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.Monoidal.Bimod | {
"line": 548,
"column": 66
} | {
"line": 548,
"column": 83
} | {
"line": 548,
"column": 83
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon C\nM : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nR S T U :... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon C\nM : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nR S T U : Mon C\nP : ... | comp_whiskerRight | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 551,
"column": 12
} | {
"line": 551,
"column": 29
} | {
"line": 551,
"column": 29
} | [
{
"pp": "case a.a.a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon C\nM : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X... | [
"case a.a.a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon C\nM : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nR S T U :... | comp_whiskerRight | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.Monoidal.Hopf_ | {
"line": 291,
"column": 2
} | {
"line": 292,
"column": 35
} | {
"line": 293,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n⊢ (Δ ⊗ₘ Δ) ≫\n (α_ A A (A ⊗ A)).hom ≫\n A ◁ (α_ A A A).inv ≫\n A ◁ (β_ A A).hom ▷ A ≫\n (α_ A (A ⊗ A) A).inv ≫ (α_ A A A).inv ▷ A ≫ μ ▷ A ▷ A ≫ �... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n⊢ (Δ ⊗ₘ Δ) ≫\n (α_ A A (A ⊗ A)).hom ≫\n A ◁ (α_ A A A).inv ≫\n A ◁ (β_ A A).hom ▷ A ≫\n (α_ A (A ⊗ A) A).inv ≫ (α_ A A A).inv ▷ A ≫ μ ▷ A ▷ A ≫ (((α_ A A A).... | slice_lhs 8 9 =>
rw [associator_naturality_left] | Mathlib.Tactic.Slice._aux_Mathlib_Tactic_CategoryTheory_Slice___macroRules_Mathlib_Tactic_Slice_sliceLHS_1 | Mathlib.Tactic.Slice.sliceLHS |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 573,
"column": 75
} | {
"line": 573,
"column": 92
} | {
"line": 573,
"column": 92
} | [
{
"pp": "case a.a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon C\nM : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\... | [
"case a.a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon C\nM : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nR S T U : M... | comp_whiskerRight | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.Monoidal.Internal.Module | {
"line": 60,
"column": 6
} | {
"line": 62,
"column": 9
} | {
"line": 63,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\nA : ModuleCat R\ninst✝ : MonObj A\nx y z : ↑A\n⊢ x * (y + z) = x * y + x * z",
"ppTerm": "?m.562",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"LinearMap.map_add",
"Mul.mk",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"out... | [] | convert! μ[A].hom.map_add (x ⊗ₜ y) (x ⊗ₜ z)
rw [← TensorProduct.tmul_add]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Internal.Module | {
"line": 60,
"column": 6
} | {
"line": 62,
"column": 9
} | {
"line": 63,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\nA : ModuleCat R\ninst✝ : MonObj A\nx y z : ↑A\n⊢ x * (y + z) = x * y + x * z",
"ppTerm": "?m.562",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"LinearMap.map_add",
"Mul.mk",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"out... | [] | convert! μ[A].hom.map_add (x ⊗ₜ y) (x ⊗ₜ z)
rw [← TensorProduct.tmul_add]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 710,
"column": 6
} | {
"line": 710,
"column": 19
} | {
"line": 710,
"column": 20
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : HasCoequalizers C\nR S : Mon C\nP : Bimod R S\n⊢ (ρ_ P.X).inv ≫ P.X ◁ η ≫ P.actRight = 𝟙 P.X",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MonoidalCategoryStruc... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : HasCoequalizers C\nR S : Mon C\nP : Bimod R S\n⊢ (ρ_ P.X).inv ≫ (ρ_ P.X).hom = 𝟙 P.X"
] | actRight_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 718,
"column": 79
} | {
"line": 725,
"column": 27
} | {
"line": 727,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : HasCoequalizers C\nR S : Mon C\nP : Bimod R S\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n⊢ (P.tens... | [] | by
dsimp; dsimp [hom, TensorBimod.actLeft, regular]
refine (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 ?_
dsimp
slice_lhs 1 4 => rw [id_tensor_π_preserves_coequalizer_inv_colimMap_desc]
slice_lhs 2 3 => rw [middle_assoc]
slice_rhs 1 2 => rw [← whiskerLeft_comp, coequalizer.π_desc]
rw [Iso.inv_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Presentable.Dense | {
"line": 37,
"column": 2
} | {
"line": 40,
"column": 35
} | {
"line": 42,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalAccessibleCategory C κ\n⊢ (isCardinalPresentable C κ).IsCardinalFilteredGenerator κ",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.ObjectPro... | [] | obtain ⟨P, _, hP⟩ := HasCardinalFilteredGenerator.exists_generator C κ
refine hP.of_le_isoClosure ?_ le_rfl
rw [ObjectProperty.isoClosure_eq_self]
exact hP.le_isCardinalPresentable | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Presentable.Dense | {
"line": 37,
"column": 2
} | {
"line": 40,
"column": 35
} | {
"line": 42,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalAccessibleCategory C κ\n⊢ (isCardinalPresentable C κ).IsCardinalFilteredGenerator κ",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.ObjectPro... | [] | obtain ⟨P, _, hP⟩ := HasCardinalFilteredGenerator.exists_generator C κ
refine hP.of_le_isoClosure ?_ le_rfl
rw [ObjectProperty.isoClosure_eq_self]
exact hP.le_isCardinalPresentable | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.MorphismProperty.Ind | {
"line": 137,
"column": 2
} | {
"line": 140,
"column": 45
} | {
"line": 142,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nP : MorphismProperty C\nhp : P ≤ isFinitelyPresentable C\ninst✝ : LocallySmall.{w, v, u} C\n⊢ P.ind.ind = P.ind",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"CategoryTheory.in... | [] | refine le_antisymm (fun X Y f hf ↦ ?_) P.ind.le_ind
have : P.underObj ≤ ObjectProperty.isFinitelyPresentable.{w} (Under X) := fun f hf ↦ hp _ hf
simpa [ind_iff_ind_underMk, underObj_ind_eq_ind_underObj,
ObjectProperty.ind_ind.{w} this] using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.MorphismProperty.Ind | {
"line": 137,
"column": 2
} | {
"line": 140,
"column": 45
} | {
"line": 142,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nP : MorphismProperty C\nhp : P ≤ isFinitelyPresentable C\ninst✝ : LocallySmall.{w, v, u} C\n⊢ P.ind.ind = P.ind",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"CategoryTheory.in... | [] | refine le_antisymm (fun X Y f hf ↦ ?_) P.ind.le_ind
have : P.underObj ≤ ObjectProperty.isFinitelyPresentable.{w} (Under X) := fun f hf ↦ hp _ hf
simpa [ind_iff_ind_underMk, underObj_ind_eq_ind_underObj,
ObjectProperty.ind_ind.{w} this] using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.MorphismProperty.Ind | {
"line": 196,
"column": 4
} | {
"line": 196,
"column": 25
} | {
"line": 197,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\nP : MorphismProperty C\ninst✝⁴ : ∀ (X : C), IsFinitelyAccessibleCategory (Under X)\ninst✝³ : HasPushouts C\ninst✝² : P.IsStableUnderComposition\ninst✝¹ : P.IsStableUnderCobaseChange\ninst✝ : P.PreIndSpreads\nH : P ≤ isFinitelyPresentable C\nX Y Z : C\nf : X ⟶ Y\n... | [
"C : Type u\ninst✝⁵ : Category.{v, u} C\nP : MorphismProperty C\ninst✝⁴ : ∀ (X : C), IsFinitelyAccessibleCategory (Under X)\ninst✝³ : HasPushouts C\ninst✝² : P.IsStableUnderComposition\ninst✝¹ : P.IsStableUnderCobaseChange\ninst✝ : P.PreIndSpreads\nH : P ≤ isFinitelyPresentable C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh... | rw [ind_iff_exists H] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Monoidal.DayConvolution | {
"line": 1262,
"column": 6
} | {
"line": 1263,
"column": 82
} | {
"line": 1264,
"column": 6
} | [
{
"pp": "C✝ : Type u₁\ninst✝¹³ : Category.{v₁, u₁} C✝\nV✝ : Type u₂\ninst✝¹² : Category.{v₂, u₂} V✝\ninst✝¹¹ : MonoidalCategory C✝\ninst✝¹⁰ : MonoidalCategory V✝\nC : Type u₁\ninst✝⁹ : Category.{v₁, u₁} C\nV : Type u₂\ninst✝⁸ : Category.{v₂, u₂} V\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : MonoidalCategory V\nD : T... | [
"C✝ : Type u₁\ninst✝¹³ : Category.{v₁, u₁} C✝\nV✝ : Type u₂\ninst✝¹² : Category.{v₂, u₂} V✝\ninst✝¹¹ : MonoidalCategory C✝\ninst✝¹⁰ : MonoidalCategory V✝\nC : Type u₁\ninst✝⁹ : Category.{v₁, u₁} C\nV : Type u₂\ninst✝⁸ : Category.{v₂, u₂} V\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : MonoidalCategory V\nD : Type u₃\ninst... | simp only [Functor.comp_obj, tensor_obj, rightUnitor,
Functor.FullyFaithful.preimageIso_hom, Functor.FullyFaithful.map_preimage] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Subobject.NoetherianObject | {
"line": 75,
"column": 2
} | {
"line": 79,
"column": 59
} | {
"line": 81,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : C\n⊢ IsNoetherianObject X ↔ ∀ (f : ℕ → Subobject X), ¬StrictMono f",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Preorder.toLT",
"StrictMono",
"CategoryTheory.ObjectProperty.is_iff",
... | [] | refine ⟨fun _ ↦ not_strictMono_of_wellFoundedGT, fun h ↦ ?_⟩
dsimp only [IsNoetherianObject]
rw [ObjectProperty.is_iff, isNoetherianObject, WellFoundedGT,
isWellFounded_iff, RelEmbedding.wellFounded_iff_isEmpty]
exact ⟨fun f ↦ h f.toFun (fun a b h ↦ f.map_rel_iff.2 h)⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Subobject.NoetherianObject | {
"line": 75,
"column": 2
} | {
"line": 79,
"column": 59
} | {
"line": 81,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : C\n⊢ IsNoetherianObject X ↔ ∀ (f : ℕ → Subobject X), ¬StrictMono f",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Preorder.toLT",
"StrictMono",
"CategoryTheory.ObjectProperty.is_iff",
... | [] | refine ⟨fun _ ↦ not_strictMono_of_wellFoundedGT, fun h ↦ ?_⟩
dsimp only [IsNoetherianObject]
rw [ObjectProperty.is_iff, isNoetherianObject, WellFoundedGT,
isWellFounded_iff, RelEmbedding.wellFounded_iff_isEmpty]
exact ⟨fun f ↦ h f.toFun (fun a b h ↦ f.map_rel_iff.2 h)⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 904,
"column": 23
} | {
"line": 904,
"column": 40
} | {
"line": 904,
"column": 40
} | [
{
"pp": "case a.a.a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon C\nM M' : B... | [
"case a.a.a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon C\nM M' : Bimod W X\nf ... | comp_whiskerRight | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.Preadditive.FreydCategory.RightFreyd | {
"line": 185,
"column": 50
} | {
"line": 185,
"column": 64
} | {
"line": 187,
"column": 0
} | [
{
"pp": "case h₁\nV : Type u_1\ninst✝² : Category.{v_1, u_1} V\ninst✝¹ : Preadditive V\ninst✝ : HasBinaryBiproducts V\nu v : Arrow V\nf : u ⟶ v\nw : Arrow V\ng : v ⟶ w\nh : RightHomotopy (f ≫ g) 0\n⊢ Hom.left (π f ≫ desc f g h) = Hom.left g",
"ppTerm": "?h₁",
"assigned": true,
"usedConstants": [
... | [] | simp [π, desc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Preadditive.FreydCategory.RightFreyd | {
"line": 185,
"column": 50
} | {
"line": 185,
"column": 64
} | {
"line": 187,
"column": 0
} | [
{
"pp": "case h₂\nV : Type u_1\ninst✝² : Category.{v_1, u_1} V\ninst✝¹ : Preadditive V\ninst✝ : HasBinaryBiproducts V\nu v : Arrow V\nf : u ⟶ v\nw : Arrow V\ng : v ⟶ w\nh : RightHomotopy (f ≫ g) 0\n⊢ Hom.right (π f ≫ desc f g h) = Hom.right g",
"ppTerm": "?h₂",
"assigned": true,
"usedConstants": [
... | [] | simp [π, desc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 979,
"column": 6
} | {
"line": 979,
"column": 23
} | {
"line": 979,
"column": 23
} | [
{
"pp": "case a.a.a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nV W X Y Z : Mon C\nM : Bi... | [
"case a.a.a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nV W X Y Z : Mon C\nM : Bimod V W\nN :... | comp_whiskerRight | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.Presentable.OrthogonalReflection | {
"line": 102,
"column": 4
} | {
"line": 102,
"column": 64
} | {
"line": 103,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nW : MorphismProperty C\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : HasCardinalFilteredColimits C κ\nhW : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), W f → IsCardinalPresentable X κ ∧ IsCardinalPresentable Y κ\nJ : Type w\nx✝¹ : SmallCategory J\nx✝ : IsCardinalFiltered J κ... | [
"C : Type u\ninst✝² : Category.{v, u} C\nW : MorphismProperty C\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : HasCardinalFilteredColimits C κ\nhW : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), W f → IsCardinalPresentable X κ ∧ IsCardinalPresentable Y κ\nJ : Type w\nx✝¹ : SmallCategory J\nx✝ : IsCardinalFiltered J κ\nthis✝ : W.... | have := HasCardinalFilteredColimits.hasColimitsOfShape C κ J | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Presentable.OrthogonalReflection | {
"line": 284,
"column": 2
} | {
"line": 290,
"column": 13
} | {
"line": 291,
"column": 2
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝⁴ : Category.{v, u} C\nW : MorphismProperty C\nZ : C\ninst✝³ : HasCoproduct D₁.obj₁\ninst✝² : HasCoproduct D₁.obj₂\ninst✝¹ : HasPushouts C\ninst✝ : HasMulticoequalizer (D₂.multispanIndex W Z)\nT : C\nhT : W.isLocal T\nφ₁ φ₂ : succ W Z ⟶ T\nh : (fun g ↦ toSucc W Z ≫ g) φ₁... | [
"case refine_2\nC : Type u\ninst✝⁴ : Category.{v, u} C\nW : MorphismProperty C\nZ : C\ninst✝³ : HasCoproduct D₁.obj₁\ninst✝² : HasCoproduct D₁.obj₂\ninst✝¹ : HasPushouts C\ninst✝ : HasMulticoequalizer (D₂.multispanIndex W Z)\nT : C\nhT : W.isLocal T\ng : Z ⟶ T\n⊢ ∃ a, (fun g ↦ toSucc W Z ≫ g) a = g"
] | · ext ⟨⟩
simp only [Category.assoc] at h
dsimp
ext d
· apply (hT d.1.1.hom d.1.2).1
simp only [← D₁.ι_comp_t_assoc, pushout.condition_assoc, h]
· exact h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Presentable.OrthogonalReflection | {
"line": 416,
"column": 4
} | {
"line": 432,
"column": 23
} | {
"line": 433,
"column": 2
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝⁷ : Category.{v, u} C\nW : MorphismProperty C\nZ : C\ninst✝⁶ : HasPushouts C\ninst✝⁵ : ∀ (Z : C), HasCoproduct D₁.obj₁\ninst✝⁴ : ∀ (Z : C), HasCoproduct D₁.obj₂\ninst✝³ : ∀ (Z : C), HasMulticoequalizer (D₂.multispanIndex W Z)\nκ : Cardinal.{w}\ninst✝² : OrderBot κ.ord.To... | [] | obtain ⟨j, g₁, g₂, rfl, rfl⟩ :
∃ (j : κ.ord.ToType) (g₁' g₂' : Y ⟶ H.F.obj j), g₁' ≫ H.incl.app j = g₁ ∧
g₂' ≫ H.incl.app j = g₂ := by
obtain ⟨j₁, g₁, rfl⟩ := IsCardinalPresentable.exists_hom_of_isColimit κ H.isColimit g₁
obtain ⟨j₂, g₂, rfl⟩ := IsCardinalPresentable.exists_hom_of_isColimit κ ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Presentable.OrthogonalReflection | {
"line": 416,
"column": 4
} | {
"line": 432,
"column": 23
} | {
"line": 433,
"column": 2
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝⁷ : Category.{v, u} C\nW : MorphismProperty C\nZ : C\ninst✝⁶ : HasPushouts C\ninst✝⁵ : ∀ (Z : C), HasCoproduct D₁.obj₁\ninst✝⁴ : ∀ (Z : C), HasCoproduct D₁.obj₂\ninst✝³ : ∀ (Z : C), HasMulticoequalizer (D₂.multispanIndex W Z)\nκ : Cardinal.{w}\ninst✝² : OrderBot κ.ord.To... | [] | obtain ⟨j, g₁, g₂, rfl, rfl⟩ :
∃ (j : κ.ord.ToType) (g₁' g₂' : Y ⟶ H.F.obj j), g₁' ≫ H.incl.app j = g₁ ∧
g₂' ≫ H.incl.app j = g₂ := by
obtain ⟨j₁, g₁, rfl⟩ := IsCardinalPresentable.exists_hom_of_isColimit κ H.isColimit g₁
obtain ⟨j₂, g₂, rfl⟩ := IsCardinalPresentable.exists_hom_of_isColimit κ ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 340,
"column": 6
} | {
"line": 340,
"column": 22
} | {
"line": 341,
"column": 4
} | [
{
"pp": "case inl.inr\nJ : Type w\ninst✝² : SmallCategory J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)\nι : Type w\nD : ι → DiagramWithUniqueTerminal J κ\nhι : HasCardinalLT ι κ\nm : J\nu : (i : ι) → (D i).top ⟶ m\nX Y : J\ni✝ : Unit\n⊢ (... | [] | exact Or.inr rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 340,
"column": 6
} | {
"line": 340,
"column": 22
} | {
"line": 341,
"column": 4
} | [
{
"pp": "case inl.inr\nJ : Type w\ninst✝² : SmallCategory J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)\nι : Type w\nD : ι → DiagramWithUniqueTerminal J κ\nhι : HasCardinalLT ι κ\nm : J\nu : (i : ι) → (D i).top ⟶ m\nX Y : J\ni✝ : Unit\n⊢ (... | [] | exact Or.inr rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 340,
"column": 6
} | {
"line": 340,
"column": 22
} | {
"line": 341,
"column": 4
} | [
{
"pp": "case inl.inr\nJ : Type w\ninst✝² : SmallCategory J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)\nι : Type w\nD : ι → DiagramWithUniqueTerminal J κ\nhι : HasCardinalLT ι κ\nm : J\nu : (i : ι) → (D i).top ⟶ m\nX Y : J\ni✝ : Unit\n⊢ (... | [] | exact Or.inr rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 348,
"column": 6
} | {
"line": 348,
"column": 22
} | {
"line": 349,
"column": 4
} | [
{
"pp": "case inl.inr\nJ : Type w\ninst✝² : SmallCategory J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)\nι : Type w\nD : ι → DiagramWithUniqueTerminal J κ\nhι : HasCardinalLT ι κ\nm : J\nu : (i : ι) → (D i).top ⟶ m\nX Y : J\ni✝ : Unit\n⊢ (... | [] | exact Or.inr rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 348,
"column": 6
} | {
"line": 348,
"column": 22
} | {
"line": 349,
"column": 4
} | [
{
"pp": "case inl.inr\nJ : Type w\ninst✝² : SmallCategory J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)\nι : Type w\nD : ι → DiagramWithUniqueTerminal J κ\nhι : HasCardinalLT ι κ\nm : J\nu : (i : ι) → (D i).top ⟶ m\nX Y : J\ni✝ : Unit\n⊢ (... | [] | exact Or.inr rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 348,
"column": 6
} | {
"line": 348,
"column": 22
} | {
"line": 349,
"column": 4
} | [
{
"pp": "case inl.inr\nJ : Type w\ninst✝² : SmallCategory J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)\nι : Type w\nD : ι → DiagramWithUniqueTerminal J κ\nhι : HasCardinalLT ι κ\nm : J\nu : (i : ι) → (D i).top ⟶ m\nX Y : J\ni✝ : Unit\n⊢ (... | [] | exact Or.inr rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 349,
"column": 6
} | {
"line": 349,
"column": 22
} | {
"line": 350,
"column": 2
} | [
{
"pp": "case inr\nJ : Type w\ninst✝² : SmallCategory J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)\nι : Type w\nD : ι → DiagramWithUniqueTerminal J κ\nhι : HasCardinalLT ι κ\nm : J\nu : (i : ι) → (D i).top ⟶ m\nX Y : J\ni : ι\nj : J\nhj :... | [] | exact Or.inr rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 349,
"column": 6
} | {
"line": 349,
"column": 22
} | {
"line": 350,
"column": 2
} | [
{
"pp": "case inr\nJ : Type w\ninst✝² : SmallCategory J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)\nι : Type w\nD : ι → DiagramWithUniqueTerminal J κ\nhι : HasCardinalLT ι κ\nm : J\nu : (i : ι) → (D i).top ⟶ m\nX Y : J\ni : ι\nj : J\nhj :... | [] | exact Or.inr rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 349,
"column": 6
} | {
"line": 349,
"column": 22
} | {
"line": 350,
"column": 2
} | [
{
"pp": "case inr\nJ : Type w\ninst✝² : SmallCategory J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)\nι : Type w\nD : ι → DiagramWithUniqueTerminal J κ\nhι : HasCardinalLT ι κ\nm : J\nu : (i : ι) → (D i).top ⟶ m\nX Y : J\ni : ι\nj : J\nhj :... | [] | exact Or.inr rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Presentable.Type | {
"line": 161,
"column": 2
} | {
"line": 164,
"column": 78
} | {
"line": 166,
"column": 0
} | [
{
"pp": "X : Type u\nκ : Cardinal.{u}\ninst✝ : Fact κ.IsRegular\n⊢ ∃ P, ∃ (_ : ObjectProperty.Small.{u, u, u + 1} P), P.IsStrongGenerator ∧ P ≤ isCardinalPresentable (Type u) κ",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"hasCardinalLT_of_finite",
"HasCardin... | [] | exact ⟨.singleton PUnit, inferInstance, isStrongGenerator_punit, by
simp only [ObjectProperty.singleton_le_iff,
CategoryTheory.isCardinalPresentable_iff, isCardinalPresentable_iff]
exact hasCardinalLT_of_finite _ _ (Cardinal.IsRegular.aleph0_le Fact.out)⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 469,
"column": 6
} | {
"line": 469,
"column": 32
} | {
"line": 470,
"column": 4
} | [
{
"pp": "case inl.inl\nJ : Type w\ninst✝² : SmallCategory J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)\nthis✝¹ : IsCardinalFiltered (DiagramWithUniqueTerminal J κ) κ\nthis✝ : IsFiltered J\nthis : IsFiltered (DiagramWithUniqueTerminal J κ)... | [] | exact (h₁ (D.tgt hf)).elim | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 469,
"column": 6
} | {
"line": 469,
"column": 32
} | {
"line": 470,
"column": 4
} | [
{
"pp": "case inl.inl\nJ : Type w\ninst✝² : SmallCategory J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)\nthis✝¹ : IsCardinalFiltered (DiagramWithUniqueTerminal J κ) κ\nthis✝ : IsFiltered J\nthis : IsFiltered (DiagramWithUniqueTerminal J κ)... | [] | exact (h₁ (D.tgt hf)).elim | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 469,
"column": 6
} | {
"line": 469,
"column": 32
} | {
"line": 470,
"column": 4
} | [
{
"pp": "case inl.inl\nJ : Type w\ninst✝² : SmallCategory J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)\nthis✝¹ : IsCardinalFiltered (DiagramWithUniqueTerminal J κ) κ\nthis✝ : IsFiltered J\nthis : IsFiltered (DiagramWithUniqueTerminal J κ)... | [] | exact (h₁ (D.tgt hf)).elim | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.CartesianMonoidal | {
"line": 35,
"column": 65
} | {
"line": 39,
"column": 46
} | {
"line": 41,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nA : Type u₂\ninst✝¹ : Category.{v₂, u₂} A\nJ : GrothendieckTopology C\ninst✝ : CartesianMonoidalCategory A\nX Y : Sheaf J A\n⊢ Presheaf.IsSheaf J (X.obj ⊗ Y.obj)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functo... | [] | by
apply isSheaf_of_isLimit (E := (Cone.postcompose (pairComp X Y (sheafToPresheaf J A)).inv).obj
(BinaryFan.mk (fst X.obj Y.obj) (snd _ _)))
exact (IsLimit.postcomposeInvEquiv _ _).invFun
(tensorProductIsBinaryProduct X.obj Y.obj) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Adjunction | {
"line": 92,
"column": 4
} | {
"line": 92,
"column": 42
} | {
"line": 93,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nE : Type u_1\ninst✝ : Category.{v_1, u_1} E\nF : D ⥤ E\nG : E ⥤ D\nadj : G ⊣ F\nP Q : Cᵒᵖ ⥤ E\nf : P ⟶ Q\nhf : J.W f\nthis : F.IsRightAdjoint\n⊢ J.W.inverseImage ((whiskeringRight Cᵒᵖ E D).... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nE : Type u_1\ninst✝ : Category.{v_1, u_1} E\nF : D ⥤ E\nG : E ⥤ D\nadj : G ⊣ F\nP Q : Cᵒᵖ ⥤ E\nf : P ⟶ Q\nhf : J.W f\nthis : F.IsRightAdjoint\n⊢ J.W (((whiskeringRight Cᵒᵖ E D).obj G).map f)"
] | rw [MorphismProperty.inverseImage_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Sites.Coherent.ReflectsPrecoherent | {
"line": 40,
"column": 4
} | {
"line": 42,
"column": 17
} | {
"line": 44,
"column": 0
} | [
{
"pp": "case refine_2\nC : Type u_1\nD : Type u_2\ninst✝⁷ : Category.{v_1, u_1} C\ninst✝⁶ : Category.{v_2, u_2} D\nF : C ⥤ D\ninst✝⁵ : F.PreservesFiniteEffectiveEpiFamilies\ninst✝⁴ : F.ReflectsFiniteEffectiveEpiFamilies\ninst✝³ : F.EffectivelyEnough\ninst✝² : Precoherent D\ninst✝¹ : F.Full\ninst✝ : F.Faithful\... | [] | · intro b
apply F.map_injective
simp [hh b] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves | {
"line": 46,
"column": 4
} | {
"line": 46,
"column": 91
} | {
"line": 47,
"column": 4
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : FinitaryPreExtensive C\nX : C\nS : Presieve X\ninst✝ : S.Extensive\n⊢ ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g",
"ppTerm": "?m.14",
"assigned": true,
"usedConstant... | [
"C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : FinitaryPreExtensive C\nX : C\nw✝³ : Type\nw✝² : Finite w✝³\nw✝¹ : w✝³ → C\nw✝ : (a : w✝³) → w✝¹ a ⟶ X\ninst✝ : (ofArrows w✝¹ w✝).Extensive\nhc : IsColimit (Cofan.mk X w✝)\n⊢ ∀ {Y Z : C} {f : Y ⟶ X}, ofArrows w✝¹ w... | obtain ⟨_, _, _, _, rfl, ⟨hc⟩⟩ := Presieve.Extensive.arrows_nonempty_isColimit (R := S) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Sites.Coherent.Comparison | {
"line": 87,
"column": 8
} | {
"line": 87,
"column": 37
} | {
"line": 88,
"column": 8
} | [
{
"pp": "case refine_2.of.a\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preregular C\ninst✝ : FinitaryPreExtensive C\nB : C\nS : Sieve B\nY : C\nI : Type\nw✝ : Finite I\nX : I → C\nf : (a : I) → X a ⟶ Y\nhT : EffectiveEpiFamily X f\n⊢ ∀ ⦃Y_1 : C⦄ ⦃f_1 : Y_1 ⟶ Y⦄,\n (generate (Presieve.ofArrows (f... | [
"case refine_2.of.a\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preregular C\ninst✝ : FinitaryPreExtensive C\nB : C\nS : Sieve B\nY✝ : C\nI : Type\nw✝ : Finite I\nX : I → C\nf : (a : I) → X a ⟶ Y✝\nhT : EffectiveEpiFamily X f\nR Y : C\ni✝ : Unit\nψ : R ⟶ ∐ fun i ↦ X i\n⊢ (extensiveCoverage C ⊔ regularCo... | rintro R g ⟨W, ψ, σ, ⟨⟩, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.CategoryTheory.Sites.EpiMono | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 69
} | {
"line": 101,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝⁵ : Category.{v', u'} A\nFA : A → A → Type u_1\nCA : A → Type w\ninst✝⁴ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)\ninst✝³ : ConcreteCategory A FA\ninst✝² : HasFunctorialSurjectiveInjectiveFactorization A\ninst✝¹ : ... | [] | apply (functorialLocallySurjectiveInjectiveFactorization J data).hp | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Sites.EpiMono | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 69
} | {
"line": 101,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝⁵ : Category.{v', u'} A\nFA : A → A → Type u_1\nCA : A → Type w\ninst✝⁴ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)\ninst✝³ : ConcreteCategory A FA\ninst✝² : HasFunctorialSurjectiveInjectiveFactorization A\ninst✝¹ : ... | [] | apply (functorialLocallySurjectiveInjectiveFactorization J data).hp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.EpiMono | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 69
} | {
"line": 101,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝⁵ : Category.{v', u'} A\nFA : A → A → Type u_1\nCA : A → Type w\ninst✝⁴ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)\ninst✝³ : ConcreteCategory A FA\ninst✝² : HasFunctorialSurjectiveInjectiveFactorization A\ninst✝¹ : ... | [] | apply (functorialLocallySurjectiveInjectiveFactorization J data).hp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.GlobalSections | {
"line": 169,
"column": 57
} | {
"line": 171,
"column": 97
} | {
"line": 173,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u₂\ninst✝² : Category.{v₂, u₂} A\ninst✝¹ : HasWeakSheafify J A\ninst✝ : HasGlobalSectionsFunctor J A\nF G : Sheaf J A\nf : F ⟶ G\nU : Cᵒᵖ\n⊢ (Γ J A).map f ≫ G.ΓRes U = F.ΓRes U ≫ f.hom.app U",
"ppTerm": "?m.65",
"assig... | [] | by
refine .trans ?_ <| congr_app (ΓHomEquiv_naturality_right_symm _ _) U
exact (congr_app (ΓHomEquiv_naturality_left_symm ((Γ J A).map f) (𝟙 _)) U).symm.trans (by simp) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.GlobalSections | {
"line": 195,
"column": 2
} | {
"line": 197,
"column": 52
} | {
"line": 199,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝¹ : HasWeakSheafify J (Type w)\ninst✝ : HasGlobalSectionsFunctor J (Type w)\nF G : Sheaf J (Type w)\nf : F ⟶ G\nx : (Γ J (Type w)).obj F\n⊢ (ΓObjEquivSections J G) ((ConcreteCategory.hom ((Γ J (Type w)).map f)) x) =\n (Concrete... | [] | dsimp [ΓObjEquivSections]
exact (congr_arg _ (ΓHomEquiv_naturality_right_symm (↾(uniqueElim x)) f)).trans
(Functor.sectionsEquivHom_naturality_symm _ _ _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.GlobalSections | {
"line": 195,
"column": 2
} | {
"line": 197,
"column": 52
} | {
"line": 199,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝¹ : HasWeakSheafify J (Type w)\ninst✝ : HasGlobalSectionsFunctor J (Type w)\nF G : Sheaf J (Type w)\nf : F ⟶ G\nx : (Γ J (Type w)).obj F\n⊢ (ΓObjEquivSections J G) ((ConcreteCategory.hom ((Γ J (Type w)).map f)) x) =\n (Concrete... | [] | dsimp [ΓObjEquivSections]
exact (congr_arg _ (ΓHomEquiv_naturality_right_symm (↾(uniqueElim x)) f)).trans
(Functor.sectionsEquivHom_naturality_symm _ _ _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Hypercover.Homotopy | {
"line": 103,
"column": 31
} | {
"line": 103,
"column": 43
} | {
"line": 103,
"column": 44
} | [
{
"pp": "case refine_3\nC : Type u\ninst✝¹ : Category.{v, u} C\nS : C\nE : PreOneHypercover S\nF : PreOneHypercover S\nA : Type u_1\ninst✝ : Category.{v_1, u_1} A\nf : E.Hom F\ng : F.Hom E\nhgf : Homotopy (g.comp f) (Hom.id F)\nG : Cᵒᵖ ⥤ A\nhE : IsLimit (E.multifork G)\nt : Multifork (F.multicospanIndex G)\nm :... | [
"case refine_3\nC : Type u\ninst✝¹ : Category.{v, u} C\nS : C\nE : PreOneHypercover S\nF : PreOneHypercover S\nA : Type u_1\ninst✝ : Category.{v_1, u_1} A\nf : E.Hom F\ng : F.Hom E\nhgf : Homotopy (g.comp f) (Hom.id F)\nG : Cᵒᵖ ⥤ A\nhE : IsLimit (E.multifork G)\nt : Multifork (F.multicospanIndex G)\nm : t.pt ⟶ (F.m... | multifork_ι, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.SheafHom | {
"line": 95,
"column": 12
} | {
"line": 95,
"column": 40
} | {
"line": 95,
"column": 40
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝ : Category.{v', u'} A\nF G : Cᵒᵖ ⥤ A\ns : ↑(presheafHom F G).sections\nX₁ X₂ : C\nf : X₂ ⟶ X₁\n⊢ F.map (op f) ≫ ((ConcreteCategory.hom ((presheafHom F G).map f.op)) (↑s (op X₁))).app (op (Over.mk (𝟙 X₂))) =\n F.m... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝ : Category.{v', u'} A\nF G : Cᵒᵖ ⥤ A\ns : ↑(presheafHom F G).sections\nX₁ X₂ : C\nf : X₂ ⟶ X₁\n⊢ F.map (op f) ≫ (↑s (op X₁)).app (op (Over.mk f)) = F.map f.op ≫ (↑s (op X₁)).app (op (Over.mk f))"
] | presheafHom_map_app_op_mk_id | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.SheafHom | {
"line": 118,
"column": 8
} | {
"line": 118,
"column": 36
} | {
"line": 118,
"column": 36
} | [
{
"pp": "case mp\nC : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u'\ninst✝ : Category.{v', u'} A\nF G : Cᵒᵖ ⥤ A\nX : C\nS : Sieve X\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\ny : (presheafHom F G).obj (op X)\nh : x.IsAmalgamation y\nY : C\ng : Y ⟶ X\nhg : S.arrows g\n⊢ y.app... | [
"case mp\nC : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u'\ninst✝ : Category.{v', u'} A\nF G : Cᵒᵖ ⥤ A\nX : C\nS : Sieve X\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\ny : (presheafHom F G).obj (op X)\nh : x.IsAmalgamation y\nY : C\ng : Y ⟶ X\nhg : S.arrows g\n⊢ y.app (op (Over.m... | presheafHom_map_app_op_mk_id | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.MayerVietorisSquare | {
"line": 134,
"column": 12
} | {
"line": 134,
"column": 81
} | {
"line": 135,
"column": 12
} | [
{
"pp": "case left.right\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝² : HasWeakSheafify J (Type v)\nsq : Square C\ninst✝¹ : Mono sq.f₂₄\ninst✝ : Mono sq.f₃₄\nh₁ : sq.IsPullback\nh₂ : Sieve.ofTwoArrows sq.f₂₄ sq.f₃₄ ∈ J sq.X₄\nthis : Mono sq.f₁₃\nF : Sheaf J (Type v)\ns : PullbackC... | [
"case left.right\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝² : HasWeakSheafify J (Type v)\nsq : Square C\ninst✝¹ : Mono sq.f₂₄\ninst✝ : Mono sq.f₃₄\nh₁ : sq.IsPullback\nh₂ : Sieve.ofTwoArrows sq.f₂₄ sq.f₃₄ ∈ J sq.X₄\nthis : Mono sq.f₁₃\nF : Sheaf J (Type v)\ns : PullbackCone (sq.op.m... | obtain ⟨φ, rfl, rfl⟩ := PullbackCone.IsLimit.lift' h₁.isLimit _ _ fac | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Sites.NonabelianCohomology.H1 | {
"line": 141,
"column": 39
} | {
"line": 143,
"column": 28
} | {
"line": 145,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nG : Cᵒᵖ ⥤ GrpCat\nI : Type w'\nU : I → C\nγ : OneCocycle G U\ni j : I\nT : C\na : T ⟶ U i\nb : T ⟶ U j\n⊢ γ.ev i j a b = (γ.ev j i b a)⁻¹",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"CancelMonoi... | [] | by
rw [← mul_left_inj (γ.ev j i b a), γ.ev_trans i j i a b a,
ev_refl, inv_mul_cancel] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Descent.DescentData | {
"line": 318,
"column": 6
} | {
"line": 319,
"column": 39
} | {
"line": 319,
"column": 40
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nS' : C\np : S' ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S'\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j) = f' j ≫ p\nβ : ι... | [
"C : Type u\ninst✝ : Category.{v, u} C\nF : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nS' : C\np : S' ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S'\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j) = f' j ≫ p\nβ : ι' → ι\np'' :... | pullHom_hom _ _ _ (q ≫ p) (by rw [w, reassoc_of% hf₂]) _ _
rfl (by cat_disch) _ _ rfl rfl, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Triangulated.Opposite.Functor | {
"line": 265,
"column": 6
} | {
"line": 266,
"column": 22
} | {
"line": 266,
"column": 22
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹³ : Category.{v_1, u_1} C\ninst✝¹² : Category.{v_2, u_2} D\ninst✝¹¹ : HasShift C ℤ\ninst✝¹⁰ : HasShift D ℤ\nF : C ⥤ D\ninst✝⁹ : F.CommShift ℤ\ninst✝⁸ : HasZeroObject C\ninst✝⁷ : Preadditive C\ninst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝⁵ : Pretriangulated C\... | [
"C : Type u_1\nD : Type u_2\ninst✝¹³ : Category.{v_1, u_1} C\ninst✝¹² : Category.{v_2, u_2} D\ninst✝¹¹ : HasShift C ℤ\ninst✝¹⁰ : HasShift D ℤ\nF : C ⥤ D\ninst✝⁹ : F.CommShift ℤ\ninst✝⁸ : HasZeroObject C\ninst✝⁷ : Preadditive C\ninst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝⁵ : Pretriangulated C\ninst✝⁴ : Ha... | distinguished_iff_of_iso ((mapTriangleOpCompTriangleOpEquivalenceFunctor F).app
(Opposite.op T)) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Triangulated.TStructure.SpectralObject | {
"line": 114,
"column": 6
} | {
"line": 114,
"column": 82
} | {
"line": 115,
"column": 6
} | [
{
"pp": "case refine_1\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 c : EInt\nhab : a ≤ b\nhbc : b ≤ c\nX : C\n⊢... | [
"case refine_1\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 c : EInt\nhab : a ≤ b\nhbc : b ≤ c\nX : C\n⊢ (t.eTruncLT... | ← cancel_epi ((t.eTruncLTGEIsoGELT a b).hom.app ((t.eTruncLT.obj c).obj X)), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Additive.AP.Three.Defs | {
"line": 127,
"column": 6
} | {
"line": 127,
"column": 18
} | {
"line": 127,
"column": 19
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : CommMonoid β\ns A : Set α\nt : Set β\nf : α → β\nhf : IsMulFreimanIso 2 s t f\nhAs : A ⊆ s\n⊢ ThreeGPFree (f '' A) ↔ ThreeGPFree A",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"M... | [
"α : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : CommMonoid β\ns A : Set α\nt : Set β\nf : α → β\nhf : IsMulFreimanIso 2 s t f\nhAs : A ⊆ s\n⊢ (∀ ⦃a : β⦄, a ∈ f '' A → ∀ ⦃b : β⦄, b ∈ f '' A → ∀ ⦃c : β⦄, c ∈ f '' A → a * c = b * b → a = b) ↔ ThreeGPFree A"
] | ThreeGPFree, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Enumerative.DoubleCounting | {
"line": 192,
"column": 2
} | {
"line": 200,
"column": 22
} | {
"line": 202,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nr : α → β → Prop\ns : Finset α\nt : Finset β\nhs : ∀ a ∈ s, ∃ b ∈ t, r a b\nht : ∀ b ∈ t, {a | a ∈ s ∧ r a b}.Subsingleton\n⊢ #s ≤ #t",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul... | [] | classical
rw [← mul_one #s, ← mul_one #t]
exact card_mul_le_card_mul r
(fun a h ↦ card_pos.2 (by
rw [← coe_nonempty, coe_bipartiteAbove]
exact hs _ h : (t.bipartiteAbove r a).Nonempty))
(fun b h ↦ card_le_one.2 (by
simp_rw [mem_bipartiteBelow]
exact ht _ h)) | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Combinatorics.Enumerative.DoubleCounting | {
"line": 192,
"column": 2
} | {
"line": 200,
"column": 22
} | {
"line": 202,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nr : α → β → Prop\ns : Finset α\nt : Finset β\nhs : ∀ a ∈ s, ∃ b ∈ t, r a b\nht : ∀ b ∈ t, {a | a ∈ s ∧ r a b}.Subsingleton\n⊢ #s ≤ #t",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul... | [] | classical
rw [← mul_one #s, ← mul_one #t]
exact card_mul_le_card_mul r
(fun a h ↦ card_pos.2 (by
rw [← coe_nonempty, coe_bipartiteAbove]
exact hs _ h : (t.bipartiteAbove r a).Nonempty))
(fun b h ↦ card_le_one.2 (by
simp_rw [mem_bipartiteBelow]
exact ht _ h)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Enumerative.DoubleCounting | {
"line": 192,
"column": 2
} | {
"line": 200,
"column": 22
} | {
"line": 202,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nr : α → β → Prop\ns : Finset α\nt : Finset β\nhs : ∀ a ∈ s, ∃ b ∈ t, r a b\nht : ∀ b ∈ t, {a | a ∈ s ∧ r a b}.Subsingleton\n⊢ #s ≤ #t",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul... | [] | classical
rw [← mul_one #s, ← mul_one #t]
exact card_mul_le_card_mul r
(fun a h ↦ card_pos.2 (by
rw [← coe_nonempty, coe_bipartiteAbove]
exact hs _ h : (t.bipartiteAbove r a).Nonempty))
(fun b h ↦ card_le_one.2 (by
simp_rw [mem_bipartiteBelow]
exact ht _ h)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Additive.AP.Three.Behrend | {
"line": 63,
"column": 10
} | {
"line": 63,
"column": 12
} | {
"line": 63,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : Field 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\ninst✝² : TopologicalSpace E\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns : Set E\nhs₀ : IsClosed s\nhs₁ : StrictConvex 𝕜 s\na : E\n⊢ a ∈ frontier s → ∀ ⦃b : E⦄, b ∈ frontier s → ∀ ⦃c : E⦄, c ... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : Field 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\ninst✝² : TopologicalSpace E\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns : Set E\nhs₀ : IsClosed s\nhs₁ : StrictConvex 𝕜 s\na : E\nha : a ∈ frontier s\n⊢ ∀ ⦃b : E⦄, b ∈ frontier s → ∀ ⦃c : E⦄, c ∈ fronti... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Combinatorics.Additive.FreimanHom | {
"line": 240,
"column": 4
} | {
"line": 240,
"column": 45
} | {
"line": 241,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : CommMonoid γ\nA : Set α\nB : Set β\nC : Set γ\nf : α → β\ng : β → γ\nn : ℕ\nhg : IsMulFreimanHom n B C g\nhf : IsMulFreimanHom n A B f\ns t : Multiset α\nhsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A\nhtA : ... | [
"case refine_2\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : CommMonoid γ\nA : Set α\nB : Set β\nC : Set γ\nf : α → β\ng : β → γ\nn : ℕ\nhg : IsMulFreimanHom n B C g\nhf : IsMulFreimanHom n A B f\ns t : Multiset α\nhsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A\nhtA : ∀ ⦃x : α⦄, x... | · simpa using fun a h ↦ hf.mapsTo (hsA h) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.Additive.CauchyDavenport | {
"line": 218,
"column": 2
} | {
"line": 219,
"column": 82
} | {
"line": 220,
"column": 2
} | [
{
"pp": "α : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : Mul α\ninst✝² : IsCancelMul α\ninst✝¹ : MulLeftMono α\ninst✝ : MulRightMono α\ns t : Finset α\nhs : s.Nonempty\nht : t.Nonempty\n⊢ s * {t.min' ht} ∩ ({s.max' hs} * t) = {s.max' hs * t.min' ht}",
"ppTerm": "?m.60",
"assigned": true,
"usedConstan... | [
"α : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : Mul α\ninst✝² : IsCancelMul α\ninst✝¹ : MulLeftMono α\ninst✝ : MulRightMono α\ns t : Finset α\nhs : s.Nonempty\nht : t.Nonempty\n⊢ ∀ x ∈ s * {t.min' ht} ∩ ({s.max' hs} * t), x = s.max' hs * t.min' ht"
] | refine eq_singleton_iff_unique_mem.2 ⟨mem_inter.2 ⟨mul_mem_mul (max'_mem _ _) <|
mem_singleton_self _, mul_mem_mul (mem_singleton_self _) <| min'_mem _ _⟩, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.SimpleGraph.Maps | {
"line": 661,
"column": 2
} | {
"line": 662,
"column": 45
} | {
"line": 664,
"column": 0
} | [
{
"pp": "V : Type u_1\nW : Type u_2\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G ≃g G'\nv : W\n⊢ (f.toHom.comp f.symm.toHom) v = Hom.id v",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"RelHom.instFunLike",
"congrArg",
"SimpleGraph.Adj",
"RelHom",
"SimpleGra... | [] | simp only [RelHom.comp_apply, RelEmbedding.coe_toRelHom, RelIso.coe_toRelEmbedding,
RelIso.apply_symm_apply, RelHom.id_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Set.Equitable | {
"line": 99,
"column": 12
} | {
"line": 99,
"column": 14
} | {
"line": 100,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_1\ns : Finset α\nf : α → ℕ\nb : ℕ\nhb : ∀ a ∈ ↑s, b ≤ f a ∧ f a ≤ b + 1\nh : ∀ a ∈ s, f a = b + 1\na : α\n⊢ a ∈ s → (∑ i ∈ s, f i) / #s ≤ f a ∧ f a ≤ (∑ i ∈ s, f i) / #s + 1",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Finset",
"Membership.mem"... | [
"case pos\nα : Type u_1\ns : Finset α\nf : α → ℕ\nb : ℕ\nhb : ∀ a ∈ ↑s, b ≤ f a ∧ f a ≤ b + 1\nh : ∀ a ∈ s, f a = b + 1\na : α\nha : a ∈ s\n⊢ (∑ i ∈ s, f i) / #s ≤ f a ∧ f a ≤ (∑ i ∈ s, f i) / #s + 1"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Combinatorics.SimpleGraph.Finite | {
"line": 315,
"column": 6
} | {
"line": 315,
"column": 37
} | {
"line": 315,
"column": 37
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nv : V\ninst✝¹ : Fintype ↑(G.neighborSet v)\ninst✝ : DecidableEq V\n⊢ #(G.incidenceFinset v) = G.degree v",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SimpleGraph.incidenceSet",
"congrArg",
"SimpleGraph.card_inc... | [
"V : Type u_1\nG : SimpleGraph V\nv : V\ninst✝¹ : Fintype ↑(G.neighborSet v)\ninst✝ : DecidableEq V\n⊢ #(G.incidenceFinset v) = Fintype.card ↑(G.incidenceSet v)"
] | ← G.card_incidenceSet_eq_degree | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Bound | {
"line": 125,
"column": 4
} | {
"line": 127,
"column": 77
} | {
"line": 129,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nP : Finpartition univ\nε : ℝ\ninst✝ : Nonempty α\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\n⊢ 4 ^ #P.parts ≤ ↑m",
"ppTerm": "?m.110",
"assigned": true,
"usedConstants": [
"SzemerediRegul... | [] | norm_cast
rwa [Nat.le_div_iff_mul_le (stepBound_pos (P.parts_nonempty <|
univ_nonempty.ne_empty).card_pos), stepBound, mul_left_comm, ← mul_pow] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Regularity.Bound | {
"line": 125,
"column": 4
} | {
"line": 127,
"column": 77
} | {
"line": 129,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nP : Finpartition univ\nε : ℝ\ninst✝ : Nonempty α\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\n⊢ 4 ^ #P.parts ≤ ↑m",
"ppTerm": "?m.110",
"assigned": true,
"usedConstants": [
"SzemerediRegul... | [] | norm_cast
rwa [Nat.le_div_iff_mul_le (stepBound_pos (P.parts_nonempty <|
univ_nonempty.ne_empty).card_pos), stepBound, mul_left_comm, ← mul_pow] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Partition.Finpartition | {
"line": 615,
"column": 4
} | {
"line": 615,
"column": 72
} | {
"line": 617,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝¹ : GeneralizedBooleanAlgebra α\ninst✝ : DecidableEq α\na b : α\nP : Finpartition a\nhab : a ≤ b\np : α\nhp : p ∈ (P.extendOfLE hab).parts\nh : ¬a < b\n⊢ p ∈ P.parts",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Finpartition.extendOfLE",
... | [] | simpa [parts_extendOfLE_of_eq _ (LE.le.eq_of_not_lt hab h)] using hp | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Order.Partition.Finpartition | {
"line": 666,
"column": 12
} | {
"line": 666,
"column": 14
} | {
"line": 666,
"column": 15
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\ns t u : Finset α\nP : Finpartition s\na✝ : α\nparts : Finset (Finset α)\nh : ∀ p ∈ parts, p ⊆ s\nh' : ∀ a ∈ s, ∃! t, t ∈ parts ∧ a ∈ t\nh'' : ∅ ∉ parts\na : Finset α\n⊢ a ∈ ↑parts → ∀ ⦃y : Finset α⦄, y ∈ ↑parts → a ≠ y → (Disjoint on id) a y",
"ppTerm": "?m.38",... | [
"α : Type u_1\ninst✝ : DecidableEq α\ns t u : Finset α\nP : Finpartition s\na✝ : α\nparts : Finset (Finset α)\nh : ∀ p ∈ parts, p ⊆ s\nh' : ∀ a ∈ s, ∃! t, t ∈ parts ∧ a ∈ t\nh'' : ∅ ∉ parts\na : Finset α\nha : a ∈ ↑parts\n⊢ ∀ ⦃y : Finset α⦄, y ∈ ↑parts → a ≠ y → (Disjoint on id) a y"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Order.Partition.Finpartition | {
"line": 677,
"column": 6
} | {
"line": 677,
"column": 28
} | {
"line": 678,
"column": 2
} | [
{
"pp": "case mpr\nα : Type u_1\ninst✝ : DecidableEq α\ns t u : Finset α\nP : Finpartition s\na : α\nparts : Finset (Finset α)\nh : ∀ p ∈ parts, p ⊆ s\nh' : ∀ a ∈ s, ∃! t, t ∈ parts ∧ a ∈ t\nh'' : ∅ ∉ parts\ni : α\nhi : i ∈ s\n⊢ ∃ i_1 ∈ parts, i ∈ i_1",
"ppTerm": "?mpr",
"assigned": true,
"usedConst... | [] | exact (h' i hi).exists | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Partition.Finpartition | {
"line": 816,
"column": 28
} | {
"line": 816,
"column": 30
} | {
"line": 816,
"column": 31
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ns✝ t u : Finset α\nP : Finpartition s✝\na : α\ns : Setoid α\nx : Finset α\ninst✝ : DecidableRel ⇑s\nthis : ∀ (a b c d : α), s a d → s b d → (s a c ↔ s b c)\na✝³ : α\na✝² : a✝³ ∈ x\na✝¹ : α\na✝ : a✝¹ ∈ x\nx✝³ : α\nx✝² : x✝³ ∈ x\nx✝¹ : α\nx✝ : x✝¹ ∈ x\n⊢ s a✝³ x✝¹ → ... | [
"α : Type u_1\ninst✝¹ : DecidableEq α\ns✝ t u : Finset α\nP : Finpartition s✝\na : α\ns : Setoid α\nx : Finset α\ninst✝ : DecidableRel ⇑s\nthis : ∀ (a b c d : α), s a d → s b d → (s a c ↔ s b c)\na✝³ : α\na✝² : a✝³ ∈ x\na✝¹ : α\na✝ : a✝¹ ∈ x\nx✝³ : α\nx✝² : x✝³ ∈ x\nx✝¹ : α\nx✝ : x✝¹ ∈ x\nha : s a✝³ x✝¹\n⊢ x✝¹ ∈ x ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Order.Partition.Finpartition | {
"line": 879,
"column": 8
} | {
"line": 879,
"column": 28
} | {
"line": 880,
"column": 8
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ u : Finset α\nP : Finpartition s✝\na : α\ns : Finset α\nF : Finset (Finset α)\nt : Finset α\nht : t ∈ image (fun Q ↦ {i ∈ s | ∀ t ∈ F, t ∈ Q ↔ i ∈ t}) F.powerset\n⊢ id t ⊆ s",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants":... | [
"case refine_1\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ u : Finset α\nP : Finpartition s✝\na : α\ns : Finset α\nF : Finset (Finset α)\nt : Finset α\nht : ∃ a ∈ F.powerset, {i ∈ s | ∀ t ∈ F, t ∈ a ↔ i ∈ t} = t\n⊢ id t ⊆ s"
] | rw [mem_image] at ht | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Partition.Equipartition | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 29
} | {
"line": 151,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\ns : Finset α\nP : Finpartition s\nhP : P.IsEquipartition\nf : ↥s ≃ (t : ↥P.parts) × Fin #↑t\nhf : ∀ (a b : ↥s), P.part ↑a = P.part ↑b ↔ (f a).fst = (f b).fst\ng : ↥P.parts ≃ Fin #P.parts\nhg : ∀ (t : ↥P.parts), #↑t = #s / #P.parts + 1 ↔ ↑(g t) < #s % #P.parts\nz : ↥... | [
"case h\nα : Type u_1\ninst✝ : DecidableEq α\ns : Finset α\nP : Finpartition s\nhP : P.IsEquipartition\nf : ↥s ≃ (t : ↥P.parts) × Fin #↑t\nhf : ∀ (a b : ↥s), P.part ↑a = P.part ↑b ↔ (f a).fst = (f b).fst\ng : ↥P.parts ≃ Fin #P.parts\nhg : ∀ (t : ↥P.parts), #↑t = #s / #P.parts + 1 ↔ ↑(g t) < #s % #P.parts\nz : ↥s → ... | use Equiv.ofBijective _ bij | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Order.Partition.Finpartition | {
"line": 909,
"column": 2
} | {
"line": 909,
"column": 84
} | {
"line": 910,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nF : Finset (Finset α)\nht : t ∈ F\nhts : t ⊆ s\na : α\n⊢ a ∈ {u ∈ (atomise s F).parts | u ⊆ t ∧ u.Nonempty}.biUnion id ↔ a ∈ t",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Finset",
"PartialOrder.toPreorder",
... | [
"α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nF : Finset (Finset α)\nht : t ∈ F\nhts : t ⊆ s\na : α\nha : a ∈ t\n⊢ ∃ a_1 ∈ {u ∈ (atomise s F).parts | u ⊆ t ∧ u.Nonempty}, a ∈ id a_1"
] | refine mem_biUnion.trans ⟨fun ⟨u, hu, ha⟩ ↦ (mem_filter.1 hu).2.1 ha, fun ha ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Order.Partition.Finpartition | {
"line": 924,
"column": 11
} | {
"line": 924,
"column": 21
} | {
"line": 924,
"column": 22
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nF : Finset (Finset α)\nht : t ∈ F\n⊢ ∀ ⦃x : Finset α⦄,\n x ∈ {u ∈ (atomise s F).parts | u ⊆ t ∧ u.Nonempty} →\n x ∈ image (fun P ↦ {i ∈ s | ∀ x ∈ F, x ∈ insert t P ↔ i ∈ x}) (F.erase t).powerset",
"ppTerm": "?m.122",
"assigned": true,... | [
"α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nF : Finset (Finset α)\nht : t ∈ F\n⊢ ∀ ⦃x : Finset α⦄,\n x ∈ {u ∈ (atomise s F).parts | u ⊆ t ∧ u.Nonempty} →\n ∃ a ∈ (F.erase t).powerset, {i ∈ s | ∀ x ∈ F, x ∈ insert t a ↔ i ∈ x} = x"
] | mem_image, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Uniform | {
"line": 438,
"column": 2
} | {
"line": 438,
"column": 56
} | {
"line": 439,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\n𝕜 : Type u_2\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : DecidableEq α\nA : Finset α\nP : Finpartition A\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\nx y : α\nhx✝ : x ∈ A\nhy✝ : y ∈ A\nh : G.Adj x y\nh' :\n ∀ x_1 ∈ P.parts,\n ∀ x_2 ∈ P.parts, x ∈ x_1 → ... | [
"case inr\nα : Type u_1\n𝕜 : Type u_2\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : DecidableEq α\nA : Finset α\nP : Finpartition A\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\nx y : α\nhx✝ : x ∈ A\nhy✝ : y ∈ A\nh : G.Adj x y\nh' :\n ∀ x_1 ∈ P.parts,\n ∀ x_2 ∈ P.parts, x ∈ x_1 → y ∈ x_2 → x_... | · exact Or.inr (Or.inl ⟨U, hU, hx, hy, G.ne_of_adj h⟩) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Copy | {
"line": 663,
"column": 2
} | {
"line": 663,
"column": 69
} | {
"line": 665,
"column": 0
} | [
{
"pp": "case e_f\nV : Type u_1\nW : Type u_2\nG : SimpleGraph V\nH : SimpleGraph W\nhH : H ≠ ⊥\nG' : (G \\ fromEdgeSet (⋃ G', ⋃ (hG' : Nonempty (H ≃g G'.coe)), {⋯.some})).Subgraph\nhHG' : Nonempty (H ≃g G'.coe)\nhG' : (Subgraph.map (Hom.ofLE ⋯) G').edgeSet.Nonempty\ne : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) G').ver... | [] | exact congr_arg _ (Equiv.Set.image_symm_apply _ _ injective_id _ _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise | {
"line": 179,
"column": 10
} | {
"line": 179,
"column": 12
} | {
"line": 179,
"column": 13
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\ns : Finset α\nm a b : ℕ\nP : Finpartition s\nh : a * m + b * (m + 1) = #s\nhm : m ≠ 0\nhunion : (equitabilise h).parts = {u ∈ (equitabilise h).parts | #u = m} ∪ {u ∈ (equitabilise h).parts | #u = m + 1}\nx : Finset α → Prop\n⊢ (x ≤ fun u ↦ #u = m) → (x ≤ fun u ↦ #u ... | [
"α : Type u_1\ninst✝ : DecidableEq α\ns : Finset α\nm a b : ℕ\nP : Finpartition s\nh : a * m + b * (m + 1) = #s\nhm : m ≠ 0\nhunion : (equitabilise h).parts = {u ∈ (equitabilise h).parts | #u = m} ∪ {u ∈ (equitabilise h).parts | #u = m + 1}\nx : Finset α → Prop\nha : x ≤ fun u ↦ #u = m\n⊢ (x ≤ fun u ↦ #u = m + 1) →... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Combinatorics.SimpleGraph.Operations | {
"line": 67,
"column": 52
} | {
"line": 67,
"column": 80
} | {
"line": 69,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ns t : V\ninst✝ : DecidableEq V\nv w : V\nhv : v ≠ t\nhw : w ≠ t\n⊢ (G.replaceVertex s t).Adj v w ↔ G.Adj v w",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Std.Irrefl",
"False",
"SimpleGraph.replaceVertex._proof_1",
"Std.Sy... | [] | simp [replaceVertex, hv, hw] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Operations | {
"line": 67,
"column": 52
} | {
"line": 67,
"column": 80
} | {
"line": 69,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ns t : V\ninst✝ : DecidableEq V\nv w : V\nhv : v ≠ t\nhw : w ≠ t\n⊢ (G.replaceVertex s t).Adj v w ↔ G.Adj v w",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Std.Irrefl",
"False",
"SimpleGraph.replaceVertex._proof_1",
"Std.Sy... | [] | simp [replaceVertex, hv, hw] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Operations | {
"line": 67,
"column": 52
} | {
"line": 67,
"column": 80
} | {
"line": 69,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ns t : V\ninst✝ : DecidableEq V\nv w : V\nhv : v ≠ t\nhw : w ≠ t\n⊢ (G.replaceVertex s t).Adj v w ↔ G.Adj v w",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Std.Irrefl",
"False",
"SimpleGraph.replaceVertex._proof_1",
"Std.Sy... | [] | simp [replaceVertex, hv, hw] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 152,
"column": 71
} | {
"line": 153,
"column": 27
} | {
"line": 154,
"column": 4
} | [
{
"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\nhunif : ¬G.IsUniform ε U V\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhε₁ :... | [] | by
rw [sub_mul, one_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 1215,
"column": 20
} | {
"line": 1215,
"column": 22
} | {
"line": 1216,
"column": 4
} | [
{
"pp": "case right\nV : Type u\nG : SimpleGraph V\nG' G'' : G.Subgraph\ns s' : Set V\nhg : G' ≤ G''\nhs : s ⊆ s'\nv w : V\nhv : v ∈ s\nhw : w ∈ s\n⊢ G'.Adj v w → v ∈ s' ∧ w ∈ s' ∧ G''.Adj v w",
"ppTerm": "?right",
"assigned": true,
"usedConstants": [
"SimpleGraph.Subgraph.Adj"
],
"use... | [
"case right\nV : Type u\nG : SimpleGraph V\nG' G'' : G.Subgraph\ns s' : Set V\nhg : G' ≤ G''\nhs : s ⊆ s'\nv w : V\nhv : v ∈ s\nhw : w ∈ s\nha : G'.Adj v w\n⊢ v ∈ s' ∧ w ∈ s' ∧ G''.Adj v w"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Combinatorics.SimpleGraph.Walk.Traversal | {
"line": 95,
"column": 6
} | {
"line": 96,
"column": 65
} | {
"line": 98,
"column": 0
} | [
{
"pp": "case cons.succ\nV : Type u\nG : SimpleGraph V\nu v v✝ : V\nh✝ : G.Adj u v✝\np✝ : G.Walk v✝ v\nn : ℕ\nh : n + 1 ≤ (cons h✝ p✝).length\n⊢ (cons h✝ p✝).getVert (n + 1) = (cons h✝ p✝).support[n + 1]",
"ppTerm": "?cons.succ",
"assigned": true,
"usedConstants": [
"Nat.succ_lt_succ_iff",
... | [] | simp_rw [support_cons, getVert_cons _ _ n.zero_ne_add_one.symm, List.getElem_cons]
exact getVert_eq_support_getElem _ (Nat.sub_le_of_le_add h) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Walk.Traversal | {
"line": 95,
"column": 6
} | {
"line": 96,
"column": 65
} | {
"line": 98,
"column": 0
} | [
{
"pp": "case cons.succ\nV : Type u\nG : SimpleGraph V\nu v v✝ : V\nh✝ : G.Adj u v✝\np✝ : G.Walk v✝ v\nn : ℕ\nh : n + 1 ≤ (cons h✝ p✝).length\n⊢ (cons h✝ p✝).getVert (n + 1) = (cons h✝ p✝).support[n + 1]",
"ppTerm": "?cons.succ",
"assigned": true,
"usedConstants": [
"Nat.succ_lt_succ_iff",
... | [] | simp_rw [support_cons, getVert_cons _ _ n.zero_ne_add_one.symm, List.getElem_cons]
exact getVert_eq_support_getElem _ (Nat.sub_le_of_le_add h) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Walk.Traversal | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 27
} | {
"line": 110,
"column": 2
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nn : ℕ\n⊢ p.getVert n = p.support.getD n v",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"SimpleGraph.Walk.length",
"SimpleGraph.Walk.support",
"List.getD",
"LE.le",
"instLENat",
"dite",
... | [
"case pos\nV : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nn : ℕ\nh : n ≤ p.length\n⊢ p.getVert n = p.support.getD n v",
"case neg\nV : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nn : ℕ\nh : ¬n ≤ p.length\n⊢ p.getVert n = p.support.getD n v"
] | by_cases h : n ≤ p.length | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Combinatorics.SimpleGraph.Walk.Basic | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 13
} | {
"line": 154,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\na b : V\np : G.Walk a b\n⊢ p.support.getLast ⋯ = b",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"List.getLast",
"SimpleGraph.Adj",
"SimpleGraph.Walk.support",
"SimpleGraph.Walk",
"SimpleGraph.Walk.head_support._proof_... | [
"case nil\nV : Type u\nG : SimpleGraph V\na b u✝ : V\n⊢ nil.support.getLast ⋯ = u✝",
"case cons\nV : Type u\nG : SimpleGraph V\na b u✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\np_ih✝ : p✝.support.getLast ⋯ = w✝\n⊢ (cons h✝ p✝).support.getLast ⋯ = w✝"
] | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Basic | {
"line": 168,
"column": 73
} | {
"line": 168,
"column": 84
} | {
"line": 168,
"column": 85
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\n⊢ v ∈ p.support",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"SimpleGraph.Adj",
"SimpleGraph.Walk.support",
"SimpleGraph.Walk",
"Membership.mem",
"List",
"List.instMembership",
"Sim... | [
"case nil\nV : Type u\nG : SimpleGraph V\nu v u✝ : V\n⊢ u✝ ∈ nil.support",
"case cons\nV : Type u\nG : SimpleGraph V\nu v u✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\np_ih✝ : w✝ ∈ p✝.support\n⊢ w✝ ∈ (cons h✝ p✝).support"
] | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Basic | {
"line": 238,
"column": 2
} | {
"line": 238,
"column": 13
} | {
"line": 238,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\n⊢ u :: List.map (fun x ↦ x.toProd.2) p.darts = p.support",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"List.map",
"SimpleGraph.Adj",
"SimpleGraph.Walk.support",
"SimpleGraph.Walk",
"SimpleGrap... | [
"case nil\nV : Type u\nG : SimpleGraph V\nu v u✝ : V\n⊢ u✝ :: List.map (fun x ↦ x.toProd.2) nil.darts = nil.support",
"case cons\nV : Type u\nG : SimpleGraph V\nu v u✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\np_ih✝ : v✝ :: List.map (fun x ↦ x.toProd.2) p✝.darts = p✝.support\n⊢ u✝ :: List.map (fun x ↦ x.toP... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Basic | {
"line": 245,
"column": 2
} | {
"line": 245,
"column": 13
} | {
"line": 245,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\n⊢ List.map (fun x ↦ x.toProd.1) p.darts ++ [v] = p.support",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"List.map",
"SimpleGraph.Adj",
"SimpleGraph.Walk.support",
"SimpleGraph.Walk",
"SimpleGr... | [
"case nil\nV : Type u\nG : SimpleGraph V\nu v u✝ : V\n⊢ List.map (fun x ↦ x.toProd.1) nil.darts ++ [u✝] = nil.support",
"case cons\nV : Type u\nG : SimpleGraph V\nu v u✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\np_ih✝ : List.map (fun x ↦ x.toProd.1) p✝.darts ++ [w✝] = p✝.support\n⊢ List.map (fun x ↦ x.toPro... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Basic | {
"line": 262,
"column": 2
} | {
"line": 262,
"column": 13
} | {
"line": 262,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\n⊢ p.support.length = p.length + 1",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"SimpleGraph.Walk.length",
"SimpleGraph.Adj",
"SimpleGraph.Walk.support",
"SimpleGraph.Walk",
"instOfNatNat",
... | [
"case nil\nV : Type u\nG : SimpleGraph V\nu v u✝ : V\n⊢ nil.support.length = nil.length + 1",
"case cons\nV : Type u\nG : SimpleGraph V\nu v u✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\np_ih✝ : p✝.support.length = p✝.length + 1\n⊢ (cons h✝ p✝).support.length = (cons h✝ p✝).length + 1"
] | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Basic | {
"line": 266,
"column": 2
} | {
"line": 266,
"column": 13
} | {
"line": 266,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\n⊢ p.darts.length = p.length",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"SimpleGraph.Walk.length",
"SimpleGraph.Adj",
"SimpleGraph.Walk",
"SimpleGraph.Dart",
"SimpleGraph.Walk.darts",
"... | [
"case nil\nV : Type u\nG : SimpleGraph V\nu v u✝ : V\n⊢ nil.darts.length = nil.length",
"case cons\nV : Type u\nG : SimpleGraph V\nu v u✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\np_ih✝ : p✝.darts.length = p✝.length\n⊢ (cons h✝ p✝).darts.length = (cons h✝ p✝).length"
] | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 178,
"column": 4
} | {
"line": 178,
"column": 95
} | {
"line": 180,
"column": 0
} | [
{
"pp": "case neg\nα : 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\nhm : m ≠ 0\nh✝ : ¬#U = m * 4 ^ #P.parts + (Fintype.card α / #P.parts - m * 4 ^ #P.parts)\n⊢ #(equitabi... | [] | rw [card_parts_equitabilise _ _ hm, tsub_add_cancel_of_le a_add_one_le_four_pow_parts_card] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 178,
"column": 4
} | {
"line": 178,
"column": 95
} | {
"line": 180,
"column": 0
} | [
{
"pp": "case neg\nα : 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\nhm : m ≠ 0\nh✝ : ¬#U = m * 4 ^ #P.parts + (Fintype.card α / #P.parts - m * 4 ^ #P.parts)\n⊢ #(equitabi... | [] | rw [card_parts_equitabilise _ _ hm, tsub_add_cancel_of_le a_add_one_le_four_pow_parts_card] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 178,
"column": 4
} | {
"line": 178,
"column": 95
} | {
"line": 180,
"column": 0
} | [
{
"pp": "case neg\nα : 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\nhm : m ≠ 0\nh✝ : ¬#U = m * 4 ^ #P.parts + (Fintype.card α / #P.parts - m * 4 ^ #P.parts)\n⊢ #(equitabi... | [] | rw [card_parts_equitabilise _ _ hm, tsub_add_cancel_of_le a_add_one_le_four_pow_parts_card] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Walk.Basic | {
"line": 296,
"column": 2
} | {
"line": 296,
"column": 13
} | {
"line": 296,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\n⊢ p.support[p.length] = v",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"SimpleGraph.Walk.length",
"SimpleGraph.Adj",
"SimpleGraph.Walk.support",
"SimpleGraph.Walk",
"GetElem.getElem",
"L... | [
"case nil\nV : Type u\nG : SimpleGraph V\nu v u✝ : V\n⊢ nil.support[nil.length] = u✝",
"case cons\nV : Type u\nG : SimpleGraph V\nu v u✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\np_ih✝ : p✝.support[p✝.length] = w✝\n⊢ (cons h✝ p✝).support[(cons h✝ p✝).length] = w✝"
] | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Basic | {
"line": 302,
"column": 4
} | {
"line": 302,
"column": 83
} | {
"line": 303,
"column": 2
} | [
{
"pp": "case refine_1\nV : Type u\nG : SimpleGraph V\nu v u' v' : V\np : G.Walk u v\nh✝¹ : G.Adj u' v'\nh✝ : { fst := u', snd := v', adj := h✝¹ } ∈ p.darts\ni : ℕ\nhi : i < p.darts.length\nh : p.darts[i] = { fst := u', snd := v', adj := h✝¹ }\n⊢ ∃ k,\n [u', v'].length + k ≤ p.support.length ∧ ∀ (i : ℕ) (h :... | [] | exact ⟨i, by grind, fun j hj ↦ by grind [fst_darts_getElem, snd_darts_getElem]⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SimpleGraph.Walk.Maps | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 13
} | {
"line": 65,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\n⊢ Walk.map Hom.id p = p",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"SimpleGraph.Walk.map",
"RelHom.instFunLike",
"SimpleGraph.Adj",
"SimpleGraph.Walk",
"SimpleGraph.Hom.id",
"SimpleGra... | [
"case nil\nV : Type u\nG : SimpleGraph V\nu v u✝ : V\n⊢ Walk.map Hom.id nil = nil",
"case cons\nV : Type u\nG : SimpleGraph V\nu v u✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\np_ih✝ : Walk.map Hom.id p✝ = p✝\n⊢ Walk.map Hom.id (cons h✝ p✝) = cons h✝ p✝"
] | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Maps | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 13
} | {
"line": 69,
"column": 14
} | [
{
"pp": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf : G →g G'\nf' : G' →g G''\nu v : V\np : G.Walk u v\n⊢ Walk.map f' (Walk.map f p) = Walk.map (f'.comp f) p",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"SimpleGraph.Walk... | [
"case nil\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf : G →g G'\nf' : G' →g G''\nu v : V\np : G.Walk u v\nu✝ : V\n⊢ Walk.map f' (Walk.map f nil) = Walk.map (f'.comp f) nil",
"case cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : Simp... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
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