module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic | {
"line": 902,
"column": 2
} | {
"line": 902,
"column": 35
} | {
"line": 904,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nH :\n ∀ ⦃X Y Z : C⦄ (f : Z ⟶ X) (g : Z ⟶ Y) [HasPushout f g],\n ∃ P inl inr, IsPushout f g inl inr ∧ IsPushout (F.map f) (F.map g) (F.map inl) (F.map inr)\nX Y Z : C\nf : Z ⟶ X\ng : Z ⟶ Y\nh✝ : HasColimi... | [] | rwa [h.preservesColimit_span_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Algebra.Module.LocalizedModule.Basic | {
"line": 76,
"column": 6
} | {
"line": 76,
"column": 53
} | {
"line": 77,
"column": 6
} | [
{
"pp": "case h\nR : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx✝⁴ x✝³ x✝² : M × ↥S\nm1 : M\ns1 : ↥S\nm2 : M\ns2 : ↥S\nx✝¹ : r S M (m1, s1) (m2, s2)\nm3 : M\ns3 : ↥S\nx✝ : r S M (m2, s2) (m3, s3)\nu1 : ↥S\nhu1 : u1 • (m2, s2).2 • (m1, s1).1 = u1 ... | [
"case h\nR : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx✝⁴ x✝³ x✝² : M × ↥S\nm1 : M\ns1 : ↥S\nm2 : M\ns2 : ↥S\nx✝¹ : r S M (m1, s1) (m2, s2)\nm3 : M\ns3 : ↥S\nx✝ : r S M (m2, s2) (m3, s3)\nu1 : ↥S\nhu1 : u1 • (m2, s2).2 • (m1, s1).1 = u1 • (m1, s1).2... | have hu1' := congr_arg ((u2 * s3) • ·) hu1.symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Adjunction.Limits | {
"line": 297,
"column": 58
} | {
"line": 297,
"column": 75
} | {
"line": 297,
"column": 76
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nJ : Type u\ninst✝ : Category.{v, u} J\nK : J ⥤ D\nX : Cᵒᵖ\nt : (F.op ⋙ (cones J D).obj K).obj X\nj j' : J\nf : j ⟶ j'\n⊢ (unop ((const J).op.obj X)).map f ≫ (adj.homEquiv (unop X) (K... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nJ : Type u\ninst✝ : Category.{v, u} J\nK : J ⥤ D\nX : Cᵒᵖ\nt : (F.op ⋙ (cones J D).obj K).obj X\nj j' : J\nf : j ⟶ j'\n⊢ (adj.homEquiv (unop X) (K.obj j')) (t.app j') =\n (adj.homEquiv ((unop... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Adjunction.Limits | {
"line": 306,
"column": 63
} | {
"line": 306,
"column": 80
} | {
"line": 306,
"column": 81
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nJ : Type u\ninst✝ : Category.{v, u} J\nK : J ⥤ D\nX : Cᵒᵖ\nt : ((cones J C).obj (K ⋙ G)).obj X\nj j' : J\nf : j ⟶ j'\n⊢ (unop ((const J).op.obj (F.op.obj X))).map f ≫ (adj.homEquiv (... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nJ : Type u\ninst✝ : Category.{v, u} J\nK : J ⥤ D\nX : Cᵒᵖ\nt : ((cones J C).obj (K ⋙ G)).obj X\nj j' : J\nf : j ⟶ j'\n⊢ (adj.homEquiv (unop X) (K.obj j')).symm (t.app j') =\n (adj.homEquiv (u... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monad.Products | {
"line": 60,
"column": 30
} | {
"line": 60,
"column": 55
} | {
"line": 60,
"column": 55
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nX : C\ninst✝ : HasBinaryProducts C\nX✝ Y✝ : (prodComonad X).Coalgebra\nf : X✝ ⟶ Y✝\n⊢ f.f ≫ (Over.mk (Y✝.a ≫ prod.fst)).hom = (Over.mk (X✝.a ≫ prod.fst)).hom",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"CategoryTheory.Comonad.Coalg... | [] | simp [← dsimp% f.h_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Monad.Products | {
"line": 60,
"column": 30
} | {
"line": 60,
"column": 55
} | {
"line": 60,
"column": 55
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nX : C\ninst✝ : HasBinaryProducts C\nX✝ Y✝ : (prodComonad X).Coalgebra\nf : X✝ ⟶ Y✝\n⊢ f.f ≫ (Over.mk (Y✝.a ≫ prod.fst)).hom = (Over.mk (X✝.a ≫ prod.fst)).hom",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"CategoryTheory.Comonad.Coalg... | [] | simp [← dsimp% f.h_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monad.Products | {
"line": 60,
"column": 30
} | {
"line": 60,
"column": 55
} | {
"line": 60,
"column": 55
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nX : C\ninst✝ : HasBinaryProducts C\nX✝ Y✝ : (prodComonad X).Coalgebra\nf : X✝ ⟶ Y✝\n⊢ f.f ≫ (Over.mk (Y✝.a ≫ prod.fst)).hom = (Over.mk (X✝.a ≫ prod.fst)).hom",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"CategoryTheory.Comonad.Coalg... | [] | simp [← dsimp% f.h_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.LocalizedModule.Basic | {
"line": 654,
"column": 21
} | {
"line": 654,
"column": 46
} | {
"line": 654,
"column": 47
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM'' : Type u_4\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M''\ninst✝¹ : Module R M\ninst✝ : Module R M''\ng : M →ₗ[R] M''\nh : ∀ (x : ↥S), IsUnit ((algebraMap R (End R M'')) ↑x)\nx y : LocalizedModule S M\na a' : M\nb b' : ↥S\... | [
"R : Type u_1\ninst✝⁴ : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM'' : Type u_4\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M''\ninst✝¹ : Module R M\ninst✝ : Module R M''\ng : M →ₗ[R] M''\nh : ∀ (x : ↥S), IsUnit ((algebraMap R (End R M'')) ↑x)\nx y : LocalizedModule S M\na a' : M\nb b' : ↥S\n⊢ ↑⋯.unit⁻¹... | LocalizedModule.lift'_mk, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.LocalizedModule.Basic | {
"line": 654,
"column": 47
} | {
"line": 654,
"column": 72
} | {
"line": 654,
"column": 73
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM'' : Type u_4\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M''\ninst✝¹ : Module R M\ninst✝ : Module R M''\ng : M →ₗ[R] M''\nh : ∀ (x : ↥S), IsUnit ((algebraMap R (End R M'')) ↑x)\nx y : LocalizedModule S M\na a' : M\nb b' : ↥S\... | [
"R : Type u_1\ninst✝⁴ : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM'' : Type u_4\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M''\ninst✝¹ : Module R M\ninst✝ : Module R M''\ng : M →ₗ[R] M''\nh : ∀ (x : ↥S), IsUnit ((algebraMap R (End R M'')) ↑x)\nx y : LocalizedModule S M\na a' : M\nb b' : ↥S\n⊢ ↑⋯.unit⁻¹... | LocalizedModule.lift'_mk, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.LocalizedModule.Basic | {
"line": 669,
"column": 8
} | {
"line": 669,
"column": 33
} | {
"line": 669,
"column": 34
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM'' : Type u_4\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M''\ninst✝¹ : Module R M\ninst✝ : Module R M''\ng : M →ₗ[R] M''\nh : ∀ (x : ↥S), IsUnit ((algebraMap R (End R M'')) ↑x)\nr : R\nm : LocalizedModule S M\na : M\nb : ↥S\n... | [
"R : Type u_1\ninst✝⁴ : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM'' : Type u_4\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M''\ninst✝¹ : Module R M\ninst✝ : Module R M''\ng : M →ₗ[R] M''\nh : ∀ (x : ↥S), IsUnit ((algebraMap R (End R M'')) ↑x)\nr : R\nm : LocalizedModule S M\na : M\nb : ↥S\n⊢ r • ↑⋯.uni... | LocalizedModule.lift'_mk, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.LocalizedModule.Basic | {
"line": 669,
"column": 60
} | {
"line": 669,
"column": 85
} | {
"line": 670,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM'' : Type u_4\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M''\ninst✝¹ : Module R M\ninst✝ : Module R M''\ng : M →ₗ[R] M''\nh : ∀ (x : ↥S), IsUnit ((algebraMap R (End R M'')) ↑x)\nr : R\nm : LocalizedModule S M\na : M\nb : ↥S\n... | [
"R : Type u_1\ninst✝⁴ : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM'' : Type u_4\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M''\ninst✝¹ : Module R M\ninst✝ : Module R M''\ng : M →ₗ[R] M''\nh : ∀ (x : ↥S), IsUnit ((algebraMap R (End R M'')) ↑x)\nr : R\nm : LocalizedModule S M\na : M\nb : ↥S\n⊢ r • ↑⋯.uni... | LocalizedModule.lift'_mk, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Adjunction.Comma | {
"line": 144,
"column": 4
} | {
"line": 145,
"column": 32
} | {
"line": 147,
"column": 0
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : Category.{v₂, u₂} D\nG : D ⥤ C\nF : C ⥤ D\nh : F ⊣ G\nA : C\ns : Cocone (Functor.empty (StructuredArrow A G))\nm : (asEmptyCocone (StructuredArrow.mk (h.unit.app A))).pt ⟶ s.pt\nx✝ : ∀ (j : Discrete PEmpty.{1}), (asEmptyCocone (StructuredA... | [] | apply StructuredArrow.ext
simp [← StructuredArrow.w m] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Adjunction.Comma | {
"line": 144,
"column": 4
} | {
"line": 145,
"column": 32
} | {
"line": 147,
"column": 0
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : Category.{v₂, u₂} D\nG : D ⥤ C\nF : C ⥤ D\nh : F ⊣ G\nA : C\ns : Cocone (Functor.empty (StructuredArrow A G))\nm : (asEmptyCocone (StructuredArrow.mk (h.unit.app A))).pt ⟶ s.pt\nx✝ : ∀ (j : Discrete PEmpty.{1}), (asEmptyCocone (StructuredA... | [] | apply StructuredArrow.ext
simp [← StructuredArrow.w m] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.LocalizedModule.Basic | {
"line": 855,
"column": 22
} | {
"line": 855,
"column": 70
} | {
"line": 857,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹² : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : AddCommMonoid M''\nA : Type u_5\ninst✝⁸ : CommSemiring A\ninst✝⁷ : Algebra R A\ninst✝⁶ : Module A M'\ninst✝⁵ : IsLocalization S A\ninst✝⁴... | [] | rw [fromLocalizedModule'_smul, RingHom.id_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Module.LocalizedModule.Basic | {
"line": 855,
"column": 22
} | {
"line": 855,
"column": 70
} | {
"line": 857,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹² : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : AddCommMonoid M''\nA : Type u_5\ninst✝⁸ : CommSemiring A\ninst✝⁷ : Algebra R A\ninst✝⁶ : Module A M'\ninst✝⁵ : IsLocalization S A\ninst✝⁴... | [] | rw [fromLocalizedModule'_smul, RingHom.id_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.LocalizedModule.Basic | {
"line": 855,
"column": 22
} | {
"line": 855,
"column": 70
} | {
"line": 857,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹² : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : AddCommMonoid M''\nA : Type u_5\ninst✝⁸ : CommSemiring A\ninst✝⁷ : Algebra R A\ninst✝⁶ : Module A M'\ninst✝⁵ : IsLocalization S A\ninst✝⁴... | [] | rw [fromLocalizedModule'_smul, RingHom.id_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.Ring.Constructions | {
"line": 395,
"column": 72
} | {
"line": 397,
"column": 19
} | {
"line": 399,
"column": 0
} | [
{
"pp": "A B : CommRingCat\nf g : A ⟶ B\n⊢ ofHom ((Hom.hom f).eqLocus (Hom.hom g)).subtype ≫ f = ofHom ((Hom.hom f).eqLocus (Hom.hom g)).subtype ≫ g",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CommRingCat.hom_ext",
"CommRingCat.Hom.hom",
"CommRingCat.... | [] | by
ext ⟨x, e⟩
simpa using e | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.Ring.Constructions | {
"line": 488,
"column": 45
} | {
"line": 488,
"column": 57
} | {
"line": 488,
"column": 57
} | [
{
"pp": "A B C : CommRingCat\nf : A ⟶ C\ng : B ⟶ C\ninst✝ : IsLocalHom (Hom.hom g)\nx : ↑(limit.cone (cospan f g)).pt\nhx : IsUnit ((ConcreteCategory.hom ((limit.cone (cospan f g)).π.app left)) x)\n⊢ (ConcreteCategory.hom ((cospan f g).map (𝟙 none))) ((ConcreteCategory.hom ((limit.cone (cospan f g)).π.app none... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Bicategory.Functor.Oplax | {
"line": 198,
"column": 4
} | {
"line": 200,
"column": 30
} | {
"line": 202,
"column": 0
} | [
{
"pp": "B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nF✝ F : B ⥤ᵒᵖᴸ C\nG : C ⥤ᵒᵖᴸ D\na✝ b✝ : B\nf : a✝ ⟶ b✝\n⊢ (F.comp G.toPrelaxFunctor).map₂ (ρ_ f).hom =\n ((G.mapFunctor (F.obj a✝) (F.obj b✝)).map (F.mapComp f (𝟙 b✝)) ≫ G.mapComp (F.map f) (F.... | [] | dsimp
simp only [map₂_rightUnitor, PrelaxFunctor.map₂_comp, mapComp_naturality_right_assoc,
whiskerLeft_comp, assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Bicategory.Functor.Oplax | {
"line": 198,
"column": 4
} | {
"line": 200,
"column": 30
} | {
"line": 202,
"column": 0
} | [
{
"pp": "B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nF✝ F : B ⥤ᵒᵖᴸ C\nG : C ⥤ᵒᵖᴸ D\na✝ b✝ : B\nf : a✝ ⟶ b✝\n⊢ (F.comp G.toPrelaxFunctor).map₂ (ρ_ f).hom =\n ((G.mapFunctor (F.obj a✝) (F.obj b✝)).map (F.mapComp f (𝟙 b✝)) ≫ G.mapComp (F.map f) (F.... | [] | dsimp
simp only [map₂_rightUnitor, PrelaxFunctor.map₂_comp, mapComp_naturality_right_assoc,
whiskerLeft_comp, assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.FinitePresentation | {
"line": 150,
"column": 4
} | {
"line": 150,
"column": 33
} | {
"line": 151,
"column": 2
} | [
{
"pp": "case mp\nR : Type w₁\nA : Type w₂\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] A\nhfs : Surjective ⇑f\nhfk : (RingHom.ker f.toRingHom).FG\nulift_var : MvPolynomial (ULift.{u_1, 0} (Fin n)) R ≃ₐ[R] MvPolynomial (Fin n) R :=\n MvPolynomial.renam... | [] | simpa using! Submodule.fg_bot | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.RingTheory.FinitePresentation | {
"line": 155,
"column": 4
} | {
"line": 155,
"column": 33
} | {
"line": 157,
"column": 0
} | [
{
"pp": "case right\nR : Type w₁\nA : Type w₂\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nι : Type u_1\nhfintype : Fintype ι\nf : MvPolynomial ι R →ₐ[R] A\nhf : Surjective ⇑f ∧ (RingHom.ker f.toRingHom).FG\nequiv : MvPolynomial ι R ≃ₐ[R] MvPolynomial (Fin (Fintype.card ι)) R\n⊢ (RingHom.ker ... | [] | simpa using! Submodule.fg_bot | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.RingTheory.FinitePresentation | {
"line": 173,
"column": 4
} | {
"line": 174,
"column": 26
} | {
"line": 175,
"column": 2
} | [
{
"pp": "case intro.refine_1\nR : Type w₁\nA : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : FinitePresentation R A\nι : Type v\ninst✝ : Finite ι\nι' : Type v\nw✝ : Fintype ι'\nf : MvPolynomial ι' R →ₐ[R] A\nhf_surj : Surjective ⇑f\nhf_ker : (RingHom.ker f.toRingHom).FG\ng : ... | [] | rw [AlgEquiv.toAlgHom_toRingHom, AlgHom.ker_coe_equiv]
exact Submodule.fg_bot | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.FinitePresentation | {
"line": 173,
"column": 4
} | {
"line": 174,
"column": 26
} | {
"line": 175,
"column": 2
} | [
{
"pp": "case intro.refine_1\nR : Type w₁\nA : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : FinitePresentation R A\nι : Type v\ninst✝ : Finite ι\nι' : Type v\nw✝ : Fintype ι'\nf : MvPolynomial ι' R →ₐ[R] A\nhf_surj : Surjective ⇑f\nhf_ker : (RingHom.ker f.toRingHom).FG\ng : ... | [] | rw [AlgEquiv.toAlgHom_toRingHom, AlgHom.ker_coe_equiv]
exact Submodule.fg_bot | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Elements | {
"line": 247,
"column": 66
} | {
"line": 247,
"column": 78
} | {
"line": 247,
"column": 78
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF✝ : C ⥤ Type w\nF : Cᵒᵖ ⥤ Type v\nX : F.Elementsᵒᵖ\n⊢ (ConcreteCategory.hom (F.map (Iso.refl ((fromCostructuredArrow F).obj (op ((toCostructuredArrow F).obj X))).fst).hom))\n ((fromCostructuredArrow F).obj (op ((toCostructuredArrow F).obj X))).snd =\n (un... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Connected | {
"line": 238,
"column": 14
} | {
"line": 238,
"column": 24
} | {
"line": 239,
"column": 12
} | [
{
"pp": "case h₁\nC : Type u₂\ninst✝³ : Category.{v₂, u₂} C\ninst✝² : HasBinaryProducts C\nJ : Type v₂\ninst✝¹ : SmallCategory J\ninst✝ : IsConnected J\nX : C\nK : J ⥤ C\nc : Cone K\nl : IsLimit c\ns : Cone (K ⋙ prod.functor.obj X)\nm : s.pt ⟶ ((prod.functor.obj X).mapCone c).pt\nL : ∀ (j : J), m ≫ ((prod.funct... | [] | simp [← L] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Limits.Connected | {
"line": 238,
"column": 14
} | {
"line": 238,
"column": 24
} | {
"line": 239,
"column": 12
} | [
{
"pp": "case h₁\nC : Type u₂\ninst✝³ : Category.{v₂, u₂} C\ninst✝² : HasBinaryProducts C\nJ : Type v₂\ninst✝¹ : SmallCategory J\ninst✝ : IsConnected J\nX : C\nK : J ⥤ C\nc : Cone K\nl : IsLimit c\ns : Cone (K ⋙ prod.functor.obj X)\nm : s.pt ⟶ ((prod.functor.obj X).mapCone c).pt\nL : ∀ (j : J), m ≫ ((prod.funct... | [] | simp [← L] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Connected | {
"line": 238,
"column": 14
} | {
"line": 238,
"column": 24
} | {
"line": 239,
"column": 12
} | [
{
"pp": "case h₁\nC : Type u₂\ninst✝³ : Category.{v₂, u₂} C\ninst✝² : HasBinaryProducts C\nJ : Type v₂\ninst✝¹ : SmallCategory J\ninst✝ : IsConnected J\nX : C\nK : J ⥤ C\nc : Cone K\nl : IsLimit c\ns : Cone (K ⋙ prod.functor.obj X)\nm : s.pt ⟶ ((prod.functor.obj X).mapCone c).pt\nL : ∀ (j : J), m ≫ ((prod.funct... | [] | simp [← L] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.WithTerminal.Basic | {
"line": 761,
"column": 12
} | {
"line": 761,
"column": 29
} | {
"line": 761,
"column": 30
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u_1\ninst✝ : Category.{v_1, u_1} D\nF : WithInitial C ⥤ D\nx y : C\nf : x ⟶ y\n⊢ 𝟙 (F.obj star) ≫ F.map (starInitial.to (of y)) = F.map (starInitial.to (of x)) ≫ F.map (incl.map f)",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u_1\ninst✝ : Category.{v_1, u_1} D\nF : WithInitial C ⥤ D\nx y : C\nf : x ⟶ y\n⊢ F.map (starInitial.to (of y)) = F.map (starInitial.to (of x)) ≫ F.map (incl.map f)"
] | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Final | {
"line": 692,
"column": 4
} | {
"line": 692,
"column": 90
} | {
"line": 693,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝³ : F.Initial\nE : Type u₃\ninst✝² : Category.{v₃, u₃} E\nG : D ⥤ E\nB : Type u₄\ninst✝¹ : Category.{v₄, u₄} B\nH : E ⥤ B\ninst✝ : CreatesLimit G H\nc : Cone ((F ⋙ G) ⋙ H)\nhc : IsLimit c\n⊢ H.mapCone ... | [
"C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝³ : F.Initial\nE : Type u₃\ninst✝² : Category.{v₃, u₃} E\nG : D ⥤ E\nB : Type u₄\ninst✝¹ : Category.{v₄, u₄} B\nH : E ⥤ B\ninst✝ : CreatesLimit G H\nc : Cone ((F ⋙ G) ⋙ H)\nhc : IsLimit c\ni : H.mapCone (liftLimit... | let i := liftedLimitMapsToOriginal ((isLimitExtendConeEquiv F (G := G ⋙ H) _).symm hc) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.CategoryTheory.Limits.Shapes.RegularMono | {
"line": 583,
"column": 4
} | {
"line": 583,
"column": 34
} | {
"line": 584,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y P Q R S : C\nf : P ⟶ Q\ng : P ⟶ R\nh : Q ⟶ S\nk : R ⟶ S\ngr : RegularEpi g\ncomm : f ≫ h = g ≫ k\nt : IsColimit (PushoutCocone.mk h k comm)\n⊢ IsColimit (Cofork.ofπ h ⋯)",
"ppTerm": "?m.74",
"assigned": true,
"usedConstants": [
"CategoryTh... | [
"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y P Q R S : C\nf : P ⟶ Q\ng : P ⟶ R\nh : Q ⟶ S\nk : R ⟶ S\ngr : RegularEpi g\ncomm : f ≫ h = g ≫ k\nt : IsColimit (PushoutCocone.mk h k comm)\n⊢ (s : Cofork (gr.left ≫ f) (gr.right ≫ f)) →\n { l // (Cofork.ofπ h ⋯).π ≫ l = s.π ∧ ∀ {m : (Cofork.ofπ h ⋯).pt ⟶ s.pt}, (Co... | apply Cofork.IsColimit.mk' _ _ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Limits.Shapes.RegularMono | {
"line": 588,
"column": 4
} | {
"line": 588,
"column": 69
} | {
"line": 589,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y P Q R S : C\nf : P ⟶ Q\ng : P ⟶ R\nh : Q ⟶ S\nk : R ⟶ S\ngr : RegularEpi g\ncomm : f ≫ h = g ≫ k\nt : IsColimit (PushoutCocone.mk h k comm)\ns : Cofork (gr.left ≫ f) (gr.right ≫ f)\nl₁ : gr.left ≫ f ≫ s.π = gr.right ≫ f ≫ s.π\nl : (Cofork.ofπ g ⋯).pt ⟶ s.pt... | [
"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y P Q R S : C\nf : P ⟶ Q\ng : P ⟶ R\nh : Q ⟶ S\nk : R ⟶ S\ngr : RegularEpi g\ncomm : f ≫ h = g ≫ k\nt : IsColimit (PushoutCocone.mk h k comm)\ns : Cofork (gr.left ≫ f) (gr.right ≫ f)\nl₁ : gr.left ≫ f ≫ s.π = gr.right ≫ f ≫ s.π\nl : (Cofork.ofπ g ⋯).pt ⟶ s.pt\nhl : (Cofo... | obtain ⟨p, hp₁, _⟩ := PushoutCocone.IsColimit.desc' t _ _ hl.symm | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.MorphismProperty.Limits | {
"line": 457,
"column": 4
} | {
"line": 471,
"column": 37
} | {
"line": 471,
"column": 37
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nW : MorphismProperty C\nJ : Type u_1\ninst✝ : Category.{v_1, u_1} J\n⊢ ∀ (f g : Arrow C) (x : f ≅ g), W.limitsOfShape J f.hom → W.limitsOfShape J g.hom",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Cat... | [] | rintro ⟨_, _, f⟩ ⟨Y₁, Y₂, g⟩ e ⟨X₁, X₂, c₁, c₂, h₁, h₂, f, hf⟩
let e₁ := Arrow.leftFunc.mapIso e
let e₂ := Arrow.rightFunc.mapIso e
have fac : g ≫ e₂.inv = e₁.inv ≫ h₂.lift (Cone.mk _ (c₁.π ≫ f)) :=
e.inv.w.symm
let c₁' : Cone X₁ := { pt := Y₁, π := (Functor.const _).map e₁.inv ≫ c₁.π }
let c₂... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.MorphismProperty.Limits | {
"line": 457,
"column": 4
} | {
"line": 471,
"column": 37
} | {
"line": 471,
"column": 37
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nW : MorphismProperty C\nJ : Type u_1\ninst✝ : Category.{v_1, u_1} J\n⊢ ∀ (f g : Arrow C) (x : f ≅ g), W.limitsOfShape J f.hom → W.limitsOfShape J g.hom",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Cat... | [] | rintro ⟨_, _, f⟩ ⟨Y₁, Y₂, g⟩ e ⟨X₁, X₂, c₁, c₂, h₁, h₂, f, hf⟩
let e₁ := Arrow.leftFunc.mapIso e
let e₂ := Arrow.rightFunc.mapIso e
have fac : g ≫ e₂.inv = e₁.inv ≫ h₂.lift (Cone.mk _ (c₁.π ≫ f)) :=
e.inv.w.symm
let c₁' : Cone X₁ := { pt := Y₁, π := (Functor.const _).map e₁.inv ≫ c₁.π }
let c₂... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts | {
"line": 196,
"column": 4
} | {
"line": 199,
"column": 9
} | {
"line": 200,
"column": 2
} | [
{
"pp": "case refine_2\nJ : Type v\ninst✝² : SmallCategory J\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nn : ℕ\nf : Fin (n + 1) → C\nc₁ : Cofan fun i ↦ f i.succ\nc₂ : BinaryCofan (f 0) c₁.pt\nt₁ : IsColimit c₁\nt₂ : IsColimit c₂\ns : Cocone (Discrete.functor f)\nj : Fin (n... | [] | · rintro i -
dsimp only [extendCofan_ι_app]
rw [Fin.cases_succ, assoc, (BinaryCofan.IsColimit.desc' t₂ _ _).2.2, t₁.fac]
rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.MorphismProperty.Limits | {
"line": 593,
"column": 12
} | {
"line": 593,
"column": 29
} | {
"line": 593,
"column": 30
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nW : MorphismProperty C\nJ : Type u_2\ninst✝¹ : Preorder J\ninst✝ : OrderBot J\nF : J ⥤ C\nc : Cocone F\nhc : IsColimit c\nh : ∀ (j : J), W (F.map (homOfLE ⋯))\nx✝² x✝¹ : J\nx✝ : x✝² ⟶ x✝¹\n⊢ 𝟙 (F.obj ⊥) ≫ F.map (homOfLE ⋯) = F.map (homOfLE ⋯) ≫ F.map x✝",
"p... | [
"C : Type u\ninst✝² : Category.{v, u} C\nW : MorphismProperty C\nJ : Type u_2\ninst✝¹ : Preorder J\ninst✝ : OrderBot J\nF : J ⥤ C\nc : Cocone F\nhc : IsColimit c\nh : ∀ (j : J), W (F.map (homOfLE ⋯))\nx✝² x✝¹ : J\nx✝ : x✝² ⟶ x✝¹\n⊢ F.map (homOfLE ⋯) = F.map (homOfLE ⋯) ≫ F.map x✝"
] | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.ObjectProperty.ColimitsOfShape | {
"line": 181,
"column": 2
} | {
"line": 183,
"column": 28
} | {
"line": 185,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP : ObjectProperty C\nJ : Type u'\ninst✝ : Category.{v', u'} J\nX : C\nh : P.isoClosure.ColimitOfShape J X\nobj : J → C\nh₁ : ∀ (j : J), P (obj j)\nh₂ : ∀ (j : J), Nonempty (h.diag.obj j ≅ obj j)\n⊢ P.colimitsOfShape J X",
"ppTerm": "?m.56",
"assign... | [] | exact
⟨{ toColimitPresentation := h.changeDiag (h.diag.isoCopyObj obj (fun j ↦ (h₂ j).some)).symm
prop_diag_obj := h₁ }⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.MorphismProperty.Limits | {
"line": 879,
"column": 16
} | {
"line": 879,
"column": 33
} | {
"line": 879,
"column": 34
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nP : MorphismProperty C\nX Y Z : C\ne : X ≅ Y\nf : Y ⟶ Z\nhf : P.universally f\nX' Z' : C\ni₁ : X' ⟶ X\ni₂ : Z' ⟶ Z\nf' : X' ⟶ Z'\nH : IsPullback f' i₁ i₂ (e.hom ≫ f)\n⊢ 𝟙 X' ≫ i₁ = (i₁ ≫ e.hom) ≫ e.inv",
"ppTerm": "?m.80",
"assigned": true,
"usedConst... | [
"C : Type u\ninst✝ : Category.{v, u} C\nP : MorphismProperty C\nX Y Z : C\ne : X ≅ Y\nf : Y ⟶ Z\nhf : P.universally f\nX' Z' : C\ni₁ : X' ⟶ X\ni₂ : Z' ⟶ Z\nf' : X' ⟶ Z'\nH : IsPullback f' i₁ i₂ (e.hom ≫ f)\n⊢ i₁ = (i₁ ≫ e.hom) ≫ e.inv"
] | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monoidal.Cartesian.Basic | {
"line": 128,
"column": 29
} | {
"line": 128,
"column": 67
} | {
"line": 130,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nℬ : (X Y : C) → LimitCone (pair X Y)\nX Y : C\n⊢ ∀ (j : Discrete WalkingPair),\n tensorHom ℬ (𝟙 X) (𝟙 Y) ≫ (ℬ X Y).cone.π.app j = 𝟙 (tensorObj ℬ X Y) ≫ (ℬ X Y).cone.π.app j",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Category... | [] | by rintro ⟨_ | _⟩ <;> simp [tensorHom] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Cartesian.Basic | {
"line": 133,
"column": 29
} | {
"line": 133,
"column": 67
} | {
"line": 135,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nℬ : (X Y : C) → LimitCone (pair X Y)\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\nf₁ : X₁ ⟶ Y₁\nf₂ : X₂ ⟶ Y₂\ng₁ : Y₁ ⟶ Z₁\ng₂ : Y₂ ⟶ Z₂\n⊢ ∀ (j : Discrete WalkingPair),\n (tensorHom ℬ f₁ f₂ ≫ tensorHom ℬ g₁ g₂) ≫ (ℬ Z₁ Z₂).cone.π.app j =\n tensorHom ℬ (f₁ ≫ g₁) (f₂ ≫ g₂) ≫ (ℬ... | [] | by rintro ⟨_ | _⟩ <;> simp [tensorHom] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Cartesian.Mon | {
"line": 197,
"column": 15
} | {
"line": 204,
"column": 8
} | {
"line": 206,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{v, u_1} C\ninst✝⁵ : CartesianMonoidalCategory C\ninst✝⁴ : Category.{w, u_2} D\ninst✝³ : CartesianMonoidalCategory D\nM N O X Y : C\ninst✝² : MonObj M\ninst✝¹ : MonObj N\ninst✝ : MonObj O\nF : Cᵒᵖ ⥤ MonCat\nα : (F ⋙ forget MonCat).RepresentableBy X\n⊢ α.hom... | [] | by
apply α.homEquiv'.injective
simp only [α.homEquiv'_comp, Equiv.apply_symm_apply, map_mul]
simp only [← α.homEquiv'_comp]
simp only [whiskerRight_fst, whiskerRight_snd, whiskerLeft_fst, associator_hom_fst,
whiskerLeft_snd, α.homEquiv'_comp, Equiv.apply_symm_apply, map_mul, _root_.mul_assoc]
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Cartesian.Basic | {
"line": 637,
"column": 2
} | {
"line": 638,
"column": 29
} | {
"line": 640,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : CartesianMonoidalCategory C\nD : Type u₁\ninst✝³ : Category.{v₁, u₁} D\ninst✝² : CartesianMonoidalCategory D\nF : C ⥤ D\nE : Type u₂\ninst✝¹ : Category.{v₂, u₂} E\ninst✝ : CartesianMonoidalCategory E\nG : D ⥤ E\nA B : C\n⊢ prodComparison (F ⋙ G) A B = G.... | [] | unfold prodComparison
ext <;> simp [← G.map_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Cartesian.Basic | {
"line": 637,
"column": 2
} | {
"line": 638,
"column": 29
} | {
"line": 640,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : CartesianMonoidalCategory C\nD : Type u₁\ninst✝³ : Category.{v₁, u₁} D\ninst✝² : CartesianMonoidalCategory D\nF : C ⥤ D\nE : Type u₂\ninst✝¹ : Category.{v₂, u₂} E\ninst✝ : CartesianMonoidalCategory E\nG : D ⥤ E\nA B : C\n⊢ prodComparison (F ⋙ G) A B = G.... | [] | unfold prodComparison
ext <;> simp [← G.map_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Closed.Basic | {
"line": 272,
"column": 23
} | {
"line": 272,
"column": 39
} | {
"line": 272,
"column": 40
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : MonoidalCategory C\nA B X Y : C\ninst✝¹ : Closed A\ninst✝ : Closed B\nf : Y ⟶ (ihom A).obj X\ng : B ⟶ A\n⊢ f ≫ (pre g).app X = curry (g ▷ Y ≫ uncurry f)",
"ppTerm": "?m.81",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Category... | [
"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : MonoidalCategory C\nA B X Y : C\ninst✝¹ : Closed A\ninst✝ : Closed B\nf : Y ⟶ (ihom A).obj X\ng : B ⟶ A\n⊢ f ≫ (pre g).app X = curry (uncurry f) ≫ (pre g).app X"
] | ← curry_pre_app, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monoidal.Closed.Basic | {
"line": 277,
"column": 2
} | {
"line": 277,
"column": 25
} | {
"line": 279,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : MonoidalCategory C\nA : C\ninst✝ : Closed A\n⊢ (conjugateEquiv (ihom.adjunction A) (ihom.adjunction A)) (𝟙 ((tensoringLeft C).obj A)) = 𝟙 (ihom A)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"CategoryTheory.MonoidalCateg... | [] | apply conjugateEquiv_id | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Monoidal.Grp | {
"line": 306,
"column": 6
} | {
"line": 306,
"column": 78
} | {
"line": 306,
"column": 78
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : CartesianMonoidalCategory C\nA : C\ninst✝ : GrpObj A\ns : PullbackCone μ μ\n⊢ lift (s.snd ≫ fst A A) (lift (s.snd ≫ fst A A ≫ ι) (s.fst ≫ fst A A) ≫ μ) ≫ μ = s.fst ≫ fst A A",
"ppTerm": "?m.202",
"assigned": true,
"usedConstants": [
... | [] | rw [← lift_lift_assoc, ← assoc, lift_comp_inv_right, lift_comp_one_left] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Monoidal.Grp | {
"line": 310,
"column": 6
} | {
"line": 313,
"column": 60
} | {
"line": 314,
"column": 6
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : CartesianMonoidalCategory C\nA : C\ninst✝ : GrpObj A\ns : PullbackCone μ μ\n⊢ lift (lift (s.snd ≫ fst A A ≫ ι) (s.fst ≫ fst A A) ≫ μ) (s.fst ≫ snd A A) ≫ μ = s.snd ≫ snd A A",
"ppTerm": "?m.253",
"assigned": true,
"usedConstants": [
... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : CartesianMonoidalCategory C\nA : C\ninst✝ : GrpObj A\ns : PullbackCone μ μ\nthis : lift (s.snd ≫ fst A A ≫ ι) (s.fst ≫ fst A A) ≫ μ = lift (s.snd ≫ snd A A) (s.fst ≫ snd A A ≫ ι) ≫ μ\n⊢ lift (lift (s.snd ≫ fst A A ≫ ι) (s.fst ≫ fst A A) ≫ μ) (s.fst ≫ snd A A) ≫ μ... | have : lift (s.snd ≫ fst _ _ ≫ ι) (s.fst ≫ fst _ _) ≫ μ =
lift (s.snd ≫ snd _ _) (s.fst ≫ snd _ _ ≫ ι) ≫ μ := by
rw [← assoc s.fst, eq_lift_inv_right, lift_lift_assoc, ← assoc s.snd, lift_inv_left_eq,
lift_comp_fst_snd, lift_comp_fst_snd, s.condition] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.Dimension.ErdosKaplansky | {
"line": 42,
"column": 2
} | {
"line": 42,
"column": 42
} | {
"line": 43,
"column": 2
} | [
{
"pp": "K : Type u\ninst✝ : DivisionRing K\naleph0_le : ℵ₀ ≤ Module.rank K (ℕ → K)\n⊢ #K ≤ Module.rank K (ℕ → K)",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Pi.Function.module",
"Preorder.toLT",
"Semiring.toModule",
"Pi.addCommMonoid",
"Cardinal",
... | [
"case inl\nK : Type u\ninst✝ : DivisionRing K\naleph0_le : ℵ₀ ≤ Module.rank K (ℕ → K)\ncard_K : #K ≤ ℵ₀\n⊢ #K ≤ Module.rank K (ℕ → K)",
"case inr\nK : Type u\ninst✝ : DivisionRing K\naleph0_le : ℵ₀ ≤ Module.rank K (ℕ → K)\ncard_K : ℵ₀ < #K\n⊢ #K ≤ Module.rank K (ℕ → K)"
] | obtain card_K | card_K := le_or_gt #K ℵ₀ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Monoidal.Rigid.Basic | {
"line": 354,
"column": 6
} | {
"line": 354,
"column": 36
} | {
"line": 354,
"column": 36
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nX Y Z : C\ninst✝² : HasRightDual X\ninst✝¹ : HasRightDual Y\ninst✝ : HasRightDual Z\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ η_ X Xᘁ ⊗≫ f ▷ Xᘁ ⊗≫ (η_ Y Yᘁ ▷ Y ⊗≫ Y ◁ ε_ Y Yᘁ) ▷ Xᘁ ⊗≫ g ▷ Xᘁ ⊗≫ 𝟙 (Z ⊗ Xᘁ) = η_ X Xᘁ ≫ f ▷ Xᘁ ≫ g ▷ Xᘁ",
"ppTerm"... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nX Y Z : C\ninst✝² : HasRightDual X\ninst✝¹ : HasRightDual Y\ninst✝ : HasRightDual Z\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ η_ X Xᘁ ⊗≫ f ▷ Xᘁ ⊗≫ ⊗𝟙.hom ▷ Xᘁ ⊗≫ g ▷ Xᘁ ⊗≫ 𝟙 (Z ⊗ Xᘁ) = η_ X Xᘁ ≫ f ▷ Xᘁ ≫ g ▷ Xᘁ"
] | rw [evaluation_coevaluation''] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Monoidal.Rigid.Basic | {
"line": 404,
"column": 8
} | {
"line": 404,
"column": 38
} | {
"line": 404,
"column": 38
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\nX Y Y' Z : C\ninst✝ : ExactPairing Y Y'\nf : X ⟶ Y ⊗ Z\n⊢ f ⊗≫ (η_ Y Y' ▷ Y ⊗≫ Y ◁ ε_ Y Y') ▷ Z ⊗≫ 𝟙 (Y ⊗ Z) = f",
"ppTerm": "?m.1073",
"assigned": true,
"usedConstants": [
"CategoryTheory.MonoidalCoherence.iso",... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\nX Y Y' Z : C\ninst✝ : ExactPairing Y Y'\nf : X ⟶ Y ⊗ Z\n⊢ f ⊗≫ ⊗𝟙.hom ▷ Z ⊗≫ 𝟙 (Y ⊗ Z) = f"
] | rw [evaluation_coevaluation''] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Monoidal.Rigid.Basic | {
"line": 420,
"column": 8
} | {
"line": 420,
"column": 38
} | {
"line": 420,
"column": 38
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\nX Y Y' Z : C\ninst✝ : ExactPairing Y Y'\nf : X ⊗ Y ⟶ Z\n⊢ 𝟙 (X ⊗ Y) ⊗≫ X ◁ (η_ Y Y' ▷ Y ⊗≫ Y ◁ ε_ Y Y') ⊗≫ f = f",
"ppTerm": "?m.251",
"assigned": true,
"usedConstants": [
"CategoryTheory.MonoidalCoherence.iso",
... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\nX Y Y' Z : C\ninst✝ : ExactPairing Y Y'\nf : X ⊗ Y ⟶ Z\n⊢ 𝟙 (X ⊗ Y) ⊗≫ X ◁ ⊗𝟙.hom ⊗≫ f = f"
] | rw [evaluation_coevaluation''] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Monoidal.Rigid.Basic | {
"line": 533,
"column": 6
} | {
"line": 533,
"column": 36
} | {
"line": 533,
"column": 36
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\nY Y' Z : C\ninst✝ : ExactPairing Y Y'\nf : Y ⟶ Z\n⊢ (η_ Y Y' ▷ Y ⊗≫ Y ◁ ε_ Y Y') ⊗≫ f = (λ_ Y).hom ≫ f",
"ppTerm": "?m.139",
"assigned": true,
"usedConstants": [
"CategoryTheory.MonoidalCoherence.iso",
"Eq.m... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\nY Y' Z : C\ninst✝ : ExactPairing Y Y'\nf : Y ⟶ Z\n⊢ ⊗𝟙.hom ⊗≫ f = (λ_ Y).hom ≫ f"
] | rw [evaluation_coevaluation''] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Monoidal.Rigid.Basic | {
"line": 544,
"column": 6
} | {
"line": 544,
"column": 36
} | {
"line": 544,
"column": 36
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\nY Z : C\ninst✝ : HasLeftDual Z\nf : Y ⟶ ᘁZ\n⊢ f ⊗≫ η_ (ᘁZ) Z ▷ ᘁZ ⊗≫ ᘁZ ◁ ε_ (ᘁZ) Z = f ≫ (ρ_ ᘁZ).inv",
"ppTerm": "?m.176",
"assigned": true,
"usedConstants": [
"CategoryTheory.MonoidalCoherence.iso",
"Eq.mp... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\nY Z : C\ninst✝ : HasLeftDual Z\nf : Y ⟶ ᘁZ\n⊢ f ⊗≫ ⊗𝟙.hom = f ≫ (ρ_ ᘁZ).inv"
] | rw [evaluation_coevaluation''] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Monoidal.Rigid.Basic | {
"line": 609,
"column": 8
} | {
"line": 609,
"column": 38
} | {
"line": 609,
"column": 38
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\nX X' Y : C\ninst✝ : ExactPairing X' Y\ni : X ≅ X'\n⊢ 𝟙 (𝟙_ C ⊗ X) ⊗≫ i.hom ⊗≫ (η_ X' Y ▷ X' ⊗≫ X' ◁ ε_ X' Y) ⊗≫ i.inv ⊗≫ 𝟙 (X ⊗ 𝟙_ C) =\n 𝟙 (𝟙_ C ⊗ X) ⊗≫ (i.hom ≫ i.inv) ⊗≫ 𝟙 (X ⊗ 𝟙_ C)",
"ppTerm": "?m.1573",
"as... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\nX X' Y : C\ninst✝ : ExactPairing X' Y\ni : X ≅ X'\n⊢ 𝟙 (𝟙_ C ⊗ X) ⊗≫ i.hom ⊗≫ ⊗𝟙.hom ⊗≫ i.inv ⊗≫ 𝟙 (X ⊗ 𝟙_ C) = 𝟙 (𝟙_ C ⊗ X) ⊗≫ (i.hom ≫ i.inv) ⊗≫ 𝟙 (X ⊗ 𝟙_ C)"
] | rw [evaluation_coevaluation''] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Monoidal.Rigid.Basic | {
"line": 635,
"column": 8
} | {
"line": 635,
"column": 38
} | {
"line": 636,
"column": 8
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\nX Y Y' : C\ninst✝ : ExactPairing X Y'\ni : Y ≅ Y'\n⊢ η_ X Y' ▷ X ⊗≫ X ◁ ε_ X Y' = (λ_ X).hom ≫ (ρ_ X).inv",
"ppTerm": "?m.1465",
"assigned": true,
"usedConstants": [
"CategoryTheory.MonoidalCoherence.iso",
"... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\nX Y Y' : C\ninst✝ : ExactPairing X Y'\ni : Y ≅ Y'\n⊢ ⊗𝟙.hom = (λ_ X).hom ≫ (ρ_ X).inv"
] | rw [evaluation_coevaluation''] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 759,
"column": 35
} | {
"line": 761,
"column": 81
} | {
"line": 763,
"column": 0
} | [
{
"pp": "n : Type u_19\nR : Type u_20\nM : Type u_21\nM₁ : Type u_22\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : AddCommMonoid M₁\ninst✝² : Module R M\ninst✝¹ : Module R M₁\nI I' : R →+* R\nB : M →ₛₗ[I] M →ₛₗ[I'] M₁\ninst✝ : Nontrivial R\nv : Basis n R M\nh : B.IsOrthoᵢ ⇑v\nhB : B.SeparatingRig... | [] | by
rw [isOrthoᵢ_flip] at h
exact h.not_isOrtho_basis_self_of_separatingLeft (flip_separatingLeft.mpr hB) i | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers | {
"line": 233,
"column": 45
} | {
"line": 239,
"column": 72
} | {
"line": 241,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D\nG : C ⥤ D\nX Y Z : C\nf✝ g✝ : X ⟶ Y\nh : Y ⟶ Z\nw : f✝ ≫ h = g✝ ≫ h\ninst✝³ : HasCoequalizer f✝ g✝\ninst✝² : HasCoequalizer (G.map f✝) (G.map g✝)\ninst✝¹ : PreservesColimit (parallelPair f✝ g✝) G\nf g : X ⟶ Y\ninst✝ :... | [] | by
apply
preservesColimit_of_preserves_colimit_cocone
(HasSplitCoequalizer.isSplitCoequalizer f g).isCoequalizer
apply
(isColimitMapCoconeCoforkEquiv G _).symm
((HasSplitCoequalizer.isSplitCoequalizer f g).map G).isCoequalizer | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers | {
"line": 69,
"column": 12
} | {
"line": 69,
"column": 29
} | {
"line": 69,
"column": 30
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : Type w\ninst✝ : SmallCategory J\nF : J ⥤ C\nc₁ : Fan F.obj\nc₂ : Fan fun f ↦ F.obj f.fst.2\ns t : c₁.pt ⟶ c₂.pt\nhs : ∀ (f : (p : J × J) × (p.1 ⟶ p.2)), s ≫ c₂.π.app { as := f } = c₁.π.app { as := f.fst.1 } ≫ F.map f.snd\nht : ∀ (f : (p : J × J) × (p.1 ⟶ p.2)... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nJ : Type w\ninst✝ : SmallCategory J\nF : J ⥤ C\nc₁ : Fan F.obj\nc₂ : Fan fun f ↦ F.obj f.fst.2\ns t : c₁.pt ⟶ c₂.pt\nhs : ∀ (f : (p : J × J) × (p.1 ⟶ p.2)), s ≫ c₂.π.app { as := f } = c₁.π.app { as := f.fst.1 } ≫ F.map f.snd\nht : ∀ (f : (p : J × J) × (p.1 ⟶ p.2)), t ≫ c₂.π.... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 404,
"column": 2
} | {
"line": 404,
"column": 87
} | {
"line": 405,
"column": 2
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nW : Subspace K V\n⊢ (dualAnnihilator W).dualCoannihilator = W",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Submodule",
"Submodule.le_dualAnnihilator_dualCoannihilator",
... | [
"K : Type u_1\nV : Type u_2\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nW : Subspace K V\nv : V\n⊢ v ∉ W → v ∉ (dualAnnihilator W).dualCoannihilator"
] | refine le_antisymm (fun v ↦ Function.mtr ?_) (le_dualAnnihilator_dualCoannihilator _) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 772,
"column": 4
} | {
"line": 772,
"column": 56
} | {
"line": 773,
"column": 4
} | [
{
"pp": "K : Type u_1\nV₁ : Type u_2\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V₁\ninst✝¹ : Module K V₁\ninst✝ : FiniteDimensional K V₁\nf g : Dual K V₁\nx : V₁\nh : LinearMap.ker f = LinearMap.ker g\nh' : f x = g x\nhx : f x ≠ 0\np : Submodule K V₁ := K ∙ x\nhp : p ≠ ⊥\nhpf : Disjoint (LinearMap.ker f) p... | [
"K : Type u_1\nV₁ : Type u_2\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V₁\ninst✝¹ : Module K V₁\ninst✝ : FiniteDimensional K V₁\nf g : Dual K V₁\nx : V₁\nh : LinearMap.ker f = LinearMap.ker g\nh' : f x = g x\nhx : f x ≠ 0\np : Submodule K V₁ := K ∙ x\nhp : p ≠ ⊥\nhpf : Disjoint (LinearMap.ker f) p\nhf : f ≠ 0... | have : v ∈ (⊤ : Submodule K V₁) := Submodule.mem_top | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Abelian.NonPreadditive | {
"line": 399,
"column": 88
} | {
"line": 400,
"column": 57
} | {
"line": 402,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : NonPreadditiveAbelian C\nX Y Z : C\nf : X ⟶ Y\ng h : Y ⟶ Z\n⊢ f ≫ (g - h) = f ≫ g - f ≫ h",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Category.assoc",
"CategoryTheory.CategoryStruct.toQ... | [] | by
rw [sub_def, ← Category.assoc, prod.comp_lift, sub_def] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.Grp.EpiMono | {
"line": 255,
"column": 4
} | {
"line": 255,
"column": 28
} | {
"line": 256,
"column": 4
} | [
{
"pp": "case refine_1\nA B : GrpCat\nf : A ⟶ B\na : ↑A\n⊢ (Hom.hom f) a ∈ {x | h x = g x}",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"GrpCat.SurjectiveOfEpiAuxs.g",
"MonoidHom.instFunLike",
"MonoidHom",
"GrpCat.str",
"Monoid.toMulOneClass",
"Gr... | [
"case refine_1\nA B : GrpCat\nf : A ⟶ B\na : ↑A\n⊢ h ((ConcreteCategory.hom f) a) = g ((ConcreteCategory.hom f) a)"
] | change h (f a) = g (f a) | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.Algebra.Homology.ShortComplex.Homology | {
"line": 1139,
"column": 4
} | {
"line": 1140,
"column": 41
} | {
"line": 1142,
"column": 0
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\ninst✝¹ : S₁.HasHomology\ninst✝ : S₂.HasHomology\nh₁ : IsIso (opcyclesMap φ)\nh₂ : Mono φ.τ₃\nh : (S₂.homologyι ≫ inv (opcyclesMap φ)) ≫ S₁.fromOpcycles = 0\nz : S₂.homology ⟶ (Kernel... | [] | rw [← cancel_mono S₂.homologyι, assoc, homologyι_naturality, reassoc_of% hz,
IsIso.inv_hom_id, comp_id, id_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.ShortComplex.Homology | {
"line": 1139,
"column": 4
} | {
"line": 1140,
"column": 41
} | {
"line": 1142,
"column": 0
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\ninst✝¹ : S₁.HasHomology\ninst✝ : S₂.HasHomology\nh₁ : IsIso (opcyclesMap φ)\nh₂ : Mono φ.τ₃\nh : (S₂.homologyι ≫ inv (opcyclesMap φ)) ≫ S₁.fromOpcycles = 0\nz : S₂.homology ⟶ (Kernel... | [] | rw [← cancel_mono S₂.homologyι, assoc, homologyι_naturality, reassoc_of% hz,
IsIso.inv_hom_id, comp_id, id_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.Homology | {
"line": 1139,
"column": 4
} | {
"line": 1140,
"column": 41
} | {
"line": 1142,
"column": 0
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\ninst✝¹ : S₁.HasHomology\ninst✝ : S₂.HasHomology\nh₁ : IsIso (opcyclesMap φ)\nh₂ : Mono φ.τ₃\nh : (S₂.homologyι ≫ inv (opcyclesMap φ)) ≫ S₁.fromOpcycles = 0\nz : S₂.homology ⟶ (Kernel... | [] | rw [← cancel_mono S₂.homologyι, assoc, homologyι_naturality, reassoc_of% hz,
IsIso.inv_hom_id, comp_id, id_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.Abelian | {
"line": 158,
"column": 51
} | {
"line": 158,
"column": 94
} | {
"line": 159,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\nS : ShortComplex C\nD : Type u'\ninst✝¹ : Category.{v', u'} D\ninst✝ : HasZeroMorphisms D\nγ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f\ng' : cokernel S.f ⟶ S.X₃ := cokernel.desc S.f S.g ⋯\nhg' : g' = cokernel.π γ ≫ cokernel.... | [] | rw [hg', kernel.condition_assoc, zero_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.ShortComplex.Abelian | {
"line": 159,
"column": 2
} | {
"line": 160,
"column": 97
} | {
"line": 161,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\nS : ShortComplex C\nD : Type u'\ninst✝¹ : Category.{v', u'} D\ninst✝ : HasZeroMorphisms D\nγ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f\ng' : cokernel S.f ⟶ S.X₃ := cokernel.desc S.f S.g ⋯\nhg' : g' = cokernel.π γ ≫ cokernel.... | [
"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\nS : ShortComplex C\nD : Type u'\ninst✝¹ : Category.{v', u'} D\ninst✝ : HasZeroMorphisms D\nγ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f\ng' : cokernel S.f ⟶ S.X₃ := cokernel.desc S.f S.g ⋯\nhg' : g' = cokernel.π γ ≫ cokernel.desc γ g' ⋯\... | let e : cokernel γ ≅ Abelian.coimage S.g :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit _) S.cokernelToAbelianCoimageIsCokernel | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Algebra.Homology.ShortComplex.PreservesHomology | {
"line": 850,
"column": 2
} | {
"line": 856,
"column": 18
} | {
"line": 858,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹¹ : Category.{v_1, u_1} C\ninst✝¹⁰ : Category.{v_2, u_2} D\ninst✝⁹ : HasZeroMorphisms C\ninst✝⁸ : HasZeroMorphisms D\nF : C ⥤ D\ninst✝⁷ : F.PreservesZeroMorphisms\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\ninst✝⁶ : S₁.HasHomology\ninst✝⁵ : S₂.HasHomology\ninst✝⁴ : (F.mapSho... | [] | have γ : RightHomologyMapData φ S₁.rightHomologyData S₂.rightHomologyData := default
rw [γ.quasiIso_iff, (γ.map F).quasiIso_iff, RightHomologyMapData.map_φH]
constructor
· intro
exact isIso_of_reflects_iso _ F
· intro
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.PreservesHomology | {
"line": 850,
"column": 2
} | {
"line": 856,
"column": 18
} | {
"line": 858,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹¹ : Category.{v_1, u_1} C\ninst✝¹⁰ : Category.{v_2, u_2} D\ninst✝⁹ : HasZeroMorphisms C\ninst✝⁸ : HasZeroMorphisms D\nF : C ⥤ D\ninst✝⁷ : F.PreservesZeroMorphisms\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\ninst✝⁶ : S₁.HasHomology\ninst✝⁵ : S₂.HasHomology\ninst✝⁴ : (F.mapSho... | [] | have γ : RightHomologyMapData φ S₁.rightHomologyData S₂.rightHomologyData := default
rw [γ.quasiIso_iff, (γ.map F).quasiIso_iff, RightHomologyMapData.map_φH]
constructor
· intro
exact isIso_of_reflects_iso _ F
· intro
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.Preadditive | {
"line": 585,
"column": 11
} | {
"line": 586,
"column": 97
} | {
"line": 588,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nH₁ : S₁.LeftHomologyData\nH₂ : S₂.LeftHomologyData\nh₀ : S₁.X₁ ⟶ S₂.X₁\nh₀_f : h₀ ≫ S₂.f = 0\nh₁ : S₁.X₂ ⟶ S₂.X₁\nh₂ : S₁.X₃ ⟶ S₂.X₂\nh₃ : S₁.X₃ ⟶ S₂.X₃\ng_h₃ : S₁.g ≫ h₃ = 0\n⊢ H₁.π ≫... | [] | by
rw [H₂.liftK_π_eq_zero_of_boundary (H₁.i ≫ h₁ ≫ S₂.f) (H₁.i ≫ h₁) (by rw [assoc]), comp_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Preadditive.LeftExact | {
"line": 171,
"column": 45
} | {
"line": 188,
"column": 45
} | {
"line": 190,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝⁶ : Category.{v₁, u₁} C\ninst✝⁵ : Preadditive C\nD : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D\ninst✝³ : Preadditive D\nF : C ⥤ D\ninst✝² : F.PreservesZeroMorphisms\ninst✝¹ : HasBinaryBiproducts C\ninst✝ : ∀ {X Y : C} (f : X ⟶ Y), PreservesColimit (parallelPair f 0) F\nX Y : C\nf g : X ⟶ ... | [] | by
letI := preservesBinaryBiproducts_of_preservesBinaryCoproducts F
haveI := additive_of_preservesBinaryBiproducts F
constructor
intro c i
let c' := isColimitCokernelCoforkOfCofork (i.ofIsoColimit (Cofork.isoCoforkOfπ c))
dsimp only [cokernelCoforkOfCofork_ofπ] at c'
let iFc := isColimitCoforkMapOfIsColim... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Adjunction.Restrict | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 37
} | {
"line": 89,
"column": 0
} | [
{
"pp": "case e_a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nC' : Type u₃\ninst✝¹ : Category.{v₃, u₃} C'\nD' : Type u₄\ninst✝ : Category.{v₄, u₄} D'\niC : C ⥤ C'\niD : D ⥤ D'\nL' : C' ⥤ D'\nR' : D' ⥤ C'\nadj : L' ⊣ R'\nhiC : iC.FullyFaithful\nhiD : iD.FullyFaithful\nL... | [] | exact (comm2.hom.naturality _).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 64,
"column": 4
} | {
"line": 65,
"column": 18
} | {
"line": 67,
"column": 0
} | [
{
"pp": "case mpr\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nS : ShortComplex C\ninst✝ : S.HasHomology\n⊢ IsZero S.homology → S.Exact",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"CategoryTheory.ShortComplex.Exact.mk",
"CategoryTheory.ShortCom... | [] | intro h
exact ⟨⟨_, h⟩⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 64,
"column": 4
} | {
"line": 65,
"column": 18
} | {
"line": 67,
"column": 0
} | [
{
"pp": "case mpr\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nS : ShortComplex C\ninst✝ : S.HasHomology\n⊢ IsZero S.homology → S.Exact",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"CategoryTheory.ShortComplex.Exact.mk",
"CategoryTheory.ShortCom... | [] | intro h
exact ⟨⟨_, h⟩⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 182,
"column": 2
} | {
"line": 183,
"column": 49
} | {
"line": 185,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS : ShortComplex C\nh : S.Exact\n⊢ S.op.Exact",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"CategoryTheory.ShortComplex.Exact.mk",
"CategoryTheory.Limits.IsZero.op",
"Opposite",
"C... | [] | obtain ⟨h, z⟩ := h
exact ⟨⟨h.op, (IsZero.of_iso z h.iso.symm).op⟩⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 182,
"column": 2
} | {
"line": 183,
"column": 49
} | {
"line": 185,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS : ShortComplex C\nh : S.Exact\n⊢ S.op.Exact",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"CategoryTheory.ShortComplex.Exact.mk",
"CategoryTheory.Limits.IsZero.op",
"Opposite",
"C... | [] | obtain ⟨h, z⟩ := h
exact ⟨⟨h.op, (IsZero.of_iso z h.iso.symm).op⟩⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Subobject.Limits | {
"line": 49,
"column": 2
} | {
"line": 49,
"column": 58
} | {
"line": 50,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nX Y W : C\nf : X ⟶ Y\ninst✝ : HasPullbacks C\ny : Subobject Y\nh : W ⟶ X\n⊢ ((Subobject.pullback f).obj y).Factors h ↔ y.Factors (h ≫ f)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"CategoryTheory.Subobject.Factors",
"Partial... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nX Y W : C\nf : X ⟶ Y\ninst✝ : HasPullbacks C\ny : Subobject Y\nh : W ⟶ X\nhf : ((Subobject.pullback f).obj y).Factors h\n⊢ y.Factors (h ≫ f)"
] | refine ⟨fun hf ↦ ?_, fun hF ↦ pullback_factors f y h hF⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Subobject.Basic | {
"line": 649,
"column": 60
} | {
"line": 651,
"column": 49
} | {
"line": 653,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX : C\nx : Subobject X\n⊢ (map (𝟙 X)).obj x = x",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"CategoryTheory.Over",
"PartialOrder.toPreorder",
"CategoryTheory.Subobject.map",
"CategoryTheory.Functor.id",
"... | [] | by
induction x using Quotient.inductionOn' with | _ f
exact Quotient.sound ⟨(MonoOver.mapId _).app f⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Subobject.Lattice | {
"line": 184,
"column": 2
} | {
"line": 186,
"column": 5
} | {
"line": 188,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nX Y Z : C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : HasImages C\ninst✝ : HasBinaryCoproducts C\nA : C\nf g : MonoOver A\n⊢ g ⟶ (sup.obj f).obj g",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.facto... | [] | refine homMk (coprod.inr ≫ factorThruImage _) ?_
erw [Category.assoc, image.fac, coprod.inr_desc]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Subobject.Lattice | {
"line": 184,
"column": 2
} | {
"line": 186,
"column": 5
} | {
"line": 188,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nX Y Z : C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : HasImages C\ninst✝ : HasBinaryCoproducts C\nA : C\nf g : MonoOver A\n⊢ g ⟶ (sup.obj f).obj g",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.facto... | [] | refine homMk (coprod.inr ≫ factorThruImage _) ?_
erw [Category.assoc, image.fac, coprod.inr_desc]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.ExactFunctor | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 77
} | {
"line": 83,
"column": 2
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁸ : Category.{v_1, u_1} C\ninst✝⁷ : Category.{v_2, u_2} D\ninst✝⁶ : Preadditive C\ninst✝⁵ : Preadditive D\nF : C ⥤ D\ninst✝⁴ : F.Additive\ninst✝³ : F.PreservesHomology\ninst✝² : HasZeroObject C\ninst✝¹ : HasFiniteCoproducts C\ninst✝ : HasCokernels C\n⊢ PreservesFiniteCo... | [
"C : Type u_1\nD : Type u_2\ninst✝⁸ : Category.{v_1, u_1} C\ninst✝⁷ : Category.{v_2, u_2} D\ninst✝⁶ : Preadditive C\ninst✝⁵ : Preadditive D\nF : C ⥤ D\ninst✝⁴ : F.Additive\ninst✝³ : F.PreservesHomology\ninst✝² : HasZeroObject C\ninst✝¹ : HasFiniteCoproducts C\ninst✝ : HasCokernels C\nthis : ∀ {X Y : C} (f : X ⟶ Y),... | have := fun {X Y : C} (f : X ⟶ Y) ↦ PreservesHomology.preservesCokernel F f | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Homology.ShortComplex.ExactFunctor | {
"line": 133,
"column": 14
} | {
"line": 133,
"column": 31
} | {
"line": 133,
"column": 32
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : Abelian C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : F.Additive\nhF : ∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Mono (F.map S.f)\nS : ShortComplex C\nhS : S.Exact\nhf : Mono S.f\nthis : F.Pr... | [
"C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : Abelian C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : F.Additive\nhF : ∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Mono (F.map S.f)\nS : ShortComplex C\nhS : S.Exact\nhf : Mono S.f\nthis : F.PreservesMonom... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.ShortComplex.ExactFunctor | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 50
} | {
"line": 141,
"column": 4
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : Abelian C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : F.Additive\ntfae_1_to_2 :\n (∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Mono (F.map S.f)) →\n ∀ (S : ShortComplex C), S.Exact ∧ Mono S... | [
"C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : Abelian C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : F.Additive\ntfae_1_to_2 :\n (∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Mono (F.map S.f)) →\n ∀ (S : ShortComplex C), S.Exact ∧ Mono S.f → (S.map ... | let hS := hF S ⟨exact_kernel f, inferInstance⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.CategoryTheory.Subobject.Lattice | {
"line": 435,
"column": 4
} | {
"line": 436,
"column": 32
} | {
"line": 437,
"column": 4
} | [
{
"pp": "case insert.inl\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : HasPullbacks C\nI : Type u_1\nB : C\nP : I → Subobject B\ns✝ : Finset I\nih : ∀ i ∈ s✝, (P i).Factors (s✝.inf P).arrow\ni✝ : I\na✝ : i✝ ∉ s✝\n⊢ (P i✝).Factors (P i✝ ⊓ s✝.inf P).arrow",
"ppTerm": "?insert.inl",
"assigned": true,... | [
"case insert.inr\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : HasPullbacks C\nI : Type u_1\nB : C\nP : I → Subobject B\na✝¹ : I\ns✝ : Finset I\na✝ : a✝¹ ∉ s✝\nih : ∀ i ∈ s✝, (P i).Factors (s✝.inf P).arrow\ni✝ : I\nm : i✝ ∈ s✝\n⊢ (P i✝).Factors (P a✝¹ ⊓ s✝.inf P).arrow"
] | · rw [← factorThru_arrow _ _ (inf_arrow_factors_left _ _)]
exact factors_comp_arrow _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Subobject.Lattice | {
"line": 477,
"column": 63
} | {
"line": 483,
"column": 39
} | {
"line": 485,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : HasPullbacks C\nX Y : C\ng : Y ⟶ X\ninst✝ : Mono g\nf₁ f₂ : Subobject Y\n⊢ (map g).obj (f₁ ⊓ f₂) = (map g).obj f₁ ⊓ (map g).obj f₂",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Over",
... | [] | by
revert f₁
apply Quotient.ind'
intro f₁
erw [inf_def, inf_def, inf_eq_map_pullback', inf_eq_map_pullback', ← map_comp]
dsimp
rw [pullback_comp, pullback_map_self] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Abelian.Exact | {
"line": 305,
"column": 12
} | {
"line": 305,
"column": 29
} | {
"line": 305,
"column": 30
} | [
{
"pp": "A : Type u₁\nB : Type u₂\ninst✝⁶ : Category.{v₁, u₁} A\ninst✝⁵ : Category.{v₂, u₂} B\ninst✝⁴ : Abelian A\ninst✝³ : Abelian B\nL : A ⥤ B\ninst✝² : L.PreservesZeroMorphisms\ninst✝¹ : L.PreservesMonomorphisms\ninst✝ : ∀ {X Y : A} (f : X ⟶ Y), PreservesColimit (parallelPair f 0) L\nS : ShortComplex A\nhS :... | [
"A : Type u₁\nB : Type u₂\ninst✝⁶ : Category.{v₁, u₁} A\ninst✝⁵ : Category.{v₂, u₂} B\ninst✝⁴ : Abelian A\ninst✝³ : Abelian B\nL : A ⥤ B\ninst✝² : L.PreservesZeroMorphisms\ninst✝¹ : L.PreservesMonomorphisms\ninst✝ : ∀ {X Y : A} (f : X ⟶ Y), PreservesColimit (parallelPair f 0) L\nS : ShortComplex A\nhS : S.Exact\n⊢ ... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Shapes.Countable | {
"line": 174,
"column": 4
} | {
"line": 174,
"column": 67
} | {
"line": 175,
"column": 4
} | [
{
"pp": "case inr\nJ : Type u_2\ninst✝² : Countable J\ninst✝¹ : Preorder J\ninst✝ : IsFiltered J\nd : J\nn : ℕ\ng : d ≤ (sequentialFunctor J).obj n\nthis✝ : Nonempty (StructuredArrow d (sequentialFunctor J))\ni j : StructuredArrow d (sequentialFunctor J)\nthis :\n ∀ (J : Type u_2) [inst : Countable J] [inst_1 ... | [
"J✝ : Type u_2\ninst✝⁵ : Countable J✝\ninst✝⁴ : Preorder J✝\ninst✝³ : IsFiltered J✝\nJ : Type u_2\ninst✝² : Countable J\ninst✝¹ : Preorder J\ninst✝ : IsFiltered J\nd : J\nn : ℕ\ng : d ≤ (sequentialFunctor J).obj n\nthis : Nonempty (StructuredArrow d (sequentialFunctor J))\ni j : StructuredArrow d (sequentialFunctor... | · exact or_comm.1 (this J d n g inferInstance j i (le_of_lt h)) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Comma.Presheaf.Basic | {
"line": 548,
"column": 23
} | {
"line": 548,
"column": 35
} | {
"line": 548,
"column": 35
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nA : Cᵒᵖ ⥤ Type v\nF : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v\ns t : (CostructuredArrow yoneda A)ᵒᵖ\nf : t ⟶ s\nx : F.obj t\nthis : (CostructuredArrow.mkPrecomp (unop t).hom f.unop.left).op = f ≫ eqToHom ⋯\n⊢ t = op (CostructuredArrow.mk (YonedaCollection.fst (cou... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.Countable | {
"line": 225,
"column": 4
} | {
"line": 225,
"column": 67
} | {
"line": 226,
"column": 4
} | [
{
"pp": "case inr\nJ : Type u_2\ninst✝² : Countable J\ninst✝¹ : Preorder J\ninst✝ : IsCofiltered J\nd : J\nn : ℕ\ng : (sequentialFunctor J).obj (op n) ≤ d\nthis✝ : Nonempty (CostructuredArrow (sequentialFunctor J) d)\ni j : CostructuredArrow (sequentialFunctor J) d\nthis :\n ∀ (J : Type u_2) [inst : Countable ... | [
"J✝ : Type u_2\ninst✝⁵ : Countable J✝\ninst✝⁴ : Preorder J✝\ninst✝³ : IsCofiltered J✝\nJ : Type u_2\ninst✝² : Countable J\ninst✝¹ : Preorder J\ninst✝ : IsCofiltered J\nd : J\nn : ℕ\ng : (sequentialFunctor J).obj (op n) ≤ d\nthis : Nonempty (CostructuredArrow (sequentialFunctor J) d)\ni j : CostructuredArrow (sequen... | · exact or_comm.1 (this J d n g inferInstance j i (le_of_lt h)) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Limits.Constructions.WeaklyInitial | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 65
} | {
"line": 66,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasWideEqualizers C\nT : C\nhT : ∀ (X : C), Nonempty (T ⟶ X)\nendos : Type v := T ⟶ T\ni : wideEqualizer id ⟶ T := wideEqualizer.ι id\nthis✝ : Nonempty endos\nthis : (X : C) → Unique (wideEqualizer id ⟶ X)\n⊢ HasInitial C",
"ppTerm": "?m.38",
"ass... | [] | exact hasInitial_of_unique (wideEqualizer (id : endos → endos)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Subobject.Comma | {
"line": 205,
"column": 2
} | {
"line": 205,
"column": 18
} | {
"line": 206,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nS : C ⥤ D\nT : D\nA : CostructuredArrow S T\nP Q : (CostructuredArrow S T)ᵒᵖ\nf : P ⟶ op A\ng : Q ⟶ op A\ninst✝¹ : Mono f.unop.left.op\ninst✝ : Mono g.unop.left.op\nh : Subobject.mk f.unop.left.op ≤ Subobject.mk g.uno... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nS : C ⥤ D\nT : D\nA : CostructuredArrow S T\nP Q : (CostructuredArrow S T)ᵒᵖ\nf : P ⟶ op A\ng : Q ⟶ op A\ninst✝¹ : Mono f.unop.left.op\ninst✝ : Mono g.unop.left.op\nh : Subobject.mk f.unop.left.op ≤ Subobject.mk g.unop.left.op\n⊢... | rw [← unop_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 552,
"column": 39
} | {
"line": 553,
"column": 63
} | {
"line": 555,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nG : C\nh : IsCoseparator G\nα : C ≌ D\n⊢ IsCoseparator (α.functor.obj G)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"CategoryTheory.ObjectProperty.strictMap",
"congrArg",
"C... | [] | by
simpa using! ObjectProperty.IsCoseparating.of_equivalence h α | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 574,
"column": 6
} | {
"line": 574,
"column": 17
} | {
"line": 574,
"column": 18
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nG : C\n⊢ IsDetector (op G) ↔ IsCodetector G",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Opposite",
"congrArg",
"CategoryTheory.ObjectProperty.singleton",
"id",
"CategoryTheory.ObjectPropert... | [
"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nG : C\n⊢ (ObjectProperty.singleton (op G)).IsDetecting ↔ IsCodetector G"
] | IsDetector, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 577,
"column": 6
} | {
"line": 577,
"column": 17
} | {
"line": 577,
"column": 18
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nG : C\n⊢ IsCodetector (op G) ↔ IsDetector G",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Opposite",
"congrArg",
"CategoryTheory.ObjectProperty.singleton",
"id",
"CategoryTheory.ObjectPropert... | [
"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nG : C\n⊢ IsCodetector (op G) ↔ (ObjectProperty.singleton G).IsDetecting"
] | IsDetector, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 580,
"column": 6
} | {
"line": 580,
"column": 17
} | {
"line": 580,
"column": 18
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nG : Cᵒᵖ\n⊢ IsCodetector (unop G) ↔ IsDetector G",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Opposite",
"congrArg",
"CategoryTheory.ObjectProperty.singleton",
"id",
"CategoryTheory.ObjectPro... | [
"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nG : Cᵒᵖ\n⊢ IsCodetector (unop G) ↔ (ObjectProperty.singleton G).IsDetecting"
] | IsDetector, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 583,
"column": 6
} | {
"line": 583,
"column": 17
} | {
"line": 583,
"column": 18
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nG : Cᵒᵖ\n⊢ IsDetector (unop G) ↔ IsCodetector G",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Opposite",
"congrArg",
"CategoryTheory.ObjectProperty.singleton",
"id",
"CategoryTheory.ObjectPro... | [
"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nG : Cᵒᵖ\n⊢ (ObjectProperty.singleton (unop G)).IsDetecting ↔ IsCodetector G"
] | IsDetector, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Presheaf | {
"line": 607,
"column": 4
} | {
"line": 607,
"column": 63
} | {
"line": 607,
"column": 63
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nℰ : Type u₂\ninst✝³ : Category.{v₂, u₂} ℰ\nA✝ A : C ⥤ ℰ\ninst✝² : uliftYoneda.{max w v₂, v₁, u₁}.HasPointwiseLeftKanExtension A\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ Type (max w v₁ v₂)), F.op.HasLeftKanExtension P\nΦ : uli... | [] | rw [natTrans_app_uliftYoneda_obj, Iso.hom_inv_id_app_assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 675,
"column": 2
} | {
"line": 675,
"column": 31
} | {
"line": 676,
"column": 2
} | [
{
"pp": "case refine_1\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nG : C\ninst✝ : ∀ (A : C), HasProduct fun x ↦ G\nh : ∀ ⦃X Y : C⦄ (f g : X ⟶ Y), (∀ (h : Y ⟶ G), f ≫ h = g ≫ h) → f = g\nA Z✝ : C\nu v : Z✝ ⟶ A\nhuv : (u ≫ Pi.lift fun f ↦ f) = v ≫ Pi.lift fun f ↦ f\ni : A ⟶ G\n⊢ u ≫ i = v ≫ i",
"ppTerm": "?re... | [
"case refine_2\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nG : C\ninst✝ : ∀ (A : C), HasProduct fun x ↦ G\nh : ∀ (A : C), Mono (Pi.lift fun f ↦ f)\nX Y : C\nf g : X ⟶ Y\nhh : ∀ (h : Y ⟶ G), f ≫ h = g ≫ h\n⊢ f = g"
] | · simpa using huv =≫ Pi.π _ i | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Category.Grp.Adjunctions | {
"line": 104,
"column": 6
} | {
"line": 105,
"column": 65
} | {
"line": 106,
"column": 4
} | [
{
"pp": "case pos\nX Y : Type u\nf : X ⟶ Y\nx✝ : Mono f\nhX : IsEmpty X\nZ✝ : AddCommGrpCat\ng✝ h✝ : Z✝ ⟶ free.obj X\na✝ : g✝ ≫ free.map f = h✝ ≫ free.map f\n⊢ g✝ = h✝",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"CategoryTheory.Limits.PreservesColimitsOfShape.pre... | [] | apply (IsInitial.isInitialObj free _
((Types.initial_iff_empty X).2 hX).some).isZero.eq_of_tgt | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Category.Grp.Injective | {
"line": 40,
"column": 4
} | {
"line": 40,
"column": 12
} | {
"line": 41,
"column": 4
} | [
{
"pp": "case inl\nA : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : DivisibleBy A ℤ\ng : ↥(ℤ ∙ 0) →ₗ[ℤ] A\nn : ℤ\nhn✝ : n ∈ ℤ ∙ 0\nhn : n ∈ ⊥\n⊢ 0 n = g ⟨n, hn✝⟩",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"AddCommGroup.toAddCommMonoid",
"Submodule.add... | [
"case inl\nA : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : DivisibleBy A ℤ\ng : ↥(ℤ ∙ 0) →ₗ[ℤ] A\nhn : 0 ∈ ℤ ∙ 0\n⊢ 0 0 = g ⟨0, hn⟩"
] | subst hn | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
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