module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.CategoryTheory.Conj | {
"line": 56,
"column": 46
} | {
"line": 56,
"column": 63
} | {
"line": 56,
"column": 64
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : C\nf : End X\n⊢ 𝟙 X ≫ f ≫ 𝟙 X = f",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryTheory.CategoryStruct.id",
"id"... | [
"C : Type u\ninst✝ : Category.{v, u} C\nX : C\nf : End X\n⊢ f ≫ 𝟙 X = f"
] | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Category.Ring.Basic | {
"line": 192,
"column": 4
} | {
"line": 192,
"column": 38
} | {
"line": 194,
"column": 0
} | [
{
"pp": "X Y : SemiRingCat\nf : X ⟶ Y\nx✝ : IsIso ((forget SemiRingCat).map f)\ni : (forget SemiRingCat).obj X ≅ (forget SemiRingCat).obj Y := asIso ((forget SemiRingCat).map f)\nff : ↑X →+* ↑Y := Hom.hom f\ne : ↑X ≃+* ↑Y :=\n let __src := i.toEquiv;\n { toFun := (↑↑ff).toFun, invFun := __src.invFun, left_inv... | [] | exact e.toSemiRingCatIso.isIso_hom | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Category.Cat | {
"line": 314,
"column": 83
} | {
"line": 314,
"column": 95
} | {
"line": 316,
"column": 0
} | [
{
"pp": "B C : Cat\nF : B ⟶ C\n⊢ toNatIso (ρ_ F) = eqToIso ⋯ ≪≫ F.toFunctor.rightUnitor ≪≫ eqToIso ⋯",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor",
"congrArg",
"CategoryTheory.Functor.category",
"CategoryTheory.Bicategory.r... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.StrongEpi | {
"line": 100,
"column": 12
} | {
"line": 100,
"column": 79
} | {
"line": 100,
"column": 79
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nP Q R : C\nf : P ⟶ Q\ng : Q ⟶ R\ninst✝ : StrongEpi (f ≫ g)\nX Y : C\nz : X ⟶ Y\nx✝ : Mono z\nu : Q ⟶ X\nv : R ⟶ Y\nsq : CommSq u g z v\nh₀ : (f ≫ u) ≫ z = (f ≫ g) ≫ v\n⊢ g ≫ ⋯.lift = u",
"ppTerm": "?m.117",
"assigned": true,
"usedConstants": [
"... | [] | simp only [← cancel_mono z, Category.assoc, CommSq.fac_right, sq.w] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Limits.Shapes.StrongEpi | {
"line": 100,
"column": 12
} | {
"line": 100,
"column": 79
} | {
"line": 100,
"column": 79
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nP Q R : C\nf : P ⟶ Q\ng : Q ⟶ R\ninst✝ : StrongEpi (f ≫ g)\nX Y : C\nz : X ⟶ Y\nx✝ : Mono z\nu : Q ⟶ X\nv : R ⟶ Y\nsq : CommSq u g z v\nh₀ : (f ≫ u) ≫ z = (f ≫ g) ≫ v\n⊢ g ≫ ⋯.lift = u",
"ppTerm": "?m.117",
"assigned": true,
"usedConstants": [
"... | [] | simp only [← cancel_mono z, Category.assoc, CommSq.fac_right, sq.w] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.StrongEpi | {
"line": 100,
"column": 12
} | {
"line": 100,
"column": 79
} | {
"line": 100,
"column": 79
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nP Q R : C\nf : P ⟶ Q\ng : Q ⟶ R\ninst✝ : StrongEpi (f ≫ g)\nX Y : C\nz : X ⟶ Y\nx✝ : Mono z\nu : Q ⟶ X\nv : R ⟶ Y\nsq : CommSq u g z v\nh₀ : (f ≫ u) ≫ z = (f ≫ g) ≫ v\n⊢ g ≫ ⋯.lift = u",
"ppTerm": "?m.117",
"assigned": true,
"usedConstants": [
"... | [] | simp only [← cancel_mono z, Category.assoc, CommSq.fac_right, sq.w] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.IsTerminal | {
"line": 349,
"column": 26
} | {
"line": 349,
"column": 43
} | {
"line": 349,
"column": 44
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nJ : Type u\ninst✝ : Category.{v, u} J\nX : J\ntX : IsInitial X\nF : J ⥤ C\nj j' : J\nk : j ⟶ j'\n⊢ 𝟙 (F.obj X) ≫ F.map (tX.to j') = F.map (tX.to j ≫ k)",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nJ : Type u\ninst✝ : Category.{v, u} J\nX : J\ntX : IsInitial X\nF : J ⥤ C\nj j' : J\nk : j ⟶ j'\n⊢ F.map (tX.to j') = F.map (tX.to j ≫ k)"
] | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.HasLimits | {
"line": 1256,
"column": 8
} | {
"line": 1256,
"column": 26
} | {
"line": 1256,
"column": 26
} | [
{
"pp": "J : Type u₁\ninst✝² : Category.{v₁, u₁} J\nK : Type u₂\ninst✝¹ : Category.{v₂, u₂} K\nC : Type u\ninst✝ : Category.{v, u} C\nF : J ⥤ C\nt : Cone F.op\nP : IsLimit t\ns : Cocone F\nm : t.unop.pt ⟶ s.pt\nw : ∀ (j : J), t.unop.ι.app j ≫ m = s.ι.app j\n⊢ m = (P.lift s.op).unop",
"ppTerm": "?m.76",
... | [
"J : Type u₁\ninst✝² : Category.{v₁, u₁} J\nK : Type u₂\ninst✝¹ : Category.{v₂, u₂} K\nC : Type u\ninst✝ : Category.{v, u} C\nF : J ⥤ C\nt : Cone F.op\nP : IsLimit t\ns : Cocone F\nm : t.unop.pt ⟶ s.pt\nw : ∀ (j : J), t.unop.ι.app j ≫ m = s.ι.app j\n⊢ m = m.op.unop",
"J : Type u₁\ninst✝² : Category.{v₁, u₁} J\nK ... | ← P.uniq s.op m.op | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.HasLimits | {
"line": 1271,
"column": 8
} | {
"line": 1271,
"column": 26
} | {
"line": 1271,
"column": 26
} | [
{
"pp": "J : Type u₁\ninst✝² : Category.{v₁, u₁} J\nK : Type u₂\ninst✝¹ : Category.{v₂, u₂} K\nC : Type u\ninst✝ : Category.{v, u} C\nF : J ⥤ C\nt : Cocone F.op\nP : IsColimit t\ns : Cone F\nm : s.pt ⟶ t.unop.pt\nw : ∀ (j : J), m ≫ t.unop.π.app j = s.π.app j\n⊢ m = (P.desc s.op).unop",
"ppTerm": "?m.76",
... | [
"J : Type u₁\ninst✝² : Category.{v₁, u₁} J\nK : Type u₂\ninst✝¹ : Category.{v₂, u₂} K\nC : Type u\ninst✝ : Category.{v, u} C\nF : J ⥤ C\nt : Cocone F.op\nP : IsColimit t\ns : Cone F\nm : s.pt ⟶ t.unop.pt\nw : ∀ (j : J), m ≫ t.unop.π.app j = s.π.app j\n⊢ m = m.op.unop",
"J : Type u₁\ninst✝² : Category.{v₁, u₁} J\n... | ← P.uniq s.op m.op | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts | {
"line": 861,
"column": 4
} | {
"line": 862,
"column": 55
} | {
"line": 863,
"column": 4
} | [
{
"pp": "case h₁\nC✝ : Type u\ninst✝⁵ : Category.{v, u} C✝\nX✝ Y✝ : C✝\nC : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nW X Y Z : C\nf : W ⟶ Y\ng : X ⟶ Z\ninst✝³ : Epi f\ninst✝² : Epi g\ninst✝¹ : HasBinaryCoproduct W X\ninst✝ : HasBinaryCoproduct Y Z\nZ✝ : C\ni₁ i₂ : Y ⨿ Z ⟶ Z✝\nh : map f g ≫ i₁ = map f g ≫ i₂\n⊢... | [
"case h₂\nC✝ : Type u\ninst✝⁵ : Category.{v, u} C✝\nX✝ Y✝ : C✝\nC : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nW X Y Z : C\nf : W ⟶ Y\ng : X ⟶ Z\ninst✝³ : Epi f\ninst✝² : Epi g\ninst✝¹ : HasBinaryCoproduct W X\ninst✝ : HasBinaryCoproduct Y Z\nZ✝ : C\ni₁ i₂ : Y ⨿ Z ⟶ Z✝\nh : map f g ≫ i₁ = map f g ≫ i₂\n⊢ inr ≫ i₁ = ... | · rw [← cancel_epi f]
simpa using congr_arg (fun f => coprod.inl ≫ f) h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts | {
"line": 1257,
"column": 2
} | {
"line": 1258,
"column": 29
} | {
"line": 1260,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nD : Type u₂\ninst✝⁵ : Category.{w, u₂} D\nE : Type u₃\ninst✝⁴ : Category.{w', u₃} E\nF : C ⥤ D\nG : D ⥤ E\nA B : C\ninst✝³ : HasBinaryProduct A B\ninst✝² : HasBinaryProduct (F.obj A) (F.obj B)\ninst✝¹ : HasBinaryProduct (G.obj (F.obj A)) (G.obj (F.obj B))\ninst✝ ... | [] | unfold prodComparison
ext <;> simp [← G.map_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts | {
"line": 1257,
"column": 2
} | {
"line": 1258,
"column": 29
} | {
"line": 1260,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nD : Type u₂\ninst✝⁵ : Category.{w, u₂} D\nE : Type u₃\ninst✝⁴ : Category.{w', u₃} E\nF : C ⥤ D\nG : D ⥤ E\nA B : C\ninst✝³ : HasBinaryProduct A B\ninst✝² : HasBinaryProduct (F.obj A) (F.obj B)\ninst✝¹ : HasBinaryProduct (G.obj (F.obj A)) (G.obj (F.obj B))\ninst✝ ... | [] | unfold prodComparison
ext <;> simp [← G.map_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts | {
"line": 1358,
"column": 36
} | {
"line": 1358,
"column": 53
} | {
"line": 1358,
"column": 54
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasBinaryCoproducts C\nA : C\nX✝ Y✝ : Over A\nk : X✝ ⟶ Y✝\ng : Over A\n⊢ coprod.desc (Hom.left k ≫ Y✝.hom) (𝟙 g.left ≫ g.hom) = coprod.desc X✝.hom g.hom",
"ppTerm": "?m.88",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Category... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasBinaryCoproducts C\nA : C\nX✝ Y✝ : Over A\nk : X✝ ⟶ Y✝\ng : Over A\n⊢ coprod.desc (Hom.left k ≫ Y✝.hom) g.hom = coprod.desc X✝.hom g.hom"
] | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono | {
"line": 246,
"column": 37
} | {
"line": 246,
"column": 54
} | {
"line": 246,
"column": 55
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nW X Y Z : C\nf✝ : X ⟶ Y\ng : X ⟶ Z\nf : X ⟶ Y\ninst✝ : Epi f\ns : PushoutCocone f f\n⊢ f ≫ 𝟙 Y ≫ s.inl = f ≫ s.inr",
"ppTerm": "?m.78",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.WalkingSpan",
"CategoryThe... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nW X Y Z : C\nf✝ : X ⟶ Y\ng : X ⟶ Z\nf : X ⟶ Y\ninst✝ : Epi f\ns : PushoutCocone f f\n⊢ f ≫ s.inl = f ≫ s.inr"
] | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Category.ModuleCat.Semi | {
"line": 434,
"column": 4
} | {
"line": 436,
"column": 20
} | {
"line": 437,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nX₁ X₂ : Type v\nA✝ B✝ : SemimoduleCat R\nf : A✝ ⟶ B✝\nx✝ : IsIso ((forget₂ (SemimoduleCat R) AddCommMonCat).map f)\n⊢ IsIso f",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"SemimoduleCat.isModule",
"SemimoduleCat.moduleCategory",
... | [
"R : Type u\ninst✝ : Semiring R\nX₁ X₂ : Type v\nA✝ B✝ : SemimoduleCat R\nf : A✝ ⟶ B✝\nx✝ : IsIso ((forget₂ (SemimoduleCat R) AddCommMonCat).map f)\nthis : IsIso ((forget (SemimoduleCat R)).map f)\n⊢ IsIso f"
] | have : IsIso ((forget _).map f) := by
change IsIso ((forget _).map ((forget₂ _ AddCommMonCat).map f))
infer_instance | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Preadditive.Basic | {
"line": 387,
"column": 20
} | {
"line": 387,
"column": 73
} | {
"line": 389,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nX Y : C\nf g : X ⟶ Y\nc : Cofork f g\n⊢ (f - g) ≫ c.π = 0 ≫ c.π",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
... | [] | by rw [sub_comp, zero_comp, sub_eq_zero, c.condition] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Preserves.Basic | {
"line": 632,
"column": 4
} | {
"line": 633,
"column": 49
} | {
"line": 634,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D\nJ : Type w\ninst✝³ : Category.{w', w} J\nF : J ⥤ C\nG : C ⥤ D\ninst✝² : G.ReflectsIsomorphisms\ninst✝¹ : HasLimit F\ninst✝ : PreservesLimit F G\nc : Cone F\nt : IsLimit (G.mapCone c)\n⊢ Nonempty (IsLimit c)",
"ppT... | [
"C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D\nJ : Type w\ninst✝³ : Category.{w', w} J\nF : J ⥤ C\nG : C ⥤ D\ninst✝² : G.ReflectsIsomorphisms\ninst✝¹ : HasLimit F\ninst✝ : PreservesLimit F G\nc : Cone F\nt : IsLimit (G.mapCone c)\n⊢ IsIso ((limit.isLimit F).lift c)"
] | suffices IsIso (IsLimit.lift (limit.isLimit F) c) from ⟨by
apply IsLimit.ofPointIso (limit.isLimit F)⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.CategoryTheory.Limits.Shapes.Kernels | {
"line": 372,
"column": 8
} | {
"line": 373,
"column": 16
} | {
"line": 373,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝¹ : HasKernel f\nX' Y' : C\nf' : X' ⟶ Y'\ninst✝ : HasKernel f'\np : X ≅ X'\nq : Y ≅ Y'\nw : f ≫ q.hom = p.hom ≫ f'\n⊢ f' ≫ q.inv = p.inv ≫ f",
"ppTerm": "?m.94",
"assigned": true,
"usedConstants": ... | [] | refine (cancel_mono q.hom).1 ?_
simp [w] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Kernels | {
"line": 372,
"column": 8
} | {
"line": 373,
"column": 16
} | {
"line": 373,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝¹ : HasKernel f\nX' Y' : C\nf' : X' ⟶ Y'\ninst✝ : HasKernel f'\np : X ≅ X'\nq : Y ≅ Y'\nw : f ≫ q.hom = p.hom ≫ f'\n⊢ f' ≫ q.inv = p.inv ≫ f",
"ppTerm": "?m.94",
"assigned": true,
"usedConstants": ... | [] | refine (cancel_mono q.hom).1 ?_
simp [w] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Preserves.Shapes.Biproducts | {
"line": 238,
"column": 2
} | {
"line": 240,
"column": 57
} | {
"line": 242,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝⁶ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : HasZeroMorphisms C\ninst✝³ : HasZeroMorphisms D\nJ : Type w₁\nF : C ⥤ D\nf : J → C\ninst✝² : HasBiproduct f\ninst✝¹ : HasBiproduct (F.obj ∘ f)\ninst✝ : F.PreservesZeroMorphisms\n⊢ F.biproductComparison' f ≫ F... | [] | classical
ext
simp [biproduct.ι_π, ← Functor.map_comp, eqToHom_map] | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.CategoryTheory.Limits.Preserves.Shapes.Biproducts | {
"line": 238,
"column": 2
} | {
"line": 240,
"column": 57
} | {
"line": 242,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝⁶ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : HasZeroMorphisms C\ninst✝³ : HasZeroMorphisms D\nJ : Type w₁\nF : C ⥤ D\nf : J → C\ninst✝² : HasBiproduct f\ninst✝¹ : HasBiproduct (F.obj ∘ f)\ninst✝ : F.PreservesZeroMorphisms\n⊢ F.biproductComparison' f ≫ F... | [] | classical
ext
simp [biproduct.ι_π, ← Functor.map_comp, eqToHom_map] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Preserves.Shapes.Biproducts | {
"line": 238,
"column": 2
} | {
"line": 240,
"column": 57
} | {
"line": 242,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝⁶ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : HasZeroMorphisms C\ninst✝³ : HasZeroMorphisms D\nJ : Type w₁\nF : C ⥤ D\nf : J → C\ninst✝² : HasBiproduct f\ninst✝¹ : HasBiproduct (F.obj ∘ f)\ninst✝ : F.PreservesZeroMorphisms\n⊢ F.biproductComparison' f ≫ F... | [] | classical
ext
simp [biproduct.ι_π, ← Functor.map_comp, eqToHom_map] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Kernels | {
"line": 1000,
"column": 8
} | {
"line": 1000,
"column": 45
} | {
"line": 1001,
"column": 8
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\nX✝ Y✝ : C\nf✝ : X✝ ⟶ Y✝\ninst✝² : HasCokernel f✝\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\ninst✝¹ : Epi f\ninst✝ : HasCokernel g\n⊢ g ≫ cokernel.π (f ≫ g) = 0",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [
"C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\nX✝ Y✝ : C\nf✝ : X✝ ⟶ Y✝\ninst✝² : HasCokernel f✝\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\ninst✝¹ : Epi f\ninst✝ : HasCokernel g\n⊢ (f ≫ g) ≫ cokernel.π (f ≫ g) = f ≫ 0"
] | rw [← cancel_epi f, ← Category.assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Preadditive.AdditiveFunctor | {
"line": 122,
"column": 16
} | {
"line": 122,
"column": 22
} | {
"line": 122,
"column": 23
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁹ : Category.{v_1, u_1} C\ninst✝⁸ : Category.{v_2, u_2} D\ninst✝⁷ : Category.{v_3, u_3} E\ninst✝⁶ : Preadditive C\ninst✝⁵ : Preadditive D\ninst✝⁴ : Preadditive E\nF : C ⥤ D\ninst✝³ : F.Additive\ninst✝² : F.Full\ninst✝¹ : F.EssSurj\nG : D ⥤ E\ninst✝ : (F ⋙ ... | [
"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁹ : Category.{v_1, u_1} C\ninst✝⁸ : Category.{v_2, u_2} D\ninst✝⁷ : Category.{v_3, u_3} E\ninst✝⁶ : Preadditive C\ninst✝⁵ : Preadditive D\ninst✝⁴ : Preadditive E\nF : C ⥤ D\ninst✝³ : F.Additive\ninst✝² : F.Full\ninst✝¹ : F.EssSurj\nG : D ⥤ E\ninst✝ : (F ⋙ G).Additive\... | ← hg', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts | {
"line": 141,
"column": 17
} | {
"line": 141,
"column": 60
} | {
"line": 142,
"column": 6
} | [
{
"pp": "J : Type w\nC : Type uC\ninst✝⁴ : Category.{uC', uC} C\ninst✝³ : HasZeroMorphisms C\nD : Type uD\ninst✝² : Category.{uD', uD} D\ninst✝¹ : HasZeroMorphisms D\nP Q : C\nF : C ⥤ D\ninst✝ : F.PreservesZeroMorphisms\nX✝ Y✝ : BinaryBicone P Q\nf : X✝ ⟶ Y✝\n⊢ { pt := F.obj X✝.pt, fst := F.map X✝.fst, snd := F... | [] | simp [-BinaryBiconeMorphism.winl, ← f.winl] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts | {
"line": 141,
"column": 17
} | {
"line": 141,
"column": 60
} | {
"line": 142,
"column": 6
} | [
{
"pp": "J : Type w\nC : Type uC\ninst✝⁴ : Category.{uC', uC} C\ninst✝³ : HasZeroMorphisms C\nD : Type uD\ninst✝² : Category.{uD', uD} D\ninst✝¹ : HasZeroMorphisms D\nP Q : C\nF : C ⥤ D\ninst✝ : F.PreservesZeroMorphisms\nX✝ Y✝ : BinaryBicone P Q\nf : X✝ ⟶ Y✝\n⊢ { pt := F.obj X✝.pt, fst := F.map X✝.fst, snd := F... | [] | simp [-BinaryBiconeMorphism.winl, ← f.winl] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts | {
"line": 141,
"column": 17
} | {
"line": 141,
"column": 60
} | {
"line": 142,
"column": 6
} | [
{
"pp": "J : Type w\nC : Type uC\ninst✝⁴ : Category.{uC', uC} C\ninst✝³ : HasZeroMorphisms C\nD : Type uD\ninst✝² : Category.{uD', uD} D\ninst✝¹ : HasZeroMorphisms D\nP Q : C\nF : C ⥤ D\ninst✝ : F.PreservesZeroMorphisms\nX✝ Y✝ : BinaryBicone P Q\nf : X✝ ⟶ Y✝\n⊢ { pt := F.obj X✝.pt, fst := F.map X✝.fst, snd := F... | [] | simp [-BinaryBiconeMorphism.winl, ← f.winl] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts | {
"line": 752,
"column": 4
} | {
"line": 753,
"column": 48
} | {
"line": 755,
"column": 0
} | [
{
"pp": "case mpr\nC : Type uC\ninst✝² : Category.{uC', uC} C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : HasBinaryBiproduct X Y\n⊢ 𝟙 (X ⊞ Y) = fst ≫ inl → IsIso inl",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | [] | intro h
exact ⟨⟨biprod.fst, biprod.inl_fst, h.symm⟩⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts | {
"line": 752,
"column": 4
} | {
"line": 753,
"column": 48
} | {
"line": 755,
"column": 0
} | [
{
"pp": "case mpr\nC : Type uC\ninst✝² : Category.{uC', uC} C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : HasBinaryBiproduct X Y\n⊢ 𝟙 (X ⊞ Y) = fst ≫ inl → IsIso inl",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | [] | intro h
exact ⟨⟨biprod.fst, biprod.inl_fst, h.symm⟩⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Biproducts | {
"line": 947,
"column": 8
} | {
"line": 947,
"column": 31
} | {
"line": 947,
"column": 31
} | [
{
"pp": "case hnc\nJ✝ : Type w\nC✝ : Type uC\ninst✝⁷ : Category.{uC', uC} C✝\ninst✝⁶ : HasZeroMorphisms C✝\nD : Type uD\ninst✝⁵ : Category.{uD', uD} D\ninst✝⁴ : HasZeroMorphisms D\nF : J✝ → C✝\nJ : Type w\nK✝ : Type u_1\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nK : Type\ninst✝¹ : Fin... | [] | · exact not_not.mpr k.2 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Preadditive.Biproducts | {
"line": 587,
"column": 6
} | {
"line": 588,
"column": 18
} | {
"line": 590,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\nX✝ Y✝ : C\ninst✝ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nb : BinaryBicone X Y\nhb : IsColimit b.inrCokernelCofork\nT : C\nf : X ⟶ T\ng : Y ⟶ T\nm : b.pt ⟶ T\nh₁ : BinaryCofan.inl b.toCocone ≫ m = f\nh₂ : BinaryCofan.inr b.toCocone ≫ m = g\nh₁... | [] | rw [← sub_eq_zero, ← hq, ← Category.id_comp q, ← b.inl_fst, Category.assoc, hq, h₁',
comp_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Category.ModuleCat.Basic | {
"line": 555,
"column": 20
} | {
"line": 555,
"column": 32
} | {
"line": 556,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nX₁ X₂ : Type v\nM N : ModuleCat R\nA : AddCommGrpCat\nφ : R →+* End A\nx✝² x✝¹ : R\nx✝ : ↑(mkOfSMul' φ)\n⊢ (x✝² + x✝¹) • x✝ = x✝² • x✝ + x✝¹ • x✝",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"ModuleCat.instSMulCarrierMkOfSMul'",
"Eq.mpr",... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Antidiag.Finsupp | {
"line": 82,
"column": 4
} | {
"line": 82,
"column": 45
} | {
"line": 83,
"column": 4
} | [
{
"pp": "case mpr\nι : Type u_1\nμ : Type u_2\ninst✝³ : DecidableEq ι\ninst✝² : AddCommMonoid μ\ninst✝¹ : HasAntidiagonal μ\ninst✝ : DecidableEq μ\na : ι\ns : Finset ι\nh : a ∉ s\nn : μ\nf : ι →₀ μ\n⊢ (∃ a_1 b, a_1 + b = n ∧ ∃ g, f = g.update a a_1 ∧ s.sum ⇑g = b ∧ g.support ⊆ s) →\n f a + ∑ x ∈ s, f x = n ∧... | [
"case mpr\nι : Type u_1\nμ : Type u_2\ninst✝³ : DecidableEq ι\ninst✝² : AddCommMonoid μ\ninst✝¹ : HasAntidiagonal μ\ninst✝ : DecidableEq μ\na : ι\ns : Finset ι\nh : a ∉ s\nn1 : μ\ng : ι →₀ μ\nhgsupp : g.support ⊆ s\n⊢ (g.update a n1) a + ∑ x ∈ s, (g.update a n1) x = n1 + s.sum ⇑g ∧ (g.update a n1).support ⊆ insert ... | rintro ⟨n1, n2, rfl, g, rfl, rfl, hgsupp⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Order.Antidiag.Finsupp | {
"line": 104,
"column": 42
} | {
"line": 104,
"column": 66
} | {
"line": 104,
"column": 66
} | [
{
"pp": "case inl\nι : Type u_1\nμ : Type u_2\nμ' : Type u_3\ninst✝³ : DecidableEq ι\ninst✝² : AddCommMonoid μ\ninst✝¹ : HasAntidiagonal μ\ninst✝ : DecidableEq μ\ns✝ : Finset ι\nn✝ : μ\nf✝ : ι →₀ μ\ns : Finset ι\nn : μ\np : μ × μ\nx✝¹ x✝ : ↥(s.finsuppAntidiag p.2)\nf : ι →₀ μ\nhf✝ : f ∈ s.finsuppAntidiag p.2\ng... | [
"case inl\nι : Type u_1\nμ : Type u_2\nμ' : Type u_3\ninst✝³ : DecidableEq ι\ninst✝² : AddCommMonoid μ\ninst✝¹ : HasAntidiagonal μ\ninst✝ : DecidableEq μ\ns✝ : Finset ι\nn✝ : μ\nf✝ : ι →₀ μ\ns : Finset ι\nn : μ\np : μ × μ\nx✝¹ x✝ : ↥(s.finsuppAntidiag p.2)\nf : ι →₀ μ\nhf✝ : f ∈ s.finsuppAntidiag p.2\ng : ι →₀ μ\nh... | notMem_support_iff.mp hg | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.Support | {
"line": 319,
"column": 2
} | {
"line": 319,
"column": 30
} | {
"line": 321,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nc : Perm α\ns : Finset α\nhcs : c.support ⊆ s\nx : α\nhx' : x ∉ c.support\n⊢ c x ∈ s ↔ x ∈ s",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.support",
"Equiv.Perm.notMem_supp... | [] | · rw [notMem_support.mp hx'] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.Enumerative.Composition | {
"line": 840,
"column": 6
} | {
"line": 840,
"column": 11
} | {
"line": 841,
"column": 6
} | [
{
"pp": "case mpr\nn✝ n : ℕ\nc : CompositionAsSet (n + 1)\nk : Fin n\ni_mem : k.castSucc.succ ∈ c.boundaries\ni_ne_last : k.castSucc.succ ≠ Fin.last (n + 1)\n⊢ ∃ j, ⟨↑j + 1, ⋯⟩ ∈ c.boundaries ∧ ↑k.castSucc.succ = ↑j + 1",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"compositionAsSet... | [
"case h\nn✝ n : ℕ\nc : CompositionAsSet (n + 1)\nk : Fin n\ni_mem : k.castSucc.succ ∈ c.boundaries\ni_ne_last : k.castSucc.succ ≠ Fin.last (n + 1)\n⊢ ⟨↑k + 1, ⋯⟩ ∈ c.boundaries ∧ ↑k.castSucc.succ = ↑k + 1"
] | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.GroupTheory.Perm.Cycle.Basic | {
"line": 195,
"column": 2
} | {
"line": 195,
"column": 31
} | {
"line": 197,
"column": 0
} | [
{
"pp": "case h\nα : Type u_2\nf : Perm α\nx : α\ninst✝ : Finite α\nk : ℤ\nh₀ : 0 < ↑(orderOf f)\nh₁ : 0 ≤ k % ↑(orderOf f)\n⊢ k % ↑(orderOf f) < ↑(orderOf f)",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"CancelMonoid.toLeftCancelMonoid",
"LeftCancelMonoid.toMonoid",
"Int... | [] | exact Int.emod_lt_of_pos _ h₀ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.Perm.Sign | {
"line": 281,
"column": 4
} | {
"line": 281,
"column": 53
} | {
"line": 282,
"column": 4
} | [
{
"pp": "n : ℕ\nx y : Fin (n + 2)\nhxy : x ≠ y\nh2n : 2 ≤ n + 2\n⊢ (swap x y).signAux = -1",
"ppTerm": "?m.81",
"assigned": true,
"usedConstants": [
"Int.instCommMonoid",
"Eq.mpr",
"Preorder.toLT",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCo... | [
"n : ℕ\nx y : Fin (n + 2)\nhxy : x ≠ y\nh2n : 2 ≤ n + 2\n⊢ IsConj (swap x y).signAux (swap ⟨0, ⋯⟩ ⟨1, ⋯⟩).signAux"
] | rw [← isConj_iff_eq, ← signAux_swap_zero_one h2n] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Set.Pairwise.List | {
"line": 34,
"column": 6
} | {
"line": 34,
"column": 21
} | {
"line": 34,
"column": 21
} | [
{
"pp": "case cons\nα : Type u_1\nr : α → α → Prop\nl✝ : List α\ninst✝ : Std.Symm r\na : α\nl : List α\nih : l.Nodup → ({a | a ∈ l}.Pairwise r ↔ Pairwise r l)\nhl : (a :: l).Nodup\n⊢ {a_1 | a_1 ∈ a :: l}.Pairwise r ↔ Pairwise r (a :: l)",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
... | [
"case cons\nα : Type u_1\nr : α → α → Prop\nl✝ : List α\ninst✝ : Std.Symm r\na : α\nl : List α\nih : l.Nodup → ({a | a ∈ l}.Pairwise r ↔ Pairwise r l)\nhl : ¬a ∈ l ∧ l.Nodup\n⊢ {a_1 | a_1 ∈ a :: l}.Pairwise r ↔ Pairwise r (a :: l)"
] | List.nodup_cons | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.Sign | {
"line": 324,
"column": 8
} | {
"line": 324,
"column": 52
} | {
"line": 324,
"column": 53
} | [
{
"pp": "α : Type u\ninst✝¹ : DecidableEq α\nβ : Type v\ninst✝ : Finite α\nf : Perm α\ns : Multiset α\nl₁ l₂ : List α\nh : l₁ ≈ l₂\nn : ℕ\ne : α ≃ Fin n\nh₁ : ∀ (x : α), x ∈ ⟦l₁⟧\nh₂ : ∀ (x : α), x ∈ ⟦l₂⟧\nx✝ : h₁ ≍ h₂\n⊢ signAux2 l₁ f ≍ signAux2 l₂ f",
"ppTerm": "?m.58",
"assigned": true,
"usedCons... | [
"α : Type u\ninst✝¹ : DecidableEq α\nβ : Type v\ninst✝ : Finite α\nf : Perm α\ns : Multiset α\nl₁ l₂ : List α\nh : l₁ ≈ l₂\nn : ℕ\ne : α ≃ Fin n\nh₁ : ∀ (x : α), x ∈ ⟦l₁⟧\nh₂ : ∀ (x : α), x ∈ ⟦l₂⟧\nx✝ : h₁ ≍ h₂\n⊢ signAux ((e.symm.trans f).trans e) ≍ signAux2 l₂ f"
] | ← signAux_eq_signAux2 _ _ e fun _ _ => h₁ _, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.Cycle.Basic | {
"line": 636,
"column": 2
} | {
"line": 644,
"column": 38
} | {
"line": 646,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ τ : Perm α\nhσ : σ.IsCycle\nhτ : τ.IsCycle\nh : #σ.support = #τ.support\n⊢ IsConj σ τ",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Iff.mpr",
"Eq.mpr",
"Equiv.Perm.support",
... | [] | refine
isConj_of_support_equiv
(hσ.zpowersEquivSupport.symm.trans <|
(zpowersEquivZPowers <| by rw [hσ.orderOf, h, hτ.orderOf]).trans hτ.zpowersEquivSupport)
?_
intro x hx
simp only [Equiv.trans_apply]
obtain ⟨n, rfl⟩ := hσ.exists_pow_eq (Classical.choose_spec hσ).1 (mem_support.1 hx)
si... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Perm.Cycle.Basic | {
"line": 636,
"column": 2
} | {
"line": 644,
"column": 38
} | {
"line": 646,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ τ : Perm α\nhσ : σ.IsCycle\nhτ : τ.IsCycle\nh : #σ.support = #τ.support\n⊢ IsConj σ τ",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Iff.mpr",
"Eq.mpr",
"Equiv.Perm.support",
... | [] | refine
isConj_of_support_equiv
(hσ.zpowersEquivSupport.symm.trans <|
(zpowersEquivZPowers <| by rw [hσ.orderOf, h, hτ.orderOf]).trans hτ.zpowersEquivSupport)
?_
intro x hx
simp only [Equiv.trans_apply]
obtain ⟨n, rfl⟩ := hσ.exists_pow_eq (Classical.choose_spec hσ).1 (mem_support.1 hx)
si... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.NoncommPiCoprod | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 48
} | {
"line": 124,
"column": 2
} | [
{
"pp": "M : Type u_1\ninst✝³ : Monoid M\nι : Type u_2\ninst✝² : Fintype ι\nN : ι → Type u_3\ninst✝¹ : (i : ι) → Monoid (N i)\nϕ : (i : ι) → N i →* M\nhcomm : Pairwise fun i j ↦ ∀ (x : N i) (y : N j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝ : DecidableEq ι\ni : ι\ny : N i\n⊢ Finset.univ.noncommProd (fun j ↦ (ϕ j) (P... | [
"M : Type u_1\ninst✝³ : Monoid M\nι : Type u_2\ninst✝² : Fintype ι\nN : ι → Type u_3\ninst✝¹ : (i : ι) → Monoid (N i)\nϕ : (i : ι) → N i →* M\nhcomm : Pairwise fun i j ↦ ∀ (x : N i) (y : N j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝ : DecidableEq ι\ni : ι\ny : N i\n⊢ (insert i (Finset.univ.erase i)).noncommProd (fun j ↦... | rw [← Finset.insert_erase (Finset.mem_univ i)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Perm.Cycle.Factors | {
"line": 76,
"column": 71
} | {
"line": 76,
"column": 93
} | {
"line": 76,
"column": 93
} | [
{
"pp": "case negSucc\nα : Type u_2\nf : Perm α\ninst✝ : DecidableRel f.SameCycle\nx : α\nz : ℕ\n⊢ (f⁻¹.cycleOf x ^ (z + 1)) x = (f⁻¹ ^ (z + 1)) x",
"ppTerm": "?negSucc",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"Equiv.Perm.instInv",
"DivInvOneMon... | [
"case negSucc\nα : Type u_2\nf : Perm α\ninst✝ : DecidableRel f.SameCycle\nx : α\nz : ℕ\n⊢ (f⁻¹ ^ (z + 1)) x = (f⁻¹ ^ (z + 1)) x"
] | cycleOf_pow_apply_self | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.Cycle.Factors | {
"line": 241,
"column": 4
} | {
"line": 257,
"column": 54
} | {
"line": 259,
"column": 0
} | [
{
"pp": "ι : 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\nf : Perm α\nx y : α\n⊢ y ∈ List.iterate (⇑f) x (Fintype.card α) ↔ f.SameCycle x y",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"List.iterate",
"Eq.mpr",
... | [] | simp only [List.mem_iterate, iterate_eq_pow, eq_comm (a := y)]
constructor
· rintro ⟨n, _, hn⟩
exact ⟨n, hn⟩
· intro hxy
by_cases hx : x ∈ f.support
case pos =>
-- we can't invoke the aux lemmas above without obtaining the decidable instance we are
-- already building; but ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Perm.Cycle.Factors | {
"line": 241,
"column": 4
} | {
"line": 257,
"column": 54
} | {
"line": 259,
"column": 0
} | [
{
"pp": "ι : 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\nf : Perm α\nx y : α\n⊢ y ∈ List.iterate (⇑f) x (Fintype.card α) ↔ f.SameCycle x y",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"List.iterate",
"Eq.mpr",
... | [] | simp only [List.mem_iterate, iterate_eq_pow, eq_comm (a := y)]
constructor
· rintro ⟨n, _, hn⟩
exact ⟨n, hn⟩
· intro hxy
by_cases hx : x ∈ f.support
case pos =>
-- we can't invoke the aux lemmas above without obtaining the decidable instance we are
-- already building; but ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Perm.Fin | {
"line": 395,
"column": 41
} | {
"line": 395,
"column": 54
} | {
"line": 395,
"column": 54
} | [
{
"pp": "n : ℕ\ni j k : Fin n\nhik : i ≤ k\nhkj : k ≤ j\ninst✝ : NeZero n\nhij : i ≤ j\nkin : k ∈ Set.range ⇑(natAdd_castLEEmb ⋯)\nthis :\n ((addNatEmb (n - (n - ↑i))).trans (finCongr ⋯).toEmbedding).toEquivRange.symm ⟨k, kin⟩ = subNat (↑i) (Fin.cast ⋯ k) ⋯\nch : ¬k = j\n⊢ ↑i ≤ ↑(Fin.cast ⋯ k)",
"ppTerm": ... | [] | by simp [hik] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Perm.Fin | {
"line": 420,
"column": 4
} | {
"line": 420,
"column": 27
} | {
"line": 422,
"column": 0
} | [
{
"pp": "case inr\nn : ℕ\ni j : Fin n\nhij✝ : i ≤ j\ninst✝ : NeZero n\nhij : i = j\n⊢ j.cycleIcc i = 1",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.instOne",
"congrArg",
"id",
"Equiv.Perm",
"Fin.cycleIcc_eq",
"One.toOfNat1",... | [] | rw [hij, ← cycleIcc_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Perm.Fin | {
"line": 420,
"column": 4
} | {
"line": 420,
"column": 27
} | {
"line": 422,
"column": 0
} | [
{
"pp": "case inr\nn : ℕ\ni j : Fin n\nhij✝ : i ≤ j\ninst✝ : NeZero n\nhij : i = j\n⊢ j.cycleIcc i = 1",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.instOne",
"congrArg",
"id",
"Equiv.Perm",
"Fin.cycleIcc_eq",
"One.toOfNat1",... | [] | rw [hij, ← cycleIcc_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Perm.Fin | {
"line": 420,
"column": 4
} | {
"line": 420,
"column": 27
} | {
"line": 422,
"column": 0
} | [
{
"pp": "case inr\nn : ℕ\ni j : Fin n\nhij✝ : i ≤ j\ninst✝ : NeZero n\nhij : i = j\n⊢ j.cycleIcc i = 1",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.instOne",
"congrArg",
"id",
"Equiv.Perm",
"Fin.cycleIcc_eq",
"One.toOfNat1",... | [] | rw [hij, ← cycleIcc_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.FunLike.Module | {
"line": 39,
"column": 21
} | {
"line": 39,
"column": 60
} | {
"line": 41,
"column": 0
} | [
{
"pp": "M : Type u_1\nM' : Type u_2\nF : Type u_3\nα : Type u_4\nβ : Type u_5\ni : FunLike F α β\ninst✝⁶ : SMul M β\ninst✝⁵ : SMul M' β\ninst✝⁴ : SMul M F\ninst✝³ : SMul M' F\ninst✝² : IsSMulApply M F α β\ninst✝¹ : IsSMulApply M' F α β\ninst✝ : SMulCommClass M M' β\nx✝² : M\nx✝¹ : M'\nx✝ : F\n⊢ x✝² • x✝¹ • x✝ ... | [] | by apply DFunLike.ext; simp [smul_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Perm.Cycle.Type | {
"line": 271,
"column": 4
} | {
"line": 272,
"column": 10
} | {
"line": 273,
"column": 2
} | [
{
"pp": "case base_one\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nτ : Perm α\nh : cycleType 1 = τ.cycleType\n⊢ IsConj 1 τ",
"ppTerm": "?base_one",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.cycleType",
"Equiv.Perm.instOne",
"congrArg",
"Mu... | [] | rw [cycleType_one, eq_comm, cycleType_eq_zero] at h
rw [h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Perm.Cycle.Type | {
"line": 271,
"column": 4
} | {
"line": 272,
"column": 10
} | {
"line": 273,
"column": 2
} | [
{
"pp": "case base_one\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nτ : Perm α\nh : cycleType 1 = τ.cycleType\n⊢ IsConj 1 τ",
"ppTerm": "?base_one",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.cycleType",
"Equiv.Perm.instOne",
"congrArg",
"Mu... | [] | rw [cycleType_one, eq_comm, cycleType_eq_zero] at h
rw [h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Perm.Cycle.Factors | {
"line": 737,
"column": 2
} | {
"line": 737,
"column": 13
} | {
"line": 738,
"column": 2
} | [
{
"pp": "α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng k c : Perm α\n⊢ ∀ {g k c : Perm α}, c ∈ g.cycleFactorsFinset → k * c * k⁻¹ ∈ (k * g * k⁻¹).cycleFactorsFinset",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"Equiv.Perm"
],
"usedFVars": [
"α"
],
... | [
"α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng✝ k✝ c✝ g k c : Perm α\n⊢ c ∈ g.cycleFactorsFinset → k * c * k⁻¹ ∈ (k * g * k⁻¹).cycleFactorsFinset"
] | intro g k c | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.GroupTheory.Perm.Cycle.Type | {
"line": 508,
"column": 2
} | {
"line": 508,
"column": 61
} | {
"line": 509,
"column": 2
} | [
{
"pp": "G : Type u_3\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ∣ Fintype.card G\nhp' : p - 1 ≠ 0\nScard : p ∣ Fintype.card ↑(vectorsProdEqOne G p)\nf : ℕ → ↑(vectorsProdEqOne G p) → ↑(vectorsProdEqOne G p) := fun k v ↦ VectorsProdEqOne.rotate v k\nhf1 : ∀ (v : ↑(vectorsProd... | [
"G : Type u_3\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ∣ Fintype.card G\nhp' : p - 1 ≠ 0\nScard : p ∣ Fintype.card ↑(vectorsProdEqOne G p)\nf : ℕ → ↑(vectorsProdEqOne G p) → ↑(vectorsProdEqOne G p) := fun k v ↦ VectorsProdEqOne.rotate v k\nhf1 : ∀ (v : ↑(vectorsProdEqOne G p)),... | have hf3 : ∀ v, f p v = v := VectorsProdEqOne.rotate_length | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.Multilinear.Basic | {
"line": 128,
"column": 30
} | {
"line": 128,
"column": 52
} | {
"line": 130,
"column": 0
} | [
{
"pp": "R : Type uR\nS : Type uS\nι : Type uι\nn : ℕ\nM : Fin n.succ → Type v\nM₁ : ι → Type v₁\nM₁' : ι → Type v₁'\nM₁'' : ι → Type v₁''\nM₂ : Type v₂\nM₃ : Type v₃\nM₄ : Type v₄\nM' : Type v'\ninst✝¹² : Semiring R\ninst✝¹¹ : (i : Fin n.succ) → AddCommMonoid (M i)\ninst✝¹⁰ : (i : ι) → AddCommMonoid (M₁ i)\nin... | [] | by convert! h₂ m i c x | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Alternating.Basic | {
"line": 658,
"column": 40
} | {
"line": 662,
"column": 94
} | {
"line": 664,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : Semiring R\nM : Type u_2\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nN : Type u_3\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type u_7\nf : M [⋀^ι]→ₗ[R] N\nv : ι → M\ninst✝ : DecidableEq ι\ni j : ι\nhij : i ≠ j\n⊢ f (v ∘ ⇑(Equiv.swap i j)) + f v = 0",
"ppTerm": "?... | [] | by
rw [Equiv.comp_swap_eq_update]
convert! f.map_update_update v hij (v i + v j)
simp [f.map_update_self _ hij, f.map_update_self _ hij.symm,
Function.update_comm hij (v i + v j) (v _) v, Function.update_comm hij.symm (v i) (v i) v] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.SemiringInverse | {
"line": 285,
"column": 2
} | {
"line": 285,
"column": 47
} | {
"line": 286,
"column": 2
} | [
{
"pp": "n✝ : Type u_1\nm : Type u_2\nR : Type u_3\ninst✝⁴ : Fintype m\ninst✝³ : Fintype n✝\ninst✝² : DecidableEq m\ninst✝¹ : DecidableEq n✝\ninst✝ : CommSemiring R\ns : ℤˣ\nA✝ B✝ : Matrix n✝ n✝ R\ni j : n✝\nn : ℕ\nA B : Matrix (Fin n) (Fin n) R\nhAB : A * B = 1\nh0 : 1 + (detp 1 A * detp (-1) B + detp (-1) A *... | [
"n✝ : Type u_1\nm : Type u_2\nR : Type u_3\ninst✝⁴ : Fintype m\ninst✝³ : Fintype n✝\ninst✝² : DecidableEq m\ninst✝¹ : DecidableEq n✝\ninst✝ : CommSemiring R\ns : ℤˣ\nA✝ B✝ : Matrix n✝ n✝ R\ni j : n✝\nn : ℕ\nA B : Matrix (Fin n) (Fin n) R\nhAB : A * B = 1\nh0 : 1 + (detp 1 A * detp (-1) B + detp (-1) A * detp 1 B) =... | replace h := congr(B * $(detp_smul_adjp hAB)) | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.CategoryTheory.Filtered.Basic | {
"line": 293,
"column": 2
} | {
"line": 306,
"column": 32
} | {
"line": 308,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : IsFiltered C\nJ : Type w\ninst✝¹ : SmallCategory J\ninst✝ : FinCategory J\nF : J ⥤ C\n⊢ Nonempty (Cocone F)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor",
"CategoryTheory.Limits... | [] | classical
let O := Finset.univ.image F.obj
let H : Finset (Σ' (X Y : C) (_ : X ∈ O) (_ : Y ∈ O), X ⟶ Y) :=
Finset.univ.biUnion fun X : J => Finset.univ.biUnion fun Y : J =>
Finset.univ.image fun f : X ⟶ Y => ⟨F.obj X, F.obj Y, by simp [O], by simp [O], F.map f⟩
obtain ⟨Z, f, w⟩ := sup_exists O H
refin... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.CategoryTheory.Filtered.Basic | {
"line": 293,
"column": 2
} | {
"line": 306,
"column": 32
} | {
"line": 308,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : IsFiltered C\nJ : Type w\ninst✝¹ : SmallCategory J\ninst✝ : FinCategory J\nF : J ⥤ C\n⊢ Nonempty (Cocone F)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor",
"CategoryTheory.Limits... | [] | classical
let O := Finset.univ.image F.obj
let H : Finset (Σ' (X Y : C) (_ : X ∈ O) (_ : Y ∈ O), X ⟶ Y) :=
Finset.univ.biUnion fun X : J => Finset.univ.biUnion fun Y : J =>
Finset.univ.image fun f : X ⟶ Y => ⟨F.obj X, F.obj Y, by simp [O], by simp [O], F.map f⟩
obtain ⟨Z, f, w⟩ := sup_exists O H
refin... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Filtered.Basic | {
"line": 293,
"column": 2
} | {
"line": 306,
"column": 32
} | {
"line": 308,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : IsFiltered C\nJ : Type w\ninst✝¹ : SmallCategory J\ninst✝ : FinCategory J\nF : J ⥤ C\n⊢ Nonempty (Cocone F)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor",
"CategoryTheory.Limits... | [] | classical
let O := Finset.univ.image F.obj
let H : Finset (Σ' (X Y : C) (_ : X ∈ O) (_ : Y ∈ O), X ⟶ Y) :=
Finset.univ.biUnion fun X : J => Finset.univ.biUnion fun Y : J =>
Finset.univ.image fun f : X ⟶ Y => ⟨F.obj X, F.obj Y, by simp [O], by simp [O], F.map f⟩
obtain ⟨Z, f, w⟩ := sup_exists O H
refin... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Multilinear.Basic | {
"line": 970,
"column": 8
} | {
"line": 970,
"column": 73
} | {
"line": 970,
"column": 74
} | [
{
"pp": "R : Type uR\nS : Type uS\nι : Type uι\nn : ℕ\nM : Fin n.succ → Type v\nM₁ : ι → Type v₁\nM₁' : ι → Type v₁'\nM₁'' : ι → Type v₁''\nM₂ : Type v₂\nM₃ : Type v₃\nM₄ : Type v₄\nM' : Type v'\ninst✝¹² : Semiring R\ninst✝¹¹ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝¹⁰ : (i : ι) → Module R (M₁ i)\ninst✝⁹ : AddCom... | [] | simp only [Function.comp, piCongrLeft'_update, f.map_update_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Multilinear.Basic | {
"line": 1106,
"column": 19
} | {
"line": 1106,
"column": 21
} | {
"line": 1107,
"column": 4
} | [
{
"pp": "R : Type uR\nS : Type uS\nι : Type uι\nn : ℕ\nM : Fin n.succ → Type v\nM₁ : ι → Type v₁\nM₁' : ι → Type v₁'\nM₁'' : ι → Type v₁''\nM₂ : Type v₂\nM₃ : Type v₃\nM₄ : Type v₄\nM' : Type v'\ninst✝⁹ : CommSemiring R\ninst✝⁸ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁷ : (i : Fin n.succ) → AddCommMonoid (M i)\ni... | [
"R : Type uR\nS : Type uS\nι : Type uι\nn : ℕ\nM : Fin n.succ → Type v\nM₁ : ι → Type v₁\nM₁' : ι → Type v₁'\nM₁'' : ι → Type v₁''\nM₂ : Type v₂\nM₃ : Type v₃\nM₄ : Type v₄\nM' : Type v'\ninst✝⁹ : CommSemiring R\ninst✝⁸ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁷ : (i : Fin n.succ) → AddCommMonoid (M i)\ninst✝⁶ : AddC... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.LinearAlgebra.Matrix.Determinant.Basic | {
"line": 640,
"column": 4
} | {
"line": 657,
"column": 45
} | {
"line": 658,
"column": 4
} | [
{
"pp": "case refine_3\nn : Type u_2\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nR : Type v\ninst✝² : CommRing R\no : Type u_3\ninst✝¹ : Fintype o\ninst✝ : DecidableEq o\nM : o → Matrix n n R\npreserving_snd : Finset (Perm (n × o)) := {σ | ∀ (x : n × o), (σ x).2 = x.2}\nmem_preserving_snd : ∀ {σ : Perm (n × o)... | [
"case refine_4\nn : Type u_2\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nR : Type v\ninst✝² : CommRing R\no : Type u_3\ninst✝¹ : Fintype o\ninst✝ : DecidableEq o\nM : o → Matrix n n R\npreserving_snd : Finset (Perm (n × o)) := {σ | ∀ (x : n × o), (σ x).2 = x.2}\nmem_preserving_snd : ∀ {σ : Perm (n × o)}, σ ∈ prese... | · intro σ hσ
rw [mem_preserving_snd] at hσ
have hσ' x : (σ.symm x).snd = x.snd := by simpa [eq_comm] using hσ (σ.symm x)
have mk_apply_eq : ∀ k x, ((σ (x, k)).fst, k) = σ (x, k) := by
intro k x
ext
· simp only
· simp only [hσ]
have mk_inv_apply_eq : ∀ k x, ((σ.sym... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Multilinear.Basic | {
"line": 1431,
"column": 6
} | {
"line": 1431,
"column": 27
} | {
"line": 1432,
"column": 6
} | [
{
"pp": "case refine_2\nR : Type uR\nS : Type uS\nι : Type uι\nn : ℕ\nM : Fin n.succ → Type v\nM₁ : ι → Type v₁\nM₁' : ι → Type v₁'\nM₁'' : ι → Type v₁''\nM₂ : Type v₂\nM₃ : Type v₃\nM₄ : Type v₄\nM' : Type v'\ninst✝⁷ : Ring R\ninst✝⁶ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : AddComm... | [
"case refine_2\nR : Type uR\nS : Type uS\nι : Type uι\nn : ℕ\nM : Fin n.succ → Type v\nM₁ : ι → Type v₁\nM₁' : ι → Type v₁'\nM₁'' : ι → Type v₁''\nM₂ : Type v₂\nM₃ : Type v₃\nM₄ : Type v₄\nM' : Type v'\ninst✝⁷ : Ring R\ninst✝⁶ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : AddCommMonoid M₂\ni... | rw [update_of_ne hij] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Category.MonCat.Limits | {
"line": 70,
"column": 14
} | {
"line": 70,
"column": 26
} | {
"line": 71,
"column": 2
} | [
{
"pp": "J : Type v\ninst✝¹ : Category.{w, v} J\nF : J ⥤ MonCat\ninst✝ : Small.{u, max u v} ↑(F ⋙ forget MonCat).sections\nj : J\n⊢ (ConcreteCategory.hom ((Types.Small.limitCone (F ⋙ forget MonCat)).π.app j)) 1 = 1",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonC... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.MonCat.Limits | {
"line": 71,
"column": 18
} | {
"line": 71,
"column": 30
} | {
"line": 73,
"column": 0
} | [
{
"pp": "J : Type v\ninst✝¹ : Category.{w, v} J\nF : J ⥤ MonCat\ninst✝ : Small.{u, max u v} ↑(F ⋙ forget MonCat).sections\nj : J\nx✝¹ x✝ : (Types.Small.limitCone (F ⋙ forget MonCat)).pt\n⊢ (ConcreteCategory.hom ((Types.Small.limitCone (F ⋙ forget MonCat)).π.app j)) (x✝¹ * x✝) =\n (ConcreteCategory.hom ((Type... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.MonCat.Limits | {
"line": 178,
"column": 20
} | {
"line": 178,
"column": 32
} | {
"line": 178,
"column": 32
} | [
{
"pp": "J : Type v\ninst✝¹ : Category.{w, v} J\nF : J ⥤ MonCat\ninst✝ : Small.{u, max u v} ↑(F ⋙ forget MonCat).sections\nc : Cone (F ⋙ forget MonCat)\nt : IsLimit c\nthis : Small.{u, max u v} ↑(F ⋙ forget MonCat).sections\ns✝ : Cone F\n⊢ (ConcreteCategory.hom ((Types.Small.limitConeIsLimit (F ⋙ forget MonCat)... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.MonCat.Limits | {
"line": 340,
"column": 2
} | {
"line": 340,
"column": 35
} | {
"line": 342,
"column": 0
} | [
{
"pp": "J : Type v\ninst✝¹ : Category.{w, v} J\nF : J ⥤ CommMonCat\ninst✝ : Small.{u, max u v} ↑(F ⋙ forget CommMonCat).sections\ne : forget CommMonCat ≅ forget₂ CommMonCat MonCat ⋙ forget MonCat :=\n NatIso.ofComponents (fun x ↦ Iso.refl ((forget CommMonCat).obj x)) ⋯\n⊢ CreatesLimit F (forget CommMonCat)",
... | [] | exact createsLimitOfNatIso e.symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Category.ModuleCat.Limits | {
"line": 89,
"column": 18
} | {
"line": 89,
"column": 30
} | {
"line": 91,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : Ring R\nJ : Type v\ninst✝¹ : Category.{t, v} J\nF : J ⥤ ModuleCat R\ninst✝ : Small.{w, max v w} ↑(F ⋙ forget (ModuleCat R)).sections\nj : J\nx✝¹ x✝ : (Types.Small.limitCone (F ⋙ forget (ModuleCat R))).pt\n⊢ (ConcreteCategory.hom ((Types.Small.limitCone (F ⋙ forget (ModuleCat R))).π... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.ModuleCat.Limits | {
"line": 88,
"column": 19
} | {
"line": 88,
"column": 31
} | {
"line": 89,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝² : Ring R\nJ : Type v\ninst✝¹ : Category.{t, v} J\nF : J ⥤ ModuleCat R\ninst✝ : Small.{w, max v w} ↑(F ⋙ forget (ModuleCat R)).sections\nj : J\nx✝¹ : R\nx✝ : (Types.Small.limitCone (F ⋙ forget (ModuleCat R))).pt\n⊢ (ConcreteCategory.hom ((Types.Small.limitCone (F ⋙ forget (ModuleCat R... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Tactic.CategoryTheory.Monoidal.PureCoherence | {
"line": 235,
"column": 18
} | {
"line": 236,
"column": 48
} | {
"line": 237,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nf g f' : C\nη θ : f ⟶ g\nη' θ' : f ≅ g\nη_f : 𝟙_ C ⊗ f ≅ f'\nη_g : 𝟙_ C ⊗ g ≅ f'\nη_hom : η'.hom = η\nΘ_hom : θ'.hom = θ\nHη : 𝟙_ C ◁ η' ≪≫ η_g = η_f\nHθ : 𝟙_ C ◁ θ' ≪≫ η_g = η_f\n⊢ (λ_ f).inv ≫ η_f.hom ≫ η_g.inv ≫ (λ_ g).hom = θ'.... | [] | by
simp [← reassoc_of% (congrArg Iso.hom Hθ)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Functor | {
"line": 421,
"column": 44
} | {
"line": 421,
"column": 48
} | {
"line": 421,
"column": 49
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : MonoidalCategory D\nF : C ⥤ D\ninst✝ : F.Monoidal\nT : D\n⊢ (η F ≫ ε F) ▷ T = 𝟙 (F.obj (𝟙_ C) ⊗ T)",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"E... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : MonoidalCategory D\nF : C ⥤ D\ninst✝ : F.Monoidal\nT : D\n⊢ 𝟙 (F.obj (𝟙_ C)) ▷ T = 𝟙 (F.obj (𝟙_ C) ⊗ T)"
] | η_ε, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monoidal.Functor | {
"line": 437,
"column": 43
} | {
"line": 437,
"column": 47
} | {
"line": 437,
"column": 48
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : MonoidalCategory D\nF : C ⥤ D\ninst✝ : F.Monoidal\nT : D\n⊢ T ◁ (η F ≫ ε F) = 𝟙 (T ⊗ F.obj (𝟙_ C))",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"E... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : MonoidalCategory D\nF : C ⥤ D\ninst✝ : F.Monoidal\nT : D\n⊢ T ◁ 𝟙 (F.obj (𝟙_ C)) = 𝟙 (T ⊗ F.obj (𝟙_ C))"
] | η_ε, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monoidal.Functor | {
"line": 564,
"column": 4
} | {
"line": 564,
"column": 37
} | {
"line": 566,
"column": 0
} | [
{
"pp": "case δ\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : MonoidalCategory D\nF : C ⥤ D\na b : F.Monoidal\neq : a.toLaxMonoidal = b.toLaxMonoidal\nx✝¹ x✝ : C\n⊢ (μIso F x✝¹ x✝).hom ≫ δ F x✝¹ x✝ = μ F x✝¹ x✝ ≫ δ F x✝¹ x✝",
"ppT... | [] | exact congr(($eq.symm).μ _ _ ≫ _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Monoidal.Braided.Basic | {
"line": 243,
"column": 4
} | {
"line": 252,
"column": 99
} | {
"line": 254,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Category.{v_2, u_2} D\ninst✝⁴ : MonoidalCategory C\ninst✝³ : MonoidalCategory D\nF : C ⥤ D\ninst✝² : F.Monoidal\ninst✝¹ : F.Faithful\ninst✝ : BraidedCategory D\nβ : (X Y : C) → X ⊗ Y ≅ Y ⊗ X\nw : ∀ (X Y : C), μ F X Y ≫ F.map (β X Y).h... | [] | intros
apply F.map_injective
refine (cancel_epi (μ F _ _)).1 ?_
refine (cancel_epi (_ ◁ μ F _ _)).1 ?_
rw [Functor.map_comp, Functor.map_comp, Functor.map_comp, Functor.map_comp, ←
μ_natural_right_assoc, ← whiskerLeft_comp_assoc, w,
whiskerLeft_comp_assoc, Functor.LaxMonoidal.associativity_i... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Braided.Basic | {
"line": 243,
"column": 4
} | {
"line": 252,
"column": 99
} | {
"line": 254,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Category.{v_2, u_2} D\ninst✝⁴ : MonoidalCategory C\ninst✝³ : MonoidalCategory D\nF : C ⥤ D\ninst✝² : F.Monoidal\ninst✝¹ : F.Faithful\ninst✝ : BraidedCategory D\nβ : (X Y : C) → X ⊗ Y ≅ Y ⊗ X\nw : ∀ (X Y : C), μ F X Y ≫ F.map (β X Y).h... | [] | intros
apply F.map_injective
refine (cancel_epi (μ F _ _)).1 ?_
refine (cancel_epi (_ ◁ μ F _ _)).1 ?_
rw [Functor.map_comp, Functor.map_comp, Functor.map_comp, Functor.map_comp, ←
μ_natural_right_assoc, ← whiskerLeft_comp_assoc, w,
whiskerLeft_comp_assoc, Functor.LaxMonoidal.associativity_i... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Opposites.Pullbacks | {
"line": 215,
"column": 4
} | {
"line": 216,
"column": 59
} | {
"line": 217,
"column": 2
} | [
{
"pp": "case f\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nJ : Type u₂\ninst✝ : Category.{v₂, u₂} J\nX Y Z : Cᵒᵖ\nf : X ⟶ Y\ng : X ⟶ Z\nc : PushoutCocone f g\nh : IsColimit c\n⊢ IsLimit c.unop",
"ppTerm": "?f",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor.op",
"CategoryT... | [] | exact ((IsColimit.precomposeHomEquiv _ _).invFun
((IsColimit.whiskerEquivalenceEquiv _).toFun h)).unop | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Monoidal.Braided.Basic | {
"line": 677,
"column": 2
} | {
"line": 678,
"column": 29
} | {
"line": 679,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX₁ X₂ : C\n⊢ (ρ_ (X₁ ⊗ X₂)).hom =\n (((((α_ X₁ X₂ (𝟙_ C)).hom ≫ X₁ ◁ (ρ_ X₂).inv ▷ 𝟙_ C) ≫ X₁ ◁ (β_ X₂ (𝟙_ C)).hom ▷ 𝟙_ C) ≫\n X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).hom) ≫\n (α_ X₁ (𝟙_ C) (... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX₁ X₂ : C\n⊢ (ρ_ (X₁ ⊗ X₂)).hom =\n (α_ X₁ X₂ (𝟙_ C)).hom ≫\n ((X₁ ◁ (λ_ X₂).inv ▷ 𝟙_ C ≫ X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).hom) ≫ (α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).inv) ≫\n ((ρ_ X₁).hom ⊗ₘ (ρ_ X₂).hom)"
] | slice_rhs 2 3 => rw [← whiskerLeft_comp, ← comp_whiskerRight,
rightUnitor_inv_braiding] | Mathlib.Tactic.Slice._aux_Mathlib_Tactic_CategoryTheory_Slice___macroRules_Mathlib_Tactic_Slice_sliceRHS_1 | Mathlib.Tactic.Slice.sliceRHS |
Mathlib.CategoryTheory.Adjunction.Mates | {
"line": 181,
"column": 41
} | {
"line": 181,
"column": 65
} | {
"line": 181,
"column": 65
} | [
{
"pp": "case a.a\nA : Type u₁\nB : Type u₂\nC : Type u₃\nD : Type u₄\nE : Type u₅\nF : Type u₆\ninst✝⁵ : Category.{v₁, u₁} A\ninst✝⁴ : Category.{v₂, u₂} B\ninst✝³ : Category.{v₃, u₃} C\ninst✝² : Category.{v₄, u₄} D\ninst✝¹ : Category.{v₅, u₅} E\ninst✝ : Category.{v₆, u₆} F\nG₁ : A ⥤ C\nG₂ : C ⥤ E\nH₁ : B ⥤ D\n... | [
"case a.a\nA : Type u₁\nB : Type u₂\nC : Type u₃\nD : Type u₄\nE : Type u₅\nF : Type u₆\ninst✝⁵ : Category.{v₁, u₁} A\ninst✝⁴ : Category.{v₂, u₂} B\ninst✝³ : Category.{v₃, u₃} C\ninst✝² : Category.{v₄, u₄} D\ninst✝¹ : Category.{v₅, u₅} E\ninst✝ : Category.{v₆, u₆} F\nG₁ : A ⥤ C\nG₂ : C ⥤ E\nH₁ : B ⥤ D\nH₂ : D ⥤ F\n... | ← unit_naturality (adj₃) | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.RingTheory.Congruence.Hom | {
"line": 375,
"column": 44
} | {
"line": 375,
"column": 59
} | {
"line": 375,
"column": 59
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝² : NonAssocSemiring M\ninst✝¹ : NonAssocSemiring N\ninst✝ : NonAssocSemiring P\nc✝ : RingCon M\nf : M →+* P\nc d : RingCon M\nh : c ≤ d\nx y : (ker (c.map d h)).Quotient\nw z : c.Quotient\na b : M\n⊢ (c.quotientQuotientEquivQuotient d.toSetoid h).toFun (↑... | [
"M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝² : NonAssocSemiring M\ninst✝¹ : NonAssocSemiring N\ninst✝ : NonAssocSemiring P\nc✝ : RingCon M\nf : M →+* P\nc d : RingCon M\nh : c ≤ d\nx y : (ker (c.map d h)).Quotient\nw z : c.Quotient\na b : M\n⊢ (c.quotientQuotientEquivQuotient d.toSetoid h).toFun (↑↑a + ↑↑b) = ... | ← d.mk'.map_add | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Category.AlgCat.TensorAlgebra | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 62
} | {
"line": 66,
"column": 2
} | [
{
"pp": "R✝ : Type u\ninst✝² : CommRing R✝\nR : Type v\ninst✝¹ : CommRing R\ninst✝ : Small.{u, v} R\ne : AlgCat R ≌ AlgCat (Shrink.{u, v} R) := restrictScalarsEquivalenceOfRingEquiv (Shrink.ringEquiv R)\nthis :\n e.inverse ⋙ forget₂ (AlgCat R) (ModuleCat R) =\n forget₂ (AlgCat (Shrink.{u, v} R)) (ModuleCat ... | [
"R✝ : Type u\ninst✝² : CommRing R✝\nR : Type v\ninst✝¹ : CommRing R\ninst✝ : Small.{u, v} R\ne : AlgCat R ≌ AlgCat (Shrink.{u, v} R) := restrictScalarsEquivalenceOfRingEquiv (Shrink.ringEquiv R)\nthis :\n e.inverse ⋙ forget₂ (AlgCat R) (ModuleCat R) =\n forget₂ (AlgCat (Shrink.{u, v} R)) (ModuleCat (Shrink.{u, ... | rw [← Functor.isRightAdjoint_comp_iff_right e.inverse, this] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.TrivSqZeroExt.Basic | {
"line": 1113,
"column": 2
} | {
"line": 1113,
"column": 75
} | {
"line": 1115,
"column": 0
} | [
{
"pp": "R' : Type u\nM : Type v\ninst✝⁴ : CommSemiring R'\ninst✝³ : AddCommMonoid M\ninst✝² : Module R' M\ninst✝¹ : Module R'ᵐᵒᵖ M\ninst✝ : IsCentralScalar R' M\n⊢ ∀ (m : M), (map LinearMap.id) (inr m) = (AlgHom.id R' (tsze R' M)) (inr m)",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
... | [] | simp only [map_inr, LinearMap.id_coe, id_eq, AlgHom.coe_id, forall_const] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Bialgebra.TensorProduct | {
"line": 175,
"column": 4
} | {
"line": 177,
"column": 90
} | {
"line": 179,
"column": 0
} | [
{
"pp": "case ha\nR : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\nC : Type u_5\nD : Type u_6\ninst✝⁵ : CommSemiring R\ninst✝⁴ : CommSemiring S\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Bialgebra R A\ninst✝ : Bialgebra R B\na : A\n⊢ (((Algebra.TensorProduct.map ↑(Algebra.TensorProduct.comm R A ... | [] | · dsimp
rw [← (ℛ R a).eq]
simp [TensorProduct.tmul_sum, TensorProduct.sum_tmul, Algebra.TensorProduct.one_def] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Bialgebra.TensorProduct | {
"line": 175,
"column": 4
} | {
"line": 177,
"column": 90
} | {
"line": 179,
"column": 0
} | [
{
"pp": "case hb\nR : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\nC : Type u_5\nD : Type u_6\ninst✝⁵ : CommSemiring R\ninst✝⁴ : CommSemiring S\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Bialgebra R A\ninst✝ : Bialgebra R B\na : B\n⊢ ((AlgHom.restrictScalars R\n ((Algebra.TensorProduc... | [] | · dsimp
rw [← (ℛ R a).eq]
simp [TensorProduct.tmul_sum, TensorProduct.sum_tmul, Algebra.TensorProduct.one_def] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Category.Pointed | {
"line": 113,
"column": 37
} | {
"line": 113,
"column": 49
} | {
"line": 115,
"column": 0
} | [
{
"pp": "X✝ Y✝ Z✝ : Type u\nx✝¹ : X✝ ⟶ Y✝\nx✝ : Y✝ ⟶ Z✝\n⊢ { toFun := Option.map ⇑(ConcreteCategory.hom (x✝¹ ≫ x✝)), map_point := ⋯ }.toFun =\n ({ toFun := Option.map ⇑(ConcreteCategory.hom x✝¹), map_point := ⋯ } ≫\n { toFun := Option.map ⇑(ConcreteCategory.hom x✝), map_point := ⋯ }).toFun",
"ppTe... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.Ring.Adjunctions | {
"line": 94,
"column": 18
} | {
"line": 94,
"column": 80
} | {
"line": 96,
"column": 0
} | [
{
"pp": "R : CommRingCat\nX✝ Y✝ Z✝ : CommMonCat\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\n⊢ Under.homMk (ofHom (MonoidAlgebra.mapDomainRingHom (↑R) (CommMonCat.Hom.hom (f ≫ g)))) ⋯ =\n Under.homMk (ofHom (MonoidAlgebra.mapDomainRingHom (↑R) (CommMonCat.Hom.hom f))) ⋯ ≫\n Under.homMk (ofHom (MonoidAlgebra.mapDomainRing... | [] | by ext : 2; apply MonoidAlgebra.ringHom_ext <;> intro <;> simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.CoalgCat.ComonEquivalence | {
"line": 166,
"column": 2
} | {
"line": 167,
"column": 50
} | {
"line": 168,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝⁶ : CommRing R\nM N : Type u\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R M\ninst✝² : Module R N\ninst✝¹ : Coalgebra R M\ninst✝ : Coalgebra R N\n⊢ ModuleCat.Hom.hom Δ = CoalgebraStruct.comul",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
... | [
"R : Type u\ninst✝⁶ : CommRing R\nM N : Type u\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R M\ninst✝² : Module R N\ninst✝¹ : Coalgebra R M\ninst✝ : Coalgebra R N\n⊢ ↑(tensorTensorTensorComm R ↑((comonEquivalence R).symm.inverse.obj (of R M)).X\n ↑((comonEquivalence R).symm.inverse.o... | simp [tensorμ_eq_tensorTensorTensorComm, TensorProduct.comul_def,
AlgebraTensorModule.tensorTensorTensorComm_eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Category.CoalgCat.ComonEquivalence | {
"line": 178,
"column": 2
} | {
"line": 179,
"column": 50
} | {
"line": 180,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝⁹ : CommRing R\nM N P : Type u\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : AddCommGroup P\ninst✝⁵ : Module R M\ninst✝⁴ : Module R N\ninst✝³ : Module R P\ninst✝² : Coalgebra R M\ninst✝¹ : Coalgebra R N\ninst✝ : Coalgebra R P\n⊢ ModuleCat.Hom.hom MonObj.mul.unop = Coalgeb... | [
"R : Type u\ninst✝⁹ : CommRing R\nM N P : Type u\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : AddCommGroup P\ninst✝⁵ : Module R M\ninst✝⁴ : Module R N\ninst✝³ : Module R P\ninst✝² : Coalgebra R M\ninst✝¹ : Coalgebra R N\ninst✝ : Coalgebra R P\n⊢ ↑(tensorTensorTensorComm R ↑((comonEquivalence R).symm.... | simp [tensorμ_eq_tensorTensorTensorComm, TensorProduct.comul_def,
AlgebraTensorModule.tensorTensorTensorComm_eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Category.CoalgCat.ComonEquivalence | {
"line": 192,
"column": 2
} | {
"line": 193,
"column": 50
} | {
"line": 194,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝⁹ : CommRing R\nM N P : Type u\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : AddCommGroup P\ninst✝⁵ : Module R M\ninst✝⁴ : Module R N\ninst✝³ : Module R P\ninst✝² : Coalgebra R M\ninst✝¹ : Coalgebra R N\ninst✝ : Coalgebra R P\n⊢ ModuleCat.Hom.hom\n (tensorμ (Opposi... | [
"R : Type u\ninst✝⁹ : CommRing R\nM N P : Type u\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : AddCommGroup P\ninst✝⁵ : Module R M\ninst✝⁴ : Module R N\ninst✝³ : Module R P\ninst✝² : Coalgebra R M\ninst✝¹ : Coalgebra R N\ninst✝ : Coalgebra R P\n⊢ ↑(tensorTensorTensorComm R (↑(of R M ⊗ of R N).toModule... | simp [tensorμ_eq_tensorTensorTensorComm, TensorProduct.comul_def,
AlgebraTensorModule.tensorTensorTensorComm_eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity | {
"line": 164,
"column": 2
} | {
"line": 168,
"column": 36
} | {
"line": 169,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_2\ninst✝¹ : CommMonoidWithZero R\ninst✝ : UniqueFactorizationMonoid R\na b : R\nha : a ≠ 0\nh : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b\nhb : ¬b = 0\nthis : StrongNormalizationMonoid R := UniqueFactorizationMonoid.strongNormalizationMonoid\nq : R\nhq : q ∈ normal... | [
"case neg\nR : Type u_2\ninst✝¹ : CommMonoidWithZero R\ninst✝ : UniqueFactorizationMonoid R\na b : R\nha : a ≠ 0\nh : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b\nhb : ¬b = 0\nthis : StrongNormalizationMonoid R := UniqueFactorizationMonoid.strongNormalizationMonoid\nq : R\nhq : q ∉ normalizedFactors ... | · have hqprime : Prime q := prime_of_normalized_factor q hq
have h1 := emultiplicity_eq_count_normalizedFactors hqprime.irreducible ha
have h2 := emultiplicity_eq_count_normalizedFactors hqprime.irreducible hb
rw [normalize_normalized_factor q hq] at h1 h2
simpa [h1, h2] using h q hqprime | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.LocalRing.Basic | {
"line": 58,
"column": 17
} | {
"line": 58,
"column": 57
} | {
"line": 60,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Semiring R\ninst✝ : IsLocalRing R\nι : Type u_3\ns : Finset ι\nf : ι → R\nh : ∀ i ∈ s, ¬IsUnit (f i)\n⊢ ¬IsUnit (∑ i ∈ s, f i)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"IsLocalRing.nonunitsAddSubmonoid",
"AddSubmonoid.sum_mem",
"Semi... | [] | exact (nonunitsAddSubmonoid R).sum_mem h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Limits.Shapes.StrictInitial | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 44
} | {
"line": 95,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasStrictInitialObjects C\nI X : C\ninst✝ : HasBinaryProduct X I\nhI : IsInitial I\n⊢ X ⨯ I ≅ I",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"CategoryTheory.IsIso",
"CategoryTheory.Limits.IsInitial.isIso_to",
"C... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasStrictInitialObjects C\nI X : C\ninst✝ : HasBinaryProduct X I\nhI : IsInitial I\nthis : IsIso prod.snd\n⊢ X ⨯ I ≅ I"
] | have := hI.isIso_to (prod.snd : X ⨯ I ⟶ I) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.LocalRing.RingHom.Basic | {
"line": 120,
"column": 20
} | {
"line": 124,
"column": 94
} | {
"line": 126,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Nontrivial S\ninst✝ : IsLocalRing R\nf : R →+* S\nhf : Function.Surjective ⇑f\n⊢ IsLocalHom f",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"RingHom.instRingHomClass",
"Semiring.toModule",... | [] | by
have := IsLocalRing.of_surjective' f ‹_›
refine ((local_hom_TFAE f).out 3 0).mp ?_
have := Ideal.comap_isMaximal_of_surjective f hf (K := maximalIdeal S)
exact ((maximal_ideal_unique R).unique (inferInstanceAs (maximalIdeal R).IsMaximal) this).le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Mon | {
"line": 851,
"column": 13
} | {
"line": 851,
"column": 33
} | {
"line": 852,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝¹⁴ : Category.{v₁, u₁} C\ninst✝¹³ : MonoidalCategory C\nM N O X✝ : C\ninst✝¹² : MonObj M\ninst✝¹¹ : MonObj N\ninst✝¹⁰ : MonObj O\nD : Type u₂\ninst✝⁹ : Category.{v₂, u₂} D\ninst✝⁸ : MonoidalCategory D\nE : Type u₃\ninst✝⁷ : Category.{v₃, u₃} E\ninst✝⁶ : MonoidalCategory E\nF F' : C ⥤ ... | [] | by simp [← map_comp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Mon | {
"line": 852,
"column": 13
} | {
"line": 852,
"column": 33
} | {
"line": 854,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝¹⁴ : Category.{v₁, u₁} C\ninst✝¹³ : MonoidalCategory C\nM N O X✝ : C\ninst✝¹² : MonObj M\ninst✝¹¹ : MonObj N\ninst✝¹⁰ : MonObj O\nD : Type u₂\ninst✝⁹ : Category.{v₂, u₂} D\ninst✝⁸ : MonoidalCategory D\nE : Type u₃\ninst✝⁷ : Category.{v₃, u₃} E\ninst✝⁶ : MonoidalCategory E\nF F' : C ⥤ ... | [] | by simp [← map_comp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic | {
"line": 91,
"column": 6
} | {
"line": 91,
"column": 28
} | {
"line": 91,
"column": 29
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP X Y Z : C\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\nh : IsPullback fst snd f g\nP' X' Y' Z' : C\nfst' : P' ⟶ X'\nsnd' : P' ⟶ Y'\nf' : X' ⟶ Z'\ng' : Y' ⟶ Z'\ne₁ : P ≅ P'\ne₂ : X ≅ X'\ne₃ : Y ≅ Y'\ne₄ : Z ≅ Z'\ncommfst : fst ≫ e₂.hom = e₁.hom ≫ fst'\ncom... | [
"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP X Y Z : C\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\nh : IsPullback fst snd f g\nP' X' Y' Z' : C\nfst' : P' ⟶ X'\nsnd' : P' ⟶ Y'\nf' : X' ⟶ Z'\ng' : Y' ⟶ Z'\ne₁ : P ≅ P'\ne₂ : X ≅ X'\ne₃ : Y ≅ Y'\ne₄ : Z ≅ Z'\ncommfst : fst ≫ e₂.hom = e₁.hom ≫ fst'\ncommsnd : snd ≫... | ← reassoc_of% commsnd, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic | {
"line": 100,
"column": 18
} | {
"line": 100,
"column": 40
} | {
"line": 100,
"column": 41
} | [
{
"pp": "case refine_2\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP X Y Z : C\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\nh : IsPullback fst snd f g\nP' X' Y' Z' : C\nfst' : P' ⟶ X'\nsnd' : P' ⟶ Y'\nf' : X' ⟶ Z'\ng' : Y' ⟶ Z'\ne₁ : P ≅ P'\ne₂ : X ≅ X'\ne₃ : Y ≅ Y'\ne₄ : Z ≅ Z'\ncommfst : fst ≫ e₂.hom = e₁.... | [
"case refine_2\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP X Y Z : C\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\nh : IsPullback fst snd f g\nP' X' Y' Z' : C\nfst' : P' ⟶ X'\nsnd' : P' ⟶ Y'\nf' : X' ⟶ Z'\ng' : Y' ⟶ Z'\ne₁ : P ≅ P'\ne₂ : X ≅ X'\ne₃ : Y ≅ Y'\ne₄ : Z ≅ Z'\ncommfst : fst ≫ e₂.hom = e₁.hom ≫ fst'\n... | ← reassoc_of% commsnd, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic | {
"line": 522,
"column": 2
} | {
"line": 522,
"column": 29
} | {
"line": 523,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nZ X Y P : C\nf : Z ⟶ X\ng : Z ⟶ Y\ninl : X ⟶ P\ninr : Y ⟶ P\nh : IsPushout f g inl inr\ninst : IsIso g\nthis : HasPushout f g\n⊢ IsIso inl",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.IsIso",
... | [
"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nZ X Y P : C\nf : Z ⟶ X\ng : Z ⟶ Y\ninl : X ⟶ P\ninr : Y ⟶ P\nh : IsPushout f g inl inr\ninst : IsIso g\nthis : HasPushout f g\n⊢ IsIso (pushout.inl f g ≫ h.isoPushout.inv)"
] | rw [← h.inl_isoPushout_inv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.FreeCommRing | {
"line": 366,
"column": 2
} | {
"line": 367,
"column": 16
} | {
"line": 368,
"column": 2
} | [
{
"pp": "α : Type u\nx : FreeRing α\n⊢ ↑x = (fun l ↦ ↑l) <$> x",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"FreeCommRing",
"Monad.toApplicative",
"Multiset",
"id",
"FreeAbelianGroup.instMonad",
"List",
"Multiset.ofList",
"Applicative.toFu... | [
"α : Type u\nx : FreeRing α\n⊢ (FreeAbelianGroup.lift ⇑(FreeMonoid.lift FreeCommRing.of)) x =\n (FreeAbelianGroup.lift ((fun α_1 ↦ FreeAbelianGroup.of α_1) ∘ fun l ↦ ↑l)) x"
] | dsimp [castFreeCommRing, toFreeCommRing, FreeRing.lift, FreeRing, FreeAbelianGroup.liftMonoid_coe,
Functor.map] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
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