module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.Algebra.Category.Ring.Colimits | {
"line": 550,
"column": 4
} | {
"line": 550,
"column": 27
} | [
{
"pp": "case a.left_distrib\nJ : Type v\ninst✝ : SmallCategory J\nF : J ⥤ CommRingCat\ns : Cocone F\nx✝ y✝ x y z : Prequotient F\n⊢ descFunLift F s (x.mul (y.add z)) = descFunLift F s ((x.mul y).add (x.mul z))",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"NonUnitalCommRing.t... | | left_distrib x y z => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity | {
"line": 117,
"column": 41
} | {
"line": 123,
"column": 50
} | [
{
"pp": "R : Type u_2\ninst✝³ : CommMonoidWithZero R\ninst✝² : UniqueFactorizationMonoid R\ninst✝¹ : NormalizationMonoid R\ninst✝ : DecidableEq R\np x : R\nhp : Irreducible p\nhnorm : normalize p = p\nn : ℕ\nhle : p ^ n ∣ x\nhlt : ¬p ^ (n + 1) ∣ x\n⊢ count p (normalizedFactors x) = n",
"usedConstants": [
... | by
by_cases hx0 : x = 0
· simp [hx0] at hlt
apply Nat.cast_injective (R := ℕ∞)
convert (emultiplicity_eq_count_normalizedFactors hp hx0).symm
· exact hnorm.symm
exact (emultiplicity_eq_coe.mpr ⟨hle, hlt⟩).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.TwoSidedIdeal.Operations | {
"line": 334,
"column": 29
} | {
"line": 334,
"column": 46
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\nI : TwoSidedIdeal R\nx : R\n⊢ x ∈ asIdeal I ↔ x ∈ I",
"usedConstants": [
"Submodule",
"SetLike.mem_coe._simp_1",
"Semiring.toModule",
"AddSubsemigroup.instSetLike",
"Ring.toNonAssocRing",
"TwoSidedIdeal.asIdeal",
"congrArg",
... | by simp [asIdeal] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Localization.Away.Basic | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 7
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommSemiring R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nx : R\ninst✝ : Away x S\na b : R\nh : (algebraMap R S) a = (algebraMap R S) b\nn : ℕ\nhx : ↑⟨(fun x_1 ↦ x ^ x_1) n, ⋯⟩ * a = ↑⟨(fun x_1 ↦ x ^ x_1) n, ⋯⟩ * b\n⊢ ∃ n, x ^ n * a = x ^ n * b",
"usedConst... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.Localization.Away.Basic | {
"line": 129,
"column": 4
} | {
"line": 129,
"column": 9
} | [
{
"pp": "case refine_3\nR : Type u_1\ninst✝³ : CommSemiring R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nr : R\ninst✝ : Away r S\nu : Rˣ\na b : R\nhab : (algebraMap R S) a = (algebraMap R S) b\nn : ℕ\nhn : r ^ n * a = r ^ n * b\n⊢ ∃ n, (r * ↑u) ^ n * a = (r * ↑u) ^ n * b",
"usedConstants"... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.Localization.Away.Basic | {
"line": 188,
"column": 6
} | {
"line": 188,
"column": 11
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nP : Type u_3\ninst✝⁵ : CommSemiring P\nx : R\ninst✝⁴ : Away x S\ng : R →+* P\nQ : Type u_4\ninst✝³ : CommSemiring Q\ninst✝² : Algebra P Q\nf : R →+* P\nr : R\ninst✝¹ : Away r S\ninst✝ : ... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.CategoryTheory.Limits.Shapes.StrictInitial | {
"line": 182,
"column": 87
} | {
"line": 185,
"column": 53
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasStrictTerminalObjects C\nI : C\nhI : IsTerminal I\nA : C\nf g : I ⟶ A\n⊢ f = g",
"usedConstants": [
"CategoryTheory.Limits.IsTerminal.isIso_from",
"CategoryTheory.eq_of_inv_eq_inv",
"CategoryTheory.Limits.IsTerminal.hom_ext",
... | by
haveI := hI.isIso_from f
haveI := hI.isIso_from g
exact eq_of_inv_eq_inv (hI.hom_ext (inv f) (inv g)) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Mon_ | {
"line": 426,
"column": 2
} | {
"line": 430,
"column": 30
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nM : C\ninst✝ : MonObj M\n⊢ (tensorμ (𝟙_ C) M (𝟙_ C) M ≫ ((λ_ (𝟙_ C)).hom ⊗ₘ μ)) ≫ (λ_ M).hom = ((λ_ M).hom ⊗ₘ (λ_ M).hom) ≫ μ",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Category.as... | rw [← Category.comp_id (λ_ (𝟙_ C)).hom, ← Category.id_comp μ, ← tensorHom_comp_tensorHom]
simp only [tensorHom_id, id_tensorHom]
slice_lhs 3 4 => rw [leftUnitor_naturality]
slice_lhs 1 3 => rw [← leftUnitor_monoidal]
simp only [Category.id_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Mon_ | {
"line": 426,
"column": 2
} | {
"line": 430,
"column": 30
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nM : C\ninst✝ : MonObj M\n⊢ (tensorμ (𝟙_ C) M (𝟙_ C) M ≫ ((λ_ (𝟙_ C)).hom ⊗ₘ μ)) ≫ (λ_ M).hom = ((λ_ M).hom ⊗ₘ (λ_ M).hom) ≫ μ",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Category.as... | rw [← Category.comp_id (λ_ (𝟙_ C)).hom, ← Category.id_comp μ, ← tensorHom_comp_tensorHom]
simp only [tensorHom_id, id_tensorHom]
slice_lhs 3 4 => rw [leftUnitor_naturality]
slice_lhs 1 3 => rw [← leftUnitor_monoidal]
simp only [Category.id_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic | {
"line": 88,
"column": 7
} | {
"line": 90,
"column": 48
} | [
{
"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... | by
rw [← cancel_epi e₁.hom, ← reassoc_of% commfst, ← commf,
← reassoc_of% commsnd, ← commg, h.w_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic | {
"line": 98,
"column": 14
} | {
"line": 99,
"column": 73
} | [
{
"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.... | change snd = e₁.hom ≫ snd' ≫ e₃.inv
rw [← reassoc_of% commsnd, e₃.hom_inv_id, Category.comp_id] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic | {
"line": 98,
"column": 14
} | {
"line": 99,
"column": 73
} | [
{
"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.... | change snd = e₁.hom ≫ snd' ≫ e₃.inv
rw [← reassoc_of% commsnd, e₃.hom_inv_id, Category.comp_id] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.BaseChange | {
"line": 208,
"column": 53
} | {
"line": 208,
"column": 97
} | [
{
"pp": "R : Type u_1\ninst✝²¹ : CommSemiring R\nS✝ : Submonoid R\nA : Type u_2\ninst✝²⁰ : CommSemiring A\ninst✝¹⁹ : Algebra R A\ninst✝¹⁸ : IsLocalization S✝ A\nM : Type u_3\ninst✝¹⁷ : AddCommMonoid M\ninst✝¹⁶ : Module R M\nM' : Type u_4\ninst✝¹⁵ : AddCommMonoid M'\ninst✝¹⁴ : Module R M'\ninst✝¹³ : Module A M'\... | simp [IsScalarTower.algebraMap_apply R A Aₚ] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Localization.BaseChange | {
"line": 208,
"column": 53
} | {
"line": 208,
"column": 97
} | [
{
"pp": "R : Type u_1\ninst✝²¹ : CommSemiring R\nS✝ : Submonoid R\nA : Type u_2\ninst✝²⁰ : CommSemiring A\ninst✝¹⁹ : Algebra R A\ninst✝¹⁸ : IsLocalization S✝ A\nM : Type u_3\ninst✝¹⁷ : AddCommMonoid M\ninst✝¹⁶ : Module R M\nM' : Type u_4\ninst✝¹⁵ : AddCommMonoid M'\ninst✝¹⁴ : Module R M'\ninst✝¹³ : Module A M'\... | simp [IsScalarTower.algebraMap_apply R A Aₚ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Localization.BaseChange | {
"line": 208,
"column": 53
} | {
"line": 208,
"column": 97
} | [
{
"pp": "R : Type u_1\ninst✝²¹ : CommSemiring R\nS✝ : Submonoid R\nA : Type u_2\ninst✝²⁰ : CommSemiring A\ninst✝¹⁹ : Algebra R A\ninst✝¹⁸ : IsLocalization S✝ A\nM : Type u_3\ninst✝¹⁷ : AddCommMonoid M\ninst✝¹⁶ : Module R M\nM' : Type u_4\ninst✝¹⁵ : AddCommMonoid M'\ninst✝¹⁴ : Module R M'\ninst✝¹³ : Module A M'\... | simp [IsScalarTower.algebraMap_apply R A Aₚ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monad.Basic | {
"line": 233,
"column": 61
} | {
"line": 235,
"column": 5
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nM N : Monad C\nf : M.toFunctor ≅ N.toFunctor\nf_η : ∀ (X : C), M.η.app X ≫ f.hom.app X = N.η.app X\nf_μ : ∀ (X : C), M.μ.app X ≫ f.hom.app X = (M.map (f.hom.app X) ≫ f.hom.app (N.obj X)) ≫ N.μ.app X\n⊢ (monadToFunctor C).mapIso (MonadIso.mk f f_η f_μ) = f",
... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monad.Basic | {
"line": 251,
"column": 65
} | {
"line": 253,
"column": 5
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nM N : Comonad C\nf : M.toFunctor ≅ N.toFunctor\nf_ε : ∀ (X : C), f.hom.app X ≫ N.ε.app X = M.ε.app X\nf_δ : ∀ (X : C), f.hom.app X ≫ N.δ.app X = M.δ.app X ≫ f.hom.app (M.obj X) ≫ N.map (f.hom.app X)\n⊢ (comonadToFunctor C).mapIso (ComonadIso.mk f f_ε f_δ) = f",... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.LocalizedModule.Basic | {
"line": 228,
"column": 6
} | {
"line": 228,
"column": 61
} | [
{
"pp": "case mk\nR : Type u\ninst✝⁷ : CommSemiring R\nS✝ : Submonoid R\nM : Type v\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\nT : Type u_1\ninst✝⁴ : CommSemiring T\ninst✝³ : Algebra R T\ninst✝² : IsLocalization S✝ T\nA : Type u_2\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Submonoid R\na✝ : LocalizedMo... | exact mk_eq.mpr ⟨1, by simp only [mul_zero, smul_zero]⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Module.LocalizedModule.Basic | {
"line": 333,
"column": 18
} | {
"line": 333,
"column": 28
} | [
{
"pp": "case h\nR : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\nx y : T\nm : M\ns : ↥S\n⊢ mk\n ((((IsLocalization.sec S x).1 * ↑(IsLocalization.sec S y)... | ← add_smul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.LocalizedModule.Basic | {
"line": 622,
"column": 8
} | {
"line": 622,
"column": 41
} | [
{
"pp": "case e_a\nR : 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\n... | rw [mul_smul, Submonoid.smul_def] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Category.Ring.Constructions | {
"line": 434,
"column": 4
} | {
"line": 434,
"column": 79
} | [
{
"pp": "case uniq\nA B C : CommRingCat\nf : A ⟶ C\ng : B ⟶ C\ns : PullbackCone f g\nm : s.pt ⟶ of ↥(((Hom.hom f).comp (RingHom.fst ↑A ↑B)).eqLocus ((Hom.hom g).comp (RingHom.snd ↑A ↑B)))\ne₁ :\n m ≫\n ofHom\n ((RingHom.fst ↑A ↑B).comp\n (((Hom.hom f).comp (RingHom.fst ↑A ↑B)).eqLocus ((Ho... | have eq1 := (congr_arg (fun f : s.pt →+* A => f x) (congrArg Hom.hom e₁) :) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Bicategory.Functor.Lax | {
"line": 184,
"column": 74
} | {
"line": 184,
"column": 85
} | [
{
"pp": "case a.a\nB : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nF✝ : B ⥤ᴸ C\nD : Type u₃\ninst✝ : Bicategory D\nF : B ⥤ᴸ C\nG : C ⥤ᴸ D\na✝ b✝ c✝ d✝ : B\nf : a✝ ⟶ b✝\ng : b✝ ⟶ c✝\nh : c✝ ⟶ d✝\n| G.map₂ (F.mapComp f g ▷ F.map h) ≫ G.map₂ (F.mapComp (f ≫ g) h ≫ F.map₂ (α_ f g h).hom)",
... | G.map₂_comp | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.Bicategory.Functor.Pseudofunctor | {
"line": 316,
"column": 38
} | {
"line": 318,
"column": 63
} | [
{
"pp": "B : Type u₁\ninst✝⁴ : Bicategory B\nC : Type u₂\ninst✝³ : Bicategory C\nD : Type u₃\ninst✝² : Bicategory D\nF✝ : B ⥤ᵖ C\nF : B ⥤ᵒᵖᴸ C\ninst✝¹ : ∀ (a : B), IsIso (F.mapId a)\ninst✝ : ∀ {a b c : B} (f : a ⟶ b) (g : b ⟶ c), IsIso (F.mapComp f g)\na✝ b✝ c✝ : B\nf : a✝ ⟶ b✝\ng h : b✝ ⟶ c✝\nη : g ⟶ h\n⊢ F.ma... | by
dsimp
rw [← assoc, IsIso.eq_comp_inv, F.mapComp_naturality_right] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.WithTerminal.Basic | {
"line": 372,
"column": 26
} | {
"line": 372,
"column": 100
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u_1\ninst✝ : Category.{v_1, u_1} D\nc c' : Comma (𝟭 (C ⥤ D)) (Functor.const C)\nφ : c ⟶ c'\nx y : WithTerminal C\nf : x ⟶ y\na : C\nx✝ : of a ⟶ star\n⊢ ((ofCommaObject c).map x✝ ≫\n match star with\n | of x => φ.left.app x\n | star => φ.r... | simp; simpa [-CommaMorphism.w] using (congrArg (fun f ↦ f.app a) φ.w).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.WithTerminal.Basic | {
"line": 372,
"column": 26
} | {
"line": 372,
"column": 100
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u_1\ninst✝ : Category.{v_1, u_1} D\nc c' : Comma (𝟭 (C ⥤ D)) (Functor.const C)\nφ : c ⟶ c'\nx y : WithTerminal C\nf : x ⟶ y\na : C\nx✝ : of a ⟶ star\n⊢ ((ofCommaObject c).map x✝ ≫\n match star with\n | of x => φ.left.app x\n | star => φ.r... | simp; simpa [-CommaMorphism.w] using (congrArg (fun f ↦ f.app a) φ.w).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.WithTerminal.Basic | {
"line": 404,
"column": 4
} | {
"line": 404,
"column": 15
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u_1\ninst✝ : Category.{v_1, u_1} D\nx : WithTerminal C ⥤ D\n⊢ mkCommaMorphism\n (NatIso.ofComponents\n (fun X ↦\n match X with\n | of x_1 => (Iso.refl (incl ⋙ x)).app x_1\n | star => Iso.refl (x.obj star))\n... | ext <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.WithTerminal.Cone | {
"line": 223,
"column": 34
} | {
"line": 223,
"column": 73
} | [
{
"pp": "case h\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nJ : Type w\ninst✝ : Category.{w', w} J\nX : C\nK : J ⥤ Under X\nF : C ⥤ D\nt✝ : Cocone K\nt : Cocone (liftFromUnder.obj K)\na b : J\nf : a ⟶ b\n⊢ (K.map f ≫ { left := 𝟙 (K.obj b).left, right := t.ι.app (of b)... | simpa using t.ι.naturality (incl.map f) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Limits.Shapes.Grothendieck | {
"line": 78,
"column": 6
} | {
"line": 79,
"column": 25
} | [
{
"pp": "case w.e_a.e_a.w_base\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nF : C ⥤ Cat\nH : Type u₂\ninst✝¹ : Category.{v₂, u₂} H\nG : Grothendieck F ⥤ H\ninst✝ : ∀ {X Y : C} (f : X ⟶ Y), HasColimit ((F.map f).toFunctor ⋙ Grothendieck.ι F Y ⋙ G)\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nd : ↑(F.obj X)\n⊢ { base := f ≫ ... | simp only [eqToHom_refl, Category.assoc, Grothendieck.comp_base,
Category.comp_id] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Limits.Shapes.Grothendieck | {
"line": 78,
"column": 6
} | {
"line": 79,
"column": 25
} | [
{
"pp": "case w.e_a.e_a.w_base\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nF : C ⥤ Cat\nH : Type u₂\ninst✝¹ : Category.{v₂, u₂} H\nG : Grothendieck F ⥤ H\ninst✝ : ∀ {X Y : C} (f : X ⟶ Y), HasColimit ((F.map f).toFunctor ⋙ Grothendieck.ι F Y ⋙ G)\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nd : ↑(F.obj X)\n⊢ { base := f ≫ ... | simp only [eqToHom_refl, Category.assoc, Grothendieck.comp_base,
Category.comp_id] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Grothendieck | {
"line": 78,
"column": 6
} | {
"line": 79,
"column": 25
} | [
{
"pp": "case w.e_a.e_a.w_base\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nF : C ⥤ Cat\nH : Type u₂\ninst✝¹ : Category.{v₂, u₂} H\nG : Grothendieck F ⥤ H\ninst✝ : ∀ {X Y : C} (f : X ⟶ Y), HasColimit ((F.map f).toFunctor ⋙ Grothendieck.ι F Y ⋙ G)\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nd : ↑(F.obj X)\n⊢ { base := f ≫ ... | simp only [eqToHom_refl, Category.assoc, Grothendieck.comp_base,
Category.comp_id] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Final | {
"line": 259,
"column": 4
} | {
"line": 260,
"column": 55
} | [
{
"pp": "case h₁\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝¹ : F.Final\nE : Type u₃\ninst✝ : Category.{v₃, u₃} E\nG : D ⥤ E\nc : Cocone (F ⋙ G)\nX : D\nY : C\nf : X ⟶ F.obj Y\n⊢ ∀ (X₁ X₂ : C) (k₁ : X ⟶ F.obj X₁) (k₂ : X ⟶ F.obj X₂) (f_1 : X₁ ⟶ X₂),\n ... | intro _ _ _ _ _ h₁ h₂
simp [← h₁, ← Functor.comp_map, c.ι.naturality, h₂] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Final | {
"line": 259,
"column": 4
} | {
"line": 260,
"column": 55
} | [
{
"pp": "case h₁\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝¹ : F.Final\nE : Type u₃\ninst✝ : Category.{v₃, u₃} E\nG : D ⥤ E\nc : Cocone (F ⋙ G)\nX : D\nY : C\nf : X ⟶ F.obj Y\n⊢ ∀ (X₁ X₂ : C) (k₁ : X ⟶ F.obj X₁) (k₂ : X ⟶ F.obj X₂) (f_1 : X₁ ⟶ X₂),\n ... | intro _ _ _ _ _ h₁ h₂
simp [← h₁, ← Functor.comp_map, c.ι.naturality, h₂] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.WithTerminal.Basic | {
"line": 699,
"column": 13
} | {
"line": 699,
"column": 31
} | [
{
"pp": "case star.of.unit\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u_1\ninst✝ : Category.{v_1, u_1} D\nZ : D\nF : C ⥤ D\nM : (x : C) → Z ⟶ F.obj x\nhM : ∀ (x y : C) (f : x ⟶ y), M x ≫ F.map f = M y\nG : WithInitial C ⥤ D\nh : incl ⋙ G ≅ F\nhG : G.obj star ≅ Z\nhh : ∀ (x : C), hG.symm.hom ≫ G.map (star... | ← Iso.eq_inv_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.WithTerminal.Basic | {
"line": 808,
"column": 4
} | {
"line": 808,
"column": 15
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u_1\ninst✝ : Category.{v_1, u_1} D\nx : WithInitial C ⥤ D\n⊢ mkCommaMorphism\n (NatIso.ofComponents\n (fun X ↦\n match X with\n | of x_1 => (Iso.refl (incl ⋙ x)).app x_1\n | star => Iso.refl (x.obj star))\n ... | ext <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Assoc | {
"line": 166,
"column": 6
} | {
"line": 166,
"column": 24
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nX₁ X₂ X₃ Y₁ Y₂ : C\nf₁ : X₁ ⟶ Y₁\nf₂ : X₂ ⟶ Y₁\nf₃ : X₂ ⟶ Y₂\nf₄ : X₃ ⟶ Y₂\ninst✝³ : HasPullback f₁ f₂\ninst✝² : HasPullback f₃ f₄\ninst✝¹ : HasPullback (pullback.snd f₁ f₂ ≫ f₃) f₄\ninst✝ : HasPullback f₁ (pullback.fst f₃ f₄ ≫ f₂)\n⊢ (pullbackAssoc f₁ f₂ f₃ f₄).... | ← Iso.eq_inv_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.RingHomProperties | {
"line": 81,
"column": 35
} | {
"line": 81,
"column": 40
} | [
{
"pp": "P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nhP : RespectsIso P\nR S R' S' : Type u\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\ninst✝³ : Algebra R R'\ninst✝² : Algebra S S'\nf : R →+* S\nr : R\ninst✝¹ : IsLocalization.Aw... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts | {
"line": 127,
"column": 4
} | {
"line": 129,
"column": 18
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nD : Type u'\ninst✝³ : Category.{v', u'} D\nF : C ⥤ D\ninst✝² : PreservesLimitsOfShape (Discrete WalkingPair) F\ninst✝¹ : PreservesLimitsOfShape (Discrete PEmpty.{1}) F\ninst✝ : HasFiniteProducts C\nf : Fin 0 → C\n⊢ PreservesLimit (Discrete.functor f) F",
"use... | letI : PreservesLimitsOfShape (Discrete (Fin 0)) F :=
preservesLimitsOfShape_of_equiv.{0, 0} (Discrete.equivalence finZeroEquiv'.symm) _
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts | {
"line": 127,
"column": 4
} | {
"line": 129,
"column": 18
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nD : Type u'\ninst✝³ : Category.{v', u'} D\nF : C ⥤ D\ninst✝² : PreservesLimitsOfShape (Discrete WalkingPair) F\ninst✝¹ : PreservesLimitsOfShape (Discrete PEmpty.{1}) F\ninst✝ : HasFiniteProducts C\nf : Fin 0 → C\n⊢ PreservesLimit (Discrete.functor f) F",
"use... | letI : PreservesLimitsOfShape (Discrete (Fin 0)) F :=
preservesLimitsOfShape_of_equiv.{0, 0} (Discrete.equivalence finZeroEquiv'.symm) _
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts | {
"line": 201,
"column": 6
} | {
"line": 201,
"column": 51
} | [
{
"pp": "case h₂\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)\nm : (extendCofan... | rw [(BinaryCofan.IsColimit.desc' t₂ _ _).2.2] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.MorphismProperty.Limits | {
"line": 470,
"column": 4
} | {
"line": 470,
"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\nY₁ Y₂ : C\ng : (𝟭 C).obj Y₁ ⟶ (𝟭 C).obj Y₂\nX Y : C\nX₁ X₂ : J ⥤ C\nc₁ : Cone X₁\nc₂ : Cone X₂\nh₁ : IsLimit c₁\nh₂ : IsLimit c₂\nf : X₁ ⟶ X₂\nhf : W.functorCategory J f\ne : { left := c₁.1, r... | exact ⟨_, _, _, _, h₁', _, _, hf⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Limits.Shapes.Diagonal | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 41
} | [
{
"pp": "case h₁\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\nX Y Z : C\ninst✝¹ : HasPullbacks C\nU V₁ V₂ : C\nf : X ⟶ Y\ni : U ⟶ Y\ni₁ : V₁ ⟶ pullback f i\ni₂ : V₂ ⟶ pullback f i\ninst✝ : HasPullback i₁ i₂\n⊢ ((snd (diagonal f) (map (i₁ ≫ snd f i) (i₂ ≫ snd f i) f f (i₁ ≫ fst f i) (i₂ ≫ fst f i) i ⋯ ⋯) ≫\n ... | · simp only [Category.assoc, condition] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.ObjectProperty.LimitsOfShape | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 15
} | [
{
"pp": "case refine_2\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP : ObjectProperty C\nJ : Type u'\ninst✝ : Category.{v', u'} J\n⊢ P.limitsOfShape J ≤ (P.strictLimitsOfShape J).isoClosure",
"usedConstants": []
}
] | intro X ⟨h⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.ObjectProperty.LimitsOfShape | {
"line": 165,
"column": 2
} | {
"line": 165,
"column": 13
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP : ObjectProperty C\nJ : Type u'\ninst✝ : Category.{v', u'} J\nQ : ObjectProperty C\nhPQ : P ≤ Q\n⊢ P.limitsOfShape J ≤ Q.limitsOfShape J",
"usedConstants": []
}
] | intro X ⟨h⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.ObjectProperty.LimitsOfShape | {
"line": 172,
"column": 2
} | {
"line": 172,
"column": 13
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP : ObjectProperty C\nJ : Type u'\ninst✝ : Category.{v', u'} J\n⊢ P.isoClosure.limitsOfShape J ≤ P.limitsOfShape J",
"usedConstants": []
}
] | intro X ⟨h⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.ObjectProperty.Retract | {
"line": 103,
"column": 2
} | {
"line": 104,
"column": 32
} | [
{
"pp": "case a\nC : Type u\ninst✝¹ : Category.{v, u} C\nP : ObjectProperty C\ninst✝ : P.IsStableUnderRetracts\n⊢ P.retractClosure ≤ P",
"usedConstants": [
"_private.Mathlib.CategoryTheory.ObjectProperty.Retract.0.CategoryTheory.ObjectProperty.retractClosure_eq_self.match_1_1",
"CategoryTheory.O... | · intro X ⟨Y, hY, ⟨e⟩⟩
exact prop_of_retract P e hY | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.ObjectProperty.ColimitsOfShape | {
"line": 163,
"column": 4
} | {
"line": 163,
"column": 15
} | [
{
"pp": "case refine_2\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP : ObjectProperty C\nJ : Type u'\ninst✝ : Category.{v', u'} J\n⊢ P.colimitsOfShape J ≤ (P.strictColimitsOfShape J).isoClosure",
"usedConstants": []
}
] | intro X ⟨h⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.ObjectProperty.ColimitsOfShape | {
"line": 171,
"column": 2
} | {
"line": 171,
"column": 13
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP : ObjectProperty C\nJ : Type u'\ninst✝ : Category.{v', u'} J\nQ : ObjectProperty C\nhPQ : P ≤ Q\n⊢ P.colimitsOfShape J ≤ Q.colimitsOfShape J",
"usedConstants": []
}
] | intro X ⟨h⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.ObjectProperty.ColimitsOfShape | {
"line": 178,
"column": 2
} | {
"line": 178,
"column": 13
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP : ObjectProperty C\nJ : Type u'\ninst✝ : Category.{v', u'} J\n⊢ P.isoClosure.colimitsOfShape J ≤ P.colimitsOfShape J",
"usedConstants": []
}
] | intro X ⟨h⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Limits.Shapes.Diagonal | {
"line": 552,
"column": 2
} | {
"line": 553,
"column": 16
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\nX✝ Y✝ Z X Y : C\nf : X ⟶ Y\ninst✝¹ : HasPushout f f\ninst✝ : Epi f\n⊢ IsIso (codiagonal f)",
"usedConstants": [
"CategoryTheory.IsIso.inv_eq_of_hom_inv_id",
"Eq.mpr",
"CategoryTheory.Limits.pushout.codiagonal",
"CategoryTheory.Is... | rw [(IsIso.inv_eq_of_hom_inv_id (inl_codiagonal f)).symm]
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Diagonal | {
"line": 552,
"column": 2
} | {
"line": 553,
"column": 16
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\nX✝ Y✝ Z X Y : C\nf : X ⟶ Y\ninst✝¹ : HasPushout f f\ninst✝ : Epi f\n⊢ IsIso (codiagonal f)",
"usedConstants": [
"CategoryTheory.IsIso.inv_eq_of_hom_inv_id",
"Eq.mpr",
"CategoryTheory.Limits.pushout.codiagonal",
"CategoryTheory.Is... | rw [(IsIso.inv_eq_of_hom_inv_id (inl_codiagonal f)).symm]
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels | {
"line": 290,
"column": 41
} | {
"line": 290,
"column": 59
} | [
{
"pp": "C : Type u₁\ninst✝¹⁰ : Category.{v₁, u₁} C\ninst✝⁹ : HasZeroMorphisms C\nD : Type u₂\ninst✝⁸ : Category.{v₂, u₂} D\ninst✝⁷ : HasZeroMorphisms D\nG : C ⥤ D\ninst✝⁶ : G.PreservesZeroMorphisms\nX Y : C\nf : X ⟶ Y\ninst✝⁵ : HasCokernel f\ninst✝⁴ : HasCokernel (G.map f)\ninst✝³ : PreservesColimit (parallelP... | ← Iso.eq_inv_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Basic | {
"line": 220,
"column": 7
} | {
"line": 223,
"column": 50
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX Y : C\ninst✝ : HasZeroMorphisms C\nP Q R S : C\nf : P ⟶ Q\ng : P ⟶ R\nh : Q ⟶ S\nk : R ⟶ S\ngn : NormalEpi g\ncomm : f ≫ h = g ≫ k\nt : IsColimit (PushoutCocone.mk h k comm)\n⊢ (NormalEpi.g ≫ f) ≫ h = 0",
"usedConstants": [
"Eq.mpr",
"Categor... | by
have reassoc' {W : C} (h' : R ⟶ W) : gn.g ≫ g ≫ h' = 0 ≫ h' := by
rw [← Category.assoc, eq_whisker gn.w]
rw [Category.assoc, comm, reassoc', zero_comp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Abelian.Basic | {
"line": 232,
"column": 10
} | {
"line": 233,
"column": 83
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasKernels C\ninst✝² : HasCokernels C\ninst✝¹ : ∀ {X Y : C} (f : X ⟶ Y), IsIso (coimageImageComparison f)\ninst✝ : HasFiniteProducts C\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nm : Epi f\nthis✝¹ : HasImages C\nthis✝ : HasEqualizers ... | refine isColimitAux _ (fun A => inv (imageMonoFactorisation f).m ≫
inv (Abelian.coimageImageComparison f) ≫ colimit.desc _ _) aux ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Abelian.Basic | {
"line": 466,
"column": 4
} | {
"line": 466,
"column": 38
} | [
{
"pp": "case w.h\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX Y : C\nf : X ⟶ Y\n⊢ (imageStrongEpiMonoFactorisation f).e ≫ equalizer.ι (cokernel.π f) 0 =\n (factorThruImage f ≫ kernel.lift (cokernel.π f) (Limits.image.ι f) ⋯) ≫ equalizer.ι (cokernel.π f) 0",
"usedConstants": [
"Cat... | imageStrongEpiMonoFactorisation_e, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.Basic | {
"line": 684,
"column": 56
} | {
"line": 684,
"column": 60
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasPullbacks C\nW X Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : Epi f\nR : C\ne : Y ⟶ R\nh : pullback.snd f g ≫ e = 0\nu : X ⊞ Y ⟶ R := biprod.desc 0 e\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g ≫ u = 0\nthis✝¹ : IsColimit (... | ← hd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.Basic | {
"line": 717,
"column": 56
} | {
"line": 717,
"column": 60
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasPullbacks C\nW X Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : Epi g\nR : C\ne : X ⟶ R\nh : pullback.fst f g ≫ e = 0\nu : X ⊞ Y ⟶ R := biprod.desc e 0\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g ≫ u = 0\nthis✝¹ : IsColimit (... | ← hd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.Basic | {
"line": 766,
"column": 58
} | {
"line": 766,
"column": 62
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasPushouts C\nW X Y Z : C\nf : X ⟶ Y\ng : X ⟶ Z\ninst✝ : Mono f\nR : C\ne : R ⟶ Z\nh : e ≫ pushout.inr f g = 0\nu : R ⟶ Y ⊞ Z := biprod.lift 0 e\nhu : u ≫ BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis✝¹ : IsLimit (Ker... | ← hd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.Basic | {
"line": 789,
"column": 58
} | {
"line": 789,
"column": 62
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasPushouts C\nW X Y Z : C\nf : X ⟶ Y\ng : X ⟶ Z\ninst✝ : Mono g\nR : C\ne : R ⟶ Y\nh : e ≫ pushout.inl f g = 0\nu : R ⟶ Y ⊞ Z := biprod.lift e 0\nhu : u ≫ BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis✝¹ : IsLimit (Ker... | ← hd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Module.Basic | {
"line": 72,
"column": 2
} | {
"line": 73,
"column": 80
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁷ : Ring R\ninst✝⁶ : TopologicalSpace R\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : AddCommGroup M\ninst✝³ : ContinuousAdd M\ninst✝² : Module R M\ninst✝¹ : ContinuousSMul R M\ninst✝ : (𝓝[{x | IsUnit x}] 0).NeBot\ns : Submodule R M\ny : M\nhy : ↑s ∈ 𝓝 y\nx : M\nthis : Tends... | obtain ⟨_, hu : y + _ • _ ∈ s, u, rfl⟩ :=
nonempty_of_mem (inter_mem (Filter.mem_map.1 (this hy)) self_mem_nhdsWithin) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.Algebra.Module.Basic | {
"line": 240,
"column": 4
} | {
"line": 240,
"column": 37
} | [
{
"pp": "case refine_2.refine_1\nι : Type u_1\nR : Type u_2\nM : ι → Type u_3\ninst✝⁴ : Semiring R\ninst✝³ : (i : ι) → AddCommMonoid (M i)\ninst✝² : (i : ι) → Module R (M i)\ninst✝¹ : (i : ι) → TopologicalSpace (M i)\ninst✝ : DecidableEq ι\ns : (i : ι) → Submodule R (M i)\nx : (i : ι) → M i\nhx : x ∈ Set.univ.p... | · simp_all [Finset.sum_pi_single] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Algebra.Module.LinearMapPiProd | {
"line": 180,
"column": 2
} | {
"line": 181,
"column": 19
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : Semiring R\nM : Type u_2\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nι : Type u_4\nφ : ι → Type u_5\ninst✝² : (i : ι) → TopologicalSpace (φ i)\ninst✝¹ : (i : ι) → AddCommMonoid (φ i)\ninst✝ : (i : ι) → Module R (φ i)\nf : (i : ι) → M →L[R] φ i\n⊢ ... | simp only [ContinuousLinearMap.ext_iff, pi_apply, funext_iff]
exact forall_comm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Module.LinearMapPiProd | {
"line": 180,
"column": 2
} | {
"line": 181,
"column": 19
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : Semiring R\nM : Type u_2\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nι : Type u_4\nφ : ι → Type u_5\ninst✝² : (i : ι) → TopologicalSpace (φ i)\ninst✝¹ : (i : ι) → AddCommMonoid (φ i)\ninst✝ : (i : ι) → Module R (φ i)\nf : (i : ι) → M →L[R] φ i\n⊢ ... | simp only [ContinuousLinearMap.ext_iff, pi_apply, funext_iff]
exact forall_comm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Module.ModuleTopology | {
"line": 422,
"column": 6
} | {
"line": 423,
"column": 59
} | [
{
"pp": "case refine_1.continuous_smul\nR : Type u_1\nS : Type u_2\nτR : TopologicalSpace R\nτS : TopologicalSpace S\ninst✝⁷ : Ring R\ninst✝⁶ : Ring S\nA : Type u_3\ninst✝⁵ : AddCommGroup A\ninst✝⁴ : Module R A\ninst✝³ : TopologicalSpace A\ninst✝² : IsModuleTopology R A\nB' : Type u_5\ninst✝¹ : AddCommGroup B'\... | have hφ2 : (fun p ↦ p.1 • p.2 : S × B' → B') ∘ (Prod.map σ φ) =
φ ∘ (fun p ↦ p.1 • p.2 : R × A → A) := by ext; simp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Algebra.Module.ModuleTopology | {
"line": 426,
"column": 6
} | {
"line": 426,
"column": 91
} | [
{
"pp": "case refine_1.continuous_smul\nR : Type u_1\nS : Type u_2\nτR : TopologicalSpace R\nτS : TopologicalSpace S\ninst✝⁷ : Ring R\ninst✝⁶ : Ring S\nA : Type u_3\ninst✝⁵ : AddCommGroup A\ninst✝⁴ : Module R A\ninst✝³ : TopologicalSpace A\ninst✝² : IsModuleTopology R A\nB' : Type u_5\ninst✝¹ : AddCommGroup B'\... | have hoq : IsOpenQuotientMap (_ : R × A → S × B') := IsOpenQuotientMap.prodMap hσ hφo | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Category.TopCat.Basic | {
"line": 206,
"column": 89
} | {
"line": 208,
"column": 5
} | [
{
"pp": "X Y : TopCat\nf : ↑X ≃ₜ ↑Y\n⊢ homeoOfIso (isoOfHomeo f) = f",
"usedConstants": [
"Homeomorph.ext",
"TopCat.str",
"TopCat.homeoOfIso",
"Homeomorph.instEquivLike",
"TopCat.carrier",
"TopCat.isoOfHomeo",
"Homeomorph",
"Eq.refl",
"DFunLike.coe",
... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Category.TopCat.Basic | {
"line": 211,
"column": 88
} | {
"line": 213,
"column": 5
} | [
{
"pp": "X Y : TopCat\nf : X ≅ Y\n⊢ isoOfHomeo (homeoOfIso f) = f",
"usedConstants": [
"CategoryTheory.ConcreteCategory.hom",
"TopCat.instCategory",
"TopCat.ext",
"ContinuousMap",
"TopCat.str",
"TopCat.homeoOfIso",
"TopCat.carrier",
"TopCat.isoOfHomeo",
... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.CommAlgCat.Monoidal | {
"line": 87,
"column": 58
} | {
"line": 87,
"column": 69
} | [
{
"pp": "case hf\nR : Type u\ninst✝ : CommRing R\nA B C D X✝ Y✝ Z✝ : CommAlgCat R\nf✝ : Y✝ ⟶ Z✝\n⊢ (↑(Algebra.TensorProduct.comm R ↑X✝ ↑Z✝)).comp (Algebra.TensorProduct.map (AlgHom.id R ↑X✝) (Hom.hom f✝)) =\n (Algebra.TensorProduct.map (Hom.hom f✝) (AlgHom.id R ↑X✝)).comp ↑(Algebra.TensorProduct.comm R ↑X✝ ↑... | ext <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Category.CommAlgCat.Monoidal | {
"line": 88,
"column": 57
} | {
"line": 88,
"column": 68
} | [
{
"pp": "case hf\nR : Type u\ninst✝ : CommRing R\nA B C D X✝ Y✝ : CommAlgCat R\nf✝ : X✝ ⟶ Y✝\nZ✝ : CommAlgCat R\n⊢ (↑(Algebra.TensorProduct.comm R ↑Y✝ ↑Z✝)).comp (Algebra.TensorProduct.map (Hom.hom f✝) (AlgHom.id R ↑Z✝)) =\n (Algebra.TensorProduct.map (AlgHom.id R ↑Z✝) (Hom.hom f✝)).comp ↑(Algebra.TensorProd... | ext <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Category.CommAlgCat.Monoidal | {
"line": 89,
"column": 46
} | {
"line": 89,
"column": 57
} | [
{
"pp": "case hf\nR : Type u\ninst✝ : CommRing R\nA B C D S T U : CommAlgCat R\n⊢ ((↑(assoc R R R ↑T ↑U ↑S)).comp ↑(Algebra.TensorProduct.comm R (↑S) (↑T ⊗[R] ↑U))).comp ↑(assoc R R R ↑S ↑T ↑U) =\n ((Algebra.TensorProduct.map (AlgHom.id R ↑T) ↑(Algebra.TensorProduct.comm R ↑S ↑U)).comp\n ↑(assoc R R... | ext <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Category.CommAlgCat.Monoidal | {
"line": 90,
"column": 46
} | {
"line": 90,
"column": 57
} | [
{
"pp": "case hf\nR : Type u\ninst✝ : CommRing R\nA B C D S T U : CommAlgCat R\n⊢ ((↑(assoc R R R ↑U ↑S ↑T).symm).comp ↑(Algebra.TensorProduct.comm R (↑S ⊗[R] ↑T) ↑U)).comp\n ↑(assoc R R R ↑S ↑T ↑U).symm =\n ((Algebra.TensorProduct.map (↑(Algebra.TensorProduct.comm R ↑S ↑U)) (AlgHom.id R ↑T)).comp\n ... | ext <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Topology.Algebra.Module.Equiv | {
"line": 350,
"column": 77
} | {
"line": 352,
"column": 5
} | [
{
"pp": "R₁ : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\ninst✝¹⁹ : Semiring R₁\ninst✝¹⁸ : Semiring R₂\ninst✝¹⁷ : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₁ : R₂ →+* R₁\ninst✝¹⁶ : RingHomInvPair σ₁₂ σ₂₁\ninst✝¹⁵ : RingHomInvPair σ₂₁ σ₁₂\nσ₂₃ : R₂ →+* R₃\nσ₃₂ : R₃ →+* R₂\ninst✝¹⁴ : RingHomInvPair σ₂₃ σ₃₂\ninst✝¹³ : RingHomIn... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ShortComplex.LeftHomology | {
"line": 314,
"column": 4
} | {
"line": 314,
"column": 12
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh₁ : S₁.LeftHomologyData\nh₂ : S₂.LeftHomologyData\nψ₁ ψ₂ : LeftHomologyMapData φ h₁ h₂\nhK : ψ₁.φK = ψ₂.φK\nhH : ψ₁.φH = ψ₂.φH\n⊢ ψ₁ = ψ₂",
"usedConstants": [
"CategoryTheory.S... | cases ψ₁ | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.Topology.Algebra.Module.LinearMap | {
"line": 255,
"column": 59
} | {
"line": 258,
"column": 39
} | [
{
"pp": "R₁ : Type u_1\nR₂ : Type u_2\ninst✝¹⁴ : Semiring R₁\ninst✝¹³ : Semiring R₂\nσ₁₂ : R₁ →+* R₂\nM₁ : Type u_4\ninst✝¹² : TopologicalSpace M₁\ninst✝¹¹ : AddCommMonoid M₁\nM₂ : Type u_6\ninst✝¹⁰ : TopologicalSpace M₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₁ M₁\ninst✝⁷ : Module R₂ M₂\ninst✝⁶ : RingHomS... | by
rw [SetLike.ext'_iff] at hs ⊢
simp only [Submodule.topologicalClosure_coe, Submodule.top_coe, ← dense_iff_closure_eq] at hs ⊢
exact hf'.dense_image f.continuous hs | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ShortComplex.RightHomology | {
"line": 409,
"column": 4
} | {
"line": 409,
"column": 12
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh₁ : S₁.RightHomologyData\nh₂ : S₂.RightHomologyData\nψ₁ ψ₂ : RightHomologyMapData φ h₁ h₂\nhQ : ψ₁.φQ = ψ₂.φQ\nhH : ψ₁.φH = ψ₂.φH\n⊢ ψ₁ = ψ₂",
"usedConstants": [
"Eq.refl",
... | cases ψ₁ | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.Algebra.Homology.ShortComplex.RightHomology | {
"line": 414,
"column": 75
} | {
"line": 414,
"column": 81
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh₁ : S₁.RightHomologyData\nh₂ : S₂.RightHomologyData\n⊢ φ.τ₁ ≫ S₂.f ≫ h₂.p = 0",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.ShortComplex.RightHomologyData.wp",
"Ca... | h₂.wp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.ShortComplex.RightHomology | {
"line": 1115,
"column": 63
} | {
"line": 1119,
"column": 66
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₁.RightHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\n⊢ S₂.RightHomologyData",
"usedConstants": [
"Opposite",
"CategoryTheory.ShortComplex.... | by
haveI : Epi (opMap φ).τ₁ := by dsimp; infer_instance
haveI : IsIso (opMap φ).τ₂ := by dsimp; infer_instance
haveI : Mono (opMap φ).τ₃ := by dsimp; infer_instance
exact (LeftHomologyData.ofEpiOfIsIsoOfMono' (opMap φ) h.op).unop | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ShortComplex.RightHomology | {
"line": 1342,
"column": 6
} | {
"line": 1342,
"column": 44
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nS : ShortComplex C\nh : S.RightHomologyData\nA : C\nk : S.X₂ ⟶ A\nhk : S.f ≫ k = 0\ninst✝ : S.HasRightHomology\n⊢ h.opcyclesIso.hom ≫ h.descQ k hk = S.descOpcycles k hk",
"usedConstants": [
"CategoryTheory.ShortComplex... | ← h.opcyclesIso_inv_comp_descOpcycles, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.ShortComplex.Homology | {
"line": 939,
"column": 2
} | {
"line": 940,
"column": 95
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\nS : ShortComplex C\ninst✝ : S.HasHomology\n⊢ S.homologyπ ≫ S.homologyι = S.iCycles ≫ S.pOpcycles",
"usedConstants": [
"CategoryTheory.ShortComplex.opcycles",
"Eq.mpr",
"CategoryTheory.Category.assoc",
"Cate... | dsimp only [homologyπ, homologyι]
simpa only [assoc, S.leftRightHomologyComparison_fac] using S.π_leftRightHomologyComparison_ι | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.Homology | {
"line": 939,
"column": 2
} | {
"line": 940,
"column": 95
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\nS : ShortComplex C\ninst✝ : S.HasHomology\n⊢ S.homologyπ ≫ S.homologyι = S.iCycles ≫ S.pOpcycles",
"usedConstants": [
"CategoryTheory.ShortComplex.opcycles",
"Eq.mpr",
"CategoryTheory.Category.assoc",
"Cate... | dsimp only [homologyπ, homologyι]
simpa only [assoc, S.leftRightHomologyComparison_fac] using S.π_leftRightHomologyComparison_ι | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.Homology | {
"line": 1143,
"column": 2
} | {
"line": 1143,
"column": 30
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\nS S₁ S₂ S₃ S₄ : ShortComplex C\nφ : S₁ ⟶ S₂\nh₁ : S₁.HomologyData\nh₂ : S₂.HomologyData\nA : C\nhf : S.f = 0\ninst✝ : S.HasHomology\n⊢ S.cycles ≅ S.homology",
"usedConstants": [
"CategoryTheory.ShortComplex.homologyπ",
... | have := S.isIso_homologyπ hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Homology.ShortComplex.Homology | {
"line": 1144,
"column": 2
} | {
"line": 1144,
"column": 25
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\nS S₁ S₂ S₃ S₄ : ShortComplex C\nφ : S₁ ⟶ S₂\nh₁ : S₁.HomologyData\nh₂ : S₂.HomologyData\nA : C\nhf : S.f = 0\ninst✝ : S.HasHomology\nthis : IsIso S.homologyπ\n⊢ S.cycles ≅ S.homology",
"usedConstants": [
"CategoryTheory.Shor... | exact asIso S.homologyπ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Homology.ShortComplex.Preadditive | {
"line": 57,
"column": 36
} | {
"line": 57,
"column": 50
} | [
{
"pp": "case h₁\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\na b : S₁ ⟶ S₂\n⊢ (a + b).τ₁ = (b + a).τ₁",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.ShortComplex.Hom.τ₁",
"AddCommGroup.... | apply add_comm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Homology.ShortComplex.Preadditive | {
"line": 57,
"column": 36
} | {
"line": 57,
"column": 50
} | [
{
"pp": "case h₂\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\na b : S₁ ⟶ S₂\n⊢ (a + b).τ₂ = (b + a).τ₂",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"AddCommGroup.toAddCommMonoid",
"CategoryTheory.Shor... | apply add_comm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Homology.ShortComplex.Preadditive | {
"line": 57,
"column": 36
} | {
"line": 57,
"column": 50
} | [
{
"pp": "case h₃\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\na b : S₁ ⟶ S₂\n⊢ (a + b).τ₃ = (b + a).τ₃",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"AddCommGroup.toAddCommMonoid",
"add_comm",
"C... | apply add_comm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Homology.ComplexShape | {
"line": 150,
"column": 14
} | {
"line": 150,
"column": 28
} | [
{
"pp": "ι : Type u_1\nc : ComplexShape ι\nj : ι\nhj : ∀ (k : ι), ¬c.Rel j k\n⊢ ¬∃ j_1, c.Rel j j_1",
"usedConstants": [
"Eq.mpr",
"not_exists._simp_1",
"Exists",
"id",
"Eq",
"Not",
"ComplexShape.Rel"
]
}
] | simpa using hj | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Homology.ComplexShape | {
"line": 150,
"column": 14
} | {
"line": 150,
"column": 28
} | [
{
"pp": "ι : Type u_1\nc : ComplexShape ι\nj : ι\nhj : ∀ (k : ι), ¬c.Rel j k\n⊢ ¬∃ j_1, c.Rel j j_1",
"usedConstants": [
"Eq.mpr",
"not_exists._simp_1",
"Exists",
"id",
"Eq",
"Not",
"ComplexShape.Rel"
]
}
] | simpa using hj | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ComplexShape | {
"line": 150,
"column": 14
} | {
"line": 150,
"column": 28
} | [
{
"pp": "ι : Type u_1\nc : ComplexShape ι\nj : ι\nhj : ∀ (k : ι), ¬c.Rel j k\n⊢ ¬∃ j_1, c.Rel j j_1",
"usedConstants": [
"Eq.mpr",
"not_exists._simp_1",
"Exists",
"id",
"Eq",
"Not",
"ComplexShape.Rel"
]
}
] | simpa using hj | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Subobject.FactorThru | {
"line": 92,
"column": 45
} | {
"line": 95,
"column": 24
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y Z : C\nP : Subobject Z\nf : X ⟶ Y\ng : Y ⟶ Z\nh : P.Factors g\n⊢ P.Factors (f ≫ g)",
"usedConstants": [
"CategoryTheory.Subobject.Factors",
"CategoryTheory.Category.assoc",
"CategoryTheory.Over",
"CategoryTheory.CategoryStruct.to... | by
induction P using Quotient.ind'
obtain ⟨g, rfl⟩ := h
exact ⟨f ≫ g, by simp⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Subobject.Basic | {
"line": 538,
"column": 4
} | {
"line": 541,
"column": 39
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX Y Z : C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nA : C\nB : D\ne : MonoOver A ≌ MonoOver B\n⊢ lower e.inverse ⋙ lower e.functor ≅ 𝟭 (Subobject B)",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Over",
"CategoryTheory.Functor",
"... | apply eqToIso
convert ThinSkeleton.map_iso_eq e.counitIso
· exact (ThinSkeleton.map_comp_eq _ _).symm
· exact ThinSkeleton.map_id_eq.symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Subobject.Basic | {
"line": 538,
"column": 4
} | {
"line": 541,
"column": 39
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX Y Z : C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nA : C\nB : D\ne : MonoOver A ≌ MonoOver B\n⊢ lower e.inverse ⋙ lower e.functor ≅ 𝟭 (Subobject B)",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Over",
"CategoryTheory.Functor",
"... | apply eqToIso
convert ThinSkeleton.map_iso_eq e.counitIso
· exact (ThinSkeleton.map_comp_eq _ _).symm
· exact ThinSkeleton.map_id_eq.symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Subobject.MonoOver | {
"line": 119,
"column": 4
} | {
"line": 119,
"column": 15
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX✝ Y Z : C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nX : C\nf g : MonoOver X\n⊢ ∀ (a b : f ⟶ g), a = b",
"usedConstants": [
"CategoryTheory.Over",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.instCategory... | intro h₁ h₂ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Subobject.Limits | {
"line": 109,
"column": 6
} | {
"line": 109,
"column": 35
} | [
{
"pp": "case refine_1.w\nC : Type u\ninst✝² : Category.{v, u} C\nX Y : C\nf g : X ⟶ Y\ninst✝¹ : HasEqualizer f g\nW : C\nh : W ⟶ X\ninst✝ : HasPullbacks C\nthis :\n pullbackπ h (equalizerSubobject f g) ≫ (equalizerSubobject f g).arrow =\n ((Subobject.pullback h).obj (equalizerSubobject f g)).arrow ≫ h\n⊢ p... | equalizerSubobject_arrow_comp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Subobject.Limits | {
"line": 107,
"column": 4
} | {
"line": 109,
"column": 36
} | [
{
"pp": "case refine_1.w\nC : Type u\ninst✝² : Category.{v, u} C\nX Y : C\nf g : X ⟶ Y\ninst✝¹ : HasEqualizer f g\nW : C\nh : W ⟶ X\ninst✝ : HasPullbacks C\nthis :\n pullbackπ h (equalizerSubobject f g) ≫ (equalizerSubobject f g).arrow =\n ((Subobject.pullback h).obj (equalizerSubobject f g)).arrow ≫ h\n⊢ (... | rw [← reassoc_of% (Subobject.isPullback h (equalizerSubobject f g)).w,
← reassoc_of% (Subobject.isPullback h (equalizerSubobject f g)).w,
equalizerSubobject_arrow_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Subobject.Limits | {
"line": 112,
"column": 22
} | {
"line": 112,
"column": 51
} | [
{
"pp": "case refine_2.hF.w\nC : Type u\ninst✝² : Category.{v, u} C\nX Y : C\nf g : X ⟶ Y\ninst✝¹ : HasEqualizer f g\nW : C\nh : W ⟶ X\ninst✝ : HasPullbacks C\n⊢ (equalizerSubobject (h ≫ f) (h ≫ g)).arrow ≫ h ≫ f = (equalizerSubobject (h ≫ f) (h ≫ g)).arrow ≫ h ≫ g",
"usedConstants": [
"Eq.mpr",
... | equalizerSubobject_arrow_comp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Subobject.Limits | {
"line": 328,
"column": 2
} | {
"line": 328,
"column": 34
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nX Y Z : C\nf : X ⟶ Y\ninst✝¹ : HasImage f\ninst✝ : HasEqualizers C\n⊢ Epi (factorThruImageSubobject f)",
"usedConstants": [
"CategoryTheory.Subobject.underlying",
"CategoryTheory.Epi",
"CategoryTheory.Limits.imageSubobject",
"PartialOr... | dsimp [factorThruImageSubobject] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.GradedObject | {
"line": 350,
"column": 2
} | {
"line": 350,
"column": 27
} | [
{
"pp": "I : Type u_1\nJ : Type u_2\nC : Type u_4\ninst✝¹ : Category.{v_1, u_4} C\nX : GradedObject I C\np : I → J\ninst✝ : X.HasMap p\nA : C\nj : J\nφ : (i : I) → p i = j → (X i ⟶ A)\ni : I\nhi : p i = j\n⊢ X.ιMapObj p i j hi ≫ X.descMapObj p φ = φ i hi",
"usedConstants": [
"CategoryTheory.CategorySt... | apply Cofan.IsColimit.fac | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.GradedObject | {
"line": 350,
"column": 2
} | {
"line": 350,
"column": 27
} | [
{
"pp": "I : Type u_1\nJ : Type u_2\nC : Type u_4\ninst✝¹ : Category.{v_1, u_4} C\nX : GradedObject I C\np : I → J\ninst✝ : X.HasMap p\nA : C\nj : J\nφ : (i : I) → p i = j → (X i ⟶ A)\ni : I\nhi : p i = j\n⊢ X.ιMapObj p i j hi ≫ X.descMapObj p φ = φ i hi",
"usedConstants": [
"CategoryTheory.CategorySt... | apply Cofan.IsColimit.fac | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.GradedObject | {
"line": 350,
"column": 2
} | {
"line": 350,
"column": 27
} | [
{
"pp": "I : Type u_1\nJ : Type u_2\nC : Type u_4\ninst✝¹ : Category.{v_1, u_4} C\nX : GradedObject I C\np : I → J\ninst✝ : X.HasMap p\nA : C\nj : J\nφ : (i : I) → p i = j → (X i ⟶ A)\ni : I\nhi : p i = j\n⊢ X.ιMapObj p i j hi ≫ X.descMapObj p φ = φ i hi",
"usedConstants": [
"CategoryTheory.CategorySt... | apply Cofan.IsColimit.fac | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Subobject.Lattice | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 50
} | [
{
"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",
"usedConstants": [
"CategoryTheory.Limits.factorThruImage",
"CategoryTheory.Over",
... | refine homMk (coprod.inr ≫ factorThruImage _) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Preadditive.Injective.Basic | {
"line": 139,
"column": 4
} | {
"line": 139,
"column": 61
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nβ : Type v\nc : β → C\ninst✝¹ : HasProduct c\ninst✝ : ∀ (b : β), Injective (c b)\nX✝ Y✝ : C\ng : X✝ ⟶ ∏ᶜ c\nf : X✝ ⟶ Y✝\nmono : Mono f\n⊢ ∃ h, f ≫ h = g",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"Categor... | refine ⟨Pi.lift fun b => factorThru (g ≫ Pi.π c _) f, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
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