module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.Valuation.LocalSubring | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 52
} | {
"line": 100,
"column": 2
} | [
{
"pp": "K : Type u_3\ninst✝ : Field K\nR : LocalSubring K\nhR : IsMax R\n⊢ ∀ x ∉ R.toSubring, x⁻¹ ∈ R.toSubring",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"DivisionCommMonoid.toDivisionMonoid",
"Subring.instSetLike",
"DivInvOneMonoid.toInvOneClass",
"Membershi... | [
"K : Type u_3\ninst✝ : Field K\nR : LocalSubring K\nhR : IsMax R\nx : K\nhx : x ∉ R.toSubring\n⊢ IsIntegral (↥R.toSubring) x⁻¹"
] | refine fun x hx ↦ mem_of_isMax_of_isIntegral hR ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Valuation.LocalSubring | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 37
} | {
"line": 157,
"column": 0
} | [
{
"pp": "K : Type u_3\ninst✝ : Field K\nA : Subring K\nI : Ideal ↥A\nhI : I ≠ ⊤\nM : Ideal ↥A\nhM : M.IsMaximal\nle : I ≤ M\nV : ValuationSubring K\nhV : LocalSubring.ofPrime A M ≤ V.toLocalSubring\na : ↥A\nha : a ∈ ↑I\n⊢ A.subtype a ∈\n Subtype.val '' ↑(map ((Subring.inclusion ⋯).comp (algebraMap ↥A ↥(Local... | [] | exact ⟨_, mem_map_of_mem _ ha, rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 682,
"column": 31
} | {
"line": 684,
"column": 62
} | {
"line": 686,
"column": 0
} | [
{
"pp": "A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nm : ℕ\nf_deg : f ∈ 𝒜 m\nhm : 0 < m\n⊢ IsIso (toSpec 𝒜 f).base",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ca... | [] | by
convert! (projIsoSpecTopComponent f_deg hm).isIso_hom
exact ConcreteCategory.hom_ext _ _ <| toSpec_base_apply_eq 𝒜 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 749,
"column": 64
} | {
"line": 749,
"column": 94
} | {
"line": 749,
"column": 94
} | [
{
"pp": "case surj.refine_2\nA : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nx : ↥(pbo f)\nm : ℕ\nf_deg : f ∈ 𝒜 m\nhm : 0 < m\nthis : Algebra (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal) := (mapId 𝒜 ⋯).toAl... | [
"case surj.refine_2\nA : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nx : ↥(pbo f)\nm : ℕ\nf_deg : f ∈ 𝒜 m\nhm : 0 < m\nthis : Algebra (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal) := (mapId 𝒜 ⋯).toAlgebra\ni : ℕ... | tsub_add_cancel_of_le (by lia) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 223,
"column": 2
} | {
"line": 226,
"column": 88
} | {
"line": 227,
"column": 2
} | [
{
"pp": "C₀ : Type u₀\nC : Type u\ninst✝² : Category.{v₀, u₀} C₀\ninst✝¹ : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\ninst✝ : IsDenseSubsite J₀ J F\nX : C\ndata : F.OneHypercoverDenseData J₀ J X\ni₁ i₂ : data.I₀\nW : C\np₁ : W ⟶ F.obj (data.X i₁)\np₂ : W ⟶ F.obj (da... | [
"C₀ : Type u₀\nC : Type u\ninst✝² : Category.{v₀, u₀} C₀\ninst✝¹ : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\ninst✝ : IsDenseSubsite J₀ J F\nX : C\ndata : F.OneHypercoverDenseData J₀ J X\ni₁ i₂ : data.I₀\nW : C\np₁ : W ⟶ F.obj (data.X i₁)\np₂ : W ⟶ F.obj (data.X i₂)\nw ... | let T := Sieve.bind S.arrows (fun Z g hg ↦ by
letI str := Presieve.getFunctorPushforwardStructure hg.bindStruct.hg
exact Sieve.pullback str.lift
(Sieve.functorPushforward F (data.sieve₁₀ str.cover.1.choose str.cover.2.choose))) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.CategoryTheory.Sites.Point.Basic | {
"line": 318,
"column": 2
} | {
"line": 319,
"column": 16
} | {
"line": 321,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nΦ : J.Point\ninst✝¹ : LocallySmall.{w, v, u} C\nU T : C\nf : U ⟶ T\ninst✝ : Mono f\n⊢ Function.Injective ⇑(ConcreteCategory.hom (Φ.fiber.map f))",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | rw [← mono_iff_injective]
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Point.Basic | {
"line": 318,
"column": 2
} | {
"line": 319,
"column": 16
} | {
"line": 321,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nΦ : J.Point\ninst✝¹ : LocallySmall.{w, v, u} C\nU T : C\nf : U ⟶ T\ninst✝ : Mono f\n⊢ Function.Injective ⇑(ConcreteCategory.hom (Φ.fiber.map f))",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | rw [← mono_iff_injective]
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.ConstantSheaf | {
"line": 207,
"column": 2
} | {
"line": 207,
"column": 65
} | {
"line": 208,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\nJ : GrothendieckTopology C\nD : Type u_2\ninst✝⁵ : Category.{v_2, u_2} D\ninst✝⁴ : HasWeakSheafify J D\nB : Type u_3\ninst✝³ : Category.{v_3, u_3} B\nU : D ⥤ B\ninst✝² : HasWeakSheafify J B\ninst✝¹ : J.PreservesSheafification U\ninst✝ : J.HasSheafCompose U\... | [
"C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\nJ : GrothendieckTopology C\nD : Type u_2\ninst✝⁵ : Category.{v_2, u_2} D\ninst✝⁴ : HasWeakSheafify J D\nB : Type u_3\ninst✝³ : Category.{v_3, u_3} B\nU : D ⥤ B\ninst✝² : HasWeakSheafify J B\ninst✝¹ : J.PreservesSheafification U\ninst✝ : J.HasSheafCompose U\nF : Sheaf J... | rw [constantCommuteCompose_hom_app_hom, assoc, Iso.inv_comp_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 669,
"column": 8
} | {
"line": 669,
"column": 41
} | {
"line": 670,
"column": 8
} | [
{
"pp": "C₀ : Type u₀\nC : Type u\ninst✝⁴ : Category.{v₀, u₀} C₀\ninst✝³ : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\nA : Type u'\ninst✝² : Category.{v', u'} A\ninst✝¹ : IsDenseSubsite J₀ J F\ndata : (X : C) → F.OneHypercoverDenseData J₀ J X\ninst✝ : HasLimitsOfSize... | [
"C₀ : Type u₀\nC : Type u\ninst✝⁴ : Category.{v₀, u₀} C₀\ninst✝³ : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\nA : Type u'\ninst✝² : Category.{v', u'} A\ninst✝¹ : IsDenseSubsite J₀ J F\ndata : (X : C) → F.OneHypercoverDenseData J₀ J X\ninst✝ : HasLimitsOfSize.{w, w, v', ... | have hb' := Sieve.ofArrows.fac hb | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 671,
"column": 26
} | {
"line": 671,
"column": 62
} | {
"line": 672,
"column": 10
} | [
{
"pp": "C₀ : Type u₀\nC : Type u\ninst✝⁴ : Category.{v₀, u₀} C₀\ninst✝³ : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\nA : Type u'\ninst✝² : Category.{v', u'} A\ninst✝¹ : IsDenseSubsite J₀ J F\ndata : (X : C) → F.OneHypercoverDenseData J₀ J X\ninst✝ : HasLimitsOfSize... | [
"C₀ : Type u₀\nC : Type u\ninst✝⁴ : Category.{v₀, u₀} C₀\ninst✝³ : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\nA : Type u'\ninst✝² : Category.{v', u'} A\ninst✝¹ : IsDenseSubsite J₀ J F\ndata : (X : C) → F.OneHypercoverDenseData J₀ J X\ninst✝ : HasLimitsOfSize.{w, w, v', ... | IsDenseSubsite.mapPreimage_comp_map, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 672,
"column": 10
} | {
"line": 672,
"column": 46
} | {
"line": 673,
"column": 10
} | [
{
"pp": "C₀ : Type u₀\nC : Type u\ninst✝⁴ : Category.{v₀, u₀} C₀\ninst✝³ : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\nA : Type u'\ninst✝² : Category.{v', u'} A\ninst✝¹ : IsDenseSubsite J₀ J F\ndata : (X : C) → F.OneHypercoverDenseData J₀ J X\ninst✝ : HasLimitsOfSize... | [
"C₀ : Type u₀\nC : Type u\ninst✝⁴ : Category.{v₀, u₀} C₀\ninst✝³ : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\nA : Type u'\ninst✝² : Category.{v', u'} A\ninst✝¹ : IsDenseSubsite J₀ J F\ndata : (X : C) → F.OneHypercoverDenseData J₀ J X\ninst✝ : HasLimitsOfSize.{w, w, v', ... | IsDenseSubsite.mapPreimage_comp_map, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Sites.SheafQuasiCompact | {
"line": 47,
"column": 4
} | {
"line": 92,
"column": 20
} | {
"line": 94,
"column": 0
} | [
{
"pp": "case refine_3\nP : MorphismProperty Scheme\ninst✝² : P.IsStableUnderBaseChange\ninst✝¹ : P.IsMultiplicative\nF : Schemeᵒᵖ ⥤ Type u_1\ninst✝ : IsZariskiLocalAtSource P\nx✝ :\n Presieve.IsSheaf zariskiTopology F ∧\n ∀ {R S : CommRingCat} (f : R ⟶ S),\n P (Spec.map f) → Surjective (Spec.map f) → ... | [] | rw [Precoverage.isSheaf_toGrothendieck_iff_of_isStableUnderBaseChange_of_small.{u}]
intro T (𝒰 : Scheme.Cover _ _)
wlog hT : ∃ (R : CommRingCat.{u}), T = Spec R generalizing T
· refine T.affineOneHypercover.isSheafFor_of_pullback hzar ?_ ?_
· intro i
rw [← Sieve.pullbackArrows_comm, ← Presiev... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Sites.SheafQuasiCompact | {
"line": 47,
"column": 4
} | {
"line": 92,
"column": 20
} | {
"line": 94,
"column": 0
} | [
{
"pp": "case refine_3\nP : MorphismProperty Scheme\ninst✝² : P.IsStableUnderBaseChange\ninst✝¹ : P.IsMultiplicative\nF : Schemeᵒᵖ ⥤ Type u_1\ninst✝ : IsZariskiLocalAtSource P\nx✝ :\n Presieve.IsSheaf zariskiTopology F ∧\n ∀ {R S : CommRingCat} (f : R ⟶ S),\n P (Spec.map f) → Surjective (Spec.map f) → ... | [] | rw [Precoverage.isSheaf_toGrothendieck_iff_of_isStableUnderBaseChange_of_small.{u}]
intro T (𝒰 : Scheme.Cover _ _)
wlog hT : ∃ (R : CommRingCat.{u}), T = Spec R generalizing T
· refine T.affineOneHypercover.isSheafFor_of_pullback hzar ?_ ?_
· intro i
rw [← Sieve.pullbackArrows_comm, ← Presiev... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.CommSq | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 67
} | {
"line": 78,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX₁ X₂ X₃ X₄ : C\nt : X₁ ⟶ X₂\nl : X₁ ⟶ X₃\nr : X₂ ⟶ X₄\nb : X₃ ⟶ X₄\nh : IsPushout t l r b\nT : C\nx₄ : T ⟶ X₄\nthis : Epi ⋯.shortComplex.g\nT' : C\nπ : T' ⟶ T\nw✝ : Epi π\nu : T' ⟶ ⋯.shortComplex.X₂\nhu : π ≫ x₄ = u ≫ ⋯.shortComplex.g\n⊢ ∃ T' ... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX₁ X₂ X₃ X₄ : C\nt : X₁ ⟶ X₂\nl : X₁ ⟶ X₃\nr : X₂ ⟶ X₄\nb : X₃ ⟶ X₄\nh : IsPushout t l r b\nT : C\nx₄ : T ⟶ X₄\nthis : Epi ⋯.shortComplex.g\nT' : C\nπ : T' ⟶ T\nw✝ : Epi π\nu : T' ⟶ ⋯.shortComplex.X₂\nhu : π ≫ x₄ = u ≫ ⋯.shortComplex.g\n⊢ π ≫ x₄ = (u ≫ bip... | refine ⟨T', π, inferInstance, u ≫ biprod.fst, u ≫ biprod.snd, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Sites.Subcanonical | {
"line": 97,
"column": 72
} | {
"line": 98,
"column": 72
} | {
"line": 100,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝ : J.Subcanonical\nX Y : Cᵒᵖ\nF : Sheaf J (Type v)\nf : J.yoneda.obj (unop X) ⟶ F\ng : X ⟶ Y\n⊢ (ConcreteCategory.hom (F.obj.map g)) (J.yonedaEquiv f) = (ConcreteCategory.hom (f.hom.app Y)) g.unop",
"ppTerm": "?m.53",
"ass... | [] | by
rw [yonedaEquiv_naturality', yonedaEquiv_comp, yonedaEquiv_yoneda_map] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Subcanonical | {
"line": 185,
"column": 2
} | {
"line": 186,
"column": 30
} | {
"line": 188,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝ : J.Subcanonical\nX : C\nF F' : Sheaf J (Type (max v v'))\nf : F ⟶ F'\nx : F.obj.obj (op X)\n⊢ J.uliftYonedaEquiv.symm x ≫ f = J.uliftYonedaEquiv.symm ((ConcreteCategory.hom (f.hom.app (op X))) x)",
"ppTerm": "?m.56",
"as... | [] | apply J.uliftYonedaEquiv.injective
simp [uliftYonedaEquiv_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Subcanonical | {
"line": 185,
"column": 2
} | {
"line": 186,
"column": 30
} | {
"line": 188,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝ : J.Subcanonical\nX : C\nF F' : Sheaf J (Type (max v v'))\nf : F ⟶ F'\nx : F.obj.obj (op X)\n⊢ J.uliftYonedaEquiv.symm x ≫ f = J.uliftYonedaEquiv.symm ((ConcreteCategory.hom (f.hom.app (op X))) x)",
"ppTerm": "?m.56",
"as... | [] | apply J.uliftYonedaEquiv.injective
simp [uliftYonedaEquiv_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Subcanonical | {
"line": 245,
"column": 2
} | {
"line": 245,
"column": 25
} | {
"line": 246,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝ : J.Subcanonical\n⊢ PreservesLimitsOfSize.{u_1, u_2, v, max u v, u, max u (v + 1)} J.yoneda",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor",
"Opposite",
"CategoryTh... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝ : J.Subcanonical\nI : Type u_2\nx✝ : Category.{u_1, u_2} I\n⊢ PreservesLimitsOfShape I J.yoneda"
] | refine ⟨fun {I} _ ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Sites.Subcanonical | {
"line": 306,
"column": 2
} | {
"line": 306,
"column": 87
} | {
"line": 307,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝³ : J.Subcanonical\nι : Type u_1\ninst✝² : CoproductsOfShapeDisjoint C ι\ninst✝¹ : HasPullbacks C\ninst✝ : HasStrictInitialObjects C\nhcov : ∀ {X : ι → C} {c : Cofan X} (x : IsColimit c), Sieve.ofArrows X c.inj ∈ J c.pt\nhtriv : ∀... | [
"C : Type u\ninst✝⁴ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝³ : J.Subcanonical\nι : Type u_1\ninst✝² : CoproductsOfShapeDisjoint C ι\ninst✝¹ : HasPullbacks C\ninst✝ : HasStrictInitialObjects C\nhcov : ∀ {X : ι → C} {c : Cofan X} (x : IsColimit c), Sieve.ofArrows X c.inj ∈ J c.pt\nhtriv : ∀ (Y : C) (a ... | apply (config := { allowSynthFailures := true }) preservesColimitsOfShape_of_discrete | Mathlib.Tactic._aux_Mathlib_Tactic_ApplyWith___elabRules_Mathlib_Tactic_applyWith_1 | Mathlib.Tactic.applyWith |
Mathlib.CategoryTheory.Limits.Shapes.Preorder.HasIterationOfShape | {
"line": 78,
"column": 4
} | {
"line": 78,
"column": 48
} | {
"line": 79,
"column": 4
} | [
{
"pp": "case neg\nJ : Type w\ninst✝⁶ : LinearOrder J\nC : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : HasIterationOfShape J C\ninst✝³ : SuccOrder J\ninst✝² : WellFoundedLT J\nα : Type u_1\ninst✝¹ : PartialOrder α\nf : α ≤i J\ninst✝ : Nonempty α\nhf : ¬Function.Surjective ⇑f\ns : α <i J := f.toPrincipalSeg hf\... | [
"case neg\nJ : Type w\ninst✝⁶ : LinearOrder J\nC : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : HasIterationOfShape J C\ninst✝³ : SuccOrder J\ninst✝² : WellFoundedLT J\nα : Type u_1\ninst✝¹ : PartialOrder α\nf : α ≤i J\ninst✝ : Nonempty α\nhf : ¬Function.Surjective ⇑f\ns : α <i J := f.toPrincipalSeg hf\ni : J\nhi₀ ... | obtain ⟨i, hi₀⟩ : ∃ i, i = s.top := ⟨_, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Order.Interval.Set.SuccOrder | {
"line": 36,
"column": 6
} | {
"line": 37,
"column": 28
} | {
"line": 37,
"column": 28
} | [
{
"pp": "J : Type u_1\ninst✝¹ : PartialOrder J\ninst✝ : SuccOrder J\nj : J\ni : ↑(Iic j)\nhi : ¬IsMax i\n⊢ Order.succ ↑i ∈ Iic j",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.OrdConnected.succOrder",
"Subtype.coe_prop",
"Order.succ",
"Set.ordC... | [] | rw [← coe_succ_of_not_isMax hi]
apply Subtype.coe_prop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Interval.Set.SuccOrder | {
"line": 36,
"column": 6
} | {
"line": 37,
"column": 28
} | {
"line": 37,
"column": 28
} | [
{
"pp": "J : Type u_1\ninst✝¹ : PartialOrder J\ninst✝ : SuccOrder J\nj : J\ni : ↑(Iic j)\nhi : ¬IsMax i\n⊢ Order.succ ↑i ∈ Iic j",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.OrdConnected.succOrder",
"Subtype.coe_prop",
"Order.succ",
"Set.ordC... | [] | rw [← coe_succ_of_not_isMax hi]
apply Subtype.coe_prop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Shrink | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 38
} | {
"line": 55,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Small.{u, u_1} α\ninst✝ : Preorder α\na b : Shrink.{u, u_1} α\n⊢ (equivShrink α) ((equivShrink α).symm a) ≤ (equivShrink α) ((equivShrink α).symm b) ↔\n (equivShrink α).symm a ≤ (equivShrink α).symm b",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\ninst✝¹ : Small.{u, u_1} α\ninst✝ : Preorder α\na b : Shrink.{u, u_1} α\n⊢ a ≤ b ↔ (equivShrink α).symm a ≤ (equivShrink α).symm b"
] | simp only [Equiv.apply_symm_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicTopology.RelativeCellComplex.AttachCells | {
"line": 95,
"column": 2
} | {
"line": 97,
"column": 26
} | {
"line": 99,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nα : Type t\nA B : α → C\ng : (a : α) → A a ⟶ B a\nX₁ X₂ : C\nf : X₁ ⟶ X₂\nc : AttachCells g f\nZ : C\nφ φ' : X₂ ⟶ Z\nh₀ : f ≫ φ = f ≫ φ'\nh : ∀ (i : c.ι), c.cell i ≫ φ = c.cell i ≫ φ'\n⊢ φ = φ'",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
... | [] | apply c.isPushout.hom_ext h₀
apply Cofan.IsColimit.hom_ext c.isColimit₂
simpa [cell_def] using h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.RelativeCellComplex.AttachCells | {
"line": 95,
"column": 2
} | {
"line": 97,
"column": 26
} | {
"line": 99,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nα : Type t\nA B : α → C\ng : (a : α) → A a ⟶ B a\nX₁ X₂ : C\nf : X₁ ⟶ X₂\nc : AttachCells g f\nZ : C\nφ φ' : X₂ ⟶ Z\nh₀ : f ≫ φ = f ≫ φ'\nh : ∀ (i : c.ι), c.cell i ≫ φ = c.cell i ≫ φ'\n⊢ φ = φ'",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
... | [] | apply c.isPushout.hom_ext h₀
apply Cofan.IsColimit.hom_ext c.isColimit₂
simpa [cell_def] using h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.SmallObject.Iteration.Basic | {
"line": 468,
"column": 2
} | {
"line": 468,
"column": 81
} | {
"line": 470,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\nJ : Type w\nΦ : SuccStruct C\ninst✝⁴ : LinearOrder J\ninst✝³ : SuccOrder J\ninst✝² : OrderBot J\ninst✝¹ : HasIterationOfShape J C\ninst✝ : WellFoundedLT J\nj₁ j₂ j₃ : J\niter₁ : Φ.Iteration j₁\niter₂ : Φ.Iteration j₂\niter₃ : Φ.Iteration j₃\nk₁ k₂ k₃ : J\nh₁₂ : k... | [] | simp [mapObj, congr_map iter₂ iter₃ h₁₂ h₂ (h₂.trans h₂₃'), ← Functor.map_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.SmallObject.Iteration.Basic | {
"line": 468,
"column": 2
} | {
"line": 468,
"column": 81
} | {
"line": 470,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\nJ : Type w\nΦ : SuccStruct C\ninst✝⁴ : LinearOrder J\ninst✝³ : SuccOrder J\ninst✝² : OrderBot J\ninst✝¹ : HasIterationOfShape J C\ninst✝ : WellFoundedLT J\nj₁ j₂ j₃ : J\niter₁ : Φ.Iteration j₁\niter₂ : Φ.Iteration j₂\niter₃ : Φ.Iteration j₃\nk₁ k₂ k₃ : J\nh₁₂ : k... | [] | simp [mapObj, congr_map iter₂ iter₃ h₁₂ h₂ (h₂.trans h₂₃'), ← Functor.map_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.SmallObject.Iteration.Basic | {
"line": 468,
"column": 2
} | {
"line": 468,
"column": 81
} | {
"line": 470,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\nJ : Type w\nΦ : SuccStruct C\ninst✝⁴ : LinearOrder J\ninst✝³ : SuccOrder J\ninst✝² : OrderBot J\ninst✝¹ : HasIterationOfShape J C\ninst✝ : WellFoundedLT J\nj₁ j₂ j₃ : J\niter₁ : Φ.Iteration j₁\niter₂ : Φ.Iteration j₂\niter₃ : Φ.Iteration j₃\nk₁ k₂ k₃ : J\nh₁₂ : k... | [] | simp [mapObj, congr_map iter₂ iter₃ h₁₂ h₂ (h₂.trans h₂₃'), ← Functor.map_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.SmallObject.IsCardinalForSmallObjectArgument | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 26
} | {
"line": 164,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nI : MorphismProperty C\nκ : Cardinal.{w}\ninst✝² : Fact κ.IsRegular\ninst✝¹ : OrderBot κ.ord.ToType\ninst✝ : I.IsCardinalForSmallObjectArgument κ\nthis✝¹ : LocallySmall.{w, v, u} C\nthis✝ : IsSmall.{w, v, u} I\nthis : ∀ (X Y : C) (p : X ⟶ Y), HasColimitsOfShape (... | [
"C : Type u\ninst✝³ : Category.{v, u} C\nI : MorphismProperty C\nκ : Cardinal.{w}\ninst✝² : Fact κ.IsRegular\ninst✝¹ : OrderBot κ.ord.ToType\ninst✝ : I.IsCardinalForSmallObjectArgument κ\nthis✝² : LocallySmall.{w, v, u} C\nthis✝¹ : IsSmall.{w, v, u} I\nthis✝ : ∀ (X Y : C) (p : X ⟶ Y), HasColimitsOfShape (Discrete (... | haveI := hasPushouts I κ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 32
} | {
"line": 93,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nG X✝ Y : C\nX : Subobject G\n⊢ MorphismProperty.monomorphisms C X.arrow",
"ppTerm": "?m.74",
"assigned": true,
"usedConstants": [
"_private.Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives.0.CategoryTheory.IsGrothendieckAbelia... | [] | exact inferInstanceAs (Mono _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Padics.PadicVal.Basic | {
"line": 222,
"column": 2
} | {
"line": 222,
"column": 51
} | {
"line": 224,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nq : ℚ\nn d : ℤ\nhqz : q ≠ 0\nqdf : q = n /. d\nhd : d ≠ 0\nc : ℤ\nhc1 : n = c * q.num\nhc2 : d = c * ↑q.den\n⊢ FiniteMultiplicity (↑p) (c * q.num)",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"Eq.mpr",
"False",
... | [] | · simpa [finite_int_prime_iff, hqz, hc2] using hd | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.Padics.PadicVal.Basic | {
"line": 245,
"column": 4
} | {
"line": 245,
"column": 11
} | {
"line": 245,
"column": 12
} | [
{
"pp": "case inl\np : ℕ\nhp : Fact (Nat.Prime p)\nk : ℕ\n⊢ padicValRat p (0 ^ k) = ↑k * padicValRat p 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"HMul.hMul",
"Rat",
"Rat.instPowNat",
"instOfNatNat",
"Int",
"padicValRat",
... | [
"case inl.zero\np : ℕ\nhp : Fact (Nat.Prime p)\n⊢ padicValRat p (0 ^ 0) = ↑0 * padicValRat p 0",
"case inl.succ\np : ℕ\nhp : Fact (Nat.Prime p)\nn✝ : ℕ\n⊢ padicValRat p (0 ^ (n✝ + 1)) = ↑(n✝ + 1) * padicValRat p 0"
] | cases k | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.CategoryTheory.SmallObject.IsCardinalForSmallObjectArgument | {
"line": 386,
"column": 2
} | {
"line": 386,
"column": 26
} | {
"line": 387,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nI : MorphismProperty C\nκ : Cardinal.{w}\ninst✝² : Fact κ.IsRegular\ninst✝¹ : OrderBot κ.ord.ToType\ninst✝ : I.IsCardinalForSmallObjectArgument κ\nX Y A B : C\ni : A ⟶ B\nhi : I i\nf : X ⟶ Y\nthis : ∀ (X Y : C) (p : X ⟶ Y), HasColimitsOfShape (Discrete (FunctorOb... | [
"C : Type u\ninst✝³ : Category.{v, u} C\nI : MorphismProperty C\nκ : Cardinal.{w}\ninst✝² : Fact κ.IsRegular\ninst✝¹ : OrderBot κ.ord.ToType\ninst✝ : I.IsCardinalForSmallObjectArgument κ\nX Y A B : C\ni : A ⟶ B\nhi : I i\nf : X ⟶ Y\nthis✝ : ∀ (X Y : C) (p : X ⟶ Y), HasColimitsOfShape (Discrete (FunctorObjIndex I.ho... | haveI := hasPushouts I κ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives | {
"line": 300,
"column": 2
} | {
"line": 300,
"column": 33
} | {
"line": 301,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nG : C\ninst✝² : Abelian C\nhG : IsSeparator G\nX : C\ninst✝¹ : IsGrothendieckAbelian.{w, v, u} C\nA : C\nf : A ⟶ X\ninst✝ : Mono f\no : Ordinal.{w}\nj : o.ToType\nhj : transfiniteIterate (largerSubobject hG) j (Subobject.mk f) = ⊤\n⊢ ∃ J x x_1 x_2,\n ∃ (x_3 : ... | [
"C : Type u\ninst✝³ : Category.{v, u} C\nG : C\ninst✝² : Abelian C\nhG : IsSeparator G\nX : C\ninst✝¹ : IsGrothendieckAbelian.{w, v, u} C\nA : C\nf : A ⟶ X\ninst✝ : Mono f\no : Ordinal.{w}\nj : o.ToType\nhj : transfiniteIterate (largerSubobject hG) j (Subobject.mk f) = ⊤\nthis : Nonempty o.ToType\n⊢ ∃ J x x_1 x_2,\... | have : Nonempty o.ToType := ⟨j⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives | {
"line": 312,
"column": 2
} | {
"line": 317,
"column": 67
} | {
"line": 319,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nG : C\ninst✝¹ : Abelian C\ninst✝ : IsGrothendieckAbelian.{w, v, u} C\nhG : IsSeparator G\n⊢ (generatingMonomorphisms G).rlp = (monomorphisms C).rlp",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"CategoryTheory.HasLiftingProperty",
... | [] | apply le_antisymm
· intro X Y p hp A B i (_ : Mono i)
obtain ⟨J, _, _, _, _, ⟨h⟩⟩ :=
generatingMonomorphisms.exists_transfiniteCompositionOfShape hG i
exact transfiniteCompositionsOfShape_le_llp_rlp _ _ _ h.mem _ (by simpa)
· exact antitone_rlp (generatingMonomorphisms_le_monomorphisms _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives | {
"line": 312,
"column": 2
} | {
"line": 317,
"column": 67
} | {
"line": 319,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nG : C\ninst✝¹ : Abelian C\ninst✝ : IsGrothendieckAbelian.{w, v, u} C\nhG : IsSeparator G\n⊢ (generatingMonomorphisms G).rlp = (monomorphisms C).rlp",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"CategoryTheory.HasLiftingProperty",
... | [] | apply le_antisymm
· intro X Y p hp A B i (_ : Mono i)
obtain ⟨J, _, _, _, _, ⟨h⟩⟩ :=
generatingMonomorphisms.exists_transfiniteCompositionOfShape hG i
exact transfiniteCompositionsOfShape_le_llp_rlp _ _ _ h.mem _ (by simpa)
· exact antitone_rlp (generatingMonomorphisms_le_monomorphisms _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Padics.PadicNorm | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 16
} | {
"line": 130,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nq : ℚ\nh : ↑p ^ (-padicValRat p q) = 0\nhq : ¬q = 0\n⊢ False",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"False",
"Rat",
"Int.instNegInt",
"Int",
"padicValRat",
"Nat.cast",
"Rat.in... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nq : ℚ\nh : ↑p ^ (-padicValRat p q) = 0\nhq : ¬q = 0\n⊢ ¬↑p ^ (-padicValRat p q) = 0"
] | apply absurd h | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.NumberTheory.Padics.PadicNorm | {
"line": 245,
"column": 31
} | {
"line": 245,
"column": 46
} | {
"line": 245,
"column": 46
} | [
{
"pp": "case neg\np : ℕ\nhp : Fact (Nat.Prime p)\nm : ℤ\nh✝ : ¬↑m = 0\nthis✝ : 1 < ↑p\nthis : 0 ≤ padicValRat p ↑m\nh : -1 < -padicValRat p ↑m\n⊢ ↑p ^ (-padicValRat p ↑m) = ↑p ^ 0",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"GroupWithZero.toDivI... | [
"case neg\np : ℕ\nhp : Fact (Nat.Prime p)\nm : ℤ\nh✝ : ¬↑m = 0\nthis✝ : 1 < ↑p\nthis : 0 ≤ padicValRat p ↑m\nh : -1 < -padicValRat p ↑m\n⊢ -padicValRat p ↑m = 0",
"case neg.ha₀\np : ℕ\nhp : Fact (Nat.Prime p)\nm : ℤ\nh✝ : ¬↑m = 0\nthis✝ : 1 < ↑p\nthis : 0 ≤ padicValRat p ↑m\nh : -1 < -padicValRat p ↑m\n⊢ 0 < ↑p",... | zpow_right_inj₀ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.PadicVal.Basic | {
"line": 494,
"column": 2
} | {
"line": 494,
"column": 79
} | {
"line": 496,
"column": 0
} | [
{
"pp": "p n : ℕ\nhp : Fact (Nat.Prime p)\nhn : n ≠ 0\n⊢ n < p ^ (padicValNat p n + 1) → p ^ padicValNat p n ≤ n",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"instPowNat",
"Nat.pos_iff_ne_zero",
"padicValNat",
"Ne",
"instOfNatNat",
"i... | [] | exact fun _ => Nat.le_of_dvd (Nat.pos_iff_ne_zero.mpr hn) pow_padicValNat_dvd | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Padics.PadicIntegers | {
"line": 510,
"column": 4
} | {
"line": 510,
"column": 57
} | {
"line": 511,
"column": 2
} | [
{
"pp": "case a\np : ℕ\nhp : Fact (Nat.Prime p)\nx : ℤ_[p]\nhx : ‖x‖ < 1\n⊢ x ∈ Ideal.span {↑p}",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",
"Submodule",
"Real",
"Dvd.dvd",
"Semiring.toModule",... | [] | rwa [Ideal.mem_span_singleton, ← norm_lt_one_iff_dvd] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 220,
"column": 4
} | {
"line": 220,
"column": 29
} | {
"line": 221,
"column": 4
} | [
{
"pp": "case mp\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : ¬f ≈ 0\nh : padicNorm p (↑f (stationaryPoint hf)) = 0\nε : ℚ\nhε : ε > 0\n⊢ ∃ i, ∀ j ≥ i, padicNorm p (↑(f - 0) j) < ε",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"padicNorm.instIsAbsoluteValueRat",
"No... | [
"case mp\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : ¬f ≈ 0\nh : padicNorm p (↑f (stationaryPoint hf)) = 0\nε : ℚ\nhε : ε > 0\n⊢ ∀ j ≥ stationaryPoint hf, padicNorm p (↑(f - 0) j) < ε"
] | exists stationaryPoint hf | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticExists_,,_1» | Lean.Parser.Tactic.«tacticExists_,,» |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 316,
"column": 79
} | {
"line": 316,
"column": 94
} | {
"line": 316,
"column": 94
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ -f.valuation = -g.valuation ↔ ↑p ^ (-f.valuation) = ↑p ^ (-g.valuation)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"PadicSeq.valuation",
"Eq.mpr",
"NonUnitalCommRing.toNonUnital... | [
"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ -f.valuation = -g.valuation ↔ -f.valuation = -g.valuation",
"case ha₀\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ 0 < ↑p",
"case ha₁\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : ... | zpow_right_inj₀ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.SheafCohomology.Basic | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 44
} | {
"line": 165,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝¹ : HasSheafify J AddCommGrpCat\ninst✝ : HasExt (Sheaf J AddCommGrpCat)\nF : Sheaf J AddCommGrpCat\nh : Limits.IsZero F\nn : ℕ\n⊢ Subsingleton ↑((functorH J n).obj F)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstan... | [
"C : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝¹ : HasSheafify J AddCommGrpCat\ninst✝ : HasExt (Sheaf J AddCommGrpCat)\nF : Sheaf J AddCommGrpCat\nh : Limits.IsZero F\nn : ℕ\n⊢ Limits.IsZero ((functorH J n).obj F)"
] | apply AddCommGrpCat.subsingleton_of_isZero | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.MorphismProperty.Representable | {
"line": 178,
"column": 21
} | {
"line": 178,
"column": 47
} | {
"line": 180,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y Z : C\nf : X ⟶ Z\nhf : F.relativelyRepresentable (F.map f)\ng : Y ⟶ Z\ninst✝¹ : F.Full\ninst✝ : F.Faithful\n⊢ F.map (hf.fst' (F.map g) ≫ f) = F.map (hf.snd (F.map g) ≫ g)",
"ppTerm": "?m.110",
"... | [] | by simp [(hf.w (F.map g))] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.DoldKan.Decomposition | {
"line": 69,
"column": 6
} | {
"line": 81,
"column": 11
} | {
"line": 83,
"column": 0
} | [
{
"pp": "case neg\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn q : ℕ\nhq : (Q q).f (n + 1) = ∑ i with ↑i < q, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ i.rev\nhqn : q ≤ n\n⊢ (Q (q + 1)).f (n + 1) = ∑ i with ↑i < q + 1, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ i.re... | [] | obtain ⟨a, ha⟩ := Nat.le.dest hqn
rw [Q_succ, HomologicalComplex.sub_f_apply, HomologicalComplex.comp_f, hq]
symm
conv_rhs => rw [sub_eq_add_neg, add_comm]
let q' : Fin (n + 1) := ⟨q, Nat.lt_succ_of_le hqn⟩
rw [← @Finset.add_sum_erase _ _ _ _ _ _ q' (by simp [q'])]
congr
· have... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.DoldKan.Decomposition | {
"line": 69,
"column": 6
} | {
"line": 81,
"column": 11
} | {
"line": 83,
"column": 0
} | [
{
"pp": "case neg\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn q : ℕ\nhq : (Q q).f (n + 1) = ∑ i with ↑i < q, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ i.rev\nhqn : q ≤ n\n⊢ (Q (q + 1)).f (n + 1) = ∑ i with ↑i < q + 1, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ i.re... | [] | obtain ⟨a, ha⟩ := Nat.le.dest hqn
rw [Q_succ, HomologicalComplex.sub_f_apply, HomologicalComplex.comp_f, hq]
symm
conv_rhs => rw [sub_eq_add_neg, add_comm]
let q' : Fin (n + 1) := ⟨q, Nat.lt_succ_of_le hqn⟩
rw [← @Finset.add_sum_erase _ _ _ _ _ _ q' (by simp [q'])]
congr
· have... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.DoldKan.Faces | {
"line": 116,
"column": 2
} | {
"line": 120,
"column": 21
} | {
"line": 121,
"column": 2
} | [
{
"pp": "case a\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn a q : ℕ\nφ : Y ⟶ X _⦋n + 1⦌\nv : HigherFacesVanish q φ\nhnaq : n = a + q\nhnaq_shift : ∀ (d : ℕ), n + d = a + d + q\nsimplif : ∀ (a b c d e f : Y ⟶ X _⦋n + 1⦌), b = f → d + e = 0 → c + a = 0 → ... | [
"case a\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn a q : ℕ\nφ : Y ⟶ X _⦋n + 1⦌\nv : HigherFacesVanish q φ\nhnaq : n = a + q\nhnaq_shift : ∀ (d : ℕ), n + d = a + d + q\nsimplif : ∀ (a b c d e f : Y ⟶ X _⦋n + 1⦌), b = f → d + e = 0 → c + a = 0 → a + b + (c +... | · -- d + e = 0
rw [X.δ_comp_σ_self' (Fin.castSucc_mk _ _ _).symm,
X.δ_comp_σ_succ' (Fin.succ_mk _ _ _).symm]
simp only [comp_id, pow_add _ (a + 1) 1, pow_one, mul_neg, mul_one, neg_mul, neg_smul,
add_neg_cancel] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 803,
"column": 4
} | {
"line": 803,
"column": 88
} | {
"line": 804,
"column": 2
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nx y : ℚ_[p]\nthis : y - x = -1 * (x - y)\n⊢ ‖x - y‖ = ‖y - x‖",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Rat.addCommMonoid",
"NormedCommRing.toNormedRing",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"NegZeroC... | [] | simp only [this, Norm.norm, map_mul, map_neg_eq_map, AbsoluteValue.map_one, one_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicTopology.DoldKan.Faces | {
"line": 162,
"column": 2
} | {
"line": 162,
"column": 13
} | {
"line": 163,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X _⦋n + 1⦌\nv : HigherFacesVanish q φ\n⊢ HigherFacesVanish (q + 1) (φ ≫ (𝟙 (AlternatingFaceMapComplex.obj X) + Hσ q).f (n + 1))",
"ppTerm": "?m.68",
"assigned": true,
"usedC... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X _⦋n + 1⦌\nv : HigherFacesVanish q φ\nj : Fin (n + 1)\nhj₁ : n + 1 ≤ ↑j + (q + 1)\n⊢ (φ ≫ (𝟙 (AlternatingFaceMapComplex.obj X) + Hσ q).f (n + 1)) ≫ X.δ j.succ = 0"
] | intro j hj₁ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 1057,
"column": 2
} | {
"line": 1060,
"column": 18
} | {
"line": 1062,
"column": 0
} | [
{
"pp": "case h\np : ℕ\nhp : Fact (Nat.Prime p)\nf : CauSeq ℚ (padicNorm p)\nhf : ⟦f⟧ ≠ 0\n⊢ ¬f ≈ 0",
"ppTerm": "?h✝",
"assigned": true,
"usedConstants": [
"padicNorm.instIsAbsoluteValueRat",
"NormedCommRing.toNormedRing",
"NormedRing.toRing",
"instZeroPadic",
"congrArg... | [] | · apply CauSeq.not_limZero_of_not_congr_zero
contrapose hf
apply Quotient.sound
simpa using hf | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicTopology.DoldKan.FunctorGamma | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 37
} | {
"line": 150,
"column": 37
} | [
{
"pp": "case neg\nC : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Preadditive C\nK : ChainComplex C ℕ\nΔ Δ' Δ'' : SimplexCategory\ni' : Δ'' ⟶ Δ'\ni : Δ' ⟶ Δ\ninst✝¹ : Mono i'\ninst✝ : Mono i\nh₁ : ¬Δ = Δ'\nh₂ : ¬Δ' = Δ''\nk : ℕ\nhk : Δ.len = Δ'.len + k + 1\nk' : ℕ\nhk' : Δ'.len = Δ''.len + k' + 1\neq : ... | [
"case neg\nC : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Preadditive C\nK : ChainComplex C ℕ\nΔ Δ' Δ'' : SimplexCategory\ni' : Δ'' ⟶ Δ'\ni : Δ' ⟶ Δ\ninst✝¹ : Mono i'\ninst✝ : Mono i\nh₁ : ¬Δ = Δ'\nh₂ : ¬Δ' = Δ''\nk : ℕ\nhk : Δ.len = Δ'.len + k + 1\nk' : ℕ\nhk' : Δ'.len = Δ''.len + k' + 1\neq : Δ.len = Δ''.... | rw [mapMono_eq_zero K (i' ≫ i) _ _] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 1147,
"column": 58
} | {
"line": 1147,
"column": 74
} | {
"line": 1147,
"column": 74
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\n⊢ ↑0 = 0",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"WithTop.coe_zero",
"Int",
"WithTop.some",
"instOfNat",
"Zero.toOfNat0",
"WithTop.zero",
"OfNat.ofNat",
... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\n⊢ 0 = 0"
] | WithTop.coe_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.DoldKan.FunctorGamma | {
"line": 207,
"column": 42
} | {
"line": 213,
"column": 7
} | {
"line": 213,
"column": 7
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nK✝ K' : ChainComplex C ℕ\nf : K✝ ⟶ K'\nΔ✝ Δ' Δ'' : SimplexCategory\ninst✝ : HasFiniteCoproducts C\nK : ChainComplex C ℕ\nΔ : SimplexCategoryᵒᵖ\nx✝ : Discrete (Splitting.IndexSet Δ)\nA : Splitting.IndexSet Δ\n⊢ colimit.ι (Discrete.fun... | [] | by
dsimp
have fac : A.e ≫ 𝟙 A.1.unop = (𝟙 Δ).unop ≫ A.e := by rw [unop_id, comp_id, id_comp]
rw [Obj.map_on_summand₀ K A fac, Obj.Termwise.mapMono_id, id_comp]
dsimp only [Obj.obj₂]
rw [comp_id]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Idempotents.HomologicalComplex | {
"line": 76,
"column": 22
} | {
"line": 76,
"column": 49
} | {
"line": 76,
"column": 49
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nι : Type u_2\nc : ComplexShape ι\nP : Karoubi (HomologicalComplex C c)\ni j : ι\nhij : ¬c.Rel i j\n⊢ { f := P.p.f i ≫ P.X.d i j, comm := ⋯ } = 0",
"ppTerm": "?m.90",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nι : Type u_2\nc : ComplexShape ι\nP : Karoubi (HomologicalComplex C c)\ni j : ι\nhij : ¬c.Rel i j\n⊢ P.p.f i ≫ P.X.d i j = 0"
] | simp only [hom_eq_zero_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicTopology.DoldKan.GammaCompN | {
"line": 57,
"column": 8
} | {
"line": 57,
"column": 66
} | {
"line": 58,
"column": 8
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nK : ChainComplex C ℕ\nn : ℕ\ni : Fin (n + 2)\nhi : i ≠ 0\n⊢ ((Γ₀.splitting K).cofan (op ⦋n + 1⦌)).inj (Splitting.IndexSet.id (op ⦋n + 1⦌)) ≫\n ((-1) ^ ↑i • (Γ₀.obj K).δ i) ≫ (Γ₀.splitting K).πSumma... | [
"C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nK : ChainComplex C ℕ\nn : ℕ\ni : Fin (n + 2)\nhi : i ≠ 0\n⊢ (-1) ^ ↑i •\n ((Γ₀.splitting K).cofan (op ⦋n + 1⦌)).inj (Splitting.IndexSet.id (op ⦋n + 1⦌)) ≫\n (Γ₀.obj K).δ i ≫ (Γ₀.splitting K).πSummand (S... | simp only [Preadditive.zsmul_comp, Preadditive.comp_zsmul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicTopology.DoldKan.GammaCompN | {
"line": 55,
"column": 8
} | {
"line": 64,
"column": 40
} | {
"line": 64,
"column": 40
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nK : ChainComplex C ℕ\nn : ℕ\n⊢ ∀ (x : Fin (n + 2)),\n x ≠ 0 →\n ((Γ₀.splitting K).cofan (op ⦋n + 1⦌)).inj (Splitting.IndexSet.id (op ⦋n + 1⦌)) ≫\n ((-1) ^ ↑x • (Γ₀.obj K).δ x) ≫ (Γ₀.spl... | [] | intro i hi
dsimp
simp only [Preadditive.zsmul_comp, Preadditive.comp_zsmul]
rw [δ, Γ₀.Obj.mapMono_on_summand_id_assoc, Γ₀.Obj.Termwise.mapMono_eq_zero, zero_comp,
zsmul_zero]
· intro h
replace h := congr_arg SimplexCategory.len h
change n + 1 = n at h
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.DoldKan.GammaCompN | {
"line": 55,
"column": 8
} | {
"line": 64,
"column": 40
} | {
"line": 64,
"column": 40
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nK : ChainComplex C ℕ\nn : ℕ\n⊢ ∀ (x : Fin (n + 2)),\n x ≠ 0 →\n ((Γ₀.splitting K).cofan (op ⦋n + 1⦌)).inj (Splitting.IndexSet.id (op ⦋n + 1⦌)) ≫\n ((-1) ^ ↑x • (Γ₀.obj K).δ x) ≫ (Γ₀.spl... | [] | intro i hi
dsimp
simp only [Preadditive.zsmul_comp, Preadditive.comp_zsmul]
rw [δ, Γ₀.Obj.mapMono_on_summand_id_assoc, Γ₀.Obj.Termwise.mapMono_eq_zero, zero_comp,
zsmul_zero]
· intro h
replace h := congr_arg SimplexCategory.len h
change n + 1 = n at h
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject | {
"line": 326,
"column": 8
} | {
"line": 327,
"column": 65
} | {
"line": 328,
"column": 8
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nS₁ S₂ : Split C\nΦ : S₁ ⟶ S₂\ni j : ℕ\nx✝ : (ComplexShape.down ℕ).Rel i j\n⊢ Φ.f i ≫\n (S₂.s.cofan (op ⦋i⦌)).inj (Splitting.IndexSet.id (op ⦋i⦌)) ≫\n K[S₂.X].d i j ≫ S₂.s.πSummand (Splitting... | [
"C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nS₁ S₂ : Split C\nΦ : S₁ ⟶ S₂\ni j : ℕ\nx✝ : (ComplexShape.down ℕ).Rel i j\n⊢ (S₁.s.cofan (op ⦋i⦌)).inj (Splitting.IndexSet.id (op ⦋i⦌)) ≫\n ((alternatingFaceMapComplex C).obj S₁.X).d i j ≫\n ((alternati... | erw [← cofan_inj_naturality_symm_assoc Φ (Splitting.IndexSet.id (op ⦋i⦌)),
((alternatingFaceMapComplex C).map Φ.F).comm_assoc i j] | Lean.Parser.Tactic._aux_Init_Meta___macroRules_Lean_Parser_Tactic_tacticErw____1 | Lean.Parser.Tactic.tacticErw___ |
Mathlib.AlgebraicTopology.ModelCategory.Bifibrant | {
"line": 49,
"column": 6
} | {
"line": 49,
"column": 19
} | {
"line": 49,
"column": 19
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : CategoryWithCofibrations C\ninst✝¹ : HasInitial C\nX : C\ninst✝ : IsCofibrant X\n⊢ cofibrantObjects C X",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"inst✝"
],
"usedGoals": []
}
] | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.ModelCategory.Bifibrant | {
"line": 106,
"column": 6
} | {
"line": 106,
"column": 19
} | {
"line": 106,
"column": 19
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : CategoryWithFibrations C\ninst✝¹ : HasTerminal C\nX : C\ninst✝ : IsFibrant X\n⊢ fibrantObjects C X",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"inst✝"
],
"usedGoals": []
}
] | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.ModelCategory.Bifibrant | {
"line": 177,
"column": 6
} | {
"line": 177,
"column": 19
} | {
"line": 177,
"column": 19
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : CategoryWithCofibrations C\ninst✝⁴ : HasInitial C\ninst✝³ : CategoryWithFibrations C\ninst✝² : HasTerminal C\nX : C\ninst✝¹ : IsCofibrant X\ninst✝ : IsFibrant X\n⊢ cofibrantObjects C X",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.ModelCategory.Bifibrant | {
"line": 177,
"column": 21
} | {
"line": 177,
"column": 34
} | {
"line": 177,
"column": 34
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : CategoryWithCofibrations C\ninst✝⁴ : HasInitial C\ninst✝³ : CategoryWithFibrations C\ninst✝² : HasTerminal C\nX : C\ninst✝¹ : IsCofibrant X\ninst✝ : IsFibrant X\n⊢ fibrantObjects C X",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [],... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Localization.CalculusOfFractions.OfAdjunction | {
"line": 46,
"column": 18
} | {
"line": 46,
"column": 31
} | {
"line": 46,
"column": 31
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\ninst✝² : Category.{v_1, u_1} C₁\ninst✝¹ : Category.{v_2, u_2} C₂\nG : C₁ ⥤ C₂\nF : C₂ ⥤ C₁\nadj : G ⊣ F\nW : MorphismProperty C₁\ninst✝ : W.IsMultiplicative\nhW : W.IsInvertedBy G\nhW' : W.functorCategory C₁ adj.unit\nX Y T : C₁\ns : T ⟶ X\nw✝ : W s\nf : T ⟶ Y\n⊢ W s",
... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Localization.CalculusOfFractions.OfAdjunction | {
"line": 55,
"column": 18
} | {
"line": 55,
"column": 31
} | {
"line": 55,
"column": 31
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\ninst✝² : Category.{v_1, u_1} C₁\ninst✝¹ : Category.{v_2, u_2} C₂\nG : C₁ ⥤ C₂\nF : C₂ ⥤ C₁\nadj : G ⊣ F\nW : MorphismProperty C₁\ninst✝ : W.IsMultiplicative\nhW : W.IsInvertedBy G\nhW' : W.functorCategory C₁ adj.unit\nX' X Y : C₁\nf₁ f₂ : X ⟶ Y\ns : X' ⟶ X\nx✝ : W s\nh : s... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.ModelCategory.Homotopy | {
"line": 250,
"column": 4
} | {
"line": 250,
"column": 27
} | {
"line": 250,
"column": 27
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : ModelCategory C\nX Y Z : C\ninst✝¹ : IsCofibrant X\ninst✝ : IsFibrant Y\nx✝¹ x✝ : X ⟶ Y\nh : RightHomotopyRel x✝¹ x✝\n⊢ LeftHomotopyRel x✝¹ x✝",
"ppTerm": "?m.75",
"assigned": true,
"usedConstants": [
"HomotopicalAlgebra.RightHomotopyRe... | [] | exact h.leftHomotopyRel | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Localization.CalculusOfFractions.OfAdjunction | {
"line": 57,
"column": 4
} | {
"line": 57,
"column": 59
} | {
"line": 58,
"column": 4
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\ninst✝² : Category.{v_1, u_1} C₁\ninst✝¹ : Category.{v_2, u_2} C₂\nG : C₁ ⥤ C₂\nF : C₂ ⥤ C₁\nadj : G ⊣ F\nW : MorphismProperty C₁\ninst✝ : W.IsMultiplicative\nhW : W.IsInvertedBy G\nhW' : W.functorCategory C₁ adj.unit\nX' X Y : C₁\nf₁ f₂ : X ⟶ Y\ns : X' ⟶ X\nx✝ : W s\nh : s... | [
"C₁ : Type u_1\nC₂ : Type u_2\ninst✝² : Category.{v_1, u_1} C₁\ninst✝¹ : Category.{v_2, u_2} C₂\nG : C₁ ⥤ C₂\nF : C₂ ⥤ C₁\nadj : G ⊣ F\nW : MorphismProperty C₁\ninst✝ : W.IsMultiplicative\nhW : W.IsInvertedBy G\nhW' : W.functorCategory C₁ adj.unit\nX' X Y : C₁\nf₁ f₂ : X ⟶ Y\ns : X' ⟶ X\nx✝ : W s\nh : s ≫ f₁ = s ≫ ... | rw [← adj.unit_naturality f₁, ← adj.unit_naturality f₂] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Localization.Resolution | {
"line": 234,
"column": 62
} | {
"line": 235,
"column": 36
} | {
"line": 237,
"column": 0
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nD₁ : Type u_3\nD₂ : Type u_4\nH : Type u_5\ninst✝⁵ : Category.{v_1, u_1} C₁\ninst✝⁴ : Category.{v_2, u_2} C₂\ninst✝³ : Category.{v_3, u_3} D₁\ninst✝² : Category.{v_4, u_4} D₂\ninst✝¹ : Category.{v_5, u_5} H\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nW₁' : Morphis... | [] | by
rwa [← hasRightResolutions_iff_op] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Localization.DerivabilityStructure.Basic | {
"line": 101,
"column": 2
} | {
"line": 102,
"column": 77
} | {
"line": 103,
"column": 2
} | [
{
"pp": "C₁ : Type u₁\nC₂ : Type u₂\ninst✝⁶ : Category.{v₁, u₁} C₁\ninst✝⁵ : Category.{v₂, u₂} C₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nΦ : LocalizerMorphism W₁ W₂\nD₁ : Type u_1\nD₂ : Type u_2\ninst✝⁴ : Category.{v_1, u_1} D₁\ninst✝³ : Category.{v_2, u_2} D₂\nL₁ : C₁ ⥤ D₁\nL₂ : C₂ ⥤ D₂\ninst✝² :... | [
"C₁ : Type u₁\nC₂ : Type u₂\ninst✝⁶ : Category.{v₁, u₁} C₁\ninst✝⁵ : Category.{v₂, u₂} C₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nΦ : LocalizerMorphism W₁ W₂\nD₁ : Type u_1\nD₂ : Type u_2\ninst✝⁴ : Category.{v_1, u_1} D₁\ninst✝³ : Category.{v_2, u_2} D₂\nL₁ : C₁ ⥤ D₁\nL₂ : C₂ ⥤ D₂\ninst✝² : L₁.IsLocali... | let e'' : (Φ.functor ⋙ W₂.Q) ⋙ E₂.functor ≅ (W₁.Q ⋙ E₁.functor) ⋙ F :=
associator _ _ _ ≪≫ isoWhiskerLeft _ e₂ ≪≫ e ≪≫ isoWhiskerRight e₁.symm F | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.CategoryTheory.Limits.Shapes.MultiequalizerPullback | {
"line": 54,
"column": 2
} | {
"line": 54,
"column": 34
} | {
"line": 55,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nJ : MultispanShape\ninst✝ : Unique J.L\nI : MultispanIndex J C\nh : {J.fst default, J.snd default} = Set.univ\nh' : J.fst default ≠ J.snd default\ns : PushoutCocone (I.fst default) (I.snd default)\n⊢ (multicofork h h' s).π (J.fst default) = s.inl",
"ppT... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nJ : MultispanShape\ninst✝ : Unique J.L\nI : MultispanIndex J C\nh : {J.fst default, J.snd default} = Set.univ\nh' : J.fst default ≠ J.snd default\ns : PushoutCocone (I.fst default) (I.snd default)\n⊢ (if hk : J.fst default = J.fst default then eqToHom ⋯ ≫ s.inl else eq... | dsimp only [multicofork, ofπ, π] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Limits.Shapes.MultiequalizerPullback | {
"line": 59,
"column": 2
} | {
"line": 59,
"column": 34
} | {
"line": 60,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nJ : MultispanShape\ninst✝ : Unique J.L\nI : MultispanIndex J C\nh : {J.fst default, J.snd default} = Set.univ\nh' : J.fst default ≠ J.snd default\ns : PushoutCocone (I.fst default) (I.snd default)\n⊢ (multicofork h h' s).π (J.snd default) = s.inr",
"ppT... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nJ : MultispanShape\ninst✝ : Unique J.L\nI : MultispanIndex J C\nh : {J.fst default, J.snd default} = Set.univ\nh' : J.fst default ≠ J.snd default\ns : PushoutCocone (I.fst default) (I.snd default)\n⊢ (if hk : J.snd default = J.fst default then eqToHom ⋯ ≫ s.inl else eq... | dsimp only [multicofork, ofπ, π] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Limits.Preserves.Shapes.Multiequalizer | {
"line": 66,
"column": 64
} | {
"line": 68,
"column": 46
} | {
"line": 68,
"column": 46
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Category.{v_2, u_2} D\nJ : MulticospanShape\nd : MulticospanIndex J C\nc : Multifork d\nF : C ⥤ D\nj : J.R\n⊢ F.map (c.ι (J.fst j)) ≫ (d.map F).fst j = F.map (c.ι (J.snd j)) ≫ (d.map F).snd j",
"ppTerm": "?m.42",
"assigned": tr... | [] | by
dsimp
rw [← F.map_comp, ← F.map_comp, condition] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Preserves.Shapes.Multiequalizer | {
"line": 119,
"column": 66
} | {
"line": 121,
"column": 46
} | {
"line": 121,
"column": 46
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Category.{v_2, u_2} D\nJ : MultispanShape\nd : MultispanIndex J C\nc : Multicofork d\nF : C ⥤ D\nj : J.L\n⊢ (d.map F).fst j ≫ F.map (c.π (J.fst j)) = (d.map F).snd j ≫ F.map (c.π (J.snd j))",
"ppTerm": "?m.42",
"assigned": true... | [] | by
dsimp
rw [← F.map_comp, ← F.map_comp, condition] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Types.Multicoequalizer | {
"line": 59,
"column": 4
} | {
"line": 59,
"column": 19
} | {
"line": 60,
"column": 4
} | [
{
"pp": "case mpr\nJ : MultispanShape\nd : MultispanIndex J (Type u)\nc : d.multispan.CoconeTypes\nthis : ∀ (x : d.multispan.ColimitType), ∃ i a, d.multispan.ιColimitType (WalkingMultispan.right i) a = x\n⊢ ((∀ (i₁ i₂ : J.R) (x₁ : d.right i₁) (x₂ : d.right i₂),\n c.ι (WalkingMultispan.right i₁) x₁ = c.ι ... | [
"case mpr\nJ : MultispanShape\nd : MultispanIndex J (Type u)\nc : d.multispan.CoconeTypes\nthis : ∀ (x : d.multispan.ColimitType), ∃ i a, d.multispan.ιColimitType (WalkingMultispan.right i) a = x\nh₁ :\n ∀ (i₁ i₂ : J.R) (x₁ : d.right i₁) (x₂ : d.right i₂),\n c.ι (WalkingMultispan.right i₁) x₁ = c.ι (WalkingMult... | rintro ⟨h₁, h₂⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.AlgebraicTopology.SimplicialSet.Boundary | {
"line": 147,
"column": 4
} | {
"line": 147,
"column": 19
} | {
"line": 148,
"column": 4
} | [
{
"pp": "case mpr\nn : ℕ\nA : Δ[n].Subcomplex\n⊢ ∂Δ[n] ≤ A ∧ A ≠ ⊤ → A = ∂Δ[n]",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"Opposite",
"CompleteLattice.toLattice",
"PartialOrder.toPreorder",
"CategoryTheory.Functor.category",
... | [
"case mpr\nn : ℕ\nA : Δ[n].Subcomplex\nh₁ : ∂Δ[n] ≤ A\nh₂ : A ≠ ⊤\n⊢ A = ∂Δ[n]"
] | rintro ⟨h₁, h₂⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.AlgebraicTopology.SimplicialSet.Boundary | {
"line": 143,
"column": 47
} | {
"line": 148,
"column": 51
} | {
"line": 150,
"column": 0
} | [
{
"pp": "n : ℕ\nA : Δ[n].Subcomplex\n⊢ A = ∂Δ[n] ↔ ∂Δ[n] ≤ A ∧ A ≠ ⊤",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"le_refl",
"Lattice.toSemilatticeSup",
"Opposite",
"CompleteLattice.toLattice",
"congrArg",
"PartialOrder.toPreorder",
... | [] | by
constructor
· rintro rfl
exact ⟨by rfl, (boundary_lt_top n).ne⟩
· rintro ⟨h₁, h₂⟩
exact le_antisymm (by rwa [le_boundary_iff]) h₁ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.Horn | {
"line": 237,
"column": 37
} | {
"line": 238,
"column": 49
} | {
"line": 239,
"column": 2
} | [
{
"pp": "n : ℕ\ni : Fin (n + 1)\nj : Fin n\nh₀ : 0 < ↑i\nhₙ : ↑i < n\n⊢ n = 2 ∨ 2 < n",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"id",
"instOfNatNat",
"LE... | [] | by
rw [eq_comm, or_comm, ← le_iff_lt_or_eq]; lia | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.Horn | {
"line": 264,
"column": 10
} | {
"line": 264,
"column": 14
} | {
"line": 264,
"column": 14
} | [
{
"pp": "case pos\nn i : ℕ\nhi : i < n + 4\nh₀ : 0 < ⟨i, hi⟩\nhₙ : ⟨i, hi⟩ < Fin.last (n + 3)\nh : 0 < n + 2\nhS : {⟨i, hi⟩, ⟨0, ⋯⟩, ⟨0 + 1, ⋯⟩, ⟨0 + 2, ⋯⟩} = univ\nthis : Fin.last (n + 3) ∈ univ\n⊢ False",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"SSet.horn.primitiveTriangle._p... | [
"case pos\nn i : ℕ\nhi : i < n + 4\nh₀ : 0 < ⟨i, hi⟩\nhₙ : ⟨i, hi⟩ < Fin.last (n + 3)\nh : 0 < n + 2\nhS : {⟨i, hi⟩, ⟨0, ⋯⟩, ⟨0 + 1, ⋯⟩, ⟨0 + 2, ⋯⟩} = univ\nthis : Fin.last (n + 3) ∈ {⟨i, hi⟩, ⟨0, ⋯⟩, ⟨0 + 1, ⋯⟩, ⟨0 + 2, ⋯⟩}\n⊢ False"
] | ← hS | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialSet.Horn | {
"line": 272,
"column": 10
} | {
"line": 272,
"column": 14
} | {
"line": 272,
"column": 14
} | [
{
"pp": "case neg\nn k : ℕ\nh : k < n + 2\ni : ℕ\nhi : i < n + 4\nh₀ : 0 < ⟨i, hi⟩\nhₙ : ⟨i, hi⟩ < Fin.last (n + 3)\nhS : {⟨i, hi⟩, ⟨k, ⋯⟩, ⟨k + 1, ⋯⟩, ⟨k + 2, ⋯⟩} = univ\nhk : ¬k = 0\nthis : 0 ∈ univ\n⊢ False",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"SSet.horn.primitiveTriang... | [
"case neg\nn k : ℕ\nh : k < n + 2\ni : ℕ\nhi : i < n + 4\nh₀ : 0 < ⟨i, hi⟩\nhₙ : ⟨i, hi⟩ < Fin.last (n + 3)\nhS : {⟨i, hi⟩, ⟨k, ⋯⟩, ⟨k + 1, ⋯⟩, ⟨k + 2, ⋯⟩} = univ\nhk : ¬k = 0\nthis : 0 ∈ {⟨i, hi⟩, ⟨k, ⋯⟩, ⟨k + 1, ⋯⟩, ⟨k + 2, ⋯⟩}\n⊢ False"
] | ← hS | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Types.Pushouts | {
"line": 203,
"column": 11
} | {
"line": 203,
"column": 13
} | {
"line": 203,
"column": 14
} | [
{
"pp": "S X₁ X₂ : Type u\nf : S ⟶ X₁\ng : S ⟶ X₂\ninst✝ : Mono f\nx₂ : X₂\n⊢ ∀ ⦃a₂ : (fun X ↦ X) X₂⦄, (ConcreteCategory.hom (inr f g)) x₂ = (ConcreteCategory.hom (inr f g)) a₂ → x₂ = a₂",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"X₂"
],
"usedGoals":... | [
"S X₁ X₂ : Type u\nf : S ⟶ X₁\ng : S ⟶ X₂\ninst✝ : Mono f\nx₂ y₂ : X₂\n⊢ (ConcreteCategory.hom (inr f g)) x₂ = (ConcreteCategory.hom (inr f g)) y₂ → x₂ = y₂"
] | y₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Limits.Types.Pushouts | {
"line": 213,
"column": 2
} | {
"line": 213,
"column": 32
} | {
"line": 214,
"column": 2
} | [
{
"pp": "S X₁ X₂ : Type u\nf : S ⟶ X₁\ng : S ⟶ X₂\nc c' : PushoutCocone f g\ne : c ≅ c'\nx₁ : X₁\nx₂ : X₂\nh : (ConcreteCategory.hom c.inl) x₁ = (ConcreteCategory.hom c.inr) x₂\n⊢ (ConcreteCategory.hom c'.inl) x₁ = (ConcreteCategory.hom c'.inr) x₂",
"ppTerm": "?m.51",
"assigned": true,
"usedConstant... | [
"case e'_2\nS X₁ X₂ : Type u\nf : S ⟶ X₁\ng : S ⟶ X₂\nc c' : PushoutCocone f g\ne : c ≅ c'\nx₁ : X₁\nx₂ : X₂\nh : (ConcreteCategory.hom c.inl) x₁ = (ConcreteCategory.hom c.inr) x₂\n⊢ (ConcreteCategory.hom c'.inl) x₁ = (ConcreteCategory.hom e.hom.hom) ((ConcreteCategory.hom c.inl) x₁)",
"case e'_3\nS X₁ X₂ : Type ... | convert! congr_arg e.hom.hom h | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.CategoryTheory.Limits.Types.Pushouts | {
"line": 273,
"column": 74
} | {
"line": 279,
"column": 32
} | {
"line": 281,
"column": 0
} | [
{
"pp": "X₁ X₂ X₃ X₄ : Type u\nt : X₁ ⟶ X₂\nr : X₂ ⟶ X₄\nl : X₁ ⟶ X₃\nb : X₃ ⟶ X₄\nh : IsPushout t l r b\nx₄ : X₄\n⊢ (∃ x₂, (ConcreteCategory.hom r) x₂ = x₄) ∨\n ∃ x₃, (ConcreteCategory.hom b) x₃ = x₄ ∧ x₃ ∉ Set.range ⇑(ConcreteCategory.hom l)",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants... | [] | by
obtain h₁ | ⟨x₃, hx₃⟩ := eq_or_eq_of_isPushout h x₄
· exact Or.inl h₁
· by_cases h₂ : x₃ ∈ Set.range l
· obtain ⟨x₁, rfl⟩ := h₂
exact Or.inl ⟨t x₁, by simpa only [← hx₃] using! ConcreteCategory.congr_hom h.w x₁⟩
· exact Or.inr ⟨x₃, hx₃, h₂⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.StrictSegal | {
"line": 217,
"column": 70
} | {
"line": 222,
"column": 43
} | {
"line": 224,
"column": 0
} | [
{
"pp": "n : ℕ\nX : Truncated (n + 1)\nsx : X.StrictSegal\nm : ℕ\nh : m ≤ n\nf : X.Path (m + 1)\ni : Fin (m + 1)\nj : Fin (m + 2)\nhij : j ≤ i.castSucc\n⊢ (X.spine m ⋯\n ((ConcreteCategory.hom (X.map (tr (SimplexCategory.δ j) ⋯ ⋯).op)) (sx.spineToSimplex (m + 1) ⋯ f))).vertex\n i =\n f.vertex i... | [] | by
rw [spine_vertex, ← Functor.map_comp_apply, ← op_comp, ← tr_comp,
SimplexCategory.const_comp, spineToSimplex_vertex]
dsimp only [SimplexCategory.δ, len_mk, mkHom, Hom.toOrderHom_mk,
Fin.succAboveOrderEmb_apply, OrderEmbedding.toOrderHom_coe]
rw [Fin.succAbove_of_le_castSucc j i hij] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.Quasicategory.StrictSegal | {
"line": 98,
"column": 6
} | {
"line": 98,
"column": 17
} | {
"line": 98,
"column": 18
} | [
{
"pp": "case h.inr.inr.succ\nX : SSet\nsx : X.StrictSegal\nn : ℕ\ni : Fin (n + 1 + 3)\nσ₀ : Λ[n + 1 + 2, i].toSSet ⟶ X\nh₀ : 0 < i\nhₙ : i < Fin.last (n + 1 + 2)\nj : Fin (n + 1 + 3)\nhj : j ≠ i\nk : Fin (n + 1 + 1)\nksucc : Fin (n + 1 + 1 + 1 + 1) := k.succ.castSucc\nheq : j = ksucc\ntriangle : Λ[n + 3, i].to... | [
"case h.inr.inr.succ.«0»\nX : SSet\nsx : X.StrictSegal\nn : ℕ\ni : Fin (n + 1 + 3)\nσ₀ : Λ[n + 1 + 2, i].toSSet ⟶ X\nh₀ : 0 < i\nhₙ : i < Fin.last (n + 1 + 2)\nj : Fin (n + 1 + 3)\nhj : j ≠ i\nk : Fin (n + 1 + 1)\nksucc : Fin (n + 1 + 1 + 1 + 1) := k.succ.castSucc\nheq : j = ksucc\ntriangle : Λ[n + 3, i].toSSet _⦋2... | fin_cases z | Lean.Elab.Tactic._aux_Mathlib_Tactic_FinCases___elabRules_Lean_Elab_Tactic_finCases_1 | Lean.Elab.Tactic.finCases |
Mathlib.AlgebraicTopology.Quasicategory.StrictSegal | {
"line": 34,
"column": 79
} | {
"line": 98,
"column": 25
} | {
"line": 100,
"column": 0
} | [
{
"pp": "X : SSet\nsx : X.StrictSegal\n⊢ X.Quasicategory",
"ppTerm": "?m.1",
"assigned": true,
"usedConstants": [
"SSet.yonedaEquiv",
"SSet.Path.map",
"_private.Mathlib.AlgebraicTopology.Quasicategory.StrictSegal.0.SSet.StrictSegal.quasicategory._proof_1_3",
"SSet.Subcomplex.... | [] | by
apply quasicategory_of_filler X
intro n i σ₀ h₀ hₙ
use sx.spineToSimplex <| Path.map (horn.spineId i h₀ hₙ) σ₀
intro j hj
apply sx.spineInjective
ext k
dsimp only [spineEquiv, spine_arrow, Function.comp_apply, Equiv.coe_fn_mk]
rw [← types_comp_apply (σ₀.app _) (X.map _), ← σ₀.naturality]
let ksucc ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Enriched.Basic | {
"line": 130,
"column": 4
} | {
"line": 130,
"column": 23
} | {
"line": 131,
"column": 4
} | [
{
"pp": "V : Type v\ninst✝⁵ : Category.{w, v} V\ninst✝⁴ : MonoidalCategory V\nC : Type u₁\ninst✝³ : EnrichedCategory V C\nW : Type v'\ninst✝² : Category.{w', v'} W\ninst✝¹ : MonoidalCategory W\nF : V ⥤ W\ninst✝ : F.LaxMonoidal\nX Y : TransportEnrichment F C\n⊢ F.map ((λ_ (X ⟶[V] Y)).inv ≫ (eId V X ▷ X ⟶[V] Y) ≫... | [
"case e'_2\nV : Type v\ninst✝⁵ : Category.{w, v} V\ninst✝⁴ : MonoidalCategory V\nC : Type u₁\ninst✝³ : EnrichedCategory V C\nW : Type v'\ninst✝² : Category.{w', v'} W\ninst✝¹ : MonoidalCategory W\nF : V ⥤ W\ninst✝ : F.LaxMonoidal\nX Y : TransportEnrichment F C\n⊢ (λ_ (X ⟶[V] Y)).inv ≫ (eId V X ▷ X ⟶[V] Y) ≫ eComp V... | convert! F.map_id _ | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.CategoryTheory.Enriched.Basic | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 23
} | {
"line": 138,
"column": 4
} | [
{
"pp": "V : Type v\ninst✝⁵ : Category.{w, v} V\ninst✝⁴ : MonoidalCategory V\nC : Type u₁\ninst✝³ : EnrichedCategory V C\nW : Type v'\ninst✝² : Category.{w', v'} W\ninst✝¹ : MonoidalCategory W\nF : V ⥤ W\ninst✝ : F.LaxMonoidal\nX Y : TransportEnrichment F C\n⊢ F.map ((ρ_ (X ⟶[V] Y)).inv ≫ (X ⟶[V] Y) ◁ eId V Y ≫... | [
"case e'_2\nV : Type v\ninst✝⁵ : Category.{w, v} V\ninst✝⁴ : MonoidalCategory V\nC : Type u₁\ninst✝³ : EnrichedCategory V C\nW : Type v'\ninst✝² : Category.{w', v'} W\ninst✝¹ : MonoidalCategory W\nF : V ⥤ W\ninst✝ : F.LaxMonoidal\nX Y : TransportEnrichment F C\n⊢ (ρ_ (X ⟶[V] Y)).inv ≫ (X ⟶[V] Y) ◁ eId V Y ≫ eComp V... | convert! F.map_id _ | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.Basic | {
"line": 127,
"column": 4
} | {
"line": 127,
"column": 11
} | {
"line": 128,
"column": 4
} | [
{
"pp": "case h.comp\nx y : SimplexCategoryGenRel\nx✝ y✝ : Paths FreeSimplexQuiver\nu✝ v✝ w✝ : FreeSimplexQuiver\np✝ : (Paths.of FreeSimplexQuiver).obj u✝ ⟶ (Paths.of FreeSimplexQuiver).obj v✝\nk : v✝ ⟶ w✝\nh : generators.multiplicativeClosure ((Quotient.functor FreeSimplexQuiver.homRel).map p✝)\n⊢ generators.m... | [
"case h.comp.δ\nx y : SimplexCategoryGenRel\nx✝ y✝ : Paths FreeSimplexQuiver\nu✝ : FreeSimplexQuiver\nn✝ : ℕ\ni✝ : Fin (n✝ + 2)\np✝ : (Paths.of FreeSimplexQuiver).obj u✝ ⟶ (Paths.of FreeSimplexQuiver).obj (FreeSimplexQuiver.mk n✝)\nh : generators.multiplicativeClosure ((Quotient.functor FreeSimplexQuiver.homRel).ma... | cases k | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.NormalForms | {
"line": 324,
"column": 2
} | {
"line": 328,
"column": 53
} | {
"line": 330,
"column": 0
} | [
{
"pp": "m : ℕ\nL : List ℕ\nhL : IsAdmissible m L\nj : ℕ\nhj₁ : j < m + L.length\nhj₂ : simplicialEvalσ L j = simplicialEvalσ L (j + 1)\n⊢ j ∈ L",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"_private.Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.NormalForms.0.Simpl... | [] | induction L generalizing m with
| nil => grind
| cons a L h_rec =>
have := simplicialEvalσ_monotone L (a := a + 1)
rcases lt_trichotomy j a with h | h | h <;> grind | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.NormalForms | {
"line": 324,
"column": 2
} | {
"line": 328,
"column": 53
} | {
"line": 330,
"column": 0
} | [
{
"pp": "m : ℕ\nL : List ℕ\nhL : IsAdmissible m L\nj : ℕ\nhj₁ : j < m + L.length\nhj₂ : simplicialEvalσ L j = simplicialEvalσ L (j + 1)\n⊢ j ∈ L",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"_private.Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.NormalForms.0.Simpl... | [] | induction L generalizing m with
| nil => grind
| cons a L h_rec =>
have := simplicialEvalσ_monotone L (a := a + 1)
rcases lt_trichotomy j a with h | h | h <;> grind | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.NormalForms | {
"line": 324,
"column": 2
} | {
"line": 328,
"column": 53
} | {
"line": 330,
"column": 0
} | [
{
"pp": "m : ℕ\nL : List ℕ\nhL : IsAdmissible m L\nj : ℕ\nhj₁ : j < m + L.length\nhj₂ : simplicialEvalσ L j = simplicialEvalσ L (j + 1)\n⊢ j ∈ L",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"_private.Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.NormalForms.0.Simpl... | [] | induction L generalizing m with
| nil => grind
| cons a L h_rec =>
have := simplicialEvalσ_monotone L (a := a + 1)
rcases lt_trichotomy j a with h | h | h <;> grind | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.HoFunctorMonoidal | {
"line": 407,
"column": 30
} | {
"line": 407,
"column": 56
} | {
"line": 408,
"column": 6
} | [
{
"pp": "X : Truncated 2\n⊢ (ρ_ (hoFunctor₂.obj X)).hom =\n hoFunctor₂.obj X ◁ (HomotopyCategory.isoTerminal (𝟙_ (Truncated 2))).symm.hom ≫\n (iso X (𝟙_ (Truncated 2))).symm.hom ≫ hoFunctor₂.map (ρ_ X).hom",
"ppTerm": "?m.94",
"assigned": true,
"usedConstants": [
"CategoryTheory.Cat.... | [] | ext; apply right_unitality | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.HoFunctorMonoidal | {
"line": 407,
"column": 30
} | {
"line": 407,
"column": 56
} | {
"line": 408,
"column": 6
} | [
{
"pp": "X : Truncated 2\n⊢ (ρ_ (hoFunctor₂.obj X)).hom =\n hoFunctor₂.obj X ◁ (HomotopyCategory.isoTerminal (𝟙_ (Truncated 2))).symm.hom ≫\n (iso X (𝟙_ (Truncated 2))).symm.hom ≫ hoFunctor₂.map (ρ_ X).hom",
"ppTerm": "?m.94",
"assigned": true,
"usedConstants": [
"CategoryTheory.Cat.... | [] | ext; apply right_unitality | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialComplex.Basic | {
"line": 278,
"column": 8
} | {
"line": 278,
"column": 23
} | {
"line": 279,
"column": 6
} | [
{
"pp": "case right.inl\nι : Type u_1\nx✝ : Set (AbstractSimplicialComplex ι)\nL : AbstractSimplicialComplex ι\nhL : L ∈ upperBounds x✝\na✝ : Finset ι\nK : AbstractSimplicialComplex ι\nhK : K ∈ x✝\nhtK : a✝ ∈ K.faces\n⊢ a✝ ∈ (fun K ↦ K.faces) L",
"ppTerm": "?right.inl",
"assigned": true,
"usedConsta... | [] | exact hL hK htK | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicTopology.SimplicialObject.II | {
"line": 134,
"column": 10
} | {
"line": 135,
"column": 53
} | {
"line": 136,
"column": 6
} | [
{
"pp": "case inl.inl.inl.right\nn m p : ℕ\nf : Fin (n + 1) →o Fin (m + 1)\ng : Fin (m + 1) →o Fin (p + 1)\nx : Fin (p + 1)\ny : Fin (m + 1)\nhy : x.castSucc ≤ (g y).castSucc ∧ ∀ i < y, (g i).castSucc < x.castSucc\nz : Fin (n + 1)\nhz : y.castSucc ≤ (f z).castSucc ∧ ∀ i < z, (f i).castSucc < y.castSucc\n⊢ ∀ i <... | [] | intro i hi
exact hy.2 (f i) (by simpa using hz.2 i hi) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialObject.II | {
"line": 134,
"column": 10
} | {
"line": 135,
"column": 53
} | {
"line": 136,
"column": 6
} | [
{
"pp": "case inl.inl.inl.right\nn m p : ℕ\nf : Fin (n + 1) →o Fin (m + 1)\ng : Fin (m + 1) →o Fin (p + 1)\nx : Fin (p + 1)\ny : Fin (m + 1)\nhy : x.castSucc ≤ (g y).castSucc ∧ ∀ i < y, (g i).castSucc < x.castSucc\nz : Fin (n + 1)\nhz : y.castSucc ≤ (f z).castSucc ∧ ∀ i < z, (f i).castSucc < y.castSucc\n⊢ ∀ i <... | [] | intro i hi
exact hy.2 (f i) (by simpa using hz.2 i hi) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialObject.II | {
"line": 143,
"column": 6
} | {
"line": 143,
"column": 14
} | {
"line": 144,
"column": 2
} | [
{
"pp": "case inl.inr\nn m p : ℕ\nf : Fin (n + 1) →o Fin (m + 1)\ng : Fin (m + 1) →o Fin (p + 1)\nx : Fin (p + 1)\nhx : ∀ (i : Fin (m + 1)), (g i).castSucc < x.castSucc\ni : Fin (n + 1)\n⊢ ((g.comp f) i).castSucc < x.castSucc",
"ppTerm": "?inl.inr",
"assigned": true,
"usedConstants": [
"Partia... | [] | apply hx | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.AlgebraicTopology.SimplicialObject.ChainHomotopy | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 55
} | {
"line": 85,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nX Y : SimplicialObject C\nf g : X ⟶ Y\nH : Homotopy f g\nn : ℕ\nα : Fin (n + 1) × Fin (n + 2) → (X _⦋n + 1⦌ ⟶ Y _⦋n + 1⦌) := fun x ↦ (-1) ^ (↑x.1 + ↑x.2) • X.δ x.2 ≫ H.h x.1\nβ : Fin (n + 3) × Fin (n + 2) → (X _⦋n + 1⦌ ⟶ Y _⦋n + 1⦌) := fun ... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nX Y : SimplicialObject C\nf g : X ⟶ Y\nH : Homotopy f g\nn : ℕ\nα : Fin (n + 1) × Fin (n + 2) → (X _⦋n + 1⦌ ⟶ Y _⦋n + 1⦌) := fun x ↦ (-1) ^ (↑x.1 + ↑x.2) • X.δ x.2 ≫ H.h x.1\nβ : Fin (n + 3) × Fin (n + 2) → (X _⦋n + 1⦌ ⟶ Y _⦋n + 1⦌) := fun x ↦ (-1) ^ (... | let γ₃ (i : Fin (n + 1)) := (i.castSucc.succ, i.succ) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.RelativeCellComplex | {
"line": 108,
"column": 47
} | {
"line": 108,
"column": 77
} | {
"line": 108,
"column": 77
} | [
{
"pp": "X : SSet\nA : X.Subcomplex\nP : A.Pairing\nι : Type v\ninst✝¹ : LinearOrder ι\nf : P.RankFunction ι\ni : ι\nc : f.Cell i\ninst✝ : P.IsProper\nh : c.horn.image c.map = (↑(P.p c.s)).subcomplex\n⊢ (↑c.s).subcomplex ≤ c.horn.image c.map",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
... | [] | simpa only [h] using! P.le c.s | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.RelativeCellComplex | {
"line": 108,
"column": 47
} | {
"line": 108,
"column": 77
} | {
"line": 108,
"column": 77
} | [
{
"pp": "X : SSet\nA : X.Subcomplex\nP : A.Pairing\nι : Type v\ninst✝¹ : LinearOrder ι\nf : P.RankFunction ι\ni : ι\nc : f.Cell i\ninst✝ : P.IsProper\nh : c.horn.image c.map = (↑(P.p c.s)).subcomplex\n⊢ (↑c.s).subcomplex ≤ c.horn.image c.map",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
... | [] | simpa only [h] using! P.le c.s | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.RelativeCellComplex | {
"line": 108,
"column": 47
} | {
"line": 108,
"column": 77
} | {
"line": 108,
"column": 77
} | [
{
"pp": "X : SSet\nA : X.Subcomplex\nP : A.Pairing\nι : Type v\ninst✝¹ : LinearOrder ι\nf : P.RankFunction ι\ni : ι\nc : f.Cell i\ninst✝ : P.IsProper\nh : c.horn.image c.map = (↑(P.p c.s)).subcomplex\n⊢ (↑c.s).subcomplex ≤ c.horn.image c.map",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
... | [] | simpa only [h] using! P.le c.s | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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