\\)th power of \\(x\\) (where \\(n \\in \\mathbb {N}\\)) is defined as the left-multiplication of \\(x\\) with the identity of \\(M\\) \\(n\\) times. It is defined inductively below \\begin{equation*} x^n := \\begin{cases} \\mathbb {e}, & n = 0 \\\\ \\mu (x^{n-1}, x), & n ~ {\\gt}~ 0 \\end{cases}\\end{equation*}","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.mpow","latex_link":"sect0003.html#definition : mpow","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.mpow","lean_decl_name":"AlgebraInLean.mpow","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter03/Sheet02.lean","lean4_code":"def mpow {M : Type*} [Monoid M] (x : M) : ℕ → M\n| Nat.zero => 𝕖   -- m⁰ = 𝕖\n| Nat.succ n => μ (mpow x n) x  -- m ^ (n + 1) = m * m ^ n\n","line":22,"end_line":24,"proof_status":"formalized"}
+{"id":"definition : order","label":"order","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : mpow"],"nlp_text":"Let \\(M\\) be a monoid and \\(x \\in M\\). Then the order of \\(x\\), denoted \\(|x|\\), is the smallest \\(n \\in \\mathbb {N}\\) satisfying \\(x^n = \\mathbb {e}\\). If no such \\(n\\) exists, then we define the order of \\(x\\) to be zero.","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.gpow_closure","latex_link":"sect0003.html#definition : order","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.gpow_closure","lean_decl_name":"AlgebraInLean.gpow_closure","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter03/Sheet02.lean","lean4_code":"theorem gpow_closure {H : Subgroup G} {n : ℤ}: x ∈ H → gpow x n ∈ H := by\n  -- EXERCISE (*)\n  intro h\n  induction n using Int.induction_on with\n  | hz => exact H.has_id\n  | hp n ih =>\n    rw [gpow_succ]\n    apply H.mul_closure\n    · exact ih\n    · exact h\n  | hn n ih =>\n    rw [←gpow_pred, gpow_neg]\n    apply H.mul_closure\n    · rw [←gpow_neg]\n      exact ih\n    · apply H.inv_closure\n      exact h\n  done\n","line":248,"end_line":265,"proof_status":"formalized"}
+{"id":"theorem : subgroupOrder","label":"subgroupOrder","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Subgroup"],"nlp_text":"Inclusion (\\(\\subseteq \\)) forms a partial order over the set of all subgroups of \\(G\\).","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.subgroupOrder","latex_link":"sect0003.html#theorem : subgroupOrder","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.subgroupOrder","lean_decl_name":"AlgebraInLean.subgroupOrder","kind_description":"theorems and lemmas","status_from_color":"statement_formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/MAO3J1m0Op/algebra-in-lean/1b93df5d65900996de7652019c55ea50f8f86ada/AlgebraInLean/Chapter03/Sheet04.lean","lean4_code":"instance subgroupOrder [Group G] : PartialOrder (Subgroup G) where\n  -- The relation in question is ⊆\n  le H K := (H : Set G) ⊆ K\n  -- Reflexivity\n  le_refl := by\n    intro H\n    /-\n    If `unfold` does not fully expand the definition as desired, try using it as a lemma in\n    `dsimp`.\n    -/\n    dsimp only [LE.le]\n    trivial\n    -- Transitivity\n  le_trans := by\n    -- EXERCISE\n    intro H₁ H₂ H₃ h12 h23 hx h1_x\n    dsimp only [LE.le] at *\n    apply h23\n    apply h12\n    exact h1_x\n  -- Here would be a good application of the `ext` tactic discussed in Sheet 1.\n  -- Antisymmetry\n  le_antisymm := by\n    -- EXERCISE\n    intro H K hH hK\n    ext x\n    apply Iff.intro\n    · intro hx\n      apply hH\n      exact hx\n    · intro hx\n      apply hK\n      exact hx","line":30,"end_line":62,"proof_status":"statement_formalized"}
+{"id":"theorem : Maximal_normal","label":"Maximal_normal","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Maximal","definition : normal"],"nlp_text":"The maximal subgroup of \\(G\\) is a normal subgroup of \\(G\\).","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Maximal_normal","latex_link":"sect0003.html#theorem : Maximal_normal","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Maximal_normal","lean_decl_name":"AlgebraInLean.Maximal_normal","kind_description":"theorems and lemmas","status_from_color":"statement_formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter03/Sheet05.lean","lean4_code":"theorem Maximal_normal : normal (Maximal G) := by\n  -- EXERCISE\n  intro _ _ _\n  trivial\n","line":104,"end_line":107,"proof_status":"statement_formalized"}
+{"id":"definition : Automorphism","label":"Automorphism","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Isomorphism","definition : Endomorphism"],"nlp_text":"An automorphism is a bijective endomorphism; i.e. an isomorphism from a group \\( G \\) to itself.","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Automorphism","latex_link":"sect0002.html#definition : Automorphism","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Automorphism","lean_decl_name":"AlgebraInLean.Automorphism","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter02/Sheet04.lean","lean4_code":"def Automorphism (φ : G → G) : Prop := Endomorphism φ ∧ Bijective φ\n","line":48,"end_line":48,"proof_status":"formalized"}
+{"id":"theorem : conj_automorphism","label":"conj_automorphism","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Automorphism"],"nlp_text":"Given a group \\( G \\) and an element \\( a \\in G \\), conjugation by \\( a \\) is an automorphism.","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.conj_automorphism","latex_link":"sect0002.html#theorem : conj_automorphism","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.conj_automorphism","lean_decl_name":"AlgebraInLean.conj_automorphism","kind_description":"theorems and lemmas","status_from_color":"statement_formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter02/Sheet04.lean","lean4_code":"theorem conj_automorphism (g : G) : ∀ x y : G, conjugate g (μ x y) = μ (conjugate g x)\n(conjugate g y) ∧ Bijective (conjugate g) := by\n  -- SAMPLE SOLUTION\n  intros x y\n  constructor\n  -- Prove homomorphism\n  · unfold conjugate\n    simp only [op_assoc]\n    rw[← op_assoc (ι g), inv_op, id_op]\n  -- Prove bijectivity\n  · rw[Bijective]\n    constructor\n    -- Injectivity\n    · intros x y h\n      unfold conjugate at h\n      have h1 : μ (μ g x) (ι g) = μ (μ g y) (ι g) → μ g x = μ g y := by\n        intro h\n        apply op_right_cancel at h\n        exact h\n      apply h1 at h\n      apply op_left_cancel at h\n      exact h\n    -- Surjectivity\n    · intro z\n      use μ (ι g) (μ z g)\n      unfold conjugate\n      simp only [op_assoc]\n      rw [← op_assoc g, op_inv, id_op, op_id]\n    -- END SAMPLE SOLUTION\n  done\n","line":161,"end_line":190,"proof_status":"statement_formalized"}
+{"id":"definition : conjugate","label":"conjugate","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Group"],"nlp_text":"Two elements \\( a,~ b \\) of a group \\( G \\) are conjugates if there exists another element \\( g \\) in \\( G \\) such that \\( b = gag^{-1} \\) (i.e.,  \\( \\mu (\\mu (g,~ a),~ \\iota (g)) \\) ).","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.conjugate","latex_link":"sect0002.html#definition : conjugate","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.conjugate","lean_decl_name":"AlgebraInLean.conjugate","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter02/Sheet04.lean","lean4_code":"def conjugate (g x : G) : G := μ (μ g x) (ι g)\n","line":134,"end_line":134,"proof_status":"formalized"}
+{"id":"definition : Centralizer","label":"Centralizer","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : conjugate","definition : Subgroup"],"nlp_text":"Let \\(G\\) be a group and \\(S \\subset G\\). The centralizer in \\(G\\) of \\(S\\), denoted \\(C_G(S)\\), is defined as \\[ C_G(S) := \\left\\{ x \\in G \\middle | gs = sg\\; \\forall \\; s \\in S\\right\\} . \\]","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Centralizer","latex_link":"sect0003.html#definition : Centralizer","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Centralizer","lean_decl_name":"AlgebraInLean.Centralizer","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter03/Sheet05.lean","lean4_code":"def Centralizer (S : Set G) : Subgroup G where\n  carrier := {g | ∀ s ∈ S, μ g s = μ s g}\n  has_id := by\n    intro s _\n    rw [id_op, op_id]\n  mul_closure := by\n    intro a b ha hb s hs\n    rw [op_assoc, hb, ←op_assoc, ha, op_assoc] <;> exact hs\n  inv_closure := by\n    intro a ha s hs\n    symm\n    rw [←op_id s, ←op_inv a]\n    repeat rw [←op_assoc]\n    apply congr <;> try rfl\n    rw [op_assoc, op_inv, op_id]\n    nth_rw 1 [←id_op s]\n    rw [←inv_op a]\n    repeat rw [op_assoc]\n    apply congr <;> try rfl\n    apply congr <;> try rfl\n    exact ha s hs\n","line":146,"end_line":166,"proof_status":"formalized"}
+{"id":"definition : Conjugate","label":"Conjugate","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : conjugate"],"nlp_text":"Let \\(G\\) be a group, \\(g \\in G\\) and \\(S \\subset G\\). Then we define the conjugate of \\(S\\) by \\(g\\) to be \\begin{equation*} gSg^{-1} := \\left\\{ gsg^{-1} \\middle | s \\in S \\right\\} \\end{equation*}","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Conjugate","latex_link":"sect0003.html#definition : Conjugate","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Conjugate","lean_decl_name":"AlgebraInLean.Conjugate","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter03/Sheet05.lean","lean4_code":"def Conjugate (g : G) (S : Set G) : Set G := conjugate g '' S\n","line":89,"end_line":89,"proof_status":"formalized"}
+{"id":"definition : Normalizer","label":"Normalizer","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Conjugate","definition : Subgroup"],"nlp_text":"Let \\(G\\) be a group and \\(S \\subset G\\). The normalizer in \\(G\\) of \\(S\\), denoted \\(N_G(S)\\), is defined as \\[ N_G(S) := \\left\\{ x \\in G \\middle | gSg^{-1} = S \\right\\} . \\]","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Normalizer","latex_link":"sect0003.html#definition : Normalizer","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Normalizer","lean_decl_name":"AlgebraInLean.Normalizer","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter03/Sheet05.lean","lean4_code":"def Normalizer (S : Set G) : Subgroup G where\n  carrier := {g | ∀ s ∈ S, Conjugate g S = S}\n  -- EXERCISES? These are hard...\n  has_id := by\n    intro s _\n    rw [Conjugate, conjugate_by_id]\n    simp\n  mul_closure := by\n    intro a b ha hb s hs\n    specialize ha s hs\n    specialize hb s hs\n    rw [Conjugate] at *\n    rw [conjugate_op, Set.image_comp, hb, ha]\n  inv_closure := by\n    intro a ha s hs\n    nth_rw 1 [←ha s hs]\n    rw [Conjugate]\n    funext x\n    rw [Conjugate, ←Set.image_comp, ←conjugate_op, inv_op, conjugate_by_id, Set.image_id]\n","line":122,"end_line":140,"proof_status":"formalized"}
+{"id":"definition : Magma","label":"Magma","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A magma is a set G with an operation \\(\\mu : G \\rightarrow G \\rightarrow G\\)","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Magma","latex_link":"sect0001.html#definition : Magma","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Magma","lean_decl_name":"AlgebraInLean.Magma","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter01/Sheet02.lean","lean4_code":"class Magma (α : Type*) where\n  /-\n  `protected` means that this definition should always be referenced as `Magma.op` to prevent\n  ambiguity with other things that may be called `op` in Lean. You should use `μ` to reference this\n  instead (see below).\n  -/\n  protected op : α → α → α\n","line":14,"end_line":20,"proof_status":"formalized"}
+{"id":"definition : Semigroup","label":"Semigroup","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Magma"],"nlp_text":"A semigroup is a magma with the associative property: \\(\\forall a,~ b,~ c \\in G, ~ \\mu (\\mu (a, b), c) = \\mu (a,~ \\mu (b, c))\\)","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Semigroup","latex_link":"sect0001.html#definition : Semigroup","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Semigroup","lean_decl_name":"AlgebraInLean.Semigroup","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter01/Sheet02.lean","lean4_code":"class Semigroup (α : Type*) extends Magma α where\n  -- extends means that it inherits the properties of a `Magma`\n  protected op_assoc : ∀ (a b c : α), μ (μ a b) c = μ a (μ b c)\n","line":28,"end_line":30,"proof_status":"formalized"}
+{"id":"definition : Monoid","label":"Monoid","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Semigroup"],"nlp_text":"A monoid is a semigroup with an identity \\(\\mathbb {e}\\) where \\(\\forall a \\in G, ~ \\mu (a, \\mathbb {e}) = \\mu (\\mathbb {e},~ a) = a\\)","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Monoid","latex_link":"sect0001.html#definition : Monoid","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Monoid","lean_decl_name":"AlgebraInLean.Monoid","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter01/Sheet02.lean","lean4_code":"class Monoid (α : Type*) extends Semigroup α where\n  protected id : α\n  protected op_id : ∀ (a : α), μ a id = a\n  protected id_op : ∀ (a : α), μ id a = a\n","line":36,"end_line":39,"proof_status":"formalized"}
+{"id":"definition : Group","label":"Group","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Monoid"],"nlp_text":"A group is a nonempty set \\(G\\) with a closed binary operation (\\(\\mu : G \\times G \\rightarrow G\\)), satisfying the following group axioms: Associativity: \\(\\forall \\, a, b, c \\in G, ~ \\mu (\\mu (a, b), c) = \\mu (a, \\mu (b, c))\\) Identity element: \\(\\mathbb {e}\\in G\\) where \\(\\forall \\, a \\in G, ~ \\mu (a, \\mathbb {e}) = \\mu (\\mathbb {e}, a) = a\\) Inverses (\\(\\iota \\)): \\(\\forall \\, a \\in G, ~ \\mu (\\iota (a), a) = \\mathbb {e}\\)","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Group","latex_link":"sect0001.html#definition : Group","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Group","lean_decl_name":"AlgebraInLean.Group","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"./AlgebraInLean/Chapter02/Sheet00.lean","lean4_code":"theorem Group.inv_bijective {G : Type*} [Group G] : Bijective (ι : G → G) := by\n  -- SAMPLE SOLUTION\n  constructor\n  · intros x y h\n    apply op_left_cancel (ι x)\n    nth_rw 2 [h]\n    rw [inv_op, inv_op]\n  · intro y\n    use ι y\n    exact inv_inv y\n  -- END SAMPLE SOLUTION\n  done\n","line":207,"end_line":218,"proof_status":"formalized"}
+{"id":"definition : CommMonoid","label":"CommMonoid","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Monoid"],"nlp_text":"A commutative monoid is a monoid with the commutative property: \\(\\forall a,~ b \\in G, ~ \\mu (a, b) = \\mu (b, a)\\)","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.CommMonoid","latex_link":"sect0001.html#definition : CommMonoid","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.CommMonoid","lean_decl_name":"AlgebraInLean.CommMonoid","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter01/Sheet02.lean","lean4_code":"class CommMonoid (α : Type*) extends Monoid α where\n  protected op_comm : ∀ (a b : α), μ a b = μ b a\n","line":51,"end_line":52,"proof_status":"formalized"}
+{"id":"theorem : id_unique","label":"id_unique","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Monoid"],"nlp_text":"In a monoid, the identity element is unique: \\(\\forall a \\in G, ~ (\\forall b \\in G, ~ \\mu (a,~ b) = \\mu (b,~ a) = b) \\Rightarrow a = \\mathbb {e}\\)","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.id_unique","latex_link":"sect0001.html#theorem : id_unique","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.id_unique","lean_decl_name":"AlgebraInLean.id_unique","kind_description":"theorems and lemmas","status_from_color":"statement_formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter01/Sheet03.lean","lean4_code":"theorem id_unique [Monoid α] (e₂ : α) (h : ∀ (a : α), (μ a e₂ = a ∧ μ e₂ a = a)) : e₂ = 𝕖 := by\n  /-\n  The `obtain` tactic may be useful. Given a hypothesis `h : P ∧ Q`, `obtain ⟨h₁, h₂⟩ := h` creates\n  hypotheses `h₁ : P` and `h₂ : Q`. This also works to destructure other types like `∃`.\n  -/\n  -- sorry\n  -- SAMPLE SOLUTION\n  specialize h 𝕖\n  obtain ⟨h, _⟩ := h -- we don't care about the property on the right, so we use `_` instead.\n  rw [id_op] at h\n  exact h\n","line":19,"end_line":29,"proof_status":"statement_formalized"}
+{"id":"definition : Int","label":"Int","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : AbelianGroup"],"nlp_text":"\\((\\mathbb {Z},~ +)\\), the integers over addition, form a group, The identity of the group is \\(0\\), and the inverse of an element \\(x \\in \\mathbb {Z}\\) is \\(-x\\).","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/Int","latex_link":"sect0001.html#definition : Int","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/Int","lean_decl_name":"Int","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Tactic/NormNum/Result.lean","lean4_code":"@[simp] def _root_.Int.rawCast [Ring α] (n : ℤ) : α := n\n\n","line":169,"end_line":170,"proof_status":"formalized"}
+{"id":"definition : Cn","label":"Cn","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : AbelianGroup"],"nlp_text":"\\(C_n\\) is the generalization of \\(C_3\\), and is therefore the group of rotations of a regular n-gon. It uses mathlib’s Fin n type, and addition on that type. Since Fin allows modular arithmetic, rotations that are seperated by a multiple of \\(360^{\\circ }\\) are treated as equal. Similarly to \\(C_3\\), the identity is a \\(0^{\\circ }\\) rotation, and the inverse of an \\(x ^{\\circ }\\) rotation is a \\((360 - x)^{\\circ }\\) rotation.","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/Fin","latex_link":"sect0001.html#definition : Cn","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/Fin","lean_decl_name":"Fin","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Order/Fin/Tuple.lean","lean4_code":"def Fin.castLEOrderIso {n m : ℕ} (h : n ≤ m) : Fin n ≃o { i : Fin m // (i : ℕ) < n } where\n  toFun i := ⟨Fin.castLE h i, by simp⟩\n  invFun i := ⟨i, i.prop⟩\n  left_inv _ := by simp\n  right_inv _ := by simp\n  map_rel_iff' := by simp [(strictMono_castLE h).le_iff_le]\n","line":183,"end_line":188,"proof_status":"formalized"}
+{"id":"definition : C₃","label":"C₃","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Cn"],"nlp_text":"\\(C_3\\) is the group of rotational symmetries of an equilateral triangle with the group operation being the composition of symmetries. This is treated as an inductive type with three elements. The identity is a \\( 0 ^{\\circ } \\) rotation, and the inverse of an \\(x ^{\\circ } \\) rotation is a \\((360 - x)^{\\circ } \\) rotation.","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.C₃","latex_link":"sect0001.html#definition : C₃","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.C₃","lean_decl_name":"AlgebraInLean.C₃","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/MAO3J1m0Op/algebra-in-lean/1b93df5d65900996de7652019c55ea50f8f86ada/AlgebraInLean/Chapter01/Sheet04.lean","lean4_code":"inductive C₃ : Type\n| rot0 : C₃\n| rot120 : C₃\n| rot240 : C₃","line":80,"end_line":83,"proof_status":"formalized"}
+{"id":"definition : Symmetric","label":"Symmetric","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Group"],"nlp_text":"\\(S_n\\), the symmetric group of order \\(n!\\), is the set of all permutations of \\(n\\) elements. It is represented as the set of all bijections from Fin n to Fin n, with function composition as the group operation. The identity is the identity function, and the inverse of a group element is its inverse function (which exists since all bijections are invertible).","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Symmetric","latex_link":"sect0001.html#definition : Symmetric","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Symmetric","lean_decl_name":"AlgebraInLean.Symmetric","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter01/Sheet07.lean","lean4_code":"def Symmetric (n : ℕ) : Type := Fin n ≃ Fin n\n","line":19,"end_line":19,"proof_status":"formalized"}
+{"id":"definition : Isomorphism","label":"Isomorphism","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Bijective","definition : Homomorphism"],"nlp_text":"An isomorphism \\( \\phi \\) from a group \\( G \\) to a group \\( H \\) is a bijective homomorphism.","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Isomorphism","latex_link":"sect0002.html#definition : Isomorphism","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Isomorphism","lean_decl_name":"AlgebraInLean.Isomorphism","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter02/Sheet01.lean","lean4_code":"def Isomorphism (φ : G → H) : Prop := (Homomorphism φ ∧ Bijective φ)\n","line":162,"end_line":162,"proof_status":"formalized"}
+{"id":"definition : Endomorphism","label":"Endomorphism","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Homomorphism"],"nlp_text":"An endomorphism is a homomorphism \\( \\phi \\) mapping from a group \\( G \\) to itself.","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Endomorphism","latex_link":"sect0002.html#definition : Endomorphism","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Endomorphism","lean_decl_name":"AlgebraInLean.Endomorphism","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter02/Sheet04.lean","lean4_code":"def Endomorphism (φ : G → G) : Prop := Homomorphism φ\n","line":29,"end_line":29,"proof_status":"formalized"}
+{"id":"definition : Maximal","label":"Maximal","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Subgroup"],"nlp_text":"For any group \\(G\\), the maximal subgroup \\(H\\) is the group that contains every element of \\(G\\).","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Maximal","latex_link":"sect0003.html#definition : Maximal","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Maximal","lean_decl_name":"AlgebraInLean.Maximal","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter03/Sheet01.lean","lean4_code":"def Maximal (G : Type*) [Group G] : Subgroup G where\n  carrier := Set.univ\n\n  -- Try to come up with one-line solutions for each of the below proofs.\n  --PROOFS BELOW ARE SOLUTIONS\n  has_id := by\n    exact trivial\n\n  mul_closure := by\n    exact λ _ _ ↦ trivial\n\n  inv_closure := by\n    exact λ _ ↦ trivial\n","line":124,"end_line":136,"proof_status":"formalized"}
+{"id":"definition : Minimal","label":"Minimal","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Subgroup"],"nlp_text":"For any group \\(G\\), the minimal subgroup \\(H\\) is the group that contains exclusively the identity of \\(G\\).","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Minimal","latex_link":"sect0003.html#definition : Minimal","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Minimal","lean_decl_name":"AlgebraInLean.Minimal","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter03/Sheet01.lean","lean4_code":"def Minimal (G : Type*) [Group G] : Subgroup G where\n  -- BELOW ARE SOLUTIONS\n  carrier := {𝕖}\n  has_id := by\n    trivial\n  mul_closure := by\n    intro a b ha hb\n    rw [ha, hb, op_id]\n    trivial\n  inv_closure := by\n    intro a ha\n    rw [ha, inv_id]\n    trivial\n","line":142,"end_line":154,"proof_status":"formalized"}
+{"id":"theorem : Minimal_normal","label":"Minimal_normal","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Minimal","definition : normal"],"nlp_text":"The minimal subgroup of \\(G\\) is a normal subgroup of \\(G\\).","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Minimal_normal","latex_link":"sect0003.html#theorem : Minimal_normal","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Minimal_normal","lean_decl_name":"AlgebraInLean.Minimal_normal","kind_description":"theorems and lemmas","status_from_color":"statement_formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter03/Sheet05.lean","lean4_code":"theorem Minimal_normal : normal (Minimal G) := by\n  -- EXERCISE\n  intro g h hh\n  rw [hh, conjugate_id]\n  trivial\n","line":98,"end_line":102,"proof_status":"statement_formalized"}
+{"id":"theorem : op_inv","label":"op_inv","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Group"],"nlp_text":"Applying the group operation to an element and its inverse returns the identity: \\(\\forall a \\in G, ~ \\mu (a, \\iota (a)) = \\mathbb {e}\\)","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.op_inv","latex_link":"sect0001.html#theorem : op_inv","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.op_inv","lean_decl_name":"AlgebraInLean.op_inv","kind_description":"theorems and lemmas","status_from_color":"statement_formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter01/Sheet01.lean","lean4_code":"theorem op_inv [Group G] (a : G) : μ a (ι a) = 𝕖 := by\n  rw [←id_op (μ a _)]\n  rw [←inv_op (ι a)]\n  rw [op_assoc]\n  rw [←op_assoc (ι a)]\n  rw [inv_op]\n  rw [id_op]\n","line":96,"end_line":102,"proof_status":"statement_formalized"}
+{"id":"theorem : inv_inv","label":"inv_inv","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["theorem : op_inv","theorem : inv_unique"],"nlp_text":"The inverse of the inverse of an element is that element: \\(\\forall a \\in G,~ \\iota (\\iota (a)) = a\\)","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.inv_inv","latex_link":"sect0001.html#theorem : inv_inv","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.inv_inv","lean_decl_name":"AlgebraInLean.inv_inv","kind_description":"theorems and lemmas","status_from_color":"statement_formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter01/Sheet03.lean","lean4_code":"theorem inv_inv (a : α) : ι (ι a) = a := by\n  --sorry\n  -- SAMPLE SOLUTION\n  symm\n  apply inv_unique\n  constructor\n  · exact inv_op a\n  · exact op_inv a\n","line":101,"end_line":108,"proof_status":"statement_formalized"}
+{"id":"definition : op_right_cancel","label":"op_right_cancel","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["theorem : op_inv"],"nlp_text":"A group has the right cancellation property \\(\\forall a, ~ b, ~ c \\in G,~ \\mu (b,~ a) = \\mu (c,~ a) \\Rightarrow b = c\\)","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.op_right_cancel","latex_link":"sect0001.html#definition : op_right_cancel","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.op_right_cancel","lean_decl_name":"AlgebraInLean.op_right_cancel","kind_description":"theorems and lemmas","status_from_color":"statement_formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter01/Sheet02.lean","lean4_code":"theorem op_right_cancel [Group α] (a b c : α) (h : μ b a = μ c a) : b = c := by\n  -- sorry\n  -- SAMPLE SOLUTION\n  rw [←op_id b, ←op_id c, ←op_inv a, ←op_assoc, ←op_assoc, h]\n","line":105,"end_line":108,"proof_status":"statement_formalized"}
+{"id":"theorem : inv_anticomm","label":"inv_anticomm","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["theorem : op_inv","theorem : inv_unique"],"nlp_text":"The inverse of a product is the product of the inverses in reverse order: \\(\\forall a,~ b \\in G,~ \\iota (\\mu (a,~ b)) = \\mu (\\iota (b),~ \\iota (a))\\)","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.inv_anticomm","latex_link":"sect0001.html#theorem : inv_anticomm","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.inv_anticomm","lean_decl_name":"AlgebraInLean.inv_anticomm","kind_description":"theorems and lemmas","status_from_color":"statement_formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter01/Sheet03.lean","lean4_code":"theorem inv_anticomm (a b : α) : ι (μ a b) = μ (ι b) (ι a) := by\n  -- sorry\n  -- SAMPLE SOLUTION\n  symm\n  apply inv_unique\n  constructor\n  · rw [op_assoc, ←op_assoc b, op_inv, id_op, op_inv]\n  · rw [op_assoc, ←op_assoc (ι a), inv_op, id_op, inv_op]\n","line":90,"end_line":97,"proof_status":"statement_formalized"}
+{"id":"definition : Image","label":"Image","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Homomorphism","definition : Subgroup"],"nlp_text":"Let \\(\\phi : G \\rightarrow G'\\) be a group homomorphism. The kernel of \\(\\phi \\), denoted \\(\\text{img}(\\phi )\\), is defined as \\[ \\left\\{ y \\in G' \\middle | \\exists \\, x,\\; \\phi (x) = y \\right\\} . \\]","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Image","latex_link":"sect0003.html#definition : Image","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Image","lean_decl_name":"AlgebraInLean.Image","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter03/Sheet05.lean","lean4_code":"def Image (φ : G → G') (h : Homomorphism φ) : Subgroup G' where\n  carrier := {x : G' | ∃ g, φ g = x}\n  -- EXERCISES\n  has_id := by\n    use 𝕖\n    rw [hom_id_to_id φ]\n    exact h\n  mul_closure := by\n    intro a b ⟨x, hx⟩ ⟨y, hy⟩\n    use μ x y\n    rw [←h, hx, hy]\n  inv_closure := by\n    intro a ⟨x, hx⟩\n    use ι x\n    rw [←hx, hom_inv_to_inv φ]\n    exact h\n","line":37,"end_line":52,"proof_status":"formalized"}
+{"id":"definition : Kernel","label":"Kernel","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Homomorphism","definition : Subgroup"],"nlp_text":"Let \\(\\phi : G \\rightarrow G'\\) be a group homomorphism. The kernel of \\(\\phi \\), denoted \\(\\text{ker}(\\phi )\\), is defined as \\[ \\left\\{ x \\in G \\middle | \\phi (x) = \\mathbb {e} \\right\\} . \\]","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Kernel","latex_link":"sect0003.html#definition : Kernel","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Kernel","lean_decl_name":"AlgebraInLean.Kernel","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter03/Sheet05.lean","lean4_code":"def Kernel (φ : G → G') (h : Homomorphism φ) : Subgroup G where\n  carrier := {g | φ g = 𝕖}\n  -- EXERCISES\n  has_id := by\n    suffices : φ 𝕖 = 𝕖\n    · exact this\n    exact hom_id_to_id φ h\n  mul_closure := by\n    intro a b ha hb\n    rw [Set.mem_setOf_eq, ←h, ha, hb, id_op]\n  inv_closure := by\n    intro a ha\n    rw [Set.mem_setOf_eq, hom_inv_to_inv φ, ha, inv_id]\n    exact h\n","line":15,"end_line":28,"proof_status":"formalized"}
+{"id":"theorem : Kernel_normal","label":"Kernel_normal","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Kernel","definition : normal"],"nlp_text":"If \\(\\phi \\) is a group homomorphism with domain \\(G\\), then \\(ker(\\phi )\\) is a normal subgroup of \\(G\\).","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Kernel_normal","latex_link":"sect0003.html#theorem : Kernel_normal","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Kernel_normal","lean_decl_name":"AlgebraInLean.Kernel_normal","kind_description":"theorems and lemmas","status_from_color":"statement_formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter03/Sheet05.lean","lean4_code":"theorem Kernel_normal (φ : G → G') (h : Homomorphism φ) : normal (Kernel φ h) := by\n  -- EXERCISE\n  intro g k hk\n  suffices : φ (conjugate g k) = 𝕖\n  · exact this\n  rw [conjugate, ←h, ←h, hk, op_id, h, op_inv, hom_id_to_id φ h]\n","line":110,"end_line":115,"proof_status":"statement_formalized"}
+{"id":"theorem : homomorphism_inj_iff_kernel_trivial","label":"homomorphism_inj_iff_kernel_trivial","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Kernel"],"nlp_text":"If \\(\\phi : G \\rightarrow G'\\) is a group homomorphism, then it is injective if and only if \\(\\text{ker}(\\phi )\\) is trivial, meaning \\(\\text{ker}(\\phi ) = \\{ \\mathbb {e}\\} \\).","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.homomorphism_inj_iff_kernel_trivial","latex_link":"sect0003.html#theorem : homomorphism_inj_iff_kernel_trivial","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.homomorphism_inj_iff_kernel_trivial","lean_decl_name":"AlgebraInLean.homomorphism_inj_iff_kernel_trivial","kind_description":"theorems and lemmas","status_from_color":"statement_formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter03/Sheet05.lean","lean4_code":"theorem homomorphism_inj_iff_kernel_trivial (φ : G → G') (h : Homomorphism φ) :\n    Function.Injective φ ↔ Kernel φ h = Minimal G := by\n  -- EXERCISE\n  apply Iff.intro\n  · intro hinj\n    apply le_antisymm\n    · intro x hx\n      change x = 𝕖\n      apply hinj\n      rw [hom_id_to_id φ h]\n      exact hx\n    · apply Minimal_smallest\n  · intro hk x y hfeq\n    have h1 : φ (μ x (ι y)) = μ (φ x) (φ (ι y)) := by\n      rw [h]\n    have h2 : (φ (ι y)) = ι (φ y):= by\n      apply hom_inv_to_inv\n      exact h\n    rw [hfeq, h2, op_inv] at h1\n    have h3 : μ x (ι y) ∈ Kernel φ h := by\n      trivial\n    rw [hk] at h3\n    have h4 : μ x (ι y) = 𝕖 := by\n      trivial\n    have h5 : μ x (ι y) = 𝕖 → μ (μ x (ι y)) y = μ 𝕖 y := by\n      intro ht\n      rw [ht]\n    apply h5 at h4\n    rw [id_op, op_assoc, inv_op, op_id] at h4\n    exact h4\n","line":196,"end_line":225,"proof_status":"statement_formalized"}
+{"id":"definition : op_left_cancel","label":"op_left_cancel","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Group"],"nlp_text":"A group has the left cancellation property \\(\\forall a,~ b,~ c \\in G, ~ \\mu (a,~ b) = \\mu (a,~ c) \\Rightarrow b = c\\)","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.op_left_cancel","latex_link":"sect0001.html#definition : op_left_cancel","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.op_left_cancel","lean_decl_name":"AlgebraInLean.op_left_cancel","kind_description":"theorems and lemmas","status_from_color":"statement_formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter01/Sheet02.lean","lean4_code":"theorem op_left_cancel [Group α] (a b c : α) (h : μ a b = μ a c) : b = c := by\n\n  -- sorry\n  -- SAMPLE SOLUTION\n  rw [←id_op b, ←id_op c, ←inv_op a, op_assoc, op_assoc, h]\n","line":97,"end_line":101,"proof_status":"statement_formalized"}
+{"id":"definition : Generate","label":"Generate","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Subgroup"],"nlp_text":"The subgroup of \\(G\\) generated by a subset \\(S \\subset G\\) is defined to be the smallest subgroup containing all of \\(S\\). Notated \\(\\langle S \\rangle \\), it is written as \\begin{equation*} \\langle S \\rangle := \\bigcap _{\\genfrac {}{}{0pt}{}{G \\le H}{S \\subset H}} H \\end{equation*}","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Generate","latex_link":"sect0003.html#definition : Generate","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Generate","lean_decl_name":"AlgebraInLean.Generate","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter03/Sheet04.lean","lean4_code":"def Generate (S : Set G) : Subgroup G where\n  carrier := {g : G | ∀ H : Subgroup G, S ⊆ H → g ∈ H}\n  -- EXERCISE\n  has_id := by\n    intro H _\n    exact H.has_id\n  mul_closure := by\n    dsimp at *\n    intro a b ha hb H hH\n    apply H.mul_closure\n    · exact ha H hH\n    · exact hb H hH\n  inv_closure := by\n    intro a ha H hH\n    apply H.inv_closure\n    exact ha H hH\n","line":150,"end_line":165,"proof_status":"formalized"}
+{"id":"theorem : hom_id_to_id","label":"hom_id_to_id","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Homomorphism"],"nlp_text":"Suppose \\( \\phi : G \\rightarrow H \\) is a homomorphism. Then \\( \\phi (\\mathbb e) = \\mathbb e \\).","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.hom_id_to_id","latex_link":"sect0002.html#theorem : hom_id_to_id","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.hom_id_to_id","lean_decl_name":"AlgebraInLean.hom_id_to_id","kind_description":"theorems and lemmas","status_from_color":"statement_formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter02/Sheet01.lean","lean4_code":"theorem hom_id_to_id (φ : G → H) (hp : Homomorphism φ) : φ 𝕖 = 𝕖 := by\n  -- SAMPLE SOLUTION\n  have h₁ : φ (μ 𝕖 𝕖) = μ (φ 𝕖) (φ 𝕖) := by\n    rw [Homomorphism_def] at hp\n    specialize hp 𝕖 𝕖\n    exact hp.symm\n  have h₂ : μ (φ 𝕖) 𝕖 = μ (φ 𝕖) (φ 𝕖) := by\n    rw [op_id]\n    nth_rewrite 1 [← op_id 𝕖]\n    exact h₁\n  have h₃ : 𝕖 = φ 𝕖 := by\n    rw [mul_left_eq (φ 𝕖) 𝕖 (φ 𝕖)]\n    exact h₂\n  exact h₃.symm\n","line":176,"end_line":189,"proof_status":"statement_formalized"}
+{"id":"theorem : D₃_nonabelian","label":"D₃_nonabelian","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Dihedral"],"nlp_text":"\\(D_3\\) is the first example we’ve seen of a nonabelian group.","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.D₃_nonabelian","latex_link":"sect0001.html#theorem : D₃_nonabelian","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.D₃_nonabelian","lean_decl_name":"AlgebraInLean.D₃_nonabelian","kind_description":"theorems and lemmas","status_from_color":"statement_formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter01/Sheet06.lean","lean4_code":"theorem D₃_nonabelian : ¬(∀ (a b : Dihedral 3), μ a b = μ b a) := by\n  -- sorry\n  -- SAMPLE SOLUTION\n  intro h\n  let a : Dihedral 3 := .rotation 1\n  let b : Dihedral 3 := .reflection 0\n  specialize h a b\n  simp [μ, Magma.op, Dihedral.op, a, b, ι, Group.inv, Fin.add] at h\n  done\n","line":90,"end_line":98,"proof_status":"statement_formalized"}
+{"id":"definition : Injective","label":"Injective","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Given a function \\( f \\) mapping from a set \\( X \\) to a set \\( Y \\): \\[ \\forall x,~ x' \\in X,~ f(x) = f(x') \\implies x = x'. \\]","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Injective","latex_link":"sect0002.html#definition : Injective","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Injective","lean_decl_name":"AlgebraInLean.Injective","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter02/Sheet00.lean","lean4_code":"def Injective (f : α → β) : Prop := ∀ (x y : α), f x = f y → x = y\n","line":35,"end_line":35,"proof_status":"formalized"}
+{"id":"definition : Bijective","label":"Bijective","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Injective","definition : Surjective"],"nlp_text":"A bijective function is one that is both injective and surjective.","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Bijective","latex_link":"sect0002.html#definition : Bijective","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Bijective","lean_decl_name":"AlgebraInLean.Bijective","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter02/Sheet00.lean","lean4_code":"def Bijective (f : α → β) : Prop := (Injective f ∧ Surjective f)\n","line":41,"end_line":41,"proof_status":"formalized"}
+{"id":"definition : AbelianGroup","label":"AbelianGroup","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Group","definition : CommMonoid"],"nlp_text":"An abelian group is a group with the commutative property: \\(\\forall a,~ b \\in G, ~ \\mu (a,~ b) = \\mu (b,~ a)\\)","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.AbelianGroup","latex_link":"sect0001.html#definition : AbelianGroup","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.AbelianGroup","lean_decl_name":"AlgebraInLean.AbelianGroup","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"./AlgebraInLean/Chapter01/Sheet05.lean","lean4_code":"instance (n : ℕ) [NeZero n] : AbelianGroup (Fin n) where\n  /-\n  Fin.add adds the two underlying integers and then considers its remainder under division by n\n  (i.e., addition modulo n). Since n itself is equivalent to a full rotation, this means that going\n  over a full rotation wraps around to being less than 360°, like in the C₃ case.\n  -/\n  op := Fin.add\n\n  -- Mathlib already has all of these proven, you just need to figure out what they are called.\n  op_assoc := add_assoc\n\n  id := 0\n\n  op_id := by\n    -- sorry\n    -- SAMPLE SOLUTION\n    exact add_zero\n    -- END OF SAMPLE SOLUTION\n\n  id_op := by\n    -- sorry\n    -- SAMPLE SOLUTION\n    exact zero_add\n    -- END OF SAMPLE SOLUTION\n\n  inv := Neg.neg\n\n  inv_op := by\n    -- sorry\n    -- SAMPLE SOLUTION\n    exact neg_add_self\n    -- END OF SAMPLE SOLUTION\n\n  op_comm := by\n    -- sorry\n    -- SAMPLE SOLUTION\n    exact add_comm\n    -- END OF SAMPLE SOLUTION\n","line":16,"end_line":53,"proof_status":"formalized"}
+{"id":"definition : Dihedral","label":"Dihedral","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Group"],"nlp_text":"\\(D_n\\) is the dihedral group of order \\(2n\\). This is the group of all symmetries of a regular n-gon. This includes n rotations and n reflections. It is represented by a pair of a boolean and an element of Fin n. The boolean represents whether the element is a rotation or not, and the element of Fin n represents how much the first vertex is rotated. The identity is a \\(0^{\\circ }\\) rotation, the inverse of an \\(x^{\\circ }\\) rotation is a \\((360 - x)^{\\circ }\\) rotation, and any reflection is its own inverse.","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Dihedral","latex_link":"sect0001.html#definition : Dihedral","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Dihedral","lean_decl_name":"AlgebraInLean.Dihedral","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter01/Sheet06.lean","lean4_code":"inductive Dihedral (n : ℕ) [NeZero n] : Type\n| rotation : Fin n → Dihedral n\n| reflection : Fin n → Dihedral n\n","line":25,"end_line":27,"proof_status":"formalized"}
+{"id":"definition : Homomorphism","label":"Homomorphism","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Group"],"nlp_text":"A homomorphism is a map \\( \\phi \\) from a group \\( G \\) to a group \\( H \\), for elements \\( g,~ h \\in G \\), \\( \\phi \\) is a homomorphism if and only if: \\[ \\mu (\\phi (g),\\phi (h)) = \\phi (\\mu (g,h)) \\]","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Homomorphism","latex_link":"sect0002.html#definition : Homomorphism","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Homomorphism","lean_decl_name":"AlgebraInLean.Homomorphism","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter02/Sheet01.lean","lean4_code":"def Homomorphism (φ : G → H) : Prop := ∀ a b : G, μ (φ a) (φ b) = φ (μ a b)\n","line":158,"end_line":158,"proof_status":"formalized"}
+{"id":"theorem : inv_unique","label":"inv_unique","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Group"],"nlp_text":"In a group, each element has a unique inverse: \\(\\forall a,~ b \\in G,~ (\\mu (a,~ b) = \\mu (b,~ a) = \\mathbb {e}) \\Rightarrow b = \\iota (a)\\)","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.inv_unique","latex_link":"sect0001.html#theorem : inv_unique","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.inv_unique","lean_decl_name":"AlgebraInLean.inv_unique","kind_description":"theorems and lemmas","status_from_color":"statement_formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter01/Sheet03.lean","lean4_code":"theorem inv_unique [Group α] (a i : α) (h : μ a i = 𝕖 ∧ μ i a = 𝕖) : i = ι a := by\n  -- sorry\n  -- SAMPLE SOLUTION\n  obtain ⟨h, _⟩ := h\n  rw [←op_id (ι a), ←h, ←op_assoc (ι a), inv_op, id_op]\n","line":37,"end_line":41,"proof_status":"statement_formalized"}
+{"id":"definition : Subgroup","label":"Subgroup","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Group"],"nlp_text":"Let \\(G\\) be a group. A subgroup \\(H\\) of \\(G\\) is a subset that is itself a group under the group operation of \\(G\\). This requires that the subgroup satisfies three operations. The identity of \\(G\\) is contained within \\(H\\). \\(H\\) must be closed under \\(G\\)’s group operation: for any \\(a,b \\in H\\), \\(\\mu (a,b) \\in H\\). \\(H\\) must be closed under \\(G\\)’s inversion operation: for any \\(a \\in H\\), \\(\\iota (a) \\in H\\).","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Subgroup","latex_link":"sect0003.html#definition : Subgroup","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Subgroup","lean_decl_name":"AlgebraInLean.Subgroup","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter03/Sheet01.lean","lean4_code":"structure Subgroup (G : Type*) [Group G] where\n  /--\n  We represent the subgroup's underlying set using Mathlib's `Set` type. Upon viewing the Mathlib\n  documentation for the set (if you are viewing this file in Visual Studio Code, you may Ctrl-Click\n  on the keyword to consult its definiton), we see that it is merely a wrapper for `G → Prop`,\n  meaning it is a function that determines what is and is not in the set.\n  -/\n  carrier : Set G\n  /--\n  This proposition asserts that the subgroup contains the identity of G. This also asserts that the\n  subgroup is not the empty set.\n  -/\n  has_id : 𝕖 ∈ carrier\n  /--\n  The below propositions assert that the subgroup is closed under the group operation μ and the\n  inverse function ι.\n  -/\n  mul_closure : ∀ {a b : G}, a ∈ carrier → b ∈ carrier → μ a b ∈ carrier\n  inv_closure : ∀ {a : G}, a ∈ carrier → ι a ∈ carrier\n","line":19,"end_line":37,"proof_status":"formalized"}
+{"id":"theorem : inv_id","label":"inv_id","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["theorem : inv_unique"],"nlp_text":"The inverse of the identity is itself: \\(\\iota (\\mathbb {e}) = \\mathbb {e}\\)","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.inv_id","latex_link":"sect0001.html#theorem : inv_id","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.inv_id","lean_decl_name":"AlgebraInLean.inv_id","kind_description":"theorems and lemmas","status_from_color":"statement_formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter01/Sheet03.lean","lean4_code":"theorem inv_id [Group α] : ι 𝕖 = (𝕖 : α) := by\n  -- sorry\n  -- SAMPLE SOLUTION\n  symm\n  apply inv_unique\n  constructor\n  · exact op_id 𝕖\n  · exact op_id 𝕖\n","line":112,"end_line":119,"proof_status":"statement_formalized"}
+{"id":"definition : normal","label":"normal","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":["definition : Subgroup"],"nlp_text":"A subgroup \\(H\\) of a group \\(G\\) is normal if a subgroup is closed under conjugation of any element in \\(G\\). In other terms, for any \\(g \\in G\\) and \\(h \\in H\\), \\(ghg^{-1} \\in H\\).","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.normal","latex_link":"sect0003.html#definition : normal","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.normal","lean_decl_name":"AlgebraInLean.normal","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter03/Sheet05.lean","lean4_code":"def normal (H : Subgroup G) : Prop :=\n  ∀ g h : G, h ∈ H → conjugate g h ∈ H\n","line":95,"end_line":96,"proof_status":"formalized"}
+{"id":"definition : Surjective","label":"Surjective","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://mao3j1m0op.github.io/algebra-in-lean/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Given a function \\( f \\) mapping from a set \\( X \\) to a set \\( Y \\): \\[ \\forall y \\in Y,~ \\exists x \\in X,~ f(x) = y \\]","doc_url":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Surjective","latex_link":"sect0002.html#definition : Surjective","lean_link":"https://MAO3J1m0Op.github.io/algebra-in-lean/docs/find/#doc/AlgebraInLean.Surjective","lean_decl_name":"AlgebraInLean.Surjective","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"algebra-in-lean","source_path":"AlgebraInLean/Chapter02/Sheet00.lean","lean4_code":"def Surjective (f : α → β) : Prop := ∀ (y : β), ∃ (x : α), f x = y\n","line":38,"end_line":38,"proof_status":"formalized"}

crawled_repos/amatelus.jsonl

@@ -0,0 +1,27 @@
+{"id":"def:dep-chapter","label":"dep-chapter","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://amatelus.com/blueprint/dep_graph_document.html","dependencies":["def:did-amt"],"nlp_text":"This chapter covers deployment models aspects of AMATELUS.","doc_url":"https://amatelus.com/docs/find/#doc/AMATELUS.Operations","latex_link":"sect0004.html#def:dep-chapter","lean_link":"https://amatelus.com/docs/find/#doc/AMATELUS.Operations","lean_decl_name":"AMATELUS.Operations","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/amatelus/amatelus/cea74f0a4b1769c90f05dc65e70d1322d368cd86/AMATELUS/Operations.lean","lean4_code":"/-\n# Wallet/Holder/Issuer/Verifier 操作定義\n\nこのファイルは、Wallet、Holder、Issuer、Verifierの具体的な操作を定義します。\n-/\n\nimport AMATELUS.DID\nimport AMATELUS.VC\nimport AMATELUS.ZKP\nimport AMATELUS.Roles\nimport AMATELUS.SecurityAssumptions\nimport AMATELUS.Cryptographic\nimport AMATELUS.TrustChain\n\n-- ## Holder操作\n\nnamespace Holder\n\n/-- WalletにVCを保存\n\n    **設計変更:**\n    Holder構造体は単なるラッパーなので、操作はWalletレベルで行います。\n    不変条件 `wallet_valid` はHolder構築時に一度だけ保証されます。\n\n    この関数は単にcredentialsリストにVCを追加するだけで、\n    identitiesは変更しません。したがって、Wallet.isValidは保持されます。\n-/\ndef storeCredential\n    (wallet : Wallet)\n    (vc : ValidVC)","line":1,"end_line":30,"proof_status":"formalized"}
+{"id":"def:did-amt","label":"did-amt","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://amatelus.com/blueprint/dep_graph_document.html","dependencies":["def:amatelus"],"nlp_text":"The did:amt method is a Decentralized Identifier (DID) that is algorithmically generated and resolved without reliance on any external Verifiable Data Registry (VDR) such as a blockchain. This method is designed for high-stakes environments (public administration, government) where data integrity and operational robustness are paramount.","doc_url":"https://amatelus.com/docs/find/#doc/AMATELUS.DID","latex_link":"sect0003.html#def:did-amt","lean_link":"https://amatelus.com/docs/find/#doc/AMATELUS.DID","lean_decl_name":"AMATELUS.DID","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/amatelus/amatelus/cea74f0a4b1769c90f05dc65e70d1322d368cd86/AMATELUS/DID.lean","lean4_code":"/-\n# DID 定義\n\nこのファイルは、AMATELUSプロトコルのDID関連の型と定義を含みます。\nW3C DID仕様を基底として、AMATELUS固有の拡張を提供します。\n-/\n\nimport AMATELUS.CryptoTypes\nimport AMATELUS.SecurityAssumptions\nimport W3C.DID\n\n-- ## 基本型定義\n\n-- ## Definition 2.1: DID and DID Document (W3C DID仕様に準拠)\n\n/-- W3C.DIDDocumentから主要な公開鍵を抽出するヘルパー関数\n\n    verificationMethodの最初のエントリから公開鍵を取得します。\n    AMATELUSでは、did:amt仕様に従い、verificationMethodに必ず公開鍵が含まれます。\n-/\ndef extractPublicKey (doc : W3C.DIDDocument) : Option PublicKey :=\n  match doc.verificationMethod.head? with\n  | none => none\n  | some vm =>\n      match vm.publicKeyMultibase with\n      | none => none\n      | some pkStr =>\n          -- multibase文字列からバイト列に変換（簡略化）\n          some { bytes := pkStr.toUTF8.data.toList }\n","line":1,"end_line":30,"proof_status":"formalized"}
+{"id":"thm:did-local-resolution","label":"did-local-resolution","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://amatelus.com/blueprint/dep_graph_document.html","dependencies":["def:did-amt","def:crypto-assumptions"],"nlp_text":"The resolution of a did:amt is completed locally by a verifier. No external Verifiable Data Registry (blockchain, centralized service) or network calls are required. Verifier receives the [AMT Version Number, Public Key] pair from the owner and executes the same local derivation steps for verification.","doc_url":"https://amatelus.com/docs/find/#doc/AMATELUS.DID","latex_link":"sect0003.html#thm:did-local-resolution","lean_link":"https://amatelus.com/docs/find/#doc/AMATELUS.DID","lean_decl_name":"AMATELUS.DID","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/amatelus/amatelus/cea74f0a4b1769c90f05dc65e70d1322d368cd86/AMATELUS/DID.lean","lean4_code":"/-\n# DID 定義\n\nこのファイルは、AMATELUSプロトコルのDID関連の型と定義を含みます。\nW3C DID仕様を基��として、AMATELUS固有の拡張を提供します。\n-/\n\nimport AMATELUS.CryptoTypes\nimport AMATELUS.SecurityAssumptions\nimport W3C.DID\n\n-- ## 基本型定義\n\n-- ## Definition 2.1: DID and DID Document (W3C DID仕様に準拠)\n\n/-- W3C.DIDDocumentから主要な公開鍵を抽出するヘルパー関数\n\n    verificationMethodの最初のエントリから公開鍵を取得します。\n    AMATELUSでは、did:amt仕様に従い、verificationMethodに必ず公開鍵が含まれます。\n-/\ndef extractPublicKey (doc : W3C.DIDDocument) : Option PublicKey :=\n  match doc.verificationMethod.head? with\n  | none => none\n  | some vm =>\n      match vm.publicKeyMultibase with\n      | none => none\n      | some pkStr =>\n          -- multibase文字列からバイト列に変換（簡略化）\n          some { bytes := pkStr.toUTF8.data.toList }\n","line":1,"end_line":30,"proof_status":"formalized"}
+{"id":"def:tech-chapter","label":"tech-chapter","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://amatelus.com/blueprint/dep_graph_document.html","dependencies":["def:did-amt"],"nlp_text":"This chapter covers technical specifications aspects of AMATELUS.","doc_url":"https://amatelus.com/docs/find/#doc/AMATELUS.Protocol","latex_link":"sect0005.html#def:tech-chapter","lean_link":"https://amatelus.com/docs/find/#doc/AMATELUS.Protocol","lean_decl_name":"AMATELUS.Protocol","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"thm:did-local-generation","label":"did-local-generation","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://amatelus.com/blueprint/dep_graph_document.html","dependencies":["def:did-amt","def:crypto-assumptions"],"nlp_text":"A did:amt identifier is generated locally on the owner’s device without network registration. The generation process consists of: Generate Ed25519 key pair Prepare information pair: AMT Version Number + Public Key Select DID Document template corresponding to AMT version Derive DID through local cryptographic operations","doc_url":"https://amatelus.com/docs/find/#doc/AMATELUS.DID","latex_link":"sect0003.html#thm:did-local-generation","lean_link":"https://amatelus.com/docs/find/#doc/AMATELUS.DID","lean_decl_name":"AMATELUS.DID","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/amatelus/amatelus/cea74f0a4b1769c90f05dc65e70d1322d368cd86/AMATELUS/DID.lean","lean4_code":"/-\n# DID 定義\n\nこのファイルは、AMATELUSプロトコルのDID関連の型と定義を含みます。\nW3C DID仕様を基底として、AMATELUS固有の拡張を提供します。\n-/\n\nimport AMATELUS.CryptoTypes\nimport AMATELUS.SecurityAssumptions\nimport W3C.DID\n\n-- ## 基本型定義\n\n-- ## Definition 2.1: DID and DID Document (W3C DID仕様に準拠)\n\n/-- W3C.DIDDocumentから主要な公開鍵を抽出するヘルパー関数\n\n    verificationMethodの最初のエントリから公開鍵を取得します。\n    AMATELUSでは、did:amt仕様に従い、verificationMethodに必ず公開鍵が含まれます。\n-/\ndef extractPublicKey (doc : W3C.DIDDocument) : Option PublicKey :=\n  match doc.verificationMethod.head? with\n  | none => none\n  | some vm =>\n      match vm.publicKeyMultibase with\n      | none => none\n      | some pkStr =>\n          -- multibase文字列からバイト列に変換（簡略化）\n          some { bytes := pkStr.toUTF8.data.toList }\n","line":1,"end_line":30,"proof_status":"formalized"}
+{"id":"def:priv-chapter","label":"priv-chapter","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://amatelus.com/blueprint/dep_graph_document.html","dependencies":["thm:did-local-generation"],"nlp_text":"This chapter covers privacy architecture aspects of AMATELUS.","doc_url":"https://amatelus.com/docs/find/#doc/AMATELUS.Privacy","latex_link":"sect0008.html#def:priv-chapter","lean_link":"https://amatelus.com/docs/find/#doc/AMATELUS.Privacy","lean_decl_name":"AMATELUS.Privacy","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:multi-chapter","label":"multi-chapter","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://amatelus.com/blueprint/dep_graph_document.html","dependencies":["thm:did-local-generation"],"nlp_text":"This chapter covers multi-device support aspects of AMATELUS.","doc_url":"https://amatelus.com/docs/find/#doc/AMATELUS.MultiDevice","latex_link":"sect0012.html#def:multi-chapter","lean_link":"https://amatelus.com/docs/find/#doc/AMATELUS.MultiDevice","lean_decl_name":"AMATELUS.MultiDevice","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/amatelus/amatelus/cea74f0a4b1769c90f05dc65e70d1322d368cd86/AMATELUS/MultiDevice.lean","lean4_code":"/-\n# マルチデバイス対応（Multi-Device Support）- クレーム個別転送版\n\nこのファイルは、AMATELUSプロトコルのマルチデバイス対応を定義します。\n単一のHolderが複数のデバイス上で異なるDIDを持つウォレットを運用し、\nデバイス間で**クレーム単位**で安全に転送する機能を提供します。\n\n**設計原則:**\n1. 秘密鍵の非共有（デバイス間で秘密鍵は転送しない）\n2. クレーム単位の転送（VCではなくクレームを転送してZKP効率化）\n3. 二重署名（issuer署名 + original subject転送署名）\n4. W3C標準準拠（holder ≠ credentialSubject をサポート）\n5. DIDComm使用（標準的なDID間通信プロトコル）\n6. エンドツーエンド暗号化（転送データは常に暗号化）\n\n**主要な概念:**\n- トラストチェーン: Schema継承、権限委譲（異なる組織のDID）\n- マルチデバイス: クレームの共有（同一Holderの異なるDID）\n\n**クレームの転送経路:**\n1. Issuer → Holder_A (初回発行)\n2. Holder_A → Holder_B (クレーム単位の転送) ← 本モジュールの対象\n\n**ZKP効率化:**\n- 必要なクレームのみをZKP回路に入力\n- 回路サイズ最小化、計算コスト削減\n- プライバシー保護（不要なクレームを扱わない）\n-/\n\nimport AMATELUS.DID","line":1,"end_line":30,"proof_status":"formalized"}
+{"id":"def:chain-chapter","label":"chain-chapter","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://amatelus.com/blueprint/dep_graph_document.html","dependencies":["thm:did-local-generation"],"nlp_text":"This chapter covers trust chain architecture aspects of AMATELUS.","doc_url":"https://amatelus.com/docs/find/#doc/AMATELUS.TrustChain","latex_link":"sect0011.html#def:chain-chapter","lean_link":"https://amatelus.com/docs/find/#doc/AMATELUS.TrustChain","lean_decl_name":"AMATELUS.TrustChain","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/amatelus/amatelus/cea74f0a4b1769c90f05dc65e70d1322d368cd86/AMATELUS/TrustChain.lean","lean4_code":"/-- VCの階層深度を取得（N階層対応）\n\n    DelegationChainから階層深度を取得：\n    - `None`: 0階層（直接発行）\n    - `Some chain`: chain.depth階層（N階層委譲発行）\n\n    **設計の利点:**\n    - 動的な階層数をサポート\n    - chain.depthはdelegationsリストの長さ\n    - O(1)の計算量（リスト長の取得のみ）\n-/\ndef getVCDepthN (vc : UnknownVC) : Nat :=\n  match UnknownVC.getDelegationChain vc with\n  | none => 0  -- 直接発行\n  | some chain => chain.depth  -- N階層委譲発行","line":23,"end_line":37,"proof_status":"formalized"}
+{"id":"def:trust-chapter","label":"trust-chapter","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://amatelus.com/blueprint/dep_graph_document.html","dependencies":["thm:did-local-generation"],"nlp_text":"This chapter covers trust architecture aspects of AMATELUS.","doc_url":"https://amatelus.com/docs/find/#doc/AMATELUS.MutualAuthentication","latex_link":"sect0006.html#def:trust-chapter","lean_link":"https://amatelus.com/docs/find/#doc/AMATELUS.MutualAuthentication","lean_decl_name":"AMATELUS.MutualAuthentication","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:crypto-assumptions","label":"crypto-assumptions","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://amatelus.com/blueprint/dep_graph_document.html","dependencies":["def:amatelus"],"nlp_text":"AMATELUS relies on the following cryptographic security assumptions: Collision-resistant hash: SHA3-512 provides 128-bit security against quantum adversaries Unforgeable signatures: Dilithium2 provides 128-bit security against quantum adversaries ZKP soundness: Standard zero-knowledge properties (completeness, soundness, zero-knowledge)","doc_url":"https://amatelus.com/docs/find/#doc/AMATELUS.Cryptographic","latex_link":"sect0007.html#def:crypto-assumptions","lean_link":"https://amatelus.com/docs/find/#doc/AMATELUS.Cryptographic","lean_decl_name":"AMATELUS.Cryptographic","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:revoke-chapter","label":"revoke-chapter","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://amatelus.com/blueprint/dep_graph_document.html","dependencies":["def:crypto-assumptions"],"nlp_text":"This chapter covers merkle tree revocation aspects of AMATELUS.","doc_url":"https://amatelus.com/docs/find/#doc/AMATELUS.RevocationMerkle","latex_link":"sect0013.html#def:revoke-chapter","lean_link":"https://amatelus.com/docs/find/#doc/AMATELUS.RevocationMerkle","lean_decl_name":"AMATELUS.RevocationMerkle","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/amatelus/amatelus/cea74f0a4b1769c90f05dc65e70d1322d368cd86/AMATELUS/RevocationMerkle.lean","lean4_code":"/-\n# Merkle Tree Revocation Specification for AMATELUS\n\nこのファイルは、AMATELUSプロトコルのMerkle Treeベースの失効確認フローを形式化します。\n\n**設計原則:**\n1. Merkle Treeの抽象化（暗号技術と同等に確立された技術）\n2. タイムスタンプ検証はVerifier側で実行（ZKP回路外）\n3. ゼロ知識性の保持（どのVCか特定されない）\n4. 型安全性によるプロトコル保証\n5. revocationEnabledフラグによる個人Issuer対応\n\n**主要な概念:**\n- ValidMerkleProof: 暗号学的に正しい包含証明\n- InvalidMerkleProof: 不正な包含証明\n- MerkleRevocationList: Issuerが管理する失効情報\n- ZKPWithRevocation: 失効確認付きZKP\n- revocationEnabled: 失効確認の有効化フラグ（Issuer署名で保護）\n\n**revocationEnabledフラグの重要性:**\n- Issuerがクレームごとに失効確認の可否を含める\n- HolderはZKPに失効確認フラグを入力\n- Verifierは数学的に失効確認の有無を判定可能\n- 個人IssuerはウェブサーバーなしでVC発行可能（revocationEnabled = false）\n\n**参考文献:**\n- RFC 6962: Certificate Transparency\n- RevocationMerkle.md: AMATELUS失効確認仕様\n-/\n","line":1,"end_line":30,"proof_status":"formalized"}
+{"id":"thm:impersonation-resistance","label":"impersonation-resistance","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://amatelus.com/blueprint/dep_graph_document.html","dependencies":["def:crypto-assumptions"],"nlp_text":"Impersonation attacks with different secret keys are cryptographically prevented. If an attacker uses a different secret key to forge a ZKP, the signature verification will fail because DIDComm makes the sender’s public key known to the recipient.","doc_url":"https://amatelus.com/docs/find/#doc/AMATELUS.Cryptographic","latex_link":"sect0007.html#thm:impersonation-resistance","lean_link":"https://amatelus.com/docs/find/#doc/AMATELUS.Cryptographic","lean_decl_name":"AMATELUS.Cryptographic","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/amatelus/amatelus/cea74f0a4b1769c90f05dc65e70d1322d368cd86/AMATELUS/Cryptographic.lean","lean4_code":"/-- Theorem 3.1: DID衝突の量子安全性\n\n    異なるValidDIDDocumentから同じDIDが生成されることは、\n    量子計算機を用いても困難です。\n\n    **元の主張（決定論的単射性）の問題点:**\n    \"異なるDIDドキュメントは異なるDIDを生成する\"は数学的に偽である。\n    なぜなら、SHA-3のような有限出力ハッシュ関数は鳩の巣原理により\n    必ず衝突を持つからである（無限入力空間 → 2^512出力空間）。\n\n    **正しい主張（確率的単射性＋量子安全性）:**\n    衝突は存在するが、量子計算機を用いても衝突を発見することは困難である。\n\n    **量子脅威下での安全性:**\n    - 衝突探索の量子コスト: 128ビット（Grover適用後）\n    - NIST最小要件: 128ビット\n    - 結論: 安全（128 ≥ 128）\n\n    **証明:**\n    SecurityAssumptions.amtHashFunction.quantum_secureにより、\n    SHA3-512の衝突探索の量子コストは128ビットであり、\n    NIST最小要件128ビットを満たす。\n    DID.fromValidDocument は内部でこのハッシュ関数を使用しているため、\n    異なるValidDIDDocumentから同じDIDが生成される確率は無視できるほど小さい。\n-/\ntheorem did_collision_quantum_secure :\n  amtHashFunction.collisionSecurity.quantumBits ≥ minSecurityLevel.quantumBits := by\n  -- 128 ≥ 128\n  exact amtHashFunction.quantum_secure","line":14,"end_line":42,"proof_status":"formalized"}
+{"id":"def:key-chapter","label":"key-chapter","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://amatelus.com/blueprint/dep_graph_document.html","dependencies":["def:crypto-assumptions"],"nlp_text":"This chapter covers secret key lifecycle management aspects of AMATELUS.","doc_url":"https://amatelus.com/docs/find/#doc/AMATELUS.KeyManagement","latex_link":"sect0015.html#def:key-chapter","lean_link":"https://amatelus.com/docs/find/#doc/AMATELUS.KeyManagement","lean_decl_name":"AMATELUS.KeyManagement","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/amatelus/amatelus/cea74f0a4b1769c90f05dc65e70d1322d368cd86/AMATELUS/KeyManagement.lean","lean4_code":"/-\n# AMATELUS Key Management Lifecycle Specification\n\nこのファイルは、AMATELUSプロトコルにおける秘密鍵のライフサイクル管理を形式化します。\n\n**設計原則:**\n1. 決定性の原則: 秘密鍵が手元にある場合のみ、暗号学的に決定的な操作が可能\n2. 連絡可能性の限界: Holderが連絡できたIssuerのみ対応可能\n3. 数学的安全性の限界: 身分証明と紐づいていないDIDの紛失は復旧不可能\n4. 本人死亡時の特性: 秘密鍵が使用不可能なため、損害は極めて限定的\n5. 被後見人保護: 後見人VCにより法的保護と技術的管理を両立\n\n**主要な概念:**\n- PrivateKeyState: 秘密鍵の状態（正常、漏洩、紛失、更新中、本人死亡、被後見人）\n- KeyManagementScenario: 対応シナリオの分類\n- RevocationRequest: 失効要求（旧秘密鍵で署名）\n- DIDMigrationCredential: DID移行証明VC（信頼継承）\n- GuardianCredential: 後見人VC（被後見人保護）\n\n**参考文献:**\n- KeyManagement.md: AMATELUS秘密鍵ライフサイクル管理仕様\n-/\n\nimport AMATELUS.DID\nimport AMATELUS.VC\nimport AMATELUS.ZKP\nimport AMATELUS.Cryptographic\nimport AMATELUS.RevocationMerkle\n\n-- ## Section 1: 秘密鍵状態の型定義","line":1,"end_line":30,"proof_status":"formalized"}
+{"id":"prop:mitm-defense","label":"mitm-defense","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://amatelus.com/blueprint/dep_graph_document.html","dependencies":["def:crypto-assumptions"],"nlp_text":"Man-in-the-Middle attacks are mitigated at the transport layer. While AMATELUS provides cryptographic identity verification, ECDH-1PU authenticated encryption and TLS/HTTPS are the responsibility of service providers.","doc_url":"https://amatelus.com/docs/find/#doc/AMATELUS.Cryptographic","latex_link":"sect0007.html#prop:mitm-defense","lean_link":"https://amatelus.com/docs/find/#doc/AMATELUS.Cryptographic","lean_decl_name":"AMATELUS.Cryptographic","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/amatelus/amatelus/cea74f0a4b1769c90f05dc65e70d1322d368cd86/AMATELUS/Cryptographic.lean","lean4_code":"/-- Theorem 3.1: DID衝突の量子安全性\n\n    異なるValidDIDDocumentから同じDIDが生成されることは、\n    量子計算機を用いても困難です。\n\n    **元の主張（決定論的単射性）の問題点:**\n    \"異なるDIDドキュメントは異なるDIDを生成する\"は数学的に偽である。\n    なぜなら、SHA-3のような有限出力ハッシュ関数は鳩の巣原理により\n    必ず衝突を持つからである（無限入力空間 → 2^512出力空間）。\n\n    **正しい主張（確率的単射性＋量子安全性）:**\n    衝突は存在するが、量子計算機を用いても衝突を発見することは困難である。\n\n    **量子脅威下での安全性:**\n    - 衝突探索の量子コスト: 128ビット（Grover適用後）\n    - NIST最小要件: 128ビット\n    - 結論: 安全（128 ≥ 128）\n\n    **証明:**\n    SecurityAssumptions.amtHashFunction.quantum_secureにより、\n    SHA3-512の衝突探索の量子コストは128ビットであり、\n    NIST最小要件128ビットを満たす。\n    DID.fromValidDocument は内部でこのハッシュ関数を使用しているため、\n    異なるValidDIDDocumentから同じDIDが生成される確率は無視できるほど小さい。\n-/\ntheorem did_collision_quantum_secure :\n  amtHashFunction.collisionSecurity.quantumBits ≥ minSecurityLevel.quantumBits := by\n  -- 128 ≥ 128\n  exact amtHashFunction.quantum_secure","line":14,"end_line":42,"proof_status":"statement_formalized"}
+{"id":"def:impl-chapter","label":"impl-chapter","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://amatelus.com/blueprint/dep_graph_document.html","dependencies":["def:crypto-assumptions"],"nlp_text":"This chapter covers implementation guidance aspects of AMATELUS.","doc_url":"https://amatelus.com/docs/find/#doc/AMATELUS.Operations","latex_link":"sect0009.html#def:impl-chapter","lean_link":"https://amatelus.com/docs/find/#doc/AMATELUS.Operations","lean_decl_name":"AMATELUS.Operations","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/amatelus/amatelus/cea74f0a4b1769c90f05dc65e70d1322d368cd86/AMATELUS/Operations.lean","lean4_code":"/-\n# Wallet/Holder/Issuer/Verifier 操作定義\n\nこのファイルは、Wallet、Holder、Issuer、Verifierの具体的な操作を定義します。\n-/\n\nimport AMATELUS.DID\nimport AMATELUS.VC\nimport AMATELUS.ZKP\nimport AMATELUS.Roles\nimport AMATELUS.SecurityAssumptions\nimport AMATELUS.Cryptographic\nimport AMATELUS.TrustChain\n\n-- ## Holder操作\n\nnamespace Holder\n\n/-- WalletにVCを保存\n\n    **設計変更:**\n    Holder構造体は単なるラッパーなので、操作はWalletレベルで行います。\n    不変条件 `wallet_valid` はHolder構築時に一度だけ保証されます。\n\n    この関数は単にcredentialsリストにVCを追加するだけで、\n    identitiesは変更しません。したがって、Wallet.isValidは保持されます。\n-/\ndef storeCredential\n    (wallet : Wallet)\n    (vc : ValidVC)","line":1,"end_line":30,"proof_status":"formalized"}
+{"id":"def:aud-chapter","label":"aud-chapter","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://amatelus.com/blueprint/dep_graph_document.html","dependencies":["def:crypto-assumptions"],"nlp_text":"This chapter covers audit mechanism: anonymous hash identifier (ahi) aspects of AMATELUS.","doc_url":"https://amatelus.com/docs/find/#doc/AMATELUS.Audit","latex_link":"sect0014.html#def:aud-chapter","lean_link":"https://amatelus.com/docs/find/#doc/AMATELUS.Audit","lean_decl_name":"AMATELUS.Audit","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/amatelus/amatelus/cea74f0a4b1769c90f05dc65e70d1322d368cd86/AMATELUS/Audit.lean","lean4_code":"/-\n# 監査メカニズムの制限性証明\n\nこのファイルは、匿名ハッシュ識別子による監査の制限性を証明します\n（Theorem 6.1, 6.2）。\n-/\n\nimport AMATELUS.DID\nimport AMATELUS.VC\nimport AMATELUS.SecurityAssumptions\n\n-- ## Theorem 6.1: Reverse Engineering Resistance (逆引き制限)\n\n/-- ハッシュの一方向性による保護（具体的なセキュリティレベル）\n\n    AMATELUSの逆引き耐性は、SHA3-512の原像攻撃コストに依存します。\n\n    **計算コスト（量子脅威下）:**\n    - 原像攻撃の量子コスト: 256ビット\n    - NIST最小要件: 128ビット\n    - 結論: 十分安全（256 ≥ 128）\n-/\ntheorem ahi_one_way_protection_quantum_secure :\n  hashPreimageSecurity.quantumBits ≥ minSecurityLevel.quantumBits := by\n  -- 256 ≥ 128\n  exact hash_preimage_quantum_secure\n\n/-- 監査区分識別子のエントロピーの指数（ビット数の指数表現）\n\n    audit_section_entropy = 2^8 = 256 ビット","line":1,"end_line":30,"proof_status":"formalized"}
+{"id":"prop:replay-application-layer","label":"replay-application-layer","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://amatelus.com/blueprint/dep_graph_document.html","dependencies":["def:crypto-assumptions"],"nlp_text":"Replay attack prevention (preventing legitimate users from reusing the same ZKP) is NOT a responsibility of AMATELUS protocol. This is an application-layer responsibility. Important distinction: Impersonation attacks (attacker with different secret key): PREVENTED by AMATELUS through DIDComm Replay attacks (legitimate user reusing same ZKP): NOT handled by AMATELUS. Applications requiring single-use semantics must implement nonce mechanisms. Applications that require ZKP single-use guarantees should implement nonce handling at the application layer: Generate unique session nonces for each verification Record and verify nonce freshness (checking that the nonce has not been used before) Maintain nonce history in application database Applications where ZKP reuse is acceptable (e.g., age verification, permission checks, one-time access grants) do not require additional replay prevention mechanisms.","doc_url":"https://amatelus.com/docs/find/#doc/AMATELUS.Cryptographic","latex_link":"sect0007.html#prop:replay-application-layer","lean_link":"https://amatelus.com/docs/find/#doc/AMATELUS.Cryptographic","lean_decl_name":"AMATELUS.Cryptographic","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/amatelus/amatelus/cea74f0a4b1769c90f05dc65e70d1322d368cd86/AMATELUS/Cryptographic.lean","lean4_code":"/-- Theorem 3.1: DID衝突の量子安全性\n\n    異なるValidDIDDocumentから同じDIDが生成されることは、\n    量子計算機を用いても困難です。\n\n    **元の主張（決定論的単射性）の問題点:**\n    \"異なるDIDドキュメントは異なるDIDを生成する\"は数学的に偽である。\n    なぜなら、SHA-3のような有限出力ハッシュ関数は鳩の巣原理により\n    必ず衝突を持つからである（無限入力空間 → 2^512出力空間）。\n\n    **正しい主張（確率的単射性＋量���安全性）:**\n    衝突は存在するが、量子計算機を用いても衝突を発見することは困難である。\n\n    **量子脅威下での安全性:**\n    - 衝突探索の量子コスト: 128ビット（Grover適用後）\n    - NIST最小要件: 128ビット\n    - 結論: 安全（128 ≥ 128）\n\n    **証明:**\n    SecurityAssumptions.amtHashFunction.quantum_secureにより、\n    SHA3-512の衝突探索の量子コストは128ビットであり、\n    NIST最小要件128ビットを満たす。\n    DID.fromValidDocument は内部でこのハッシュ関数を使用しているため、\n    異なるValidDIDDocumentから同じDIDが生成される確率は無視できるほど小さい。\n-/\ntheorem did_collision_quantum_secure :\n  amtHashFunction.collisionSecurity.quantumBits ≥ minSecurityLevel.quantumBits := by\n  -- 128 ≥ 128\n  exact amtHashFunction.quantum_secure","line":14,"end_line":42,"proof_status":"statement_formalized"}
+{"id":"prop:design-philosophy","label":"design-philosophy","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://amatelus.com/blueprint/dep_graph_document.html","dependencies":["prop:amatelus-non-scope","prop:amatelus-capabilities"],"nlp_text":"AMATELUS is a cryptographic authentication mechanism, not a communication infrastructure.","doc_url":"https://amatelus.com/docs/find/#doc/AMATELUS.Protocol","latex_link":"sect0002.html#prop:design-philosophy","lean_link":"https://amatelus.com/docs/find/#doc/AMATELUS.Protocol","lean_decl_name":"AMATELUS.Protocol","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/amatelus/amatelus/cea74f0a4b1769c90f05dc65e70d1322d368cd86/AMATELUS/Protocol.lean","lean4_code":"/-\n# プロトコル全体の一貫性と実装考慮事項\n\nこのファイルは、AMATELUSプロトコル全体の安全性と実装の実現可能性を証明します\n（Theorem 7.1, 7.2, 8.1）。\n-/\n\nimport AMATELUS.DID\nimport AMATELUS.VC\nimport AMATELUS.ZKP\nimport AMATELUS.SecurityAssumptions\nimport AMATELUS.Cryptographic\nimport AMATELUS.TrustChain\nimport AMATELUS.Privacy\nimport AMATELUS.Audit\nimport AMATELUS.Network\n\n-- ## Definition 7.1: Protocol State\n\n/-- Wallet内の通信用DID管理情報\n\n    通信相手ごとの通信用DIDとVCのアソシエーション。\n-/\nstructure WalletCommunicationRecord where\n  -- 通信用DID\n  commDID : ValidDID\n  -- 通信相手のDID\n  counterpartyDID : UnknownDID\n  -- 発行されたVC（VC発行ありの場合のみ保存）\n  vc : Option ValidVC","line":1,"end_line":30,"proof_status":"formalized"}
+{"id":"thm:boundary-clarity","label":"boundary-clarity","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://amatelus.com/blueprint/dep_graph_document.html","dependencies":["prop:design-philosophy"],"nlp_text":"The boundary between cryptographic verification (AMATELUS responsibility) and service provision (service provider responsibility) eliminates distributed infrastructure complexity while preserving cryptographic security properties.","doc_url":"https://amatelus.com/docs/find/#doc/AMATELUS.Protocol","latex_link":"sect0002.html#thm:boundary-clarity","lean_link":"https://amatelus.com/docs/find/#doc/AMATELUS.Protocol","lean_decl_name":"AMATELUS.Protocol","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/amatelus/amatelus/cea74f0a4b1769c90f05dc65e70d1322d368cd86/AMATELUS/Protocol.lean","lean4_code":"/-\n# プロトコル全体の一貫性と実装考慮事項\n\nこのファイルは、AMATELUSプロトコル全体の安全性と実装の実現可能性を証明します\n（Theorem 7.1, 7.2, 8.1）。\n-/\n\nimport AMATELUS.DID\nimport AMATELUS.VC\nimport AMATELUS.ZKP\nimport AMATELUS.SecurityAssumptions\nimport AMATELUS.Cryptographic\nimport AMATELUS.TrustChain\nimport AMATELUS.Privacy\nimport AMATELUS.Audit\nimport AMATELUS.Network\n\n-- ## Definition 7.1: Protocol State\n\n/-- Wallet内の通信用DID管理情報\n\n    通信相手ごとの通信用DIDとVCのアソシエーション。\n-/\nstructure WalletCommunicationRecord where\n  -- 通信用DID\n  commDID : ValidDID\n  -- 通信相手のDID\n  counterpartyDID : UnknownDID\n  -- 発行されたVC（VC発行ありの場合のみ保存）\n  vc : Option ValidVC","line":1,"end_line":30,"proof_status":"formalized"}
+{"id":"def:amatelus","label":"amatelus","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://amatelus.com/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The AMATELUS protocol is a cryptographic authentication mechanism integrating: Decentralized Identifiers (DIDs) Verifiable Credentials (VCs) Zero-Knowledge Proofs (ZKPs)","doc_url":"https://amatelus.com/docs/find/#doc/AMATELUS.Protocol","latex_link":"sect0002.html#def:amatelus","lean_link":"https://amatelus.com/docs/find/#doc/AMATELUS.Protocol","lean_decl_name":"AMATELUS.Protocol","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/amatelus/amatelus/cea74f0a4b1769c90f05dc65e70d1322d368cd86/AMATELUS/Protocol.lean","lean4_code":"/-\n# プロトコル全体の一貫性と実装考慮事項\n\nこのファイルは、AMATELUSプロトコル全体の安全性と実装の実現可能性を証明します\n（Theorem 7.1, 7.2, 8.1）。\n-/\n\nimport AMATELUS.DID\nimport AMATELUS.VC\nimport AMATELUS.ZKP\nimport AMATELUS.SecurityAssumptions\nimport AMATELUS.Cryptographic\nimport AMATELUS.TrustChain\nimport AMATELUS.Privacy\nimport AMATELUS.Audit\nimport AMATELUS.Network\n\n-- ## Definition 7.1: Protocol State\n\n/-- Wallet内の通信用DID管理情報\n\n    通信相手ごとの通信用DIDとVCのアソシエーション。\n-/\nstructure WalletCommunicationRecord where\n  -- 通信用DID\n  commDID : ValidDID\n  -- 通信相手のDID\n  counterpartyDID : UnknownDID\n  -- 発行されたVC（VC発行ありの場合のみ保存）\n  vc : Option ValidVC","line":1,"end_line":30,"proof_status":"formalized"}
+{"id":"prop:sybil-resilience","label":"sybil-resilience","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://amatelus.com/blueprint/dep_graph_document.html","dependencies":["def:amatelus"],"nlp_text":"Multiple DID possession is intentional protocol design for privacy protection. While a single entity can control multiple DIDs, Anonymous Hash Identifiers (AHI) restrict per-audit-domain abuse through cryptographic binding to national identity systems.","doc_url":"https://amatelus.com/docs/find/#doc/AMATELUS.Protocol","latex_link":"sect0007.html#prop:sybil-resilience","lean_link":"https://amatelus.com/docs/find/#doc/AMATELUS.Protocol","lean_decl_name":"AMATELUS.Protocol","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/amatelus/amatelus/cea74f0a4b1769c90f05dc65e70d1322d368cd86/AMATELUS/Protocol.lean","lean4_code":"/-\n# プロトコル全体の一貫性と実装考慮事項\n\nこのファイルは、AMATELUSプロトコル全体の安全性と実装の実現可能性を証明します\n（Theorem 7.1, 7.2, 8.1）。\n-/\n\nimport AMATELUS.DID\nimport AMATELUS.VC\nimport AMATELUS.ZKP\nimport AMATELUS.SecurityAssumptions\nimport AMATELUS.Cryptographic\nimport AMATELUS.TrustChain\nimport AMATELUS.Privacy\nimport AMATELUS.Audit\nimport AMATELUS.Network\n\n-- ## Definition 7.1: Protocol State\n\n/-- Wallet内の通信用DID管理情報\n\n    通信相手ごとの通信用DIDとVCのアソシエーション。\n-/\nstructure WalletCommunicationRecord where\n  -- 通信用DID\n  commDID : ValidDID\n  -- 通信相手のDID\n  counterpartyDID : UnknownDID\n  -- 発行されたVC（VC発行ありの場合のみ保存）\n  vc : Option ValidVC","line":1,"end_line":30,"proof_status":"statement_formalized"}
+{"id":"def:schema-chapter","label":"schema-chapter","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://amatelus.com/blueprint/dep_graph_document.html","dependencies":["def:amatelus"],"nlp_text":"This chapter covers formally verifiable json schema subset aspects of AMATELUS.","doc_url":"https://amatelus.com/docs/find/#doc/AMATELUS.JSONSchemaSubset","latex_link":"sect0010.html#def:schema-chapter","lean_link":"https://amatelus.com/docs/find/#doc/AMATELUS.JSONSchemaSubset","lean_decl_name":"AMATELUS.JSONSchemaSubset","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/amatelus/amatelus/cea74f0a4b1769c90f05dc65e70d1322d368cd86/AMATELUS/JSONSchemaSubset.lean","lean4_code":"/-\n# AMATELUS JSON Schema Subset Definition\n\nこのファイルは、AMATELUS プロトコル用の JSON Schema サブセットの Lean 形式化を含みます。\n\n## 参考文献\n- AMATELUS JSON Schema Subset: AMATELUS/JSONSchemaSubset.md\n- JSON Schema 2020-12 (Base): https://json-schema.org/draft/2020-12/json-schema-core\n- JSON Schema Validation: https://json-schema.org/draft/2020-12/json-schema-validation\n\n## 設計方針\n1. **形式検証可能**: すべての機能が Lean で証明可能\n2. **参照なし**: $ref, $defs を除外し、循環参照を防止\n3. **停止性保証**: ネスト深さ制限により停止性を保証\n4. **プロトコル保証**: サブセットのみがプロトコルレベルで保証される\n\n## サブセットの制限\n- ❌ 参照系: $ref, $defs, $dynamicRef (循環参照防止)\n- ❌ 動的検証: additionalProperties, patternProperties (プロトコルレベ���で無視)\n- ❌ 条件分岐: if/then/else (oneOf で代替可能)\n- ❌ フォーマット: format (pattern で代替可能)\n- ⚠️ ネスト制限: composition keywords は最大3レベル\n-/\n\n-- ## JSON データモデル\n\n/-- JSON値の型定義\n\n    JSON仕様（RFC 8259）に基づく6つのプリミティブ型：\n    - null: ヌル値","line":1,"end_line":30,"proof_status":"formalized"}
+{"id":"prop:amatelus-non-scope","label":"amatelus-non-scope","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://amatelus.com/blueprint/dep_graph_document.html","dependencies":["def:amatelus"],"nlp_text":"AMATELUS does not provide: Centralized directory services (DID resolution is local-only) Mediation or relay infrastructure User authorization decisions Communication endpoint management Message delivery guarantees","doc_url":"https://amatelus.com/docs/find/#doc/AMATELUS.Protocol","latex_link":"sect0002.html#prop:amatelus-non-scope","lean_link":"https://amatelus.com/docs/find/#doc/AMATELUS.Protocol","lean_decl_name":"AMATELUS.Protocol","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/amatelus/amatelus/cea74f0a4b1769c90f05dc65e70d1322d368cd86/AMATELUS/Protocol.lean","lean4_code":"/-\n# プロトコル全体の一貫性と実装考慮事項\n\nこのファイルは、AMATELUSプロトコル全体の安全性と実装の実現可能性を証明します\n（Theorem 7.1, 7.2, 8.1）。\n-/\n\nimport AMATELUS.DID\nimport AMATELUS.VC\nimport AMATELUS.ZKP\nimport AMATELUS.SecurityAssumptions\nimport AMATELUS.Cryptographic\nimport AMATELUS.TrustChain\nimport AMATELUS.Privacy\nimport AMATELUS.Audit\nimport AMATELUS.Network\n\n-- ## Definition 7.1: Protocol State\n\n/-- Wallet内の通信用DID管理情報\n\n    通信相手ごとの通信用DIDとVCのアソシエーション。\n-/\nstructure WalletCommunicationRecord where\n  -- 通信用DID\n  commDID : ValidDID\n  -- 通信相手のDID\n  counterpartyDID : UnknownDID\n  -- 発行されたVC（VC発行ありの場合のみ保存）\n  vc : Option ValidVC","line":1,"end_line":30,"proof_status":"statement_formalized"}
+{"id":"prop:amatelus-capabilities","label":"amatelus-capabilities","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://amatelus.com/blueprint/dep_graph_document.html","dependencies":["def:amatelus"],"nlp_text":"AMATELUS delivers the following cryptographic capabilities: Public key infrastructure (PKI) based distributed identifiers Zero-knowledge proof-based attribute ownership verification Verifiable credentials for claim issuance and storage Cryptographic trust guarantees","doc_url":"https://amatelus.com/docs/find/#doc/AMATELUS.Protocol","latex_link":"sect0002.html#prop:amatelus-capabilities","lean_link":"https://amatelus.com/docs/find/#doc/AMATELUS.Protocol","lean_decl_name":"AMATELUS.Protocol","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/amatelus/amatelus/cea74f0a4b1769c90f05dc65e70d1322d368cd86/AMATELUS/Protocol.lean","lean4_code":"/-\n# プロトコル全体の一貫性と実装考慮事項\n\nこのファイルは、AMATELUSプロトコル全体の安全性と実装の実現可能性を証明します\n（Theorem 7.1, 7.2, 8.1）。\n-/\n\nimport AMATELUS.DID\nimport AMATELUS.VC\nimport AMATELUS.ZKP\nimport AMATELUS.SecurityAssumptions\nimport AMATELUS.Cryptographic\nimport AMATELUS.TrustChain\nimport AMATELUS.Privacy\nimport AMATELUS.Audit\nimport AMATELUS.Network\n\n-- ## Definition 7.1: Protocol State\n\n/-- Wallet内の通信用DID管理情報\n\n    通信相手ごとの通信用DIDとVCのアソシエーション。\n-/\nstructure WalletCommunicationRecord where\n  -- 通信用DID\n  commDID : ValidDID\n  -- 通信相手のDID\n  counterpartyDID : UnknownDID\n  -- 発行されたVC（VC発行ありの場合のみ保存）\n  vc : Option ValidVC","line":1,"end_line":30,"proof_status":"formalized"}
+{"id":"thm:privacy-with-didcomm","label":"privacy-with-didcomm","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://amatelus.com/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Privacy is maintained despite public key transmission via DIDComm due to: Ephemeral Communication DIDs: Each session uses distinct DIDs with different key pairs Different services, different keys: User generates new keys for each service Cryptographic unlinkability: Public keys from different sessions cannot be linked (requires \\(2^{128}\\) quantum operations to link, assuming quantum-resistant hash functions) Service-specific communication DIDs: Verifier knows only the communication DID, not the user’s persistent identity DID Result: While DIDComm requires public key disclosure for impersonation prevention, the use of ephemeral DIDs with distinct key pairs per service ensures cross-service unlinkability and privacy.","doc_url":"https://amatelus.com/docs/find/#doc/AMATELUS.Privacy","latex_link":"sect0008.html#thm:privacy-with-didcomm","lean_link":"https://amatelus.com/docs/find/#doc/AMATELUS.Privacy","lean_decl_name":"AMATELUS.Privacy","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/amatelus/amatelus/cea74f0a4b1769c90f05dc65e70d1322d368cd86/AMATELUS/Privacy.lean","lean4_code":"/-\n# プライバシー保護機構の完全性証明\n\nこのファイルは、複数DID使用による名寄せ防止と\nZKPの零知識性を証明します（Theorem 5.1, 5.3）。\n-/\n\nimport AMATELUS.DID\nimport AMATELUS.ZKP\nimport AMATELUS.SecurityAssumptions\n\n-- ## サービスとプロトコル実行履歴の定義\n\n/-- サービスを表す型 -/\nstructure Service where\n  id : String\n  deriving Repr, DecidableEq\n\n/-- ZKP提示イベント\n\n    HolderがVerifier（サービス提供者）にZKPを提示したイベントを表します。\n    このイベントにより、プロトコルの実行履歴を構造的に記録できます。\n\n    **重要な設計:**\n    - ZKPには**HolderのDIDは含まれない**（ゼロ知識性の保証）\n    - ZKPは属性のみを証明し、身元は明かさない\n    - presentedDIDは別チャネル（DID提示）で送信されることを想定\n-/\nstructure ZKPPresentationEvent where\n  presentedDID : UnknownDID                -- 別チャネルで送信されたDID（ZKPとは独立）","line":1,"end_line":30,"proof_status":"formalized"}
+{"id":"thm:didcomm-impersonation-prevention","label":"didcomm-impersonation-prevention","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://amatelus.com/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Impersonation attacks (attacker with different secret key) are cryptographically prevented through DIDComm. Mechanism: Verifier receives ZKP along with sender’s public key via DIDComm (senderDoc) ZKP must be cryptographically signed with secret key corresponding to this public key If attacker uses different secret key, the signature verification fails Attacker cannot reuse legitimate ZKP with different SK (cryptographically impossible) Security guarantee: Impersonation attacks are prevented by DIDComm alone.","doc_url":"https://amatelus.com/docs/find/#doc/AMATELUS.Cryptographic","latex_link":"sect0005.html#thm:didcomm-impersonation-prevention","lean_link":"https://amatelus.com/docs/find/#doc/AMATELUS.Cryptographic","lean_decl_name":"AMATELUS.Cryptographic","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/amatelus/amatelus/cea74f0a4b1769c90f05dc65e70d1322d368cd86/AMATELUS/Cryptographic.lean","lean4_code":"/-- Theorem 3.1: DID衝突の量子安全性\n\n    異なるValidDIDDocumentから同じDIDが生成されることは、\n    量子計算機を用いても困難です。\n\n    **元の主張（決定論的単射性）の問題点:**\n    \"異なるDIDドキュメントは異なるDIDを生成する\"は数学的に偽である。\n    なぜなら、SHA-3のような有限出力ハッシュ関数は鳩の巣原理により\n    必ず衝突を持つからである（無限入力空間 → 2^512出力空間）。\n\n    **正しい主張（確率的単射性＋量子安全性）:**\n    衝突は存在するが、量子計算機を用いても衝突を発見することは困難である。\n\n    **量子脅威下での安全性:**\n    - 衝突探索の量子コスト: 128ビット（Grover適用後）\n    - NIST最小要件: 128ビット\n    - 結論: 安全（128 ≥ 128）\n\n    **証明:**\n    SecurityAssumptions.amtHashFunction.quantum_secureにより、\n    SHA3-512の衝突探索の量子コストは128ビットであり、\n    NIST最小要件128ビットを満たす。\n    DID.fromValidDocument は内部でこのハッシュ関数を使用しているため、\n    異なるValidDIDDocumentから同じDIDが生成される確率は無視できるほど小さい。\n-/\ntheorem did_collision_quantum_secure :\n  amtHashFunction.collisionSecurity.quantumBits ≥ minSecurityLevel.quantumBits := by\n  -- 128 ≥ 128\n  exact amtHashFunction.quantum_secure","line":14,"end_line":42,"proof_status":"formalized"}
+{"id":"thm:secret-key-correspondence-certainty","label":"secret-key-correspondence-certainty","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://amatelus.com/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"DIDComm establishes cryptographic certainty between ZKP and secret key correspondence. By requiring explicit transmission of DIDDocument (containing public key), the protocol ensures: Verifier definitively knows which public key corresponds to the ZKP ZKP is bound to exactly one secret key (the corresponding private key) Any ZKP signed with different secret key will fail verification Result: Cryptographic correspondence is certain, making impersonation impossible.","doc_url":"https://amatelus.com/docs/find/#doc/AMATELUS.Cryptographic","latex_link":"sect0005.html#thm:secret-key-correspondence-certainty","lean_link":"https://amatelus.com/docs/find/#doc/AMATELUS.Cryptographic","lean_decl_name":"AMATELUS.Cryptographic","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/amatelus/amatelus/cea74f0a4b1769c90f05dc65e70d1322d368cd86/AMATELUS/Cryptographic.lean","lean4_code":"/-- Theorem 3.1: DID衝突の量子安全性\n\n    異なるValidDIDDocumentから同じDIDが生成されることは、\n    量子計算機を用いても困難です。\n\n    **元の主張（決定論的単射性）の問題点:**\n    \"異なるDIDドキュメントは異なるDIDを生成する\"は数学的に偽である。\n    なぜなら、SHA-3のような有限出力ハッシュ関数は鳩の巣原理により\n    必ず衝突を持つからである（無限入力空間 → 2^512出力空間）。\n\n    **正しい主張（確率的単射性＋量子安全性）:**\n    衝突は存在するが、量子計算機を用いても衝突を発見することは困難である。\n\n    **量子脅威下での安全性:**\n    - 衝突探索の量子コスト: 128ビット（Grover適用後）\n    - NIST最小要件: 128ビット\n    - 結論: 安全（128 ≥ 128）\n\n    **証明:**\n    SecurityAssumptions.amtHashFunction.quantum_secureにより、\n    SHA3-512の衝突探索の量子コストは128ビットであり、\n    NIST最小要件128ビットを満たす。\n    DID.fromValidDocument は内部でこのハッシュ関数を使用しているため、\n    異なるValidDIDDocumentから同じDIDが生成される確率は無視できるほど小さい。\n-/\ntheorem did_collision_quantum_secure :\n  amtHashFunction.collisionSecurity.quantumBits ≥ minSecurityLevel.quantumBits := by\n  -- 128 ≥ 128\n  exact amtHashFunction.quantum_secure","line":14,"end_line":42,"proof_status":"formalized"}

crawled_repos/ananas-web.jsonl

@@ -0,0 +1,3 @@
+{"id":"condensed:profinite:inj-in-prod-f2","label":"inj-in-prod-f2","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://Waschbaar.github.io/ananas-web/dep_graph_document.html","dependencies":[],"nlp_text":"If \\( S \\) is a profinite set, then the continuous map \\( S\\to \\prod _{f\\in \\mathrm{Cont}(S, \\mathbb {F}_{2})} \\mathbb {F}_{2} \\) determined by the maps \\( f\\in \\mathrm{Cont}(S, \\mathbb {F}_{2}) \\) is injective.","doc_url":null,"latex_link":"sect0001.html#condensed:profinite:inj-in-prod-f2","lean_link":null,"lean_decl_name":"inj-in-prod-f2","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"profinite:noebeling","label":"noebeling","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://Waschbaar.github.io/ananas-web/dep_graph_document.html","dependencies":[],"nlp_text":"If \\( S \\) is a profinite set, then the abelian group \\( \\mathrm{Cont}(S, \\mathbb {Z}) \\) is free.","doc_url":null,"latex_link":"sect0001.html#profinite:noebeling","lean_link":null,"lean_decl_name":"noebeling","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"condensed:profinite:extend-loc-const-from-closed","label":"extend-loc-const-from-closed","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://Waschbaar.github.io/ananas-web/dep_graph_document.html","dependencies":[],"nlp_text":"Suppose that \\( S \\) is a profinite set, \\( T\\subset S \\) a closed subset and \\( M \\) a discrete topological space. Then the map \\( \\mathrm{Cont}(S, M)\\to \\mathrm{Cont}(T, M) \\) given by precomposition with the inclusion map is surjective.","doc_url":null,"latex_link":"sect0001.html#condensed:profinite:extend-loc-const-from-closed","lean_link":null,"lean_decl_name":"extend-loc-const-from-closed","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}

crawled_repos/autth.jsonl

@@ -0,0 +1,4 @@
+{"id":"def:mygrammar","label":"mygrammar","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://shetzl.github.io/autth/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Define the context-free grammar \\(G_1\\) as \\(S \\to \\varepsilon | a S b\\).","doc_url":"https://shetzl.github.io/autth/docs/find/#doc/mygrammar","latex_link":"sect0002.html#def:mygrammar","lean_link":"https://shetzl.github.io/autth/docs/find/#doc/mygrammar","lean_decl_name":"mygrammar","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"autth","source_path":"autth/Example.lean","lean4_code":"def mygrammar : ( ContextFreeGrammar Alphabet ) where\n  NT := Nonterminals\n  initial := Nonterminals.S\n  rules := [ myfirstrule, mysecondrule ]\n","line":53,"end_line":56,"proof_status":"formalized"}
+{"id":"thm:my_second_theorem","label":"my_second_theorem","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://shetzl.github.io/autth/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\(\\varepsilon \\in L(G_1)\\)","doc_url":"https://shetzl.github.io/autth/docs/find/#doc/my_second_theorem","latex_link":"sect0002.html#thm:my_second_theorem","lean_link":"https://shetzl.github.io/autth/docs/find/#doc/my_second_theorem","lean_decl_name":"my_second_theorem","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"autth","source_path":"autth/Example.lean","lean4_code":"theorem my_second_theorem : List.nil ∈ ( ContextFreeGrammar.language mygrammar ) := by\n  apply (ContextFreeGrammar.mem_language_iff mygrammar []).mpr\n  apply ContextFreeGrammar.Produces.single\n  use myfirstrule\n  constructor\n  · unfold mygrammar\n    simp\n  · refine (ContextFreeRule.rewrites_iff [Symbol.nonterminal mygrammar.initial]\n      (List.map Symbol.terminal [])).mpr\n      ?h.right.a\n    use [], []\n    simp\n    unfold mygrammar myfirstrule\n    simp\n","line":66,"end_line":79,"proof_status":"formalized"}
+{"id":"thm:my_first_theorem","label":"my_first_theorem","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://shetzl.github.io/autth/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\(aab \\in a^* b^*\\)","doc_url":"https://shetzl.github.io/autth/docs/find/#doc/my_first_theorem","latex_link":"sect0001.html#thm:my_first_theorem","lean_link":"https://shetzl.github.io/autth/docs/find/#doc/my_first_theorem","lean_decl_name":"my_first_theorem","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"autth","source_path":"autth/Example.lean","lean4_code":"theorem my_first_theorem: [a,a,b] ∈ matches' ( comp ( star ( char a ) ) ( star ( char b ) ) ) := by\n  simp\n  refine Language.mem_mul.mpr ?_\n  use [a,a]\n  constructor\n  refine Language.mem_kstar_iff_exists_nonempty.mpr ?h.left.a\n  use [[a],[a]]\n  simp\n  rfl\n  use [b]\n  constructor\n  refine Language.mem_kstar_iff_exists_nonempty.mpr ?_\n  use [[b]]\n  simp only [List.join_cons, List.join_nil, List.singleton_append, List.mem_singleton, ne_eq,\n    forall_eq, List.cons_ne_self, not_false_eq_true, and_true, true_and]\n  rfl\n  simp\n","line":15,"end_line":31,"proof_status":"formalized"}
+{"id":"thm:my_third_theorem","label":"my_third_theorem","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://shetzl.github.io/autth/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\(ab \\in L(G_1)\\)","doc_url":"https://shetzl.github.io/autth/docs/find/#doc/my_third_theorem","latex_link":"sect0002.html#thm:my_third_theorem","lean_link":"https://shetzl.github.io/autth/docs/find/#doc/my_third_theorem","lean_decl_name":"my_third_theorem","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"autth","source_path":"autth/Example.lean","lean4_code":"theorem my_third_theorem : [a,b] ∈ ( ContextFreeGrammar.language mygrammar ) := by\n  apply (ContextFreeGrammar.mem_language_iff mygrammar [a, b]).mpr\n  have fst : mygrammar.Derives [Symbol.nonterminal mygrammar.initial]\n    [ ( Symbol.terminal Alphabet.a ) ,\n    ( Symbol.nonterminal  Nonterminals.S ),\n    ( Symbol.terminal Alphabet.b ) ] := by\n    apply ContextFreeGrammar.Produces.single\n    use mysecondrule\n    constructor\n    · unfold mygrammar\n      simp\n    · apply (ContextFreeRule.rewrites_iff [Symbol.nonterminal mygrammar.initial]\n        [Symbol.terminal a, Symbol.nonterminal Nonterminals.S, Symbol.terminal b]).mpr\n      use [], []\n      simp\n      unfold mygrammar mysecondrule\n      simp\n  have snd : mygrammar.Derives [ ( Symbol.terminal Alphabet.a ) ,\n    ( Symbol.nonterminal  Nonterminals.S ),\n    ( Symbol.terminal Alphabet.b ) ] [ ( Symbol.terminal Alphabet.a ) ,\n    ( Symbol.terminal Alphabet.b ) ] := by\n    apply ContextFreeGrammar.Produces.single\n    use myfirstrule\n    constructor\n    · unfold mygrammar\n      simp\n    · apply (ContextFreeRule.rewrites_iff [Symbol.terminal a, Symbol.nonterminal Nonterminals.S, Symbol.terminal b]\n      [Symbol.terminal a, Symbol.terminal b]).mpr\n      use [Symbol.terminal a], [Symbol.terminal b]\n      unfold myfirstrule\n      simp\n  exact ContextFreeGrammar.Derives.trans fst snd\n","line":82,"end_line":113,"proof_status":"formalized"}

crawled_repos/blossoming-principle.jsonl

@@ -0,0 +1,8 @@
+{"id":"def:CoordinateSystem","label":"CoordinateSystem","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://matijasteblaj.github.io/blossoming-principle/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A coordinate system on a given affine space \\((P,V,+)\\) consists of an origin point in \\(P\\) and a vector space basis for \\(V\\).","doc_url":"https://matijasteblaj.github.io/blossoming-principle/docs/find/#doc/CoordinateSystem","latex_link":"index.html#def:CoordinateSystem","lean_link":"https://matijasteblaj.github.io/blossoming-principle/docs/find/#doc/CoordinateSystem","lean_decl_name":"CoordinateSystem","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"blossoming-principle","source_path":"BlossomingPrinciple/PolynomialMaps.lean","lean4_code":"open Affine\nnotation \"VectorSpace\" => Module\n\nsection PolynomialMaps\n\n","line":16,"end_line":20,"proof_status":"formalized"}
+{"id":"a0000000004","label":"a0000000004","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://matijasteblaj.github.io/blossoming-principle/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Polynomial map (for any basis)","doc_url":null,"latex_link":"index.html#a0000000004","lean_link":null,"lean_decl_name":"a0000000004","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000008","label":"a0000000008","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://matijasteblaj.github.io/blossoming-principle/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Symmetric map","doc_url":null,"latex_link":"index.html#a0000000008","lean_link":null,"lean_decl_name":"a0000000008","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000007","label":"a0000000007","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://matijasteblaj.github.io/blossoming-principle/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Multiaffine map","doc_url":null,"latex_link":"index.html#a0000000007","lean_link":null,"lean_decl_name":"a0000000007","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:PolynomialMapInCoords","label":"PolynomialMapInCoords","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://matijasteblaj.github.io/blossoming-principle/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A polynomial map between two affine spaces with chosen coordinate systems ...","doc_url":null,"latex_link":"index.html#def:PolynomialMapInCoords","lean_link":null,"lean_decl_name":"PolynomialMapInCoords","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"blossoming-principle","source_path":"./BlossomingPrinciple/PolynomialMaps.lean","lean4_code":"structure PolynomialMapInCoords (map : P → R) (cs₁ : CoordinateSystem ι F V P)\n  (cs₂ : CoordinateSystem κ F W R) where\n\n  /-- The polynomials representing the map.-/\n  polys : κ  → MvPolynomial ι F\n\n  /-- The coordinates of a point mapped with `map` must be equal to values\n  of the polynomials in `polys` in the coordinates of the initial point -/\n  map_poly_eq: ∀ (p : P) (i : κ),\n   (cs₂.coordinates (map p)) i = (polys i).eval (cs₁.coordinates p)\n\n","line":42,"end_line":52,"proof_status":"statement_formalized"}
+{"id":"a0000000003","label":"a0000000003","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://matijasteblaj.github.io/blossoming-principle/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Polynomial map definition is independent of chosen affine space basis.","doc_url":null,"latex_link":"index.html#a0000000003","lean_link":null,"lean_decl_name":"a0000000003","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000005","label":"a0000000005","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://matijasteblaj.github.io/blossoming-principle/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Affine maps are polynomial of degree 1.","doc_url":null,"latex_link":"index.html#a0000000005","lean_link":null,"lean_decl_name":"a0000000005","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"a0000000009","label":"a0000000009","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://matijasteblaj.github.io/blossoming-principle/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Blossoming principle","doc_url":null,"latex_link":"index.html#a0000000009","lean_link":null,"lean_decl_name":"a0000000009","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}

crawled_repos/finalized.jsonl

@@ -0,0 +1,17 @@
+{"id":"def:subgroup-of-semigroup","label":"subgroup-of-semigroup","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(S\\) be a semigroup. A subgroup of \\(S\\) is a subset \\(H \\subseteq S\\) equipped with: A carrier set \\(H \\subseteq S\\); Closure under multiplication: For all \\(a, b \\in H\\), we have \\(a \\cdot b \\in H\\); An identity element \\(e \\in H\\) such that: \\(e \\in H\\) (the identity is in the carrier); For all \\(x \\in H\\), \\(e \\cdot x = x\\) (left identity); For all \\(x \\in H\\), \\(x \\cdot e = x\\) (right identity); An inverse function \\(\\mathrm{inv} : S \\to S\\) such that: For all \\(x \\in H\\), \\(\\mathrm{inv}(x) \\in H\\) (closure under inverses); For all \\(x \\in H\\), \\(\\mathrm{inv}(x) \\cdot x = e\\) (left inverse); For all \\(x \\in H\\), \\(x \\cdot \\mathrm{inv}(x) = e\\) (right inverse). With these properties, the carrier \\(H\\) forms a group under the multiplication inherited from \\(S\\). Note that the identity element \\(e\\) is uniquely determined by the carrier set: if \\(e_1\\) and \\(e_2\\) are both identities for the same carrier \\(H\\), then \\(e_1 = e_1 \\cdot e_2 = e_2\\). Similarly, the inverse function is uniquely determined by the carrier and identity.","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.Subgroup","latex_link":"sect0006.html#def:subgroup-of-semigroup","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.Subgroup","lean_decl_name":"Semigroup.Subgroup","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"finalized","source_path":"MyProject/Substructures.lean","lean4_code":"/-- A Subgroup of a Semigroup `S` is a subset closed under multiplication and containing\nand identity element and an inverse function -/\nstructure Subgroup (S : Type*) [Semigroup S] extends Submonoid S where\n  /-- The inverse function -/\n  inv (x : S) : S\n  /-- The inverse function is the identity if `x ∉ carrier`. This is to ensure\n  injectivity with `Set` -/\n  inv_not_mem {x : S} (hin : x ∉ carrier) : inv x = x\n  /-- The inverses are in the carrier -/\n  inv_mem {x : S} (hin : x ∈ carrier) : inv x ∈ carrier\n  /-- The inverse function provides right inverses -/\n  inv_mul {x : S} (hin : x ∈ carrier) : inv x * x = one\n  /-- The inverse function provides left inverses -/\n  mul_inv {x : S} (hin : x ∈ carrier) : x * inv x = one\n","line":178,"end_line":191,"proof_status":"formalized"}
+{"id":"def:maximal-subgroup","label":"maximal-subgroup","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":["def:subgroup-of-semigroup"],"nlp_text":"A subgroup \\(H\\) of a semigroup \\(S\\) is called maximal if it is not properly contained in any other subgroup of \\(S\\). That is, \\(H\\) is maximal if for every subgroup \\(K\\) of \\(S\\) with \\(H \\subseteq K\\), we have \\(H = K\\).","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.Subgroup.isMaximal","latex_link":"sect0006.html#def:maximal-subgroup","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.Subgroup.isMaximal","lean_decl_name":"Semigroup.Subgroup.isMaximal","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"finalized","source_path":"MyProject/Substructures.lean","lean4_code":"/-- A maximal subgroup is a subgroup that is not properly contained in any other subgroup. -/\ndef isMaximal (H : Subgroup S) : Prop := ∀ (K : Subgroup S), H ≤ K → H = K\n","line":327,"end_line":328,"proof_status":"formalized"}
+{"id":"lem:hclass-subgroup","label":"hclass-subgroup","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":["def:maximal-subgroup","lem:location-theorem"],"nlp_text":"Let \\(S\\) be a semigroup and let \\(e \\in S\\) be an idempotent element. Then: The \\(\\mathcal{H}\\)-class \\([e]_{\\mathcal{H}}\\) is closed under multiplication. The element \\(e\\) acts as an identity on \\([e]_{\\mathcal{H}}\\): for all \\(x \\in [e]_{\\mathcal{H}}\\), \\(e \\cdot x = x\\) and \\(x \\cdot e = x\\). Every element \\(x \\in [e]_{\\mathcal{H}}\\) has an inverse \\(y \\in [e]_{\\mathcal{H}}\\) satisfying \\(x \\cdot y = e\\) and \\(y \\cdot x = e\\). The \\(\\mathcal{H}\\)-class \\([e]_{\\mathcal{H}}\\) forms a subgroup of \\(S\\) with identity \\(e\\). Every maximal subgroup of \\(S\\) is the \\(\\mathcal{H}\\)-class of some idempotent.","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.HEquiv.subgroup_of_idempotent","latex_link":"sect0007.html#lem:hclass-subgroup","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.HEquiv.subgroup_of_idempotent","lean_decl_name":"Semigroup.HEquiv.subgroup_of_idempotent","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"finalized","source_path":"./WorkInProgress/Location.lean","lean4_code":"noncomputable def HEquiv.subgroup_of_idempotent {e : S} (he : IsIdempotentElem e) : Subgroup S where\n  carrier := ⟦e⟧𝓗\n  mul_mem := HEquiv.mul_closed_of_idempotent he\n  one := e\n  one_mem := by simp\n  one_mul {x : S} (hx : x 𝓗 e) := HEquiv.idempotent_mul he hx\n  mul_one {x : S} (hx : x 𝓗 e) := HEquiv.mul_idempotent he hx\n  inv (x : S) := by\n    have hd : Decidable (x ∈ ⟦e⟧𝓗) := by exact Classical.propDecidable (x ∈ ⟦e⟧𝓗)\n    exact (if hx : x ∈ ⟦e⟧𝓗\n      then Exists.choose (HEquiv.exists_inverse_of_idempotent he hx)\n      else x )\n  inv_not_mem := by simp_all\n  inv_mem := by\n    simp_all\n    intros x hx\n    have h := Classical.choose_spec (HEquiv.exists_inverse_of_idempotent he hx)\n    exact h.1\n  inv_mul := by\n    simp_all\n    intros x hx\n    have h := Classical.choose_spec (HEquiv.exists_inverse_of_idempotent he hx)\n    exact h.2.2\n  mul_inv := by\n    simp_all\n    intros x hx\n    have h := Classical.choose_spec (HEquiv.exists_inverse_of_idempotent he hx)\n    exact h.2.1\n\n","line":254,"end_line":282,"proof_status":"formalized"}
+{"id":"lem:maximal-subgroups-isomorphic","label":"maximal-subgroups-isomorphic","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":["lem:hclass-subgroup"],"nlp_text":"Let \\(S\\) be a semigroup and let \\(H\\) and \\(K\\) be maximal subgroups of \\(S\\). If there exist \\(x \\in H\\) and \\(y \\in K\\) with \\(x \\mathcal{D} y\\), then \\(H\\) and \\(K\\) are isomorphic as groups.","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.DEquiv.maximal_subgroups_equiv","latex_link":"sect0007.html#lem:maximal-subgroups-isomorphic","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.DEquiv.maximal_subgroups_equiv","lean_decl_name":"Semigroup.DEquiv.maximal_subgroups_equiv","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/lean-summer-research/finalized/ebb5a308adbcdfed2c51c782d323ab1b588df041/MyProject/Green/Location.lean","lean4_code":"/-- Two maximal subgroups of a 𝓓-class are isomorphic. -/\ntheorem DEquiv.maximal_subgroups_equiv {x y : S} {H K : Subgroup S}\n  (hH : H.isMaximal) (hK : K.isMaximal) (hx : x ∈ H) (hy : y ∈ K) (hd : x 𝓓 y) :\n    Nonempty (H ≃* K) := by\n  obtain ⟨e₁, hi₁, h₁⟩ := HEquiv.hClass_of_subgroup hH\n  obtain ⟨e₂, hi₂, h₂⟩ := HEquiv.hClass_of_subgroup hK\n  simp [Subgroup.mem_def, h₁, h₂] at hx hy\n  have he : e₁ 𝓓 e₂ := by\n    apply HEquiv.to_dEquiv at hx\n    apply HEquiv.to_dEquiv at hy\n    apply DEquiv.trans hx.symm\n    apply DEquiv.trans hd hy\n  have heq₁ : H = HEquiv.subgroup_of_idempotent hi₁ := by\n    apply SetLike.ext\n    simp [Subgroup.mem_def, h₁]\n  have heq₂ : K = HEquiv.subgroup_of_idempotent hi₂ := by\n    apply SetLike.ext\n    simp [Subgroup.mem_def, h₂]\n  rw [heq₁, heq₂]\n  exact DEquiv.hClass_equiv hi₁ hi₂ he","line":505,"end_line":524,"proof_status":"formalized"}
+{"id":"lem:exists-pow-sandwich","label":"exists-pow-sandwich","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":["lem:exists-unique-idempotent-pow"],"nlp_text":"In a finite monoid \\(M\\), if an element \\(a\\) satisfies the property \\(a = xay\\) for some \\(x, y \\in M\\), then there exist non-zero powers of \\(x\\) and \\(y\\), say \\(n_1\\) and \\(n_2\\), such that \\(x^{n_1}a = a\\) and \\(ay^{n_2} = a\\).","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Monoid.exists_pow_sandwich_eq_self","latex_link":"sect0001.html#lem:exists-pow-sandwich","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Monoid.exists_pow_sandwich_eq_self","lean_decl_name":"Monoid.exists_pow_sandwich_eq_self","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"finalized","source_path":"MyProject/Idempotent.lean","lean4_code":"/-- In finite monoids, elements satisfying a sandwich property have powers that left-cancel\nand right-cancel the sandwich factors. -/\ntheorem exists_pow_sandwich_eq_self [Finite M] {x a y : M} (h : x * a * y = a) :\n    ∃ n₁ n₂ : ℕ, n₁ ≠ 0 ∧ n₂ ≠ 0 ∧ x ^ n₁ * a = a ∧ a * y ^ n₂ = a := by\n  have loop : ∀ k : ℕ, x ^ k * a * y ^ k = a := by\n    intro k; induction k with\n    | zero => simp\n    | succ n ih =>\n      rw [pow_succ, pow_succ']\n      rw [← mul_assoc, mul_assoc _ a, mul_assoc _ x, ← mul_assoc x a y, h, ih]\n  have ⟨n₁, ⟨hn₁, hneq₁⟩⟩ := Monoid.exists_idempotent_pow x\n  have ⟨n₂, ⟨hn₂, hneq₂⟩⟩ := Monoid.exists_idempotent_pow y\n  use n₁, n₂\n  constructor\n  · exact hneq₁\n  constructor\n  · exact hneq₂\n  constructor\n  · rw [← (loop n₁), ← mul_assoc, ← mul_assoc, hn₁]\n  · rw [← (loop n₂), mul_assoc, hn₂]\n","line":122,"end_line":141,"proof_status":"formalized"}
+{"id":"lem:d-j-theorem","label":"d-j-theorem","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":["lem:exists-pow-sandwich","def:greens-relations"],"nlp_text":"In a finite semigroup, the \\(\\mathcal{D}\\) and \\(\\mathcal{J}\\) relations are equivalent. That is, for any two elements \\(x, y \\in S\\), \\(x \\mathcal{D} y\\) if and only if \\(x \\mathcal{J} y\\).","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.JEquiv.to_dEquiv","latex_link":"sect0004.html#lem:d-j-theorem","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.JEquiv.to_dEquiv","lean_decl_name":"Semigroup.JEquiv.to_dEquiv","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/lean-summer-research/finalized/ebb5a308adbcdfed2c51c782d323ab1b588df041/MyProject/Green/Finite.lean","lean4_code":"/-- In finite semigroups, 𝓙-equivalence implies 𝓓-equivalence. -/\n@[simp] lemma JEquiv.to_dEquiv (hj : x 𝓙 y) : x 𝓓 y := by\n  have hj₁ := hj\n  obtain ⟨⟨s, t, ha⟩, ⟨u, v, hb⟩⟩ := hj₁\n  have hab : s * u * x * (v * t) = ↑x := by\n    have hrw : s * u * ↑x * (v * t) = s * (u * ↑x * v * t) := by simp [mul_assoc]\n    rw [hrw, hb, ← mul_assoc, ha]\n  obtain ⟨k, ⟨l, ⟨hkne, hlne, heq₁, heq₂⟩⟩⟩ := Monoid.exists_pow_sandwich_eq_self hab\n  cases v with\n  | one =>\n    use x\n    simp at ⊢ hb heq₂ hab\n    constructor -- we prove `x 𝓛 y`\n    · use (s * u)^(k-1) * s\n      have hk : k - 1 + 1 = k := by exact Nat.succ_pred_eq_of_ne_zero hkne\n      simp_rw [← hb, ← mul_assoc, mul_assoc _ s u, ← _root_.pow_succ, hk]\n      simp [heq₁]\n    · use u\n  | coe v =>\n    use x * v\n    simp [REquiv, LEquiv]\n    constructor\n    · use t * (v * t) ^ (l - 1) -- `x ≤𝓡 x * v`\n      rw [WithOne.coe_mul, ← mul_assoc, mul_assoc ↑x ↑v t]\n      rw [mul_assoc ↑x (↑v * t), ← pow_succ']\n      have hl : l - 1 + 1 = l := by exact Nat.succ_pred_eq_of_ne_zero hlne\n      rw [hl, heq₂]\n    · constructor\n      · use (s * u)^(k-1) * s -- `x * v ≤𝓛 y`\n        rw [← hb]\n        have hk : k - 1 + 1 = k := by exact Nat.succ_pred_eq_of_ne_zero hkne\n        conv => lhs; rw [← mul_assoc, ← mul_assoc, mul_assoc _ s u]\n        rw [WithOne.coe_mul, ← _root_.pow_succ, hk, heq₁]\n      · use u -- `y ≤𝓛 x * v`\n        simp [← mul_assoc, hb]","line":66,"end_line":100,"proof_status":"formalized"}
+{"id":"lem:j-strengthening","label":"j-strengthening","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":["lem:d-j-theorem"],"nlp_text":"In a finite semigroup, if two elements are \\(\\mathcal{J}\\)-equivalent, then a one-sided preorder implies the corresponding one-sided equivalence. Specifically, if \\(x \\mathcal{J} y\\) and \\(x \\leq _{\\mathcal{R}} y\\), then \\(x \\mathcal{R} y\\). Similarly, if \\(x \\mathcal{J} y\\) and \\(x \\leq _{\\mathcal{L}} y\\), then \\(x \\mathcal{L} y\\).","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.REquiv.of_rPreorder_and_jEquiv","latex_link":"sect0004.html#lem:j-strengthening","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.REquiv.of_rPreorder_and_jEquiv","lean_decl_name":"Semigroup.REquiv.of_rPreorder_and_jEquiv","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/lean-summer-research/finalized/ebb5a308adbcdfed2c51c782d323ab1b588df041/MyProject/Green/Finite.lean","lean4_code":"/-- In finite semigroups, 𝓙-equivalence strengthens the 𝓡-preorder to 𝓡-equivalence. -/\ntheorem REquiv.of_rPreorder_and_jEquiv (hr : x ≤𝓡 y) (hj : x 𝓙 y) : x 𝓡 y := by\n  obtain ⟨z, hz⟩ := hr\n  cases z with\n  | one =>\n    have heq : x = y := by simp_all\n    subst x; simp\n  | coe z =>\n    have heq : y * z = x := by\n      rw [← WithOne.coe_inj, WithOne.coe_mul]\n      exact hz\n    subst x\n    symm\n    apply REquiv.of_jEquiv_mul_right hj.symm","line":137,"end_line":150,"proof_status":"formalized"}
+{"id":"lem:h-of-sandwich","label":"h-of-sandwich","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":["lem:j-strengthening"],"nlp_text":"In a finite semigroup, if an element \\(x\\) can be written as \\(x = uxv\\) for some \\(u, v \\in S\\), then \\(x\\) is \\(\\mathcal{H}\\)-equivalent to both \\(ux\\) and \\(xv\\).","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.HEquiv.of_eq_sandwich","latex_link":"sect0004.html#lem:h-of-sandwich","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.HEquiv.of_eq_sandwich","lean_decl_name":"Semigroup.HEquiv.of_eq_sandwich","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/lean-summer-research/finalized/ebb5a308adbcdfed2c51c782d323ab1b588df041/MyProject/Green/Finite.lean","lean4_code":"/-- In finite semigroups, an element sandwiched between two factors is 𝓗-related to its\nleft and right partial products. -/\ntheorem HEquiv.of_eq_sandwich (h : u * x * v = x) : x 𝓗 u * x ∧ x 𝓗 x * v := by\n  simp [HEquiv.iff_rEquiv_and_lEquiv]\n  constructor <;> constructor\n  · apply REquiv.of_rPreorder_and_jEquiv\n    · use v\n      simpa [← WithOne.coe_mul]\n    · simp [JEquiv]\n      use 1, ↑v\n      simpa [← WithOne.coe_mul]\n  · apply LEquiv.of_jEquiv_mul_left\n    simp [JEquiv]\n    use 1, v\n    simpa [← WithOne.coe_mul]\n  · apply REquiv.of_jEquiv_mul_right\n    simp [JEquiv]\n    use u, 1\n    simpa [← WithOne.coe_mul, ← mul_assoc]\n  · apply LEquiv.of_lPreorder_and_jEquiv\n    · use u\n      simpa [← WithOne.coe_mul, ← mul_assoc]\n    · simp [JEquiv]\n      use u, 1\n      simpa [← WithOne.coe_mul, ← mul_assoc]","line":168,"end_line":192,"proof_status":"formalized"}
+{"id":"lem:location-theorem","label":"location-theorem","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":["lem:j-strengthening","lem:greens-lemma","lem:le-idempotent"],"nlp_text":"Let \\(S\\) be a semigroup and let \\(x, y \\in S\\). The following conditions are equivalent: \\(x \\cdot y \\in [x]_{\\mathcal{R}} \\cap [y]_{\\mathcal{L}}\\) (i.e., \\(x \\cdot y \\mathcal{R} x\\) and \\(x \\cdot y \\mathcal{L} y\\)). There exists an idempotent \\(e \\in [y]_{\\mathcal{R}} \\cap [x]_{\\mathcal{L}}\\). Moreover, in a finite semigroup, these conditions are equivalent to: \\(x \\mathcal{D} y\\) and \\(x \\cdot y \\mathcal{D} x\\).","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.DEquiv.mul_in_inter_iff_equiv","latex_link":"sect0007.html#lem:location-theorem","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.DEquiv.mul_in_inter_iff_equiv","lean_decl_name":"Semigroup.DEquiv.mul_in_inter_iff_equiv","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/lean-summer-research/finalized/ebb5a308adbcdfed2c51c782d323ab1b588df041/MyProject/Green/Location.lean","lean4_code":"/-- In Finite semigroups, `x * y` is in the intersection of the 𝓡-class of `x` and the 𝓛-class\nof `y` iff `x`, `y`, and `x * y` are 𝓓-Equivalent. -/\ntheorem DEquiv.mul_in_inter_iff_equiv [Finite S] : x * y ∈ ⟦x⟧𝓡 ∩ ⟦y⟧𝓛 ↔ x 𝓓 y ∧ x * y 𝓓 x := by\n  simp_all\n  constructor\n  · rintro ⟨hr, hl⟩\n    constructor\n    · use x * y\n      exact ⟨hr.symm, hl⟩\n    · exact JEquiv.to_dEquiv <| REquiv.to_jEquiv hr\n  · rintro ⟨hj₁, hj₂⟩\n    apply DEquiv.to_jEquiv at hj₁\n    apply DEquiv.to_jEquiv at hj₂\n    constructor\n    · refine REquiv.of_rPreorder_and_jEquiv ?_ hj₂\n      simp\n    · refine LEquiv.of_lPreorder_and_jEquiv ?_ ?_\n      · simp\n      · apply JEquiv.trans hj₂ hj₁","line":84,"end_line":102,"proof_status":"formalized"}
+{"id":"lem:greens-relations-mul-compat","label":"greens-relations-mul-compat","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":["def:greens-relations"],"nlp_text":"The \\(\\mathcal{R}\\) and \\(\\mathcal{L}\\) relations exhibit compatibility with semigroup multiplication on one side. Specifically, the \\(\\mathcal{R}\\)-preorder is compatible with left multiplication, and the \\(\\mathcal{L}\\)-preorder is compatible with right multiplication. If \\(x \\leq _{\\mathcal{R}} y\\), then for any \\(z \\in S\\), we have \\(zx \\leq _{\\mathcal{R}} zy\\). This property extends to the equivalence relation \\(\\mathcal{R}\\). If \\(x \\mathcal{R} y\\), then \\(zx \\mathcal{R} zy\\). A similar argument holds for the \\(\\mathcal{L}\\)-preorder and \\(\\mathcal{L}\\)-equivalence, which are compatible with right multiplication. If \\(x \\leq _{\\mathcal{L}} y\\), then \\(xz \\leq _{\\mathcal{L}} yz\\) for any \\(z \\in S\\).","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.RPreorder.lmult_compat","latex_link":"sect0002.html#lem:greens-relations-mul-compat","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.RPreorder.lmult_compat","lean_decl_name":"Semigroup.RPreorder.lmult_compat","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/lean-summer-research/finalized/ebb5a308adbcdfed2c51c782d323ab1b588df041/MyProject/Green/Defs.lean","lean4_code":"/-- The 𝓡-preorder is compatible with left multiplication. -/\n@[simp] lemma RPreorder.lmult_compat {x y : S} (h : x ≤𝓡 y) (z : S) : z * x ≤𝓡 z * y := by\n  obtain ⟨u, hu⟩ := h; use u\n  simp [mul_assoc, hu]","line":394,"end_line":397,"proof_status":"formalized"}
+{"id":"lem:greens-lemma","label":"greens-lemma","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":["lem:greens-relations-mul-compat"],"nlp_text":"Let \\(S\\) be a semigroup and suppose \\(x \\mathcal{R} y\\) with witnesses \\(u, v \\in S\\) such that \\(x \\cdot v = y\\) and \\(y \\cdot u = x\\). Then: The map \\(\\rho _v : w \\mapsto w \\cdot v\\) is a bijection from the \\(\\mathcal{L}\\)-class of \\(x\\) to the \\(\\mathcal{L}\\)-class of \\(y\\). The map \\(\\rho _u : w \\mapsto w \\cdot u\\) is the inverse of \\(\\rho _v\\). These bijections preserve \\(\\mathcal{H}\\)-classes: for any \\(a, b\\) in the \\(\\mathcal{L}\\)-class of \\(x\\), we have \\(a \\mathcal{H} b\\) if and only if \\(a \\cdot v \\mathcal{H} b \\cdot v\\). Dually, if \\(x \\mathcal{L} y\\) with witnesses \\(u, v\\) such that \\(v \\cdot x = y\\) and \\(u \\cdot y = x\\), then left translation \\(\\lambda _v : w \\mapsto v \\cdot w\\) is a bijection from the \\(\\mathcal{R}\\)-class of \\(x\\) to the \\(\\mathcal{R}\\)-class of \\(y\\), with inverse \\(\\lambda _u\\), and these bijections preserve \\(\\mathcal{H}\\)-classes.","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.REquiv.bijOn_lClass","latex_link":"sect0005.html#lem:greens-lemma","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.REquiv.bijOn_lClass","lean_decl_name":"Semigroup.REquiv.bijOn_lClass","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/lean-summer-research/finalized/ebb5a308adbcdfed2c51c782d323ab1b588df041/MyProject/Green/GreensLemma.lean","lean4_code":"/-- If `x 𝓡 y` such that `x * v = y`, then the map `w ↦ w * v` is a bijection from\nthe 𝓛-class of `x` to that of `y`. -/\ntheorem REquiv.bijOn_lClass (hr : x 𝓡 y) (hv : x * v = y) :\n    Set.BijOn (fun w ↦ w * v) ⟦x⟧𝓛 ⟦y⟧𝓛 := by\n  refine Set.BijOn.mk ?_ ?_ ?_\n  · apply RPreorder.mapsTo_lClass hv\n  · apply hr.injOn_lClass hv\n  · apply hr.surjOn_lClass hv","line":131,"end_line":138,"proof_status":"formalized"}
+{"id":"lem:r-l-comm","label":"r-l-comm","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":["lem:greens-relations-mul-compat"],"nlp_text":"The composition of the relations \\(\\mathcal{R}\\) and \\(\\mathcal{L}\\) is commutative. That is, for any \\(x, y \\in S\\), there exists a \\(z\\) such that \\(x \\mathcal{R} z\\) and \\(z \\mathcal{L} y \\iff \\) there exists a \\(w\\) such that \\(x \\mathcal{L} w\\) and \\(w \\mathcal{R} y\\). This can be written as \\(\\mathcal{R} \\circ \\mathcal{L} = \\mathcal{L} \\circ \\mathcal{R}\\). This property is crucial for proving that the relation \\(\\mathcal{D} = \\mathcal{R} \\circ \\mathcal{L}\\) is symmetric, and therefore an equivalence relation.","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.rEquiv_lEquiv_comm","latex_link":"sect0002.html#lem:r-l-comm","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.rEquiv_lEquiv_comm","lean_decl_name":"Semigroup.rEquiv_lEquiv_comm","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"finalized","source_path":"MyProject/Green/Defs.lean","lean4_code":"/-- The 𝓡 and 𝓛 relations commute under composition. -/\ntheorem rEquiv_lEquiv_comm (x y : S) : (∃ z, x 𝓡 z ∧ z 𝓛 y) ↔ (∃ z, x 𝓛 z ∧ z 𝓡 y) := by\n  constructor\n  · rintro ⟨z, ⟨hr, hl⟩⟩\n    have hr₁ := hr\n    have hl₁ := hl\n    rcases hr₁ with ⟨⟨w₁, hw₁⟩, ⟨v₁, hv₁⟩⟩\n    rcases hl₁ with ⟨⟨w₂, hw₂⟩, ⟨v₂, hv₂⟩⟩\n    -- We have `x = w₂ * y * w₁` and `y = v₂ * z * v₁`\n    -- We want to use `v₂ * z * w₁`, but we need to destruct on `v₂, w₁ : WithOne S`\n    cases hv₂one : v₂ with\n    | one =>\n      subst v₂ -- trivial case where `v₂ = 1`\n      simp at hv₂\n      subst z -- `z = y`\n      use x; simp [hr]\n    | coe v₂ =>\n      subst hv₂one\n      cases hw₁one : w₁ with\n      | one =>\n        subst w₁ -- trivial case where `w₁ = 1` and `z = x`\n        simp at hw₁\n        subst hw₁\n        use y; simp [hl]\n      | coe w₁ => -- Non-trivial case, use `v₂ * z * w₁`\n        subst hw₁one\n        use v₂ * z * w₁\n        constructor\n          <;> constructor\n        · use w₂ -- `x ≤𝓛 v₂ * z * w₁`\n          simp [← mul_assoc]\n          rw [mul_assoc w₂, hv₂, hw₂, ← hw₁]\n        · use v₂ -- `v₂ * z * w₁ ≤𝓛 x`\n          simp [mul_assoc]\n          rw [← hw₁]\n        · use w₁ -- `v₂ * z * w₁ ≤𝓡 y`\n          simp [hv₂]\n        · use v₁ -- `y ≤𝓡 v₂ * z * w₁`\n          simp [mul_assoc v₂]\n          rw [hw₁, ← hv₂, ← hv₁]\n          simp [mul_assoc]\n  · rintro ⟨z, ⟨hl, hr⟩⟩\n    have hr₁ := hr\n    have hl₁ := hl\n    rcases hr₁ with ⟨⟨w₁, hw₁⟩, ⟨v₁, hv₁⟩⟩\n    rcases hl₁ with ⟨⟨w₂, hw₂⟩, ⟨v₂, hv₂⟩⟩\n    -- We have `y = v₂ * x * v₁` and `x = w₂ * y * w₁`\n    -- We want to use `w₂ * z * v₁`, but we need to destruct on `w₂, v₁ : WithOne S`\n    cases hw₂one : w₂ with\n    | one =>\n      subst w₂ -- trivial case where `w₂ = 1`\n      simp at hw₂\n      subst z -- `z = x`\n      use y; simp [hr]\n    | coe w₂ =>\n      subst hw₂one\n      cases hv₁one : v₁ with\n      | one =>\n        subst v₁ -- trivial case where `v₁ = 1` and `z = y`\n        simp at hv₁\n        subst hv₁\n        use x; simp [hl]\n      | coe v₁ => -- Non-trivial case, use `w₂ * z * v₁`\n        subst hv₁one\n        use w₂ * z * v₁\n        constructor\n          <;> constructor\n        · use w₁; simp -- `x ≤𝓡 w₂ * z * v₁`\n          nth_rw 1 [← hw₂]\n          conv => lhs; lhs; rw [mul_assoc, hv₁]\n          rw [mul_assoc, hw₁]\n        · use v₁ -- `w₂ * z * v₁ ≤𝓡 x`\n          simp [hw₂]\n        · use w₂ -- `w₂ * z * v₁ ≤𝓛 y`\n          simp [← hv₁, ← mul_assoc]\n        · use v₂ -- `y ≤𝓛 w₂ * z * v₁`\n          simp [hw₂, ← mul_assoc, hv₂, hv₁]\n","line":413,"end_line":489,"proof_status":"formalized"}
+{"id":"lem:d-equiv-closed","label":"d-equiv-closed","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":["lem:r-l-comm"],"nlp_text":"The \\(\\mathcal{D}\\) relation is closed under composition with \\(\\mathcal{R}\\) and \\(\\mathcal{L}\\). If \\(x \\mathcal{D} y\\) and \\(y \\mathcal{L} z\\), then \\(x \\mathcal{D} z\\). Similarly, if \\(x \\mathcal{D} y\\) and \\(y \\mathcal{R} z\\), then \\(x \\mathcal{D} z\\). This property is used to prove the transitivity of \\(\\mathcal{D}\\).","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.DEquiv.closed_under_lEquiv","latex_link":"sect0002.html#lem:d-equiv-closed","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.DEquiv.closed_under_lEquiv","lean_decl_name":"Semigroup.DEquiv.closed_under_lEquiv","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"finalized","source_path":"MyProject/Green/Defs.lean","lean4_code":"/-- The 𝓓 relation is closed under composition with 𝓛. -/\nlemma closed_under_lEquiv {x y z : S} (hd : x 𝓓 y) (hl : y 𝓛 z) : x 𝓓 z := by\n  rcases hd with ⟨w, ⟨hw1, hw2⟩⟩\n  use w\n  simp_all\n  apply LEquiv.trans hw2 hl\n","line":511,"end_line":516,"proof_status":"formalized"}
+{"id":"def:greens-relations","label":"greens-relations","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"In a semigroup \\(S\\), Green’s relations are a set of five equivalence relations that characterize the elements of \\(S\\) in terms of the principal ideals they generate. These relations are fundamental to the study of semigroups. First, we define four preorder relations: \\(\\leq _{\\mathcal{R}}\\), \\(\\leq _{\\mathcal{L}}\\), \\(\\leq _{\\mathcal{J}}\\), and \\(\\leq _{\\mathcal{H}}\\). Let \\(S^1\\) be the monoid obtained by adjoining an identity element to \\(S\\) if it does not already have one. For any two elements \\(x, y \\in S\\): \\(x \\leq _{\\mathcal{R}} y\\) iff the principal right ideal generated by \\(x\\) is a subset of the principal right ideal generated by \\(y\\) (\\(xS^1 \\subseteq yS^1\\)). In Lean, we use the equivalent definition that there must exist some \\(z \\in S^1\\) such that \\(x = yz\\). \\(x \\leq _{\\mathcal{L}} y\\) iff the principal left ideal generated by \\(x\\) is a subset of the principal left ideal generated by \\(y\\) (\\(S^1x \\subseteq S^1y\\)). In Lean, we use the equivalent definition that there must exist some \\(z \\in S^1\\) such that \\(x = zy\\). \\(x \\leq _{\\mathcal{J}} y\\) iff the principal two-sided ideal generated by \\(x\\) is a subset of the principal two-sided ideal generated by \\(y\\) (\\(S^1xS^1 \\subseteq S^1yS^1\\)). In Lean, we use the equivalent definition that there must exist some \\(u, v \\in S^1\\) such that \\(x = uyv\\). \\(x \\leq _{\\mathcal{H}} y \\iff x \\leq _{\\mathcal{R}} y\\) and \\(x \\leq _{\\mathcal{L}} y\\). These four relations are preorders (they are reflexive and transitive). From these preorders, we define the equivalence relations \\(\\mathcal{R}\\), \\(\\mathcal{L}\\), \\(\\mathcal{J}\\), and \\(\\mathcal{H}\\) as the symmetric closures of their corresponding preorders. For example, \\(x \\mathcal{R} y \\iff x \\leq _{\\mathcal{R}} y\\) and \\(y \\leq _{\\mathcal{R}} x\\). The equivalence classes are denoted by \\([x]_{\\mathcal{R}}\\), \\([x]_{\\mathcal{L}}\\), etc. Finally, the \\(\\mathcal{D}\\) relation is defined by composing \\(\\mathcal{R}\\) and \\(\\mathcal{L}\\): \\(x \\mathcal{D} y\\) if there exists an element \\(z \\in S\\) such that \\(x \\mathcal{R} z\\) and \\(z \\mathcal{L} y\\). It can be shown that \\(\\mathcal{D}\\) is an equivalence relation and that it can also be defined by composing \\(\\mathcal{L}\\) and \\(\\mathcal{R}\\).","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.RPreorder","latex_link":"sect0002.html#def:greens-relations","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.RPreorder","lean_decl_name":"Semigroup.RPreorder","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"finalized","source_path":"./MyProject/Green/GreensLemma.lean","lean4_code":"lemma RPreorder.mapsTo_lClass (hv : x * v = y) :\n    Set.MapsTo (fun w ↦ w * v) ⟦x⟧𝓛 ⟦y⟧𝓛 := by\n  simp [Set.MapsTo]\n  intros z hz\n  rw [← hv]\n  apply LEquiv.rmult_compat\n  exact hz\n\n/-- If `x 𝓡 y` such that `x * v = y` then the map `w ↦ w * v` is injective on the\n𝓛-class of `x`. -/\n","line":65,"end_line":74,"proof_status":"formalized"}
+{"id":"lem:le-idempotent","label":"le-idempotent","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":["def:greens-relations"],"nlp_text":"Let \\(e\\) be an idempotent element in a semigroup \\(S\\). An element \\(x \\in S\\) is \\(\\mathcal{R}\\)-below \\(e\\) if and only if \\(x = ex\\). Similarly, \\(x\\) is \\(\\mathcal{L}\\)-below \\(e\\) if and only if \\(x = xe\\). It follows that \\(x\\) is \\(\\mathcal{H}\\)-below \\(e\\) if and only if both conditions hold, i.e., \\(x = ex\\) and \\(x = xe\\).","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.RPreorder.le_idempotent","latex_link":"sect0003.html#lem:le-idempotent","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.RPreorder.le_idempotent","lean_decl_name":"Semigroup.RPreorder.le_idempotent","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/lean-summer-research/finalized/ebb5a308adbcdfed2c51c782d323ab1b588df041/MyProject/Green/Basic.lean","lean4_code":"/-- An element `x` is 𝓡-below an idempotent `e` if and only if `x = e * x`. -/\ntheorem RPreorder.le_idempotent {e : S} (h : IsIdempotentElem e) (x : S) :\n    (x ≤𝓡 e) ↔ (e * x = x) := by\n  constructor\n  · rintro ⟨u, hru⟩\n    unfold IsIdempotentElem at h\n    rw [← WithOne.coe_inj, WithOne.coe_mul] at h ⊢\n    rw [← hru, ← mul_assoc, h]\n  · intro hl; use x\n    rw [← WithOne.coe_inj] at hl\n    exact hl","line":65,"end_line":75,"proof_status":"formalized"}
+{"id":"lem:greens-relations-hom-pres","label":"greens-relations-hom-pres","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":["def:greens-relations"],"nlp_text":"Green’s relations are preserved under semigroup morphisms. Let \\(f: S \\to T\\) be a semigroup morphism. If two elements \\(x, y \\in S\\) are related by any of Green’s preorders or equivalence relations, then their images \\(f(x), f(y)\\) are related by the same relation in \\(T\\).","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.RPreorder.hom_pres","latex_link":"sect0003.html#lem:greens-relations-hom-pres","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.RPreorder.hom_pres","lean_decl_name":"Semigroup.RPreorder.hom_pres","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/lean-summer-research/finalized/ebb5a308adbcdfed2c51c782d323ab1b588df041/MyProject/Green/Basic.lean","lean4_code":"/-- The 𝓡-preorder is preserved by semigroup morphisms. -/\ntheorem RPreorder.hom_pres (f : F) (x y : S) (h : x ≤𝓡 y) : f x ≤𝓡 f y := by\n  obtain ⟨z, hz⟩ := h\n  cases z with\n  | one => simp_all\n  | coe z =>\n    have heq : x = y * z := by\n      rw [← WithOne.coe_inj, WithOne.coe_mul]\n      exact hz.symm\n    rw [← WithOne.coe_mul, WithOne.coe_inj] at hz\n    subst x\n    simp","line":115,"end_line":126,"proof_status":"formalized"}
+{"id":"lem:exists-unique-idempotent-pow","label":"exists-unique-idempotent-pow","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://lean-summer-research.github.io/finalized/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"In a finite semigroup, for any element \\(x\\), there exists a positive integer \\(m\\) such that \\(x^m\\) is an idempotent. This idempotent power is unique.","doc_url":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.exists_repeating_pow","latex_link":"sect0001.html#lem:exists-unique-idempotent-pow","lean_link":"https://lean-summer-research.github.io/finalized/docs/find/#doc/Semigroup.exists_repeating_pow","lean_decl_name":"Semigroup.exists_repeating_pow","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"finalized","source_path":"MyProject/Idempotent.lean","lean4_code":"/-- In a finite semigroup, powers of any element eventually repeat. -/\nlemma exists_repeating_pow [Finite S] (x : S) : ∃ (m n : ℕ+), x ^ m * x ^ n = x ^ m := by\n  obtain ⟨o, p, hop, heq⟩ : ∃ o p : ℕ+, o ≠ p ∧ x ^ o = x ^ p := by\n    apply Finite.exists_ne_map_eq_of_infinite\n  simp_all only [ne_eq, pow_add]; pnat_to_nat\n  rcases (Nat.lt_or_gt_of_ne hop) with (o_lt_p | p_lt_o)\n  · use o, p - o; simp_all\n  · use p, o - p; simp_all\n","line":51,"end_line":58,"proof_status":"formalized"}

crawled_repos/flt-regular.jsonl

@@ -0,0 +1,45 @@
+{"id":"lemma:alt_definition_of_norm","label":"alt_definition_of_norm","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(K\\) be a number field, \\(\\alpha \\in K\\) and let \\(\\sigma _i\\) be the embeddings of \\(K\\) into \\(\\mathbb {C}\\). Then \\[ N_{K/\\mathbb {Q}}(\\alpha )=\\prod _i \\sigma _i(\\alpha ) \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.norm_eq_prod_embeddings","latex_link":"sect0002.html#lemma:alt_definition_of_norm","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.norm_eq_prod_embeddings","lean_decl_name":"Algebra.norm_eq_prod_embeddings","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Norm/Basic.lean","lean4_code":"theorem norm_eq_prod_embeddings [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsAlgClosed E]\n    (x : L) : algebraMap K E (norm K x) = ∏ σ : L →ₐ[K] E, σ x := by\n  have hx := Algebra.IsSeparable.isIntegral K x\n  rw [norm_eq_norm_adjoin K x, RingHom.map_pow, ← adjoin.powerBasis_gen hx,\n    norm_eq_prod_embeddings_gen E (adjoin.powerBasis hx) (IsAlgClosed.splits_codomain _)]\n  · exact (prod_embeddings_eq_finrank_pow L (L := K⟮x⟯) E (adjoin.powerBasis hx)).symm\n  · haveI := Algebra.isSeparable_tower_bot_of_isSeparable K K⟮x⟯ L\n    exact Algebra.IsSeparable.isSeparable K _\n\n","line":219,"end_line":227,"proof_status":"formalized"}
+{"id":"lem:lin_indep_iff_disc_ne_zero","label":"lin_indep_iff_disc_ne_zero","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["lemma:alt_definition_of_norm","defn_of_disc"],"nlp_text":"Let \\(L/K\\) be an extension of fields and let \\(B=\\{ b_1,\\dots ,b_n\\} \\) be a \\(K\\)-basis of \\(L\\). Then \\(\\Delta (B) \\neq 0\\).","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.discr_not_zero_of_basis","latex_link":"sect0002.html#lem:lin_indep_iff_disc_ne_zero","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.discr_not_zero_of_basis","lean_decl_name":"Algebra.discr_not_zero_of_basis","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Discriminant.lean","lean4_code":"theorem discr_not_zero_of_basis [Algebra.IsSeparable K L] (b : Basis ι K L) :\n    discr K b ≠ 0 := by\n  rw [discr_def, traceMatrix_of_basis, ← LinearMap.BilinForm.nondegenerate_iff_det_ne_zero]\n  exact traceForm_nondegenerate _ _\n\n/-- If `b` is a basis of a finite separable field extension `L/K`,\n  then `Algebra.discr K b` is a unit. -/\n","line":127,"end_line":133,"proof_status":"formalized"}
+{"id":"lemma:num_field_disc_in_terms_of_norm","label":"num_field_disc_in_terms_of_norm","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["lemma:alt_definition_of_norm","lemma:disc_of_prim_elt_basis","lemma:diff_of_irr_pol"],"nlp_text":"Let \\(K=\\mathbb {Q}(\\alpha )\\) be a number field with \\(n=[K:\\mathbb {Q}]\\) and let \\(B=\\{ 1,\\alpha ,\\alpha ^2,\\dots ,\\alpha ^{n-1}\\} \\). Then \\[ \\Delta (B)=(-1)^{\\frac{n(n-1)}{2}}N_{K/\\mathbb {Q}}(m_\\alpha '(\\alpha )) \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.discr_powerBasis_eq_norm","latex_link":"sect0002.html#lemma:num_field_disc_in_terms_of_norm","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.discr_powerBasis_eq_norm","lean_decl_name":"Algebra.discr_powerBasis_eq_norm","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Discriminant.lean","lean4_code":"theorem discr_powerBasis_eq_norm [Algebra.IsSeparable K L] :\n    discr K pb.basis =\n      (-1) ^ (n * (n - 1) / 2) *\n      norm K (aeval pb.gen (derivative (R := K) (minpoly K pb.gen))) := by\n  let E := AlgebraicClosure L\n  letI := fun a b : E => Classical.propDecidable (Eq a b)\n  have e : Fin pb.dim ≃ (L →ₐ[K] E) := by\n    refine equivOfCardEq ?_\n    rw [Fintype.card_fin, AlgHom.card]\n    exact (PowerBasis.finrank pb).symm\n  have hnodup : ((minpoly K pb.gen).aroots E).Nodup :=\n    nodup_roots (Separable.map (Algebra.IsSeparable.isSeparable K pb.gen))\n  have hroots : ∀ σ : L →ₐ[K] E, σ pb.gen ∈ (minpoly K pb.gen).aroots E := by\n    intro σ\n    rw [mem_roots, IsRoot.def, eval_map, ← aeval_def, aeval_algHom_apply]\n    repeat' simp [minpoly.ne_zero (Algebra.IsSeparable.isIntegral K pb.gen)]\n  apply (algebraMap K E).injective\n  rw [RingHom.map_mul, RingHom.map_pow, RingHom.map_neg, RingHom.map_one,\n    discr_powerBasis_eq_prod'' _ _ _ e]\n  congr\n  rw [norm_eq_prod_embeddings, prod_prod_Ioi_mul_eq_prod_prod_off_diag]\n  conv_rhs =>\n    congr\n    rfl\n    ext σ\n    rw [← aeval_algHom_apply,\n      aeval_root_derivative_of_splits (minpoly.monic (Algebra.IsSeparable.isIntegral K pb.gen))\n        (IsAlgClosed.splits_codomain _) (hroots σ),\n      ← Finset.prod_mk _ (hnodup.erase _)]\n  rw [Finset.prod_sigma', Finset.prod_sigma']\n  refine prod_bij' (fun i _ ↦ ⟨e i.2, e i.1 pb.gen⟩)\n    (fun σ hσ ↦ ⟨e.symm (PowerBasis.lift pb σ.2 ?_), e.symm σ.1⟩) ?_ ?_ ?_ ?_ (fun i _ ↦ by simp)\n  -- Porting note: `@mem_compl` was not necessary.\n    <;> simp only [mem_sigma, mem_univ, Finset.mem_mk, hnodup.mem_erase_iff, IsRoot.def,\n      mem_roots',\n      mem_singleton, true_and, @mem_compl _ _ _ (_), Sigma.forall, Equiv.apply_symm_apply,\n      PowerBasis.lift_gen, implies_true, Equiv.symm_apply_apply,\n      Sigma.ext_iff, Equiv.symm_apply_eq, heq_eq_eq, and_true] at *\n  · simpa only [aeval_def, eval₂_eq_eval_map] using hσ.2.2\n  · exact fun a b hba ↦ ⟨fun h ↦ hba <| e.injective <| pb.algHom_ext h.symm, hroots _⟩\n  · rintro a b hba ha\n    rw [ha, PowerBasis.lift_gen] at hba\n    exact hba.1 rfl\n  · exact fun a b _ ↦ pb.algHom_ext <| pb.lift_gen _ _\n\nsection Integral\n\nvariable {R : Type z} [CommRing R] [Algebra R K] [Algebra R L] [IsScalarTower R K L]\n\n","line":201,"end_line":249,"proof_status":"formalized"}
+{"id":"lemma:alt_definition_of_trace","label":"alt_definition_of_trace","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(K\\) be a number field, \\(\\alpha \\in K\\) and let \\(\\sigma _i\\) be the embeddings of \\(K\\) into \\(\\mathbb {C}\\). Then \\[ \\operatorname {Tr}_{K/\\mathbb {Q}}(\\alpha ) =\\sum _i \\sigma _i(\\alpha ). \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/trace_eq_sum_embeddings","latex_link":"sect0002.html#lemma:alt_definition_of_trace","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/trace_eq_sum_embeddings","lean_decl_name":"trace_eq_sum_embeddings","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Trace/Basic.lean","lean4_code":"theorem trace_eq_sum_embeddings [FiniteDimensional K L] [Algebra.IsSeparable K L] {x : L} :\n    algebraMap K E (Algebra.trace K L x) = ∑ σ : L →ₐ[K] E, σ x := by\n  have hx := Algebra.IsSeparable.isIntegral K x\n  let pb := adjoin.powerBasis hx\n  rw [trace_eq_trace_adjoin K x, Algebra.smul_def, RingHom.map_mul, ← adjoin.powerBasis_gen hx,\n    trace_eq_sum_embeddings_gen E pb (IsAlgClosed.splits_codomain _), ← Algebra.smul_def,\n    algebraMap_smul]\n  · exact (sum_embeddings_eq_finrank_mul L E pb).symm\n  · haveI := Algebra.isSeparable_tower_bot_of_isSeparable K K⟮x⟯ L\n    exact Algebra.IsSeparable.isSeparable K _\n\n","line":254,"end_line":264,"proof_status":"formalized"}
+{"id":"lemma:disc_via_embs","label":"disc_via_embs","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["lemma:alt_definition_of_trace","defn_of_disc"],"nlp_text":"Let \\(K\\) be a number field with basis \\(B=\\{ b_1,\\dots ,b_n\\} \\) and let \\(\\sigma _1,\\dots ,\\sigma _n\\) be the embeddings of \\(K\\) into \\(\\mathbb {C}\\). Now let \\(M\\) be the matrix \\[ \\left(\\begin{matrix} \\sigma _1(b_1) & \\cdots & \\sigma _1(b_n) \\\\ \\vdots & & \\vdots \\\\ \\sigma _n(b_1) & \\cdots & \\sigma _n(b_n) \\end{matrix} \\right). \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.discr_eq_det_embeddingsMatrixReindex_pow_two","latex_link":"sect0002.html#lemma:disc_via_embs","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.discr_eq_det_embeddingsMatrixReindex_pow_two","lean_decl_name":"Algebra.discr_eq_det_embeddingsMatrixReindex_pow_two","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Discriminant.lean","lean4_code":"theorem discr_eq_det_embeddingsMatrixReindex_pow_two\n    [Algebra.IsSeparable K L] (e : ι ≃ (L →ₐ[K] E)) :\n    algebraMap K E (discr K b) = (embeddingsMatrixReindex K E b e).det ^ 2 := by\n  rw [discr_def, RingHom.map_det, RingHom.mapMatrix_apply,\n    traceMatrix_eq_embeddingsMatrixReindex_mul_trans, det_mul, det_transpose, pow_two]\n\n/-- The discriminant of a power basis. -/\n","line":143,"end_line":149,"proof_status":"formalized"}
+{"id":"lemma:disc_int_basis","label":"disc_int_basis","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["lemma:int_basis_int_disc","lem:disc_change_of_basis"],"nlp_text":"Let \\(K = \\mathbb {Q}(\\alpha )\\) be a number field, where \\(\\alpha \\) is an algebraic integer. Let \\(B = \\{ 1, \\alpha , \\ldots , \\alpha ^{[K : \\mathbb {Q}] - 1} \\} \\) be the basis given by \\(\\alpha \\) and let \\(x \\in \\mathcal{O}_K\\). Then \\(\\Delta (B)x \\in \\mathbb {Z}[\\alpha ]\\).","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.discr_mul_isIntegral_mem_adjoin","latex_link":"sect0002.html#lemma:disc_int_basis","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.discr_mul_isIntegral_mem_adjoin","lean_decl_name":"Algebra.discr_mul_isIntegral_mem_adjoin","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Discriminant.lean","lean4_code":"theorem discr_mul_isIntegral_mem_adjoin [Algebra.IsSeparable K L] [IsIntegrallyClosed R]\n    [IsFractionRing R K] {B : PowerBasis K L} (hint : IsIntegral R B.gen) {z : L}\n    (hz : IsIntegral R z) : discr K B.basis • z ∈ adjoin R ({B.gen} : Set L) := by\n  have hinv : IsUnit (traceMatrix K B.basis).det := by\n    simpa [← discr_def] using discr_isUnit_of_basis _ B.basis\n  have H :\n    (traceMatrix K B.basis).det • (traceMatrix K B.basis) *ᵥ (B.basis.equivFun z) =\n      (traceMatrix K B.basis).det • fun i => trace K L (z * B.basis i) := by\n    congr; exact traceMatrix_of_basis_mulVec _ _\n  have cramer := mulVec_cramer (traceMatrix K B.basis) fun i => trace K L (z * B.basis i)\n  suffices ∀ i, ((traceMatrix K B.basis).det • B.basis.equivFun z) i ∈ (⊥ : Subalgebra R K) by\n    rw [← B.basis.sum_repr z, Finset.smul_sum]\n    refine Subalgebra.sum_mem _ fun i _ => ?_\n    replace this := this i\n    rw [← discr_def, Pi.smul_apply, mem_bot] at this\n    obtain ⟨r, hr⟩ := this\n    rw [Basis.equivFun_apply] at hr\n    rw [← smul_assoc, ← hr, algebraMap_smul]\n    refine Subalgebra.smul_mem _ ?_ _\n    rw [B.basis_eq_pow i]\n    exact Subalgebra.pow_mem _ (subset_adjoin (Set.mem_singleton _)) _\n  intro i\n  rw [← H, ← mulVec_smul] at cramer\n  replace cramer := congr_arg (mulVec (traceMatrix K B.basis)⁻¹) cramer\n  rw [mulVec_mulVec, nonsing_inv_mul _ hinv, mulVec_mulVec, nonsing_inv_mul _ hinv, one_mulVec,\n    one_mulVec] at cramer\n  rw [← congr_fun cramer i, cramer_apply, det_apply]\n  refine\n    Subalgebra.sum_mem _ fun σ _ => Subalgebra.zsmul_mem _ (Subalgebra.prod_mem _ fun j _ => ?_) _\n  by_cases hji : j = i\n  · simp only [updateCol_apply, hji, PowerBasis.coe_basis]\n    exact mem_bot.2 (IsIntegrallyClosed.isIntegral_iff.1 <| isIntegral_trace (hz.mul <| hint.pow _))\n  · simp only [updateCol_apply, hji, PowerBasis.coe_basis]\n    exact mem_bot.2\n      (IsIntegrallyClosed.isIntegral_iff.1 <| isIntegral_trace <| (hint.pow _).mul (hint.pow _))\n\nend Integral\n\nend Field\n\nsection Int\n\n/-- Two (finite) ℤ-bases have the same discriminant. -/\n","line":261,"end_line":303,"proof_status":"formalized"}
+{"id":"theorem:ring_of_ints_of_cyclo","label":"ring_of_ints_of_cyclo","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["lemma:disc_int_basis","lem:discr_of_cyclo","lemma:eis_crit_and_alg_ints"],"nlp_text":"Let \\(\\zeta _p\\) be a \\(p\\)-th root of unity for \\(p\\) an odd prime, let \\(\\lambda _p=1-\\zeta _p\\) and \\(K=\\mathbb {Q}(\\zeta _p)\\). Then \\(\\mathcal{O}_K=\\mathbb {Z}[\\zeta _p]=\\mathbb {Z}[\\lambda _p]\\).","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/IsCyclotomicExtension.Rat.isIntegralClosure_adjoin_singleton_of_prime_pow","latex_link":"sect0003.html#theorem:ring_of_ints_of_cyclo","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/IsCyclotomicExtension.Rat.isIntegralClosure_adjoin_singleton_of_prime_pow","lean_decl_name":"IsCyclotomicExtension.Rat.isIntegralClosure_adjoin_singleton_of_prime_pow","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/NumberTheory/Cyclotomic/Rat.lean","lean4_code":"theorem isIntegralClosure_adjoin_singleton_of_prime_pow [hcycl : IsCyclotomicExtension {p ^ k} ℚ K]\n    (hζ : IsPrimitiveRoot ζ (p ^ k)) : IsIntegralClosure (adjoin ℤ ({ζ} : Set K)) ℤ K := by\n  refine ⟨Subtype.val_injective, @fun x => ⟨fun h => ⟨⟨x, ?_⟩, rfl⟩, ?_⟩⟩\n  swap\n  · rintro ⟨y, rfl⟩\n    exact\n      IsIntegral.algebraMap\n        ((le_integralClosure_iff_isIntegral.1\n          (adjoin_le_integralClosure (hζ.isIntegral (NeZero.pos _)))).isIntegral _)\n  let B := hζ.subOnePowerBasis ℚ\n  have hint : IsIntegral ℤ B.gen := (hζ.isIntegral (NeZero.pos _)).sub isIntegral_one\n  -- Porting note: the following `letI` was not needed because the locale `cyclotomic` set it\n  -- as instances.\n  letI := IsCyclotomicExtension.finiteDimensional {p ^ k} ℚ K\n  have H := discr_mul_isIntegral_mem_adjoin ℚ hint h\n  obtain ⟨u, n, hun⟩ := discr_prime_pow_eq_unit_mul_pow' hζ\n  rw [hun] at H\n  replace H := Subalgebra.smul_mem _ H u.inv\n  rw [← smul_assoc, ← smul_mul_assoc, Units.inv_eq_val_inv, zsmul_eq_mul, ← Int.cast_mul,\n    Units.inv_mul, Int.cast_one, one_mul, smul_def, map_pow] at H\n  cases k\n  · haveI : IsCyclotomicExtension {1} ℚ K := by simpa using hcycl\n    have : x ∈ (⊥ : Subalgebra ℚ K) := by\n      rw [singleton_one ℚ K]\n      exact mem_top\n    obtain ⟨y, rfl⟩ := mem_bot.1 this\n    replace h := (isIntegral_algebraMap_iff (algebraMap ℚ K).injective).1 h\n    obtain ⟨z, hz⟩ := IsIntegrallyClosed.isIntegral_iff.1 h\n    rw [← hz, ← IsScalarTower.algebraMap_apply]\n    exact Subalgebra.algebraMap_mem _ _\n  · have hmin : (minpoly ℤ B.gen).IsEisensteinAt (Submodule.span ℤ {(p : ℤ)}) := by\n      have h₁ := minpoly.isIntegrallyClosed_eq_field_fractions' ℚ hint\n      have h₂ := hζ.minpoly_sub_one_eq_cyclotomic_comp (cyclotomic.irreducible_rat (NeZero.pos _))\n      rw [IsPrimitiveRoot.subOnePowerBasis_gen] at h₁\n      rw [h₁, ← map_cyclotomic_int, show Int.castRingHom ℚ = algebraMap ℤ ℚ by rfl,\n        show X + 1 = map (algebraMap ℤ ℚ) (X + 1) by simp, ← map_comp] at h₂\n      rw [IsPrimitiveRoot.subOnePowerBasis_gen,\n        map_injective (algebraMap ℤ ℚ) (algebraMap ℤ ℚ).injective_int h₂]\n      exact cyclotomic_prime_pow_comp_X_add_one_isEisensteinAt p _\n    refine\n      adjoin_le ?_\n        (mem_adjoin_of_smul_prime_pow_smul_of_minpoly_isEisensteinAt (n := n)\n          (Nat.prime_iff_prime_int.1 hp.out) hint h (by simpa using H) hmin)\n    simp only [Set.singleton_subset_iff, SetLike.mem_coe]\n    exact Subalgebra.sub_mem _ (self_mem_adjoin_singleton ℤ _) (Subalgebra.one_mem _)\n\n","line":72,"end_line":117,"proof_status":"formalized"}
+{"id":"lemma:ideals_mult_to_power","label":"ideals_mult_to_power","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(R\\) be a Dedekind domain, \\(p\\) a prime and \\(\\mathfrak {a},\\mathfrak {b},\\mathfrak {c}\\) ideals such that \\[ \\mathfrak {a}\\mathfrak {b}=\\mathfrak {c}^p \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/ideal.exists_eq_pow_of_mul_eq_pow","latex_link":"sect0003.html#lemma:ideals_mult_to_power","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/ideal.exists_eq_pow_of_mul_eq_pow","lean_decl_name":"ideal.exists_eq_pow_of_mul_eq_pow","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Algebra/GCDMonoid/Basic.lean","lean4_code":"theorem exists_eq_pow_of_mul_eq_pow [GCDMonoid α] [Subsingleton αˣ]\n    {a b c : α} (hab : IsUnit (gcd a b)) {k : ℕ} (h : a * b = c ^ k) : ∃ d : α, a = d ^ k :=\n  let ⟨d, hd⟩ := exists_associated_pow_of_mul_eq_pow hab h\n  ⟨d, (associated_iff_eq.mp hd).symm⟩\n\n","line":590,"end_line":594,"proof_status":"formalized"}
+{"id":"lemma:may_assume_coprime","label":"may_assume_coprime","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["lemma:ideals_mult_to_power","theorem:ring_of_ints_of_cyclo","lemma:unit_lemma","lemma:zeta_pow_sub_eq_unit_zeta_sub_one"],"nlp_text":"Let \\(p \\geq 3\\) be an prime number, \\(\\zeta _p\\) a \\(p\\)-th root of unity and \\(K=\\mathbb {Q}(\\zeta _p)\\). Assume that we have \\(x,y,z \\in \\mathbb {Z}\\) with \\(\\gcd (xyz,p)=1\\) and such that \\[ x^p+y^p=z^p. \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/FltRegular.CaseI.may_assume","latex_link":"sect0004.html#lemma:may_assume_coprime","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/FltRegular.CaseI.may_assume","lean_decl_name":"FltRegular.CaseI.may_assume","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"flt-regular","source_path":"FltRegular/CaseI/Statement.lean","lean4_code":"    rw [← Int.even_pow' two_ne_zero, ← H]\n    exact (Odd.of_mul_left (Odd.of_mul_left hI)).pow.add_odd\n      (Odd.of_mul_right (Odd.of_mul_left hI)).pow\n  have hprod : a * b * c ≠ 0 := by\n    intro h\n    simp [h] at hI\n  have hp5 : 5 ≤ p := by\n    by_contra! habs\n    have : 2 < p := Nat.lt_of_le_of_ne hpri.out.two_le hodd.symm\n    interval_cases p\n    · exact fermatLastTheoremFor_iff_int.1 fermatLastTheoremThree a b c\n        (fun ha ↦ hprod <| by simp [ha]) (fun hb ↦ hprod <| by simp [hb])\n        (fun hc ↦ hprod <| by simp [hc]) H\n    · exact Nat.not_prime_mul one_lt_two.ne' one_lt_two.ne' hpri.out\n  let d := ({a, b, c} : Finset ℤ).gcd id\n  have hd : d ≠ 0 := Finset.gcd_ne_zero_iff.mpr ⟨c, by simp, fun hc ↦ hprod <| by simp_all⟩\n  have hdiv : ¬↑p ∣ a / d * (b / d) * (c / d) := by\n    contrapose! hI with hdiv\n    have hadiv : d ∣ a := gcd_dvd (by simp)\n    have hbdiv : d ∣ b := gcd_dvd (by simp)\n    have hcdiv : d ∣ c := gcd_dvd (by simp)\n    rw [← Int.ediv_mul_cancel hadiv, ← Int.ediv_mul_cancel hbdiv, ← Int.ediv_mul_cancel hcdiv]\n    convert dvd_mul_of_dvd_right hdiv (d * d * d) using 1\n    grind\n  rcases MayAssume.coprime H hprod with ⟨Hxyz, hunit, hprodxyx⟩\n  obtain ⟨X, Y, Z, H1, H2, H3, _, H5⟩ := a_not_cong_b hpri.out hp5 hprodxyx Hxyz hunit hdiv\n  exact Heasy hreg hp5 H2 H3 H5 H1\n\nend CaseI\n\ntheorem ab_coprime {a b c : ℤ} (H : a ^ p + b ^ p = c ^ p) (hpzero : p ≠ 0)\n    (hgcd : ({a, b, c} : Finset ℤ).gcd id = 1) : IsCoprime a b := by\n  rw [isCoprime_iff_gcd_eq_one]\n  by_contra! h\n  obtain ⟨q, hqpri, hq⟩ := exists_prime_and_dvd h\n  replace hqpri : Prime (q : ℤ) := prime_iff_natAbs_prime.2 (by simp [hqpri])\n  obtain ⟨n, hn⟩ := hq\n  have haq : ↑q ∣ a := by\n","line":33,"end_line":70,"proof_status":"formalized"}
+{"id":"lem:dvd_coeff_cycl_integer","label":"dvd_coeff_cycl_integer","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(p\\) be an prime number, \\(\\zeta _p\\) a \\(p\\)-th root of unity and \\( \\alpha \\in \\mathbb {Z}[\\zeta _p]\\) with \\(\\alpha =\\sum _i a_i \\zeta _p^i\\). Let us suppose that there is \\(i\\) such that \\(a_i = 0\\). If \\(n\\) is an integer that divides \\(\\alpha \\) in \\(\\mathbb {Z}[\\zeta _p]\\), then \\(n\\) divides each \\(a_i\\).","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/dvd_coeff_cycl_integer","latex_link":"sect0004.html#lem:dvd_coeff_cycl_integer","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/dvd_coeff_cycl_integer","lean_decl_name":"dvd_coeff_cycl_integer","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"flt-regular","source_path":"FltRegular/NumberTheory/Cyclotomic/CyclRat.lean","lean4_code":"\nend IntFacts\n","line":377,"end_line":423,"proof_status":"formalized"}
+{"id":"theorem:FLT_case_one","label":"FLT_case_one","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["lem:dvd_coeff_cycl_integer","lem:exists_int_sum_eq_zero","lem:flt_ideals_coprime","defn:is_regular_number","lemma:may_assume_coprime"],"nlp_text":"Let \\(p\\) be an odd regular prime. Then \\[ x^p+y^p=z^p \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/FltRegular.caseI","latex_link":"sect0004.html#theorem:FLT_case_one","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/FltRegular.caseI","lean_decl_name":"FltRegular.caseI","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:gen_ideal_coprimality","label":"gen_ideal_coprimality","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(p\\) be a regular odd prime, \\(x,y,z,\\epsilon \\in \\mathbb {Z}[\\zeta _p]\\), \\(\\epsilon \\) a unit, and \\(n \\in \\mathbb {Z}_{\\geq 1}\\). Assume \\(x,y,z\\) are coprime to \\((1-\\zeta _p)\\), \\(x^p+y^p+\\epsilon (1-\\zeta _p)^{pn}z^p=0\\) and \\(x+y\\) is divisible by \\(\\mathfrak {p}^2\\) and \\(x+ \\zeta _p^ky\\) is only divisible by \\(\\mathfrak {p}=(1-\\zeta _p)\\) (for \\(0 \\lt k \\leq p-1\\)). Let \\(\\mathfrak {m}= \\mathrm{gcd}((x),(y))\\). Then: We can write \\[ (x+y)=\\mathfrak {p}^{p(m-1)+1}\\mathfrak {m}\\mathfrak {c}_0 \\]","doc_url":null,"latex_link":"sect0004.html#lem:gen_ideal_coprimality","lean_link":null,"lean_decl_name":"gen_ideal_coprimality","kind_description":"theorems and lemmas","status_from_color":"statement_formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"thm:gen_flt_eqn","label":"gen_flt_eqn","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["lem:gen_ideal_coprimality","lem:gen_dvd_by_frak_p"],"nlp_text":"Let \\(p\\) be a regular odd prime, \\(\\epsilon \\in \\mathbb {Z}[\\zeta _p]^\\times \\) and \\(n \\in \\mathbb {Z}_{\\geq 1}\\). Then the equation \\(x^p+y^p+\\epsilon (1-\\zeta _p)^{pn}z^p=0\\) has no solutions with \\(x,y,z \\in \\mathbb {Z}[\\zeta _p]\\), all non-zero and \\(xyz\\) coprime to \\((1-\\zeta _p)\\).","doc_url":null,"latex_link":"sect0004.html#thm:gen_flt_eqn","lean_link":null,"lean_decl_name":"gen_flt_eqn","kind_description":"theorems and lemmas","status_from_color":"statement_formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"defn_of_disc","label":"defn_of_disc","background_color":"#b0eca3","border_color":"green","kind":"box","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(A,K\\) be commutative rings with \\(K\\) and \\(A\\)-algebra. let \\(B=\\{ b_1,\\dots ,b_n\\} \\) be a set of elements in \\(K\\). The discriminant of \\(B\\) is defined as \\[ \\Delta (B)= \\det \\left(\\begin{matrix} \\operatorname {Tr}_{K/A}(b_1b_1) & \\cdots & \\operatorname {Tr}_{K/A}(b_1b_n) \\\\ \\vdots & & \\vdots \\\\ \\operatorname {Tr}_{K/A}(b_nb_1) & \\cdots & \\operatorname {Tr}_{K/A}(b_nb_n) \\end{matrix} \\right). \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.discr","latex_link":"sect0002.html#defn_of_disc","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.discr","lean_decl_name":"Algebra.discr","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Discriminant.lean","lean4_code":"noncomputable def discr (A : Type u) {B : Type v} [CommRing A] [CommRing B] [Algebra A B]\n    [Fintype ι] (b : ι → B) := (traceMatrix A b).det\n\n","line":66,"end_line":68,"proof_status":"formalized"}
+{"id":"lemma:disc_of_prim_elt_basis","label":"disc_of_prim_elt_basis","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["defn_of_disc","lemma:disc_via_embs"],"nlp_text":"Let \\(K\\) be a number field and \\(B=\\{ 1,\\alpha ,\\alpha ^2,\\dots ,\\alpha ^{n-1}\\} \\) for some \\(\\alpha \\in K\\). Then \\[ \\Delta (B)=\\prod _{i \\lt j} (\\sigma _i(\\alpha )-\\sigma _j(\\alpha ))^2 \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.discr_powerBasis_eq_prod","latex_link":"sect0002.html#lemma:disc_of_prim_elt_basis","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.discr_powerBasis_eq_prod","lean_decl_name":"Algebra.discr_powerBasis_eq_prod","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Discriminant.lean","lean4_code":"theorem discr_powerBasis_eq_prod (e : Fin pb.dim ≃ (L →ₐ[K] E)) [Algebra.IsSeparable K L] :\n    algebraMap K E (discr K pb.basis) =\n      ∏ i : Fin pb.dim, ∏ j ∈ Ioi i, (e j pb.gen - e i pb.gen) ^ 2 := by\n  rw [discr_eq_det_embeddingsMatrixReindex_pow_two K E pb.basis e,\n    embeddingsMatrixReindex_eq_vandermonde, det_transpose, det_vandermonde, ← prod_pow]\n  congr; ext i\n  rw [← prod_pow]\n\n/-- A variation of `Algebra.discr_powerBasis_eq_prod`. -/\n","line":150,"end_line":158,"proof_status":"formalized"}
+{"id":"lemma:int_basis_int_disc","label":"int_basis_int_disc","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["defn_of_disc","lemma:trace_of_alg_int_is_int"],"nlp_text":"Let \\(K\\) be a number field and \\(B=\\{ b_1,\\dots ,b_n\\} \\) be elements in \\(\\mathcal{O}_K\\), then \\(\\Delta (B) \\in \\mathbb {Z}\\).","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.discr_isIntegral","latex_link":"sect0002.html#lemma:int_basis_int_disc","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.discr_isIntegral","lean_decl_name":"Algebra.discr_isIntegral","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Discriminant.lean","lean4_code":"theorem discr_isIntegral {b : ι → L} (h : ∀ i, IsIntegral R (b i)) : IsIntegral R (discr K b) := by\n  classical\n  rw [discr_def]\n  exact IsIntegral.det fun i j ↦ isIntegral_trace ((h i).mul (h j))\n\n/-- Let `K` be the fraction field of an integrally closed domain `R` and let `L` be a finite\nseparable extension of `K`. Let `B : PowerBasis K L` be such that `IsIntegral R B.gen`.\nThen for all, `z : L` that are integral over `R`, we have\n`(discr K B.basis) • z ∈ adjoin R ({B.gen} : Set L)`. -/\n","line":252,"end_line":260,"proof_status":"formalized"}
+{"id":"lem:disc_change_of_basis","label":"disc_change_of_basis","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["defn_of_disc"],"nlp_text":"Let \\(K\\) be a number field and \\(B,B'\\) bases for \\(K/\\mathbb {Q}\\). If \\(P\\) denotes the change of basis matrix, then \\[ \\Delta (B)=\\det (P)^2 \\Delta (B'). \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.discr_of_matrix_mulVec","latex_link":"sect0002.html#lem:disc_change_of_basis","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.discr_of_matrix_mulVec","lean_decl_name":"Algebra.discr_of_matrix_mulVec","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Discriminant.lean","lean4_code":"theorem discr_of_matrix_mulVec (b : ι → B) (P : Matrix ι ι A) :\n    discr A (P.map (algebraMap A B) *ᵥ b) = P.det ^ 2 * discr A b := by\n  rw [discr_def, traceMatrix_of_matrix_mulVec, det_mul, det_mul, det_transpose, mul_comm, ←\n    mul_assoc, discr_def, pow_two]\n\nend Basic\n\nsection Field\n\nvariable (K : Type u) {L : Type v} (E : Type z) [Field K] [Field L] [Field E]\nvariable [Algebra K L] [Algebra K E]\nvariable [Module.Finite K L] [IsAlgClosed E]\n\n/-- If `b` is a basis of a finite separable field extension `L/K`, then `Algebra.discr K b ≠ 0`. -/\n","line":113,"end_line":126,"proof_status":"formalized"}
+{"id":"lem:exists_int_sum_eq_zero","label":"exists_int_sum_eq_zero","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["lem:exists_int_sub_pow_prime_dvd","lemma:unit_lemma"],"nlp_text":"Let \\(p \\geq 3\\) be an prime number, \\(\\zeta _p\\) a \\(p\\)-th root of unity and \\( \\alpha \\in \\mathbb {Z}[\\zeta _p]\\). Let \\(x\\) and \\(y\\) be integers such that \\(x+y\\zeta _p^i=u \\alpha ^p\\) with \\(u \\in \\mathbb {Z}[\\zeta _p]^\\times \\) and \\(\\alpha \\in \\mathbb {Z}[\\zeta _p]\\). Then there is an integer \\(k\\) such that \\[ x+y\\zeta _p^i-\\zeta _p^{2k}x-\\zeta _p^{2k-i}y \\equiv 0 \\pmod p. \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/FltRegular.CaseI.exists_int_sum_eq_zero","latex_link":"sect0004.html#lem:exists_int_sum_eq_zero","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/FltRegular.CaseI.exists_int_sum_eq_zero","lean_decl_name":"FltRegular.CaseI.exists_int_sum_eq_zero","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"flt-regular","source_path":"FltRegular/NumberTheory/Cyclotomic/CaseI.lean","lean4_code":"    ringOfIntegersComplexConj K (x + y * ↑(hζ.unit' ^ i) : 𝓞 K) =\n      x + y * (hζ.unit' ^ (-i) : (𝓞 K)ˣ) := by\n  haveI := IsCyclotomicExtension.IsCMField K hp\n  ext1\n  simp only [map_add, map_intCast, map_mul, coe_ringOfIntegersComplexConj, zpow_neg, map_units_inv,\n    add_right_inj, mul_eq_mul_left_iff, Int.cast_eq_zero]\n  simp_rw [NumberField.Units.coe_zpow]\n  left\n  simp only [map_zpow₀]\n  rw [← inv_zpow]\n  congr\n  suffices hζ.unit' ∈ Units.torsion K by\n","line":30,"end_line":41,"proof_status":"formalized"}
+{"id":"lemma:cyclo_poly_deg","label":"cyclo_poly_deg","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For \\(n\\) any integer, \\(\\Phi _n\\) (the \\(n\\)-th cyclotomic polynomial) is a polynomial of degree \\(\\varphi (n)\\) (where \\(\\varphi \\) is Euler’s Totient function).","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Polynomial.degree_cyclotomic","latex_link":"sect0003.html#lemma:cyclo_poly_deg","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Polynomial.degree_cyclotomic","lean_decl_name":"Polynomial.degree_cyclotomic","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean","lean4_code":"theorem degree_cyclotomic' {ζ : R} {n : ℕ} (h : IsPrimitiveRoot ζ n) :\n    (cyclotomic' n R).degree = Nat.totient n := by\n  simp only [degree_eq_natDegree (cyclotomic'_ne_zero n R), natDegree_cyclotomic' h]\n\n/-- The roots of `cyclotomic' n R` are the primitive `n`-th roots of unity. -/\n","line":105,"end_line":109,"proof_status":"formalized"}
+{"id":"lem:discr_of_cyclo","label":"discr_of_cyclo","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["lemma:cyclo_poly_deg","lemma:num_field_disc_in_terms_of_norm","lemma:cyclo_poly_irr"],"nlp_text":"Let \\(\\zeta _p\\) be a \\(p\\)-th root of unity for \\(p\\) an odd prime, let \\(\\lambda _p=1-\\zeta _p\\) and \\(K=\\mathbb {Q}(\\zeta _p)\\). Then \\[ \\Delta (\\{ 1,\\zeta _p,\\dots ,\\zeta _p^{p-2}\\} )=\\Delta (\\{ 1,\\lambda _p,\\dots ,\\lambda _p^{p-2}\\} )=(-1)^{\\frac{(p-1)}{2}}p^{p-2}. \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/IsCyclotomicExtension.Rat.discr_prime_pow'","latex_link":"sect0003.html#lem:discr_of_cyclo","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/IsCyclotomicExtension.Rat.discr_prime_pow'","lean_decl_name":"IsCyclotomicExtension.Rat.discr_prime_pow'","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:exists_int_sub_pow_prime_dvd","label":"exists_int_sub_pow_prime_dvd","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(p\\) be an prime number, \\(\\zeta _p\\) a \\(p\\)-th root of unity and \\(\\alpha \\in \\mathbb {Z}[\\zeta _p]\\). Then \\(\\alpha ^p\\) is congruent to an integer modulo \\(p\\).","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/exists_int_sub_pow_prime_dvd","latex_link":"sect0004.html#lem:exists_int_sub_pow_prime_dvd","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/exists_int_sub_pow_prime_dvd","lean_decl_name":"exists_int_sub_pow_prime_dvd","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"flt-regular","source_path":"FltRegular/NumberTheory/Cyclotomic/CyclRat.lean","lean4_code":"  isPrimitiveRoot_of_mem_primitiveRoots <| mem_primitiveRoots_of_mem_nthRootsFinset hp hη hne1\n\nend Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots\n\nopen FiniteDimensional Polynomial Algebra Nat Finset Fintype\n\nvariable (p : ℕ) (L : Type*) [Field L] [CharZero L] [IsCyclotomicExtension {p} ℚ L]\n\ninstance Ring.toSubtractionMonoid {S : Type*} [Ring S] : SubtractionMonoid S := inferInstance\n\nsection IntFacts\n\nopen scoped NumberField\n\ninstance {K : Type*} [Field K] : Module (𝓞 K) (𝓞 K) := Semiring.toModule\n\nopen Ideal IsCyclotomicExtension\n\ntheorem exists_int_sub_pow_prime_dvd {A : Type*} [CommRing A] [IsCyclotomicExtension {p} ℤ A]\n    [hp : Fact p.Prime] (a : A) : ∃ m : ℤ, a ^ p - m ∈ span ({(p : A)} : Set A) := by\n  refine Algebra.adjoin_induction ?_ ?_ ?_ ?_ (adjoin_roots (S := {p}) (A := ℤ) a)\n  · intro x hx\n    use 1\n    simp_all\n  · intro r\n    use r ^ p\n    simp\n  · rintro x y _ _ ⟨b, hb⟩ ⟨c, hc⟩\n    obtain ⟨r, hr⟩ := exists_add_pow_prime_eq hp.out x y\n    rw [mul_assoc, mul_assoc] at hr\n    rw [hr]\n    use c + b\n    push_cast\n    rw [sub_add_eq_sub_sub, sub_eq_add_neg, sub_eq_add_neg, add_comm (x ^ p + _), add_assoc,\n      add_assoc]\n    apply Ideal.add_mem _ _\n","line":27,"end_line":62,"proof_status":"formalized"}
+{"id":"lem:loc_ramification","label":"loc_ramification","background_color":"#a3d6ff","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(K\\) be a number field, \\(p\\) be a rational prime below \\(\\mathfrak {p}\\) and let \\(\\alpha _0, \\xi \\) be in \\(\\mathcal{O}_\\mathfrak {p}^\\times \\) (the units of the ring of integers of the completion of \\(K\\) at \\(\\mathfrak {p}\\)) be such that \\[ \\alpha _0^p \\equiv \\xi \\mod \\mathfrak {p}^{m+r} \\]","doc_url":null,"latex_link":"sect0006.html#lem:loc_ramification","lean_link":null,"lean_decl_name":"loc_ramification","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:ramification_lem","label":"ramification_lem","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["lem:loc_ramification","lem:diff_ideal_eqn","lem:diff_ram"],"nlp_text":"Let \\(F=\\mathbb {Q}(\\zeta _p)\\) with \\(\\zeta _p\\) a primitive \\(p\\)-th root of unity. If \\(\\xi \\in \\mathcal{O}_F\\) and coprime to \\(\\lambda _p:=1-\\zeta _p\\) and if \\[ \\xi \\equiv \\alpha _0^p \\mod \\lambda _p^p \\]","doc_url":null,"latex_link":"sect0006.html#lem:ramification_lem","lean_link":null,"lean_decl_name":"ramification_lem","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"theorem:FLT_case_two","label":"FLT_case_two","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["thm:gen_flt_eqn","thm:Kummers_lemma"],"nlp_text":"Let \\(p\\) be an odd regular prime. Then \\[ x^p+y^p=z^p \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/FltRegular.caseII","latex_link":"sect0004.html#theorem:FLT_case_two","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/FltRegular.caseII","lean_decl_name":"FltRegular.caseII","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"flt-regular","source_path":"FltRegular/CaseII/Statement.lean","lean4_code":"section Mathlib.Data.Int.Init\n\nlemma Int.gcd_left_comm (a b c : ℤ) : Int.gcd a (Int.gcd b c) = Int.gcd b (Int.gcd a c) := by\n  rw [← Int.gcd_assoc, ← Int.gcd_assoc, Int.gcd_comm a b]\n\nend Mathlib.Data.Int.Init\n\n/-- CaseII. -/\ntheorem caseII {a b c : ℤ} {p : ℕ} [hpri : Fact p.Prime] (hreg : IsRegularPrime p) (hodd : p ≠ 2)\n    (hprod : a * b * c ≠ 0) (hgcd : ({a, b, c} : Finset ℤ).gcd id = 1)\n    (caseII : ↑p ∣ a * b * c) : a ^ p + b ^ p ≠ c ^ p := by\n  intro e\n  simp only [ne_eq, mul_eq_zero, not_or] at hprod\n  obtain ⟨⟨a0, b0⟩, c0⟩ := hprod\n  have hodd' := Nat.Prime.odd_of_ne_two hpri.out hodd\n  obtain hab | hc := (Nat.prime_iff_prime_int.mp hpri.out).dvd_or_dvd caseII\n  · obtain ha| hb := (Nat.prime_iff_prime_int.mp hpri.out).dvd_or_dvd hab\n    · refine not_exists_Int_solution' hreg hodd ⟨b, -c, -a, ?_, ?_, ?_, ?_⟩\n      · simp only [← hgcd, Finset.gcd_insert,id_eq, ← Int.coe_gcd, Int.neg_gcd,\n          ← LawfulSingleton.insert_empty_eq, Finset.gcd_empty, Int.gcd_left_comm _ a]\n      · rwa [dvd_neg]\n      · rwa [ne_eq, neg_eq_zero]\n      · simp [hodd'.neg_pow, ← e]\n    · refine not_exists_Int_solution' hreg hodd ⟨-c, a, -b, ?_, ?_, ?_, ?_⟩\n","line":80,"end_line":103,"proof_status":"formalized"}
+{"id":"FLT_regular","label":"FLT_regular","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["theorem:FLT_case_two","theorem:FLT_case_one"],"nlp_text":"Let \\(p\\) be an odd regular prime. Then \\[ x^p+y^p=z^p \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/flt_regular","latex_link":"sect0004.html#FLT_regular","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/flt_regular","lean_decl_name":"flt_regular","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"flt-regular","source_path":"FltRegular/FltRegular.lean","lean4_code":"theorem flt_regular {p : ℕ} [Fact p.Prime] (hreg : IsRegularPrime p) (hodd : p ≠ 2) :\n    FermatLastTheoremFor p := by\n  apply fermatLastTheoremFor_iff_int.mpr\n  intro a b c ha hb hc e\n  have hprod := mul_ne_zero (mul_ne_zero ha hb) hc\n  obtain ⟨e', hgcd, hprod'⟩ := MayAssume.coprime e hprod\n  let d := ({a, b, c} : Finset ℤ).gcd id\n  by_cases case : ↑p ∣ (a / d) * (b / d) * (c / d)\n  · exact caseII hreg hodd hprod' hgcd case e'\n  · exact caseI hreg case e'\n","line":8,"end_line":17,"proof_status":"formalized"}
+{"id":"lemma:unit_lemma","label":"unit_lemma","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["lemma:alg_int_abs_val_one","lem:roots_of_unity_in_cyclo"],"nlp_text":"Any unit \\(u\\) in \\(\\mathbb {Z}[\\zeta _p]\\) can be written in the form \\(\\beta \\zeta _p^k \\) with \\(k\\) an integer and \\(\\beta \\in \\mathbb {R}\\).","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/unit_lemma_gal_conj","latex_link":"sect0003.html#lemma:unit_lemma","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/unit_lemma_gal_conj","lean_decl_name":"unit_lemma_gal_conj","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:flt_ideals_coprime","label":"flt_ideals_coprime","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["lemma:zeta_pow_sub_eq_unit_zeta_sub_one"],"nlp_text":"Let \\(p \\geq 5\\) be an prime number, \\(\\zeta _p\\) a \\(p\\)-th root of unity and \\(x, y \\in \\mathbb {Z}\\) coprime. For \\(i \\neq j\\) with \\(i,j \\in {0,\\dots ,p-1}\\) we can write \\[ (\\zeta _p^i-\\zeta _p^j)=u(1-\\zeta _p) \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/fltIdeals_coprime","latex_link":"sect0004.html#lem:flt_ideals_coprime","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/fltIdeals_coprime","lean_decl_name":"fltIdeals_coprime","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"flt-regular","source_path":"FltRegular/NumberTheory/Cyclotomic/CyclRat.lean","lean4_code":"    ← (Fin.castOrderIso hdim).toEquiv.sum_comp] at hy\n  simp only [RelIso.coe_fn_toEquiv, Fin.val_castSucc, Fin.coe_orderIso_apply] at hy\n  conv_lhs at hy =>\n    congr; rfl; ext x\n    rw [smul_neg]\n    congr; congr; rfl; congr\n    rw [show ζ = hζ'.toInteger from rfl, ← hζ'.integralPowerBasis_gen, ← hb]\n    rfl; rfl; congr; congr; rfl; congr\n    rw [show ζ = hζ'.toInteger from rfl, ← hζ'.integralPowerBasis_gen, ← hb]\n  conv_lhs at hy =>\n    congr; rfl; ext x\n    rw [← show ∀ y, _ = _ from fun y => congr_fun b.coe_basis y, ← sub_eq_add_neg]\n  rw [sum_sub_distrib] at hy\n  replace hy := congr_arg (b.basis.coord ((Fin.castOrderIso hdim.symm) ⟨i, hi⟩)) hy\n  rw [← b.basis.equivFun_symm_apply, ← b.basis.equivFun_symm_apply, LinearMap.map_sub,\n    b.basis.coord_equivFun_symm, b.basis.coord_equivFun_symm, ← smul_eq_mul,\n","line":308,"end_line":323,"proof_status":"formalized"}
+{"id":"Kummer_alt","label":"Kummer_alt","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["lem:ramification_lem","Hilbert90","lem:Hilbert92"],"nlp_text":"Let \\(u \\in F\\) with \\(F=\\mathbb {Q}(\\zeta _p)\\) be a unit such that \\[ u \\equiv a^p \\mod \\lambda _p^p \\]","doc_url":null,"latex_link":"sect0007.html#Kummer_alt","lean_link":null,"lean_decl_name":"Kummer_alt","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lemma:diff_of_irr_pol","label":"diff_of_irr_pol","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(f\\) be a monic irreducible polynomial over a number field \\(K\\) and let \\(\\alpha \\) be one of its roots in \\(\\mathbb {C}\\). Then \\[ f'(\\alpha )=\\prod _{\\beta \\neq \\alpha } (\\alpha -\\beta ), \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Polynomial.aeval_root_derivative_of_splits","latex_link":"sect0002.html#lemma:diff_of_irr_pol","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Polynomial.aeval_root_derivative_of_splits","lean_decl_name":"Polynomial.aeval_root_derivative_of_splits","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Algebra/Polynomial/Splits.lean","lean4_code":"theorem aeval_root_derivative_of_splits [Algebra K L] [DecidableEq L] {P : K[X]} (hmo : P.Monic)\n    (hP : P.Splits (algebraMap K L)) {r : L} (hr : r ∈ P.aroots L) :\n    aeval r (Polynomial.derivative P) =\n    (((P.aroots L).erase r).map fun a => r - a).prod := by\n  replace hmo := hmo.map (algebraMap K L)\n  replace hP := (splits_id_iff_splits (algebraMap K L)).2 hP\n  rw [aeval_def, ← eval_map, ← derivative_map]\n  nth_rw 1 [eq_prod_roots_of_monic_of_splits_id hmo hP]\n  rw [eval_multiset_prod_X_sub_C_derivative hr]\n\n/-- If `P` is a monic polynomial that splits, then `coeff P 0` equals the product of the roots. -/\n","line":537,"end_line":547,"proof_status":"formalized"}
+{"id":"lemma:alg_int_abs_val_one","label":"alg_int_abs_val_one","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(\\alpha \\) be an algebraic integer all of whose conjugates have absolute value one. Then \\(\\alpha \\) is a root of unity.","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/mem_roots_of_unity_of_abs_eq_one","latex_link":"sect0003.html#lemma:alg_int_abs_val_one","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/mem_roots_of_unity_of_abs_eq_one","lean_decl_name":"mem_roots_of_unity_of_abs_eq_one","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"thm:Kummers_lemma","label":"Kummers_lemma","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["Kummer_alt"],"nlp_text":"Let \\(p\\) be a regular prime and let \\(u \\in \\mathbb {Z}[\\zeta _p]^\\times \\). If \\(u \\equiv a \\mod p\\) for some \\(a \\in \\mathbb {Z}\\), then there exists \\(v \\in \\mathbb {Z}[\\zeta _p]^\\times \\) such that \\(u=v^p\\).","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/eq_pow_prime_of_unit_of_congruent","latex_link":"sect0004.html#thm:Kummers_lemma","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/eq_pow_prime_of_unit_of_congruent","lean_decl_name":"eq_pow_prime_of_unit_of_congruent","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"flt-regular","source_path":"FltRegular/NumberTheory/KummersLemma/KummersLemma.lean","lean4_code":"      refine Int.modEq_iff_dvd.mp (Int.ModEq.pow_card_sub_one_eq_one hpri.out ?_).symm\n      rw [isCoprime_comm, (Nat.prime_iff_prime_int.mp hpri.out).coprime_iff_not_dvd]\n      intro h\n      replace h := Int.cast_dvd_cast (α := 𝓞 K) _ _ h\n      simp only [Int.cast_natCast, ← dvd_iff_dvd_of_dvd_sub hn] at h\n      refine hζ.zeta_sub_one_prime'.not_unit ((isUnit_pow_iff ?_).mp\n        (isUnit_of_dvd_unit ((associated_zeta_sub_one_pow_prime hζ).dvd.trans h) u.isUnit))\n      simpa only [ne_eq, tsub_eq_zero_iff_le, not_le] using hpri.out.one_lt\n    replace hn' := Int.cast_dvd_cast (α := 𝓞 K) _ _ hn'\n    simp only [Int.cast_natCast, Int.cast_sub, Int.cast_pow, Int.cast_one] at hn'\n    rw [← Ideal.mem_span_singleton, ← Ideal.Quotient.eq_zero_iff_mem,\n      RingHom.map_sub, sub_eq_zero] at hn hn' ⊢\n    rw [Units.val_pow_eq_pow_val, RingHom.map_pow, hn, ← RingHom.map_pow, hn']\n  have : (hζ.unit' - 1 : 𝓞 K) ^ p ∣ (↑(u ^ (p - 1)) : 𝓞 K) - 1 := by\n    rw [hx]\n    rw [sub_eq_iff_eq_add, add_comm] at hx\n    have H : Algebra.norm ℤ (1 + p • x) = 1 := norm_add_one_smul_of_isUnit hpri.out\n      hp x (by rw [nsmul_eq_mul, ← hx]; exact Units.isUnit _)\n    have := H ▸ zeta_sub_one_pow_dvd_norm_sub_pow hζ x\n    simpa [ge_iff_le, Int.cast_one, sub_self, nsmul_eq_mul, Nat.cast_mul, PNat.pos,\n      Nat.cast_pred, zero_sub, IsUnit.mul_iff, ne_eq, tsub_eq_zero_iff_le, not_le, dvd_neg,\n      Units.isUnit, and_true, zero_add] using this\n  obtain ⟨v, hv⟩ := exists_pow_eq_of_zeta_sub_one_pow_dvd_sub_one hp hreg hζ this\n  have hv' : IsIntegral ℤ v := by\n    apply IsIntegral.of_pow (NeZero.pos p); rw [hv]\n    exact NumberField.RingOfIntegers.isIntegral_coe _\n  set w : 𝓞 K := ⟨v, hv'⟩\n  have : IsUnit w := by\n    rw [← isUnit_pow_iff (NeZero.pos p).ne.symm]; convert (u ^ (p - 1) : (𝓞 K)ˣ).isUnit; ext\n    exact hv\n  have hv'' : this.unit ^ p = u ^ (p - 1) := by\n    ext; simpa only [Units.val_pow_eq_pow_val, IsUnit.unit_spec, SubmonoidClass.coe_pow] using hv\n  use u / this.unit\n  rw [div_pow, hv'', div_eq_mul_inv, ← pow_sub _ tsub_le_self,\n    tsub_tsub_cancel_of_le (Nat.Prime.one_lt hpri.out).le, pow_one]\n","line":48,"end_line":90,"proof_status":"formalized"}
+{"id":"Hilbert90","label":"Hilbert90","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["lem:exists_alg_int"],"nlp_text":"Let \\(K/F\\) be a Galois extension of number fields whose Galois group \\(\\operatorname{Gal}(K/F)\\) is cyclic with generator \\(\\sigma \\). If \\(\\alpha \\in K\\) is such that \\(N_{K/F}(\\alpha )=1\\), then \\[ \\alpha =\\beta / \\sigma (\\beta ) \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Hilbert90","latex_link":"sect0005.html#Hilbert90","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Hilbert90","lean_decl_name":"Hilbert90","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:Hilbert92","label":"Hilbert92","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(K/F\\) be a Galois extension of \\(F=\\mathbb {Q}(\\zeta _p)\\) with Galois group \\(\\operatorname{Gal}(K/F)\\) cyclic with generator \\(\\sigma \\). Then there exists a unit \\(\\eta \\in \\mathcal{O}_K\\) such that \\(N_{K/F}(\\eta )=1\\) but does not have the form \\(\\epsilon /\\sigma (\\epsilon )\\) for any unit \\(\\epsilon \\in \\mathcal{O}_K\\).","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Hilbert92","latex_link":"sect0005.html#lem:Hilbert92","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Hilbert92","lean_decl_name":"Hilbert92","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"flt-regular","source_path":"./FltRegular/NumberTheory/Hilbert92.lean","lean4_code":"lemma Hilbert92 [Algebra k K] [IsGalois k K] [NumberField k] [NumberField K]\n    (hKL : Nat.Prime (finrank k K)) (hpodd : finrank k K ≠ 2)\n    (σ : K ≃ₐ[k] K) (hσ : ∀ x, x ∈ Subgroup.zpowers σ) :\n    ∃ η : (𝓞 K)ˣ, Algebra.norm k (η : K) = 1 ∧ ∀ ε : (𝓞 K)ˣ, (η : K) ≠ ε / (σ ε : K) :=\n  have := IsUnramifiedAtInfinitePlaces_of_odd_finrank (hKL.odd_of_ne_two hpodd)\n  have : IsCyclic (K ≃ₐ[k] K) := ⟨σ, hσ⟩\n  almostHilbert92 (finrank k K) hKL rfl σ hσ hpodd\n\n","line":867,"end_line":874,"proof_status":"formalized"}
+{"id":"defn:is_regular_number","label":"is_regular_number","background_color":"#b0eca3","border_color":"green","kind":"box","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A prime number \\(p\\) is called regular if it does not divide the class number of \\(\\mathbb {Q}(\\zeta _p)\\).","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/IsRegularNumber","latex_link":"sect0004.html#defn:is_regular_number","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/IsRegularNumber","lean_decl_name":"IsRegularNumber","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"flt-regular","source_path":"FltRegular/NumberTheory/RegularPrimes.lean","lean4_code":"  (L : Type*) [Field L] [CharZero L] [IsCyclotomicExtension {2} ℚ L] :\n    IsPrincipalIdealRing (𝓞 L) := by\n","line":51,"end_line":52,"proof_status":"formalized"}
+{"id":"lem:roots_of_unity_in_cyclo","label":"roots_of_unity_in_cyclo","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(p\\) be a prime, \\(K=\\mathbb {Q}(\\zeta _p)\\) \\(\\alpha \\in K\\) such that there exists \\(n \\in \\mathbb {N}\\) such that \\(\\alpha ^n=1\\), then \\(\\alpha =\\pm \\zeta _p^k\\) for some \\(k\\).","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/roots_of_unity_in_cyclo","latex_link":"sect0003.html#lem:roots_of_unity_in_cyclo","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/roots_of_unity_in_cyclo","lean_decl_name":"roots_of_unity_in_cyclo","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"flt-regular","source_path":"FltRegular/NumberTheory/Cyclotomic/UnitLemmas.lean","lean4_code":"  apply_fun (fun n : ℕ => (n : 𝓞 K)) at hk\n  rw [Nat.cast_add, Nat.cast_mul, Nat.cast_two, Nat.cast_one, add_comm] at hk\n  intro h2m\n  have := Ideal.sub_mem I hpm (Ideal.mul_mem_right (↑k) I h2m)\n  rw [sub_eq_of_eq_add hk] at this\n  exact hζ.isPrime_one_sub_zeta.ne_top (Ideal.eq_top_of_isUnit_mem I this isUnit_one)\n\nlemma neg_one_eq_one_iff_two_eq_zero {M : Type*} [AddGroupWithOne M] :\n    (-1 : M) = 1 ↔ (2 : M) = 0 := by rw [neg_eq_iff_add_eq_zero, one_add_one_eq_two]\n\nlemma Units.coe_map_inv' {M N F : Type*} [Monoid M] [Monoid N] [FunLike F M N]\n    [MonoidHomClass F M N] (f : F) (m : Mˣ) :\n    ↑((Units.map (f : M →* N) m)⁻¹) = f ↑(m⁻¹ : Mˣ) :=\n  m.coe_map_inv (f : M →* N)\n\nvariable (K) in\ntheorem IsCyclotomicExtension.IsCMField (hp : 2 < p) :\n    IsCMField K :=\n  haveI := IsCyclotomicExtension.isAbelianGalois {p} ℚ K\n  haveI := nrRealPlaces_eq_zero_iff.1 (Rat.nrRealPlaces_eq_zero K hp)\n  ⟨⟩\n\nlemma unit_inv_conj_not_neg_zeta_runity_aux (u : (𝓞 K)ˣ) [Fact (p.Prime)] (hp : 2 < p) :\n    haveI := IsCyclotomicExtension.IsCMField K hp\n    algebraMap (𝓞 K) (𝓞 K ⧸ I) (unitsMulComplexConjInv K u).1 = 1 := by\n  haveI := IsCyclotomicExtension.IsCMField K hp\n  have := Units.coe_map_inv' (N := 𝓞 K ⧸ I) (algebraMap (𝓞 K) (𝓞 K ⧸ I)) (unitsComplexConj K u)\n  rw [unitsMulComplexConjInv_apply, Units.val_mul, map_mul, ← this, Units.mul_inv_eq_one,\n    Units.coe_map , MonoidHom.coe_coe]\n  haveI := Fact.mk hp\n  have hu := hζ.integralPowerBasis.basis.sum_repr u\n  let a := hζ.integralPowerBasis.basis.repr\n  let φn := hζ.integralPowerBasis.dim\n  simp_rw [PowerBasis.basis_eq_pow, IsPrimitiveRoot.integralPowerBasis_gen] at hu\n  have hu' := congr_arg (ringOfIntegersComplexConj K) hu\n  replace hu' : ∑ x : Fin φn, (a u) x • (ringOfIntegersComplexConj K)\n      (⟨ζ, hζ.isIntegral (NeZero.pos p)⟩ ^ (x : ℕ)) = unitsComplexConj K u := by\n    refine Eq.trans ?_ hu'\n    rw [map_sum]\n    congr 1\n    ext x\n    congr 1\n    rw [map_zsmul]\n  have : ∀ x : Fin φn, ringOfIntegersComplexConj K (⟨ζ, hζ.isIntegral (NeZero.pos p)⟩ ^ (x : ℕ)) =\n      ⟨ζ⁻¹, hζ.inv.isIntegral (NeZero.pos p)⟩ ^ (x : ℕ) := by\n    intro x\n    ext\n    simp only [map_pow, coe_ringOfIntegersComplexConj, RingOfIntegers.map_mk, inv_pow]\n    suffices hζ.unit' ∈ Units.torsion K by\n      have H := RingOfIntegers.ext_iff.1 <|\n        Units.ext_iff.1 <| unitsComplexConj_torsion K ⟨hζ.unit', ‹_›⟩\n      have : ↑↑hζ.unit' = ζ := rfl\n      simp only [Units.coe_mapEquiv, AlgEquiv.toRingEquiv_eq_coe, RingEquiv.coe_toMulEquiv,\n","line":206,"end_line":258,"proof_status":"formalized"}
+{"id":"lem:exists_alg_int","label":"exists_alg_int","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(K\\) be a number fields and \\(\\alpha \\in K\\). Then there exists a \\(n \\in \\mathbb {Z}\\backslash \\{ 0\\} \\) such that \\(n \\alpha \\) is an algebraic integer.","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/exists_alg_int","latex_link":"sect0005.html#lem:exists_alg_int","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/exists_alg_int","lean_decl_name":"exists_alg_int","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:diff_ideal_eqn","label":"diff_ideal_eqn","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["def:rel_different"],"nlp_text":"Let \\(K/F\\) be an extension of number fields and let \\(S\\) denote the set of \\(\\alpha \\in \\mathcal{O}_K\\) be such that \\(K=F(\\alpha )\\). Then \\[ \\mathfrak {d}_{K/F} = \\left(m_{\\alpha }'(\\alpha ) : \\alpha \\in S \\right) \\]","doc_url":null,"latex_link":"sect0006.html#lem:diff_ideal_eqn","lean_link":null,"lean_decl_name":"diff_ideal_eqn","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lemma:norm_of_alg_int_is_int","label":"norm_of_alg_int_is_int","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(K\\) is a number field and \\(\\alpha \\in \\mathcal{O}_K\\) then \\(N_{K/\\mathbb {Q}}(\\alpha )\\) is in \\(\\mathbb {Z}\\).","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.isIntegral_norm","latex_link":"sect0002.html#lemma:norm_of_alg_int_is_int","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.isIntegral_norm","lean_decl_name":"Algebra.isIntegral_norm","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Norm/Basic.lean","lean4_code":"theorem isIntegral_norm [Algebra R L] [Algebra R K] [IsScalarTower R K L] [Algebra.IsSeparable K L]\n    [FiniteDimensional K L] {x : L} (hx : IsIntegral R x) : IsIntegral R (norm K x) := by\n  have hx' : IsIntegral K x := hx.tower_top\n  rw [← isIntegral_algebraMap_iff (algebraMap K (AlgebraicClosure L)).injective, norm_eq_prod_roots]\n  · refine (IsIntegral.multiset_prod fun y hy => ?_).pow _\n    rw [mem_roots_map (minpoly.ne_zero hx')] at hy\n    use minpoly R x, minpoly.monic hx\n    rw [← aeval_def] at hy ⊢\n    exact minpoly.aeval_of_isScalarTower R x y hy\n  · apply IsAlgClosed.splits_codomain\n\n","line":238,"end_line":248,"proof_status":"formalized"}
+{"id":"lemma:eis_crit_and_alg_ints","label":"eis_crit_and_alg_ints","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["lemma:norm_of_alg_int_is_int","lemma:trace_of_alg_int_is_int"],"nlp_text":"Let \\(K = \\mathbb {Q}(\\alpha )\\) be a number field, where \\(\\alpha \\) is an algebraic integer with minimal polynomial that is Eisenstein at \\(p\\). Let \\(x \\in \\mathcal{O}_K\\) such that \\(p^n x \\in \\mathbb {Z}[\\alpha ]\\) for some \\(n\\). Then \\(x \\in \\mathbb {Z}[\\alpha ]\\).","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/mem_adjoin_of_smul_prime_pow_smul_of_minpoly_isEisensteinAt","latex_link":"sect0002.html#lemma:eis_crit_and_alg_ints","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/mem_adjoin_of_smul_prime_pow_smul_of_minpoly_isEisensteinAt","lean_decl_name":"mem_adjoin_of_smul_prime_pow_smul_of_minpoly_isEisensteinAt","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Polynomial/Eisenstein/IsIntegral.lean","lean4_code":"theorem mem_adjoin_of_smul_prime_pow_smul_of_minpoly_isEisensteinAt {B : PowerBasis K L}\n    (hp : Prime p) (hBint : IsIntegral R B.gen) {n : ℕ} {z : L} (hzint : IsIntegral R z)\n    (hz : p ^ n • z ∈ adjoin R ({B.gen} : Set L)) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) :\n    z ∈ adjoin R ({B.gen} : Set L) := by\n  induction' n with n hn\n  · simpa using hz\n  · rw [_root_.pow_succ', mul_smul] at hz\n    exact\n      hn (mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt hp hBint (hzint.smul _) hz hei)\n\n","line":360,"end_line":369,"proof_status":"formalized"}
+{"id":"lemma:zeta_pow_sub_eq_unit_zeta_sub_one","label":"zeta_pow_sub_eq_unit_zeta_sub_one","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(p\\) be an odd prime, \\(\\zeta _p\\) a primitive \\(p\\)-th root of unity and let \\(K=\\mathbb {Q}(\\zeta _p)\\). Then for any \\(i,j \\in {0,\\dots ,p-1}\\) with \\(i \\ne j\\), there exists a unit \\(u \\in \\mathcal{O}_K^\\times \\) such that \\(\\zeta _p^i-\\zeta _p^j = u (\\zeta _p-1)\\).","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/is_primitive_root.zeta_pow_sub_eq_unit_zeta_sub_one","latex_link":"sect0003.html#lemma:zeta_pow_sub_eq_unit_zeta_sub_one","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/is_primitive_root.zeta_pow_sub_eq_unit_zeta_sub_one","lean_decl_name":"is_primitive_root.zeta_pow_sub_eq_unit_zeta_sub_one","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:diff_ram","label":"diff_ram","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(K/F\\) be an extension of number fields with \\(\\mathfrak {p}_F, \\mathfrak {p}_K\\) prime ideas in \\(K,F\\) respectively, with \\(\\mathfrak {p}_K^e \\parallel \\mathfrak {p}_F\\) for \\(e \\gt 0\\). Then \\(\\mathfrak {p}_K^{e-1} \\mid \\mathfrak {d}_{K/F}\\) and \\(\\mathfrak {p}_K^{e} \\mid \\mathfrak {d}_{K/F}\\) iff \\(\\mathfrak {p}_F \\mid e\\).","doc_url":null,"latex_link":"sect0006.html#lem:diff_ram","lean_link":null,"lean_decl_name":"diff_ram","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"def:rel_different","label":"rel_different","background_color":"#b0eca3","border_color":"green","kind":"box","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(K, F\\) be number fields with \\(F \\subseteq K\\). Let \\(A\\) be an additive subgroup of \\(K\\). Let \\[ A^{-1}=\\{ \\alpha \\in K | \\alpha A \\in \\mathcal{O}_K\\} \\]","doc_url":null,"latex_link":"sect0006.html#def:rel_different","lean_link":null,"lean_decl_name":"rel_different","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lemma:trace_of_alg_int_is_int","label":"trace_of_alg_int_is_int","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(K\\) is a number field and \\(\\alpha \\in \\mathcal{O}_K\\) then \\(\\operatorname {Tr}_{K/\\mathbb {Q}}(\\alpha )\\) is in \\(\\mathbb {Z}\\).","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.isIntegral_trace","latex_link":"sect0002.html#lemma:trace_of_alg_int_is_int","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.isIntegral_trace","lean_decl_name":"Algebra.isIntegral_trace","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lemma:cyclo_poly_irr","label":"cyclo_poly_irr","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For \\(n\\) any integer, \\(\\Phi _n\\) (the \\(n\\)-th cyclotomic polynomial) is an irreducible polynomial .","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Polynomial.cyclotomic.irreducible","latex_link":"sect0003.html#lemma:cyclo_poly_irr","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Polynomial.cyclotomic.irreducible","lean_decl_name":"Polynomial.cyclotomic.irreducible","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/FieldTheory/Minpoly/Basic.lean","lean4_code":"theorem irreducible (hx : IsIntegral A x) : Irreducible (minpoly A x) := by\n  refine (irreducible_of_monic (monic hx) <| ne_one A x).2 fun f g hf hg he => ?_\n  rw [← hf.isUnit_iff, ← hg.isUnit_iff]\n  by_contra! h\n  have heval := congr_arg (Polynomial.aeval x) he\n  rw [aeval A x, aeval_mul, mul_eq_zero] at heval\n  rcases heval with heval | heval\n  · exact aeval_ne_zero_of_dvdNotUnit_minpoly hx hf ⟨hf.ne_zero, g, h.2, he.symm⟩ heval\n  · refine aeval_ne_zero_of_dvdNotUnit_minpoly hx hg ⟨hg.ne_zero, f, h.1, ?_⟩ heval\n    rw [mul_comm, he]\n\nend IsDomain\n\nend CommRing\n\n","line":273,"end_line":287,"proof_status":"formalized"}
+{"id":"lem:gen_dvd_by_frak_p","label":"gen_dvd_by_frak_p","background_color":"#a3d6ff","border_color":"blue","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(p\\) be a regular odd prime, \\(x,y,z,\\epsilon \\in \\mathbb {Z}[\\zeta _p]\\), \\(\\epsilon \\) a unit, and \\(n \\in \\mathbb {Z}_{\\geq 1}\\). Assume \\(x,y,z\\) are coprime to \\((1-\\zeta _p)\\) and that \\(x^p+y^p+\\epsilon (1-\\zeta _p)^{pn}z^p=0\\). Then each of \\((x+\\zeta _p^i y)\\) is divisible by \\((1-\\zeta _p)\\) and there is a unique \\(i_0\\) such that \\((x+\\zeta _p^{i_0})\\) is divisible by \\((1-\\zeta _p)^2\\).","doc_url":null,"latex_link":"sect0004.html#lem:gen_dvd_by_frak_p","lean_link":null,"lean_decl_name":"gen_dvd_by_frak_p","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}

crawled_repos/infinite-primes-via-log.jsonl

@@ -0,0 +1,16 @@
+{"id":"thm:primeCountingReal_unbounded","label":"primeCountingReal_unbounded","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://rwst.github.io/infinite-primes-via-log/blueprint/dep_graph_document.html","dependencies":["thm:log_le_primeCountingReal_add_one"],"nlp_text":"The prime counting function in the real domain, \\(\\pi :\\ \\mathbb {R} \\mapsto \\mathbb {N} \\ := \\# \\big\\{ p\\mid p \\in \\mathbb {P}, p\\le x \\big\\} \\) is unbounded above.","doc_url":"https://rwst.github.io/infinite-primes-via-log/docs/find/#doc/primeCountingReal_unbounded","latex_link":"sect0001.html#thm:primeCountingReal_unbounded","lean_link":"https://rwst.github.io/infinite-primes-via-log/docs/find/#doc/primeCountingReal_unbounded","lean_decl_name":"primeCountingReal_unbounded","kind_description":"theorems and lemmas","status_from_color":"statement_formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"infinite-primes-via-log","source_path":"InfinitePrimesViaLog/Basic.lean","lean4_code":"  rw [h <| subset_of_ssubset <| hnx.symm ▸ (Icc_ssubset_smoothNumbers n hn)]\n","line":251,"end_line":251,"proof_status":"statement_formalized"}
+{"id":"thm:infinite_primes","label":"infinite_primes","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://rwst.github.io/infinite-primes-via-log/blueprint/dep_graph_document.html","dependencies":["thm:primeCountingReal_unbounded"],"nlp_text":"The set \\(\\mathbb {P} = \\big\\{ p \\in \\mathbb {N} \\ \\mid \\ p \\ \\mathrm{prime} \\big\\} \\) is infinite.","doc_url":"https://rwst.github.io/infinite-primes-via-log/docs/find/#doc/infinite_setOf_prime","latex_link":"sect0001.html#thm:infinite_primes","lean_link":"https://rwst.github.io/infinite-primes-via-log/docs/find/#doc/infinite_setOf_prime","lean_decl_name":"infinite_setOf_prime","kind_description":"theorems and lemmas","status_from_color":"statement_formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"infinite-primes-via-log","source_path":"InfinitePrimesViaLog/Basic.lean","lean4_code":"@[simp]\nlemma getbang_natCast_eq_get {α : Type*} [Inhabited α] (l : List α) (i : Fin l.length) :\n","line":254,"end_line":255,"proof_status":"statement_formalized"}
+{"id":"lem:lemma4-2","label":"lemma4-2","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://rwst.github.io/infinite-primes-via-log/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For all \\(3\\le k\\le \\pi (x)\\), the \\(k\\)th prime \\(p_k \\ge k+2\\).","doc_url":"https://rwst.github.io/infinite-primes-via-log/docs/find/#doc/H_P4_4b","latex_link":"sect0001.html#lem:lemma4-2","lean_link":"https://rwst.github.io/infinite-primes-via-log/docs/find/#doc/H_P4_4b","lean_decl_name":"H_P4_4b","kind_description":"theorems and lemmas","status_from_color":"statement_formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"infinite-primes-via-log","source_path":"InfinitePrimesViaLog/Basic.lean","lean4_code":"    match m, hm with\n    | 0, _ =>\n      intro h\n","line":115,"end_line":117,"proof_status":"statement_formalized"}
+{"id":"lem:lemma4","label":"lemma4","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://rwst.github.io/infinite-primes-via-log/blueprint/dep_graph_document.html","dependencies":["lem:lemma4-2","lem:lemma4-1"],"nlp_text":"\\[ \\prod _{k=1}^{\\pi (x)}\\frac{p_k}{p_k-1} \\le \\prod _{k=1}^{\\pi (x)}\\frac{k+1}{k}. \\]","doc_url":"https://rwst.github.io/infinite-primes-via-log/docs/find/#doc/H_P4_4","latex_link":"sect0001.html#lem:lemma4","lean_link":"https://rwst.github.io/infinite-primes-via-log/docs/find/#doc/H_P4_4","lean_decl_name":"H_P4_4","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"infinite-primes-via-log","source_path":"InfinitePrimesViaLog/Basic.lean","lean4_code":"    · unfold PrimeBelow\n      unfold primesBelow\n      rw [Finset.mem_filter, Finset.mem_range]\n      exact ⟨ hn, by decide ⟩\n  · exact hn\n\nlemma le_PrimeBelow (n : ℕ) (hn : n < primeCountingReal x) (hxg4 : 4 ≤ x) : 1 ≤ nth (PrimeBelow ⌊x⌋₊) n := by\n  induction n with\n","line":130,"end_line":137,"proof_status":"statement_formalized"}
+{"id":"lem:lemma1","label":"lemma1","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://rwst.github.io/infinite-primes-via-log/blueprint/dep_graph_document.html","dependencies":["lem:lemma4","lem:lemma5","lem:lemma2","lem:lemma0","lem:lemma3a","lem:lemma3"],"nlp_text":"\\[ \\sum _{k=1}^n\\frac1k \\le \\sum _{m\\in S_x}\\frac{1}{m}. \\]","doc_url":"https://rwst.github.io/infinite-primes-via-log/docs/find/#doc/H_P4_1","latex_link":"sect0001.html#lem:lemma1","lean_link":"https://rwst.github.io/infinite-primes-via-log/docs/find/#doc/H_P4_1","lean_decl_name":"H_P4_1","kind_description":"theorems and lemmas","status_from_color":"statement_formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"infinite-primes-via-log","source_path":"InfinitePrimesViaLog/Basic.lean","lean4_code":"  · apply H_P4_4e\n    · exact h_x\n    · exact hxg4\n  · apply H_P4_4b\n\nlemma prod_Icc_succ_div (n : ℕ) (hn : 1 ≤ n) : (∏ x ∈ Icc 1 n, ((x + 1) : ℝ) / x) = n + 1 := by\n  rw [← Nat.Ico_succ_right]\n","line":190,"end_line":196,"proof_status":"statement_formalized"}
+{"id":"thm:log_le_primeCountingReal_add_one","label":"log_le_primeCountingReal_add_one","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://rwst.github.io/infinite-primes-via-log/blueprint/dep_graph_document.html","dependencies":["lem:lemma1"],"nlp_text":"For \\(x\\in \\mathbb {R}\\), \\(\\log x \\le \\pi (x) +1\\).","doc_url":"https://rwst.github.io/infinite-primes-via-log/docs/find/#doc/log_le_primeCountingReal_add_one","latex_link":"index.html#thm:log_le_primeCountingReal_add_one","lean_link":"https://rwst.github.io/infinite-primes-via-log/docs/find/#doc/log_le_primeCountingReal_add_one","lean_decl_name":"log_le_primeCountingReal_add_one","kind_description":"theorems and lemmas","status_from_color":"statement_formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"infinite-primes-via-log","source_path":"InfinitePrimesViaLog/Basic.lean","lean4_code":"  use n * 2\n  constructor\n  . unfold S₁\n    exact hnx ▸ two_n_smooth n hn\n  . rw [Set.mem_Icc, not_and, not_le]\n    exact fun _ => (lt_mul_iff_one_lt_right (zero_lt_of_lt hn)).mpr one_lt_two\n-/\n\nlemma H_P4_1 (n : ℕ) (hn : 1 < n) (hnx : n = ⌊x⌋₊) :\n    (∑ k ∈ Icc 1 n, (k : ℝ)⁻¹) ≤ (∑' m : (S₁ x), (m : ℝ)⁻¹) := by\n  have h (hs : Set.Icc 1 n ⊆ S₁ x) : (∑' m : (S₁ x), 1 / (m : ℝ)) = (∑ k ∈ Icc 1 n, 1 / (k : ℝ)) + (∑' m : ((S₁ x) \\ (Icc 1 n) : Set ℕ), 1 / (m : ℝ)):= by sorry\n  have h' : 0 ≤ (∑' m : ((S₁ x) \\ (Icc 1 n) : Set ℕ), 1 / (m : ℝ)) := by sorry\n","line":238,"end_line":249,"proof_status":"statement_formalized"}
+{"id":"lem:lemma5-2-1","label":"lemma5-2-1","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://rwst.github.io/infinite-primes-via-log/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\(\\pi (x)\\) is monotone.","doc_url":"https://rwst.github.io/infinite-primes-via-log/docs/find/#doc/monotone_primeCountingReal","latex_link":"index.html#lem:lemma5-2-1","lean_link":"https://rwst.github.io/infinite-primes-via-log/docs/find/#doc/monotone_primeCountingReal","lean_decl_name":"monotone_primeCountingReal","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"infinite-primes-via-log","source_path":"InfinitePrimesViaLog/Basic.lean","lean4_code":"theorem monotone_primeCountingReal : Monotone primeCountingReal := by\n  have count_primes_upto_three : 1 ≤ count Nat.Prime ⌊3⌋₊ := by rw [floor_nat]; norm_num; decide\n  intro a b hab\n  unfold primeCountingReal\n  by_cases ha : a < 3\n  · by_cases hb : b < 3\n    · simp only [ha, hb]\n      exact le_of_ble_eq_true rfl\n    · simp [ha, hb]\n      unfold primeCounting'\n      exact le_trans count_primes_upto_three <| count_monotone Nat.Prime\n        <| le_floor <| le_of_not_lt hb\n  · by_cases hb : b < 3\n    · linarith\n    · simp only [ha, hb, not_lt, ite_false]\n      simp_all only [not_lt]\n      exact monotone_primeCounting' <| Nat.floor_le_floor a b hab\n","line":67,"end_line":83,"proof_status":"formalized"}
+{"id":"lem:lemma5","label":"lemma5","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://rwst.github.io/infinite-primes-via-log/blueprint/dep_graph_document.html","dependencies":["lem:lemma5-1"],"nlp_text":"For \\(x\\ge 3\\), \\[ \\prod _{k=1}^{\\pi (x)}\\frac{k+1}{k} = \\pi (x)+1. \\]","doc_url":"https://rwst.github.io/infinite-primes-via-log/docs/find/#doc/H_P4_5","latex_link":"index.html#lem:lemma5","lean_link":"https://rwst.github.io/infinite-primes-via-log/docs/find/#doc/H_P4_5","lean_decl_name":"H_P4_5","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"infinite-primes-via-log","source_path":"InfinitePrimesViaLog/Basic.lean","lean4_code":"        split_ifs\n        · contradiction\n        · unfold primeCountingReal at hm\n","line":155,"end_line":157,"proof_status":"formalized"}
+{"id":"def:S","label":"S","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://rwst.github.io/infinite-primes-via-log/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\[ S_x = \\{ 1\\} \\ \\cup \\ \\{ m\\, \\big|\\, \\text{$m$ has only prime factors $p\\le x$}\\} \\]","doc_url":"https://rwst.github.io/infinite-primes-via-log/docs/find/#doc/S₁","latex_link":"index.html#def:S","lean_link":"https://rwst.github.io/infinite-primes-via-log/docs/find/#doc/S₁","lean_decl_name":"S₁","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/rwst/infinite-primes-via-log/63d31d052e7205d86f9acdede0cec0cfe3c672bf/InfinitePrimesViaLog/Basic.lean","lean4_code":"def S₁ (x : ℝ) : Set ℕ := smoothNumbers ((Nat.floor x) + 1)","line":12,"end_line":12,"proof_status":"formalized"}
+{"id":"lem:lemma2","label":"lemma2","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://rwst.github.io/infinite-primes-via-log/blueprint/dep_graph_document.html","dependencies":["def:S"],"nlp_text":"\\[ \\sum _{m\\in S_x}\\frac{1}{m} = \\underset {p\\le x}{\\prod _{p\\in \\mathbb {P}}}\\Big(\\sum _{k\\in \\mathbb {N}}\\frac{1}{p^k}\\Big). \\]","doc_url":"https://rwst.github.io/infinite-primes-via-log/docs/find/#doc/H_P4_2","latex_link":"sect0001.html#lem:lemma2","lean_link":"https://rwst.github.io/infinite-primes-via-log/docs/find/#doc/H_P4_2","lean_decl_name":"H_P4_2","kind_description":"theorems and lemmas","status_from_color":"statement_formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"infinite-primes-via-log","source_path":"InfinitePrimesViaLog/Basic.lean","lean4_code":"lemma H_P4_2 : (∑' m : (S₁ x), (m : ℝ)⁻¹)\n    = (∏ p ∈ primesBelow ⌊x⌋₊, (∑' k : ℕ, (p ^ k : ℝ)⁻¹)) := by sorry\n","line":38,"end_line":39,"proof_status":"statement_formalized"}
+{"id":"lem:lemma0","label":"lemma0","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://rwst.github.io/infinite-primes-via-log/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\[ \\log x \\le \\sum _{i=1}^n\\frac1i. \\]","doc_url":"https://rwst.github.io/infinite-primes-via-log/docs/find/#doc/log_le_harmonic","latex_link":"index.html#lem:lemma0","lean_link":"https://rwst.github.io/infinite-primes-via-log/docs/find/#doc/log_le_harmonic","lean_decl_name":"log_le_harmonic","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"infinite-primes-via-log","source_path":"InfinitePrimesViaLog/Basic.lean","lean4_code":"lemma log_le_harmonic (n : ℕ) (hn : 0 < n) (hnx : n ≤ x) (hxn : x < n + 1) :\n    Real.log x ≤ ∑ k ∈ Icc 1 n, (k : ℝ)⁻¹ :=\n  have hxpos : 0 < x := LT.lt.trans_le (cast_pos.mpr hn) hnx\n  calc\n    Real.log x ≤ Real.log (n + 1 : ℝ) := by\n        rw [Real.log_le_log_iff hxpos]; linarith; linarith\n    _ ≤ harmonic n := by\n        norm_cast; exact log_add_one_le_harmonic n\n    _ = ∑ k ∈ range n, (k + 1 : ℝ)⁻¹ := by\n        unfold harmonic\n        simp_all only [cast_add, cast_one, Rat.cast_sum, Rat.cast_inv,\n          Rat.cast_add, Rat.cast_natCast, Rat.cast_one]\n    _ = ∑ k ∈ Ico 1 (n + 1), (k : ℝ)⁻¹ := by\n        simp_rw [sum_Ico_eq_sum_range, Nat.add_one_sub_one, add_comm]\n        simp_all only [cast_add, cast_one]\n    _ = ∑ k ∈ Icc 1 n, (k : ℝ)⁻¹ := by rfl\n","line":21,"end_line":36,"proof_status":"formalized"}
+{"id":"lem:lemma4-1","label":"lemma4-1","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://rwst.github.io/infinite-primes-via-log/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For \\(k,p \\in \\mathbb {R}\\), \\(k\\ge 0\\), \\(p \\ge k+1\\), \\[ \\frac{p}{p-1} \\le \\frac{k+1}{k}. \\]","doc_url":"https://rwst.github.io/infinite-primes-via-log/docs/find/#doc/H_P4_4a","latex_link":"sect0001.html#lem:lemma4-1","lean_link":"https://rwst.github.io/infinite-primes-via-log/docs/find/#doc/H_P4_4a","lean_decl_name":"H_P4_4a","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"infinite-primes-via-log","source_path":"InfinitePrimesViaLog/Basic.lean","lean4_code":"      exact not_prime_zero h.2\n    | 1, _ =>\n      intro h\n      simp_all only [mem_range, ofNat_pos, not_and]\n      exact not_prime_one h.2\n  rw [← this]\n  apply count_eq_card_filter_range\n\nlemma nth_zero_PrimeBelow  (n : ℕ) (hn : 2 < n) : nth (PrimeBelow n) 0 = 2 := by\n  rw [← count_PrimeBelow]\n","line":119,"end_line":128,"proof_status":"formalized"}
+{"id":"lem:lemma3a","label":"lemma3a","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://rwst.github.io/infinite-primes-via-log/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\[ \\underset {p\\le x}{\\prod _{p\\in \\mathbb {P}}}\\frac{p}{p-1} = \\prod _{k=1}^{\\pi (x)}\\frac{p_k}{p_k-1}. \\]","doc_url":"https://rwst.github.io/infinite-primes-via-log/docs/find/#doc/H_P4_3a","latex_link":"sect0001.html#lem:lemma3a","lean_link":"https://rwst.github.io/infinite-primes-via-log/docs/find/#doc/H_P4_3a","lean_decl_name":"H_P4_3a","kind_description":"theorems and lemmas","status_from_color":"statement_formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"infinite-primes-via-log","source_path":"InfinitePrimesViaLog/Basic.lean","lean4_code":"--lemma H_P4_5' : (∏ k in Icc 1 (primeCountingReal x), (nth primesBelow k : ℝ) / ((nth primesBelow k) - 1))\n--    ≤ (∏ k in Icc 1 (primeCountingReal x), (k + 1 : ℝ) / k) := by\n--  sorry\n\nlemma two_n_smooth (n : ℕ) (hn : 1 < n) : n * 2 ∈ (n + 1).smoothNumbers := by\n  have h1 : n ∈ (n + 1).smoothNumbers := by\n    apply mem_smoothNumbers_of_lt (zero_lt_of_lt hn); linarith\n  have h2 : 2 ∈ (n + 1).smoothNumbers := by\n    apply mem_smoothNumbers_of_lt ofNat_pos; linarith\n  apply mul_mem_smoothNumbers\n  assumption'\n\n","line":215,"end_line":226,"proof_status":"statement_formalized"}
+{"id":"lem:lemma5-1","label":"lemma5-1","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://rwst.github.io/infinite-primes-via-log/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For \\(n\\ge 2\\), \\[ \\prod _{k=1}^{n}\\frac{k+1}{k} = n+1. \\]","doc_url":"https://rwst.github.io/infinite-primes-via-log/docs/find/#doc/prod_Icc_succ_div","latex_link":"sect0001.html#lem:lemma5-1","lean_link":"https://rwst.github.io/infinite-primes-via-log/docs/find/#doc/prod_Icc_succ_div","lean_decl_name":"prod_Icc_succ_div","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"infinite-primes-via-log","source_path":"InfinitePrimesViaLog/Basic.lean","lean4_code":"    rw [nth_zero_PrimeBelow ⌊x⌋₊ h_x]\n    exact NeZero.one_le\n  | succ n' ih =>\n    have h₁ (m : ℕ) (hm : m + 1 < primeCountingReal x) :\n        nth (PrimeBelow ⌊x⌋₊) m ≤ nth (PrimeBelow ⌊x⌋₊) (m + 1) := by\n      apply nth_le_nth_of_lt_card (p := PrimeBelow ⌊x⌋₊)\n      · exact le_add_right m 1\n      · have : ¬x < 3 := by linarith\n        have h₂ : (primesBelow ⌊x⌋₊).card = primeCountingReal x := by\n          unfold primeCountingReal\n          split_ifs\n          · contradiction\n          · rw [primesBelow_card_eq_primeCounting']\n        simp_rw [setOf_mem, Set.toFinite_toFinset, toFinset_coe]\n        rw [h₂]\n","line":139,"end_line":153,"proof_status":"formalized"}
+{"id":"def:nx","label":"nx","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://rwst.github.io/infinite-primes-via-log/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"In the following, the variables \\(n\\in \\mathbb {N}\\), \\(x\\in \\mathbb {R}\\), always satisfy \\(x\\ge 1\\), \\(n\\le x {\\lt} n+1\\), which is equivalent to \\(n=\\lfloor x\\rfloor \\).","doc_url":null,"latex_link":"index.html#def:nx","lean_link":null,"lean_decl_name":"nx","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
+{"id":"lem:lemma3","label":"lemma3","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://rwst.github.io/infinite-primes-via-log/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\[ \\underset {p\\le x}{\\prod _{p\\in \\mathbb {P}}}\\Big(\\sum _{k\\in \\mathbb {N}}\\frac{1}{p^k}\\Big) = \\underset {p\\le x}{\\prod _{p\\in \\mathbb {P}}}\\frac{p}{p-1} \\]","doc_url":"https://rwst.github.io/infinite-primes-via-log/docs/find/#doc/H_P4_3","latex_link":"index.html#lem:lemma3","lean_link":"https://rwst.github.io/infinite-primes-via-log/docs/find/#doc/H_P4_3","lean_decl_name":"H_P4_3","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"infinite-primes-via-log","source_path":"InfinitePrimesViaLog/Basic.lean","lean4_code":"lemma H_P4_3 : (∏ p ∈ primesBelow ⌊x⌋₊, (∑' k : ℕ, ((p : ℝ) ^ k)⁻¹)) =\n    (∏ p ∈ primesBelow ⌊x⌋₊, ((p : ℝ) / (p - 1))) := by\n  have h_p_nonzero (p' : ℕ) (hp : p' ∈ ⌊x⌋₊.primesBelow) : p' ≠ 0 :=\n      Nat.Prime.ne_zero (mem_primesBelow.mp hp).2\n  refine Finset.prod_congr rfl fun x hx ↦ ?_\n  rw [(inv_div (_ - 1 : ℝ) (_ : ℝ)).symm]\n  rw [sub_div, div_self (by exact_mod_cast h_p_nonzero x hx), one_div]\n  rw [← tsum_geometric_of_lt_one]\n  simp_rw [inv_pow]\n  simp only [ne_eq, forall_mem_not_eq', inv_nonneg, cast_nonneg]\n  rw [mem_primesBelow] at hx\n  exact inv_lt_one <| one_lt_cast.mpr <| Prime.one_lt hx.2\n","line":41,"end_line":52,"proof_status":"formalized"}

crawled_repos/lean-ga_sec1.jsonl

@@ -0,0 +1,16 @@
+{"id":"QuadraticForm","label":"QuadraticForm","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/lean-ga/blueprint/dep_graph_section_1.html","dependencies":["BilinForm"],"nlp_text":"Let \\(R\\) be a commutative ring, \\(M\\) a \\(R\\)-module. An quadratic form \\(Q\\) over \\(M\\) is a map \\(Q : M \\to R\\), satisfying: \\( Q(a \\bullet x) = a * a * Q(x)\\) for all \\(a \\in R, x \\in M\\). there exists a companion bilinear form \\(B : M \\to M \\to R\\), such that \\(Q(x + y) = Q(x) + Q(y) + B(x, y)\\) In some literatures, the bilinear form is denoted \\(\\Phi \\), and called the polar form associated with the quadratic form \\(Q\\), or simply the polar form of \\(Q\\).","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/QuadraticForm","latex_link":"sec-forms.html#QuadraticForm","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/QuadraticForm","lean_decl_name":"QuadraticForm","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean","lean4_code":"instance : CommRing (CliffordAlgebra (0 : QuadraticForm R Unit)) :=\n  { CliffordAlgebra.instRing _ with\n    mul_comm := fun x y => by\n      induction x using CliffordAlgebra.induction with\n      | algebraMap r => apply Algebra.commutes\n      | ι x => simp\n      | add x₁ x₂ hx₁ hx₂ => rw [mul_add, add_mul, hx₁, hx₂]\n      | mul x₁ x₂ hx₁ hx₂ => rw [mul_assoc, hx₂, ← mul_assoc, hx₁, ← mul_assoc] }\n\n","line":77,"end_line":85,"proof_status":"formalized"}
+{"id":"LinearMap","label":"LinearMap","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/lean-ga/blueprint/dep_graph_section_1.html","dependencies":["Module","RingHom"],"nlp_text":"Let \\(R, S\\) be rings, \\(M\\) an \\(R\\)-module, \\(N\\) an \\(S\\)-module. A linear map from \\(M\\) to \\(N\\) is a function \\(f : M \\to _{l} N\\) over a ring homomorphism \\(\\sigma : R \\to _{+*} S\\), satisfying: \\(f(x + y) = f(x) + f(y)\\) for all \\(x, y \\in M\\). \\(f(r \\bullet x) = \\sigma (r) \\bullet f(x)\\) for all \\(r \\in R\\), \\(x \\in M\\).","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/LinearMap","latex_link":"sec-algebras.html#LinearMap","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/LinearMap","lean_decl_name":"LinearMap","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Topology/Algebra/Module/Basic.lean","lean4_code":"theorem LinearMap.continuous_on_pi {ι : Type*} {R : Type*} {M : Type*} [Finite ι] [Semiring R]\n    [TopologicalSpace R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [ContinuousAdd M]\n    [ContinuousSMul R M] (f : (ι → R) →ₗ[R] M) : Continuous f := by\n  cases nonempty_fintype ι\n  classical\n    -- for the proof, write `f` in the standard basis, and use that each coordinate is a continuous\n    -- function.\n    have : (f : (ι → R) → M) = fun x => ∑ i : ι, x i • f fun j => if i = j then 1 else 0 := by\n      ext x\n      exact f.pi_apply_eq_sum_univ x\n    rw [this]\n    refine continuous_finset_sum _ fun i _ => ?_\n    exact (continuous_apply i).smul continuous_const\n\nend Pi\n\nsection PointwiseLimits\n\nvariable {M₁ M₂ α R S : Type*} [TopologicalSpace M₂] [T2Space M₂] [Semiring R] [Semiring S]\n  [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M₁] [Module S M₂] [ContinuousConstSMul S M₂]\n\nvariable [ContinuousAdd M₂] {σ : R →+* S} {l : Filter α}\n\n/-- Constructs a bundled linear map from a function and a proof that this function belongs to the\nclosure of the set of linear maps. -/\n","line":245,"end_line":269,"proof_status":"formalized"}
+{"id":"Dual","label":"Dual","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/lean-ga/blueprint/dep_graph_section_1.html","dependencies":["LinearMap"],"nlp_text":"The dual module \\(M^* : M \\to _{l[R]} R\\) is the \\(R\\)-module of all linear maps from \\(M\\) to \\(R\\).","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Module.Dual","latex_link":"sec-basics.html#Dual","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Module.Dual","lean_decl_name":"Module.Dual","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RepresentationTheory/Basic.lean","lean4_code":"def dual : Representation k G (Module.Dual k V) where\n  toFun g :=\n    { toFun := fun f => f ∘ₗ ρV g⁻¹\n      map_add' := fun f₁ f₂ => by simp only [add_comp]\n      map_smul' := fun r f => by\n        ext\n        simp only [coe_comp, Function.comp_apply, smul_apply, RingHom.id_apply] }\n  map_one' := by ext; simp\n  map_mul' g h := by ext; simp\n\n","line":675,"end_line":684,"proof_status":"formalized"}
+{"id":"TensorAlgebra","label":"TensorAlgebra","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/lean-ga/blueprint/dep_graph_section_1.html","dependencies":["LinearMap","FreeAlgebra"],"nlp_text":"Let \\(A\\) be a free \\(R\\)-algebra generated by module \\(M\\), let \\(\\iota : M \\to A\\) denote the map from \\(M\\) to \\(A\\). An tensor algebra over \\(M\\) (or “of \\(M\\) \") \\(T\\) is the ring quotient of the free \\(R\\)-algebra generated by \\(M\\), by the equivalence relation satisfying: for all \\(a, b\\) in \\(M\\), \\(\\iota (a + b) \\sim \\iota (a) + \\iota (b)\\). for all \\(r\\) in \\(R\\), \\(a\\) in \\(M\\), \\(\\iota (r \\bullet a) \\sim r * \\iota (a)\\). i.e. making the inclusion of \\(M\\) into an \\(R\\)-linear map.","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/TensorAlgebra","latex_link":"sec-algebras.html#TensorAlgebra","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/TensorAlgebra","lean_decl_name":"TensorAlgebra","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/LinearAlgebra/TensorAlgebra/Basic.lean","lean4_code":"def TensorAlgebra :=\n  RingQuot (TensorAlgebra.Rel R M)\n\n-- The `Inhabited, Semiring, Algebra` instances should be constructed by a deriving handler.\n-- https://github.com/leanprover-community/mathlib4/issues/380\n","line":59,"end_line":63,"proof_status":"formalized"}
+{"id":"Group","label":"Group","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/lean-ga/blueprint/dep_graph_section_1.html","dependencies":[],"nlp_text":"A group is a pair \\((G, *)\\), satisfying: \\((a * b) * c = a * (b * c)\\) for all \\(a, b, c \\in G\\) (associativity). there exists \\(1 \\in G\\) such that \\(1 * a = a * 1 = a\\) for all \\(a \\in G\\). for each \\(a \\in G\\) there exists \\(a^{-1} \\in G\\) such that \\(a * a^{-1} = a^{-1} * a = 1\\).","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Group","latex_link":"sec-basics.html#Group","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Group","lean_decl_name":"Group","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Order/Filter/Germ/Basic.lean","lean4_code":"instance instGroup [Group G] : Group (Germ l G) :=\n  { inv_mul_cancel := Quotient.ind' fun _ => congrArg ofFun <| inv_mul_cancel _ }\n\n","line":505,"end_line":507,"proof_status":"formalized"}
+{"id":"Ring","label":"Ring","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/lean-ga/blueprint/dep_graph_section_1.html","dependencies":["Group","Monoid"],"nlp_text":"A ring is a triple \\((R, +, *)\\), satisfying: \\(R\\) is a commutative group under \\(+\\). \\(R\\) is a monoid under \\(*\\). for all \\(a, b, c \\in R\\), we have \\(a * (b + c) = a * b + a * c\\) \\((a + b) * c = a * c + b * c\\) (left and right distributivity over \\(+\\)).","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Ring","latex_link":"sec-basics.html#Ring","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Ring","lean_decl_name":"Ring","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Tactic/FastInstance.lean","lean4_code":"instance instRing : Ring X := fast_instance% Function.Injective.ring ..\n```\nthis will define `instRing` as a nested constructor application that refers to `instSemiring`\nrather than applications of `Function.Injective.ring` or other non-canonical constructors.\nThe advantage is then that `instRing.toSemiring` unifies almost immediately with `instSemiring`,\nrather than having to break it down into smaller pieces.\n-/\nsyntax (name := fastInstance) \"fast_instance%\" term : term\n\n","line":111,"end_line":119,"proof_status":"formalized"}
+{"id":"Module","label":"Module","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/lean-ga/blueprint/dep_graph_section_1.html","dependencies":["Ring"],"nlp_text":"Let \\(R\\) be a commutative ring. A module over \\(R\\) (in short \\(R\\)-module) is a pair \\((M, \\bullet )\\), satisfying: M is a group under \\(+\\). \\(\\bullet : R \\to M \\to M\\) is called scalar multiplication, for every \\(a, b \\in R\\), \\(x, y \\in M\\), we have \\(a \\bullet (x + y) = a \\bullet x + b \\bullet y\\) \\((a + b) \\bullet x = a \\bullet x + b \\bullet x\\) \\(a * (b \\bullet x)=(a * b) \\bullet x\\) \\(1_R \\bullet x = x\\)","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Module","latex_link":"sec-basics.html#Module","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Module","lean_decl_name":"Module","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Order/Filter/Germ/Basic.lean","lean4_code":"instance instModule [Semiring R] [AddCommMonoid M] [Module R M] : Module R (Germ l M) where\n  add_smul c₁ c₂ f :=\n    inductionOn f fun f => by\n      norm_cast\n      simp [add_smul]\n  zero_smul f :=\n    inductionOn f fun f => by\n      norm_cast\n      simp [zero_smul]\n\n","line":652,"end_line":661,"proof_status":"formalized"}
+{"id":"DivisionRing","label":"DivisionRing","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/lean-ga/blueprint/dep_graph_section_1.html","dependencies":["Ring"],"nlp_text":"A division ring is a ring \\((R, +, *)\\), satisfying: \\(R\\) contains at least 2 elements. for all \\(a \\neq 0\\) in \\(R\\), there exists a multiplicative inverse \\(a^{-1} \\in R\\) such that \\[ a * a^{-1} = a^{-1} * a = 1 \\]","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/DivisionRing","latex_link":"sec-basics.html#DivisionRing","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/DivisionRing","lean_decl_name":"DivisionRing","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Order/Filter/FilterProduct.lean","lean4_code":"instance instDivisionRing [DivisionRing β] : DivisionRing β* where\n  __ := instRing\n  __ := instDivisionSemiring\n  qsmul := _\n  qsmul_def := fun _ _ => rfl\n\n","line":51,"end_line":56,"proof_status":"formalized"}
+{"id":"RingHom","label":"RingHom","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/lean-ga/blueprint/dep_graph_section_1.html","dependencies":["Ring"],"nlp_text":"Let \\((\\alpha , +_\\alpha , *_\\alpha )\\) and \\((\\beta , +_\\beta , *_\\beta )\\) be rings. A ring homomorphism from \\(\\alpha \\) to \\(\\beta \\) is a map \\(\\mathit{1}: \\alpha \\to _{+*}\\beta \\) such that \\(\\mathit{1}(x +_{\\alpha } y) = \\mathit{1}(x) +_{\\beta } \\mathit{1}(y)\\) for each \\(x,y \\in \\alpha \\). \\(\\mathit{1}(x *_{\\alpha } y) = \\mathit{1}(x) *_{\\beta } \\mathit{1}(y)\\) for each \\(x,y \\in \\alpha \\). \\(\\mathit{1}(1_{\\alpha }) = 1_{\\beta }\\).","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/RingHom","latex_link":"sec-algebras.html#RingHom","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/RingHom","lean_decl_name":"RingHom","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Tactic/Algebraize.lean","lean4_code":"def RingHom.FiniteType (f : A →+* B) : Prop :=\n  @Algebra.FiniteType A B _ _ f.toAlgebra\n```\nAn example when the `Name` is provided (as the `Algebra` does not have the expected name):\n```\n","line":51,"end_line":55,"proof_status":"formalized"}
+{"id":"VectorSpace","label":"VectorSpace","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/lean-ga/blueprint/dep_graph_section_1.html","dependencies":["Module","DivisionRing"],"nlp_text":"If \\(R\\) is a division ring, then a module \\(M\\) over \\(R\\) is called a vector space.","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Module","latex_link":"sec-basics.html#VectorSpace","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Module","lean_decl_name":"Module","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Order/Filter/Germ/Basic.lean","lean4_code":"instance instModule [Semiring R] [AddCommMonoid M] [Module R M] : Module R (Germ l M) where\n  add_smul c₁ c₂ f :=\n    inductionOn f fun f => by\n      norm_cast\n      simp [add_smul]\n  zero_smul f :=\n    inductionOn f fun f => by\n      norm_cast\n      simp [zero_smul]\n\n","line":652,"end_line":661,"proof_status":"formalized"}
+{"id":"Algebra","label":"Algebra","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/lean-ga/blueprint/dep_graph_section_1.html","dependencies":["Module","RingHom"],"nlp_text":"Let \\(R\\) be a commutative ring. An algebra \\(A\\) over \\(R\\) is a pair \\((A, \\bullet )\\), satisfying: \\(A\\) is a ring under \\(*\\). there exists a ring homomorphism from \\(R\\) to \\(A\\), denoted \\(\\mathit{1}: R \\to _{+*} A\\). \\(\\bullet : R \\to M \\to M\\) is a scalar multiplication for every \\(r \\in R\\), \\(x \\in A\\), we have \\(r * x = x * r\\) (commutativity between \\(R\\) and \\(A\\)) \\(r \\bullet x = r * x\\) (definition of scalar multiplication) where we omitted that the ring homomorphism \\(\\mathit{1}\\) is applied to \\(r\\) before each multiplication.","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Algebra","latex_link":"sec-algebras.html#Algebra","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Algebra","lean_decl_name":"Algebra","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Topology/LocallyConstant/Algebra.lean","lean4_code":"instance : Algebra R (LocallyConstant X Y) where\n  algebraMap := constRingHom.comp <| algebraMap R Y\n  commutes' := by\n    intros\n    ext\n    exact Algebra.commutes' _ _\n  smul_def' := by\n    intros\n    ext\n    exact Algebra.smul_def' _ _\n\n","line":250,"end_line":260,"proof_status":"formalized"}
+{"id":"RingQuot","label":"RingQuot","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/lean-ga/blueprint/dep_graph_section_1.html","dependencies":["Module","RingHom"],"nlp_text":"Let \\(R\\) be a non-commutative ring, \\(r\\) an arbitrary equivalence relation on \\(R\\). The ring quotient of \\(R\\) by \\(r\\) is the quotient of \\(R\\) by the strengthen equivalence relation of \\(r\\) such that for all \\(a, b, c\\) in \\(R\\): \\(a + c \\sim b + c\\) if \\(a \\sim b\\) \\(a * c \\sim b * c\\) if \\(a \\sim b\\) \\(a * b \\sim a * c\\) if \\(b \\sim c\\) i.e. the equivalence relation is compatible with the ring operations \\(+\\) and \\(*\\).","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/RingQuot","latex_link":"sec-algebras.html#RingQuot","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/RingQuot","lean_decl_name":"RingQuot","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean","lean4_code":"instance instInhabited : Inhabited (CliffordAlgebra Q) := RingQuot.instInhabited _\n","line":73,"end_line":73,"proof_status":"formalized"}
+{"id":"BilinForm","label":"BilinForm","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/lean-ga/blueprint/dep_graph_section_1.html","dependencies":["Module"],"nlp_text":"Let \\(R\\) be a ring, \\(M\\) an \\(R\\)-module. An bilinear form \\(B\\) over \\(M\\) is a map \\(B : M \\to M \\to R\\), satisfying: \\( B(x + y, z) = B(x, z) +B(y, z) \\) \\( B(x, y + z) = B(x, y) +B(x, z) \\) \\( B(a \\bullet x, y) = a * B(x, y)\\) \\( B(x, a \\bullet y) = a * B(x, y)\\) for all \\(a \\in R, x, y, z \\in M\\).","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/BilinForm","latex_link":"sec-forms.html#BilinForm","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/BilinForm","lean_decl_name":"BilinForm","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/LinearAlgebra/RootSystem/Finite/CanonicalBilinear.lean","lean4_code":"def RootForm : LinearMap.BilinForm R M :=\n  ∑ i, (P.coroot' i).smulRight (P.coroot' i)\n\n/-- An invariant inner product on the coweight space. -/\n","line":80,"end_line":83,"proof_status":"formalized"}
+{"id":"FreeAlgebra","label":"FreeAlgebra","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/lean-ga/blueprint/dep_graph_section_1.html","dependencies":["Algebra","RingQuot"],"nlp_text":"Let \\(X\\) be an arbitrary set. An free \\(R\\)-algebra \\(A\\) on \\(X\\) (or “ generated by \\(X\\) \") is the ring quotient of the following inductively constructed set \\(A_{\\mathtt{pre}}\\) for all \\(x\\) in \\(X\\), there exists a map \\(X \\to A_{\\mathtt{pre}}\\). for all \\(r\\) in \\(R\\), there exists a map \\(R \\to A_{\\mathtt{pre}}\\). for all \\(a, b\\) in \\(A_{\\mathtt{pre}}\\), \\(a + b\\) is in \\(A_{\\mathtt{pre}}\\). for all \\(a, b\\) in \\(A_{\\mathtt{pre}}\\), \\(a * b\\) is in \\(A_{\\mathtt{pre}}\\). by that equivalence relation that makes \\(A\\) an \\(R\\)-algebra, namely: there exists a ring homomorphism from \\(R\\) to \\(A_{\\mathtt{pre}}\\), denoted \\(R \\to _{+*} A_{\\mathtt{pre}}\\). \\(A\\) is a commutative group under \\(+\\). \\(A\\) is a monoid under \\(*\\). left and right distributivity under \\(*\\) over \\(+\\). \\(0 * a \\sim a * 0 \\sim 0\\). for all \\(a, b, c\\) in \\(A\\), if \\(a \\sim b\\), we have \\(a + c \\sim b + c\\) \\(c + a \\sim c + b\\) \\(a * c \\sim b * c\\) \\(c * a \\sim c * b\\) (compatibility with the ring operations \\(+\\) and \\(*\\))","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/FreeAlgebra","latex_link":"sec-algebras.html#FreeAlgebra","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/FreeAlgebra","lean_decl_name":"FreeAlgebra","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Algebra/FreeAlgebra.lean","lean4_code":"def FreeAlgebra :=\n  Quot (FreeAlgebra.Rel R X)\n\nnamespace FreeAlgebra\n\nattribute [local instance] Pre.hasCoeGenerator Pre.hasCoeSemiring Pre.hasMul Pre.hasAdd\n  Pre.hasZero Pre.hasOne Pre.hasSMul\n\n/-! Define the basic operations -/\n\n","line":157,"end_line":166,"proof_status":"formalized"}
+{"id":"AlgHom","label":"AlgHom","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/lean-ga/blueprint/dep_graph_section_1.html","dependencies":["RingHom"],"nlp_text":"Let \\(A\\) and \\(B\\) be \\(R\\)-algebras. \\(\\mathit{1}_A\\) and \\(\\mathit{1}_B\\) are ring homomorphisms from \\(R\\) to \\(A\\) and \\(B\\), respectively. A algebra homomorphism from \\(A\\) to \\(B\\) is a map \\(f : \\alpha \\to _{a}\\beta \\) such that \\(f\\) is a ring homomorphism \\(f(\\mathit{1}_{A}(r)) = \\mathit{1}_{B}(r)\\) for each \\(r \\in R\\)","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/AlgHom","latex_link":"sec-algebras.html#AlgHom","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/AlgHom","lean_decl_name":"AlgHom","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean","lean4_code":"theorem AlgHom.closure_ker_inter {F S K A : Type*} [CommRing K] [Ring A] [Algebra K A]\n    [TopologicalSpace K] [T1Space K] [TopologicalSpace A] [ContinuousSub A] [ContinuousSMul K A]\n    [FunLike F A K] [AlgHomClass F K A K] [SetLike S A] [OneMemClass S A] [AddSubgroupClass S A]\n    [SMulMemClass S K A] (φ : F) (hφ : Continuous φ) (s : S) :\n    closure (s ∩ RingHom.ker φ) = closure s ∩ (ker φ : Set A) := by\n  refine subset_antisymm ?_ ?_\n  · simpa only [ker_eq, (isClosed_singleton.preimage hφ).closure_eq]\n      using closure_inter_subset_inter_closure s (ker φ : Set A)\n  · intro x ⟨hxs, (hxφ : φ x = 0)⟩\n    rw [mem_closure_iff_clusterPt, ClusterPt] at hxs\n    have : Tendsto (fun y ↦ y - φ y • 1) (𝓝 x ⊓ 𝓟 s) (𝓝 x) := by\n      conv => congr; rfl; rfl; rw [← sub_zero x, ← zero_smul K 1, ← hxφ]\n      exact Filter.tendsto_inf_left (Continuous.tendsto (by fun_prop) x)\n    refine mem_closure_of_tendsto this <| eventually_inf_principal.mpr ?_\n    filter_upwards [] with g hg using\n      ⟨sub_mem hg (SMulMemClass.smul_mem _ <| one_mem _), by simp [RingHom.mem_ker]⟩\n\n","line":582,"end_line":598,"proof_status":"formalized"}
+{"id":"Monoid","label":"Monoid","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://utensil.github.io/lean-ga/blueprint/dep_graph_section_1.html","dependencies":[],"nlp_text":"A monoid is a pair \\((R, *)\\), satisfying: \\((a * b) * c = a * (b * c)\\) for all \\(a, b, c \\in R\\) (associativity). there exists an element \\(1 \\in R\\) such that \\(1 * a = a * 1 = a\\) for all \\(a \\in R\\) i.e. \\(1\\) is the multiplicative identity (neutral element).","doc_url":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Monoid","latex_link":"sec-basics.html#Monoid","lean_link":"https://leanprover-community.github.io/mathlib4_docs/find/#doc/Monoid","lean_decl_name":"Monoid","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Order/Filter/Germ/Basic.lean","lean4_code":"instance instMonoid [Monoid M] : Monoid (Germ l M) :=\n  { Function.Surjective.monoid ofFun Quot.mk_surjective (by rfl)\n      (fun _ _ => by rfl) fun _ _ => by rfl with\n    toSemigroup := instSemigroup\n    toOne := instOne\n    npow := fun n a => a ^ n }\n\n/-- Coercion from functions to germs as a monoid homomorphism. -/\n","line":404,"end_line":411,"proof_status":"formalized"}