module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 415,
"column": 2
} | {
"line": 415,
"column": 13
} | {
"line": 415,
"column": 14
} | [
{
"pp": "C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\ninst✝ : IsTriangulated C\na b : EInt\nhab : b ≤ a\nX : C\n⊢ (t.eTruncLT.obj b).map ((t.eT... | [
"C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\ninst✝ : IsTriangulated C\na b : EInt\nhab : b ≤ a\nX : C\n⊢ (t.eTruncLTLTIsoLT a b hab).inv.app ((t.e... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 415,
"column": 2
} | {
"line": 415,
"column": 58
} | {
"line": 417,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\ninst✝ : IsTriangulated C\na b : EInt\nhab : b ≤ a\nX : C\n⊢ (t.eTruncLT.obj b).map ((t.eT... | [] | simpa using (t.eTruncLTLTIsoLT a b hab).hom_inv_id_app X | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 415,
"column": 2
} | {
"line": 415,
"column": 58
} | {
"line": 417,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\ninst✝ : IsTriangulated C\na b : EInt\nhab : b ≤ a\nX : C\n⊢ (t.eTruncLT.obj b).map ((t.eT... | [] | simpa using (t.eTruncLTLTIsoLT a b hab).hom_inv_id_app X | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 415,
"column": 2
} | {
"line": 415,
"column": 58
} | {
"line": 417,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\ninst✝ : IsTriangulated C\na b : EInt\nhab : b ≤ a\nX : C\n⊢ (t.eTruncLT.obj b).map ((t.eT... | [] | simpa using (t.eTruncLTLTIsoLT a b hab).hom_inv_id_app X | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 421,
"column": 2
} | {
"line": 421,
"column": 13
} | {
"line": 421,
"column": 14
} | [
{
"pp": "C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\ninst✝ : IsTriangulated C\na b : EInt\nhab : b ≤ a\nX : C\n⊢ (t.eTruncLTLTIsoLT a b hab).i... | [
"C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\ninst✝ : IsTriangulated C\na b : EInt\nhab : b ≤ a\nX : C\n⊢ (t.eTruncLTLTIsoLT a b hab).inv.app X ≫ (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 463,
"column": 13
} | {
"line": 463,
"column": 24
} | {
"line": 463,
"column": 25
} | [
{
"pp": "case coe.bot\nC : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\ninst✝ : IsTriangulated C\na b✝ : EInt\nX : C\nb : ℤ\n⊢ IsIso ((t.eTruncLTGE... | [
"case coe.bot\nC : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\ninst✝ : IsTriangulated C\na b✝ : EInt\nX : C\nb : ℤ\n⊢ IsIso ((t.truncLT b).map ((t.tru... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 473,
"column": 11
} | {
"line": 473,
"column": 22
} | {
"line": 473,
"column": 23
} | [
{
"pp": "case top\nC : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\ninst✝ : IsTriangulated C\na b : EInt\nX : C\n⊢ IsIso ((t.eTruncLTGELTSelfToLTGE... | [
"case top\nC : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\ninst✝ : IsTriangulated C\na b : EInt\nX : C\n⊢ IsIso (𝟙 ((t.eTruncGE.obj a).obj X))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 482,
"column": 11
} | {
"line": 482,
"column": 22
} | {
"line": 482,
"column": 23
} | [
{
"pp": "case bot\nC : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\ninst✝ : IsTriangulated C\na b✝ b : EInt\nX✝ X : C\n⊢ IsIso ((t.eTruncLTGELTSelf... | [
"case bot\nC : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\ninst✝ : IsTriangulated C\na b✝ b : EInt\nX✝ X : C\n⊢ IsIso ((t.eTruncLTι b).app ((t.eTruncL... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 485,
"column": 13
} | {
"line": 485,
"column": 61
} | {
"line": 486,
"column": 8
} | [
{
"pp": "case coe.bot\nC : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\ninst✝ : IsTriangulated C\na✝ b✝ b : EInt\nX✝ X : C\na : ℤ\n⊢ IsIso ((t.eTru... | [
"case coe.bot\nC : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\ninst✝ : IsTriangulated C\na✝ b✝ b : EInt\nX✝ X : C\na : ℤ\n⊢ IsZero ((t.truncGE a).obj ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 491,
"column": 13
} | {
"line": 491,
"column": 24
} | {
"line": 491,
"column": 25
} | [
{
"pp": "case coe.top\nC : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\ninst✝ : IsTriangulated C\na✝ b✝ b : EInt\nX✝ X : C\na : ℤ\n⊢ IsIso ((t.eTru... | [
"case coe.top\nC : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\ninst✝ : IsTriangulated C\na✝ b✝ b : EInt\nX✝ X : C\na : ℤ\n⊢ IsIso (𝟙 ((t.truncGE a).o... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Uniform | {
"line": 99,
"column": 8
} | {
"line": 99,
"column": 19
} | {
"line": 99,
"column": 20
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_2\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\ns t : Finset α\nhG : G.IsUniform ε s t\nhε : ε ≤ 0\n⊢ ↑(#s) * ε ≤ ↑(#∅)",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\n𝕜 : Type u_2\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\ns t : Finset α\nhG : G.IsUniform ε s t\nhε : ε ≤ 0\n⊢ ↑(#s) * ε ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Uniform | {
"line": 100,
"column": 8
} | {
"line": 100,
"column": 19
} | {
"line": 100,
"column": 20
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_2\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\ns t : Finset α\nhG : G.IsUniform ε s t\nhε : ε ≤ 0\n⊢ ↑(#t) * ε ≤ ↑(#∅)",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\n𝕜 : Type u_2\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\ns t : Finset α\nhG : G.IsUniform ε s t\nhε : ε ≤ 0\n⊢ ↑(#t) * ε ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Uniform | {
"line": 106,
"column": 29
} | {
"line": 106,
"column": 40
} | {
"line": 106,
"column": 41
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_2\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\na b : α\nhε : 0 < ε\nt' : Finset α\nht' : t' ⊆ {b}\nht : ε ≤ ↑(#t')\nhs' : ∅ ⊆ {a}\nhs : ε ≤ ↑(#∅)\n⊢ ε ≤ 0",
"ppTerm": "?m.129",
"ass... | [
"α : Type u_1\n𝕜 : Type u_2\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\na b : α\nhε : 0 < ε\nt' : Finset α\nht' : t' ⊆ {b}\nht : ε ≤ ↑(#t')\nhs' : ∅ ⊆ {a}\nhs : ε ≤ ↑(#∅)\n⊢ ε ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Uniform | {
"line": 109,
"column": 29
} | {
"line": 109,
"column": 40
} | {
"line": 109,
"column": 41
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_2\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\na b : α\nhε : 0 < ε\nhs' : {a} ⊆ {a}\nhs : ε ≤ ↑(#{a})\nht' : ∅ ⊆ {b}\nht : ε ≤ ↑(#∅)\n⊢ ε ≤ 0",
"ppTerm": "?m.186",
"assigned": false... | [
"α : Type u_1\n𝕜 : Type u_2\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\na b : α\nhε : 0 < ε\nhs' : {a} ⊆ {a}\nhs : ε ≤ ↑(#{a})\nht' : ∅ ⊆ {b}\nht : ε ≤ ↑(#∅)\n⊢ ε ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Density | {
"line": 208,
"column": 4
} | {
"line": 208,
"column": 96
} | {
"line": 208,
"column": 97
} | [
{
"pp": "case inl\n𝕜 : Type u_1\nα : Type u_4\nβ : Type u_5\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns₁ : Finset α\nt₁ t₂ : Finset β\nδ : 𝕜\nht : t₂ ⊆ t₁\nhδ₀ : 0 ≤ δ\nhδ₁ : 0 < 1 - δ\nht₂ : (1 - δ) * ↑(#t₁) ≤ ↑(#t₂)... | [
"case inl\n𝕜 : Type u_1\nα : Type u_4\nβ : Type u_5\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns₁ : Finset α\nt₁ t₂ : Finset β\nδ : 𝕜\nht : t₂ ⊆ t₁\nhδ₀ : 0 ≤ δ\nhδ₁ : 0 < 1 - δ\nht₂ : (1 - δ) * ↑(#t₁) ≤ ↑(#t₂)\nhδ' : 0 ≤ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Density | {
"line": 211,
"column": 4
} | {
"line": 211,
"column": 96
} | {
"line": 211,
"column": 97
} | [
{
"pp": "case inr.inl\n𝕜 : Type u_1\nα : Type u_4\nβ : Type u_5\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns₁ s₂ : Finset α\nt₁ : Finset β\nδ : 𝕜\nhs : s₂ ⊆ s₁\nhδ₀ : 0 ≤ δ\nhδ₁ : 0 < 1 - δ\nhs₂ : (1 - δ) * ↑(#s₁) ≤ ↑(... | [
"case inr.inl\n𝕜 : Type u_1\nα : Type u_4\nβ : Type u_5\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns₁ s₂ : Finset α\nt₁ : Finset β\nδ : 𝕜\nhs : s₂ ⊆ s₁\nhδ₀ : 0 ≤ δ\nhδ₁ : 0 < 1 - δ\nhs₂ : (1 - δ) * ↑(#s₁) ≤ ↑(#s₂)\nhδ' : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Uniform | {
"line": 337,
"column": 4
} | {
"line": 337,
"column": 15
} | {
"line": 337,
"column": 16
} | [
{
"pp": "case inr.calc_1\nα : Type u_1\n𝕜 : Type u_2\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\ninst✝ : DecidableEq α\nA : Finset α\nP : Finpartition A\nhP : P.IsEquipartition\nh : P.parts.Nonempty\n⊢ ↑(#A / #P.parts + 1) ≤ ↑(#A) / ↑(#P.parts) + 1",
"ppTerm": "?inr.calc_1... | [
"case inr.calc_1\nα : Type u_1\n𝕜 : Type u_2\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\ninst✝ : DecidableEq α\nA : Finset α\nP : Finpartition A\nhP : P.IsEquipartition\nh : P.parts.Nonempty\n⊢ ↑(#A / #P.parts) ≤ ↑(#A) / ↑(#P.parts)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Uniform | {
"line": 377,
"column": 4
} | {
"line": 377,
"column": 53
} | {
"line": 379,
"column": 0
} | [
{
"pp": "case hbc\nα : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : DecidableEq α\nA : Finset α\nP : Finpartition A\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\nhA : A.Nonempty\nhε : 0 < ε\nhP : P.IsEquipartition\nhG : P.IsUniform G ε\... | [] | exact aux (P.parts_nonempty hA.ne_empty).card_pos | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SimpleGraph.Regularity.Increment | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 75
} | {
"line": 72,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPG : ¬P.IsUniform G ε\nhPα' : stepBound #P.parts ≤ Fintype.card α\nhPpos : 0 < stepBound #P.pa... | [
"α : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPG : ¬P.IsUniform G ε\nhPα' : stepBound #P.parts ≤ Fintype.card α\nhPpos : 0 < stepBound #P.parts\n⊢ ∑ x, ... | simp_rw [chunk, apply_dite Finpartition.parts, apply_dite card, sum_dite] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Combinatorics.SimpleGraph.Regularity.Lemma | {
"line": 107,
"column": 4
} | {
"line": 109,
"column": 10
} | {
"line": 110,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ #univ\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nh... | [
"case refine_2\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nl : ℕ\nhε : 0 < ε\nhl : l ≤ Fintype.card α\nhα : bound ε l ≤ Fintype.card α\nt : ℕ := initialBound ε l\nhtα : t ≤ #univ\ndum : Finpartition univ\nhdum₁ : dum.IsEquipartition\nhdum₂ : #dum.... | · rw [iterate_succ_apply', stepBound, bound]
gcongr
simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise | {
"line": 127,
"column": 6
} | {
"line": 127,
"column": 17
} | {
"line": 127,
"column": 18
} | [
{
"pp": "case neg.refine_2.refine_1.inr\nα : Type u_1\ninst✝ : DecidableEq α\nm : ℕ\nm_pos : m > 0\ns : Finset α\nih :\n ∀ t ⊂ s,\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = #t →\n ∃ Q,\n (∀ x ∈ Q.parts, #x = m ∨ #x = m + 1) ∧\n (∀ x ∈ P.parts, #(x \\ {y ∈ Q.p... | [
"case neg.refine_2.refine_1.inr\nα : Type u_1\ninst✝ : DecidableEq α\nm : ℕ\nm_pos : m > 0\ns : Finset α\nih :\n ∀ t ⊂ s,\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = #t →\n ∃ Q,\n (∀ x ∈ Q.parts, #x = m ∨ #x = m + 1) ∧\n (∀ x ∈ P.parts, #(x \\ {y ∈ Q.parts | y ⊆ x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.DeleteEdges | {
"line": 90,
"column": 39
} | {
"line": 90,
"column": 50
} | {
"line": 90,
"column": 51
} | [
{
"pp": "V : Type u_1\ns : Set (Sym2 V)\nG : SimpleGraph V\nhs : s ⊆ Sym2.diagSet\nu v : V\n⊢ (G.deleteEdges s).Adj u v ↔ G.Adj u v",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"SimpleGraph.deleteEdges",
"Eq.mpr",
"Sym2.mk",
"congrArg",
"SimpleGraph.Adj",
... | [
"V : Type u_1\ns : Set (Sym2 V)\nG : SimpleGraph V\nhs : s ⊆ Sym2.diagSet\nu v : V\n⊢ G.Adj u v → s(u, v) ∉ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.DeleteEdges | {
"line": 237,
"column": 4
} | {
"line": 238,
"column": 28
} | {
"line": 238,
"column": 29
} | [
{
"pp": "case refine_2\nV : Type u_1\nG : SimpleGraph V\n𝕜 : Type u_2\ninst✝³ : Ring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : Fintype ↑G.edgeSet\np : SimpleGraph V → Prop\nr : 𝕜\ninst✝ : Fintype (Sym2 V)\nh : ∀ ⦃H : SimpleGraph V⦄ [inst : DecidableRel H.Adj], H ≤ G → p H → r ≤ ↑(#G.edgeFinset) - ↑(#H.edgeFinset... | [
"case refine_2\nV : Type u_1\nG : SimpleGraph V\n𝕜 : Type u_2\ninst✝³ : Ring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : Fintype ↑G.edgeSet\np : SimpleGraph V → Prop\nr : 𝕜\ninst✝ : Fintype (Sym2 V)\nh : ∀ ⦃H : SimpleGraph V⦄ [inst : DecidableRel H.Adj], H ≤ G → p H → r ≤ ↑(#G.edgeFinset) - ↑(#H.edgeFinset)\ns : Finse... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 346,
"column": 8
} | {
"line": 347,
"column": 28
} | {
"line": 348,
"column": 6
} | [
{
"pp": "ι : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : G.Subgraph\na b : V\ns : Set G.Subgraph\n⊢ ∀ {v w : V}, (∃ G' ∈ s, G'.Adj v w) → G.Adj v w",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"SimpleGraph.Subgraph",
"SimpleGraph.Adj",
"SimpleGraph.Sub... | [] | rintro a b ⟨G', -, hab⟩
exact G'.adj_sub hab | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 346,
"column": 8
} | {
"line": 347,
"column": 28
} | {
"line": 348,
"column": 6
} | [
{
"pp": "ι : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : G.Subgraph\na b : V\ns : Set G.Subgraph\n⊢ ∀ {v w : V}, (∃ G' ∈ s, G'.Adj v w) → G.Adj v w",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"SimpleGraph.Subgraph",
"SimpleGraph.Adj",
"SimpleGraph.Sub... | [] | rintro a b ⟨G', -, hab⟩
exact G'.adj_sub hab | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 351,
"column": 28
} | {
"line": 351,
"column": 50
} | {
"line": 351,
"column": 51
} | [
{
"pp": "ι : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : G.Subgraph\na✝ b✝ : V\ns : Set G.Subgraph\na b : V\nh : ∃ G' ∈ s, G'.Adj a b\n⊢ ∃ G' ∈ s, G'.Adj b a",
"ppTerm": "?m.96",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"SimpleGraph.Subgraph",
... | [
"ι : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : G.Subgraph\na✝ b✝ : V\ns : Set G.Subgraph\na b : V\nh : ∃ G' ∈ s, G'.Adj a b\n⊢ ∃ G' ∈ s, G'.Adj a b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 613,
"column": 32
} | {
"line": 613,
"column": 43
} | {
"line": 613,
"column": 44
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nH₁ H₂ : G.Subgraph\nh : Disjoint H₁ H₂\n⊢ H₁.edgeSet ⊓ H₂.edgeSet ⊆ ⊥",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"CompleteLattice.toLattice",
"OrderBot.toBot",
... | [
"V : Type u\nG : SimpleGraph V\nH₁ H₂ : G.Subgraph\nh : Disjoint H₁ H₂\n⊢ H₁.edgeSet ∩ H₂.edgeSet = ∅"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Copy | {
"line": 186,
"column": 4
} | {
"line": 186,
"column": 28
} | {
"line": 186,
"column": 29
} | [
{
"pp": "case mp\nα : Type u_4\nβ : Type u_5\nA : SimpleGraph α\nB : SimpleGraph β\nf : A.Copy B\n⊢ f.toSubgraph ∈ {B' | Nonempty (A ≃g B'.coe)}",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"SimpleGraph.Iso",
"Set.ofPred",
"SimpleGraph.Subgraph",
"Membership.mem",
... | [
"case mp\nα : Type u_4\nβ : Type u_5\nA : SimpleGraph α\nB : SimpleGraph β\nf : A.Copy B\n⊢ Nonempty (A ≃g (Subgraph.map f.toHom ⊤).coe)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Copy | {
"line": 207,
"column": 44
} | {
"line": 207,
"column": 55
} | {
"line": 207,
"column": 56
} | [
{
"pp": "V : Type u_1\nW : Type u_2\nX : Type u_3\nα : Type u_4\nβ : Type u_5\nγ : Type u_6\nG G₁ G₂ G₃ : SimpleGraph V\nH : SimpleGraph W\nI : SimpleGraph X\nA : SimpleGraph α\nB : SimpleGraph β\nC : SimpleGraph γ\nf : ⊤.Copy G\nv w : α\nh : G.Adj (f.toEmbedding v) (f.toEmbedding w)\n⊢ ⊤.Adj v w",
"ppTerm"... | [
"V : Type u_1\nW : Type u_2\nX : Type u_3\nα : Type u_4\nβ : Type u_5\nγ : Type u_6\nG G₁ G₂ G₃ : SimpleGraph V\nH : SimpleGraph W\nI : SimpleGraph X\nA : SimpleGraph α\nB : SimpleGraph β\nC : SimpleGraph γ\nf : ⊤.Copy G\nv w : α\nh : G.Adj (f.toEmbedding v) (f.toEmbedding w)\n⊢ ¬v = w"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Copy | {
"line": 317,
"column": 2
} | {
"line": 317,
"column": 42
} | {
"line": 318,
"column": 4
} | [
{
"pp": "V : Type u_1\nW : Type u_2\nG : SimpleGraph V\nH : SimpleGraph W\nf : G.Copy H\nv : V\ninst✝¹ : Fintype ↑(G.neighborSet v)\ninst✝ : Fintype ↑(H.neighborSet (f v))\n⊢ G.degree v ≤ H.degree (f v)",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals"... | [
"V : Type u_1\nW : Type u_2\nG : SimpleGraph V\nH : SimpleGraph W\nf : G.Copy H\nv : V\ninst✝¹ : Fintype ↑(G.neighborSet v)\ninst✝ : Fintype ↑(H.neighborSet (f v))\n⊢ G.degree v ≤ H.degree (f v)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Copy | {
"line": 539,
"column": 2
} | {
"line": 539,
"column": 38
} | {
"line": 539,
"column": 39
} | [
{
"pp": "V : Type u_1\nW : Type u_2\nG : SimpleGraph V\nH : SimpleGraph W\ninst✝² : Fintype V\ninst✝¹ : Fintype { f // Injective ⇑f }\ninst✝ : DecidableEq G.Subgraph\n⊢ ↑{G' | Nonempty (H ≃g G'.coe)} = ↑(image Copy.toSubgraph univ)",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.... | [
"V : Type u_1\nW : Type u_2\nG : SimpleGraph V\nH : SimpleGraph W\ninst✝² : Fintype V\ninst✝¹ : Fintype { f // Injective ⇑f }\ninst✝ : DecidableEq G.Subgraph\n⊢ {G' | Nonempty (H ≃g G'.coe)} = Set.range Copy.toSubgraph"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 118,
"column": 2
} | {
"line": 119,
"column": 69
} | {
"line": 121,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nU : Finset α\nhU : U ∈ P.parts\nV : Finset α\nhV : V ∈ P.parts\nhUV : U ≠ V\nh₂ : ¬G.IsUniform ε U V\nhX : G.nonuniformWitness ε U V ∈ P.nonunif... | [] | grw [sum_const, smul_eq_mul, card_filter_atomise_le_two_pow (s := U) hX,
Finpartition.card_nonuniformWitnesses_le, filter_subset] <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 720,
"column": 66
} | {
"line": 720,
"column": 77
} | {
"line": 720,
"column": 78
} | [
{
"pp": "ι : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : G.Subgraph\na✝ b✝ : V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nH : G.Subgraph\na b : V\n⊢ (H.verts.toFinset, fun a b ↦ decide (H.Adj a b)).2 a b = true → G.Adj a b",
"ppTerm": "?m.131",
"assigned": ... | [
"ι : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : G.Subgraph\na✝ b✝ : V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nH : G.Subgraph\na b : V\n⊢ H.Adj a b → G.Adj a b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 721,
"column": 19
} | {
"line": 721,
"column": 30
} | {
"line": 721,
"column": 31
} | [
{
"pp": "ι : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : G.Subgraph\na✝ b✝ : V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nH : G.Subgraph\na b : V\n⊢ (H.verts.toFinset, fun a b ↦ decide (H.Adj a b)).2 a b = true → a ∈ (H.verts.toFinset, fun a b ↦ decide (H.Adj a b))... | [
"ι : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : G.Subgraph\na✝ b✝ : V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nH : G.Subgraph\na b : V\n⊢ H.Adj a b → a ∈ H.verts"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 835,
"column": 54
} | {
"line": 840,
"column": 35
} | {
"line": 842,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nG' : G.Subgraph\nv : V\ninst✝ : Fintype ↑(G'.neighborSet v)\nhG : G'.verts.Subsingleton\n⊢ G'.degree v = 0",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"congrArg",
"SimpleGraph.Subgraph.coe_degree",
... | [] | by
by_cases hv : v ∈ G'.verts
· rw [← G'.coe_degree ⟨v, hv⟩]
have := (Set.subsingleton_coe _).mpr hG
exact G'.coe.degree_eq_zero_of_subsingleton ⟨v, hv⟩
· exact degree_of_notMem_verts hv | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 973,
"column": 28
} | {
"line": 973,
"column": 53
} | {
"line": 973,
"column": 54
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv w : V\nhvw : G.Adj v w\nu : V\nthis : w = u ↔ u = w\n⊢ u ∈ (G.subgraphOfAdj hvw).neighborSet v ↔ u ∈ {w}",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"SimpleGraph.Subgraph.mem_neighborSet._simp_1",
"Eq.mpr",
"False",
"Sym... | [
"V : Type u\nG : SimpleGraph V\nv w : V\nhvw : G.Adj v w\nu : V\nthis : w = u ↔ u = w\n⊢ w = u ↔ u = w"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 1046,
"column": 4
} | {
"line": 1046,
"column": 15
} | {
"line": 1046,
"column": 16
} | [
{
"pp": "case Adj\nV : Type u\nG : SimpleGraph V\nG' : G.Subgraph\nG'' : G'.coe.Subgraph\nx✝¹ x✝ : ↑G'.verts\n⊢ (G'.Adj ↑x✝¹ ↑x✝ ∧ ∃ (hv : ↑x✝¹ ∈ G'.verts) (hw : ↑x✝ ∈ G'.verts), G''.Adj ⟨↑x✝¹, hv⟩ ⟨↑x✝, hw⟩) ↔ G''.Adj x✝¹ x✝",
"ppTerm": "?Adj",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"case Adj\nV : Type u\nG : SimpleGraph V\nG' : G.Subgraph\nG'' : G'.coe.Subgraph\nx✝¹ x✝ : ↑G'.verts\n⊢ G''.Adj x✝¹ x✝ → G'.Adj ↑x✝¹ ↑x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 1205,
"column": 43
} | {
"line": 1208,
"column": 22
} | {
"line": 1210,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nG' : G.Subgraph\n⊢ G'.IsInduced ↔ ∃ s, G' = ⊤.induce s",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"SimpleGraph.Subgraph",
"SimpleGraph.Adj",
"Membership.mem",
"Exists",
"Eq.rec",
... | [] | by
refine ⟨fun h ↦ ⟨G'.verts, h.induce_top_verts.symm⟩, fun ⟨s, h⟩ _ hu _ hv hadj ↦ ?_⟩
rw [h, (h ▸ rfl : s = G'.verts)]
exact ⟨hu, hv, hadj⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Walk.Traversal | {
"line": 153,
"column": 2
} | {
"line": 153,
"column": 13
} | {
"line": 153,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv w : V\np : G.Walk v w\nhp : ¬p.Nil\n⊢ G.Adj v p.snd",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u\nG : SimpleGraph V\nv w : V\np : G.Walk v w\nhp : ¬p.Nil\n⊢ G.Adj v p.snd"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Traversal | {
"line": 153,
"column": 37
} | {
"line": 153,
"column": 72
} | {
"line": 153,
"column": 73
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv w : V\np : G.Walk v w\nhp : ¬p.Nil\n⊢ 0 < p.length",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u\nG : SimpleGraph V\nv w : V\np : G.Walk v w\nhp : ¬p.Nil\n⊢ 0 < p.length"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Basic | {
"line": 241,
"column": 2
} | {
"line": 241,
"column": 13
} | {
"line": 241,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\n⊢ List.map (fun x ↦ x.toProd.2) p.darts = p.support.tail",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\n⊢ List.map (fun x ↦ x.toProd.2) p.darts = p.support.tail"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Basic | {
"line": 343,
"column": 58
} | {
"line": 343,
"column": 81
} | {
"line": 343,
"column": 82
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu✝ v✝ u c v : V\nh₁ : G.Adj u v\nw₁ : G.Walk v c\nv' : V\nh₂ : G.Adj u v'\nw₂ : G.Walk v' c\nh : (cons' u v c h₁ w₁).edges = (cons' u v' c h₂ w₂).edges\n⊢ v = v' ∧ w₁.edges = w₂.edges",
"ppTerm": "?m.351",
"assigned": false,
"usedConstants": [],
"usedFVars... | [
"V : Type u\nG : SimpleGraph V\nu✝ v✝ u c v : V\nh₁ : G.Adj u v\nw₁ : G.Walk v c\nv' : V\nh₂ : G.Adj u v'\nw₂ : G.Walk v' c\nh : (cons' u v c h₁ w₁).edges = (cons' u v' c h₂ w₂).edges\n⊢ v = v' ∧ w₁.edges = w₂.edges"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Basic | {
"line": 499,
"column": 4
} | {
"line": 499,
"column": 15
} | {
"line": 499,
"column": 16
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nl✝ : List V\nhead✝ v : V\nl : List V\nhne : head✝ :: v :: l ≠ []\nhchain : List.IsChain G.Adj (head✝ :: v :: l)\n⊢ (ofSupport (head✝ :: v :: l) hne hchain).support = head✝ :: v :: l",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"List.head",
... | [
"V : Type u\nG : SimpleGraph V\nl✝ : List V\nhead✝ v : V\nl : List V\nhne : head✝ :: v :: l ≠ []\nhchain : List.IsChain G.Adj (head✝ :: v :: l)\n⊢ (ofSupport (v :: l) ⋯ ⋯).support = v :: l"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Basic | {
"line": 536,
"column": 4
} | {
"line": 536,
"column": 33
} | {
"line": 536,
"column": 34
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nl✝ : List G.Dart\nd₁ d₂ : G.Dart\nl : List G.Dart\nhne : d₁ :: d₂ :: l ≠ []\nhchain : List.IsChain G.DartAdj (d₁ :: d₂ :: l)\n⊢ (ofDarts (d₁ :: d₂ :: l) hne hchain).darts = d₁ :: d₂ :: l",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"List.hea... | [
"V : Type u\nG : SimpleGraph V\nl✝ : List G.Dart\nd₁ d₂ : G.Dart\nl : List G.Dart\nhne : d₁ :: d₂ :: l ≠ []\nhchain : List.IsChain G.DartAdj (d₁ :: d₂ :: l)\n⊢ (ofDarts (d₂ :: l) ⋯ ⋯).darts = d₂ :: l"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Operations | {
"line": 265,
"column": 6
} | {
"line": 266,
"column": 12
} | {
"line": 268,
"column": 0
} | [
{
"pp": "case inr\nV : Type u\nG : SimpleGraph V\nu v : V\ni : ℕ\nu✝ v✝ w✝ : V\nh : G.Adj u✝ v✝\np : G.Walk v✝ w✝\nih : p.reverse.getVert i = p.getVert (p.length - i)\nhi : ¬i < p.length\nhi' : p.length < i\n⊢ (cons ⋯ nil).getVert (i - p.length) = (cons h p).getVert (p.length + 1 - i)",
"ppTerm": "?inr",
... | [] | · rw [Nat.eq_add_of_sub_eq (Nat.sub_pos_of_lt hi') rfl, Nat.sub_eq_zero_of_le hi']
simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Walk.Decomp | {
"line": 254,
"column": 2
} | {
"line": 254,
"column": 13
} | {
"line": 254,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v w : V\np : G.Walk v w\nh : u ∈ p.support\n⊢ (p.takeUntil u h).length ≤ p.length",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v w : V\np : G.Walk v w\nh : u ∈ p.support\n⊢ (p.takeUntil u h).length ≤ p.length"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Decomp | {
"line": 260,
"column": 2
} | {
"line": 260,
"column": 13
} | {
"line": 260,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v w : V\np : G.Walk v w\nh : u ∈ p.support\n⊢ (p.dropUntil u h).length ≤ p.length",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v w : V\np : G.Walk v w\nh : u ∈ p.support\n⊢ (p.dropUntil u h).length ≤ p.length"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Decomp | {
"line": 280,
"column": 2
} | {
"line": 280,
"column": 34
} | {
"line": 280,
"column": 35
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv w u : V\ninst✝ : DecidableEq V\np : G.Walk u v\nh : w ∈ p.support\nhsu : ¬1 ≤ (p.takeUntil w h).length\n⊢ u = w",
"ppTerm": "?m.40",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u\nG : SimpleGraph V\nv w u : V\ninst✝ : DecidableEq V\np : G.Walk u v\nh : w ∈ p.support\nhsu : ¬1 ≤ (p.takeUntil w h).length\n⊢ u = w"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Operations | {
"line": 490,
"column": 2
} | {
"line": 490,
"column": 13
} | {
"line": 490,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nd : G.Dart\nh : d ∈ p.darts\n⊢ d.toProd.2 ∈ p.support",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nd : G.Dart\nh : d ∈ p.darts\n⊢ d.toProd.2 ∈ p.support"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Subwalks | {
"line": 237,
"column": 2
} | {
"line": 244,
"column": 46
} | {
"line": 246,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v : V\nn k : ℕ\np : G.Walk u v\nh : n ≤ k\n⊢ (p.drop k).IsSubwalk (p.drop n)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Nat.recAux",
"SimpleGraph.Walk.drop_zero",
"HEq.refl",
"SimpleGraph.Walk.IsSubwalk.copy",
... | [] | induction k, h using Nat.le_induction with
| base => rfl
| succ k h ih =>
apply IsSubwalk.trans ?_ ih
clear h ih
induction k generalizing p u with
| zero => exact p.drop_zero ▸ (p.isSubwalk_rfl.copy rfl rfl p.getVert_zero.symm rfl).tail
| succ _ ih => cases p <;> simp [drop, ih] | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Subwalks | {
"line": 237,
"column": 2
} | {
"line": 244,
"column": 46
} | {
"line": 246,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v : V\nn k : ℕ\np : G.Walk u v\nh : n ≤ k\n⊢ (p.drop k).IsSubwalk (p.drop n)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Nat.recAux",
"SimpleGraph.Walk.drop_zero",
"HEq.refl",
"SimpleGraph.Walk.IsSubwalk.copy",
... | [] | induction k, h using Nat.le_induction with
| base => rfl
| succ k h ih =>
apply IsSubwalk.trans ?_ ih
clear h ih
induction k generalizing p u with
| zero => exact p.drop_zero ▸ (p.isSubwalk_rfl.copy rfl rfl p.getVert_zero.symm rfl).tail
| succ _ ih => cases p <;> simp [drop, ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Walk.Subwalks | {
"line": 237,
"column": 2
} | {
"line": 244,
"column": 46
} | {
"line": 246,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v : V\nn k : ℕ\np : G.Walk u v\nh : n ≤ k\n⊢ (p.drop k).IsSubwalk (p.drop n)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Nat.recAux",
"SimpleGraph.Walk.drop_zero",
"HEq.refl",
"SimpleGraph.Walk.IsSubwalk.copy",
... | [] | induction k, h using Nat.le_induction with
| base => rfl
| succ k h ih =>
apply IsSubwalk.trans ?_ ih
clear h ih
induction k generalizing p u with
| zero => exact p.drop_zero ▸ (p.isSubwalk_rfl.copy rfl rfl p.getVert_zero.symm rfl).tail
| succ _ ih => cases p <;> simp [drop, ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Walk.Decomp | {
"line": 304,
"column": 25
} | {
"line": 304,
"column": 36
} | {
"line": 304,
"column": 37
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v w : V\np : G.Walk v w\nh : u ∈ p.support\nhuw : u ≠ w\nhl : (p.takeUntil u h).length = p.length\n⊢ u = w",
"ppTerm": "?m.38",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v w : V\np : G.Walk v w\nh : u ∈ p.support\nhuw : u ≠ w\nhl : (p.takeUntil u h).length = p.length\n⊢ u = w"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Decomp | {
"line": 357,
"column": 2
} | {
"line": 357,
"column": 13
} | {
"line": 357,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv : V\ninst✝ : DecidableEq V\nc : G.Walk v v\nu : V\nh : u ∈ c.support\n⊢ (c.rotate u h).length = c.length",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u\nG : SimpleGraph V\nv : V\ninst✝ : DecidableEq V\nc : G.Walk v v\nu : V\nh : u ∈ c.support\n⊢ (c.rotate u h).length = c.length"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Operations | {
"line": 582,
"column": 2
} | {
"line": 582,
"column": 78
} | {
"line": 583,
"column": 2
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v w : V\nh : G.Adj u v\np : G.Walk v w\nn : ℕ\nhn : n ≠ 0\n⊢ ((cons h p).drop n).support = ((p.drop (n - 1)).copy ⋯ ⋯).support",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"SimpleGraph.Walk.getVert_cons",
"Nat.ne_zer... | [
"V : Type u\nG : SimpleGraph V\nu v w : V\nh : G.Adj u v\np : G.Walk v w\nw✝ : ℕ\nhn : w✝ + 1 ≠ 0\n⊢ ((cons h p).drop (w✝ + 1)).support = ((p.drop (w✝ + 1 - 1)).copy ⋯ ⋯).support"
] | obtain ⟨_, rfl⟩ := Nat.exists_add_one_eq.mpr (Nat.ne_zero_iff_zero_lt.mp hn) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 27
} | {
"line": 144,
"column": 28
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nh : p.IsTrail\n⊢ p.reverse.IsTrail",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"_private.Mathlib.Combinatorics.SimpleGraph.Paths.0.SimpleGraph.Walk.IsTrail.reverse._simp... | [
"V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nh : p.IsTrail\n⊢ p.edges.Nodup"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 184,
"column": 4
} | {
"line": 184,
"column": 23
} | {
"line": 184,
"column": 24
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝ : Fintype ↑G.edgeSet\nu v : V\nw : G.Walk u v\nh✝ : w.IsTrail\nedges : Finset (Sym2 V) := ⋯\nthis : edges.card = w.length\ne : Sym2 V\nh : e ∈ edges\n⊢ e ∈ w.edges",
"ppTerm": "?m.75",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"use... | [
"V : Type u\nG : SimpleGraph V\ninst✝ : Fintype ↑G.edgeSet\nu v : V\nw : G.Walk u v\nh✝ : w.IsTrail\nedges : Finset (Sym2 V) := ⋯\nthis : edges.card = w.length\ne : Sym2 V\nh : e ∈ edges\n⊢ e ∈ w.edges"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 26
} | {
"line": 218,
"column": 27
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nh : p.IsPath\n⊢ p.reverse.IsPath",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"SimpleGraph.Walk.support",
"id",
"_private.Mathlib.Combinatorics.SimpleGraph.Paths.0.Simple... | [
"V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nh : p.IsPath\n⊢ p.support.Nodup"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 319,
"column": 31
} | {
"line": 319,
"column": 42
} | {
"line": 319,
"column": 43
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv : V\np : G.Walk v v\nv✝ : V\nh✝¹ : G.Adj v v✝\nh✝ : G.Adj v✝ v\nhp : (cons h✝¹ (cons h✝ nil)).IsCircuit\n⊢ 3 ≤ (cons h✝¹ (cons h✝ nil)).length",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"congrArg",
"S... | [
"V : Type u\nG : SimpleGraph V\nv : V\np : G.Walk v v\nv✝ : V\nh✝¹ : G.Adj v v✝\nh✝ : G.Adj v✝ v\nhp : (cons h✝¹ (cons h✝ nil)).IsCircuit\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 343,
"column": 13
} | {
"line": 343,
"column": 24
} | {
"line": 343,
"column": 25
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu : V\np : G.Walk u u\nh : p.reverse.IsCycle\n⊢ p.IsCycle",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u\nG : SimpleGraph V\nu : V\np : G.Walk u u\nh : p.reverse.IsCycle\n⊢ p.IsCycle"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Operations | {
"line": 667,
"column": 2
} | {
"line": 667,
"column": 13
} | {
"line": 667,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\n⊢ p.reverse.snd = p.penultimate",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\n⊢ p.reverse.snd = p.penultimate"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 438,
"column": 6
} | {
"line": 438,
"column": 55
} | {
"line": 439,
"column": 6
} | [
{
"pp": "case pos\nV : Type u\nG : SimpleGraph V\nu✝ v✝ v w u : V\nh : G.Adj v w\np : G.Walk w u\nihp :\n p.IsPath → ∀ ⦃n : ℕ⦄, n ∈ {i | i ≤ p.length} → ∀ ⦃m : ℕ⦄, m ∈ {i | i ≤ p.length} → p.getVert n = p.getVert m → n = m\nhp : (cons h p).IsPath\nn : ℕ\nhn : n ≤ p.length + 1\nm : ℕ\nhm : m ≤ p.length + 1\nhnm... | [
"case pos\nV : Type u\nG : SimpleGraph V\nu✝ v✝ v w u : V\nh : G.Adj v w\np : G.Walk w u\nihp :\n p.IsPath → ∀ ⦃n : ℕ⦄, n ∈ {i | i ≤ p.length} → ∀ ⦃m : ℕ⦄, m ∈ {i | i ≤ p.length} → p.getVert n = p.getVert m → n = m\nhp : (cons h p).IsPath\nn : ℕ\nhn : n ≤ p.length + 1\nm : ℕ\nhm : m ≤ p.length + 1\nhn0 : ¬n = 0\nh... | simp only [hm0, Walk.getVert_cons p h hn0] at hnm | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 459,
"column": 4
} | {
"line": 459,
"column": 15
} | {
"line": 459,
"column": 16
} | [
{
"pp": "case nil\nV : Type u\nG : SimpleGraph V\nu : V\ni : ℕ\nhp : Walk.nil.IsPath\nhi : i ≤ Walk.nil.length\n⊢ Walk.nil.getVert i = u ↔ i = 0",
"ppTerm": "?nil",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"true_iff",
"id",
"instOfNatNat",
"Iff",... | [
"case nil\nV : Type u\nG : SimpleGraph V\nu : V\ni : ℕ\nhp : Walk.nil.IsPath\nhi : i ≤ Walk.nil.length\n⊢ i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 64,
"column": 4
} | {
"line": 64,
"column": 15
} | {
"line": 64,
"column": 16
} | [
{
"pp": "case refine_2\nα : Type u_1\nG : SimpleGraph α\ns : Set α\nh : s.Pairwise G.Adj\nv : α\nhv : v ∈ s\nw : α\nhw : w ∈ s\n⊢ (induce s G).Adj ⟨v, hv⟩ ⟨w, hw⟩ ↔ ⊤.Adj ⟨v, hv⟩ ⟨w, hw⟩",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SimpleGraph.comap_adj._simp_... | [
"case refine_2\nα : Type u_1\nG : SimpleGraph α\ns : Set α\nh : s.Pairwise G.Adj\nv : α\nhv : v ∈ s\nw : α\nhw : w ∈ s\n⊢ G.Adj v w ↔ ¬v = w"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 483,
"column": 12
} | {
"line": 483,
"column": 38
} | {
"line": 483,
"column": 39
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v u✝ v✝ w✝ : V\nh : G.Adj u✝ v✝\nq : G.Walk v✝ w✝\nih : Set.InjOn q.getVert {i | i ≤ q.length} → q.IsPath\nhinj : Set.InjOn (cons h q).getVert {i | i ≤ (cons h q).length}\nn : ℕ\nhn : n ≤ q.length\nm : ℕ\nhm : m ≤ q.length\nhnm : q.getVert n = q.getVert m\n⊢ (cons h q).... | [
"V : Type u\nG : SimpleGraph V\nu v u✝ v✝ w✝ : V\nh : G.Adj u✝ v✝\nq : G.Walk v✝ w✝\nih : Set.InjOn q.getVert {i | i ≤ q.length} → q.IsPath\nhinj : Set.InjOn (cons h q).getVert {i | i ≤ (cons h q).length}\nn : ℕ\nhn : n ≤ q.length\nm : ℕ\nhm : m ≤ q.length\nhnm : q.getVert n = q.getVert m\n⊢ q.getVert n = q.getVert... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 185,
"column": 2
} | {
"line": 185,
"column": 13
} | {
"line": 185,
"column": 14
} | [
{
"pp": "case mpr\nα : Type u_1\nβ : Type u_2\nG : SimpleGraph α\nf : α ↪ β\ns : Finset α\nhs : G.IsClique ↑s\nht : (map f s).Nontrivial\n⊢ (SimpleGraph.map (⇑f) G).IsClique ↑(map f s)",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"S... | [
"case mpr\nα : Type u_1\nβ : Type u_2\nG : SimpleGraph α\nf : α ↪ β\ns : Finset α\nhs : G.IsClique ↑s\nht : (map f s).Nontrivial\n⊢ G.IsClique ↑s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Operations | {
"line": 903,
"column": 2
} | {
"line": 903,
"column": 13
} | {
"line": 903,
"column": 14
} | [
{
"pp": "case cons\nV : Type u\nG : SimpleGraph V\nu v✝ : V\nh : G.Adj u v✝\np : G.Walk v✝ u\n⊢ (cons h p).support.tail ~ (cons h p).support.dropLast",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"SimpleGraph.Walk.support",
"id",
"List.Perm",
"List.dropLast",
... | [
"case cons\nV : Type u\nG : SimpleGraph V\nu v✝ : V\nh : G.Adj u v✝\np : G.Walk v✝ u\n⊢ p.support ~ (u :: p.support).dropLast"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 211,
"column": 2
} | {
"line": 211,
"column": 26
} | {
"line": 212,
"column": 2
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\nv w : α\ns : Set α\nhc : (G ⊔ edge v w).IsClique s\nx✝ : α\nhx : x✝ ∈ s \\ {v}\ny✝ : α\nhy : y✝ ∈ s \\ {v}\nhxy : x✝ ≠ y✝\n⊢ G.Adj x✝ y✝",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"SimpleGraph.edge",
"SimpleGraph.Adj",
"Membe... | [
"α : Type u_1\nG : SimpleGraph α\nv w : α\ns : Set α\nhc : (G ⊔ edge v w).IsClique s\nx✝ : α\nhx : x✝ ∈ s \\ {v}\ny✝ : α\nhy : y✝ ∈ s \\ {v}\nhxy : x✝ ≠ y✝\nthis : (G ⊔ edge v w).Adj x✝ y✝\n⊢ G.Adj x✝ y✝"
] | have := hc hx.1 hy.1 hxy | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 501,
"column": 2
} | {
"line": 501,
"column": 20
} | {
"line": 501,
"column": 21
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v w : V\np : G.Walk u v\nhp : p.IsPath\nhmem : s(v, w) ∈ p.edges\n⊢ w = p.penultimate",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u\nG : SimpleGraph V\nu v w : V\np : G.Walk u v\nhp : p.IsPath\nhmem : s(v, w) ∈ p.edges\n⊢ w = p.penultimate"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 605,
"column": 38
} | {
"line": 605,
"column": 66
} | {
"line": 605,
"column": 67
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\ninst✝ : DecidableEq V\nx : V\nw : G.Walk u v\nhw : w.IsTrail\nhx : x ∈ w.support\n⊢ ((w.takeUntil x hx).edges ++ (w.dropUntil x hx).edges).Nodup",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"SimpleG... | [
"V : Type u\nG : SimpleGraph V\nu v : V\ninst✝ : DecidableEq V\nx : V\nw : G.Walk u v\nhw : w.IsTrail\nhx : x ∈ w.support\n⊢ w.edges.Nodup"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 466,
"column": 2
} | {
"line": 466,
"column": 37
} | {
"line": 466,
"column": 38
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\nn : ℕ\ninst✝ : Fintype α\nhc : Nonempty ((completeGraph (Fin n)).Copy G)\n⊢ n ≤ Fintype.card α",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nG : SimpleGraph α\nn : ℕ\ninst✝ : Fintype α\nhc : Nonempty ((completeGraph (Fin n)).Copy G)\n⊢ n ≤ Fintype.card α"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Basic | {
"line": 131,
"column": 64
} | {
"line": 131,
"column": 75
} | {
"line": 131,
"column": 76
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\nthis :\n ∀ (a b : α),\n a ≠ b → {s | s ∈ G.cliqueSet 3 ∧ s(a, b) ∈ s.sym2} = {s | G.Adj a b ∧ ∃ c, G.Adj a c ∧ G.Adj b c ∧ s = {a, b, c}}\nhG :\n ∀ ⦃e : Sym2 α⦄,\n ¬e.IsDiag → {s | (∃ a b c, G.Adj a b ∧ G.Adj a c ∧ G.Adj b c ∧ s = {a, b, c}) ∧ e ∈ s.sym2}.Subsin... | [
"α : Type u_1\nG : SimpleGraph α\nthis :\n ∀ (a b : α),\n a ≠ b → {s | s ∈ G.cliqueSet 3 ∧ s(a, b) ∈ s.sym2} = {s | G.Adj a b ∧ ∃ c, G.Adj a c ∧ G.Adj b c ∧ s = {a, b, c}}\nhG :\n ∀ ⦃e : Sym2 α⦄,\n ¬e.IsDiag → {s | (∃ a b c, G.Adj a b ∧ G.Adj a c ∧ G.Adj b c ∧ s = {a, b, c}) ∧ e ∈ s.sym2}.Subsingleton\na b ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Basic | {
"line": 132,
"column": 25
} | {
"line": 132,
"column": 36
} | {
"line": 132,
"column": 37
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\nthis :\n ∀ (a b : α),\n a ≠ b → {s | s ∈ G.cliqueSet 3 ∧ s(a, b) ∈ s.sym2} = {s | G.Adj a b ∧ ∃ c, G.Adj a c ∧ G.Adj b c ∧ s = {a, b, c}}\nhG :\n ∀ ⦃e : Sym2 α⦄,\n ¬e.IsDiag → {s | (∃ a b c, G.Adj a b ∧ G.Adj a c ∧ G.Adj b c ∧ s = {a, b, c}) ∧ e ∈ s.sym2}.Subsin... | [
"α : Type u_1\nG : SimpleGraph α\nthis :\n ∀ (a b : α),\n a ≠ b → {s | s ∈ G.cliqueSet 3 ∧ s(a, b) ∈ s.sym2} = {s | G.Adj a b ∧ ∃ c, G.Adj a c ∧ G.Adj b c ∧ s = {a, b, c}}\nhG :\n ∀ ⦃e : Sym2 α⦄,\n ¬e.IsDiag → {s | (∃ a b c, G.Adj a b ∧ G.Adj a c ∧ G.Adj b c ∧ s = {a, b, c}) ∧ e ∈ s.sym2}.Subsingleton\na b ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 541,
"column": 2
} | {
"line": 542,
"column": 68
} | {
"line": 543,
"column": 2
} | [
{
"pp": "case mp\nα : Type u_1\nG : SimpleGraph α\n⊢ G.CliqueFree 2 → G = ⊥",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Finset.coe_singleton",
"eq_false",
"SimpleGraph.Adj.ne",
"Sym2.mk",
"congrArg",
"Finset",
"_pr... | [
"case mpr\nα : Type u_1\nG : SimpleGraph α\n⊢ G = ⊥ → G.CliqueFree 2"
] | · simp_rw [← edgeSet_eq_empty, Set.eq_empty_iff_forall_notMem, Sym2.forall, mem_edgeSet]
exact fun h a b hab => h _ ⟨by simpa [hab.ne], card_pair hab.ne⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 573,
"column": 6
} | {
"line": 573,
"column": 51
} | {
"line": 573,
"column": 51
} | [
{
"pp": "case right\nα : Type u_1\nG : SimpleGraph α\nn : ℕ\ninst✝ : DecidableEq α\nh : Maximal (fun H ↦ H.CliqueFree (n + 1)) G\nx y : α\nhne : x ≠ y\nhn : ¬G.Adj x y\nt : Finset α\nhc : (G ⊔ edge x y).IsNClique (n + 1) t\nh1 : x ∈ t\nh2 : y ∈ t\n⊢ G.IsNClique n (insert x ((t.erase y).erase x)) ∧ G.IsNClique n... | [
"case right\nα : Type u_1\nG : SimpleGraph α\nn : ℕ\ninst✝ : DecidableEq α\nh : Maximal (fun H ↦ H.CliqueFree (n + 1)) G\nx y : α\nhne : x ≠ y\nhn : ¬G.Adj x y\nt : Finset α\nhc : (G ⊔ edge x y).IsNClique (n + 1) t\nh1 : x ∈ t\nh2 : y ∈ t\n⊢ G.IsNClique n (t.erase y) ∧ G.IsNClique n (t.erase x)"
] | insert_erase <| mem_erase_of_ne_of_mem hne h1 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Basic | {
"line": 157,
"column": 4
} | {
"line": 159,
"column": 11
} | {
"line": 159,
"column": 12
} | [
{
"pp": "case refine_2\nα : Type u_1\nG : SimpleGraph α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : DecidableRel G.Adj\nhG : G.EdgeDisjointTriangles\ne : Sym2 α\nhe : e ∈ G.edgeFinset\n⊢ #(bipartiteBelow (fun s e ↦ e ∈ s.sym2) (G.cliqueFinset 3) e) ≤ 1",
"ppTerm": "?refine_2✝",
"assigned": true... | [
"case refine_2\nα : Type u_1\nG : SimpleGraph α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : DecidableRel G.Adj\nhG : G.EdgeDisjointTriangles\ne : Sym2 α\nhe : e ∈ G.edgeFinset\n⊢ ∀ (a : Finset α), G.IsNClique 3 a → e ∈ a.sym2 → ∀ (b : Finset α), G.IsNClique 3 b → e ∈ b.sym2 → a = b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Basic | {
"line": 166,
"column": 4
} | {
"line": 168,
"column": 11
} | {
"line": 168,
"column": 12
} | [
{
"pp": "case refine_1\nα : Type u_1\nG : SimpleGraph α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : DecidableRel G.Adj\nhG : G.LocallyLinear\n⊢ ∀ a ∈ G.edgeFinset, 1 ≤ #(bipartiteAbove (fun e s ↦ e ∈ s.sym2) (G.cliqueFinset 3) a)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [... | [
"case refine_1\nα : Type u_1\nG : SimpleGraph α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : DecidableRel G.Adj\nhG : G.LocallyLinear\n⊢ ∀ (x y : α), G.Adj x y → ∃ x_1, G.IsNClique 3 x_1 ∧ x ∈ x_1 ∧ y ∈ x_1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 710,
"column": 4
} | {
"line": 710,
"column": 15
} | {
"line": 710,
"column": 16
} | [
{
"pp": "case inr\nα : Type u_1\nβ : Type u_2\nG : SimpleGraph α\ne : α ≃ β\nn : ℕ\nhn : n ≠ 1\n⊢ (SimpleGraph.map (⇑e) G).cliqueSet n = map e.toEmbedding '' G.cliqueSet n",
"ppTerm": "?inr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case inr\nα : Type u_1\nβ : Type u_2\nG : SimpleGraph α\ne : α ≃ β\nn : ℕ\nhn : n ≠ 1\n⊢ (SimpleGraph.map (⇑e) G).cliqueSet n = map e.toEmbedding '' G.cliqueSet n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 816,
"column": 2
} | {
"line": 816,
"column": 37
} | {
"line": 816,
"column": 38
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : DecidableRel G.Adj\nn : ℕ\ns : Finset α\n⊢ s ∈ G.cliqueFinset n → s ∈ powersetCard n univ",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SimpleGraph.IsNClique",
"Fins... | [
"α : Type u_1\nG : SimpleGraph α\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : DecidableRel G.Adj\nn : ℕ\ns : Finset α\n⊢ G.IsNClique n s → #s = n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 915,
"column": 2
} | {
"line": 915,
"column": 13
} | {
"line": 915,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\ninst✝ : DecidableEq V\np : G.Walk u v\n⊢ p.bypass.length ≤ p.length",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u\nG : SimpleGraph V\nu v : V\ninst✝ : DecidableEq V\np : G.Walk u v\n⊢ p.bypass.length ≤ p.length"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 921,
"column": 72
} | {
"line": 921,
"column": 83
} | {
"line": 921,
"column": 84
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\ninst✝ : DecidableEq V\np : G.Walk u v\nh : p.length ≤ p.bypass.length\n⊢ p.support.length ≤ p.bypass.support.length",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"SimpleGraph.Walk.length",
"Sim... | [
"V : Type u\nG : SimpleGraph V\nu v : V\ninst✝ : DecidableEq V\np : G.Walk u v\nh : p.length ≤ p.bypass.length\n⊢ p.length ≤ p.bypass.length"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 972,
"column": 2
} | {
"line": 972,
"column": 13
} | {
"line": 972,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv : V\ninst✝ : DecidableEq V\nw : G.Walk v v\n⊢ w.cycleBypass.length ≤ w.length",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u\nG : SimpleGraph V\nv : V\ninst✝ : DecidableEq V\nw : G.Walk v v\n⊢ w.cycleBypass.length ≤ w.length"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 981,
"column": 6
} | {
"line": 981,
"column": 17
} | {
"line": 981,
"column": 18
} | [
{
"pp": "case refine_2\nV : Type u\nG : SimpleGraph V\nv : V\ninst✝ : DecidableEq V\nv' : V\nhvv' : G.Adj v v'\nw : G.Walk v' v\nhw : (cons hvv' w).IsCircuit\n⊢ (cons hvv' w.bypass).support.tail.Nodup",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"SimpleGraph.Walk.support",
... | [
"case refine_2\nV : Type u\nG : SimpleGraph V\nv : V\ninst✝ : DecidableEq V\nv' : V\nhvv' : G.Adj v v'\nw : G.Walk v' v\nhw : (cons hvv' w).IsCircuit\n⊢ w.bypass.support.Nodup"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 999,
"column": 2
} | {
"line": 999,
"column": 39
} | {
"line": 1000,
"column": 2
} | [
{
"pp": "α : Type u_3\nG : SimpleGraph α\ninst✝ : Finite α\nt : Finset α\ntc : Gᶜ.IsClique ↑t\n⊢ #t ≤ G.indepNum",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Combinatorics.SimpleGraph.Clique.0.SimpleGraph.IsIndepSet.card_le_indepNum._simp_1_2",
"Eq.mpr",
... | [
"α : Type u_3\nG : SimpleGraph α\ninst✝ : Finite α\nt : Finset α\ntc : Gᶜ.IsClique ↑t\n⊢ #t ≤ sSup {n | ∃ s, Gᶜ.IsNClique n s}"
] | simp_rw [indepNum, ← isNClique_compl] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 1003,
"column": 2
} | {
"line": 1003,
"column": 39
} | {
"line": 1004,
"column": 2
} | [
{
"pp": "α : Type u_3\nG : SimpleGraph α\n⊢ ∃ s, G.IsNIndepSet G.indepNum s",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Combinatorics.SimpleGraph.Clique.0.SimpleGraph.exists_isNIndepSet_indepNum._simp_1_2",
"SimpleGraph.IsNClique",
"co... | [
"α : Type u_3\nG : SimpleGraph α\n⊢ ∃ s, Gᶜ.IsNClique (sSup {n | ∃ s, Gᶜ.IsNClique n s}) s"
] | simp_rw [indepNum, ← isNClique_compl] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 1042,
"column": 2
} | {
"line": 1042,
"column": 37
} | {
"line": 1043,
"column": 2
} | [
{
"pp": "α : Type u_3\nG : SimpleGraph α\ninst✝ : Finite α\nt : Finset α\ntmc : G.IsMaximumIndepSet t\n⊢ #t = G.indepNum",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"SimpleGraph.IsMaximumIndepSet",
"congrArg",
"Compl.compl",
"Eq.mp",
"SimpleGraph",
"S... | [
"α : Type u_3\nG : SimpleGraph α\ninst✝ : Finite α\nt : Finset α\ntmc : Gᶜ.IsMaximumClique t\n⊢ #t = G.indepNum"
] | rw [← isMaximumClique_compl] at tmc | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 1043,
"column": 2
} | {
"line": 1043,
"column": 39
} | {
"line": 1044,
"column": 2
} | [
{
"pp": "α : Type u_3\nG : SimpleGraph α\ninst✝ : Finite α\nt : Finset α\ntmc : Gᶜ.IsMaximumClique t\n⊢ #t = G.indepNum",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SimpleGraph.IsNClique",
"congrArg",
"Set.ofPred",
"Compl.compl",
"Finset",
... | [
"α : Type u_3\nG : SimpleGraph α\ninst✝ : Finite α\nt : Finset α\ntmc : Gᶜ.IsMaximumClique t\n⊢ #t = sSup {n | ∃ s, Gᶜ.IsNClique n s}"
] | simp_rw [indepNum, ← isNClique_compl] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Combinatorics.SimpleGraph.Triangle.Removal | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 13
} | {
"line": 67,
"column": 14
} | [
{
"pp": "n k : ℕ\nhk : 0 < k\nhn : k ≤ n\n⊢ k ≤ k * (n / k)",
"ppTerm": "?m.87",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n k : ℕ\nhk : 0 < k\nhn : k ≤ n\n⊢ k ≤ k * (n / k)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Basic | {
"line": 264,
"column": 28
} | {
"line": 264,
"column": 39
} | {
"line": 264,
"column": 40
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_3\ninst✝⁵ : Field 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\nG : SimpleGraph α\nε : 𝕜\ninst✝² : Fintype α\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Nonempty α\nhε : G.FarFromTriangleFree ε\n⊢ ε * ↑(Fintype.card α) ^ 2 ≤ ↑(#G.edgeFinset)",
"ppTerm": "?m.65",... | [
"α : Type u_1\n𝕜 : Type u_3\ninst✝⁵ : Field 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\nG : SimpleGraph α\nε : 𝕜\ninst✝² : Fintype α\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Nonempty α\nhε : G.FarFromTriangleFree ε\n⊢ ε * ↑(Fintype.card α) ^ 2 ≤ ↑(#G.edgeFinset)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Basic | {
"line": 267,
"column": 6
} | {
"line": 267,
"column": 36
} | {
"line": 267,
"column": 37
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_3\ninst✝⁵ : Field 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\nG : SimpleGraph α\nε : 𝕜\ninst✝² : Fintype α\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Nonempty α\nhε : G.FarFromTriangleFree ε\n⊢ ↑((Fintype.card α).choose 2) < 2⁻¹ * ↑(Fintype.card α) ^ 2",
"ppT... | [
"α : Type u_1\n𝕜 : Type u_3\ninst✝⁵ : Field 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\nG : SimpleGraph α\nε : 𝕜\ninst✝² : Fintype α\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Nonempty α\nhε : G.FarFromTriangleFree ε\n⊢ ↑((Fintype.card α).choose 2) < ↑(Fintype.card α) ^ 2 / 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 344,
"column": 4
} | {
"line": 344,
"column": 45
} | {
"line": 345,
"column": 4
} | [
{
"pp": "case inl.refine_1\nα : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\nU V : Finset α\ninst✝ : Nonempty α\nhP : P.IsEquipartition\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhU : ... | [
"case inl.refine_2\nα : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\nU V : Finset α\ninst✝ : Nonempty α\nhP : P.IsEquipartition\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhU : U ∈ P.parts\... | · exact mod_cast G.edgeDensity_nonneg _ _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Triangle.Tripartite | {
"line": 161,
"column": 2
} | {
"line": 163,
"column": 48
} | {
"line": 165,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nt : Finset (α × β × γ)\ninst✝² : DecidableEq α\ninst✝¹ : DecidableEq β\ninst✝ : DecidableEq γ\nx y z : α ⊕ β ⊕ γ\n⊢ (graph t).Adj x y →\n (graph t).Adj x z →\n (graph t).Adj y z →\n ∃ a b c,\n {in₀ a, in₁ b, in₂ c} = {x, y, z} ∧\n ... | [] | rintro (_ | _ | _) (_ | _ | _) (_ | _ | _) <;>
refine ⟨_, _, _, by ext; simp only [Finset.mem_insert, Finset.mem_singleton]; try tauto,
?_, ?_, ?_⟩ <;> constructor <;> assumption | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Combinatorics.SimpleGraph.Triangle.Tripartite | {
"line": 161,
"column": 2
} | {
"line": 163,
"column": 48
} | {
"line": 165,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nt : Finset (α × β × γ)\ninst✝² : DecidableEq α\ninst✝¹ : DecidableEq β\ninst✝ : DecidableEq γ\nx y z : α ⊕ β ⊕ γ\n⊢ (graph t).Adj x y →\n (graph t).Adj x z →\n (graph t).Adj y z →\n ∃ a b c,\n {in₀ a, in₁ b, in₂ c} = {x, y, z} ∧\n ... | [] | rintro (_ | _ | _) (_ | _ | _) (_ | _ | _) <;>
refine ⟨_, _, _, by ext; simp only [Finset.mem_insert, Finset.mem_singleton]; try tauto,
?_, ?_, ?_⟩ <;> constructor <;> assumption | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Triangle.Tripartite | {
"line": 161,
"column": 2
} | {
"line": 163,
"column": 48
} | {
"line": 165,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nt : Finset (α × β × γ)\ninst✝² : DecidableEq α\ninst✝¹ : DecidableEq β\ninst✝ : DecidableEq γ\nx y z : α ⊕ β ⊕ γ\n⊢ (graph t).Adj x y →\n (graph t).Adj x z →\n (graph t).Adj y z →\n ∃ a b c,\n {in₀ a, in₁ b, in₂ c} = {x, y, z} ∧\n ... | [] | rintro (_ | _ | _) (_ | _ | _) (_ | _ | _) <;>
refine ⟨_, _, _, by ext; simp only [Finset.mem_insert, Finset.mem_singleton]; try tauto,
?_, ?_, ?_⟩ <;> constructor <;> assumption | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Triangle.Tripartite | {
"line": 168,
"column": 42
} | {
"line": 170,
"column": 72
} | {
"line": 170,
"column": 73
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\n𝕜 : Type u_4\ninst✝⁵ : Field 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\nt : Finset (α × β × γ)\na✝ a'✝ : α\nb✝ b'✝ : β\nc✝ c'✝ : γ\nx : α × β × γ\ninst✝² : DecidableEq α\ninst✝¹ : DecidableEq β\ninst✝ : DecidableEq γ\nx✝¹ x✝ : α × β × γ\na :... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\n𝕜 : Type u_4\ninst✝⁵ : Field 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\nt : Finset (α × β × γ)\na✝ a'✝ : α\nb✝ b'✝ : β\nc✝ c'✝ : γ\nx : α × β × γ\ninst✝² : DecidableEq α\ninst✝¹ : DecidableEq β\ninst✝ : DecidableEq γ\nx✝¹ x✝ : α × β × γ\na : α\nb : β\nc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 407,
"column": 41
} | {
"line": 407,
"column": 51
} | {
"line": 407,
"column": 52
} | [
{
"pp": "α : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\nU V : Finset α\ninst✝ : Nonempty α\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhε₁ : ε ≤ 1\nhU : U ∈ P.... | [
"α : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\nU V : Finset α\ninst✝ : Nonempty α\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhε₁ : ε ≤ 1\nhU : U ∈ P.parts\nhV : ... | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Removal | {
"line": 150,
"column": 18
} | {
"line": 150,
"column": 50
} | {
"line": 150,
"column": 51
} | [
{
"pp": "α : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nhG : G.FarFromTriangleFree ε\nh✝ : Nonempty α\nhε : 0 < ε\nl : ℕ := ⌈4 / ε⌉₊\nhl : 4 / ε ≤ ↑l\nhl' : Fintype.card α ≤ l\n⊢ 1 ≤ ↑(#(G.cliqueFinset 3))",
"ppTerm": "?m.192",
"assigned":... | [
"α : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nhG : G.FarFromTriangleFree ε\nh✝ : Nonempty α\nhε : 0 < ε\nl : ℕ := ⌈4 / ε⌉₊\nhl : 4 / ε ≤ ↑l\nhl' : Fintype.card α ≤ l\n⊢ ¬G.CliqueFree 3"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.Corner.Roth | {
"line": 61,
"column": 2
} | {
"line": 61,
"column": 29
} | {
"line": 61,
"column": 30
} | [
{
"pp": "G : Type u_1\ninst✝² : AddCommGroup G\nA : Finset (G × G)\nε : ℝ\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nhε : ε * ↑(Fintype.card G) ^ 2 ≤ ↑(#A)\n⊢ ε * ↑(Fintype.card G) ^ 2 ≤ ↑(#A)",
"ppTerm": "?m.59",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
... | [
"G : Type u_1\ninst✝² : AddCommGroup G\nA : Finset (G × G)\nε : ℝ\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nhε : ε * ↑(Fintype.card G) ^ 2 ≤ ↑(#A)\n⊢ ε * ↑(Fintype.card G) ^ 2 ≤ ↑(#A)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Removal | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 10
} | {
"line": 166,
"column": 11
} | [
{
"pp": "α : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nhG : ↑(#(G.cliqueFinset 3)) < triangleRemovalBound ε * ↑(Fintype.card α) ^ 3\nh :\n ∀ G' ≤ G,\n ∀ (x : DecidableRel G'.Adj), ↑(#G.edgeFinset) - ↑(#G'.edgeFinset) < ε * ↑(Fintype.card α ^ ... | [
"α : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nhG : ↑(#(G.cliqueFinset 3)) < triangleRemovalBound ε * ↑(Fintype.card α) ^ 3\nh :\n ∀ G' ≤ G,\n ∀ (x : DecidableRel G'.Adj), ↑(#G.edgeFinset) - ↑(#G'.edgeFinset) < ε * ↑(Fintype.card α ^ 2) → ¬G'.Cli... | intro G' | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 449,
"column": 4
} | {
"line": 449,
"column": 15
} | {
"line": 449,
"column": 16
} | [
{
"pp": "α : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\nU V : Finset α\ninst✝ : Nonempty α\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhε₁ : ε ≤ 1\nhU : U ∈ P.... | [
"α : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\nU V : Finset α\ninst✝ : Nonempty α\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhε₁ : ε ≤ 1\nhU : U ∈ P.parts\nhV : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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