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989cdb8 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 | {"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"}
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