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 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.IsTensorProduct | {
"line": 731,
"column": 2
} | {
"line": 740,
"column": 33
} | {
"line": 742,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type v₃\ninst✝¹² : CommSemiring R\ninst✝¹¹ : CommSemiring S\ninst✝¹⁰ : Algebra R S\nR' : Type u_6\nS' : Type u_7\ninst✝⁹ : CommSemiring R'\ninst✝⁸ : CommSemiring S'\ninst✝⁷ : Algebra R R'\ninst✝⁶ : Algebra S S'\ninst✝⁵ : Algebra R' S'\ninst✝⁴ : Algebra R S'\ninst✝³ : IsScalarTower R R... | [] | ext x
refine H.1.inductionOn x _ ?_ ?_ ?_ ?_
· simp only [map_zero]
· exact AlgHom.congr_fun h₁
· intro s s' e
rw [Algebra.smul_def, map_mul, map_mul, e]
congr 1
exact (AlgHom.congr_fun h₂ s :)
· intro s₁ s₂ e₁ e₂
rw [map_add, map_add, e₁, e₂] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.ConcreteCategory.Basic | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 36
} | {
"line": 166,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type w\ninst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝ : ConcreteCategory C FC\nX : C\nx : ToType X\n⊢ (hom (𝟙 X)) x = x",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"congrArg",
"Ca... | [] | simp [ConcreteCategory.id_apply _] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.ConcreteCategory.Basic | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 36
} | {
"line": 166,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type w\ninst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝ : ConcreteCategory C FC\nX : C\nx : ToType X\n⊢ (hom (𝟙 X)) x = x",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"congrArg",
"Ca... | [] | simp [ConcreteCategory.id_apply _] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.ConcreteCategory.Basic | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 36
} | {
"line": 166,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type w\ninst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝ : ConcreteCategory C FC\nX : C\nx : ToType X\n⊢ (hom (𝟙 X)) x = x",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"congrArg",
"Ca... | [] | simp [ConcreteCategory.id_apply _] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Quiver.Push | {
"line": 62,
"column": 6
} | {
"line": 62,
"column": 23
} | {
"line": 63,
"column": 6
} | [
{
"pp": "V : Type u_1\ninst✝¹ : Quiver V\nW : Type u_2\nσ : V → W\nW' : Type u_3\ninst✝ : Quiver W'\nφ : V ⥤q W'\nτ : W → W'\nh : ∀ (x : V), φ.obj x = τ (σ x)\nX Y : V\nf : X ⟶ Y\n⊢ τ (σ X) ⟶ τ (σ Y)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Quiver.Hom",
... | [
"V : Type u_1\ninst✝¹ : Quiver V\nW : Type u_2\nσ : V → W\nW' : Type u_3\ninst✝ : Quiver W'\nφ : V ⥤q W'\nτ : W → W'\nh : ∀ (x : V), φ.obj x = τ (σ x)\nX Y : V\nf : X ⟶ Y\n⊢ φ.obj X ⟶ φ.obj Y"
] | rw [← h X, ← h Y] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.ObjectProperty.ClosedUnderIsomorphisms | {
"line": 100,
"column": 4
} | {
"line": 100,
"column": 26
} | {
"line": 101,
"column": 4
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nP : ObjectProperty C\nF : C ⥤ D\n⊢ (P.strictMap F).isoClosure ≤ P.map F",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.ObjectProperty.strictM... | [
"case refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nP : ObjectProperty C\nF : C ⥤ D\n⊢ P.strictMap F ≤ P.map F"
] | rw [isoClosure_le_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Equivalence | {
"line": 278,
"column": 35
} | {
"line": 278,
"column": 54
} | {
"line": 278,
"column": 55
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\ne : C ≌ D\nY : D\n⊢ e.unit.app (e.inverse.obj Y) ≫ 𝟙 (e.inverse.obj (e.functor.obj (e.inverse.obj Y))) ≫ e.inverse.map (e.counit.app Y) =\n 𝟙 (e.inverse.obj Y)",
"ppTerm": "?m.75",
"assigned": true,
"u... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\ne : C ≌ D\nY : D\n⊢ e.unit.app (e.inverse.obj Y) ≫ e.inverse.map (𝟙 (e.functor.obj (e.inverse.obj Y))) ≫ e.inverse.map (e.counit.app Y) =\n 𝟙 (e.inverse.obj Y)"
] | ← map_id e.inverse, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.MorphismProperty.Basic | {
"line": 710,
"column": 6
} | {
"line": 710,
"column": 20
} | {
"line": 712,
"column": 0
} | [
{
"pp": "case hprecomp\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z : C\ne : X ≅ Y\nf : Y ⟶ Z\nx✝ : Epi f\n⊢ Epi (e.hom ≫ f)",
"ppTerm": "?hprecomp",
"assigned": true,
"usedConstants": [
"CategoryTheory.epi_comp",
"CategoryTheory.Iso.isIso_hom",
"CategoryTheory.IsIso.epi_of_iso",
... | [] | apply epi_comp | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.MorphismProperty.Basic | {
"line": 710,
"column": 6
} | {
"line": 710,
"column": 20
} | {
"line": 712,
"column": 0
} | [
{
"pp": "case hpostcomp\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z : C\ne : Y ≅ Z\nf : X ⟶ Y\nx✝ : Epi f\n⊢ Epi (f ≫ e.hom)",
"ppTerm": "?hpostcomp",
"assigned": true,
"usedConstants": [
"CategoryTheory.epi_comp",
"CategoryTheory.Iso.isIso_hom",
"CategoryTheory.IsIso.epi_of_iso"... | [] | apply epi_comp | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Comma.Basic | {
"line": 304,
"column": 12
} | {
"line": 305,
"column": 64
} | {
"line": 306,
"column": 12
} | [
{
"pp": "A : Type u₁\ninst✝¹⁰ : Category.{v₁, u₁} A\nB : Type u₂\ninst✝⁹ : Category.{v₂, u₂} B\nT : Type u₃\ninst✝⁸ : Category.{v₃, u₃} T\nA' : Type u₄\ninst✝⁷ : Category.{v₄, u₄} A'\nB' : Type u₅\ninst✝⁶ : Category.{v₅, u₅} B'\nT' : Type u₆\ninst✝⁵ : Category.{v₆, u₆} T'\nL : A ⥤ T\nR : B ⥤ T\nL₁ L₂ L₃ : A ⥤ T... | [
"A : Type u₁\ninst✝¹⁰ : Category.{v₁, u₁} A\nB : Type u₂\ninst✝⁹ : Category.{v₂, u₂} B\nT : Type u₃\ninst✝⁸ : Category.{v₃, u₃} T\nA' : Type u₄\ninst✝⁷ : Category.{v₄, u₄} A'\nB' : Type u₅\ninst✝⁶ : Category.{v₅, u₅} B'\nT' : Type u₆\ninst✝⁵ : Category.{v₆, u₆} T'\nL : A ⥤ T\nR : B ⥤ T\nL₁ L₂ L₃ : A ⥤ T\nR₁ R₂ R₃ :... | simp only [NatIso.isIso_inv_app, Functor.comp_obj, Functor.map_preimage, assoc,
IsIso.inv_hom_id, comp_id, IsIso.hom_inv_id_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Yoneda | {
"line": 784,
"column": 62
} | {
"line": 785,
"column": 72
} | {
"line": 787,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y : Cᵒᵖ\nF : Cᵒᵖ ⥤ Type v₁\nf : yoneda.obj (unop X) ⟶ F\ng : X ⟶ Y\n⊢ (ConcreteCategory.hom (F.map g)) (yonedaEquiv f) = (ConcreteCategory.hom (f.app Y)) g.unop",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Categ... | [] | by
rw [yonedaEquiv_naturality', yonedaEquiv_comp, yonedaEquiv_yoneda_map] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Yoneda | {
"line": 944,
"column": 2
} | {
"line": 944,
"column": 34
} | {
"line": 945,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y : Cᵒᵖ\nf : X ⟶ Y\nF : Cᵒᵖ ⥤ Type (max w v₁)\nu : uliftYoneda.{w, v₁, u₁}.obj (unop X) ⟶ F\n⊢ uliftYonedaEquiv.symm ((ConcreteCategory.hom (F.map f)) (uliftYonedaEquiv u)) =\n uliftYoneda.{w, v₁, u₁}.map f.unop ≫ uliftYonedaEquiv.symm (uliftYonedaEquiv u)... | [
"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y : Cᵒᵖ\nf : X ⟶ Y\nF : Cᵒᵖ ⥤ Type (max w v₁)\nu : uliftYoneda.{w, v₁, u₁}.obj (unop X) ⟶ F\n⊢ uliftYonedaEquiv.symm (uliftYonedaEquiv (uliftYoneda.{w, v₁, u₁}.map f.unop ≫ u)) =\n uliftYoneda.{w, v₁, u₁}.map f.unop ≫ uliftYonedaEquiv.symm (uliftYonedaEquiv u)"
] | rw [uliftYonedaEquiv_naturality] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.LiftingProperties.Adjunction | {
"line": 55,
"column": 18
} | {
"line": 55,
"column": 68
} | {
"line": 56,
"column": 6
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Category.{v_2, u_2} D\nG : C ⥤ D\nF : D ⥤ C\nA B : C\nX Y : D\ni : A ⟶ B\np : X ⟶ Y\nu : G.obj A ⟶ X\nv : G.obj B ⟶ Y\nsq : CommSq u (G.map i) p v\nadj : G ⊣ F\nl : sq.LiftStruct\n⊢ i ≫ (adj.homEquiv B X) l.l = (adj.homEquiv A X) u",
... | [] | by rw [← adj.homEquiv_naturality_left, l.fac_left] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.LiftingProperties.Adjunction | {
"line": 101,
"column": 22
} | {
"line": 101,
"column": 76
} | {
"line": 101,
"column": 77
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Category.{v_2, u_2} D\nG : C ⥤ D\nF : D ⥤ C\nA B : C\nX Y : D\ni : A ⟶ B\np : X ⟶ Y\nu : A ⟶ F.obj X\nv : B ⟶ F.obj Y\nsq : CommSq u i (F.map p) v\nadj : G ⊣ F\nl : sq.LiftStruct\n⊢ (adj.homEquiv B X).symm l.l ≫ p = (adj.homEquiv B Y).... | [] | rw [← adj.homEquiv_naturality_right_symm, l.fac_right] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.LiftingProperties.Adjunction | {
"line": 101,
"column": 22
} | {
"line": 101,
"column": 76
} | {
"line": 101,
"column": 77
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Category.{v_2, u_2} D\nG : C ⥤ D\nF : D ⥤ C\nA B : C\nX Y : D\ni : A ⟶ B\np : X ⟶ Y\nu : A ⟶ F.obj X\nv : B ⟶ F.obj Y\nsq : CommSq u i (F.map p) v\nadj : G ⊣ F\nl : sq.LiftStruct\n⊢ (adj.homEquiv B X).symm l.l ≫ p = (adj.homEquiv B Y).... | [] | rw [← adj.homEquiv_naturality_right_symm, l.fac_right] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.LiftingProperties.Adjunction | {
"line": 101,
"column": 22
} | {
"line": 101,
"column": 76
} | {
"line": 101,
"column": 77
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Category.{v_2, u_2} D\nG : C ⥤ D\nF : D ⥤ C\nA B : C\nX Y : D\ni : A ⟶ B\np : X ⟶ Y\nu : A ⟶ F.obj X\nv : B ⟶ F.obj Y\nsq : CommSq u i (F.map p) v\nadj : G ⊣ F\nl : sq.LiftStruct\n⊢ (adj.homEquiv B X).symm l.l ≫ p = (adj.homEquiv B Y).... | [] | rw [← adj.homEquiv_naturality_right_symm, l.fac_right] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Functor.EpiMono | {
"line": 265,
"column": 2
} | {
"line": 268,
"column": 22
} | {
"line": 270,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y : C\nf : X ⟶ Y\nhF₁ : F.PreservesMonomorphisms\nhF₂ : F.ReflectsMonomorphisms\n⊢ Mono (F.map f) ↔ Mono f",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"CategoryTheory.Mono",... | [] | constructor
· exact F.mono_of_mono_map
· intro h
exact F.map_mono f | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Functor.EpiMono | {
"line": 265,
"column": 2
} | {
"line": 268,
"column": 22
} | {
"line": 270,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y : C\nf : X ⟶ Y\nhF₁ : F.PreservesMonomorphisms\nhF₂ : F.ReflectsMonomorphisms\n⊢ Mono (F.map f) ↔ Mono f",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"CategoryTheory.Mono",... | [] | constructor
· exact F.mono_of_mono_map
· intro h
exact F.map_mono f | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Products | {
"line": 836,
"column": 44
} | {
"line": 840,
"column": 48
} | {
"line": 842,
"column": 0
} | [
{
"pp": "β : Type w\nα : Type w₂\nγ : Type w₃\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y : C\ne : X ≅ Y\nJ : Type u_1\ninst✝ : Unique J\n⊢ IsLimit (mk X fun x ↦ e.hom)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Category.assoc",
"Inhabited.... | [] | by
refine Fan.IsLimit.mk _ (fun s ↦ s.proj default ≫ e.inv) (fun s j ↦ ?_) fun s m hm ↦ ?_
· obtain rfl : j = default := Subsingleton.elim _ _
simp
· simpa [← cancel_mono e.hom] using hm default | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Iso | {
"line": 228,
"column": 2
} | {
"line": 228,
"column": 87
} | {
"line": 230,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : C\nf : X ⟶ Y\ng : X ⟶ Z\ninst✝ : IsIso g\n⊢ pushout.inr f g ≫ inv (pushout.inl f g) = inv g ≫ f",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | [] | rw [IsIso.eq_inv_comp, ← pushout.condition_assoc, IsIso.hom_inv_id, Category.comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Iso | {
"line": 228,
"column": 2
} | {
"line": 228,
"column": 87
} | {
"line": 230,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : C\nf : X ⟶ Y\ng : X ⟶ Z\ninst✝ : IsIso g\n⊢ pushout.inr f g ≫ inv (pushout.inl f g) = inv g ≫ f",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | [] | rw [IsIso.eq_inv_comp, ← pushout.condition_assoc, IsIso.hom_inv_id, Category.comp_id] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Iso | {
"line": 228,
"column": 2
} | {
"line": 228,
"column": 87
} | {
"line": 230,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : C\nf : X ⟶ Y\ng : X ⟶ Z\ninst✝ : IsIso g\n⊢ pushout.inr f g ≫ inv (pushout.inl f g) = inv g ≫ f",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | [] | rw [IsIso.eq_inv_comp, ← pushout.condition_assoc, IsIso.hom_inv_id, Category.comp_id] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Equalizers | {
"line": 378,
"column": 2
} | {
"line": 378,
"column": 59
} | {
"line": 380,
"column": 0
} | [
{
"pp": "C : Type u\nX Y : C\ninst✝ : Category.{v, u} C\nf g : X ⟶ Y\ns : Fork f g\n⊢ s.π.app one = s.ι ≫ f",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.Cone.π",
"CategoryTheory.Functor",
"CategoryTheory.CategoryStruct.toQuiver",
... | [] | rw [← s.app_zero_eq_ι, ← s.w left, parallelPair_map_left] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Limits.Shapes.Equalizers | {
"line": 378,
"column": 2
} | {
"line": 378,
"column": 59
} | {
"line": 380,
"column": 0
} | [
{
"pp": "C : Type u\nX Y : C\ninst✝ : Category.{v, u} C\nf g : X ⟶ Y\ns : Fork f g\n⊢ s.π.app one = s.ι ≫ f",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.Cone.π",
"CategoryTheory.Functor",
"CategoryTheory.CategoryStruct.toQuiver",
... | [] | rw [← s.app_zero_eq_ι, ← s.w left, parallelPair_map_left] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Equalizers | {
"line": 378,
"column": 2
} | {
"line": 378,
"column": 59
} | {
"line": 380,
"column": 0
} | [
{
"pp": "C : Type u\nX Y : C\ninst✝ : Category.{v, u} C\nf g : X ⟶ Y\ns : Fork f g\n⊢ s.π.app one = s.ι ≫ f",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.Cone.π",
"CategoryTheory.Functor",
"CategoryTheory.CategoryStruct.toQuiver",
... | [] | rw [← s.app_zero_eq_ι, ← s.w left, parallelPair_map_left] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Images | {
"line": 515,
"column": 2
} | {
"line": 515,
"column": 16
} | {
"line": 517,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nX Y : C\nf : X ⟶ Y\ninst✝² : HasImage f\ninst✝¹ : Epi (image.ι f)\ninst✝ : Epi (factorThruImage f)\n⊢ Epi (factorThruImage f ≫ image.ι f)",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.factorThruImage",
"C... | [] | apply epi_comp | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.ObjectProperty.CompleteLattice | {
"line": 92,
"column": 4
} | {
"line": 92,
"column": 26
} | {
"line": 93,
"column": 4
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝ : Category.{v, u} C\nα : Sort u_1\nP : α → ObjectProperty C\na : α\n⊢ (P a).isoClosure ≤ (⨆ a, P a).isoClosure",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"iSup",
"Prop.le",
"PartialOrder.... | [
"case refine_2\nC : Type u\ninst✝ : Category.{v, u} C\nα : Sort u_1\nP : α → ObjectProperty C\na : α\n⊢ P a ≤ (⨆ a, P a).isoClosure"
] | rw [isoClosure_le_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Preadditive.Biproducts | {
"line": 365,
"column": 2
} | {
"line": 365,
"column": 38
} | {
"line": 365,
"column": 38
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nX Y : C\nt : BinaryBicone X Y\nht : IsLimit t.toCone\n⊢ t.inr = BinaryFan.IsLimit.lift ht 0 (𝟙 Y)",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.BinaryBicone.toCone",
"CategoryTheor... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nX Y : C\nt : BinaryBicone X Y\nht : IsLimit t.toCone\n⊢ ∀ (j : Discrete WalkingPair), t.inr ≫ t.toCone.π.app j = (BinaryFan.mk 0 (𝟙 Y)).π.app j"
] | apply ht.uniq (BinaryFan.mk 0 (𝟙 Y)) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Limits.Shapes.Biproducts | {
"line": 1042,
"column": 2
} | {
"line": 1044,
"column": 79
} | {
"line": 1046,
"column": 0
} | [
{
"pp": "J : Type w\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\nf : J → C\ninst✝ : HasBiproduct f\nb : Bicone f\nhb : b.IsBilimit\n⊢ (hb.isLimit.conePointUniqueUpToIso (isLimit f)).inv = desc b.ι",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | refine biproduct.hom_ext' _ _ fun j => hb.isLimit.hom_ext fun j' => ?_
rw [Category.assoc, IsLimit.conePointUniqueUpToIso_inv_comp, Bicone.toCone_π_app,
biproduct.bicone_π, biproduct.ι_desc, biproduct.ι_π, b.toCone_π_app, b.ι_π] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Biproducts | {
"line": 1042,
"column": 2
} | {
"line": 1044,
"column": 79
} | {
"line": 1046,
"column": 0
} | [
{
"pp": "J : Type w\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\nf : J → C\ninst✝ : HasBiproduct f\nb : Bicone f\nhb : b.IsBilimit\n⊢ (hb.isLimit.conePointUniqueUpToIso (isLimit f)).inv = desc b.ι",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | refine biproduct.hom_ext' _ _ fun j => hb.isLimit.hom_ext fun j' => ?_
rw [Category.assoc, IsLimit.conePointUniqueUpToIso_inv_comp, Bicone.toCone_π_app,
biproduct.bicone_π, biproduct.ι_desc, biproduct.ι_π, b.toCone_π_app, b.ι_π] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Biproducts | {
"line": 1067,
"column": 23
} | {
"line": 1070,
"column": 42
} | {
"line": 1072,
"column": 0
} | [
{
"pp": "J✝¹ : Type w\nC✝¹ : Type uC\ninst✝¹⁰ : Category.{uC', uC} C✝¹\ninst✝⁹ : HasZeroMorphisms C✝¹\nD✝ : Type uD\ninst✝⁸ : Category.{uD', uD} D✝\ninst✝⁷ : HasZeroMorphisms D✝\nF : J✝¹ → C✝¹\nJ✝ : Type w\nK : Type u_1\nC✝ : Type u\ninst✝⁶ : Category.{v, u} C✝\ninst✝⁵ : HasZeroMorphisms C✝\nJ : Type w\nC : Typ... | [] | by
refine ⟨⟨biproduct Empty.elim, fun X => ⟨⟨⟨0⟩, ?_⟩⟩, fun X => ⟨⟨⟨0⟩, ?_⟩⟩⟩⟩
· intro a; apply biproduct.hom_ext'; simp
· intro a; apply biproduct.hom_ext; simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Preadditive.Biproducts | {
"line": 929,
"column": 4
} | {
"line": 929,
"column": 13
} | {
"line": 930,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : Preadditive C\nD : Type u'\ninst✝⁶ : Category.{v', u'} D\ninst✝⁵ : Preadditive D\nF : C ⥤ D\ninst✝⁴ : F.PreservesZeroMorphisms\nJ : Type u_1\ninst✝³ : Finite J\nf : J → C\ninst✝² : HasBiproduct f\ninst✝¹ : HasBiproduct (F.obj ∘ f)\ninst✝ : Mono (F.biprod... | [
"C : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : Preadditive C\nD : Type u'\ninst✝⁶ : Category.{v', u'} D\ninst✝⁵ : Preadditive D\nF : C ⥤ D\ninst✝⁴ : F.PreservesZeroMorphisms\nJ : Type u_1\ninst✝³ : Finite J\nf : J → C\ninst✝² : HasBiproduct f\ninst✝¹ : HasBiproduct (F.obj ∘ f)\ninst✝ : Mono (F.biproductCompariso... | rw [that] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Enumerative.Composition | {
"line": 281,
"column": 17
} | {
"line": 283,
"column": 67
} | {
"line": 285,
"column": 0
} | [
{
"pp": "n : ℕ\nc : Composition n\n⊢ Fin.last n ∈ c.boundaries",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.univ",
"Composition.length",
"congrArg",
"_private.Mathlib.Combinatorics.Enumerative.Composition.0.Composition.toCompositionAsSet._... | [] | by
simp only [boundaries, Finset.mem_univ, Finset.mem_map]
exact ⟨Fin.last c.length, And.intro True.intro c.boundary_last⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Enumerative.Composition | {
"line": 350,
"column": 2
} | {
"line": 350,
"column": 53
} | {
"line": 352,
"column": 0
} | [
{
"pp": "n : ℕ\nc : Composition n\nj : Fin n\n⊢ ↑((c.embedding (c.index j)) (c.invEmbedding j)) = ↑j",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Nat.instCanonicallyOrderedAdd",
"Nat.instOrderedSub",
"Nat.instIsOrderedAddMonoid",
"Composition.length",
"Par... | [] | apply add_tsub_cancel_of_le (c.sizeUpTo_index_le j) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Combinatorics.Enumerative.Composition | {
"line": 364,
"column": 4
} | {
"line": 366,
"column": 17
} | {
"line": 367,
"column": 4
} | [
{
"pp": "case mpr.refine_1\nn : ℕ\nc : Composition n\nj : Fin n\ni : Fin c.length\nh : c.sizeUpTo ↑i ≤ ↑j ∧ ↑j < c.sizeUpTo (↑i).succ\n⊢ ↑j - c.sizeUpTo ↑i < c.blocksFun i",
"ppTerm": "?mpr.refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"... | [
"case mpr.refine_2\nn : ℕ\nc : Composition n\nj : Fin n\ni : Fin c.length\nh : c.sizeUpTo ↑i ≤ ↑j ∧ ↑j < c.sizeUpTo (↑i).succ\n⊢ (c.embedding i) ⟨↑j - c.sizeUpTo ↑i, ⋯⟩ = j"
] | · rw [tsub_lt_iff_left, ← sizeUpTo_succ']
· exact h.2
· exact h.1 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Preadditive.Biproducts | {
"line": 1044,
"column": 4
} | {
"line": 1044,
"column": 13
} | {
"line": 1045,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝⁷ : Category.{v, u} C\ninst✝⁶ : Preadditive C\nD : Type u'\ninst✝⁵ : Category.{v', u'} D\ninst✝⁴ : Preadditive D\nF : C ⥤ D\ninst✝³ : F.PreservesZeroMorphisms\nX Y : C\ninst✝² : HasBinaryBiproduct X Y\ninst✝¹ : HasBinaryBiproduct (F.obj X) (F.obj Y)\ninst✝ : Mono (F.biprodComparison X ... | [
"C : Type u\ninst✝⁷ : Category.{v, u} C\ninst✝⁶ : Preadditive C\nD : Type u'\ninst✝⁵ : Category.{v', u'} D\ninst✝⁴ : Preadditive D\nF : C ⥤ D\ninst✝³ : F.PreservesZeroMorphisms\nX Y : C\ninst✝² : HasBinaryBiproduct X Y\ninst✝¹ : HasBinaryBiproduct (F.obj X) (F.obj Y)\ninst✝ : Mono (F.biprodComparison X Y)\nthat :\n... | rw [that] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Perm.Basic | {
"line": 62,
"column": 32
} | {
"line": 62,
"column": 51
} | {
"line": 64,
"column": 0
} | [
{
"pp": "α : Type u_1\nf : Perm α\ns : Set α\n⊢ BijOn (⇑f) s s → ∀ (n : ℕ), BijOn (⇑f)^[n] s s",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Equiv.instEquivLike",
"Equiv.Perm",
"Set.BijOn.iterate",
"DFunLike.coe",
"EquivLike.toFunLike"
],
"usedFVars... | [] | exact BijOn.iterate | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.Perm.Support | {
"line": 231,
"column": 2
} | {
"line": 231,
"column": 34
} | {
"line": 232,
"column": 2
} | [
{
"pp": "α : Type u_1\np : Perm α\nn : ℤ\nx : α\n⊢ x ∈ {x | (p ^ n) x ≠ x} → x ∈ {x | p x ≠ x}",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Equiv.instEquivLike",
"setOf",
"DivInvMonoid.toZPow",
"Membership.mem",
"id",
"Ne",
"Int",
"Group.... | [
"α : Type u_1\np : Perm α\nn : ℤ\nx : α\n⊢ ¬(p ^ n) x = x → ¬p x = x"
] | simp only [Set.mem_setOf_eq, Ne] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Fintype.Inv | {
"line": 96,
"column": 2
} | {
"line": 98,
"column": 79
} | {
"line": 100,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq β\nf : α ↪ β\ninst✝ : Nonempty α\n⊢ (Set.range ⇑f).restrict (invFun ⇑f) = f.invOfMemRange",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Function.invFun",
"Function.Embedding.invOfMemRange",
"... | [] | ext ⟨b, h⟩
apply f.injective
simp [f.left_inv_of_invOfMemRange, @invFun_eq _ _ _ f b (Set.mem_range.mp h)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Fintype.Inv | {
"line": 96,
"column": 2
} | {
"line": 98,
"column": 79
} | {
"line": 100,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq β\nf : α ↪ β\ninst✝ : Nonempty α\n⊢ (Set.range ⇑f).restrict (invFun ⇑f) = f.invOfMemRange",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Function.invFun",
"Function.Embedding.invOfMemRange",
"... | [] | ext ⟨b, h⟩
apply f.injective
simp [f.left_inv_of_invOfMemRange, @invFun_eq _ _ _ f b (Set.mem_range.mp h)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Antidiag.Pi | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 51
} | {
"line": 218,
"column": 2
} | [
{
"pp": "ι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : DecidableEq (ι → ℕ)\ns : Finset ι\nm n : ℕ\nhn : n ≠ 0\nf : ι → ℕ\n⊢ (∃ y ∈ s.piAntidiag m, n • y = f) ↔ f ∈ {f ∈ s.piAntidiag (n * m) | ∀ i ∈ s, n ∣ f i}",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul... | [
"ι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : DecidableEq (ι → ℕ)\ns : Finset ι\nm n : ℕ\nhn : n ≠ 0\nf : ι → ℕ\n⊢ (∃ y, s.sum y = m ∧ (∀ (i : ι), y i ≠ 0 → i ∈ s) ∧ n • y = f) ↔\n s.sum f = n * m ∧ (∀ (i : ι), f i ≠ 0 → i ∈ s) ∧ ∀ i ∈ s, n ∣ f i"
] | simp only [mem_filter, mem_piAntidiag, and_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Logic.Equiv.Fintype | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 45
} | {
"line": 176,
"column": 2
} | [
{
"pp": "β : Type u_2\ns t : Finset β\nh : s.card = t.card\nσ : Perm β\nhσ : ∀ (a : ↥s), σ ↑a = ↑((Finset.equivOfCardEq h) a)\nb : β\nhb : b ∈ Finset.map (Equiv.toEmbedding σ) s\n⊢ b ∈ t",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"Finset",
"Finset.map",
"Membership.m... | [
"β : Type u_2\ns t : Finset β\nh : s.card = t.card\nσ : Perm β\nhσ : ∀ (a : ↥s), σ ↑a = ↑((Finset.equivOfCardEq h) a)\na : β\nha : a ∈ s\nhb : (Equiv.toEmbedding σ) a ∈ Finset.map (Equiv.toEmbedding σ) s\n⊢ (Equiv.toEmbedding σ) a ∈ t"
] | obtain ⟨a, ha, rfl⟩ := Finset.mem_map.mp hb | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.Enumerative.Composition | {
"line": 1006,
"column": 2
} | {
"line": 1006,
"column": 13
} | {
"line": 1007,
"column": 2
} | [
{
"pp": "n : ℕ\nc : CompositionAsSet n\n⊢ c.toComposition.boundaries = c.boundaries",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Fin.casesOn",
"Finset",
"Finset.ext",
"Membership.mem",
"instOfNatNat",
"CompositionAsSet.boundaries",
"instHAdd",
... | [
"n : ℕ\nc : CompositionAsSet n\nj : ℕ\nhj : j < n + 1\n⊢ ⟨j, hj⟩ ∈ c.toComposition.boundaries ↔ ⟨j, hj⟩ ∈ c.boundaries"
] | ext ⟨j, hj⟩ | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.GroupTheory.Perm.List | {
"line": 86,
"column": 2
} | {
"line": 88,
"column": 48
} | {
"line": 90,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nl l' : List α\n⊢ (zipWith swap l l').prod.support ≤ l.toFinset ⊔ l'.toFinset",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.support",
"List.zipWith",
"Lattice.toSemilatticeSup",... | [] | intro x hx
have hx' : x ∈ { x | (zipWith swap l l').prod x ≠ x } := by simpa using hx
simpa using zipWith_swap_prod_support' _ _ hx' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Perm.List | {
"line": 86,
"column": 2
} | {
"line": 88,
"column": 48
} | {
"line": 90,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nl l' : List α\n⊢ (zipWith swap l l').prod.support ≤ l.toFinset ⊔ l'.toFinset",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.support",
"List.zipWith",
"Lattice.toSemilatticeSup",... | [] | intro x hx
have hx' : x ∈ { x | (zipWith swap l l').prod x ≠ x } := by simpa using hx
simpa using zipWith_swap_prod_support' _ _ hx' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Perm.List | {
"line": 169,
"column": 6
} | {
"line": 169,
"column": 16
} | {
"line": 170,
"column": 4
} | [
{
"pp": "case succ.nil\nα : Type u_1\ninst✝ : DecidableEq α\nn : ℕ\nIH : ∀ (xs : List α), xs.Nodup → ∀ (hn : n + 1 < xs.length), xs.formPerm xs[n] = xs[n + 1]\nh : [].Nodup\nhn : n + 1 + 1 < [].length\n⊢ [].formPerm [][n + 1] = [][n + 1 + 1]",
"ppTerm": "?succ.nil",
"assigned": true,
"usedConstants"... | [] | simp at hn | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.Perm.List | {
"line": 169,
"column": 6
} | {
"line": 169,
"column": 16
} | {
"line": 170,
"column": 4
} | [
{
"pp": "case succ.nil\nα : Type u_1\ninst✝ : DecidableEq α\nn : ℕ\nIH : ∀ (xs : List α), xs.Nodup → ∀ (hn : n + 1 < xs.length), xs.formPerm xs[n] = xs[n + 1]\nh : [].Nodup\nhn : n + 1 + 1 < [].length\n⊢ [].formPerm [][n + 1] = [][n + 1 + 1]",
"ppTerm": "?succ.nil",
"assigned": true,
"usedConstants"... | [] | simp at hn | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Perm.List | {
"line": 169,
"column": 6
} | {
"line": 169,
"column": 16
} | {
"line": 170,
"column": 4
} | [
{
"pp": "case succ.nil\nα : Type u_1\ninst✝ : DecidableEq α\nn : ℕ\nIH : ∀ (xs : List α), xs.Nodup → ∀ (hn : n + 1 < xs.length), xs.formPerm xs[n] = xs[n + 1]\nh : [].Nodup\nhn : n + 1 + 1 < [].length\n⊢ [].formPerm [][n + 1] = [][n + 1 + 1]",
"ppTerm": "?succ.nil",
"assigned": true,
"usedConstants"... | [] | simp at hn | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Perm.Sign | {
"line": 256,
"column": 6
} | {
"line": 256,
"column": 75
} | {
"line": 257,
"column": 4
} | [
{
"pp": "case inr.inl\nn : ℕ\na₁ : Fin (n + 2)\nH : 0 < a₁\nha₁ : 0 < a₁\nha₂ : ¬a₁ = 1\nthis : 1 < a₁\nh01 : (swap 0 1) 0 = 1\n⊢ (if (swap 0 1) a₁ ≤ (swap 0 1) 0 then -1 else 1) = 1",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
"Int.instCommMonoid",
"Eq.mpr",
"instN... | [] | rw [swap_apply_of_ne_of_ne (ne_of_gt H) ha₂, h01, if_neg this.not_ge] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Perm.Closure | {
"line": 85,
"column": 4
} | {
"line": 86,
"column": 19
} | {
"line": 87,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ : Perm α\nh1 : σ.IsCycle\nh2 : σ.support = univ\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (σ x)}\nh3 : σ ∈ H\nh4 : swap x (σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap ((σ ^ n) x) ((σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x ((σ ^ ... | [
"case neg\nα : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ : Perm α\nh1 : σ.IsCycle\nh2 : σ.support = univ\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (σ x)}\nh3 : σ ∈ H\nh4 : swap x (σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap ((σ ^ n) x) ((σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x ((σ ^ n) x) ∈ H\ns... | · rw [h5, swap_comm]
exact step3 y | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.Perm.Cycle.Basic | {
"line": 337,
"column": 6
} | {
"line": 343,
"column": 50
} | {
"line": 344,
"column": 6
} | [
{
"pp": "case left\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nf g : Perm α\nx y : α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ : Perm α\nhσ : σ.IsCycle\n⊢ Injective fun τ ↦ ⟨↑τ (Classical.choose hσ), ⋯⟩",
"ppTerm": "?left",
"assigned": true,
"usedConstants": [
"Subtype.coe_mk",
"Eq.mp... | [
"case right\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nf g : Perm α\nx y : α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ : Perm α\nhσ : σ.IsCycle\n⊢ Surjective fun τ ↦ ⟨↑τ (Classical.choose hσ), ⋯⟩"
] | · rintro ⟨a, m, rfl⟩ ⟨b, n, rfl⟩ h
ext y
by_cases hy : σ y = y
· simp_rw [zpow_apply_eq_self_of_apply_eq_self hy]
· obtain ⟨i, rfl⟩ := (Classical.choose_spec hσ).2 hy
rw [Subtype.coe_mk, Subtype.coe_mk, zpow_apply_comm σ m i, zpow_apply_comm σ n i]
exact congr_arg _ (... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.Perm.Cycle.Basic | {
"line": 432,
"column": 2
} | {
"line": 432,
"column": 84
} | {
"line": 433,
"column": 2
} | [
{
"pp": "case refine_2\nα : Type u_4\ninst✝ : DecidableEq α\nf : Perm α\nhf : f.IsCycle\nx : α\nhx : f x ≠ x\nhffx : f (f x) ≠ x\ny : α\nhy : (swap x (f x) * f) y ≠ y\n⊢ (swap x (f x) * f).SameCycle (f x) y",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Equiv.instEquivLike",
... | [
"case refine_2\nα : Type u_4\ninst✝ : DecidableEq α\nf : Perm α\nhf : f.IsCycle\nx : α\nhx : f x ≠ x\nhffx : f (f x) ≠ x\ni : ℤ\nhy : (swap x (f x) * f) ((f ^ i) x) ≠ (f ^ i) x\n⊢ (swap x (f x) * f).SameCycle (f x) ((f ^ i) x)"
] | obtain ⟨i, rfl⟩ := hf.exists_zpow_eq hx (ne_and_ne_of_swap_mul_apply_ne_self hy).1 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.GroupTheory.Perm.Cycle.Basic | {
"line": 518,
"column": 2
} | {
"line": 528,
"column": 19
} | {
"line": 530,
"column": 0
} | [
{
"pp": "case mpr\nα : Type u_2\nf : Perm α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhf : f.IsCycle\nn : ℕ\n⊢ ¬f ^ n = 1 → (f ^ n).support = f.support",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.support",
"MulOne.toOne",
"Equiv.Perm.IsCyc... | [] | · intro H
apply le_antisymm (support_pow_le _ n) _
intro x hx
contrapose H
ext z
by_cases hz : f z = z
· rw [pow_apply_eq_self_of_apply_eq_self hz, one_apply]
· obtain ⟨k, rfl⟩ := hf.exists_pow_eq hz (mem_support.mp hx)
apply (f ^ k).injective
rw [← mul_apply, (Commute.pow_pow_se... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.Perm.Sign | {
"line": 609,
"column": 2
} | {
"line": 609,
"column": 26
} | {
"line": 610,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ : Perm α\nhσ : σ ∈ ofSign 1\nhτ : σ ∈ ofSign (-1)\n⊢ False",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"MonoidHom.instFunLike",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalComm... | [
"α : Type u\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ : Perm α\nhσ : sign σ = 1\nhτ : sign σ = -1\n⊢ False"
] | rw [mem_ofSign] at hσ hτ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Perm.Cycle.Factors | {
"line": 169,
"column": 6
} | {
"line": 169,
"column": 33
} | {
"line": 169,
"column": 34
} | [
{
"pp": "α : Type u_2\nf : Perm α\ninst✝ : DecidableRel f.SameCycle\nn : ℕ\nx : α\n⊢ (f ^ (n % orderOf (f.cycleOf x))) x = (f ^ n) x",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"congrArg",
"Equiv.Perm.instPowNat",
"Equiv.Pe... | [
"α : Type u_2\nf : Perm α\ninst✝ : DecidableRel f.SameCycle\nn : ℕ\nx : α\n⊢ (f.cycleOf x ^ (n % orderOf (f.cycleOf x))) x = (f ^ n) x"
] | ← cycleOf_pow_apply_self f, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.Cycle.Factors | {
"line": 169,
"column": 34
} | {
"line": 169,
"column": 61
} | {
"line": 169,
"column": 62
} | [
{
"pp": "α : Type u_2\nf : Perm α\ninst✝ : DecidableRel f.SameCycle\nn : ℕ\nx : α\n⊢ (f.cycleOf x ^ (n % orderOf (f.cycleOf x))) x = (f ^ n) x",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"congrArg",
"Equiv.Perm.instPowNat",
... | [
"α : Type u_2\nf : Perm α\ninst✝ : DecidableRel f.SameCycle\nn : ℕ\nx : α\n⊢ (f.cycleOf x ^ (n % orderOf (f.cycleOf x))) x = (f.cycleOf x ^ n) x"
] | ← cycleOf_pow_apply_self f, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.Cycle.Factors | {
"line": 269,
"column": 53
} | {
"line": 269,
"column": 70
} | {
"line": 269,
"column": 70
} | [
{
"pp": "α : Type u_2\nf : Perm α\nx : α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n⊢ 0 < #(f.cycleOf x).support ↔ 2 ≤ #(f.cycleOf x).support",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Equiv.Perm.instDecidableRelSameCycle",
"Eq.mpr",
"Equiv.Perm.support",
"co... | [
"α : Type u_2\nf : Perm α\nx : α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n⊢ Nat.succ 0 ≤ #(f.cycleOf x).support ↔ 2 ≤ #(f.cycleOf x).support"
] | ← Nat.succ_le_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.Cycle.Factors | {
"line": 315,
"column": 34
} | {
"line": 315,
"column": 61
} | {
"line": 315,
"column": 62
} | [
{
"pp": "case neg\nα : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf : Perm α\nn : ℕ\nx : α\nhx : ¬f x = x\n⊢ (f.cycleOf x ^ (n % #(f.cycleOf x).support)) x = (f ^ n) x",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Equiv.Perm.instDecidableRelSameCycle",
"Eq.mpr",
... | [
"case neg\nα : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf : Perm α\nn : ℕ\nx : α\nhx : ¬f x = x\n⊢ (f.cycleOf x ^ (n % #(f.cycleOf x).support)) x = (f.cycleOf x ^ n) x"
] | ← cycleOf_pow_apply_self f, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.Fin | {
"line": 237,
"column": 4
} | {
"line": 238,
"column": 28
} | {
"line": 239,
"column": 2
} | [
{
"pp": "case succ.inl.h\nn✝ : ℕ\ni j : Fin (n✝ + 1)\nhlt : j < i\nthis : (j + 1).castSucc = j.succ\n⊢ (j + 1).castSucc < i.succ",
"ppTerm": "?succ.inl.h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instNeZeroNatHAdd_1",
"Preorder.toLT",
"Nat.instOne",
"Fin.succ",
... | [] | · rw [Fin.lt_def]
simpa [this] using hlt | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.Perm.Cycle.Factors | {
"line": 620,
"column": 4
} | {
"line": 623,
"column": 33
} | {
"line": 624,
"column": 2
} | [
{
"pp": "case mp\nα : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\nx : α\n⊢ x ∈ g.support → ∃ c ∈ g.cycleFactorsFinset, x ∈ c.support",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Equiv.Perm.instDecidableRelSameCycle",
"Eq.mpr",
"Equi... | [] | intro h
use g.cycleOf x, cycleOf_mem_cycleFactorsFinset_iff.mpr h
rw [mem_support_cycleOf_iff]
exact ⟨SameCycle.refl g x, h⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Perm.Cycle.Factors | {
"line": 620,
"column": 4
} | {
"line": 623,
"column": 33
} | {
"line": 624,
"column": 2
} | [
{
"pp": "case mp\nα : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\nx : α\n⊢ x ∈ g.support → ∃ c ∈ g.cycleFactorsFinset, x ∈ c.support",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Equiv.Perm.instDecidableRelSameCycle",
"Eq.mpr",
"Equi... | [] | intro h
use g.cycleOf x, cycleOf_mem_cycleFactorsFinset_iff.mpr h
rw [mem_support_cycleOf_iff]
exact ⟨SameCycle.refl g x, h⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Perm.Cycle.Factors | {
"line": 624,
"column": 4
} | {
"line": 625,
"column": 49
} | {
"line": 627,
"column": 0
} | [
{
"pp": "case mpr\nα : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\nx : α\n⊢ (∃ c ∈ g.cycleFactorsFinset, x ∈ c.support) → x ∈ g.support",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Equiv.Perm.support",
"Finset",
"Equiv.Perm.cycleFactorsFinset",
... | [] | rintro ⟨c, hc, hx⟩
exact mem_cycleFactorsFinset_support_le hc hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Perm.Cycle.Factors | {
"line": 624,
"column": 4
} | {
"line": 625,
"column": 49
} | {
"line": 627,
"column": 0
} | [
{
"pp": "case mpr\nα : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\nx : α\n⊢ (∃ c ∈ g.cycleFactorsFinset, x ∈ c.support) → x ∈ g.support",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Equiv.Perm.support",
"Finset",
"Equiv.Perm.cycleFactorsFinset",
... | [] | rintro ⟨c, hc, hx⟩
exact mem_cycleFactorsFinset_support_le hc hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Perm.Cycle.Factors | {
"line": 734,
"column": 71
} | {
"line": 734,
"column": 76
} | {
"line": 735,
"column": 4
} | [
{
"pp": "α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng k c : Perm α\nimp_lemma : ∀ {g k c : Perm α}, c ∈ g.cycleFactorsFinset → k * c * k⁻¹ ∈ (k * g * k⁻¹).cycleFactorsFinset\nh : k * c * k⁻¹ ∈ (k * g * k⁻¹).cycleFactorsFinset\nx✝ : Perm α\n⊢ x✝ = k⁻¹ * (k * x✝ * k⁻¹) * k",
"ppTerm": "?m.99",
... | [] | group | Mathlib.Tactic.Group._aux_Mathlib_Tactic_Group___macroRules_Mathlib_Tactic_Group_group_1 | Mathlib.Tactic.Group.group |
Mathlib.GroupTheory.Perm.Cycle.Factors | {
"line": 734,
"column": 71
} | {
"line": 734,
"column": 76
} | {
"line": 735,
"column": 4
} | [
{
"pp": "α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng k c : Perm α\nimp_lemma : ∀ {g k c : Perm α}, c ∈ g.cycleFactorsFinset → k * c * k⁻¹ ∈ (k * g * k⁻¹).cycleFactorsFinset\nh : k * c * k⁻¹ ∈ (k * g * k⁻¹).cycleFactorsFinset\nx✝ : Perm α\n⊢ x✝ = k⁻¹ * (k * x✝ * k⁻¹) * k",
"ppTerm": "?m.99",
... | [] | group | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Perm.Cycle.Factors | {
"line": 734,
"column": 71
} | {
"line": 734,
"column": 76
} | {
"line": 735,
"column": 4
} | [
{
"pp": "α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng k c : Perm α\nimp_lemma : ∀ {g k c : Perm α}, c ∈ g.cycleFactorsFinset → k * c * k⁻¹ ∈ (k * g * k⁻¹).cycleFactorsFinset\nh : k * c * k⁻¹ ∈ (k * g * k⁻¹).cycleFactorsFinset\nx✝ : Perm α\n⊢ x✝ = k⁻¹ * (k * x✝ * k⁻¹) * k",
"ppTerm": "?m.99",
... | [] | group | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Perm.Fin | {
"line": 461,
"column": 2
} | {
"line": 461,
"column": 68
} | {
"line": 462,
"column": 2
} | [
{
"pp": "case inl\nn : ℕ\ni j k : Fin n\ninst✝ : NeZero n\nhij : i ≤ j\nhjk : j ≤ k\nx : Fin n\nch : x < i\n⊢ ↑((⇑(i.cycleIcc j) ∘ ⇑(j.cycleIcc k)) x) = ↑((i.cycleIcc k) x)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Equiv.instEquivLike",
"congrArg",
"PartialOrder.toP... | [
"case inr\nn : ℕ\ni j k : Fin n\ninst✝ : NeZero n\nhij : i ≤ j\nhjk : j ≤ k\nx : Fin n\nch : i ≤ x\n⊢ ↑((⇑(i.cycleIcc j) ∘ ⇑(j.cycleIcc k)) x) = ↑((i.cycleIcc k) x)"
] | · simp [cycleIcc_of_lt (lt_of_lt_of_le ch hij), cycleIcc_of_lt ch] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.Perm.Cycle.Type | {
"line": 593,
"column": 8
} | {
"line": 593,
"column": 46
} | {
"line": 593,
"column": 47
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ τ : Perm α\nh : σ.partition = τ.partition\n⊢ σ.cycleType = τ.cycleType",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.cycleType",
"congrArg",
"Nat.Partition... | [
"case refine_2\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ τ : Perm α\nh : σ.partition = τ.partition\n⊢ Multiset.filter (fun n ↦ 2 ≤ n) σ.partition.parts = τ.cycleType"
] | ← filter_parts_partition_eq_cycleType, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.Cycle.Type | {
"line": 593,
"column": 47
} | {
"line": 593,
"column": 85
} | {
"line": 593,
"column": 86
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ τ : Perm α\nh : σ.partition = τ.partition\n⊢ Multiset.filter (fun n ↦ 2 ≤ n) σ.partition.parts = τ.cycleType",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.cycleType",
... | [
"case refine_2\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ τ : Perm α\nh : σ.partition = τ.partition\n⊢ Multiset.filter (fun n ↦ 2 ≤ n) σ.partition.parts = Multiset.filter (fun n ↦ 2 ≤ n) τ.partition.parts"
] | ← filter_parts_partition_eq_cycleType, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Multilinear.DFinsupp | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 34
} | {
"line": 153,
"column": 2
} | [
{
"pp": "ι : Type uι\nκ : ι → Type uκ\nR : Type uR\nM : (i : ι) → κ i → Type uM\nN : ((i : ι) → κ i) → Type uN\ninst✝⁷ : DecidableEq ι\ninst✝⁶ : Fintype ι\ninst✝⁵ : Semiring R\ninst✝⁴ : (i : ι) → (k : κ i) → AddCommMonoid (M i k)\ninst✝³ : (p : (i : ι) → κ i) → AddCommMonoid (N p)\ninst✝² : (i : ι) → (k : κ i) ... | [
"case inl\nι : Type uι\nκ : ι → Type uκ\nR : Type uR\nM : (i : ι) → κ i → Type uM\nN : ((i : ι) → κ i) → Type uN\ninst✝⁷ : DecidableEq ι\ninst✝⁶ : Fintype ι\ninst✝⁵ : Semiring R\ninst✝⁴ : (i : ι) → (k : κ i) → AddCommMonoid (M i k)\ninst✝³ : (p : (i : ι) → κ i) → AddCommMonoid (N p)\ninst✝² : (i : ι) → (k : κ i) → ... | obtain rfl | hp := eq_or_ne p p' | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.LinearAlgebra.Matrix.SemiringInverse | {
"line": 49,
"column": 2
} | {
"line": 49,
"column": 24
} | {
"line": 50,
"column": 2
} | [
{
"pp": "n : Type u_1\nR : Type u_3\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommSemiring R\nd : n → R\n⊢ detp (-1) (diagonal d) = 0",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Equiv.instEquivLike... | [
"n : Type u_1\nR : Type u_3\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommSemiring R\nd : n → R\n⊢ ∀ x ∈ ofSign (-1), ∏ k, diagonal d k (x k) = 0"
] | rw [detp, sum_eq_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Matrix.SemiringInverse | {
"line": 127,
"column": 4
} | {
"line": 127,
"column": 50
} | {
"line": 128,
"column": 4
} | [
{
"pp": "case refine_1\nn : Type u_1\nR : Type u_3\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommSemiring R\nA : Matrix n n R\ni j : n\nh : i ≠ j\nf : (_ : n) × Perm n → (_ : n) × Perm n :=\n fun x ↦\n match x with\n | ⟨x, σ⟩ => ⟨σ i, σ * swap i j⟩\nt : ℤˣ → Finset ((_ : n) × Perm n) := fun s ... | [
"case refine_1\nn : Type u_1\nR : Type u_3\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommSemiring R\nA : Matrix n n R\ni j : n\nh : i ≠ j\nf : (_ : n) × Perm n → (_ : n) × Perm n :=\n fun x ↦\n match x with\n | ⟨x, σ⟩ => ⟨σ i, σ * swap i j⟩\nt : ℤˣ → Finset ((_ : n) × Perm n) := fun s ↦ univ.sigma... | rw [mem_sigma, mem_filter, mem_ofSign] at hp ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Matrix.SemiringInverse | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 50
} | {
"line": 132,
"column": 4
} | [
{
"pp": "case refine_2\nn : Type u_1\nR : Type u_3\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommSemiring R\nA : Matrix n n R\ni j : n\nh : i ≠ j\nf : (_ : n) × Perm n → (_ : n) × Perm n :=\n fun x ↦\n match x with\n | ⟨x, σ⟩ => ⟨σ i, σ * swap i j⟩\nt : ℤˣ → Finset ((_ : n) × Perm n) := fun s ... | [
"case refine_2\nn : Type u_1\nR : Type u_3\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommSemiring R\nA : Matrix n n R\ni j : n\nh : i ≠ j\nf : (_ : n) × Perm n → (_ : n) × Perm n :=\n fun x ↦\n match x with\n | ⟨x, σ⟩ => ⟨σ i, σ * swap i j⟩\nt : ℤˣ → Finset ((_ : n) × Perm n) := fun s ↦ univ.sigma... | rw [mem_sigma, mem_filter, mem_ofSign] at hp ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Multilinear.Basic | {
"line": 500,
"column": 6
} | {
"line": 500,
"column": 76
} | {
"line": 501,
"column": 6
} | [
{
"pp": "R : Type uR\nι : Type uι\nM₁ : ι → Type v₁\nM₂ : Type v₂\ninst✝⁶ : Semiring R\ninst✝⁵ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : (i : ι) → Module R (M₁ i)\ninst✝² : Module R M₂\nf : MultilinearMap R M₁ M₂\nα : ι → Type u_1\ng : (i : ι) → α i → M₁ i\ninst✝¹ : DecidableEq ι\nin... | [
"R : Type uR\nι : Type uι\nM₁ : ι → Type v₁\nM₂ : Type v₂\ninst✝⁶ : Semiring R\ninst✝⁵ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : (i : ι) → Module R (M₁ i)\ninst✝² : Module R M₂\nf : MultilinearMap R M₁ M₂\nα : ι → Type u_1\ng : (i : ι) → α i → M₁ i\ninst✝¹ : DecidableEq ι\ninst✝ : Fintyp... | have pos : #(A i) ≠ 0 := by rw [Finset.card_ne_zero]; exact Ai_empty i | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.Matrix.SemiringInverse | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 26
} | {
"line": 167,
"column": 2
} | [
{
"pp": "n : Type u_1\nR : Type u_3\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommSemiring R\nA B : Matrix n n R\nd : n → R\nhAB : A * B = diagonal d\ns t : ℤˣ\nh : s ≠ t\nσ : Perm n\nhσ : σ ∈ ofSign s\nτ : Perm n\nhτ : τ ∈ ofSign t\n⊢ IsAddUnit ((∏ k, A k (σ k)) * ∏ k, B k (τ k))",
"ppTerm": "?m... | [
"n : Type u_1\nR : Type u_3\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommSemiring R\nA B : Matrix n n R\nd : n → R\nhAB : A * B = diagonal d\ns t : ℤˣ\nh : s ≠ t\nσ : Perm n\nhσ : sign σ = s\nτ : Perm n\nhτ : sign τ = t\n⊢ IsAddUnit ((∏ k, A k (σ k)) * ∏ k, B k (τ k))"
] | rw [mem_ofSign] at hσ hτ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Category.Ring.Colimits | {
"line": 281,
"column": 18
} | {
"line": 281,
"column": 27
} | {
"line": 282,
"column": 4
} | [
{
"pp": "case neg\nJ : Type v\ninst✝ : SmallCategory J\nF : J ⥤ RingCat\ns : Cocone F\nm : (colimitCocone F).pt ⟶ s.pt\nw : ∀ (j : J), (colimitCocone F).ι.app j ≫ m = s.ι.app j\nx✝ : ↑(colimitCocone F).pt\nx : Prequotient F\nih : (Hom.hom m) (Quot.mk (⇑(colimitSetoid F)) x) = (Hom.hom (descMorphism F s)) (Quot.... | [] | simp [ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Category.Ring.Colimits | {
"line": 281,
"column": 18
} | {
"line": 281,
"column": 27
} | {
"line": 282,
"column": 4
} | [
{
"pp": "case neg\nJ : Type v\ninst✝ : SmallCategory J\nF : J ⥤ RingCat\ns : Cocone F\nm : (colimitCocone F).pt ⟶ s.pt\nw : ∀ (j : J), (colimitCocone F).ι.app j ≫ m = s.ι.app j\nx✝ : ↑(colimitCocone F).pt\nx : Prequotient F\nih : (Hom.hom m) (Quot.mk (⇑(colimitSetoid F)) x) = (Hom.hom (descMorphism F s)) (Quot.... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.Ring.Colimits | {
"line": 281,
"column": 18
} | {
"line": 281,
"column": 27
} | {
"line": 282,
"column": 4
} | [
{
"pp": "case neg\nJ : Type v\ninst✝ : SmallCategory J\nF : J ⥤ RingCat\ns : Cocone F\nm : (colimitCocone F).pt ⟶ s.pt\nw : ∀ (j : J), (colimitCocone F).ι.app j ≫ m = s.ι.app j\nx✝ : ↑(colimitCocone F).pt\nx : Prequotient F\nih : (Hom.hom m) (Quot.mk (⇑(colimitSetoid F)) x) = (Hom.hom (descMorphism F s)) (Quot.... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.SemiringInverse | {
"line": 187,
"column": 2
} | {
"line": 187,
"column": 26
} | {
"line": 188,
"column": 2
} | [
{
"pp": "n : Type u_1\nR : Type u_3\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommSemiring R\nA B : Matrix n n R\nd : n → R\nhAB : A * B = diagonal d\ns t : ℤˣ\nh : s ≠ t\ni j k : n\nhk : k ∈ univ\nσ : Perm n\nhσ : σ ∈ ofSign t ∧ σ j = k\nτ : Perm n\nhτ : τ ∈ ofSign s\n⊢ IsAddUnit ((∏ k, A k (τ k)) *... | [
"n : Type u_1\nR : Type u_3\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommSemiring R\nA B : Matrix n n R\nd : n → R\nhAB : A * B = diagonal d\ns t : ℤˣ\nh : s ≠ t\ni j k : n\nhk : k ∈ univ\nσ : Perm n\nhσ : sign σ = t ∧ σ j = k\nτ : Perm n\nhτ : sign τ = s\n⊢ IsAddUnit ((∏ k, A k (τ k)) * (B i k * ∏ k ∈ ... | rw [mem_ofSign] at hσ hτ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Matrix.SemiringInverse | {
"line": 195,
"column": 2
} | {
"line": 195,
"column": 28
} | {
"line": 196,
"column": 2
} | [
{
"pp": "n : Type u_1\nR : Type u_3\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommSemiring R\nA B : Matrix n n R\nd : n → R\nhAB : A * B = diagonal d\ns t : ℤˣ\ni j k : n\nhk : k ∈ univ\nσ : Perm n\nhσ : sign σ = t ∧ σ j = k\nτ : Perm n\nhτ : sign τ = s\nh : ¬σ * τ = 1\nl : n\nhl1 : l ≠ (σ * τ) l\nhl... | [
"n : Type u_1\nR : Type u_3\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommSemiring R\nA B : Matrix n n R\nd : n → R\nhAB : A * B = diagonal d\ns t : ℤˣ\ni j k : n\nhk : k ∈ univ\nσ : Perm n\nhσ : sign σ = t ∧ σ j = k\nτ : Perm n\nhτ : sign τ = s\nh : ¬σ * τ = 1\nl : n\nhl1 : l ≠ (σ * τ) l\nhl2 : l ∈ {τ⁻¹... | apply IsAddUnit.smul_right | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Category.Ring.Colimits | {
"line": 582,
"column": 18
} | {
"line": 582,
"column": 27
} | {
"line": 583,
"column": 4
} | [
{
"pp": "case neg\nJ : Type v\ninst✝ : SmallCategory J\nF : J ⥤ CommRingCat\ns : Cocone F\nm : (colimitCocone F).pt ⟶ s.pt\nw : ∀ (j : J), (colimitCocone F).ι.app j ≫ m = s.ι.app j\nx✝ : ↑(colimitCocone F).pt\nx : Prequotient F\nih : (Hom.hom m) (Quot.mk (⇑(colimitSetoid F)) x) = (Hom.hom (descMorphism F s)) (Q... | [] | simp [ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Category.Ring.Colimits | {
"line": 582,
"column": 18
} | {
"line": 582,
"column": 27
} | {
"line": 583,
"column": 4
} | [
{
"pp": "case neg\nJ : Type v\ninst✝ : SmallCategory J\nF : J ⥤ CommRingCat\ns : Cocone F\nm : (colimitCocone F).pt ⟶ s.pt\nw : ∀ (j : J), (colimitCocone F).ι.app j ≫ m = s.ι.app j\nx✝ : ↑(colimitCocone F).pt\nx : Prequotient F\nih : (Hom.hom m) (Quot.mk (⇑(colimitSetoid F)) x) = (Hom.hom (descMorphism F s)) (Q... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.Ring.Colimits | {
"line": 582,
"column": 18
} | {
"line": 582,
"column": 27
} | {
"line": 583,
"column": 4
} | [
{
"pp": "case neg\nJ : Type v\ninst✝ : SmallCategory J\nF : J ⥤ CommRingCat\ns : Cocone F\nm : (colimitCocone F).pt ⟶ s.pt\nw : ∀ (j : J), (colimitCocone F).ι.app j ≫ m = s.ι.app j\nx✝ : ↑(colimitCocone F).pt\nx : Prequotient F\nih : (Hom.hom m) (Quot.mk (⇑(colimitSetoid F)) x) = (Hom.hom (descMorphism F s)) (Q... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Filtered.Basic | {
"line": 447,
"column": 24
} | {
"line": 447,
"column": 53
} | {
"line": 447,
"column": 54
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : IsFilteredOrEmpty C\ni j j' : C\nf : i ⟶ j\nf' : i ⟶ j'\nK : C\nG : j ⟶ K\nG' : j' ⟶ K\nh✝ : True\nk : C\ne : K ⟶ k\nhe : (f ≫ G) ≫ e = (f' ≫ G') ≫ e\n⊢ f ≫ G ≫ e = f' ≫ G' ≫ e",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"E... | [] | simpa only [← Category.assoc] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Filtered.Basic | {
"line": 447,
"column": 24
} | {
"line": 447,
"column": 53
} | {
"line": 447,
"column": 54
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : IsFilteredOrEmpty C\ni j j' : C\nf : i ⟶ j\nf' : i ⟶ j'\nK : C\nG : j ⟶ K\nG' : j' ⟶ K\nh✝ : True\nk : C\ne : K ⟶ k\nhe : (f ≫ G) ≫ e = (f' ≫ G') ≫ e\n⊢ f ≫ G ≫ e = f' ≫ G' ≫ e",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"E... | [] | simpa only [← Category.assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Filtered.Basic | {
"line": 447,
"column": 24
} | {
"line": 447,
"column": 53
} | {
"line": 447,
"column": 54
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : IsFilteredOrEmpty C\ni j j' : C\nf : i ⟶ j\nf' : i ⟶ j'\nK : C\nG : j ⟶ K\nG' : j' ⟶ K\nh✝ : True\nk : C\ne : K ⟶ k\nhe : (f ≫ G) ≫ e = (f' ≫ G') ≫ e\n⊢ f ≫ G ≫ e = f' ≫ G' ≫ e",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"E... | [] | simpa only [← Category.assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Types.ColimitType | {
"line": 358,
"column": 4
} | {
"line": 361,
"column": 7
} | {
"line": 363,
"column": 0
} | [
{
"pp": "J : Type u\ninst✝ : Category.{v, u} J\nF : J ⥤ Type w₀\n⊢ Function.Bijective (F.descColimitType F.coconeTypes)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor.ιColimitType_jointly_surjective",
"Function.bijective_id",
"HEq.... | [] | convert! Function.bijective_id
ext y
obtain ⟨j, x, rfl⟩ := F.ιColimitType_jointly_surjective y
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Types.ColimitType | {
"line": 358,
"column": 4
} | {
"line": 361,
"column": 7
} | {
"line": 363,
"column": 0
} | [
{
"pp": "J : Type u\ninst✝ : Category.{v, u} J\nF : J ⥤ Type w₀\n⊢ Function.Bijective (F.descColimitType F.coconeTypes)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor.ιColimitType_jointly_surjective",
"Function.bijective_id",
"HEq.... | [] | convert! Function.bijective_id
ext y
obtain ⟨j, x, rfl⟩ := F.ιColimitType_jointly_surjective y
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.MonCat.FilteredColimits | {
"line": 186,
"column": 24
} | {
"line": 188,
"column": 58
} | {
"line": 189,
"column": 4
} | [
{
"pp": "J : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nx : M F\n⊢ 1 * x = x",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonCat.FilteredColimits.colimit_mul_mk_eq'",
"MulOne.toOne",
"Mul.mk",
"One.mk",
"HMul.hM... | [] | by
obtain ⟨j, x, rfl⟩ := x.mk_surjective
rw [colimit_one_eq F j, colimit_mul_mk_eq', one_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.Grp.FilteredColimits | {
"line": 77,
"column": 2
} | {
"line": 80,
"column": 18
} | {
"line": 82,
"column": 0
} | [
{
"pp": "J : Type v\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF : J ⥤ GrpCat\nj : J\nx y : ↑(F.obj j)\n⊢ G.mk F ⟨j, x⟩ * G.mk F ⟨j, y⟩ = G.mk F ⟨j, x * y⟩",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"GrpCat.instConcreteCategoryMonoidHomCarrier",
"GrpCat",
"Mono... | [] | #adaptation_note /-- Prior to leanprover/lean4#12564, this was just
`simpa using colimit_mul_mk_eq F ⟨j, x⟩ ⟨j, y⟩ j (𝟙 _) (𝟙 _)` -/
have := colimit_mul_mk_eq F ⟨j, x⟩ ⟨j, y⟩ j (𝟙 _) (𝟙 _)
simpa using this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.Grp.FilteredColimits | {
"line": 77,
"column": 2
} | {
"line": 80,
"column": 18
} | {
"line": 82,
"column": 0
} | [
{
"pp": "J : Type v\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF : J ⥤ GrpCat\nj : J\nx y : ↑(F.obj j)\n⊢ G.mk F ⟨j, x⟩ * G.mk F ⟨j, y⟩ = G.mk F ⟨j, x * y⟩",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"GrpCat.instConcreteCategoryMonoidHomCarrier",
"GrpCat",
"Mono... | [] | #adaptation_note /-- Prior to leanprover/lean4#12564, this was just
`simpa using colimit_mul_mk_eq F ⟨j, x⟩ ⟨j, y⟩ j (𝟙 _) (𝟙 _)` -/
have := colimit_mul_mk_eq F ⟨j, x⟩ ⟨j, y⟩ j (𝟙 _) (𝟙 _)
simpa using this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Types.Images | {
"line": 117,
"column": 12
} | {
"line": 117,
"column": 22
} | {
"line": 117,
"column": 22
} | [
{
"pp": "F : ℕᵒᵖ ⥤ Type u\nc : Cone F\nhc : IsLimit c\nhF : ∀ (n : ℕ), Function.Surjective ⇑(ConcreteCategory.hom (F.map (homOfLE ⋯).op))\ni : c.pt ≅ (limitCone F).pt := hc.conePointUniqueUpToIso (limitConeIsLimit F)\nthis : c.π.app (Opposite.op 0) = i.hom ≫ (limitCone F).π.app (Opposite.op 0)\n⊢ Function.Surje... | [
"F : ℕᵒᵖ ⥤ Type u\nc : Cone F\nhc : IsLimit c\nhF : ∀ (n : ℕ), Function.Surjective ⇑(ConcreteCategory.hom (F.map (homOfLE ⋯).op))\ni : c.pt ≅ (limitCone F).pt := hc.conePointUniqueUpToIso (limitConeIsLimit F)\nthis : c.π.app (Opposite.op 0) = i.hom ≫ (limitCone F).π.app (Opposite.op 0)\n⊢ Function.Surjective (⇑(Con... | types_comp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Determinant.Basic | {
"line": 773,
"column": 2
} | {
"line": 776,
"column": 98
} | {
"line": 777,
"column": 2
} | [
{
"pp": "R : Type v\ninst✝ : CommRing R\nn : ℕ\nA : Matrix (Fin n.succ) (Fin n.succ) R\ni : Fin n.succ\n⊢ A.det = (-1) ^ ↑i * ∑ i_1, (-1) ^ ↑i_1 * (A i i_1 * (A.submatrix i.succAbove i_1.succAbove).det)",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
... | [
"R : Type v\ninst✝ : CommRing R\nn : ℕ\nA : Matrix (Fin n.succ) (Fin n.succ) R\ni : Fin n.succ\nthis : A.det = (-1) ^ ↑i * ↑↑(sign i.cycleRange⁻¹) * A.det\n⊢ A.det = (-1) ^ ↑i * ∑ i_1, (-1) ^ ↑i_1 * (A i i_1 * (A.submatrix i.succAbove i_1.succAbove).det)"
] | have : det A = (-1 : R) ^ (i : ℕ) * (Perm.sign i.cycleRange⁻¹) * det A := by
calc
det A = ↑((-1 : ℤˣ) ^ (i : ℕ) * (-1 : ℤˣ) ^ (i : ℕ) : ℤˣ) * det A := by simp
_ = (-1 : R) ^ (i : ℕ) * (Perm.sign i.cycleRange⁻¹) * det A := by simp [-Int.units_mul_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Limits.FunctorCategory.Basic | {
"line": 439,
"column": 4
} | {
"line": 443,
"column": 35
} | {
"line": 443,
"column": 35
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\nJ : Type u₁\ninst✝¹ : Category.{v₁, u₁} J\nK : Type u₂\ninst✝ : Category.{v₂, u₂} K\nF : D ⥤ K ⥤ C\nG : J ⥤ D\nH : ∀ (k : K), PreservesColimit G (F ⋙ (evaluation K C).obj k)\nc : Cocone G\nhc : IsColimit c\n⊢ IsColimit (... | [] | apply evaluationJointlyReflectsColimits
intro X
haveI := H X
change IsColimit ((F ⋙ (evaluation K C).obj X).mapCocone c)
exact isColimitOfPreserves _ hc | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.FunctorCategory.Basic | {
"line": 439,
"column": 4
} | {
"line": 443,
"column": 35
} | {
"line": 443,
"column": 35
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\nJ : Type u₁\ninst✝¹ : Category.{v₁, u₁} J\nK : Type u₂\ninst✝ : Category.{v₂, u₂} K\nF : D ⥤ K ⥤ C\nG : J ⥤ D\nH : ∀ (k : K), PreservesColimit G (F ⋙ (evaluation K C).obj k)\nc : Cocone G\nhc : IsColimit c\n⊢ IsColimit (... | [] | apply evaluationJointlyReflectsColimits
intro X
haveI := H X
change IsColimit ((F ⋙ (evaluation K C).obj X).mapCocone c)
exact isColimitOfPreserves _ hc | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.ShrinkYoneda | {
"line": 191,
"column": 4
} | {
"line": 191,
"column": 58
} | {
"line": 192,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : LocallySmall.{w, v, u} C\nX✝ Y✝ : C\nf : shrinkYoneda.{w, v, u}.obj X✝ ⟶ shrinkYoneda.{w, v, u}.obj Y✝\n⊢ shrinkYoneda.{w, v, u}.map (shrinkYonedaObjObjEquiv (shrinkYonedaEquiv f)) = f",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : LocallySmall.{w, v, u} C\nX✝ Y✝ : C\nf : (shrinkYoneda.{w, v, u}.obj Y✝).obj (op X✝)\n⊢ shrinkYoneda.{w, v, u}.map (shrinkYonedaObjObjEquiv (shrinkYonedaEquiv (shrinkYonedaEquiv.symm f))) =\n shrinkYonedaEquiv.symm f"
] | obtain ⟨f, rfl⟩ := shrinkYonedaEquiv.symm.surjective f | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Functor.Trifunctor | {
"line": 71,
"column": 8
} | {
"line": 73,
"column": 60
} | {
"line": 73,
"column": 61
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nC₄ : Type u_4\nC₁₂ : Type u_5\nC₂₃ : Type u_6\ninst✝⁵ : Category.{v_1, u_1} C₁\ninst✝⁴ : Category.{v_2, u_2} C₂\ninst✝³ : Category.{v_3, u_3} C₃\ninst✝² : Category.{v_4, u_4} C₄\ninst✝¹ : Category.{v_5, u_5} C₁₂\ninst✝ : Category.{v_6, u_6} C₂₃\nF₁₂ : C₁ ⥤ C... | [] | ext X₂ X₃
dsimp
simp only [← NatTrans.comp_app, NatTrans.naturality] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Functor.Trifunctor | {
"line": 71,
"column": 8
} | {
"line": 73,
"column": 60
} | {
"line": 73,
"column": 61
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nC₄ : Type u_4\nC₁₂ : Type u_5\nC₂₃ : Type u_6\ninst✝⁵ : Category.{v_1, u_1} C₁\ninst✝⁴ : Category.{v_2, u_2} C₂\ninst✝³ : Category.{v_3, u_3} C₃\ninst✝² : Category.{v_4, u_4} C₄\ninst✝¹ : Category.{v_5, u_5} C₁₂\ninst✝ : Category.{v_6, u_6} C₂₃\nF₁₂ : C₁ ⥤ C... | [] | ext X₂ X₃
dsimp
simp only [← NatTrans.comp_app, NatTrans.naturality] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Functor.Trifunctor | {
"line": 158,
"column": 8
} | {
"line": 160,
"column": 60
} | {
"line": 160,
"column": 61
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nC₄ : Type u_4\nC₁₂ : Type u_5\nC₂₃ : Type u_6\ninst✝⁵ : Category.{v_1, u_1} C₁\ninst✝⁴ : Category.{v_2, u_2} C₂\ninst✝³ : Category.{v_3, u_3} C₃\ninst✝² : Category.{v_4, u_4} C₄\ninst✝¹ : Category.{v_5, u_5} C₁₂\ninst✝ : Category.{v_6, u_6} C₂₃\nF F' : C₁ ⥤ ... | [] | ext X₂ X₃
dsimp
simp only [← NatTrans.comp_app, NatTrans.naturality] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Functor.Trifunctor | {
"line": 158,
"column": 8
} | {
"line": 160,
"column": 60
} | {
"line": 160,
"column": 61
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nC₄ : Type u_4\nC₁₂ : Type u_5\nC₂₃ : Type u_6\ninst✝⁵ : Category.{v_1, u_1} C₁\ninst✝⁴ : Category.{v_2, u_2} C₂\ninst✝³ : Category.{v_3, u_3} C₃\ninst✝² : Category.{v_4, u_4} C₄\ninst✝¹ : Category.{v_5, u_5} C₁₂\ninst✝ : Category.{v_6, u_6} C₂₃\nF F' : C₁ ⥤ ... | [] | ext X₂ X₃
dsimp
simp only [← NatTrans.comp_app, NatTrans.naturality] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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