module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.CategoryTheory.Localization.Pi | {
"line": 38,
"column": 4
} | {
"line": 54,
"column": 52
} | [
{
"pp": "case of_equiv\n⊢ ∀ {α β : Type w} (a : α ≃ β),\n (∀ {C : α → Type u₁} {D : α → Type u₂} [inst : (j : α) → Category.{v₁, u₁} (C j)]\n [inst_1 : (j : α) → Category.{v₂, u₂} (D j)] (L : (j : α) → C j ⥤ D j) (W : (j : α) → MorphismProperty (C j))\n [∀ (j : α), (W j).ContainsIdentities] [∀ ... | intro J₁ J₂ e hJ₁ C₂ D₂ _ _ L₂ W₂ _ _
let L₁ := fun j => (L₂ (e j))
let E := Pi.equivalenceOfEquiv C₂ e
let E' := Pi.equivalenceOfEquiv D₂ e
haveI : CatCommSq E.functor (Functor.pi L₁) (Functor.pi L₂) E'.functor :=
(CatCommSq.hInvEquiv E (Functor.pi L₁) (Functor.pi L₂) E').symm ⟨Iso.refl _⟩
re... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Localization.Pi | {
"line": 38,
"column": 4
} | {
"line": 54,
"column": 52
} | [
{
"pp": "case of_equiv\n⊢ ∀ {α β : Type w} (a : α ≃ β),\n (∀ {C : α → Type u₁} {D : α → Type u₂} [inst : (j : α) → Category.{v₁, u₁} (C j)]\n [inst_1 : (j : α) → Category.{v₂, u₂} (D j)] (L : (j : α) → C j ⥤ D j) (W : (j : α) → MorphismProperty (C j))\n [∀ (j : α), (W j).ContainsIdentities] [∀ ... | intro J₁ J₂ e hJ₁ C₂ D₂ _ _ L₂ W₂ _ _
let L₁ := fun j => (L₂ (e j))
let E := Pi.equivalenceOfEquiv C₂ e
let E' := Pi.equivalenceOfEquiv D₂ e
haveI : CatCommSq E.functor (Functor.pi L₁) (Functor.pi L₂) E'.functor :=
(CatCommSq.hInvEquiv E (Functor.pi L₁) (Functor.pi L₂) E').symm ⟨Iso.refl _⟩
re... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Localization.Pi | {
"line": 55,
"column": 18
} | {
"line": 55,
"column": 19
} | [
{
"pp": "case h_empty\nC : PEmpty.{w + 1} → Type u₁\nD : PEmpty.{w + 1} → Type u₂\ninst✝¹ : (j : PEmpty.{w + 1}) → Category.{v₁, u₁} (C j)\ninst✝ : (j : PEmpty.{w + 1}) → Category.{v₂, u₂} (D j)\n⊢ ∀ (L : (j : PEmpty.{w + 1}) → C j ⥤ D j) (W : (j : PEmpty.{w + 1}) → MorphismProperty (C j))\n [∀ (j : PEmpty.{... | L | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Localization.Pi | {
"line": 60,
"column": 25
} | {
"line": 60,
"column": 26
} | [
{
"pp": "case h_option\nJ : Type w\ninst✝² : Fintype J\nhJ :\n ∀ {C : J → Type u₁} {D : J → Type u₂} [inst : (j : J) → Category.{v₁, u₁} (C j)]\n [inst_1 : (j : J) → Category.{v₂, u₂} (D j)] (L : (j : J) → C j ⥤ D j) (W : (j : J) → MorphismProperty (C j))\n [∀ (j : J), (W j).ContainsIdentities] [∀ (j : J... | L | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Localization.LocallySmall | {
"line": 54,
"column": 29
} | {
"line": 68,
"column": 46
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nW : MorphismProperty C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : LocallySmall.{w, v₂, u₂} D\nL : C ⥤ D\ninst✝ : L.IsLocalization W\n⊢ W.HasLocalization",
"usedConstants": [
"CategoryTheory.MorphismProperty.definition._proof_5._@.Mathlib.Ca... | by
have : LocallySmall.{w} (InducedCategory _ L.obj) :=
⟨fun X Y ↦ small_of_injective InducedCategory.homEquiv.injective⟩
let L' : C ⥤ (InducedCategory _ L.obj) :=
{ obj X := X
map f := InducedCategory.homMk (L.map f) }
have := Localization.essSurj L W
have : (inducedFunctor L.obj).EssSurj := ⟨fun... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Localization.Monoidal.Functor | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 18
} | [
{
"pp": "case refine_1\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝¹⁰ : Category.{v_1, u_1} C\ninst✝⁹ : Category.{v_2, u_2} D\ninst✝⁸ : Category.{v_3, u_3} E\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : MonoidalCategory D\ninst✝⁵ : MonoidalCategory E\nL : C ⥤ D\nW : MorphismProperty C\ninst✝⁴ : L.IsLocalization W... | monoidal_simps | Mathlib.Tactic.Coherence._aux_Mathlib_Tactic_CategoryTheory_Coherence___elabRules_Mathlib_Tactic_Coherence_monoidal_simps_1 | Mathlib.Tactic.Coherence.monoidal_simps |
Mathlib.CategoryTheory.Localization.Monoidal.Functor | {
"line": 115,
"column": 4
} | {
"line": 115,
"column": 18
} | [
{
"pp": "case refine_2\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝¹⁰ : Category.{v_1, u_1} C\ninst✝⁹ : Category.{v_2, u_2} D\ninst✝⁸ : Category.{v_3, u_3} E\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : MonoidalCategory D\ninst✝⁵ : MonoidalCategory E\nL : C ⥤ D\nW : MorphismProperty C\ninst✝⁴ : L.IsLocalization W... | monoidal_simps | Mathlib.Tactic.Coherence._aux_Mathlib_Tactic_CategoryTheory_Coherence___elabRules_Mathlib_Tactic_Coherence_monoidal_simps_1 | Mathlib.Tactic.Coherence.monoidal_simps |
Mathlib.CategoryTheory.Localization.Monoidal.Functor | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 18
} | [
{
"pp": "case refine_3\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝¹⁰ : Category.{v_1, u_1} C\ninst✝⁹ : Category.{v_2, u_2} D\ninst✝⁸ : Category.{v_3, u_3} E\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : MonoidalCategory D\ninst✝⁵ : MonoidalCategory E\nL : C ⥤ D\nW : MorphismProperty C\ninst✝⁴ : L.IsLocalization W... | monoidal_simps | Mathlib.Tactic.Coherence._aux_Mathlib_Tactic_CategoryTheory_Coherence___elabRules_Mathlib_Tactic_Coherence_monoidal_simps_1 | Mathlib.Tactic.Coherence.monoidal_simps |
Mathlib.CategoryTheory.LocallyCartesianClosed.Sections | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 88
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : CartesianMonoidalCategory C\nI : C\ninst✝² : Closed I\ninst✝¹ : ChosenPullbacksAlong (curryRightUnitorHom I)\ninst✝ : BraidedCategory C\nX : Over I\nA : C\nv : A ⟶ (sections I).obj X\nv₂ : A ⟶ (ihom I).obj X.left := v ≫ ChosenPullbacksAlong.fst ((ihom... | rw [Category.assoc, ← w', whiskerLeft_toUnit_comp_rightUnitor_hom, braiding_hom_fst] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Monoidal.Cartesian.ShrinkYoneda | {
"line": 31,
"column": 2
} | {
"line": 32,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : LocallySmall.{w, v, u} C\ninst✝ : CartesianMonoidalCategory C\nM : Mon C\nX : Cᵒᵖ\n⊢ Small.{w, v} ↑((yonedaMon.obj M).obj X)",
"usedConstants": [
"CategoryTheory.Functor",
"Opposite",
"CategoryTheory.CategoryStruct.toQuiver",
... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Cartesian.ShrinkYoneda | {
"line": 31,
"column": 2
} | {
"line": 32,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : LocallySmall.{w, v, u} C\ninst✝ : CartesianMonoidalCategory C\nM : Mon C\nX : Cᵒᵖ\n⊢ Small.{w, v} ↑((yonedaMon.obj M).obj X)",
"usedConstants": [
"CategoryTheory.Functor",
"Opposite",
"CategoryTheory.CategoryStruct.toQuiver",
... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Cartesian.ShrinkYoneda | {
"line": 35,
"column": 2
} | {
"line": 36,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : LocallySmall.{w, v, u} C\ninst✝ : CartesianMonoidalCategory C\nM : Grp C\nX : Cᵒᵖ\n⊢ Small.{w, v} ↑((yonedaGrp.obj M).obj X)",
"usedConstants": [
"CategoryTheory.Functor",
"GrpCat",
"Opposite",
"CategoryTheory.CategoryStruct.t... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Cartesian.ShrinkYoneda | {
"line": 35,
"column": 2
} | {
"line": 36,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : LocallySmall.{w, v, u} C\ninst✝ : CartesianMonoidalCategory C\nM : Grp C\nX : Cᵒᵖ\n⊢ Small.{w, v} ↑((yonedaGrp.obj M).obj X)",
"usedConstants": [
"CategoryTheory.Functor",
"GrpCat",
"Opposite",
"CategoryTheory.CategoryStruct.t... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Limits.Basic | {
"line": 72,
"column": 6
} | {
"line": 75,
"column": 47
} | [
{
"pp": "J : Type w\ninst✝³ : SmallCategory J\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : MonoidalCategory C\nF G H : J ⥤ C\nj : J\n⊢ ((limit.π F j ⊗ₘ limit.π G j) ⊗ₘ limit.π H j) ≫ (α_ (F.obj j) (G.obj j) (H.obj j)).hom =\n limit.π F j ▷ limit G ▷ limit H ≫ (α_ (F.obj j) ... | conv_rhs => rw [tensorHom_def, whiskerLeft_comp,
← associator_naturality_middle_assoc,
← associator_naturality_right, ← comp_whiskerRight_assoc,
← tensorHom_def, ← tensorHom_def_assoc] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convRHS_1 | Mathlib.Tactic.Conv.convRHS |
Mathlib.CategoryTheory.Monoidal.Closed.Functor | {
"line": 97,
"column": 89
} | {
"line": 100,
"column": 6
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nD : Type u'\ninst✝⁵ : Category.{v, u'} D\ninst✝⁴ : CartesianMonoidalCategory C\ninst✝³ : CartesianMonoidalCategory D\nF : C ⥤ D\ninst✝² : MonoidalClosed C\ninst✝¹ : MonoidalClosed D\ninst✝ : Limits.PreservesLimitsOfShape (Discrete Limits.WalkingPair) F\nA B : C\n... | by
convert unit_mateEquiv _ _ (prodComparisonNatIso F A).inv B using 3
apply IsIso.inv_eq_of_hom_inv_id -- Porting note (https://github.com/leanprover-community/mathlib4/issues/11041): was `ext`
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.DayConvolution.Closed | {
"line": 239,
"column": 2
} | {
"line": 239,
"column": 18
} | [
{
"pp": "C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\nV : Type u₂\ninst✝⁴ : Category.{v₂, u₂} V\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalCategory V\ninst✝¹ : MonoidalClosed V\nF H G : C ⥤ V\ninst✝ : DayConvolution F G\nℌ : DayConvolutionInternalHom F (F ⊛ G) H\nc j : C\n⊢ ℌ.coev_app.app c ≫ ℌ.π c j = Monoid... | dsimp [coev_app] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Monoidal.Free.Coherence | {
"line": 239,
"column": 26
} | {
"line": 239,
"column": 52
} | [
{
"pp": "case mk.h.whiskerLeft\nC : Type u\nX Y X✝ Y₁✝ Y₂✝ : F C\nf✝ : Y₁✝ ⟶ᵐ Y₂✝\nih : (fun n ↦ Y₁✝.normalizeObj n) = Y₂✝.normalizeObj\n⊢ (fun n ↦ (X✝.tensor Y₁✝).normalizeObj n) = (X✝.tensor Y₂✝).normalizeObj",
"usedConstants": [
"CategoryTheory.FreeMonoidalCategory.normalizeObj",
"funext",
... | funext; apply congr_fun ih | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Free.Coherence | {
"line": 239,
"column": 26
} | {
"line": 239,
"column": 52
} | [
{
"pp": "case mk.h.whiskerLeft\nC : Type u\nX Y X✝ Y₁✝ Y₂✝ : F C\nf✝ : Y₁✝ ⟶ᵐ Y₂✝\nih : (fun n ↦ Y₁✝.normalizeObj n) = Y₂✝.normalizeObj\n⊢ (fun n ↦ (X✝.tensor Y₁✝).normalizeObj n) = (X✝.tensor Y₂✝).normalizeObj",
"usedConstants": [
"CategoryTheory.FreeMonoidalCategory.normalizeObj",
"funext",
... | funext; apply congr_fun ih | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 863,
"column": 19
} | {
"line": 863,
"column": 41
} | [
{
"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... | rw [comp_whiskerRight] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 863,
"column": 19
} | {
"line": 863,
"column": 41
} | [
{
"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... | rw [comp_whiskerRight] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 863,
"column": 19
} | {
"line": 863,
"column": 41
} | [
{
"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... | rw [comp_whiskerRight] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.CategoryTheory.Monoidal.Free.Coherence | {
"line": 299,
"column": 4
} | {
"line": 299,
"column": 61
} | [
{
"pp": "C : Type u\nX Y : F C\nf g : X ⟶ Y\nhfg : (fullNormalize C).map f = (fullNormalize C).map g\n⊢ f = g",
"usedConstants": [
"CategoryTheory.Functor",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.Functor.category",
"CategoryTheory.FreeMonoidalCa... | have hf := NatIso.naturality_2 (fullNormalizeIso.{u} C) f | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Monoidal.Hopf_ | {
"line": 217,
"column": 42
} | {
"line": 217,
"column": 75
} | [
{
"pp": "case a.a.a.a.a.a.a.a.a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n| ((α_ A A (A ⊗ A)).inv ≫ (A ⊗ A) ◁ μ) ≫ μ ▷ A",
"usedConstants": [
"CategoryTheory.MonoidalCategoryStruct.whiskerLeft",
"CategoryTheor... | ← associator_inv_naturality_right | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.Monoidal.Hopf_ | {
"line": 221,
"column": 4
} | {
"line": 221,
"column": 34
} | [
{
"pp": "case a.a.a.a.a.a.a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n| A ◁ A ◁ (β_ A A).hom ≫ A ◁ (α_ A A A).inv ≫ A ◁ (β_ A A).hom ▷ A ≫ A ◁ (α_ A A A).hom ≫ A ◁ A ◁ μ",
"usedConstants": [
"CategoryTheory.Monoidal... | simp only [← whiskerLeft_comp] | Lean.Elab.Tactic.Conv.evalSimp | Lean.Parser.Tactic.Conv.simp |
Mathlib.CategoryTheory.Monoidal.Hopf_ | {
"line": 227,
"column": 4
} | {
"line": 227,
"column": 34
} | [
{
"pp": "case a.a.a.a.a.a.a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n| A ◁ Δ ≫ A ◁ A ◁ Δ",
"usedConstants": [
"CategoryTheory.ComonObj.comul",
"CategoryTheory.MonoidalCategoryStruct.whiskerLeft",
"Categ... | simp only [← whiskerLeft_comp] | Lean.Elab.Tactic.Conv.evalSimp | Lean.Parser.Tactic.Conv.simp |
Mathlib.CategoryTheory.Monoidal.Hopf_ | {
"line": 231,
"column": 4
} | {
"line": 231,
"column": 34
} | [
{
"pp": "case a.a.a.a.a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n| A ◁ Δ ▷ A ≫ A ◁ (α_ A A A).hom ≫ A ◁ A ◁ 𝒮 ▷ A ≫ A ◁ (α_ A A A).inv ≫ A ◁ μ ▷ A",
"usedConstants": [
"CategoryTheory.ComonObj.comul",
"Categ... | simp only [← whiskerLeft_comp] | Lean.Elab.Tactic.Conv.evalSimp | Lean.Parser.Tactic.Conv.simp |
Mathlib.CategoryTheory.Monoidal.Hopf_ | {
"line": 238,
"column": 4
} | {
"line": 238,
"column": 34
} | [
{
"pp": "case a.a.a.a.a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n| A ◁ Δ ≫ A ◁ ε ▷ A",
"usedConstants": [
"CategoryTheory.ComonObj.comul",
"CategoryTheory.MonoidalCategoryStruct.whiskerLeft",
"CategoryT... | simp only [← whiskerLeft_comp] | Lean.Elab.Tactic.Conv.evalSimp | Lean.Parser.Tactic.Conv.simp |
Mathlib.CategoryTheory.Monoidal.Hopf_ | {
"line": 242,
"column": 4
} | {
"line": 242,
"column": 34
} | [
{
"pp": "case a.a.a.a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n| A ◁ η ▷ A ≫ A ◁ (β_ A A).hom",
"usedConstants": [
"CategoryTheory.MonoidalCategoryStruct.whiskerLeft",
"CategoryTheory.CategoryStruct.toQuiver"... | simp only [← whiskerLeft_comp] | Lean.Elab.Tactic.Conv.evalSimp | Lean.Parser.Tactic.Conv.simp |
Mathlib.CategoryTheory.Monoidal.Hopf_ | {
"line": 246,
"column": 4
} | {
"line": 246,
"column": 34
} | [
{
"pp": "case a.a.a.a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n| A ◁ A ◁ η ≫ A ◁ 𝒮 ▷ A",
"usedConstants": [
"CategoryTheory.MonoidalCategoryStruct.whiskerLeft",
"CategoryTheory.CategoryStruct.toQuiver",
... | simp only [← whiskerLeft_comp] | Lean.Elab.Tactic.Conv.evalSimp | Lean.Parser.Tactic.Conv.simp |
Mathlib.CategoryTheory.Monoidal.Hopf_ | {
"line": 358,
"column": 4
} | {
"line": 358,
"column": 34
} | [
{
"pp": "case a.a.a.a.a.a.a.a.a.a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n| A ◁ (β_ A A).hom ▷ A ≫ A ◁ (α_ A A A).hom ≫ A ◁ A ◁ (β_ A A).hom",
"usedConstants": [
"CategoryTheory.MonoidalCategoryStruct.whiskerLeft"... | simp only [← whiskerLeft_comp] | Lean.Elab.Tactic.Conv.evalSimp | Lean.Parser.Tactic.Conv.simp |
Mathlib.CategoryTheory.Monoidal.Hopf_ | {
"line": 364,
"column": 4
} | {
"line": 364,
"column": 34
} | [
{
"pp": "case a.a.a.a.a.a.a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n| A ◁ A ◁ Δ ≫ A ◁ (β_ A (A ⊗ A)).hom",
"usedConstants": [
"CategoryTheory.ComonObj.comul",
"CategoryTheory.MonoidalCategoryStruct.whiskerLe... | simp only [← whiskerLeft_comp] | Lean.Elab.Tactic.Conv.evalSimp | Lean.Parser.Tactic.Conv.simp |
Mathlib.CategoryTheory.Monoidal.Hopf_ | {
"line": 370,
"column": 4
} | {
"line": 370,
"column": 34
} | [
{
"pp": "case a.a.a.a.a.a.a.a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n| A ◁ (α_ A A A).hom ≫ A ◁ A ◁ A ◁ 𝒮",
"usedConstants": [
"CategoryTheory.MonoidalCategoryStruct.whiskerLeft",
"CategoryTheory.CategoryS... | simp only [← whiskerLeft_comp] | Lean.Elab.Tactic.Conv.evalSimp | Lean.Parser.Tactic.Conv.simp |
Mathlib.CategoryTheory.Monoidal.Hopf_ | {
"line": 374,
"column": 4
} | {
"line": 374,
"column": 34
} | [
{
"pp": "case a.a.a.a.a.a.a.a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n| A ◁ Δ ▷ A ≫ A ◁ (A ⊗ A) ◁ 𝒮",
"usedConstants": [
"CategoryTheory.ComonObj.comul",
"CategoryTheory.MonoidalCategoryStruct.whiskerLeft",... | simp only [← whiskerLeft_comp] | Lean.Elab.Tactic.Conv.evalSimp | Lean.Parser.Tactic.Conv.simp |
Mathlib.CategoryTheory.Monoidal.Hopf_ | {
"line": 378,
"column": 4
} | {
"line": 378,
"column": 34
} | [
{
"pp": "case a.a.a.a.a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n| A ◁ Δ ▷ A ≫ A ◁ (α_ A A A).hom ≫ A ◁ A ◁ 𝒮 ▷ A ≫ A ◁ (α_ A A A).inv ≫ A ◁ μ ▷ A",
"usedConstants": [
"CategoryTheory.ComonObj.comul",
"Categ... | simp only [← whiskerLeft_comp] | Lean.Elab.Tactic.Conv.evalSimp | Lean.Parser.Tactic.Conv.simp |
Mathlib.CategoryTheory.Monoidal.Hopf_ | {
"line": 385,
"column": 4
} | {
"line": 385,
"column": 34
} | [
{
"pp": "case a.a.a.a.a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n| A ◁ η ▷ A ≫ A ◁ μ",
"usedConstants": [
"CategoryTheory.MonoidalCategoryStruct.whiskerLeft",
"CategoryTheory.CategoryStruct.toQuiver",
"... | simp only [← whiskerLeft_comp] | Lean.Elab.Tactic.Conv.evalSimp | Lean.Parser.Tactic.Conv.simp |
Mathlib.CategoryTheory.Monoidal.Hopf_ | {
"line": 389,
"column": 4
} | {
"line": 389,
"column": 34
} | [
{
"pp": "case a.a.a.a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n| A ◁ (β_ A A).hom ≫ A ◁ A ◁ 𝒮",
"usedConstants": [
"CategoryTheory.MonoidalCategoryStruct.whiskerLeft",
"CategoryTheory.CategoryStruct.toQuiver... | simp only [← whiskerLeft_comp] | Lean.Elab.Tactic.Conv.evalSimp | Lean.Parser.Tactic.Conv.simp |
Mathlib.CategoryTheory.Monoidal.Hopf_ | {
"line": 393,
"column": 4
} | {
"line": 393,
"column": 34
} | [
{
"pp": "case a.a.a.a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n| A ◁ (β_ A A).hom ≫ A ◁ ε ▷ A",
"usedConstants": [
"CategoryTheory.MonoidalCategoryStruct.whiskerLeft",
"CategoryTheory.CategoryStruct.toQuiver"... | simp only [← whiskerLeft_comp] | Lean.Elab.Tactic.Conv.evalSimp | Lean.Parser.Tactic.Conv.simp |
Mathlib.CategoryTheory.Monoidal.Hopf_ | {
"line": 399,
"column": 4
} | {
"line": 399,
"column": 34
} | [
{
"pp": "case a.a.a.a.a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n| A ◁ (λ_ A).inv ≫ A ◁ (λ_ A).hom",
"usedConstants": [
"CategoryTheory.MonoidalCategoryStruct.whiskerLeft",
"CategoryTheory.CategoryStruct.toQu... | simp only [← whiskerLeft_comp] | Lean.Elab.Tactic.Conv.evalSimp | Lean.Parser.Tactic.Conv.simp |
Mathlib.CategoryTheory.MorphismProperty.Ind | {
"line": 124,
"column": 92
} | {
"line": 127,
"column": 33
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nP Q : MorphismProperty C\ninst✝¹ : P.IsStableUnderCobaseChange\ninst✝ : HasPushouts C\n⊢ P.ind.IsStableUnderCobaseChange",
"usedConstants": [
"CategoryTheory.MorphismProperty.instRespectsIsoInd",
"Eq.mpr",
"CategoryTheory.instCategoryUnder",... | by
refine .mk' fun A B A' f g _ hf ↦ ?_
rw [ind_iff_ind_underMk] at hf ⊢
exact ind_underObj_pushout g hf | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.MorphismProperty.Ind | {
"line": 194,
"column": 19
} | {
"line": 194,
"column": 22
} | [
{
"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... | hpu | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Preadditive.EndoFunctor | {
"line": 95,
"column": 18
} | {
"line": 98,
"column": 22
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Algebra F\n⊢ ∀ (a b : A₁ ⟶ A₂), a + b = b + a",
"usedConstants": [
"CategoryTheory.Endofunctor.Algebra",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"AddComm... | by
intros
apply Algebra.Hom.ext
apply add_comm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Preadditive.EndoFunctor | {
"line": 148,
"column": 8
} | {
"line": 150,
"column": 24
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Coalgebra F\n⊢ ∀ (n : ℕ) (x : A₁ ⟶ A₂), { f := (n + 1) • x.f, h := ⋯ } = { f := n • x.f, h := ⋯ } + x",
"usedConstants": [
"instHSMul",
"CategoryTheory.Endofunctor.Coalgebra.instCat... | intros
apply Coalgebra.Hom.ext
apply succ_nsmul | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Preadditive.EndoFunctor | {
"line": 148,
"column": 8
} | {
"line": 150,
"column": 24
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Coalgebra F\n⊢ ∀ (n : ℕ) (x : A₁ ⟶ A₂), { f := (n + 1) • x.f, h := ⋯ } = { f := n • x.f, h := ⋯ } + x",
"usedConstants": [
"instHSMul",
"CategoryTheory.Endofunctor.Coalgebra.instCat... | intros
apply Coalgebra.Hom.ext
apply succ_nsmul | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.CartesianMonoidal | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 18
} | [
{
"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\n⊢ Discrete PEmpty.{1} ⥤ Sheaf J A",
"usedConstants": [
"CategoryTheory.Functor",
"Opposite",
"CategoryTheory.Functor.category",
... | · exact .empty _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Sites.Coherent.CoherentSheaves | {
"line": 51,
"column": 2
} | {
"line": 51,
"column": 83
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Precoherent C\nW X : C\nα : Type\ninst✝ : Finite α\nY : α → C\nπ : (a : α) → Y a ⟶ X\nH : EffectiveEpiFamily Y π\n⊢ Presieve.IsSheafFor (yoneda.obj W) (Presieve.ofArrows Y π)",
"usedConstants": [
"CategoryTheory.Over",
"CategoryTheo... | have h_colim := isColimitOfEffectiveEpiFamilyStruct Y π H.effectiveEpiFamily.some | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Preadditive.Mat | {
"line": 572,
"column": 4
} | {
"line": 572,
"column": 21
} | [
{
"pp": "case h\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nR : Type\ninst✝ : Ring R\nX : Mat_ (SingleObj Rᵐᵒᵖ)\ni✝ j✝ : (FintypeCat.of X.ι).obj\n⊢ MulOpposite.unop (if h : i✝ = j✝ then eqToHom ⋯ else 0) = if i✝ = j✝ then 1 else 0",
"usedConstants": [
"Eq.mpr",
"NonAssocS... | split_ifs <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Sites.Coherent.Comparison | {
"line": 40,
"column": 4
} | {
"line": 40,
"column": 65
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Precoherent C\ninst✝ : HasFiniteCoproducts C\nX Y Z : C\nf : X ⟶ Y\ng : Z ⟶ Y\nx✝ : EffectiveEpi g\nhp : EffectiveEpi g → ∃ β, ∃ (_ : Finite β), ∃ X₂ π₂, EffectiveEpiFamily X₂ π₂ ∧ ∃ ι, ∀ (b : β), ι b ≫ g = π₂ b ≫ f\nβ : Type\nw✝ : Finite β\nX₂ : β... | refine ⟨∐ X₂, Sigma.desc π₂, inferInstance, Sigma.desc ι, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves | {
"line": 102,
"column": 6
} | {
"line": 103,
"column": 15
} | [
{
"pp": "case refine_1.refine_1.a.refine_1\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : FinitaryPreExtensive C\ninst✝ : FinitaryExtensive C\nF : Cᵒᵖ ⥤ Type w\nhF : ∀ {X : C}, ∀ R ∈ (extensiveCoverage C).coverings X, IsSheafFor F R\nn : ℕ\nK : Discrete (Fin n) ⥤ Cᵒᵖ\nZ : Fin n → C := fun i ↦ unop (K.o... | · ext b
cases b | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Sites.Coherent.RegularTopology | {
"line": 39,
"column": 4
} | {
"line": 39,
"column": 70
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preregular C\nX : C\nS : Sieve X\nY : C\nπ : Y ⟶ X\nh : EffectiveEpi π ∧ S.arrows π\n⊢ (Presieve.ofArrows (fun x ↦ Y) fun x ↦ π) ≤ S.arrows",
"usedConstants": [
"CategoryTheory.Presieve.le_of_factorsThru_sieve",
"Unit",
"Catego... | apply Presieve.le_of_factorsThru_sieve (Presieve.ofArrows _ _) S _ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 491,
"column": 8
} | {
"line": 492,
"column": 65
} | [
{
"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 ⟨φ ⟨_, D.src hf⟩, φ ⟨_, D.tgt hf⟩,
Or.inr ⟨_⟩, Or.inr ⟨_⟩, D.isTerminal.comm_assoc _ hf _⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 491,
"column": 8
} | {
"line": 492,
"column": 65
} | [
{
"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 ⟨φ ⟨_, D.src hf⟩, φ ⟨_, D.tgt hf⟩,
Or.inr ⟨_⟩, Or.inr ⟨_⟩, D.isTerminal.comm_assoc _ hf _⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 491,
"column": 8
} | {
"line": 492,
"column": 65
} | [
{
"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 ⟨φ ⟨_, D.src hf⟩, φ ⟨_, D.tgt hf⟩,
Or.inr ⟨_⟩, Or.inr ⟨_⟩, D.isTerminal.comm_assoc _ hf _⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 495,
"column": 2
} | {
"line": 496,
"column": 27
} | [
{
"pp": "J : 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)\nj : J\nD : D... | have lift_eq (j : J) (hj : D.P j) : hm₁.lift (Or.inl hj) = φ ⟨_, hj⟩ :=
hm₁.uniq _ (Or.inr ⟨_⟩) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology | {
"line": 59,
"column": 4
} | {
"line": 59,
"column": 48
} | [
{
"pp": "case mpr.h\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : FinitaryPreExtensive C\nX : C\nS : Sieve X\nα : Type\nw✝ : Finite α\nY : α → C\nπ : (a : α) → Y a ⟶ X\nh : Nonempty (IsColimit (Cofan.mk X π))\nh' : ∀ (a : α), S.arrows (π a)\n⊢ Presieve.ofArrows Y π ≤ S.arrows",
"usedConstants": [
... | · exact fun _ _ hh ↦ by cases hh; exact h' _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Sites.Coherent.LocallySurjective | {
"line": 47,
"column": 39
} | {
"line": 58,
"column": 24
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\nFD : D → D → Type u_3\nCD : D → Type w\ninst✝² : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)\ninst✝¹ : ConcreteCategory D FD\ninst✝ : Preregular C\nF G : Cᵒᵖ ⥤ D\nf : F ⟶ G\n⊢ Presheaf.IsLocallySurjective (regula... | by
constructor
· intro ⟨h⟩ X y
specialize h y
rw [regularTopology.mem_sieves_iff_hasEffectiveEpi] at h
obtain ⟨X', π, h, h'⟩ := h
exact ⟨X', π, h, h'⟩
· intro h
refine ⟨fun y ↦ ?_⟩
obtain ⟨X', π, h, h'⟩ := h _ y
rw [regularTopology.mem_sieves_iff_hasEffectiveEpi]
exact ⟨X', π, h, h... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 303,
"column": 4
} | {
"line": 303,
"column": 58
} | [
{
"pp": "case refine_1.refine_2\nC₀ : Type u₀\nC : Type u\ninst✝² : Category.{v₀, u₀} C₀\ninst✝¹ : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\ninst✝ : IsDenseSubsite J₀ J F\nX : C\ndata : F.OneHypercoverDenseData J₀ J X\nX₀ : C₀\nf : F.obj X₀ ⟶ X\nthis✝ : F.IsCoverDe... | · rw [w₁, assoc, ← reassoc_of% fac, hb.some.fac_assoc] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 575,
"column": 4
} | {
"line": 576,
"column": 71
} | [
{
"pp": "C₀ : Type u₀\nC : Type u\ninst✝⁴ : Category.{v₀, u₀} C₀\ninst✝³ : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\nA : Type u'\ninst✝² : Category.{v', u'} A\ninst✝¹ : IsDenseSubsite J₀ J F\ndata : (X : C) → F.OneHypercoverDenseData J₀ J X\ninst✝ : HasLimitsOfSize... | refine Presheaf.IsSheaf.hom_ext G₀.property
⟨_, cover_lift F J₀ _ (J.pullback_stable a (data Y).mem₀)⟩ _ _ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 748,
"column": 2
} | {
"line": 749,
"column": 12
} | [
{
"pp": "C₀ : Type u₀\nC : Type u\ninst✝⁴ : Category.{v₀, u₀} C₀\ninst✝³ : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\nA : Type u'\ninst✝² : Category.{v', u'} A\ninst✝¹ : IsDenseSubsite J₀ J F\ndata : (X : C) → F.OneHypercoverDenseData J₀ J X\ninst✝ : HasLimitsOfSize... | rw [← cancel_mono (presheafObjObjIso data G₀ ((data X).X i)).inv, assoc, Iso.hom_inv_id,
comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Sites.Point.Conservative | {
"line": 228,
"column": 6
} | {
"line": 228,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nP : ObjectProperty J.Point\ninst✝¹ : LocallySmall.{w, v, u} C\nhP :\n ∀ ⦃X : C⦄ (S : Sieve X),\n (∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X), ∃ Y g, ∃ (_ : S.arrows g), ∃ y, Φ.obj.fiber.map g y = x) →\n S ∈ J X\nF₁ : Cᵒ... | rw [← hx₁] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Topos.Classifier | {
"line": 388,
"column": 2
} | {
"line": 390,
"column": 47
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasPullbacks C\nΩ : C\nh : SubobjectRepresentableBy Ω\nU X : C\nm : U ⟶ X\ninst✝ : Mono m\nχ' : X ⟶ Ω\nπ : U ⟶ underlying.obj h.Ω₀\nsq : IsPullback m π χ' h.Ω₀.arrow\n⊢ χ' = h.χ m",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Subobject.a... | apply h.homEquiv.injective
simp only [χ, Equiv.apply_symm_apply, homEquiv_eq]
simpa using Subobject.pullback_obj_mk sq.flip | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Topos.Classifier | {
"line": 388,
"column": 2
} | {
"line": 390,
"column": 47
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasPullbacks C\nΩ : C\nh : SubobjectRepresentableBy Ω\nU X : C\nm : U ⟶ X\ninst✝ : Mono m\nχ' : X ⟶ Ω\nπ : U ⟶ underlying.obj h.Ω₀\nsq : IsPullback m π χ' h.Ω₀.arrow\n⊢ χ' = h.χ m",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Subobject.a... | apply h.homEquiv.injective
simp only [χ, Equiv.apply_symm_apply, homEquiv_eq]
simpa using Subobject.pullback_obj_mk sq.flip | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Topos.Sheaf | {
"line": 106,
"column": 2
} | {
"line": 108,
"column": 18
} | [
{
"pp": "case w.h.h.mp\nC : Type u\ninst✝ : Category.{v, u} C\nF G : Cᵒᵖ ⥤ Type (max u v)\nm : F ⟶ G\nχ' : G ⟶ Functor.sieves C\nX : Cᵒᵖ\nx : G.obj X\nh₁ : ∀ (x : Cᵒᵖ), m.app x ≫ χ'.app x = Types.isTerminalPUnit.from (F.obj x) ≫ (truth C).app x\nh₂ : ∀ (x : Cᵒᵖ) (x₁ y₁ : F.obj x), m.app x x₁ = m.app x y₁ → x₁ =... | · intro h
obtain ⟨z, hz⟩ := h₃ _ _ h
use z, hz.symm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Triangulated.Opposite.OpOp | {
"line": 110,
"column": 4
} | {
"line": 118,
"column": 71
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasShift C ℤ\np q : ℤ\n⊢ iso C (p + q) = Functor.CommShift.isoAdd (iso C p) (iso C q)",
"usedConstants": [
"CategoryTheory.Functor.CommShift.isoAdd_hom_app",
"CategoryTheory.Pretriangulated.Opposite.OpOpCommShift.iso_hom_app._proof_4... | ext X
refine Quiver.Hom.unop_inj (Quiver.Hom.unop_inj ?_)
simp [← shiftFunctorAdd'_eq_shiftFunctorAdd, ← unop_comp_assoc, ← Functor.map_comp,
fun X n ↦ iso_hom_app X n (-n) (add_neg_cancel n),
shiftFunctor_op_map _ q (-q),
shiftFunctorAdd'_op_inv_app _ p q (p + q) rfl (-p) (-q) (-(p + q))
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Triangulated.Opposite.OpOp | {
"line": 110,
"column": 4
} | {
"line": 118,
"column": 71
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasShift C ℤ\np q : ℤ\n⊢ iso C (p + q) = Functor.CommShift.isoAdd (iso C p) (iso C q)",
"usedConstants": [
"CategoryTheory.Functor.CommShift.isoAdd_hom_app",
"CategoryTheory.Pretriangulated.Opposite.OpOpCommShift.iso_hom_app._proof_4... | ext X
refine Quiver.Hom.unop_inj (Quiver.Hom.unop_inj ?_)
simp [← shiftFunctorAdd'_eq_shiftFunctorAdd, ← unop_comp_assoc, ← Functor.map_comp,
fun X n ↦ iso_hom_app X n (-n) (add_neg_cancel n),
shiftFunctor_op_map _ q (-q),
shiftFunctorAdd'_op_inv_app _ p q (p + q) rfl (-p) (-q) (-(p + q))
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Triangulated.Adjunction | {
"line": 106,
"column": 10
} | {
"line": 109,
"column": 66
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹⁵ : Category.{v_1, u_1} C\ninst✝¹⁴ : Category.{v_2, u_2} D\ninst✝¹³ : HasZeroObject C\ninst✝¹² : HasZeroObject D\ninst✝¹¹ : Preadditive C\ninst✝¹⁰ : Preadditive D\ninst✝⁹ : HasShift C ℤ\ninst✝⁸ : HasShift D ℤ\ninst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝⁶ : ∀... | rw [hα, sub_comp, ← cancel_mono ((Functor.commShiftIso G (1 : ℤ)).hom.app T.obj₁),
assoc, sub_comp, assoc, assoc, hψ, zero_comp, sub_eq_zero,
← cancel_mono ((Functor.commShiftIso G (1 : ℤ)).inv.app T.obj₁), assoc,
assoc, assoc, assoc, h₂', Iso.hom_inv_id_app, comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Triangulated.TStructure.Induced | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 25
} | [
{
"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\nP : ObjectProperty C\nt : TStructure C\ninst✝ : P.IsTriangulated\nh : P.HasInducedTStructure t\nX Y : P.Ful... | rw [Functor.map_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 61
} | [
{
"pp": "C : Type u_1\nA : Type u_2\ninst✝⁹ : Category.{v_1, u_1} C\ninst✝⁸ : HasZeroObject C\ninst✝⁷ : Preadditive C\ninst✝⁶ : HasShift C ℤ\ninst✝⁵ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝⁴ : Pretriangulated C\ninst✝³ : Category.{v_2, u_2} A\nι : A ⥤ C\nhι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunc... | obtain ⟨m, hm⟩ := Triangle.yoneda_exact₃ _ hT' (ι.map k) hl | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE | {
"line": 289,
"column": 2
} | {
"line": 290,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nX : C\nn : ℤ\n⊢ t.IsLE ((t.triangleLTGE n).obj X).obj₁ (n - 1)",
"usedConstants": [
"C... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE | {
"line": 289,
"column": 2
} | {
"line": 290,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nX : C\nn : ℤ\n⊢ t.IsLE ((t.triangleLTGE n).obj X).obj₁ (n - 1)",
"usedConstants": [
"C... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE | {
"line": 293,
"column": 2
} | {
"line": 294,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nX : C\nn : ℤ\n⊢ t.IsGE ((t.triangleLTGE n).obj X).obj₃ n",
"usedConstants": [
"Categor... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE | {
"line": 293,
"column": 2
} | {
"line": 294,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nX : C\nn : ℤ\n⊢ t.IsGE ((t.triangleLTGE n).obj X).obj₃ n",
"usedConstants": [
"Categor... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE | {
"line": 622,
"column": 4
} | {
"line": 622,
"column": 30
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nT : Triangle C\nhT : T ∈ distinguishedTriangles\n⊢ t.minus T.obj₁ → t.minus T.obj₃ → t.minus T.o... | rintro ⟨i₁, hi₁⟩ ⟨i₃, hi₃⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE | {
"line": 629,
"column": 4
} | {
"line": 629,
"column": 30
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nT : Triangle C\nhT : T ∈ distinguishedTriangles\n⊢ t.plus T.obj₁ → t.plus T.obj₃ → t.plus T.obj₂... | rintro ⟨i₁, hi₁⟩ ⟨i₃, hi₃⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Combinatorics.Additive.AP.Three.Behrend | {
"line": 205,
"column": 2
} | {
"line": 206,
"column": 54
} | [
{
"pp": "n d : ℕ\n⊢ (2 * d + 1) ^ n - 1 = (∑ i, d * (2 * d + 1) ^ ↑i) * 2",
"usedConstants": [
"Eq.mpr",
"Finset.mul_sum",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.instOrderedSub",
"Nat.instIsOrderedAddMonoid",
"HMul.hMul",
"Finset.univ",
"CommSemiring.... | rw [← sum_range fun i => d * (2 * d + 1) ^ (i : ℕ), ← mul_sum, mul_right_comm, mul_comm d, ←
geom_sum_mul_add, add_tsub_cancel_right, mul_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Additive.CauchyDavenport | {
"line": 85,
"column": 83
} | {
"line": 86,
"column": 47
} | [
{
"pp": "α : Type u_2\ninst✝¹ : Group α\ninst✝ : DecidableEq α\nx y : Finset α × Finset α\n⊢ DevosMulRel x y ↔\n #(x.1 * x.2) < #(y.1 * y.2) ∨\n #(x.1 * x.2) = #(y.1 * y.2) ∧ #y.1 + #y.2 < #x.1 + #x.2 ∨\n #(x.1 * x.2) = #(y.1 * y.2) ∧ #x.1 + #x.2 = #y.1 + #y.2 ∧ #x.1 < #y.1",
"usedConstants":... | by
simp [DevosMulRel, Prod.lex_iff, and_or_left] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Additive.SmallTripling | {
"line": 42,
"column": 2
} | {
"line": 42,
"column": 45
} | [
{
"pp": "G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : Group G\nA : Finset G\nk : ℝ\nm : ℕ\nhm : 3 ≤ m\nh : ∀ (ε : Fin 3 → ℤ), (∀ (i : Fin 3), |ε i| = 1) → ↑(#(List.map (fun i ↦ A ^ ε i) (finRange 3)).prod) ≤ k * ↑(#A)\nε : Fin m → ℤ\nhε : ∀ (i : Fin m), |ε i| = 1\n⊢ ↑(#(List.map (fun i ↦ A ^ ε i) (finRange m))... | induction m, hm using Nat.le_induction with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Order.Partition.Equipartition | {
"line": 88,
"column": 4
} | {
"line": 88,
"column": 12
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\ns : Finset α\nP : Finpartition s\nhP : P.IsEquipartition\nz :\n #({x ∈ P.parts | #x = #s / #P.parts + 1}) * (#s / #P.parts + 1) +\n #({p ∈ P.parts | ¬#p = #s / #P.parts + 1}) * (#s / #P.parts) =\n #s\n⊢ #({p ∈ P.parts | #p = #s / #P.parts + 1}) = #s % #P.pa... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Partition.Finpartition | {
"line": 139,
"column": 10
} | {
"line": 139,
"column": 25
} | [
{
"pp": "case h.e'_4\nα : Type u_1\ninst✝³ : Lattice α\ninst✝² : OrderBot α\nβ : Type u_2\ninst✝¹ : Lattice β\ninst✝ : OrderBot β\na : α\ne : α ≃o β\nP : Finpartition a\nu : Finset β\nhu : Finset.map (↑e).symm.toEmbedding u ⊆ P.parts\nx✝¹ : β\nhbu : x✝¹ ∉ u\nx✝ : β\nhx : x✝ ≤ id x✝¹\nhxu : x✝ ≤ u.sup id\nhb : (... | map_finset_sup, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Density | {
"line": 145,
"column": 53
} | {
"line": 145,
"column": 62
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns : Finset α\n⊢ ↑(#(interedges r s ∅)) / (↑(#s) * 0) = 0",
"usedConstants": [
"Rat.instOfNat",
"Eq.mpr",
"Rat.instMul",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Density | {
"line": 171,
"column": 2
} | {
"line": 174,
"column": 29
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns₁ s₂ : Finset α\nt₁ t₂ : Finset β\nhs : s₂ ⊆ s₁\nht : t₂ ⊆ t₁\nhs₂ : s₂.Nonempty\nht₂ : t₂.Nonempty\n⊢ ↑(#s₂) / ↑(#s₁) * (↑(#t₂) / ↑(#t₁)) * edgeDensity r s₂ t₂ ≤ edgeDensity r s₁ t₁",
"usedConstants": [
"Ra... | have hst : (#s₂ : ℚ) * #t₂ ≠ 0 := by simp [hs₂.ne_empty, ht₂.ne_empty]
rw [edgeDensity, edgeDensity, div_mul_div_comm, mul_comm, div_mul_div_cancel₀ hst]
gcongr
exact interedges_mono hs ht | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Density | {
"line": 171,
"column": 2
} | {
"line": 174,
"column": 29
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns₁ s₂ : Finset α\nt₁ t₂ : Finset β\nhs : s₂ ⊆ s₁\nht : t₂ ⊆ t₁\nhs₂ : s₂.Nonempty\nht₂ : t₂.Nonempty\n⊢ ↑(#s₂) / ↑(#s₁) * (↑(#t₂) / ↑(#t₁)) * edgeDensity r s₂ t₂ ≤ edgeDensity r s₁ t₁",
"usedConstants": [
"Ra... | have hst : (#s₂ : ℚ) * #t₂ ≠ 0 := by simp [hs₂.ne_empty, ht₂.ne_empty]
rw [edgeDensity, edgeDensity, div_mul_div_comm, mul_comm, div_mul_div_cancel₀ hst]
gcongr
exact interedges_mono hs ht | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Regularity.Bound | {
"line": 144,
"column": 14
} | {
"line": 144,
"column": 22
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nP : Finpartition univ\nu : Finset α\nhucard : #u = m * 4 ^ #P.parts + a\n⊢ (4 ^ #P.parts - a) * m + a * (m + 1) = m * 4 ^ #P.parts + a",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"SzemerediRegularity.stepBound... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Bound | {
"line": 153,
"column": 57
} | {
"line": 153,
"column": 65
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nP : Finpartition univ\nu : Finset α\nhP : P.IsEquipartition\nhu : u ∈ P.parts\nhucard : #u ≠ Fintype.card α / #P.parts\nthis : m * 4 ^ #P.parts ≤ Fintype.card α / #P.parts\n⊢ (4 ^ #P.parts - (a + 1)) * m + (a + 1) * (m + 1) = #univ / #P.parts + 1... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise | {
"line": 139,
"column": 4
} | {
"line": 140,
"column": 80
} | [
{
"pp": "case neg\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}.biUnion ... | rw [card_insert_of_notMem, hR₃, if_neg h, Nat.sub_add_cancel (hab.resolve_left h)]
intro H; exact ht.ne_empty (le_sdiff_right.1 <| R.le <| filter_subset _ _ H) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise | {
"line": 139,
"column": 4
} | {
"line": 140,
"column": 80
} | [
{
"pp": "case neg\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}.biUnion ... | rw [card_insert_of_notMem, hR₃, if_neg h, Nat.sub_add_cancel (hab.resolve_left h)]
intro H; exact ht.ne_empty (le_sdiff_right.1 <| R.le <| filter_subset _ _ H) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise | {
"line": 199,
"column": 18
} | {
"line": 199,
"column": 26
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\ns : Finset α\nn : ℕ\nhn : 0 < n\nhs : n ≤ #s\n⊢ n * (#s / n) - #s % n * (#s / n) + #s % n * (#s / n + 1) = #s",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"instHDiv",
"HMul.hMul",
"congrArg",
"HSub.hSub",
"... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Uniform | {
"line": 359,
"column": 2
} | {
"line": 373,
"column": 53
} | [
{
"pp": "α : 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\n⊢ ∑ i ∈ P... | calc
_ ≤ #(P.nonUniforms G ε) • (↑(#A / #P.parts + 1) : 𝕜) ^ 2 :=
sum_le_card_nsmul _ _ _ ?_
_ = _ := nsmul_eq_mul _ _
_ ≤ _ := mul_le_mul_of_nonneg_right hG <| by positivity
_ < _ := ?_
· simp only [Prod.forall, Finpartition.mk_mem_nonUniforms, and_imp]
rintro U V hU hV - -
rw [sq, ← N... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Regularity.Uniform | {
"line": 359,
"column": 2
} | {
"line": 373,
"column": 53
} | [
{
"pp": "α : 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\n⊢ ∑ i ∈ P... | calc
_ ≤ #(P.nonUniforms G ε) • (↑(#A / #P.parts + 1) : 𝕜) ^ 2 :=
sum_le_card_nsmul _ _ _ ?_
_ = _ := nsmul_eq_mul _ _
_ ≤ _ := mul_le_mul_of_nonneg_right hG <| by positivity
_ < _ := ?_
· simp only [Prod.forall, Finpartition.mk_mem_nonUniforms, and_imp]
rintro U V hU hV - -
rw [sq, ← N... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Regularity.Lemma | {
"line": 154,
"column": 8
} | {
"line": 154,
"column": 22
} | [
{
"pp": "case neg.refine_3\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.IsEquipartitio... | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Lemma | {
"line": 154,
"column": 23
} | {
"line": 154,
"column": 31
} | [
{
"pp": "case neg.refine_3\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.IsEquipartitio... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Walks.Traversal | {
"line": 189,
"column": 2
} | {
"line": 190,
"column": 76
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nh : ¬p.Nil\n⊢ p.penultimate ∈ p.support.dropLast",
"usedConstants": [
"SimpleGraph.Walk.adj_penultimate",
"SimpleGraph.Adj.ne",
"Ne",
"_private.Mathlib.Combinatorics.SimpleGraph.Walks.Traversal.0.SimpleGraph.Walk.penult... | have := adj_penultimate h |>.ne
grind [getVert_mem_support, List.dropLast_concat_getLast, getLast_support] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Walks.Traversal | {
"line": 189,
"column": 2
} | {
"line": 190,
"column": 76
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nh : ¬p.Nil\n⊢ p.penultimate ∈ p.support.dropLast",
"usedConstants": [
"SimpleGraph.Walk.adj_penultimate",
"SimpleGraph.Adj.ne",
"Ne",
"_private.Mathlib.Combinatorics.SimpleGraph.Walks.Traversal.0.SimpleGraph.Walk.penult... | have := adj_penultimate h |>.ne
grind [getVert_mem_support, List.dropLast_concat_getLast, getLast_support] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Walks.Traversal | {
"line": 223,
"column": 2
} | {
"line": 224,
"column": 27
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv w : V\np : G.Walk v w\nh₁ : ¬p.Nil\nh₂ : 0 < p.darts.length\n⊢ p.lastDart h₁ = p.darts[p.darts.length - 1]",
"usedConstants": [
"SimpleGraph.Walk.lastDart",
"_private.Mathlib.Combinatorics.SimpleGraph.Walks.Traversal.0.SimpleGraph.Walk.lastDart_eq._simp_... | simp (disch := grind) [Dart.ext_iff, lastDart_toProd, darts_getElem_eq_getVert,
p.getVert_of_length_le] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Walks.Traversal | {
"line": 223,
"column": 2
} | {
"line": 224,
"column": 27
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv w : V\np : G.Walk v w\nh₁ : ¬p.Nil\nh₂ : 0 < p.darts.length\n⊢ p.lastDart h₁ = p.darts[p.darts.length - 1]",
"usedConstants": [
"SimpleGraph.Walk.lastDart",
"_private.Mathlib.Combinatorics.SimpleGraph.Walks.Traversal.0.SimpleGraph.Walk.lastDart_eq._simp_... | simp (disch := grind) [Dart.ext_iff, lastDart_toProd, darts_getElem_eq_getVert,
p.getVert_of_length_le] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Walks.Traversal | {
"line": 223,
"column": 2
} | {
"line": 224,
"column": 27
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv w : V\np : G.Walk v w\nh₁ : ¬p.Nil\nh₂ : 0 < p.darts.length\n⊢ p.lastDart h₁ = p.darts[p.darts.length - 1]",
"usedConstants": [
"SimpleGraph.Walk.lastDart",
"_private.Mathlib.Combinatorics.SimpleGraph.Walks.Traversal.0.SimpleGraph.Walk.lastDart_eq._simp_... | simp (disch := grind) [Dart.ext_iff, lastDart_toProd, darts_getElem_eq_getVert,
p.getVert_of_length_le] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 1199,
"column": 9
} | {
"line": 1199,
"column": 33
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nG' : G.Subgraph\nx✝² : ∃ s, G' = ⊤.induce s\nx✝¹ : V\nhu : x✝¹ ∈ G'.verts\nx✝ : V\nhv : x✝ ∈ G'.verts\nhadj : G.Adj x✝¹ x✝\ns : Set V\nh : G' = ⊤.induce s\n⊢ (⊤.induce s).Adj x✝¹ x✝",
"usedConstants": [
"Eq.mpr",
"congrArg",
"SimpleGraph.Subgraph",
... | (h ▸ rfl : s = G'.verts) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Walks.Operations | {
"line": 369,
"column": 66
} | {
"line": 373,
"column": 38
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nt u v w : V\np : G.Walk u v\np' : G.Walk v w\n⊢ t ∈ (p.append p').support ↔ t ∈ p.support ∨ t ∈ p'.support",
"usedConstants": [
"Eq.mpr",
"False",
"eq_false",
"congrArg",
"true_or",
"SimpleGraph.Walk.support",
"SimpleGraph.Wal... | by
simp only [mem_support_iff, mem_tail_support_append_iff]
obtain rfl | h := eq_or_ne t v <;> obtain rfl | h' := eq_or_ne t u <;>
-- this `have` triggers the unusedHavesSuffices linter:
(try have := h'.symm) <;> simp [*] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Walks.Operations | {
"line": 388,
"column": 2
} | {
"line": 388,
"column": 43
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v w : V\np : G.Walk u v\nq : G.Walk v w\na✝ : V\n⊢ a✝ ∈ q.support → a✝ ∈ (p.append q).support",
"usedConstants": [
"congrArg",
"SimpleGraph.Walk.support",
"Membership.mem",
"List",
"List.instMembership",
"True",
"eq_true",
... | simp +contextual [mem_support_append_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Walks.Operations | {
"line": 510,
"column": 81
} | {
"line": 511,
"column": 59
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nn : ℕ\n⊢ (p.drop n).length = p.length - n",
"usedConstants": [
"of_decide_eq_true",
"congrArg",
"Nat.Simproc.add_sub_add_le",
"SimpleGraph.Walk.length",
"SimpleGraph.Adj",
"HSub.hSub",
"SimpleGraph.Wal... | by
induction p generalizing n <;> cases n <;> simp [*, drop] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Walks.Operations | {
"line": 528,
"column": 79
} | {
"line": 529,
"column": 59
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nn : ℕ\n⊢ (p.drop n).Nil ↔ p.length ≤ n",
"usedConstants": [
"False",
"Nat.instOne",
"instReflLe",
"congrArg",
"SimpleGraph.Walk.length",
"SimpleGraph.Adj",
"AddMonoid.toAddZeroClass",
"Nat.add_eq... | by
induction p generalizing n <;> cases n <;> simp [*, drop] | [anonymous] | Lean.Parser.Term.byTactic |
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