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.AlgebraicGeometry.Sites.ConstantSheaf | {
"line": 61,
"column": 2
} | {
"line": 65,
"column": 70
} | [
{
"pp": "T : Type v\ninst✝ : TopologicalSpace T\n⊢ Presheaf.IsSheaf Scheme.zariskiTopology (continuousMapPresheaf T)",
"usedConstants": [
"ULift.topologicalSpace",
"CategoryTheory.Presieve.IsSheaf",
"CategoryTheory.Functor.op",
"Eq.mpr",
"CategoryTheory.Functor",
"Algebra... | rw [Presheaf.isSheaf_of_iso_iff (continuousMapPresheafIsoUlift T)]
apply Scheme.forgetToTop.op_comp_isSheaf_of_isSheaf _ TopCat.grothendieckTopology
apply TopCat.uliftFunctor.op_comp_isSheaf_of_isSheaf _ TopCat.grothendieckTopology
rw [isSheaf_iff_isSheaf_of_type]
exact GrothendieckTopology.Subcanonical.isSheaf... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Sites.ConstantSheaf | {
"line": 61,
"column": 2
} | {
"line": 65,
"column": 70
} | [
{
"pp": "T : Type v\ninst✝ : TopologicalSpace T\n⊢ Presheaf.IsSheaf Scheme.zariskiTopology (continuousMapPresheaf T)",
"usedConstants": [
"ULift.topologicalSpace",
"CategoryTheory.Presieve.IsSheaf",
"CategoryTheory.Functor.op",
"Eq.mpr",
"CategoryTheory.Functor",
"Algebra... | rw [Presheaf.isSheaf_of_iso_iff (continuousMapPresheafIsoUlift T)]
apply Scheme.forgetToTop.op_comp_isSheaf_of_isSheaf _ TopCat.grothendieckTopology
apply TopCat.uliftFunctor.op_comp_isSheaf_of_isSheaf _ TopCat.grothendieckTopology
rw [isSheaf_iff_isSheaf_of_type]
exact GrothendieckTopology.Subcanonical.isSheaf... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Sites.ConstantSheaf | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 38
} | [
{
"pp": "case refine_2\nT : Type v\ninst✝ : TopologicalSpace T\nR S : CommRingCat\nf : R ⟶ S\nhf₁ : Flat (Spec.map f)\nhf₂ : Surjective (Spec.map f)\n⊢ Presieve.IsSheafFor (continuousMapPresheaf T) (Presieve.singleton (Spec.map f))",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Spec",
"... | rw [Presieve.isSheafFor_singleton] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Sites.ConstantSheaf | {
"line": 84,
"column": 6
} | {
"line": 85,
"column": 98
} | [
{
"pp": "case refine_2.refine_3\nT : Type v\ninst✝ : TopologicalSpace T\nR S : CommRingCat\nf : R ⟶ S\nhf₁ : Flat (Spec.map f)\nhf₂ : Surjective (Spec.map f)\nthis : Topology.IsQuotientMap ⇑(Spec.map f)\nx : C(↥(Spec S), T)\nh :\n ∀ {Z : Scheme} (p₁ p₂ : Z ⟶ Spec S),\n p₁ ≫ Spec.map f = p₂ ≫ Spec.map f → (c... | intro y hy
rwa [← ContinuousMap.cancel_right (Spec.map f).surjective, Topology.IsQuotientMap.lift_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Sites.ConstantSheaf | {
"line": 84,
"column": 6
} | {
"line": 85,
"column": 98
} | [
{
"pp": "case refine_2.refine_3\nT : Type v\ninst✝ : TopologicalSpace T\nR S : CommRingCat\nf : R ⟶ S\nhf₁ : Flat (Spec.map f)\nhf₂ : Surjective (Spec.map f)\nthis : Topology.IsQuotientMap ⇑(Spec.map f)\nx : C(↥(Spec S), T)\nh :\n ∀ {Z : Scheme} (p₁ p₂ : Z ⟶ Spec S),\n p₁ ≫ Spec.map f = p₂ ≫ Spec.map f → (c... | intro y hy
rwa [← ContinuousMap.cancel_right (Spec.map f).surjective, Topology.IsQuotientMap.lift_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.EffectiveEpimorphic | {
"line": 164,
"column": 4
} | {
"line": 167,
"column": 58
} | [
{
"pp": "case mp\nC : Type u\ninst✝ : Category.{v, u} C\nX Y : C\nf : Y ⟶ X\n⊢ (Presieve.singleton f).EffectiveEpimorphic → EffectiveEpi f",
"usedConstants": [
"CategoryTheory.Over",
"CategoryTheory.Sieve.generateSingleton_eq",
"congrArg",
"CategoryTheory.effectiveEpiStructOfIsColimi... | intro (h : Nonempty _)
rw [Sieve.generateSingleton_eq] at h
constructor
apply Nonempty.map (effectiveEpiStructOfIsColimit _) h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.EffectiveEpimorphic | {
"line": 164,
"column": 4
} | {
"line": 167,
"column": 58
} | [
{
"pp": "case mp\nC : Type u\ninst✝ : Category.{v, u} C\nX Y : C\nf : Y ⟶ X\n⊢ (Presieve.singleton f).EffectiveEpimorphic → EffectiveEpi f",
"usedConstants": [
"CategoryTheory.Over",
"CategoryTheory.Sieve.generateSingleton_eq",
"congrArg",
"CategoryTheory.effectiveEpiStructOfIsColimi... | intro (h : Nonempty _)
rw [Sieve.generateSingleton_eq] at h
constructor
apply Nonempty.map (effectiveEpiStructOfIsColimit _) h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Sites.SheafQuasiCompact | {
"line": 84,
"column": 4
} | {
"line": 84,
"column": 43
} | [
{
"pp": "P : 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) → Presieve.IsShea... | obtain ⟨φ, hφ⟩ := Spec.map_surjective f | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Limits.Elements | {
"line": 78,
"column": 2
} | {
"line": 80,
"column": 30
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nA : C ⥤ Type w\nI : Type u₁\ninst✝³ : Category.{v₁, u₁} I\ninst✝² : Small.{w, u₁} I\nF : I ⥤ A.Elements\ninst✝¹ : HasLimitsOfShape I C\ninst✝ : PreservesLimitsOfShape I A\ni : I\n⊢ A.map (limit.π (F ⋙ π A) i) (liftedConeElement F) = (F.obj i).snd",
"usedConst... | have := congrFun
(preservesLimitIso_inv_π A (F ⋙ π A) i) (liftedConeElement' F)
simp_all [liftedConeElement] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Elements | {
"line": 78,
"column": 2
} | {
"line": 80,
"column": 30
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nA : C ⥤ Type w\nI : Type u₁\ninst✝³ : Category.{v₁, u₁} I\ninst✝² : Small.{w, u₁} I\nF : I ⥤ A.Elements\ninst✝¹ : HasLimitsOfShape I C\ninst✝ : PreservesLimitsOfShape I A\ni : I\n⊢ A.map (limit.π (F ⋙ π A) i) (liftedConeElement F) = (F.obj i).snd",
"usedConst... | have := congrFun
(preservesLimitIso_inv_π A (F ⋙ π A) i) (liftedConeElement' F)
simp_all [liftedConeElement] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Filtered.FinallySmall | {
"line": 68,
"column": 4
} | {
"line": 70,
"column": 50
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : IsFiltered C\ninst✝⁴ : LocallySmall.{w, v, u} C\ninst✝³ : FinallySmall C\nC₀ : Type u\ninst✝² : Category.{w, u} C₀\ninst✝¹ : IsFiltered C₀\ninst✝ : FinallySmall C₀\nP : ObjectProperty C₀ := ⊤.strictMap (fromFinalModel C₀)\nhP : ∀ (X : C₀), ∃ Y, ∃ (_ : P ... | exact ⟨P.FullSubcategory, small_of_surjective (f := G.obj)
(by rintro ⟨_, Y, _, rfl⟩; exact ⟨Y, rfl⟩), inferInstance, inferInstance, P.ι,
Functor.final_of_comp_full_faithful' G P.ι ⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Sites.Small | {
"line": 108,
"column": 75
} | {
"line": 124,
"column": 50
} | [
{
"pp": "P : MorphismProperty Scheme\nS : Scheme\ninst✝² : P.IsStableUnderBaseChange\ninst✝¹ : P.IsMultiplicative\ninst✝ : P.RespectsIso\n⊢ overGrothendieckTopology P S = (overPretopology P S).toGrothendieck",
"usedConstants": [
"Set.ext",
"Eq.mpr",
"CategoryTheory.Over",
"AlgebraicG... | by
ext X R
rw [GrothendieckTopology.mem_over_iff]
constructor
· intro hR
obtain ⟨𝒰, hle⟩ := exists_cover_of_mem_grothendieckTopology hR
rw [mem_grothendieckTopology_iff] at hR
letI (i : 𝒰.I₀) : (𝒰.X i).Over S := { hom := 𝒰.f i ≫ X.hom }
letI : 𝒰.Over S :=
{ over := inferInstance
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Point.Basic | {
"line": 229,
"column": 4
} | {
"line": 229,
"column": 28
} | [
{
"pp": "case mp\nC : Type u\ninst✝⁶ : Category.{v, u} C\nJ : GrothendieckTopology C\nΦ : J.Point\nA : Type u'\ninst✝⁵ : Category.{v', u'} A\ninst✝⁴ : HasColimitsOfSize.{w, w, v', u'} A\nFC : A → A → Type u_1\nCC : A → Type w'\ninst✝³ : (X Y : A) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝² : ConcreteCategory A FC\... | exact ⟨Y, f, y, hf, hf'⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper | {
"line": 144,
"column": 18
} | {
"line": 144,
"column": 40
} | [
{
"pp": "σ : Type u_1\nA : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : SetLike σ A\ninst✝² : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝¹ : GradedRing 𝒜\ninst✝ : Algebra.FiniteType (↥(𝒜 0)) A\nx : Finset A\nhx : Algebra.adjoin ↥(𝒜 0) ↑x = ⊤\nd : (i : A) → i ∈ x → ℕ\nhd : ∀ (i : A) (a : i ∈ x), d i a ≠ 0\nhxd : ∀ (i ... | Proj.awayι_toSpecZero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Colim | {
"line": 154,
"column": 45
} | {
"line": 157,
"column": 52
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nJ : Type u'\ninst✝² : Category.{v', u'} J\ninst✝¹ : HasColimitsOfShape J C\ninst✝ : colim.PreservesMonomorphisms\nX₁ X₂ : J ⥤ C\nc₁ : Cocone X₁\nc₂ : Cocone X₂\nhc₁ : IsColimit c₁\nhc₂ : IsColimit c₂\nf : X₁ ⟶ X₂\nhf : (monomorphisms C).functorCategory J f\nφ : c... | by
have (j : J) : Mono (f.app j) := hf _
have := NatTrans.mono_of_mono_app f
apply colim.map_mono' f hc₁ hc₂ φ (by simp [hφ]) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.Subobject | {
"line": 56,
"column": 62
} | {
"line": 56,
"column": 76
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\ninst✝² : IsGrothendieckAbelian.{w, v, u} C\nX : C\nJ : Type w\ninst✝¹ : SmallCategory J\nF : J ⥤ MonoOver X\ninst✝ : IsFiltered J\nc : Cocone (F ⋙ MonoOver.forget X ⋙ Over.forget X)\nhc : IsColimit c\nf : c.pt ⟶ X\nhf : ∀ (j : J), c.ι.app j ≫ ... | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.Subobject | {
"line": 56,
"column": 62
} | {
"line": 56,
"column": 76
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\ninst✝² : IsGrothendieckAbelian.{w, v, u} C\nX : C\nJ : Type w\ninst✝¹ : SmallCategory J\nF : J ⥤ MonoOver X\ninst✝ : IsFiltered J\nc : Cocone (F ⋙ MonoOver.forget X ⋙ Over.forget X)\nhc : IsColimit c\nf : c.pt ⟶ X\nhf : ∀ (j : J), c.ι.app j ≫ ... | simpa using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.ColimCoyoneda | {
"line": 96,
"column": 16
} | {
"line": 96,
"column": 34
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\ninst✝² : IsGrothendieckAbelian.{w, v, u} C\nX : C\nJ : Type w\ninst✝¹ : SmallCategory J\nY : J ⥤ C\nc : Cocone Y\nhc : IsColimit c\nj₀ : J\ny : X ⟶ Y.obj j₀\nhy : y ≫ c.ι.app j₀ = 0\ninst✝ : IsFiltered J\nj : Under j₀\n⊢ (colimit.cocone\n ... | simpa using hf y j | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.Subobject | {
"line": 56,
"column": 62
} | {
"line": 56,
"column": 76
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\ninst✝² : IsGrothendieckAbelian.{w, v, u} C\nX : C\nJ : Type w\ninst✝¹ : SmallCategory J\nF : J ⥤ MonoOver X\ninst✝ : IsFiltered J\nc : Cocone (F ⋙ MonoOver.forget X ⋙ Over.forget X)\nhc : IsColimit c\nf : c.pt ⟶ X\nhf : ∀ (j : J), c.ι.app j ≫ ... | simpa using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.ColimCoyoneda | {
"line": 96,
"column": 16
} | {
"line": 96,
"column": 34
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\ninst✝² : IsGrothendieckAbelian.{w, v, u} C\nX : C\nJ : Type w\ninst✝¹ : SmallCategory J\nY : J ⥤ C\nc : Cocone Y\nhc : IsColimit c\nj₀ : J\ny : X ⟶ Y.obj j₀\nhy : y ≫ c.ι.app j₀ = 0\ninst✝ : IsFiltered J\nj : Under j₀\n⊢ (colimit.cocone\n ... | simpa using hf y j | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.ColimCoyoneda | {
"line": 96,
"column": 16
} | {
"line": 96,
"column": 34
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\ninst✝² : IsGrothendieckAbelian.{w, v, u} C\nX : C\nJ : Type w\ninst✝¹ : SmallCategory J\nY : J ⥤ C\nc : Cocone Y\nhc : IsColimit c\nj₀ : J\ny : X ⟶ Y.obj j₀\nhy : y ≫ c.ι.app j₀ = 0\ninst✝ : IsFiltered J\nj : Under j₀\n⊢ (colimit.cocone\n ... | simpa using hf y j | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.Subobject | {
"line": 53,
"column": 2
} | {
"line": 56,
"column": 77
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\ninst✝² : IsGrothendieckAbelian.{w, v, u} C\nX : C\nJ : Type w\ninst✝¹ : SmallCategory J\nF : J ⥤ MonoOver X\ninst✝ : IsFiltered J\nc : Cocone (F ⋙ MonoOver.forget X ⋙ Over.forget X)\nhc : IsColimit c\nf : c.pt ⟶ X\nhf : ∀ (j : J), c.ι.app j ≫ ... | let α : F ⋙ MonoOver.forget _ ⋙ Over.forget _ ⟶ (Functor.const _).obj X :=
{ app j := (F.obj j).obj.hom }
have := NatTrans.mono_of_mono_app α
exact colim.map_mono' α hc (isColimitConstCocone J X) f (by simpa using hf) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.Subobject | {
"line": 53,
"column": 2
} | {
"line": 56,
"column": 77
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\ninst✝² : IsGrothendieckAbelian.{w, v, u} C\nX : C\nJ : Type w\ninst✝¹ : SmallCategory J\nF : J ⥤ MonoOver X\ninst✝ : IsFiltered J\nc : Cocone (F ⋙ MonoOver.forget X ⋙ Over.forget X)\nhc : IsColimit c\nf : c.pt ⟶ X\nhf : ∀ (j : J), c.ι.app j ≫ ... | let α : F ⋙ MonoOver.forget _ ⋙ Over.forget _ ⟶ (Functor.const _).obj X :=
{ app j := (F.obj j).obj.hom }
have := NatTrans.mono_of_mono_app α
exact colim.map_mono' α hc (isColimitConstCocone J X) f (by simpa using hf) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.Subobject | {
"line": 68,
"column": 4
} | {
"line": 68,
"column": 20
} | [
{
"pp": "case a\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\ninst✝² : IsGrothendieckAbelian.{w, v, u} C\nX : C\nJ : Type w\ninst✝¹ : SmallCategory J\nF : J ⥤ MonoOver X\ninst✝ : IsFiltered J\nc : Cocone (F ⋙ MonoOver.forget X ⋙ Over.forget X)\nhc : IsColimit c\nf : c.pt ⟶ X\nhf : ∀ (j : J), c.ι.... | rw [le_iSup_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.ColimCoyoneda | {
"line": 122,
"column": 59
} | {
"line": 122,
"column": 77
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\ninst✝² : IsGrothendieckAbelian.{w, v, u} C\nX : C\nJ : Type w\ninst✝¹ : SmallCategory J\nY : J ⥤ C\nc : Cocone Y\nhc : IsColimit c\nκ : Cardinal.{w}\nhκ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhXκ : HasCardinalLT (Subobject X) κ\nj... | simpa using hf y j | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.ColimCoyoneda | {
"line": 122,
"column": 59
} | {
"line": 122,
"column": 77
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\ninst✝² : IsGrothendieckAbelian.{w, v, u} C\nX : C\nJ : Type w\ninst✝¹ : SmallCategory J\nY : J ⥤ C\nc : Cocone Y\nhc : IsColimit c\nκ : Cardinal.{w}\nhκ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhXκ : HasCardinalLT (Subobject X) κ\nj... | simpa using hf y j | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.ColimCoyoneda | {
"line": 122,
"column": 59
} | {
"line": 122,
"column": 77
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\ninst✝² : IsGrothendieckAbelian.{w, v, u} C\nX : C\nJ : Type w\ninst✝¹ : SmallCategory J\nY : J ⥤ C\nc : Cocone Y\nhc : IsColimit c\nκ : Cardinal.{w}\nhκ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhXκ : HasCardinalLT (Subobject X) κ\nj... | simpa using hf y j | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.Subobject | {
"line": 100,
"column": 31
} | {
"line": 100,
"column": 56
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Abelian C\ninst✝³ : IsGrothendieckAbelian.{w, v, u} C\nX : C\nJ : Type w\ninst✝² : SmallCategory J\nF : J ⥤ MonoOver X\ninst✝¹ : IsFiltered J\nc : Cocone (F ⋙ MonoOver.forget X)\ninst✝ : Mono c.pt.hom\nh : Subobject.mk c.pt.hom = ⨆ j, Subobject.mk (F.obj... | by simp [MonoOver.forget] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.Subobject | {
"line": 106,
"column": 6
} | {
"line": 106,
"column": 30
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Abelian C\ninst✝³ : IsGrothendieckAbelian.{w, v, u} C\nX : C\nJ : Type w\ninst✝² : SmallCategory J\nF : J ⥤ MonoOver X\ninst✝¹ : IsFiltered J\nc : Cocone (F ⋙ MonoOver.forget X)\ninst✝ : Mono c.pt.hom\nh : Subobject.mk c.pt.hom = ⨆ j, Subobject.mk (F.obj... | ← cancel_mono (c.pt.hom) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Preorder | {
"line": 207,
"column": 19
} | {
"line": 210,
"column": 46
} | [
{
"pp": "C : Type u\ninst✝ : SemilatticeSup C\nF : Discrete WalkingPair ⥤ C\n⊢ HasColimit F",
"usedConstants": [
"le_sup_left",
"PartialOrder.toPreorder",
"CategoryTheory.Limits.hasColimit_of_iso",
"SemilatticeSup.toMax",
"CategoryTheory.Limits.WalkingPair.right",
"le_sup... | by
have : HasColimit (pair (F.obj ⟨WalkingPair.left⟩) (F.obj ⟨WalkingPair.right⟩)) :=
⟨⟨⟨_, isColimitBinaryCofan (F.obj ⟨WalkingPair.left⟩) (F.obj ⟨WalkingPair.right⟩)⟩⟩⟩
apply hasColimit_of_iso (diagramIsoPair F) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper | {
"line": 320,
"column": 2
} | {
"line": 320,
"column": 24
} | [
{
"pp": "σ : Type u_1\nA : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : SetLike σ A\ninst✝² : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝¹ : GradedRing 𝒜\ninst✝ : Algebra.FiniteType (↥(𝒜 0)) A\nO : Type u_2\ncommRing✝ : CommRing O\ndomain✝ : IsDomain O\nvaluationRing✝ : ValuationRing O\nK : Type u_2\nfield✝ : Field K\... | obtain ⟨i, hi⟩ := this | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.SmallObject.Iteration.ExtendToSucc | {
"line": 184,
"column": 33
} | {
"line": 186,
"column": 44
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\nJ : Type u\ninst✝¹ : LinearOrder J\ninst✝ : SuccOrder J\nj : J\nhj : ¬IsMax j\nF : ↑(Set.Iic j) ⥤ C\nX : C\nτ : F.obj ⟨j, ⋯⟩ ⟶ X\ni₁ i₂ : J\nhi : i₁ ≤ i₂\nhi₂ : i₂ ≤ j\n⊢ arrowMap (extendToSucc hj F τ) i₁ i₂ hi ⋯ = arrowMap F i₁ i₂ hi hi₂",
"usedConstan... | by
simp [arrowMap, extendToSucc_map hj F τ i₁ i₂ hi hi₂,
extendToSuccObjIso, extendToSucc.objIso] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.SmallObject.Construction | {
"line": 137,
"column": 39
} | {
"line": 137,
"column": 66
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nI : Type w\nA B : I → C\nf : (i : I) → A i ⟶ B i\nS X : C\nπX : X ⟶ S\ninst✝¹ : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C\ninst✝ : HasPushout (functorObjTop f πX) (functorObjLeft f πX)\n⊢ functorObjTop f πX ≫ πX = functorObjLeft f πX ≫ π'FunctorObj f... | by ext; simp [π'FunctorObj] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.SmallObject.Construction | {
"line": 183,
"column": 2
} | {
"line": 184,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nI : Type w\nA B : I → C\nf : (i : I) → A i ⟶ B i\nS X : C\nπX : X ⟶ S\ninst✝³ : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C\ninst✝² : HasPushout (functorObjTop f πX) (functorObjLeft f πX)\ninst✝¹ : LocallySmall.{t, v, u} C\ninst✝ : Small.{t, w} I\n⊢ Sm... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.SmallObject.Construction | {
"line": 183,
"column": 2
} | {
"line": 184,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nI : Type w\nA B : I → C\nf : (i : I) → A i ⟶ B i\nS X : C\nπX : X ⟶ S\ninst✝³ : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C\ninst✝² : HasPushout (functorObjTop f πX) (functorObjLeft f πX)\ninst✝¹ : LocallySmall.{t, v, u} C\ninst✝ : Small.{t, w} I\n⊢ Sm... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.SmallObject.WellOrderInductionData | {
"line": 233,
"column": 6
} | {
"line": 233,
"column": 32
} | [
{
"pp": "case h.e'_3.e_x\nJ : Type u\ninst✝³ : LinearOrder J\ninst✝² : SuccOrder J\nF : Jᵒᵖ ⥤ Type v\nd : F.WellOrderInductionData\ninst✝¹ : OrderBot J\nval₀ : F.obj (op ⊥)\ninst✝ : WellFoundedLT J\nj : J\nhj : Order.IsSuccLimit j\ne : (i : J) → i < j → d.Extension val₀ i\ni : J\nhi : i < j\n⊢ F.map (homOfLE ⋯)... | rw [d.map_lift _ _ _ _ hi] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Padics.PadicNorm | {
"line": 77,
"column": 64
} | {
"line": 78,
"column": 59
} | [
{
"pp": "p : ℕ\nhp : 1 < p\n⊢ padicNorm p ↑p = (↑p)⁻¹",
"usedConstants": [
"Rat.instOfNat",
"Nat.instCanonicallyOrderedAdd",
"False",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"eq_false",
"congrArg",
"AddMonoid.toAddZeroClass",
... | by
simp [padicNorm, (pos_of_gt hp).ne', padicValNat.self hp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Padics.PadicNorm | {
"line": 238,
"column": 4
} | {
"line": 238,
"column": 23
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nz : ℤ\nhz : ¬↑z = 0\n⊢ z ≠ 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Rat.instOfNat",
"Int.cast",
"AddGroupWithOne.toAddGroup",
"congrArg",
"Int.cast_zero",
"AddMonoid.toAddZeroClass",
"Rat",
"... | · exact_mod_cast hz | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.Padics.PadicVal.Basic | {
"line": 588,
"column": 2
} | {
"line": 588,
"column": 69
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nhn : n ≠ 0\n⊢ 0 < (p.digits n).sum",
"usedConstants": [
"Iff.mpr",
"Ne",
"instOfNatNat",
"List",
"Nat",
"Nat.digits_ne_nil_iff_ne_zero",
"Nat.digits",
"OfNat.ofNat",
"List.nil"
]
}
] | have hnil : p.digits n ≠ [] := Nat.digits_ne_nil_iff_ne_zero.mpr hn | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives | {
"line": 254,
"column": 27
} | {
"line": 254,
"column": 52
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nG : C\ninst✝⁵ : Abelian C\nhG : IsSeparator G\nX : C\ninst✝⁴ : IsGrothendieckAbelian.{w, v, u} C\nA₀ : Subobject X\nJ : Type w\ninst✝³ : LinearOrder J\ninst✝² : OrderBot J\ninst✝¹ : SuccOrder J\ninst✝ : WellFoundedLT J\nj : J\nk k' : ↑(Set.Iic j)\nh : k ⟶ k'\n⊢ (... | by simp [MonoOver.forget] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives | {
"line": 368,
"column": 2
} | {
"line": 368,
"column": 54
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nG : C\ninst✝¹ : Abelian C\ninst✝ : IsGrothendieckAbelian.{w, v, u} C\nX : C\n⊢ Injective (monoMapFactorizationDataRlp 0).Z",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"CategoryTheory.IsGrothendieckAbelian.monoMapFactorizationDataR... | let fac := (monoMapFactorizationDataRlp (0 : X ⟶ 0)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 148,
"column": 2
} | {
"line": 148,
"column": 22
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nx : ℚ\n⊢ 0 < (Int.padicValuation p) ↑x.den",
"usedConstants": [
"Int.instAddCommMonoid",
"Multiplicative.linearOrder",
"False",
"Int.instIsStrictOrderedRing",
"Preorder.toLT",
"instConditionallyCompleteLinearOrder",
"Int.p... | · simp [zero_lt_iff] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicTopology.DoldKan.PInfty | {
"line": 54,
"column": 4
} | {
"line": 55,
"column": 85
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn : ℕ\n⊢ (fun n ↦ (P n).f n) (n + 1) ≫ AlternatingFaceMapComplex.objD X n =\n AlternatingFaceMapComplex.objD X n ≫ (fun n ↦ (P n).f n) n",
"usedConstants": [
"Eq.mpr",
"le_refl",
"instH... | simpa only [← P_is_eventually_constant (show n ≤ n by rfl),
AlternatingFaceMapComplex.obj_d_eq] using (P (n + 1) : K[X] ⟶ _).comm (n + 1) n | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.AlgebraicTopology.DoldKan.PInfty | {
"line": 54,
"column": 4
} | {
"line": 55,
"column": 85
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn : ℕ\n⊢ (fun n ↦ (P n).f n) (n + 1) ≫ AlternatingFaceMapComplex.objD X n =\n AlternatingFaceMapComplex.objD X n ≫ (fun n ↦ (P n).f n) n",
"usedConstants": [
"Eq.mpr",
"le_refl",
"instH... | simpa only [← P_is_eventually_constant (show n ≤ n by rfl),
AlternatingFaceMapComplex.obj_d_eq] using (P (n + 1) : K[X] ⟶ _).comm (n + 1) n | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.DoldKan.PInfty | {
"line": 54,
"column": 4
} | {
"line": 55,
"column": 85
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn : ℕ\n⊢ (fun n ↦ (P n).f n) (n + 1) ≫ AlternatingFaceMapComplex.objD X n =\n AlternatingFaceMapComplex.objD X n ≫ (fun n ↦ (P n).f n) n",
"usedConstants": [
"Eq.mpr",
"le_refl",
"instH... | simpa only [← P_is_eventually_constant (show n ≤ n by rfl),
AlternatingFaceMapComplex.obj_d_eq] using (P (n + 1) : K[X] ⟶ _).comm (n + 1) n | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.MorphismProperty.Representable | {
"line": 528,
"column": 83
} | {
"line": 529,
"column": 28
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝² : F.Full\nA₁ A₂ A₃ : C\nX : D\nf₁ : F.obj A₁ ⟶ X\nhf₁ : F.relativelyRepresentable f₁\nf₂ : F.obj A₂ ⟶ X\nf₃ : F.obj A₃ ⟶ X\ninst✝¹ : HasPullback (hf₁.fst' f₂) (hf₁.fst' f₃)\ninst✝ : F.Faithful\nZ : C... | by
simp [lift₃, pullback₃.p₂] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.DoldKan.PInfty | {
"line": 112,
"column": 65
} | {
"line": 114,
"column": 30
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\n⊢ PInfty ≫ QInfty = 0",
"usedConstants": [
"HomologicalComplex.hom_ext",
"ChainComplex",
"HomologicalComplex.instCategory",
"Nat.instOne",
"CategoryTheory.CategoryStruct.toQuiv... | by
ext n
apply PInfty_f_comp_QInfty_f | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 960,
"column": 60
} | {
"line": 964,
"column": 34
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : ℤ\nn : ℕ\n⊢ ‖↑k‖ ≤ ↑p ^ (-↑n) ↔ ↑p ^ n ∣ k",
"usedConstants": [
"zpow_natCast",
"NormedCommRing.toNormedRing",
"Norm.norm",
"Int.cast",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Padic.eq_padicNorm",
"Rat... | by
have : (p : ℝ) ^ (-n : ℤ) = (p : ℚ) ^ (-n : ℤ) := by simp
rw [show (k : ℚ_[p]) = ((k : ℚ) : ℚ_[p]) by norm_cast, eq_padicNorm, this]
norm_cast
rw [← padicNorm.dvd_iff_norm_le] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 1060,
"column": 4
} | {
"line": 1060,
"column": 18
} | [
{
"pp": "case h.hf.a\np : ℕ\nhp : Fact (Nat.Prime p)\nf : CauSeq ℚ (padicNorm p)\nhf : f - 0 ≈ 0\n⊢ f ≈ const (padicNorm p) 0",
"usedConstants": [
"padicNorm.instIsAbsoluteValueRat",
"NormedCommRing.toNormedRing",
"NormedRing.toRing",
"Ring.toNonAssocRing",
"congrArg",
"s... | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 1155,
"column": 26
} | {
"line": 1155,
"column": 35
} | [
{
"pp": "case pos\np : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhx : ¬x = 0\nhy : y = 0\n⊢ (if x * 0 = 0 then ⊤ else ↑(x * 0).valuation) = (if x = 0 then ⊤ else ↑x.valuation) + ⊤",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"instZeroPad... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject | {
"line": 139,
"column": 2
} | {
"line": 144,
"column": 61
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nX : SimplicialObject C\ns : Splitting X\ninst✝ : Preadditive C\nn : ℕ\n⊢ s.πSummand (IndexSet.id (op ⦋n⦌)) ≫ (s.cofan (op ⦋n⦌)).inj (IndexSet.id (op ⦋n⦌)) ≫ PInfty.f n = PInfty.f n",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Category.asso... | conv_rhs => rw [← id_comp (PInfty.f n)]
dsimp only [AlternatingFaceMapComplex.obj_X]
rw [s.decomposition_id, Preadditive.sum_comp]
rw [Fintype.sum_eq_single (IndexSet.id (op ⦋n⦌)), assoc]
rintro A (hA : ¬A.EqId)
rw [assoc, s.cofan_inj_comp_PInfty_eq_zero A hA, comp_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject | {
"line": 139,
"column": 2
} | {
"line": 144,
"column": 61
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nX : SimplicialObject C\ns : Splitting X\ninst✝ : Preadditive C\nn : ℕ\n⊢ s.πSummand (IndexSet.id (op ⦋n⦌)) ≫ (s.cofan (op ⦋n⦌)).inj (IndexSet.id (op ⦋n⦌)) ≫ PInfty.f n = PInfty.f n",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Category.asso... | conv_rhs => rw [← id_comp (PInfty.f n)]
dsimp only [AlternatingFaceMapComplex.obj_X]
rw [s.decomposition_id, Preadditive.sum_comp]
rw [Fintype.sum_eq_single (IndexSet.id (op ⦋n⦌)), assoc]
rintro A (hA : ¬A.EqId)
rw [assoc, s.cofan_inj_comp_PInfty_eq_zero A hA, comp_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.DoldKan.FunctorGamma | {
"line": 181,
"column": 81
} | {
"line": 188,
"column": 48
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Preadditive C\nK : ChainComplex C ℕ\ninst✝² : HasFiniteCoproducts C\nΔ Δ' : SimplexCategoryᵒᵖ\nA : Splitting.IndexSet Δ\nθ : Δ ⟶ Δ'\nΔ'' : SimplexCategory\ne : unop Δ' ⟶ Δ''\ni : Δ'' ⟶ unop A.fst\ninst✝¹ : Epi e\ninst✝ : Mono i\nfac : e ≫ i = θ.uno... | by
simp only [map, colimit.ι_desc, Cofan.mk_ι_app]
obtain rfl := SimplexCategory.image_eq fac
congr
· exact SimplexCategory.image_ι_eq fac
· dsimp only [SimplicialObject.Splitting.IndexSet.pull]
congr
exact SimplexCategory.factorThruImage_eq fac | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject | {
"line": 243,
"column": 13
} | {
"line": 243,
"column": 82
} | [
{
"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... | ← cofan_inj_naturality_symm_assoc Φ (Splitting.IndexSet.id (op ⦋i⦌)), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.DoldKan.HomotopyEquivalence | {
"line": 67,
"column": 4
} | {
"line": 68,
"column": 70
} | [
{
"pp": "case zero\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\n⊢ PInfty.f 0 =\n (((dNext 0) fun i j ↦ (homotopyPToId X (j + 1)).hom i j) + (prevD 0) fun i j ↦ (homotopyPToId X (j + 1)).hom i j) +\n (𝟙 K[X]).f 0",
"usedConstants": [
"AlgebraicT... | · simpa only [Homotopy.dNext_zero_chainComplex, Homotopy.prevD_chainComplex,
PInfty_f, P_f_0_eq, zero_add] using (homotopyPToId X 2).comm 0 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Homotopy.Product | {
"line": 71,
"column": 6
} | {
"line": 71,
"column": 28
} | [
{
"pp": "I : Type u_1\nA : Type u_2\nX : I → Type u_3\ninst✝¹ : (i : I) → TopologicalSpace (X i)\ninst✝ : TopologicalSpace A\nf g : (i : I) → C(A, X i)\nS : Set A\nhomotopies : (i : I) → (f i).HomotopyRel (g i) S\nt : ↑unitInterval\nx : A\nhx : x ∈ S\n⊢ (Homotopy.pi fun i ↦ (homotopies i).toHomotopy).toContinuo... | simp only [funext_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicTopology.FundamentalGroupoid.InducedMaps | {
"line": 158,
"column": 44
} | {
"line": 158,
"column": 88
} | [
{
"pp": "X₁ X₂ Y : TopCat\nf : C(↑X₁, ↑Y)\ng : C(↑X₂, ↑Y)\nx₀ x₁ : ↑X₁\nx₂ x₃ : ↑X₂\np : Path x₀ x₁\nq : Path x₂ x₃\nhfg : ∀ (t : ↑I), f (p t) = g (q t)\n⊢ f x₀ = g x₂",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"Real.partialOrder",
"Real",
"Set.Icc.instZero",
... | by convert hfg 0 <;> simp only [Path.source] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.ModelCategory.CategoryWithCofibrations | {
"line": 282,
"column": 2
} | {
"line": 283,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nX✝ Y✝ : C\nf✝ : X✝ ⟶ Y✝\ninst✝¹ : CategoryWithWeakEquivalences C\nP : ObjectProperty C\nX Y : P.FullSubcategory\nf : X ⟶ Y\ninst✝ : WeakEquivalence f\n⊢ WeakEquivalence (P.ι.map f)",
"usedConstants": [
"CategoryTheory.ObjectProperty.ι",
"inferInst... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.ModelCategory.CategoryWithCofibrations | {
"line": 282,
"column": 2
} | {
"line": 283,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nX✝ Y✝ : C\nf✝ : X✝ ⟶ Y✝\ninst✝¹ : CategoryWithWeakEquivalences C\nP : ObjectProperty C\nX Y : P.FullSubcategory\nf : X ⟶ Y\ninst✝ : WeakEquivalence f\n⊢ WeakEquivalence (P.ι.map f)",
"usedConstants": [
"CategoryTheory.ObjectProperty.ι",
"inferInst... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.ModelCategory.IsCofibrant | {
"line": 98,
"column": 19
} | {
"line": 101,
"column": 16
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : CategoryWithFibrations C\ninst✝² : HasTerminal C\ninst✝¹ : (fibrations C).IsStableUnderComposition\nX Y : C\np : X ⟶ Y\ninst✝ : Fibration p\nhY : IsFibrant Y\n⊢ IsFibrant X",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStru... | by
rw [isFibrant_iff] at hY ⊢
rw [Subsingleton.elim (terminal.from X) (p ≫ terminal.from Y)]
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.FundamentalGroupoid.Basic | {
"line": 64,
"column": 20
} | {
"line": 70,
"column": 39
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\np : Path x₀ x₁\nx : ↑I\n⊢ p ⟨reflTransSymmAux (1, x), ⋯⟩ = (p.trans p.symm).toContinuousMap x",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAs... | by
simp only [reflTransSymmAux, Path.trans]
cases le_or_gt (x : ℝ) 2⁻¹ with
| inl hx => simp [hx, ← extend_apply]
| inr hx =>
have : p.extend (2 - 2 * ↑x) = p.extend (1 - (2 * ↑x - 1)) := by ring_nf
simpa [hx.not_ge, ← extend_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.ModelCategory.Cylinder | {
"line": 98,
"column": 44
} | {
"line": 98,
"column": 52
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : C\nP : Precylinder A\ninst✝ : HasBinaryCoproduct A A\n⊢ coprod.inl ≫ P.i = P.i₀",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"HomotopicalAlgebra.Precylinder.I",
"HomotopicalAlgebra.... | simp [i] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicTopology.ModelCategory.Cylinder | {
"line": 98,
"column": 44
} | {
"line": 98,
"column": 52
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : C\nP : Precylinder A\ninst✝ : HasBinaryCoproduct A A\n⊢ coprod.inl ≫ P.i = P.i₀",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"HomotopicalAlgebra.Precylinder.I",
"HomotopicalAlgebra.... | simp [i] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.ModelCategory.Cylinder | {
"line": 98,
"column": 44
} | {
"line": 98,
"column": 52
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : C\nP : Precylinder A\ninst✝ : HasBinaryCoproduct A A\n⊢ coprod.inl ≫ P.i = P.i₀",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"HomotopicalAlgebra.Precylinder.I",
"HomotopicalAlgebra.... | simp [i] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.ModelCategory.Cylinder | {
"line": 102,
"column": 44
} | {
"line": 102,
"column": 52
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : C\nP : Precylinder A\ninst✝ : HasBinaryCoproduct A A\n⊢ coprod.inr ≫ P.i = P.i₁",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"HomotopicalAlgebra.Precylinder.I",
"HomotopicalAlgebra.... | simp [i] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicTopology.ModelCategory.Cylinder | {
"line": 102,
"column": 44
} | {
"line": 102,
"column": 52
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : C\nP : Precylinder A\ninst✝ : HasBinaryCoproduct A A\n⊢ coprod.inr ≫ P.i = P.i₁",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"HomotopicalAlgebra.Precylinder.I",
"HomotopicalAlgebra.... | simp [i] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.ModelCategory.Cylinder | {
"line": 102,
"column": 44
} | {
"line": 102,
"column": 52
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : C\nP : Precylinder A\ninst✝ : HasBinaryCoproduct A A\n⊢ coprod.inr ≫ P.i = P.i₁",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"HomotopicalAlgebra.Precylinder.I",
"HomotopicalAlgebra.... | simp [i] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.ModelCategory.Homotopy | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 46
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : ModelCategory C\nX Y : C\nf g : X ⟶ Y\ninst✝ : IsFibrant Y\nh : RightHomotopyRel f g\n⊢ LeftHomotopyRel f g",
"usedConstants": [
"HomotopicalAlgebra.Cylinder.exists_very_good"
]
}
] | obtain ⟨P, _⟩ := Cylinder.exists_very_good X | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Localization.Resolution | {
"line": 300,
"column": 6
} | {
"line": 300,
"column": 48
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nD₂ : Type u_3\nH : Type u_4\ninst✝⁵ : Category.{v_1, u_1} C₁\ninst✝⁴ : Category.{v_2, u_2} C₂\ninst✝³ : Category.{v_3, u_3} D₂\ninst✝² : Category.{v_4, u_4} H\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nΦ : LocalizerMorphism W₁ W₂\nL₂ : C₂ ⥤ D₂\ninst✝¹ : L₂.IsLoca... | have := Φ.essSurj_of_hasLeftResolutions L₂ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicTopology.ModelCategory.BifibrantObjectHomotopy | {
"line": 98,
"column": 2
} | {
"line": 106,
"column": 63
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : ModelCategory C\nD : Type u_1\ninst✝ : Category.{v_1, u_1} D\nF : BifibrantObject C ⥤ D\nH : (weakEquivalences (BifibrantObject C)).IsInvertedBy F\nK L : BifibrantObject C\nf g : K ⟶ L\nh : homRel C f g\n⊢ F.map f = F.map g",
"usedCons... | · obtain ⟨P, _, ⟨h⟩⟩ := h.exists_very_good_pathObject
have := isCofibrant_of_cofibration P.ι
have : IsIso (F.map (homMk P.ι)) := H _ (by
rw [← weakEquivalence_iff, weakEquivalence_iff_of_objectProperty]
exact inferInstanceAs (WeakEquivalence P.ι))
simp only [show f = homMk h.h ≫ homMk P.p₀ by ca... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicTopology.ModelCategory.BifibrantObjectHomotopy | {
"line": 123,
"column": 12
} | {
"line": 123,
"column": 13
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : ModelCategory C\nD : Type u_1\ninst✝ : Category.{v_1, u_1} D\nK : BifibrantObject C\n⊢ ∀ ⦃L : BifibrantObject C⦄ (f g : K ⟶ L), homRel C f g → toHoCat.map f = toHoCat.map g",
"usedConstants": [
"HomotopicalAlgebra.BifibrantObject",
"Categ... | L | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Localization.DerivabilityStructure.Basic | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 39
} | [
{
"pp": "C₁ : Type u₁\nC₂ : Type u₂\ninst✝¹ : Category.{v₁, u₁} C₁\ninst✝ : Category.{v₂, u₂} C₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nΦ : LocalizerMorphism W₁ W₂\n⊢ Φ.IsLeftDerivabilityStructure ↔ Φ.op.IsRightDerivabilityStructure",
"usedConstants": [
"CategoryTheory.Functor",
"C... | let F := Φ.localizedFunctor W₁.Q W₂.Q | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.AlgebraicTopology.ModelCategory.FundamentalLemma | {
"line": 107,
"column": 4
} | {
"line": 107,
"column": 64
} | [
{
"pp": "case h\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : ModelCategory C\nH : Type u_2\ninst✝³ : Category.{v_2, u_2} H\nL : C ⥤ H\ninst✝² : L.IsLocalization (weakEquivalences C)\nX✝ Y✝ Y : C\ninst✝¹ : IsFibrant Y\nh✝¹ : IsCofibrant Y\nX : C\ninst✝ : IsCofibrant X\nh✝ : IsFibrant X\nE : BifibrantO... | exact (NatIso.naturality_1 e (BifibrantObject.homMk f)).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.LiftingProperties.Over | {
"line": 44,
"column": 14
} | {
"line": 44,
"column": 25
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nS : C\nX₁ X₂ X₃ X₄ : Over S\nt : X₁ ⟶ X₂\nl : X₁ ⟶ X₃\nr : X₂ ⟶ X₄\nb : X₃ ⟶ X₄\nsq : CommSq t l r b\ninst✝ : ⋯.HasLift\nsq' : CommSq t.left l.left r.left b.left := map (Over.forget S) sq\n⊢ sq'.lift ≫ X₂.hom = X₃.hom",
"usedConstants": [
"Eq.mpr",
... | ← Over.w b, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialSet.StrictSegal | {
"line": 168,
"column": 52
} | {
"line": 173,
"column": 51
} | [
{
"pp": "n : ℕ\nX : Truncated (n + 1)\nsx : X.StrictSegal\nm : ℕ\nh : m ≤ n + 1\nf : X.Path m\nj l : ℕ\nhjl : j + l ≤ m\n⊢ X.map (tr (subinterval j l hjl) ⋯ h).op (sx.spineToSimplex m h f) = sx.spineToSimplex l ⋯ (f.interval j l hjl)",
"usedConstants": [
"SSet.Truncated.spine_map_subinterval._proof_3"... | by
apply sx.spineInjective l
dsimp only [spineEquiv, Equiv.coe_fn_mk]
rw [spine_spineToSimplex_apply]
convert spine_map_subinterval X m h j l hjl <| sx.spineToSimplex m h f
exact sx.spine_spineToSimplex_apply m h f |>.symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Bicategory.CatEnriched | {
"line": 310,
"column": 10
} | {
"line": 310,
"column": 66
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : EnrichedOrdinaryCategory Cat C\na b c d : CatEnrichedOrdinary C\nf f' : a ⟶ b\ng g' : b ⟶ c\nh h' : c ⟶ d\nη : f ⟶ f'\nθ : g ⟶ g'\nκ : h ⟶ h'\n| CatEnriched.hComp (eqToHom ⋯ ≫ CatEnriched.hComp (Hom.base η) (Hom.base θ) ≫ eqToHom ⋯)\n (𝟙 (homEquiv h... | enter [2, 1]; exact ((id_comp _).trans (comp_id _)).symm | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.CategoryTheory.Bicategory.CatEnriched | {
"line": 310,
"column": 10
} | {
"line": 310,
"column": 66
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : EnrichedOrdinaryCategory Cat C\na b c d : CatEnrichedOrdinary C\nf f' : a ⟶ b\ng g' : b ⟶ c\nh h' : c ⟶ d\nη : f ⟶ f'\nθ : g ⟶ g'\nκ : h ⟶ h'\n| CatEnriched.hComp (eqToHom ⋯ ≫ CatEnriched.hComp (Hom.base η) (Hom.base θ) ≫ eqToHom ⋯)\n (𝟙 (homEquiv h... | enter [2, 1]; exact ((id_comp _).trans (comp_id _)).symm | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.AlgebraicTopology.SimplicialSet.Coskeletal | {
"line": 91,
"column": 52
} | {
"line": 94,
"column": 5
} | [
{
"pp": "X : SSet\nsx : X.StrictSegal\nn : ℕ\ns : Cone (proj (op ⦋n⦌) (inclusion 2).op ⋙ (inclusion 2).op ⋙ X)\nx : s.pt\ni : ℕ\nhi : i < n\n⊢ X.map (mkOfSucc ⟨i, hi⟩).op (lift sx s x) = s.π.app (strArrowMk₂ (mkOfSucc ⟨i, hi⟩) fac_aux₁._proof_1) x",
"usedConstants": [
"CategoryTheory.Functor.op",
... | by
dsimp [lift]
rw [spineToSimplex_arrow]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.HomotopyCat | {
"line": 293,
"column": 2
} | {
"line": 293,
"column": 37
} | [
{
"pp": "V : Truncated 2\nmotive : V.HomotopyCategory → Prop\nh : ∀ (x : V.obj (op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })), motive (mk x)\nx : V.HomotopyCategory\n⊢ motive x",
"usedConstants": [
"SSet.Truncated.HomotopyCategory.mk_surjective"
]
}
] | obtain ⟨x', rfl⟩ := mk_surjective x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.EpiMono | {
"line": 114,
"column": 6
} | {
"line": 114,
"column": 36
} | [
{
"pp": "case inl.inl\nn : ℕ\ni i' : Fin (n + 2)\nh✝ : i.succ ≤ i'.succ\nh : i.succ < i'.succ\n⊢ δ i'.succ ≫ σ i = 𝟙 (mk (n + 1)) ∨ ∃ j j', δ i'.succ ≫ σ i = σ j ≫ δ j'",
"usedConstants": [
"Fin.succ",
"congrArg",
"Fin.succ_lt_succ_iff",
"Eq.mp",
"instOfNatNat",
"instHAd... | rw [Fin.succ_lt_succ_iff] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.EpiMono | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 42
} | [
{
"pp": "case succ.inr\nn : ℕ\ni : Fin (n + 1 + 1)\ni' : Fin (n + 1 + 2)\nj : Fin (n + 1)\nj' : Fin (n + 2)\nh : δ i' ≫ σ i = σ j ≫ δ j'\n⊢ ∃ z e m, ∃ (_ : P_σ e) (_ : P_δ m), δ i' ≫ σ i = e ≫ m",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"SimplexCategoryGe... | · exact ⟨_, _, _, P_σ.σ _, P_δ.δ _, h⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.EpiMono | {
"line": 198,
"column": 10
} | {
"line": 198,
"column": 67
} | [
{
"pp": "case comp_σ.comp_of.succ.inr\nx y : SimplexCategoryGenRel\nn✝ : ℕ\nz : SimplexCategoryGenRel\ne : mk n✝ ⟶ z\nhe : P_σ e\nn : ℕ\nj : Fin (n + 1 + 1)\nX Y : SimplexCategoryGenRel\ni : Fin (n + 1 + 2)\nf : z ⟶ mk (n + 1)\nhf : faces.multiplicativeClosure f\nj' : Fin (n + 1)\nj'' : Fin (n + 2)\nh' : δ i ≫ ... | obtain ⟨_, _, m₁, ⟨he₁, hm₁, h₁⟩⟩ := factor_P_δ_σ j' f hf | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicTopology.SimplexCategory.MorphismProperty | {
"line": 79,
"column": 2
} | {
"line": 86,
"column": 44
} | [
{
"pp": "W : MorphismProperty SimplexCategory\ninst✝ : W.IsMultiplicative\nδ_mem : ∀ {n : ℕ} (i : Fin (n + 2)), W (δ i)\nσ_mem : ∀ {n : ℕ} (i : Fin (n + 1)), W (σ i)\n⊢ W = ⊤",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"CategoryTheory.ObjectProperty.FullSubcategory.mk... | have hW (d : ℕ) : W.inverseImage (Truncated.inclusion d) = ⊤ :=
Truncated.morphismProperty_eq_top _ (fun _ _ i ↦ δ_mem i)
(fun _ _ i ↦ σ_mem i)
ext a b f
simp only [MorphismProperty.top_apply, iff_true]
change W.inverseImage (Truncated.inclusion (max a.len b.len))
(Truncated.Hom.tr f (ha := by simp)... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplexCategory.MorphismProperty | {
"line": 79,
"column": 2
} | {
"line": 86,
"column": 44
} | [
{
"pp": "W : MorphismProperty SimplexCategory\ninst✝ : W.IsMultiplicative\nδ_mem : ∀ {n : ℕ} (i : Fin (n + 2)), W (δ i)\nσ_mem : ∀ {n : ℕ} (i : Fin (n + 1)), W (σ i)\n⊢ W = ⊤",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"CategoryTheory.ObjectProperty.FullSubcategory.mk... | have hW (d : ℕ) : W.inverseImage (Truncated.inclusion d) = ⊤ :=
Truncated.morphismProperty_eq_top _ (fun _ _ i ↦ δ_mem i)
(fun _ _ i ↦ σ_mem i)
ext a b f
simp only [MorphismProperty.top_apply, iff_true]
change W.inverseImage (Truncated.inclusion (max a.len b.len))
(Truncated.Hom.tr f (ha := by simp)... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Rank | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 14
} | [
{
"pp": "X : SSet\nA : X.Subcomplex\nP : A.Pairing\nα : Type v\ninst✝² : PartialOrder α\ninst✝¹ : WellFoundedLT α\ninst✝ : P.IsProper\nf : P.WeakRankFunction α\nx✝ : { f // ∀ (n : ℕ), P.AncestralRel (f (n + 1)) (f n) }\ng : ℕ → ↑P.II\nhg : ∀ (n : ℕ), P.AncestralRel (g (n + 1)) (g n)\nn₀ : ℕ\nhn₀ :\n ∀ (m : ℕ),... | dsimp at hn₀ | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Rank | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 82
} | [
{
"pp": "X : SSet\nA : X.Subcomplex\nP : A.Pairing\nα : Type v\ninst✝² : PartialOrder α\ninst✝¹ : WellFoundedLT α\ninst✝ : P.IsProper\nf : P.WeakRankFunction α\nx✝ : { f // ∀ (n : ℕ), P.AncestralRel (f (n + 1)) (f n) }\ng : ℕ → ↑P.II\nhg : ∀ (n : ℕ), P.AncestralRel (g (n + 1)) (g n)\nn₀ : ℕ\nhn₀ : ∀ (m : ℕ), n₀... | exact f.lt (hg _) (by rw [← hn₀ (n₀ + n + 1) (by lia), ← hn₀ (n₀ + n) (by lia)]) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicTopology.SimplicialSet.ProdStdSimplex | {
"line": 155,
"column": 4
} | {
"line": 155,
"column": 90
} | [
{
"pp": "case refine_1\np q n : ℕ\nz : (Δ[p] ⊗ Δ[q]) _⦋n⦌\nhn : p + q = n\nh : z ∈ (Δ[p] ⊗ Δ[q]).nonDegenerate n\n⊢ orderHomOfSimplex z hn = OrderHom.id",
"usedConstants": [
"Opposite",
"CategoryTheory.typesCartesianMonoidalCategory",
"PartialOrder.toPreorder",
"CategoryTheory.Functo... | exact OrderHom.eq_id_of_injective _ (strictMono_orderHomOfSimplex ⟨z, h⟩ hn).injective | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicTopology.SimplicialSet.ProdStdSimplex | {
"line": 155,
"column": 4
} | {
"line": 155,
"column": 90
} | [
{
"pp": "case refine_1\np q n : ℕ\nz : (Δ[p] ⊗ Δ[q]) _⦋n⦌\nhn : p + q = n\nh : z ∈ (Δ[p] ⊗ Δ[q]).nonDegenerate n\n⊢ orderHomOfSimplex z hn = OrderHom.id",
"usedConstants": [
"Opposite",
"CategoryTheory.typesCartesianMonoidalCategory",
"PartialOrder.toPreorder",
"CategoryTheory.Functo... | exact OrderHom.eq_id_of_injective _ (strictMono_orderHomOfSimplex ⟨z, h⟩ hn).injective | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.ProdStdSimplex | {
"line": 155,
"column": 4
} | {
"line": 155,
"column": 90
} | [
{
"pp": "case refine_1\np q n : ℕ\nz : (Δ[p] ⊗ Δ[q]) _⦋n⦌\nhn : p + q = n\nh : z ∈ (Δ[p] ⊗ Δ[q]).nonDegenerate n\n⊢ orderHomOfSimplex z hn = OrderHom.id",
"usedConstants": [
"Opposite",
"CategoryTheory.typesCartesianMonoidalCategory",
"PartialOrder.toPreorder",
"CategoryTheory.Functo... | exact OrderHom.eq_id_of_injective _ (strictMono_orderHomOfSimplex ⟨z, h⟩ hn).injective | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.ProdStdSimplexOne | {
"line": 42,
"column": 6
} | {
"line": 54,
"column": 11
} | [
{
"pp": "p : ℕ\ni : Fin (p + 1)\n⊢ (stdSimplex.objEquiv.symm (SimplexCategory.σ i), objMk₁ i.succ.castSucc) ∈ (Δ[p] ⊗ Δ[1]).nonDegenerate (p + 1)",
"usedConstants": [
"OrderHom.id",
"Eq.mpr",
"instNeZeroNatHAdd_1",
"Preorder.toLT",
"Opposite",
"Equiv.instEquivLike",
... | rw [nonDegenerate_max_dim_iff _ rfl]
ext j
dsimp
by_cases hj : j ≤ i.castSucc
· rw [objMk₁_of_castSucc_lt _ _ (by simpa),
Fin.coe_ofNat_eq_mod, Nat.zero_mod, add_zero]
change (i.predAbove j : ℕ) = _
simp [Fin.predAbove_of_le_castSucc _ _ hj]
· simp only [not_le] a... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.ProdStdSimplexOne | {
"line": 42,
"column": 6
} | {
"line": 54,
"column": 11
} | [
{
"pp": "p : ℕ\ni : Fin (p + 1)\n⊢ (stdSimplex.objEquiv.symm (SimplexCategory.σ i), objMk₁ i.succ.castSucc) ∈ (Δ[p] ⊗ Δ[1]).nonDegenerate (p + 1)",
"usedConstants": [
"OrderHom.id",
"Eq.mpr",
"instNeZeroNatHAdd_1",
"Preorder.toLT",
"Opposite",
"Equiv.instEquivLike",
... | rw [nonDegenerate_max_dim_iff _ rfl]
ext j
dsimp
by_cases hj : j ≤ i.castSucc
· rw [objMk₁_of_castSucc_lt _ _ (by simpa),
Fin.coe_ofNat_eq_mod, Nat.zero_mod, add_zero]
change (i.predAbove j : ℕ) = _
simp [Fin.predAbove_of_le_castSucc _ _ hj]
· simp only [not_le] a... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Presentable.CardinalFilteredPresentation | {
"line": 117,
"column": 19
} | {
"line": 117,
"column": 81
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nP : ObjectProperty C\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\nh : P.IsCardinalFilteredGenerator κ\ninst✝ : LocallySmall.{w, v, u} C\nX : C\nJ : Type w\nw✝¹ : SmallCategory J\nw✝ : IsCardinalFiltered J κ\nhX : P.ColimitOfShape J X\nκ' : Cardinal.{w}\nh₁ : κ'.... | simpa [hasCardinalLT_iff_cardinal_mk_lt] using h₂ (Sum.inr ⟨⟩) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Presentable.CardinalFilteredPresentation | {
"line": 117,
"column": 19
} | {
"line": 117,
"column": 81
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nP : ObjectProperty C\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\nh : P.IsCardinalFilteredGenerator κ\ninst✝ : LocallySmall.{w, v, u} C\nX : C\nJ : Type w\nw✝¹ : SmallCategory J\nw✝ : IsCardinalFiltered J κ\nhX : P.ColimitOfShape J X\nκ' : Cardinal.{w}\nh₁ : κ'.... | simpa [hasCardinalLT_iff_cardinal_mk_lt] using h₂ (Sum.inr ⟨⟩) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Presentable.CardinalFilteredPresentation | {
"line": 117,
"column": 19
} | {
"line": 117,
"column": 81
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nP : ObjectProperty C\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\nh : P.IsCardinalFilteredGenerator κ\ninst✝ : LocallySmall.{w, v, u} C\nX : C\nJ : Type w\nw✝¹ : SmallCategory J\nw✝ : IsCardinalFiltered J κ\nhX : P.ColimitOfShape J X\nκ' : Cardinal.{w}\nh₁ : κ'.... | simpa [hasCardinalLT_iff_cardinal_mk_lt] using h₂ (Sum.inr ⟨⟩) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.NerveAdjunction | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 35
} | [
{
"pp": "X Y : Truncated 2\nf₀ : X.obj (op { obj := ⦋0⦌, property := _proof_11 }) → Y.obj (op { obj := ⦋0⦌, property := _proof_11 })\nf₁ : X.obj (op { obj := ⦋1⦌, property := _proof_12 }) → Y.obj (op { obj := ⦋1⦌, property := _proof_12 })\nhδ₁ :\n ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),\n ... | apply (hY.spineEquiv 2).injective | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Presentable.Presheaf | {
"line": 92,
"column": 21
} | {
"line": 94,
"column": 43
} | [
{
"pp": "A : Type u'\ninst✝³ : Category.{v', u'} A\ninst✝² : IsLocallyPresentable.{w, v', u'} A\ninst✝¹ : HasPullbacks A\nC : Type w\ninst✝ : SmallCategory C\n⊢ ∃ κ, ∃ (x : Fact κ.IsRegular), IsCardinalLocallyPresentable (Cᵒᵖ ⥤ A) κ",
"usedConstants": [
"CategoryTheory.Functor",
"Opposite",
... | by
obtain ⟨κ, _, _⟩ := IsLocallyPresentable.exists_cardinal.{w} A
exact ⟨κ, inferInstance, inferInstance⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.NerveAdjunction | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 35
} | [
{
"pp": "X Y : Truncated 2\nf₀ : X.obj (op { obj := ⦋0⦌, property := _proof_11 }) → Y.obj (op { obj := ⦋0⦌, property := _proof_11 })\nf₁ : X.obj (op { obj := ⦋1⦌, property := _proof_12 }) → Y.obj (op { obj := ⦋1⦌, property := _proof_12 })\nhδ₁ :\n ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),\n ... | apply (hY.spineEquiv 2).injective | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Normed.Field.WithAbs | {
"line": 111,
"column": 87
} | {
"line": 112,
"column": 66
} | [
{
"pp": "K : Type u_3\ninst✝¹ : Field K\nv : AbsoluteValue K ℝ\nL : Type u_4\ninst✝ : NormedField L\nf : WithAbs v →+* L\nh : ∀ (x : WithAbs v), ‖f x‖ = v x.ofAbs\n⊢ PseudoMetricSpace.induced (⇑f) inferInstance = (normedField v).toPseudoMetricSpace",
"usedConstants": [
"NormedCommRing.toSeminormedComm... | by
ext; exact AddMonoidHomClass.isometry_of_norm _ h |>.dist_eq _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.AbsoluteValue.Equivalence | {
"line": 109,
"column": 6
} | {
"line": 109,
"column": 30
} | [
{
"pp": "case inr\nR : Type u_1\nS : Type u_2\ninst✝³ : Field R\ninst✝² : Semifield S\ninst✝¹ : LinearOrder S\nv w : AbsoluteValue R S\ninst✝ : IsStrictOrderedRing S\nh : ∀ (x : R), v x < 1 ↔ w x < 1\nx y : R\nhy₀ : v x ≠ 0\n⊢ v x ≤ v y ↔ w x ≤ w y",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",... | le_iff_le_iff_lt_iff_lt, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.AbsoluteValue.Equivalence | {
"line": 250,
"column": 4
} | {
"line": 250,
"column": 70
} | [
{
"pp": "case refine_1\nR : Type u_1\nS : Type u_2\ninst✝⁷ : Field R\ninst✝⁶ : Field S\ninst✝⁵ : LinearOrder S\ninst✝⁴ : TopologicalSpace S\ninst✝³ : IsStrictOrderedRing S\ninst✝² : Archimedean S\ninst✝¹ : OrderTopology S\nι✝ : Type u_3\ninst✝ : Finite ι✝\nv✝ : ι✝ → AbsoluteValue R S\nthis : Fintype ι✝\nP : (ι ... | exact ⟨a, ha, fun j hij ↦ absurd (Subsingleton.elim i j) hij.symm⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
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