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.CategoryTheory.FiberedCategory.Cartesian | {
"line": 394,
"column": 4
} | {
"line": 394,
"column": 39
} | {
"line": 395,
"column": 2
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝³ : Category.{v₁, u₁} 𝒮\ninst✝² : Category.{v₂, u₂} 𝒳\np : 𝒳 ⥤ 𝒮\nR R' S : 𝒮\na a' b : 𝒳\nf : R ⟶ S\nf' : R' ⟶ S\ng : R' ≅ R\nh : f' = g.hom ≫ f\nφ : a ⟶ b\nφ' : a' ⟶ b\ninst✝¹ : p.IsStronglyCartesian f φ\ninst✝ : p.IsStronglyCartesian f' φ'\n⊢ p.IsHomLift ((fun x... | [] | simpa using IsCartesian.toIsHomLift | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.FiberedCategory.Cartesian | {
"line": 394,
"column": 4
} | {
"line": 394,
"column": 39
} | {
"line": 395,
"column": 2
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝³ : Category.{v₁, u₁} 𝒮\ninst✝² : Category.{v₂, u₂} 𝒳\np : 𝒳 ⥤ 𝒮\nR R' S : 𝒮\na a' b : 𝒳\nf : R ⟶ S\nf' : R' ⟶ S\ng : R' ≅ R\nh : f' = g.hom ≫ f\nφ : a ⟶ b\nφ' : a' ⟶ b\ninst✝¹ : p.IsStronglyCartesian f φ\ninst✝ : p.IsStronglyCartesian f' φ'\n⊢ p.IsHomLift ((fun x... | [] | simpa using IsCartesian.toIsHomLift | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.FiberedCategory.Cocartesian | {
"line": 367,
"column": 5
} | {
"line": 367,
"column": 66
} | {
"line": 367,
"column": 66
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝³ : Category.{v₁, u₁} 𝒮\ninst✝² : Category.{v₂, u₂} 𝒳\np : 𝒳 ⥤ 𝒮\nR S S' : 𝒮\na b b' : 𝒳\nf : R ⟶ S\nf' : R ⟶ S'\ng : S ≅ S'\nh : f' = f ≫ g.hom\nφ : a ⟶ b\nφ' : a ⟶ b'\ninst✝¹ : p.IsStronglyCocartesian f φ\ninst✝ : p.IsStronglyCocartesian f' φ'\n⊢ p.IsHomLift ((f... | [] | by simp only [assoc, Iso.hom_inv_id, comp_id]; infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Enriched.FunctorCategory | {
"line": 272,
"column": 8
} | {
"line": 272,
"column": 34
} | {
"line": 272,
"column": 34
} | [
{
"pp": "V : Type u₁\ninst✝⁷ : Category.{v₁, u₁} V\ninst✝⁶ : MonoidalCategory V\nC : Type u₂\ninst✝⁵ : Category.{v₂, u₂} C\nJ : Type u₃\ninst✝⁴ : Category.{v₃, u₃} J\nK : Type u₄\ninst✝³ : Category.{v₄, u₄} K\ninst✝² : EnrichedOrdinaryCategory V C\nF₁ F₂ F₃ F₄ : J ⥤ C\nG : K ⥤ J\ninst✝¹ : HasEnrichedHom V F₁ F₂... | [
"V : Type u₁\ninst✝⁷ : Category.{v₁, u₁} V\ninst✝⁶ : MonoidalCategory V\nC : Type u₂\ninst✝⁵ : Category.{v₂, u₂} C\nJ : Type u₃\ninst✝⁴ : Category.{v₃, u₃} J\nK : Type u₄\ninst✝³ : Category.{v₄, u₄} K\ninst✝² : EnrichedOrdinaryCategory V C\nF₁ F₂ F₃ F₄ : J ⥤ C\nG : K ⥤ J\ninst✝¹ : HasEnrichedHom V F₁ F₂\nF₁' F₂' : ... | eHomWhiskerLeft_comp_assoc | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.EpiMono | {
"line": 73,
"column": 2
} | {
"line": 77,
"column": 48
} | {
"line": 79,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX Y : C\nf : X ⟶ Y\n⊢ Mono f ↔ IsPullback (𝟙 X) (𝟙 X) f f",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"CategoryTheory.Limits.PullbackCone.isLimitMkIdId",
"CategoryTheory.Mono",
"CategoryTheory.Ca... | [] | constructor
· intro
exact IsPullback.of_isLimit (PullbackCone.isLimitMkIdId f)
· intro hf
exact (mono_iff_fst_eq_snd hf.isLimit).2 rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.EpiMono | {
"line": 73,
"column": 2
} | {
"line": 77,
"column": 48
} | {
"line": 79,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX Y : C\nf : X ⟶ Y\n⊢ Mono f ↔ IsPullback (𝟙 X) (𝟙 X) f f",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"CategoryTheory.Limits.PullbackCone.isLimitMkIdId",
"CategoryTheory.Mono",
"CategoryTheory.Ca... | [] | constructor
· intro
exact IsPullback.of_isLimit (PullbackCone.isLimitMkIdId f)
· intro hf
exact (mono_iff_fst_eq_snd hf.isLimit).2 rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Galois.GaloisObjects | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 37
} | {
"line": 79,
"column": 0
} | [
{
"pp": "case e₂\nC : Type u₁\ninst✝³ : Category.{u₂, u₁} C\ninst✝² : GaloisCategory C\nF : C ⥤ FintypeCat\ninst✝¹ : FiberFunctor F\nX : C\ninst✝ : IsConnected X\nJ : SingleObj (Aut X) ⥤ C := ⋯\ne : (F ⋙ FintypeCat.incl).obj (colimit J) ≅\n MulAction.orbitRel.Quotient (Aut X) ((J ⋙ F ⋙ FintypeCat.incl).obj (Si... | [] | exact Types.isTerminalEquivUnique _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Functor.TypeValuedFlat | {
"line": 51,
"column": 4
} | {
"line": 59,
"column": 44
} | {
"line": 61,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nF : C ⥤ Type w\ninst✝¹ : HasFiniteLimits C\ninst✝ : PreservesFiniteLimits F\n⊢ ∀ ⦃X Y : F.Elements⦄ (f g : X ⟶ Y), ∃ W h, h ≫ f = h ≫ g",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"CategoryTheory.categoryOfElements",
"Categor... | [] | rintro ⟨X, x⟩ ⟨Y, y⟩ ⟨f, hf⟩ ⟨g, hg⟩
dsimp at f g hf hg
rw [← hg] at hf
let h := isLimitForkMapOfIsLimit F _ (equalizerIsEqualizer f g)
let h' := (Types.equalizerLimit (g := F.map f) (h := F.map g)).isLimit
exact ⟨⟨equalizer f g, (h'.conePointUniqueUpToIso h).hom ⟨x, hf⟩⟩,
⟨equalizer.ι f g, Co... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Functor.TypeValuedFlat | {
"line": 51,
"column": 4
} | {
"line": 59,
"column": 44
} | {
"line": 61,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nF : C ⥤ Type w\ninst✝¹ : HasFiniteLimits C\ninst✝ : PreservesFiniteLimits F\n⊢ ∀ ⦃X Y : F.Elements⦄ (f g : X ⟶ Y), ∃ W h, h ≫ f = h ≫ g",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"CategoryTheory.categoryOfElements",
"Categor... | [] | rintro ⟨X, x⟩ ⟨Y, y⟩ ⟨f, hf⟩ ⟨g, hg⟩
dsimp at f g hf hg
rw [← hg] at hf
let h := isLimitForkMapOfIsLimit F _ (equalizerIsEqualizer f g)
let h' := (Types.equalizerLimit (g := F.map f) (h := F.map g)).isLimit
exact ⟨⟨equalizer f g, (h'.conePointUniqueUpToIso h).hom ⟨x, hf⟩⟩,
⟨equalizer.ι f g, Co... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Galois.Prorepresentability | {
"line": 242,
"column": 2
} | {
"line": 242,
"column": 33
} | {
"line": 243,
"column": 2
} | [
{
"pp": "case h\nC : Type u₁\ninst✝¹ : Category.{u₂, u₁} C\ninst✝ : GaloisCategory C\nF : C ⥤ FintypeCat\nA B : PointedGaloisObject F\nf : A ⟶ B\nφ : Aut B.obj\nψ : Aut A.obj\nhψ : autMap f.val ψ = φ\n⊢ (ConcreteCategory.hom ((autGaloisSystem F).map f)) ψ = φ",
"ppTerm": "?h",
"assigned": true,
"use... | [
"case h\nC : Type u₁\ninst✝¹ : Category.{u₂, u₁} C\ninst✝ : GaloisCategory C\nF : C ⥤ FintypeCat\nA B : PointedGaloisObject F\nf : A ⟶ B\nφ : Aut B.obj\nψ : Aut A.obj\nhψ : autMap f.val ψ = φ\n⊢ (ConcreteCategory.hom (GrpCat.ofHom (autMapHom f.val))) ψ = φ"
] | simp only [autGaloisSystem_map] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Galois.IsFundamentalgroup | {
"line": 147,
"column": 57
} | {
"line": 150,
"column": 42
} | {
"line": 152,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝⁷ : Category.{u₂, u₁} C\nF : C ⥤ FintypeCat\nG : Type u_1\ninst✝⁶ : Group G\ninst✝⁵ : (X : C) → MulAction G (F.obj X).obj\ninst✝⁴ : IsNaturalSMul F G\ninst✝³ : GaloisCategory C\ninst✝² : FiberFunctor F\nt : Aut F\nX : C\ninst✝¹ : IsGalois X\ninst✝ : MulAction.IsPretransitive G (F.obj ... | [] | by
obtain ⟨a⟩ := nonempty_fiber_of_isConnected F X
obtain ⟨g, hg⟩ := MulAction.exists_smul_eq G a (t.hom.app X a)
exact ⟨g, action_ext_of_isGalois F _ hg⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Groupoid.Subgroupoid | {
"line": 250,
"column": 67
} | {
"line": 251,
"column": 51
} | {
"line": 253,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Groupoid C\nS T : Subgroupoid C\nh : S ≤ T\nx✝¹ x✝ : ↑S.objs\ns : C\nhs : s ∈ S.objs\nt : C\nht : t ∈ S.objs\n⊢ (inclusion h).obj ⟨s, hs⟩ = (inclusion h).obj ⟨t, ht⟩ → ⟨s, hs⟩ = ⟨t, ht⟩",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Memb... | [] | by
simpa only [inclusion, Subtype.mk_eq_mk] using id | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.FormalCoproducts.Cech | {
"line": 65,
"column": 18
} | {
"line": 79,
"column": 24
} | {
"line": 79,
"column": 24
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nU : FormalCoproduct C\nα : Type\ninst✝ : HasProductsOfShape α C\ns : Fan fun x ↦ U\nm : s.pt ⟶ (U.powerFan α).pt\nhm : ∀ (j : α), m ≫ (U.powerFan α).proj j = s.proj j\n⊢ m = { f := fun i a ↦ (s.proj a).f i, φ := fun i ↦ Pi.lift fun a ↦ (s.proj a).φ i }",
"ppT... | [] | by
obtain ⟨f, φ⟩ := m
obtain rfl : f = fun i a ↦ (s.proj a).f i := by
ext i
dsimp
ext a
exact congr_fun (congr_arg FormalCoproduct.Hom.f (hm a)) i
ext i
· rfl
· dsimp
ext a
specialize hm a
rw [hom_ext_iff] at hm
obtain ⟨_, hm⟩... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Preserves.BifunctorCokernel | {
"line": 83,
"column": 2
} | {
"line": 83,
"column": 26
} | {
"line": 85,
"column": 0
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nC : Type u_3\ninst✝¹¹ : Category.{v_1, u_1} C₁\ninst✝¹⁰ : Category.{v_2, u_2} C₂\ninst✝⁹ : Category.{v_3, u_3} C\ninst✝⁸ : HasZeroMorphisms C₁\ninst✝⁷ : HasZeroMorphisms C₂\ninst✝⁶ : HasZeroMorphisms C\nX₁ Y₁ : C₁\nf₁ : X₁ ⟶ Y₁\nc₁ : CokernelCofork f₁\nhc₁ : IsColimit c₁\n... | [] | exact ⟨l', by cat_disch⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.ChosenPullback | {
"line": 117,
"column": 37
} | {
"line": 117,
"column": 76
} | {
"line": 119,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX S : C\nf : X ⟶ S\nh : ChosenPullback f f\n⊢ Nonempty h.Diagonal",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryTheory.CategoryStruct.id",
... | [] | apply LiftStruct.nonempty <;> cat_disch | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.ChosenPullback | {
"line": 117,
"column": 37
} | {
"line": 117,
"column": 76
} | {
"line": 119,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX S : C\nf : X ⟶ S\nh : ChosenPullback f f\n⊢ Nonempty h.Diagonal",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryTheory.CategoryStruct.id",
... | [] | apply LiftStruct.nonempty <;> cat_disch | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.ChosenPullback | {
"line": 117,
"column": 37
} | {
"line": 117,
"column": 76
} | {
"line": 119,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX S : C\nf : X ⟶ S\nh : ChosenPullback f f\n⊢ Nonempty h.Diagonal",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryTheory.CategoryStruct.id",
... | [] | apply LiftStruct.nonempty <;> cat_disch | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.SequentialProduct | {
"line": 193,
"column": 32
} | {
"line": 193,
"column": 48
} | {
"line": 193,
"column": 48
} | [
{
"pp": "C : Type u_1\nM N : ℕ → C\ninst✝¹ : Category.{v_1, u_1} C\nf : (n : ℕ) → M n ⟶ N n\ninst✝ : HasCountableProducts C\ns : Cone (Functor.ofOpSequence (functorMap f))\nn✝ m n : ℕ\nhh : m + 1 ≤ n\nih :\n ∀ (h : m < n),\n (Functor.ofOpSequence (functorMap f)).map (homOfLE hh).op ≫\n Pi.π (fun i ↦ ... | [
"C : Type u_1\nM N : ℕ → C\ninst✝¹ : Category.{v_1, u_1} C\nf : (n : ℕ) → M n ⟶ N n\ninst✝ : HasCountableProducts C\ns : Cone (Functor.ofOpSequence (functorMap f))\nn✝ m n : ℕ\nhh : m + 1 ≤ n\nih :\n ∀ (h : m < n),\n (Functor.ofOpSequence (functorMap f)).map (homOfLE hh).op ≫\n Pi.π (fun i ↦ if i < m + 1... | dif_pos (by lia) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Shapes.SequentialProduct | {
"line": 195,
"column": 25
} | {
"line": 195,
"column": 41
} | {
"line": 195,
"column": 41
} | [
{
"pp": "C : Type u_1\nM N : ℕ → C\ninst✝¹ : Category.{v_1, u_1} C\nf : (n : ℕ) → M n ⟶ N n\ninst✝ : HasCountableProducts C\ns : Cone (Functor.ofOpSequence (functorMap f))\nn✝ m n : ℕ\nhh : m + 1 ≤ n\nih :\n ∀ (h : m < n),\n (Functor.ofOpSequence (functorMap f)).map (homOfLE hh).op ≫\n Pi.π (fun i ↦ ... | [
"C : Type u_1\nM N : ℕ → C\ninst✝¹ : Category.{v_1, u_1} C\nf : (n : ℕ) → M n ⟶ N n\ninst✝ : HasCountableProducts C\ns : Cone (Functor.ofOpSequence (functorMap f))\nn✝ m n : ℕ\nhh : m + 1 ≤ n\nih :\n ∀ (h : m < n),\n (Functor.ofOpSequence (functorMap f)).map (homOfLE hh).op ≫\n Pi.π (fun i ↦ if i < m + 1... | dif_pos (by lia) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.CatCospanTransform | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 38
} | {
"line": 159,
"column": 39
} | [
{
"pp": "A : Type u₁\nB : Type u₂\nC : Type u₃\nA' : Type u₄\nB' : Type u₅\nC' : Type u₆\ninst✝⁵ : Category.{v₁, u₁} A\ninst✝⁴ : Category.{v₂, u₂} B\ninst✝³ : Category.{v₃, u₃} C\nF : A ⥤ B\nG : C ⥤ B\ninst✝² : Category.{v₄, u₄} A'\ninst✝¹ : Category.{v₅, u₅} B'\ninst✝ : Category.{v₆, u₆} C'\nF' : A' ⥤ B'\nG' :... | [
"case left\nA : Type u₁\nB : Type u₂\nC : Type u₃\nA' : Type u₄\nB' : Type u₅\nC' : Type u₆\ninst✝⁵ : Category.{v₁, u₁} A\ninst✝⁴ : Category.{v₂, u₂} B\ninst✝³ : Category.{v₃, u₃} C\nF : A ⥤ B\nG : C ⥤ B\ninst✝² : Category.{v₄, u₄} A'\ninst✝¹ : Category.{v₅, u₅} B'\ninst✝ : Category.{v₆, u₆} C'\nF' : A' ⥤ B'\nG' : ... | apply CatCospanTransformMorphism.ext | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Limits.Shapes.SequentialProduct | {
"line": 197,
"column": 29
} | {
"line": 197,
"column": 40
} | {
"line": 197,
"column": 40
} | [
{
"pp": "case a.a\nC : Type u_1\nM N : ℕ → C\ninst✝¹ : Category.{v_1, u_1} C\nf : (n : ℕ) → M n ⟶ N n\ninst✝ : HasCountableProducts C\ns : Cone (Functor.ofOpSequence (functorMap f))\nn✝ m n : ℕ\nhh : m + 1 ≤ n\nih :\n ∀ (h : m < n),\n (Functor.ofOpSequence (functorMap f)).map (homOfLE hh).op ≫\n Pi.π... | [
"case a.a\nC : Type u_1\nM N : ℕ → C\ninst✝¹ : Category.{v_1, u_1} C\nf : (n : ℕ) → M n ⟶ N n\ninst✝ : HasCountableProducts C\ns : Cone (Functor.ofOpSequence (functorMap f))\nn✝ m n : ℕ\nhh : m + 1 ≤ n\nih :\n ∀ (h : m < n),\n (Functor.ofOpSequence (functorMap f)).map (homOfLE hh).op ≫\n Pi.π (fun i ↦ if... | ih (by lia) | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.Limits.Shapes.SequentialProduct | {
"line": 200,
"column": 12
} | {
"line": 200,
"column": 28
} | {
"line": 200,
"column": 28
} | [
{
"pp": "case e_a.le_succ_of_le\nC : Type u_1\nM N : ℕ → C\ninst✝¹ : Category.{v_1, u_1} C\nf : (n : ℕ) → M n ⟶ N n\ninst✝ : HasCountableProducts C\ns : Cone (Functor.ofOpSequence (functorMap f))\nn✝ m n : ℕ\nhh : m + 1 ≤ n\nih :\n ∀ (h : m < n),\n (Functor.ofOpSequence (functorMap f)).map (homOfLE hh).op ≫... | [
"case e_a.le_succ_of_le\nC : Type u_1\nM N : ℕ → C\ninst✝¹ : Category.{v_1, u_1} C\nf : (n : ℕ) → M n ⟶ N n\ninst✝ : HasCountableProducts C\ns : Cone (Functor.ofOpSequence (functorMap f))\nn✝ m n : ℕ\nhh : m + 1 ≤ n\nih :\n ∀ (h : m < n),\n (Functor.ofOpSequence (functorMap f)).map (homOfLE hh).op ≫\n Pi... | dif_pos (by lia) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.SmallComplete | {
"line": 77,
"column": 10
} | {
"line": 77,
"column": 17
} | {
"line": 78,
"column": 10
} | [
{
"pp": "C : Type u\ninst✝¹ : SmallCategory C\ninst✝ : HasProducts C\nX Y : C\nr s : X ⟶ Y\nr_ne_s : ¬r = s\nz : 2 ≤ #(X ⟶ Y)\nmd : Type u := (Z : C) × (W : C) × (Z ⟶ W)\nα : Cardinal.{u} := #md\nyp : C := ∏ᶜ fun x ↦ Y\nf g : X ⟶ yp\nk : (fun f ↦ ⟨X, ⟨yp, f⟩⟩) f = (fun f ↦ ⟨X, ⟨yp, f⟩⟩) g\n⊢ f = g",
"ppTerm... | [
"case refl\nC : Type u\ninst✝¹ : SmallCategory C\ninst✝ : HasProducts C\nX Y : C\nr s : X ⟶ Y\nr_ne_s : ¬r = s\nz : 2 ≤ #(X ⟶ Y)\nmd : Type u := (Z : C) × (W : C) × (Z ⟶ W)\nα : Cardinal.{u} := #md\nyp : C := ∏ᶜ fun x ↦ Y\nf : X ⟶ yp\n⊢ f = f"
] | cases k | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.CategoryTheory.Limits.Types.ColimitTypeFiltered | {
"line": 52,
"column": 6
} | {
"line": 52,
"column": 29
} | {
"line": 53,
"column": 4
} | [
{
"pp": "case mp.symm\nJ : Type u\ninst✝¹ : Category.{v, u} J\ninst✝ : IsFiltered J\nF : J ⥤ Type w₀\nx y x✝ y✝ : (j : J) × F.obj j\na✝ : Relation.EqvGen F.ColimitTypeRel x✝ y✝\nk : J\nf : x✝.fst ⟶ k\ng : y✝.fst ⟶ k\nh : (ConcreteCategory.hom (F.map f)) x✝.snd = (ConcreteCategory.hom (F.map g)) y✝.snd\n⊢ ∃ k f ... | [] | exact ⟨k, g, f, h.symm⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Limits.Types.End | {
"line": 97,
"column": 33
} | {
"line": 97,
"column": 74
} | {
"line": 97,
"column": 74
} | [
{
"pp": "case mk\nJ : Type u\ninst✝ : Category.{v, u} J\nF : Jᵒᵖ ⥤ J ⥤ Type (max w u)\nX : Type (max w u)\nf✝ : (j : J) → (F.obj (op j)).obj j ⟶ X\nhf : ∀ ⦃i j : J⦄ (g : i ⟶ j), (F.map g.op).app i ≫ f✝ i = (F.obj (op j)).map g ≫ f✝ j\nx✝ : chosenCoend F\nj✝ j'✝ : J\nf : j✝ ⟶ j'✝\nx : (F.obj (op j'✝)).obj j✝\n⊢ ... | [] | exact ConcreteCategory.congr_hom (hf f) _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Limits.Types.End | {
"line": 97,
"column": 33
} | {
"line": 97,
"column": 74
} | {
"line": 97,
"column": 74
} | [
{
"pp": "case mk\nJ : Type u\ninst✝ : Category.{v, u} J\nF : Jᵒᵖ ⥤ J ⥤ Type (max w u)\nX : Type (max w u)\nf✝ : (j : J) → (F.obj (op j)).obj j ⟶ X\nhf : ∀ ⦃i j : J⦄ (g : i ⟶ j), (F.map g.op).app i ≫ f✝ i = (F.obj (op j)).map g ≫ f✝ j\nx✝ : chosenCoend F\nj✝ j'✝ : J\nf : j✝ ⟶ j'✝\nx : (F.obj (op j'✝)).obj j✝\n⊢ ... | [] | exact ConcreteCategory.congr_hom (hf f) _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Types.End | {
"line": 97,
"column": 33
} | {
"line": 97,
"column": 74
} | {
"line": 97,
"column": 74
} | [
{
"pp": "case mk\nJ : Type u\ninst✝ : Category.{v, u} J\nF : Jᵒᵖ ⥤ J ⥤ Type (max w u)\nX : Type (max w u)\nf✝ : (j : J) → (F.obj (op j)).obj j ⟶ X\nhf : ∀ ⦃i j : J⦄ (g : i ⟶ j), (F.map g.op).app i ≫ f✝ i = (F.obj (op j)).map g ≫ f✝ j\nx✝ : chosenCoend F\nj✝ j'✝ : J\nf : j✝ ⟶ j'✝\nx : (F.obj (op j'✝)).obj j✝\n⊢ ... | [] | exact ConcreteCategory.congr_hom (hf f) _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Localization.Pi | {
"line": 52,
"column": 6
} | {
"line": 52,
"column": 29
} | {
"line": 53,
"column": 6
} | [
{
"pp": "J₁ J₂ : Type w\ne : J₁ ≃ J₂\nhJ₁ :\n ∀ {C : J₁ → Type u₁} {D : J₁ → Type u₂} [inst : (j : J₁) → Category.{v₁, u₁} (C j)]\n [inst_1 : (j : J₁) → Category.{v₂, u₂} (D j)] (L : (j : J₁) → C j ⥤ D j) (W : (j : J₁) → MorphismProperty (C j))\n [∀ (j : J₁), (W j).ContainsIdentities] [∀ (j : J₁), (L j).... | [
"J₁ J₂ : Type w\ne : J₁ ≃ J₂\nhJ₁ :\n ∀ {C : J₁ → Type u₁} {D : J₁ → Type u₂} [inst : (j : J₁) → Category.{v₁, u₁} (C j)]\n [inst_1 : (j : J₁) → Category.{v₂, u₂} (D j)] (L : (j : J₁) → C j ⥤ D j) (W : (j : J₁) → MorphismProperty (C j))\n [∀ (j : J₁), (W j).ContainsIdentities] [∀ (j : J₁), (L j).IsLocalizati... | rintro j _ rfl _ _ g hg | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.CategoryTheory.Monoidal.Free.Basic | {
"line": 268,
"column": 2
} | {
"line": 268,
"column": 21
} | {
"line": 268,
"column": 22
} | [
{
"pp": "case h.comp\nC : Type u\nmotive : {X Y : F C} → (X ⟶ Y) → Prop\nX Y : F C\nid : ∀ (X : F C), motive (𝟙 X)\nα_hom : ∀ (X Y Z : F C), motive (α_ X Y Z).hom\nα_inv : ∀ (X Y Z : F C), motive (α_ X Y Z).inv\nl_hom : ∀ (X : F C), motive (λ_ X).hom\nl_inv : ∀ (X : F C), motive (λ_ X).inv\nρ_hom : ∀ (X : F C)... | [] | | comp f g hf hg => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.CategoryTheory.MarkovCategory.Positive | {
"line": 75,
"column": 37
} | {
"line": 75,
"column": 74
} | {
"line": 75,
"column": 74
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : MonoidalCategory C\ninst✝¹ : PositiveCategory C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsIso f\n⊢ Deterministic (f ≫ inv f)",
"ppTerm": "?m.264",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
... | [] | rw [IsIso.hom_inv_id]; infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.MarkovCategory.Positive | {
"line": 75,
"column": 37
} | {
"line": 75,
"column": 74
} | {
"line": 75,
"column": 74
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : MonoidalCategory C\ninst✝¹ : PositiveCategory C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsIso f\n⊢ Deterministic (f ≫ inv f)",
"ppTerm": "?m.264",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
... | [] | rw [IsIso.hom_inv_id]; infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Action.LinearFunctor | {
"line": 81,
"column": 49
} | {
"line": 85,
"column": 26
} | {
"line": 87,
"column": 0
} | [
{
"pp": "D : Type u_1\nD' : Type u_2\ninst✝⁶ : Category.{v_1, u_1} D\ninst✝⁵ : Category.{v_2, u_2} D'\nF : D ⥤ D'\nC : Type u_3\ninst✝⁴ : Category.{v_3, u_3} C\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalLeftAction C D\ninst✝¹ : MonoidalLeftAction C D'\ninst✝ : F.LaxLeftLinear C\nc c' : C\nd : D\n⊢ c ⊴ₗ μₗ F... | [] | by
simpa [-μₗ_associativity, -μₗ_associativity_assoc] using
(αₗ _ _ _).inv ≫=
(μₗ_associativity F c c' d).symm =≫
F.map (αₗ _ _ _).inv | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Action.End | {
"line": 136,
"column": 4
} | {
"line": 138,
"column": 50
} | {
"line": 139,
"column": 4
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : MonoidalCategory C\ninst✝¹ : Category.{v_2, u_2} D\nF : C ⥤ (D ⥤ D)ᴹᵒᵖ\ninst✝ : F.Monoidal\nc : C\nd : D\n⊢ (F.map (ρ_ c).hom).unmop.app d =\n ((Functor.Monoidal.μIso F c (𝟙_ C)).unmop.app d).symm.hom ≫\n (F.obj c).unmop.map ... | [
"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : MonoidalCategory C\ninst✝¹ : Category.{v_2, u_2} D\nF : C ⥤ (D ⥤ D)ᴹᵒᵖ\ninst✝ : F.Monoidal\nc : C\nd : D\ne :\n (F.map (ρ_ c).hom).unmop.app d ≫ (ρ_ (F.obj c)).inv.unmop.app d =\n (F.map (ρ_ c).hom).unmop.app d ≫\n (F.map (ρ_ c).inv ≫ Fu... | have e := (F.map (ρ_ c).hom).unmop.app d ≫=
(congrArg (fun t ↦ t.unmop.app d) <|
Functor.OplaxMonoidal.right_unitality F c) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Monoidal.Braided.Reflection | {
"line": 114,
"column": 6
} | {
"line": 114,
"column": 69
} | {
"line": 115,
"column": 4
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Category.{v_2, u_2} D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : SymmetricCategory D\ninst✝² : MonoidalClosed D\nR : C ⥤ D\ninst✝¹ : R.Faithful\ninst✝ : R.Full\nL : D ⥤ C\nadj : L ⊣ R\nx✝ : ∀ (d d' : D), IsIso (L.map (adj.unit.app d ▷ d')... | [] | simp [← tensorHom_id, ← id_tensorHom, tensorHom_comp_tensorHom] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Basic | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 52
} | {
"line": 121,
"column": 0
} | [
{
"pp": "case e_a\nC : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\ninst✝⁴ : MonoidalCategory C\ninst✝³ : MonoidalClosed C\nJ : Type u₂\ninst✝² : Category.{v₂, u₂} J\ninst✝¹ : ∀ (F₁ F₂ : J ⥤ C), HasFunctorEnrichedHom C F₁ F₂\nF₁ F₂ F₃ F₃' : J ⥤ C\ninst✝ : ∀ (F₁ F₂ : J ⥤ C), HasEnrichedHom C F₁ F₂\nf : F₁ ⊗ F₂ ⟶ F₃\nf... | [] | apply enrichedOrdinaryCategorySelf_eHomWhiskerLeft | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Monoidal.Closed.Ideal | {
"line": 74,
"column": 2
} | {
"line": 77,
"column": 85
} | {
"line": 79,
"column": 0
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₁, u₂} D\ni : D ⥤ C\ninst✝¹ : CartesianMonoidalCategory C\ninst✝ : MonoidalClosed C\n⊢ ExponentialIdeal (subterminalInclusion C)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"CategoryTheory.Object... | [] | apply ExponentialIdeal.mk'
intro B A
refine ⟨⟨A ⟹ B.1, fun Z g h => ?_⟩, ⟨Iso.refl _⟩⟩
exact uncurry_injective (B.2 (MonoidalClosed.uncurry g) (MonoidalClosed.uncurry h)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Closed.Ideal | {
"line": 74,
"column": 2
} | {
"line": 77,
"column": 85
} | {
"line": 79,
"column": 0
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₁, u₂} D\ni : D ⥤ C\ninst✝¹ : CartesianMonoidalCategory C\ninst✝ : MonoidalClosed C\n⊢ ExponentialIdeal (subterminalInclusion C)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"CategoryTheory.Object... | [] | apply ExponentialIdeal.mk'
intro B A
refine ⟨⟨A ⟹ B.1, fun Z g h => ?_⟩, ⟨Iso.refl _⟩⟩
exact uncurry_injective (B.2 (MonoidalClosed.uncurry g) (MonoidalClosed.uncurry h)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Closed.Ideal | {
"line": 73,
"column": 56
} | {
"line": 77,
"column": 85
} | {
"line": 79,
"column": 0
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₁, u₂} D\ni : D ⥤ C\ninst✝¹ : CartesianMonoidalCategory C\ninst✝ : MonoidalClosed C\n⊢ ExponentialIdeal (subterminalInclusion C)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"CategoryTheory.Object... | [] | by
apply ExponentialIdeal.mk'
intro B A
refine ⟨⟨A ⟹ B.1, fun Z g h => ?_⟩, ⟨Iso.refl _⟩⟩
exact uncurry_injective (B.2 (MonoidalClosed.uncurry g) (MonoidalClosed.uncurry h)) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Closed.Ideal | {
"line": 316,
"column": 29
} | {
"line": 318,
"column": 49
} | {
"line": 318,
"column": 49
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝⁷ : Category.{v₁, u₁} C\ninst✝⁶ : Category.{v₁, u₂} D\ni : D ⥤ C\ninst✝⁵ : CartesianMonoidalCategory C\ninst✝⁴ : Reflective i\ninst✝³ : MonoidalClosed C\ninst✝² : CartesianMonoidalCategory D\ninst✝¹ : ExponentialIdeal i\ninst✝ : BraidedCategory C\nA B : C\n⊢ prodCompariso... | [] | by
rw [← (bijection i _ _ _).injective.eq_iff, bijection_natural, ← bijection_symm_apply_id,
Equiv.apply_symm_apply, Category.id_comp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.DayConvolution.Closed | {
"line": 212,
"column": 10
} | {
"line": 212,
"column": 60
} | {
"line": 213,
"column": 10
} | [
{
"pp": "C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\nV : Type u₂\ninst✝⁴ : Category.{v₂, u₂} V\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalCategory V\ninst✝¹ : MonoidalClosed V\nF G✝ H G : C ⥤ V\ninst✝ : DayConvolution F G\nℌ : DayConvolutionInternalHom F (F ⊛ G) H\nc c' c'' : C\nf : c' ⟶ c''\n⊢ MonoidalClose... | [
"C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\nV : Type u₂\ninst✝⁴ : Category.{v₂, u₂} V\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalCategory V\ninst✝¹ : MonoidalClosed V\nF G✝ H G : C ⥤ V\ninst✝ : DayConvolution F G\nℌ : DayConvolutionInternalHom F (F ⊛ G) H\nc c' c'' : C\nf : c' ⟶ c''\nthis :\n (F.map f ⊗ₘ G.map... | have := DayConvolution.unit_naturality F G f (𝟙 c) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Monoidal.Free.Coherence | {
"line": 215,
"column": 10
} | {
"line": 215,
"column": 52
} | {
"line": 215,
"column": 52
} | [
{
"pp": "C : Type u\nX Y : F C\nn : (Discrete ∘ NormalMonoidalObject) C\n⊢ (α_ (inclusion.obj { as := n.as }) X Y).symm ≪≫\n normalizeIsoApp' C X n.as ▷ᵢ Y ≪≫ normalizeIsoApp C Y { as := X.normalizeObj n.as } =\n (α_ (inclusionObj n.as) X Y).symm ≪≫ normalizeIsoApp' C X n.as ▷ᵢ Y ≪≫ normalizeIsoApp' C Y... | [
"C : Type u\nX Y : F C\nn : (Discrete ∘ NormalMonoidalObject) C\n⊢ (α_ (inclusion.obj { as := n.as }) X Y).symm ≪≫\n normalizeIsoApp' C X n.as ▷ᵢ Y ≪≫ normalizeIsoApp' C Y { as := X.normalizeObj n.as }.as =\n (α_ (inclusionObj n.as) X Y).symm ≪≫ normalizeIsoApp' C X n.as ▷ᵢ Y ≪≫ normalizeIsoApp' C Y (X.norm... | normalizeIsoApp_eq Y ⟨normalizeObj X n.as⟩ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monoidal.Limits.Cokernels | {
"line": 55,
"column": 43
} | {
"line": 55,
"column": 56
} | {
"line": 56,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝⁷ : Category.{v_1, u_1} C\ninst✝⁶ : Preadditive C\ninst✝⁵ : MonoidalCategory C\ninst✝⁴ : MonoidalPreadditive C\nX₁ Y₁ : C\nf₁ : X₁ ⟶ Y₁\nc₁ : CokernelCofork f₁\nhc₁ : IsColimit c₁\nX₂ Y₂ : C\nf₂ : X₂ ⟶ Y₂\nc₂ : CokernelCofork f₂\nhc₂ : IsColimit c₂\ninst✝³ : HasBinaryCoproduct (X₁ ⊗ ... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Rigid.Braided | {
"line": 47,
"column": 37
} | {
"line": 47,
"column": 68
} | {
"line": 47,
"column": 68
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX Y : C\ninst : ExactPairing X Y\n⊢ 𝟙 X ⊗≫ ((β_ X (𝟙_ C)).hom ≫ η_ X Y ▷ X) ⊗≫ ((β_ (Y ⊗ X) X).inv ≫ ε_ X Y ▷ X) ⊗≫ 𝟙 X = 𝟙 X",
"ppTerm": "?m.885",
"assigned": true,
"usedConstants": [
... | [
"C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX Y : C\ninst : ExactPairing X Y\n⊢ 𝟙 X ⊗≫ ((β_ X (𝟙_ C)).hom ≫ η_ X Y ▷ X) ⊗≫ (X ◁ ε_ X Y ≫ (β_ (𝟙_ C) X).inv) ⊗≫ 𝟙 X = 𝟙 X"
] | ← braiding_inv_naturality_right | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Presentable.ColimitPresentation | {
"line": 119,
"column": 6
} | {
"line": 121,
"column": 60
} | {
"line": 123,
"column": 0
} | [
{
"pp": "case hom\nC : Type u\ninst✝⁷ : Category.{v, u} C\nJ✝ : Type u_1\nI✝ : J✝ → Type u_2\ninst✝⁶ : Category.{v_1, u_1} J✝\ninst✝⁵ : (j : J✝) → Category.{?u.14, u_2} (I✝ j)\nD✝ : J✝ ⥤ C\nP✝ : (j : J✝) → ColimitPresentation (I✝ j) (D✝.obj j)\nJ : Type w\nI : J → Type w\ninst✝⁴ : SmallCategory J\ninst✝³ : (j :... | [] | rw [Category.assoc] at hpq
simp only [Functor.map_comp, comp_hom, reassoc_of% hpq]
simp [← Functor.map_comp, ← IsFiltered.coeq_condition] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Presentable.ColimitPresentation | {
"line": 119,
"column": 6
} | {
"line": 121,
"column": 60
} | {
"line": 123,
"column": 0
} | [
{
"pp": "case hom\nC : Type u\ninst✝⁷ : Category.{v, u} C\nJ✝ : Type u_1\nI✝ : J✝ → Type u_2\ninst✝⁶ : Category.{v_1, u_1} J✝\ninst✝⁵ : (j : J✝) → Category.{?u.14, u_2} (I✝ j)\nD✝ : J✝ ⥤ C\nP✝ : (j : J✝) → ColimitPresentation (I✝ j) (D✝.obj j)\nJ : Type w\nI : J → Type w\ninst✝⁴ : SmallCategory J\ninst✝³ : (j :... | [] | rw [Category.assoc] at hpq
simp only [Functor.map_comp, comp_hom, reassoc_of% hpq]
simp [← Functor.map_comp, ← IsFiltered.coeq_condition] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.MorphismProperty.LocalEpi | {
"line": 127,
"column": 78
} | {
"line": 129,
"column": 56
} | {
"line": 131,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝² : Category.{v_2, u_2} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\ninst✝¹ : G.Faithful\ninst✝ : G.Full\n⊢ G.essImage.localEpi = (MorphismProperty.epimorphisms D).inverseImage F",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants"... | [] | by
rw [← Functor.isoClosure_eq_essImage, localEpi_isoClosure,
localEpi_mem_range_eq_inverseImage_epimorphisms adj] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.DayConvolution | {
"line": 1199,
"column": 2
} | {
"line": 1199,
"column": 70
} | {
"line": 1200,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝⁹ : Category.{v₁, u₁} C\nV : Type u₂\ninst✝⁸ : Category.{v₂, u₂} V\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : MonoidalCategory V\nD : Type u₃\ninst✝⁵ : Category.{v₃, u₃} D\ninst✝⁴ : InducedLawfulDayConvolutionMonoidalCategoryStructCore C V D\ninst✝³ : ∀ (v : V) (d : C), Limits.PreservesCo... | [
"C : Type u₁\ninst✝⁹ : Category.{v₁, u₁} C\nV : Type u₂\ninst✝⁸ : Category.{v₂, u₂} V\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : MonoidalCategory V\nD : Type u₃\ninst✝⁵ : Category.{v₃, u₃} D\ninst✝⁴ : InducedLawfulDayConvolutionMonoidalCategoryStructCore C V D\ninst✝³ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShap... | apply (convolutions C V d₁ d₁').corepresentableBy.homEquiv.injective | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Preadditive.HomOrthogonal | {
"line": 102,
"column": 4
} | {
"line": 107,
"column": 20
} | {
"line": 109,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nι : Type u_1\ns : ι → C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasFiniteBiproducts C\no : HomOrthogonal s\nα β : Type\ninst✝¹ : Finite α\ninst✝ : Finite β\nf : α → ι\ng : β → ι\nz : (i : ι) → Matrix (↑(g ⁻¹' {i})) (↑(f ⁻¹' {i})) (End (s i))\n⊢ (fun z i j k ↦ eqTo... | [] | ext i ⟨j, w⟩ ⟨k, ⟨⟩⟩
simp only [eqToHom_refl, biproduct.matrix_components, Category.id_comp]
split_ifs with h
· simp
· exfalso
exact h w.symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Preadditive.HomOrthogonal | {
"line": 102,
"column": 4
} | {
"line": 107,
"column": 20
} | {
"line": 109,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nι : Type u_1\ns : ι → C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasFiniteBiproducts C\no : HomOrthogonal s\nα β : Type\ninst✝¹ : Finite α\ninst✝ : Finite β\nf : α → ι\ng : β → ι\nz : (i : ι) → Matrix (↑(g ⁻¹' {i})) (↑(f ⁻¹' {i})) (End (s i))\n⊢ (fun z i j k ↦ eqTo... | [] | ext i ⟨j, w⟩ ⟨k, ⟨⟩⟩
simp only [eqToHom_refl, biproduct.matrix_components, Category.id_comp]
split_ifs with h
· simp
· exfalso
exact h w.symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Category.PartOrdEmb | {
"line": 303,
"column": 4
} | {
"line": 303,
"column": 80
} | {
"line": 304,
"column": 4
} | [
{
"pp": "J : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF : J ⥤ PartOrdEmb\nc : Cocone (F ⋙ forget PartOrdEmb)\nhc : IsColimit c\ns : Cocone F\nj : J\nx' y' : (F ⋙ forget PartOrdEmb).obj j\n⊢ { toFun := ⇑(ConcreteCategory.hom (hc.desc ((forget PartOrdEmb).mapCocone s))), inj' := ⋯ }\n ((Conc... | [
"J : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF : J ⥤ PartOrdEmb\nc : Cocone (F ⋙ forget PartOrdEmb)\nhc : IsColimit c\ns : Cocone F\nj : J\nx' y' : (F ⋙ forget PartOrdEmb).obj j\nhx :\n (ConcreteCategory.hom (c.ι.app j ≫ hc.desc ((forget PartOrdEmb).mapCocone s))) x' =\n (ConcreteCategory.hom ((... | have hx := ConcreteCategory.congr_hom (hc.fac ((forget _).mapCocone s) j) x' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Presentable.OrthogonalReflection | {
"line": 83,
"column": 4
} | {
"line": 90,
"column": 74
} | {
"line": 91,
"column": 4
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝⁴ : Category.{v, u} C\nW : MorphismProperty C\nJ : Type u'\ninst✝³ : Category.{v', u'} J\ninst✝² : EssentiallySmall.{w, v', u'} J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhW : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), W f → IsCardinalPresentable X κ ∧... | [
"case refine_2\nC : Type u\ninst✝⁴ : Category.{v, u} C\nW : MorphismProperty C\nJ : Type u'\ninst✝³ : Category.{v', u'} J\ninst✝² : EssentiallySmall.{w, v', u'} J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhW : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), W f → IsCardinalPresentable X κ ∧ IsCardinalP... | · obtain ⟨j₁, g₁, rfl⟩ := IsCardinalPresentable.exists_hom_of_isColimit κ p.isColimit g₁
obtain ⟨j₂, g₂, rfl⟩ := IsCardinalPresentable.exists_hom_of_isColimit κ p.isColimit g₂
dsimp at h ⊢
obtain ⟨j₃, u, v, huv⟩ :=
IsCardinalPresentable.exists_eq_of_isColimit κ p.isColimit (f ≫ g₁) (f ≫ g₂)
... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Presentable.Type | {
"line": 111,
"column": 2
} | {
"line": 112,
"column": 23
} | {
"line": 114,
"column": 0
} | [
{
"pp": "case refine_2\nX : Type u\nκ : Cardinal.{u}\nhκ : Cardinal.aleph0 ≤ κ\nthis : IsFiltered (HasCardinalLT.Set X κ)\n⊢ ∀ (i : HasCardinalLT.Set X κ) (x y : (fun X ↦ X) ((functor X κ).obj i)),\n (hom ((cocone X κ).ι.app i)) x = (hom ((cocone X κ).ι.app i)) y →\n ∃ k f, (hom ((functor X κ).map f)) x... | [] | · rintro A ⟨x, hx⟩ ⟨y, hy⟩ rfl
exact ⟨A, 𝟙 _, rfl⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.RepresentedBy | {
"line": 61,
"column": 27
} | {
"line": 61,
"column": 93
} | {
"line": 62,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nX : C\nx : F.obj (op X)\nY : C\n⊢ (Function.Bijective fun f ↦ (ConcreteCategory.hom (F.map f.op)) x) ↔\n Function.Bijective ⇑(ConcreteCategory.hom ((uliftYonedaEquiv.symm { down := x }).app (op Y)))",
"ppTerm": "?m.58",
"assigned": tru... | [
"C : Type u\ninst✝ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nX : C\nx : F.obj (op X)\nY : C\n⊢ (Function.Bijective fun f ↦ (ConcreteCategory.hom (F.map f.op)) x) ↔\n Function.Bijective (⇑(ConcreteCategory.hom ((uliftYonedaEquiv.symm { down := x }).app (op Y))) ∘ ⇑Equiv.ulift.symm)"
] | ← Function.Bijective.of_comp_iff _ Equiv.ulift.{w}.symm.bijective, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.Coherent.CoherentSheaves | {
"line": 59,
"column": 2
} | {
"line": 59,
"column": 20
} | {
"line": 60,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Precoherent C\nW X : C\nα : Type\ninst✝ : Finite α\nY : α → C\nπ : (a : α) → Y a ⟶ X\nH : EffectiveEpiFamily Y π\nh_colim : Limits.IsColimit (Sieve.generate (Presieve.ofArrows Y π)).arrows.cocone\nx : Presieve.FamilyOfElements (yoneda.obj W) (Presi... | [
"case refine_1\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Precoherent C\nW X : C\nα : Type\ninst✝ : Finite α\nY : α → C\nπ : (a : α) → Y a ⟶ X\nH : EffectiveEpiFamily Y π\nh_colim : Limits.IsColimit (Sieve.generate (Presieve.ofArrows Y π)).arrows.cocone\nx : Presieve.FamilyOfElements (yoneda.obj W) (Pr... | refine ⟨t, ?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Sites.Coherent.Comparison | {
"line": 72,
"column": 6
} | {
"line": 72,
"column": 60
} | {
"line": 73,
"column": 6
} | [
{
"pp": "case refine_1.of\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preregular C\ninst✝ : FinitaryPreExtensive C\nB : C\nS : Sieve B\nY : C\nT : Presieve Y\nhT : T ∈ (extensiveCoverage C ⊔ regularCoverage C).coverings Y\n⊢ T ∈ (coherentCoverage C).coverings Y",
"ppTerm": "?refine_1.of",
"a... | [
"case refine_1.of\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preregular C\ninst✝ : FinitaryPreExtensive C\nB : C\nS : Sieve B\nY : C\nT : Presieve Y\nhT : T ∈ (extensiveCoverage C).coverings Y ∨ T ∈ (regularCoverage C).coverings Y\n⊢ T ∈ (coherentCoverage C).coverings Y"
] | simp only [Coverage.sup_covering, Set.mem_union] at hT | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Sites.Coherent.RegularTopology | {
"line": 38,
"column": 8
} | {
"line": 38,
"column": 55
} | {
"line": 38,
"column": 55
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preregular C\nX : C\nS : Sieve X\nY : C\nπ : Y ⟶ X\nh : EffectiveEpi π ∧ S.arrows π\n⊢ Sieve.generate (Presieve.ofArrows (fun x ↦ Y) fun x ↦ π) ≤ S",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Category... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preregular C\nX : C\nS : Sieve X\nY : C\nπ : Y ⟶ X\nh : EffectiveEpi π ∧ S.arrows π\n⊢ (Presieve.ofArrows (fun x ↦ Y) fun x ↦ π) ≤ S.arrows"
] | Sieve.generate_le_iff (Presieve.ofArrows _ _) S | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.Coherent.RegularTopology | {
"line": 49,
"column": 63
} | {
"line": 64,
"column": 88
} | {
"line": 66,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Preregular C\nX Y Y' : C\nπ : Y ⟶ X\ninst✝¹ : EffectiveEpi π\nπ' : Y' ⟶ Y\ninst✝ : EffectiveEpi π'\n⊢ EffectiveEpi (π' ≫ π)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Unit.unit",
"CategoryTheo... | [] | by
rw [effectiveEpi_iff_effectiveEpiFamily, ← Sieve.effectiveEpimorphic_family]
suffices h₂ : (Sieve.generate (Presieve.ofArrows _ _)) ∈ (regularTopology C) X by
change Nonempty _
rw [← Sieve.forallYonedaIsSheaf_iff_colimit]
exact fun W => regularTopology.isSheaf_yoneda_obj W _ h₂
apply Coverage.Satur... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Coherent.SheafComparison | {
"line": 182,
"column": 4
} | {
"line": 185,
"column": 37
} | {
"line": 187,
"column": 0
} | [
{
"pp": "case refine_2.refine_2\nC : Type u_1\nD : Type u_2\ninst✝⁷ : Category.{v_1, u_1} C\ninst✝⁶ : Category.{v_2, u_2} D\nF : C ⥤ D\ninst✝⁵ : F.PreservesEffectiveEpis\ninst✝⁴ : F.ReflectsEffectiveEpis\ninst✝³ : F.Full\ninst✝² : F.Faithful\ninst✝¹ : F.EffectivelyEnough\ninst✝ : Preregular D\nX : C\nS : Sieve ... | [] | · obtain ⟨W, g₁, g₂, h₁, h₂⟩ := H₂
rw [h₂]
convert! S.downward_closed h₁ (F.preimage (g₀ ≫ g₂))
exact F.map_injective (by simp) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves | {
"line": 292,
"column": 2
} | {
"line": 292,
"column": 20
} | {
"line": 293,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preregular C\nW X w✝ Y : C\nf : Y ⟶ X\nhf : EffectiveEpi f\nhS : ∃ f_1, ((ofArrows (fun x ↦ Y) fun x ↦ f) = ofArrows (fun x ↦ w✝) fun x ↦ f_1) ∧ EffectiveEpi f_1\nthis : (ofArrows (fun x ↦ Y) fun x ↦ f).regular\nh_colim : IsColimit (Sieve.generate (... | [
"case refine_1\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preregular C\nW X w✝ Y : C\nf : Y ⟶ X\nhf : EffectiveEpi f\nhS : ∃ f_1, ((ofArrows (fun x ↦ Y) fun x ↦ f) = ofArrows (fun x ↦ w✝) fun x ↦ f_1) ∧ EffectiveEpi f_1\nthis : (ofArrows (fun x ↦ Y) fun x ↦ f).regular\nh_colim : IsColimit (Sieve.generat... | refine ⟨t, ?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Sites.Descent.DescentDataAsCoalgebra | {
"line": 163,
"column": 34
} | {
"line": 166,
"column": 10
} | {
"line": 166,
"column": 10
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Adj Cat\nι : Type u_1\ninst✝ : Unique ι\nX S : C\nf : X ⟶ S\nD₁ D₂ : F.DescentDataAsCoalgebra fun x ↦ f\nα : D₁ ⟶ D₂\n⊢ (𝟭 (F.DescentDataAsCoalgebra fun x ↦ f)).map α ≫ (isoMk (fun i ↦ eqToIso ⋯) ⋯).hom =\n (isoMk (fun i ↦ eqToIso ⋯... | [] | by
ext i
obtain rfl := Subsingleton.elim i default
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Hypercover.Homotopy | {
"line": 101,
"column": 4
} | {
"line": 101,
"column": 16
} | {
"line": 102,
"column": 4
} | [
{
"pp": "case refine_3\nC : Type u\ninst✝¹ : Category.{v, u} C\nS : C\nE : PreOneHypercover S\nF : PreOneHypercover S\nA : Type u_1\ninst✝ : Category.{v_1, u_1} A\nf : E.Hom F\ng : F.Hom E\nhgf : Homotopy (g.comp f) (Hom.id F)\nG : Cᵒᵖ ⥤ A\nhE : IsLimit (E.multifork G)\n⊢ ∀ (E_1 : Multifork (F.multicospanIndex ... | [
"case refine_3\nC : Type u\ninst✝¹ : Category.{v, u} C\nS : C\nE : PreOneHypercover S\nF : PreOneHypercover S\nA : Type u_1\ninst✝ : Category.{v_1, u_1} A\nf : E.Hom F\ng : F.Hom E\nhgf : Homotopy (g.comp f) (Hom.id F)\nG : Cᵒᵖ ⥤ A\nhE : IsLimit (E.multifork G)\nt : Multifork (F.multicospanIndex G)\nm : t.pt ⟶ (F.m... | intro t m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Sites.Descent.Precoverage | {
"line": 210,
"column": 61
} | {
"line": 213,
"column": 33
} | {
"line": 213,
"column": 33
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝ : F.IsPrestack J\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j... | [] | by
convert! hq
ext
simpa using (Over.w q).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.LocalProperties | {
"line": 63,
"column": 8
} | {
"line": 63,
"column": 27
} | {
"line": 63,
"column": 28
} | [
{
"pp": "case refine_2\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nK : GrothendieckTopology C\nA : Type u_2\ninst✝ : Category.{v_2, u_2} A\nι : Type u_3\nX : ι → C\nhX : K.CoversTop X\nF G : Sheaf K A\nf : F ⟶ G\nh : ∀ (i : ι), IsIso ((K.overPullback A (X i)).map f)\nhiso : ∀ (Z : C) (i : ι) (g : Z ⟶ X i), I... | [
"case refine_2\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nK : GrothendieckTopology C\nA : Type u_2\ninst✝ : Category.{v_2, u_2} A\nι : Type u_3\nX : ι → C\nhX : K.CoversTop X\nF G : Sheaf K A\nf : F ⟶ G\nh : ∀ (i : ι), IsIso ((K.overPullback A (X i)).map f)\nhiso : ∀ (Z : C) (i : ι) (g : Z ⟶ X i), IsIso (f.hom.... | ← f.hom.naturality, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.MayerVietorisSquare | {
"line": 71,
"column": 2
} | {
"line": 75,
"column": 88
} | {
"line": 76,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝ : HasWeakSheafify J (Type v)\nF : Sheaf J (Type v)\nsq : Square C\n⊢ (sq.op.map ((yoneda ⋙ presheafToSheaf J (Type v)).op ⋙ yoneda.obj F)).IsPullback ↔ (sq.op.map F.obj).IsPullback",
"ppTerm": "?m.57",
"assigned": true,
... | [
"case refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝ : HasWeakSheafify J (Type v)\nF : Sheaf J (Type v)\nsq : Square C\n⊢ ⇑(((sheafificationAdjunction J (Type v)).homEquiv (yoneda.obj (unop sq.op.X₂)) F).trans yonedaEquiv) ∘\n ⇑(ConcreteCategory.hom (sq.op.map ((yoneda ⋙ pr... | refine Square.IsPullback.iff_of_equiv _ _
(((sheafificationAdjunction J (Type v)).homEquiv _ _).trans yonedaEquiv)
(((sheafificationAdjunction J (Type v)).homEquiv _ _).trans yonedaEquiv)
(((sheafificationAdjunction J (Type v)).homEquiv _ _).trans yonedaEquiv)
(((sheafificationAdjunction J (Type v)).hom... | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Sites.Descent.Precoverage | {
"line": 291,
"column": 4
} | {
"line": 291,
"column": 55
} | {
"line": 292,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝ : F.IsPrestack J\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝ : F.IsPrestack J\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j) = f' j\nhf... | mor_eq _ _ _ _ (by grind) (p ≫ q) _ (by grind) rfl, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.Descent.Precoverage | {
"line": 292,
"column": 4
} | {
"line": 292,
"column": 55
} | {
"line": 293,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝ : F.IsPrestack J\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝ : F.IsPrestack J\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j) = f' j\nhf... | mor_eq _ _ _ _ (by grind) (p ≫ q) _ (by grind) rfl, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.Descent.Precoverage | {
"line": 323,
"column": 2
} | {
"line": 323,
"column": 34
} | {
"line": 324,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝ : F.IsPrestack J\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝ : F.IsPrestack J\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j) = f' j\nhf... | have := full_pullFunctor F w hf' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Sites.SheafHom | {
"line": 198,
"column": 13
} | {
"line": 198,
"column": 15
} | {
"line": 198,
"column": 16
} | [
{
"pp": "case hunique\nC : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u'\ninst✝ : Category.{v', u'} A\nF G : Cᵒᵖ ⥤ A\nX : C\nS : Sieve X\nhG : ⦃Y : C⦄ → (f : Y ⟶ X) → IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\ny₁ : (p... | [
"case hunique\nC : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u'\ninst✝ : Category.{v', u'} A\nF G : Cᵒᵖ ⥤ A\nX : C\nS : Sieve X\nhG : ⦃Y : C⦄ → (f : Y ⟶ X) → IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\ny₁ y₂ : (presheafHo... | y₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Sites.Point.Conservative | {
"line": 150,
"column": 4
} | {
"line": 151,
"column": 43
} | {
"line": 152,
"column": 2
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝⁴ : Category.{v, u} C\nJ : GrothendieckTopology C\nP : ObjectProperty J.Point\ninst✝³ : LocallySmall.{w, v, u} C\ninst✝² : HasSheafify J (Type w)\ninst✝¹ : J.WEqualsLocallyBijective (Type w)\nhP : P.IsConservativeFamilyOfPoints\nX : C\nι : Type u_1\ninst✝ : Small.{w, u_1... | [] | obtain ⟨Z, _, ⟨_, p, _, ⟨i⟩, rfl⟩, z, rfl⟩ := Φ.obj.jointly_surjective _ hf x
exact ⟨i, Φ.obj.fiber.map p z, by simp⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Point.Conservative | {
"line": 150,
"column": 4
} | {
"line": 151,
"column": 43
} | {
"line": 152,
"column": 2
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝⁴ : Category.{v, u} C\nJ : GrothendieckTopology C\nP : ObjectProperty J.Point\ninst✝³ : LocallySmall.{w, v, u} C\ninst✝² : HasSheafify J (Type w)\ninst✝¹ : J.WEqualsLocallyBijective (Type w)\nhP : P.IsConservativeFamilyOfPoints\nX : C\nι : Type u_1\ninst✝ : Small.{w, u_1... | [] | obtain ⟨Z, _, ⟨_, p, _, ⟨i⟩, rfl⟩, z, rfl⟩ := Φ.obj.jointly_surjective _ hf x
exact ⟨i, Φ.obj.fiber.map p z, by simp⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Point.Conservative | {
"line": 172,
"column": 4
} | {
"line": 173,
"column": 43
} | {
"line": 174,
"column": 2
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝⁴ : Category.{v, u} C\nJ : GrothendieckTopology C\nP : ObjectProperty J.Point\ninst✝³ : LocallySmall.{w, v, u} C\ninst✝² : HasSheafify J (Type w)\ninst✝¹ : J.WEqualsLocallyBijective (Type w)\nhP : P.IsConservativeFamilyOfPoints\ninst✝ : ObjectProperty.Small.{w, max u w, ... | [] | obtain ⟨Z, _, ⟨_, p, _, ⟨i⟩, rfl⟩, z, rfl⟩ := Φ.obj.jointly_surjective _ hf x
exact ⟨i, Φ.obj.fiber.map p z, by simp⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Point.Conservative | {
"line": 172,
"column": 4
} | {
"line": 173,
"column": 43
} | {
"line": 174,
"column": 2
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝⁴ : Category.{v, u} C\nJ : GrothendieckTopology C\nP : ObjectProperty J.Point\ninst✝³ : LocallySmall.{w, v, u} C\ninst✝² : HasSheafify J (Type w)\ninst✝¹ : J.WEqualsLocallyBijective (Type w)\nhP : P.IsConservativeFamilyOfPoints\ninst✝ : ObjectProperty.Small.{w, max u w, ... | [] | obtain ⟨Z, _, ⟨_, p, _, ⟨i⟩, rfl⟩, z, rfl⟩ := Φ.obj.jointly_surjective _ hf x
exact ⟨i, Φ.obj.fiber.map p z, by simp⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Topos.Sheaf | {
"line": 215,
"column": 2
} | {
"line": 215,
"column": 33
} | {
"line": 217,
"column": 0
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nF G : Sheaf J (Type (max u v))\nm : F ⟶ G\ninst✝ : Mono m\nχ' : G ⟶ Ω J\nhχ' : IsPullback m ((isTerminalTerminal J Types.isTerminalPUnit).from F) χ' (truth J)\npb : IsPullback (𝟙 G.obj) χ'.hom (χ'.hom ≫ (closedSieves J)... | [] | simpa using this.paste_horiz pb | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Triangulated.Opposite.Triangulated | {
"line": 48,
"column": 6
} | {
"line": 48,
"column": 38
} | {
"line": 49,
"column": 6
} | [
{
"pp": "case refine_2\nC : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : HasShift C ℤ\ninst✝⁴ : HasZeroObject C\ninst✝³ : Preadditive C\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\ninst✝ : IsTriangulated C\nX₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : Cᵒᵖ\nu₁₂ : X₁ ⟶ X₂\nu₂₃ : X₂ ⟶ X₃\nu₁₃ : X₁... | [
"case refine_2\nC : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : HasShift C ℤ\ninst✝⁴ : HasZeroObject C\ninst✝³ : Preadditive C\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\ninst✝ : IsTriangulated C\nX₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : Cᵒᵖ\nu₁₂ : X₁ ⟶ X₂\nu₂₃ : X₂ ⟶ X₃\nu₁₃ : X₁ ⟶ X₃\ncomm ... | have eq₂ := congr($(o.comm₂).op) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory | {
"line": 275,
"column": 2
} | {
"line": 275,
"column": 38
} | {
"line": 276,
"column": 2
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nD₁ : Type u_3\nD₂ : Type u_4\ninst✝¹³ : Category.{v_1, u_1} C\ninst✝¹² : Category.{v_2, u_2} D\ninst✝¹¹ : Category.{v_3, u_3} D₁\ninst✝¹⁰ : Category.{v_4, u_4} D₂\nA B : ObjectProperty C\ninst✝⁹ : HasZeroObject C\ninst✝⁸ : HasShift C ℤ\ninst✝⁷ : Preadditive C\ninst✝⁶ : ∀ (n ... | [
"C : Type u_1\nD : Type u_2\nD₁ : Type u_3\nD₂ : Type u_4\ninst✝¹³ : Category.{v_1, u_1} C\ninst✝¹² : Category.{v_2, u_2} D\ninst✝¹¹ : Category.{v_3, u_3} D₁\ninst✝¹⁰ : Category.{v_4, u_4} D₂\nA B : ObjectProperty C\ninst✝⁹ : HasZeroObject C\ninst✝⁸ : HasShift C ℤ\ninst✝⁷ : Preadditive C\ninst✝⁶ : ∀ (n : ℤ), (shift... | let L₁ := (B.inverseImage A.ι).trW.Q | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory | {
"line": 309,
"column": 30
} | {
"line": 339,
"column": 89
} | {
"line": 339,
"column": 89
} | [
{
"pp": "C : Type u_1\nA : Type u_2\ninst✝¹² : Category.{v_1, u_1} C\ninst✝¹¹ : HasZeroObject C\ninst✝¹⁰ : Preadditive C\ninst✝⁹ : HasShift C ℤ\ninst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝⁷ : Pretriangulated C\ninst✝⁶ : Category.{v_2, u_2} A\nι : A ⥤ C\nhι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftF... | [] | by
obtain ⟨X₃, f₂, f₃, hT⟩ := distinguished_cocone_triangle (ι.map f₁)
have : admissibleMorphism ι f₁ := by simp [hA]
obtain ⟨K, Q, α, β, γ, hT'⟩ := this f₂ f₃ hT
have := epi_πQ hι hT hT'
obtain ⟨I, i, δ, hI⟩ := exists_distinguished_triangle_of_epi hι hA (πQ f₂ β)
have H := someOctahedron (show ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Pigeonhole | {
"line": 261,
"column": 2
} | {
"line": 262,
"column": 63
} | {
"line": 264,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nM : Type w\ninst✝³ : DecidableEq β\ns : Finset α\nt : Finset β\nf : α → β\nb : M\ninst✝² : CommSemiring M\ninst✝¹ : LinearOrder M\ninst✝ : IsStrictOrderedRing M\nhf : ∀ a ∈ s, f a ∈ t\nht : t.Nonempty\nhb : #t • b ≤ ↑(#s)\n⊢ ∃ y ∈ t, b ≤ ↑(#({x ∈ s | f x = y}))",
"ppTerm": "... | [] | simp_rw [cast_card] at hb ⊢
exact exists_le_sum_fiber_of_maps_to_of_nsmul_le_sum hf ht hb | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Pigeonhole | {
"line": 261,
"column": 2
} | {
"line": 262,
"column": 63
} | {
"line": 264,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nM : Type w\ninst✝³ : DecidableEq β\ns : Finset α\nt : Finset β\nf : α → β\nb : M\ninst✝² : CommSemiring M\ninst✝¹ : LinearOrder M\ninst✝ : IsStrictOrderedRing M\nhf : ∀ a ∈ s, f a ∈ t\nht : t.Nonempty\nhb : #t • b ≤ ↑(#s)\n⊢ ∃ y ∈ t, b ≤ ↑(#({x ∈ s | f x = y}))",
"ppTerm": "... | [] | simp_rw [cast_card] at hb ⊢
exact exists_le_sum_fiber_of_maps_to_of_nsmul_le_sum hf ht hb | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Additive.AP.Three.Behrend | {
"line": 323,
"column": 52
} | {
"line": 323,
"column": 70
} | {
"line": 323,
"column": 70
} | [
{
"pp": "x : ℝ\nhx : 2 / (1 - 2 / rexp 1) ≤ x\nthis : 0 < 1 - 2 / rexp 1\n⊢ 2 ≤ x * (1 - 2 / rexp 1)",
"ppTerm": "?m.137",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"div_le_iff₀",
"MulOne.toOne",
"Real.partialOrder",
"Real",
... | [
"x : ℝ\nhx : 2 / (1 - 2 / rexp 1) ≤ x\nthis : 0 < 1 - 2 / rexp 1\n⊢ 2 / (1 - 2 / rexp 1) ≤ x",
"x : ℝ\nhx : 2 / (1 - 2 / rexp 1) ≤ x\nthis : 0 < 1 - 2 / rexp 1\n⊢ 0 < 2",
"x : ℝ\nhx : 2 / (1 - 2 / rexp 1) ≤ x\nthis : 0 < 1 - 2 / rexp 1\n⊢ 2 ≠ 0"
] | ← div_le_iff₀ this | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Additive.PluenneckeRuzsa | {
"line": 243,
"column": 2
} | {
"line": 243,
"column": 67
} | {
"line": 244,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : CommGroup G\nA : Finset G\nhA : A.Nonempty\nB : Finset G\nm n : ℕ\nhA' : A ∈ A.powerset.erase ∅\nC : Finset G\nhCmin : ∀ x' ∈ A.powerset.erase ∅, ↑(#(C * B)) / ↑(#C) ≤ ↑(#(x' * B)) / ↑(#x')\nhC : C.Nonempty\nhCA : C ⊆ A\n⊢ ↑(#(B ^ m / B ^ n)) ≤ (↑(#(A * B))... | [
"G : Type u_1\ninst✝¹ : DecidableEq G\ninst✝ : CommGroup G\nA : Finset G\nhA : A.Nonempty\nB : Finset G\nm n : ℕ\nhA' : A ∈ A.powerset.erase ∅\nC : Finset G\nhCmin : ∀ x' ∈ A.powerset.erase ∅, ↑(#(C * B)) / ↑(#C) ≤ ↑(#(x' * B)) / ↑(#x')\nhC : C.Nonempty\nhCA : C ⊆ A\n⊢ ↑(#(B ^ m / B ^ n)) * ↑(#C) ≤ (↑(#(A * B)) / ↑... | refine le_of_mul_le_mul_right ?_ (by positivity : (0 : ℚ≥0) < #C) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.Additive.ApproximateSubgroup | {
"line": 197,
"column": 4
} | {
"line": 204,
"column": 13
} | {
"line": 205,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nA : Set G\nhA : IsApproximateSubgroup 1 A\n⊢ A * A ⊆ A",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Combinatorics.Additive.ApproximateSubgroup.0.isApproximateSubgroup_one._simp_1_2",
"MulOne.toOne",
"False",
... | [
"G : Type u_1\ninst✝ : Group G\nA : Set G\nhA : IsApproximateSubgroup 1 A\nx : G\nhx : A * A ⊆ x • A\n⊢ A * A ⊆ A"
] | obtain ⟨x, hx⟩ : ∃ x : G, A * A ⊆ x • A := by
obtain ⟨K, hK, hKA⟩ := hA.sq_covBySMul
simp only [Nat.cast_le_one, Finset.card_le_one_iff_subset_singleton,
Finset.subset_singleton_iff] at hK
obtain ⟨x, rfl | rfl⟩ := hK
· simp [hA.nonempty.ne_empty] at hKA
· rw [Finset.coe_singleton, ... | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.Additive.ApproximateSubgroup | {
"line": 211,
"column": 26
} | {
"line": 211,
"column": 39
} | {
"line": 212,
"column": 6
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nA : Set G\nhA : IsApproximateSubgroup 1 A\nx : G\nhx : A * A ⊆ x • A\nhx' : x⁻¹ • (A * A) ⊆ A\nhx_inv : x⁻¹ ∈ A\nhx_sq : x * x ∈ A\n⊢ A * A ⊆ x • A",
"ppTerm": "?m.280",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"hx"
],
"usedGoals... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Partition.Finpartition | {
"line": 279,
"column": 4
} | {
"line": 279,
"column": 57
} | {
"line": 281,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝² : Lattice α\ninst✝¹ : OrderBot α\na : α\nP✝ : Finpartition a\ninst✝ : Decidable (a = ⊥)\nP : Finpartition a\nh : ¬a = ⊥\n⊢ P ≤ indiscrete h",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Finset.mem_singleton_self",
"Finset",
"Part... | [] | · exact fun b hb ↦ ⟨a, mem_singleton_self _, P.le hb⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.Partition.Finpartition | {
"line": 276,
"column": 4
} | {
"line": 279,
"column": 57
} | {
"line": 281,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : Lattice α\ninst✝¹ : OrderBot α\na : α\nP✝ : Finpartition a\ninst✝ : Decidable (a = ⊥)\nP : Finpartition a\n⊢ P ≤ if ha : a = ⊥ then (Finpartition.empty α).copy ⋯ else indiscrete ha",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Finpartition.empty_par... | [] | split_ifs with h
· intro x hx
simpa [h, P.ne_bot hx] using P.le hx
· exact fun b hb ↦ ⟨a, mem_singleton_self _, P.le hb⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Partition.Finpartition | {
"line": 276,
"column": 4
} | {
"line": 279,
"column": 57
} | {
"line": 281,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : Lattice α\ninst✝¹ : OrderBot α\na : α\nP✝ : Finpartition a\ninst✝ : Decidable (a = ⊥)\nP : Finpartition a\n⊢ P ≤ if ha : a = ⊥ then (Finpartition.empty α).copy ⋯ else indiscrete ha",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Finpartition.empty_par... | [] | split_ifs with h
· intro x hx
simpa [h, P.ne_bot hx] using P.le hx
· exact fun b hb ↦ ⟨a, mem_singleton_self _, P.le hb⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise | {
"line": 52,
"column": 31
} | {
"line": 52,
"column": 78
} | {
"line": 52,
"column": 78
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\ns : Finset α\na b : ℕ\nP : Finpartition s\nhs : a * 0 + b * (0 + 1) = #s\n⊢ #({i ∈ ⊥.parts | #i = 0 + 1}) = b",
"ppTerm": "?m.127",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"HMul.hMul",
"Iff.of_eq",
... | [] | simpa [Finset.filter_true_of_mem] using hs.symm | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise | {
"line": 52,
"column": 31
} | {
"line": 52,
"column": 78
} | {
"line": 52,
"column": 78
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\ns : Finset α\na b : ℕ\nP : Finpartition s\nhs : a * 0 + b * (0 + 1) = #s\n⊢ #({i ∈ ⊥.parts | #i = 0 + 1}) = b",
"ppTerm": "?m.127",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"HMul.hMul",
"Iff.of_eq",
... | [] | simpa [Finset.filter_true_of_mem] using hs.symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise | {
"line": 52,
"column": 31
} | {
"line": 52,
"column": 78
} | {
"line": 52,
"column": 78
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\ns : Finset α\na b : ℕ\nP : Finpartition s\nhs : a * 0 + b * (0 + 1) = #s\n⊢ #({i ∈ ⊥.parts | #i = 0 + 1}) = b",
"ppTerm": "?m.127",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"HMul.hMul",
"Iff.of_eq",
... | [] | simpa [Finset.filter_true_of_mem] using hs.symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Partition.Finpartition | {
"line": 497,
"column": 2
} | {
"line": 502,
"column": 51
} | {
"line": 504,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : Lattice α\ninst✝³ : OrderBot α\ninst✝² : IsModularLattice α\ninst✝¹ : DecidableEq α\nι : Type u_2\nI : Finset ι\ns : ι → α\nP : (i : ι) → Finpartition (s i)\nha : I.SupIndep s\nM : Type u_3\ninst✝ : AddCommMonoid M\nf : α → M\n⊢ ∑ p ∈ (combine P ha).parts, f p = ∑ i ∈ I, ∑ p ∈ (P... | [] | simp_rw [combine]
refine Finset.sum_biUnion fun i hi j hj hij => ?_
rw [Function.onFun, Finset.disjoint_left]
intro p hpi hpj
have hp_disj : Disjoint p p := (ha.pairwiseDisjoint hi hj hij).mono ((P i).le hpi) ((P j).le hpj)
exact (P i).ne_bot hpi (disjoint_self.mp hp_disj) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Partition.Finpartition | {
"line": 497,
"column": 2
} | {
"line": 502,
"column": 51
} | {
"line": 504,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : Lattice α\ninst✝³ : OrderBot α\ninst✝² : IsModularLattice α\ninst✝¹ : DecidableEq α\nι : Type u_2\nI : Finset ι\ns : ι → α\nP : (i : ι) → Finpartition (s i)\nha : I.SupIndep s\nM : Type u_3\ninst✝ : AddCommMonoid M\nf : α → M\n⊢ ∑ p ∈ (combine P ha).parts, f p = ∑ i ∈ I, ∑ p ∈ (P... | [] | simp_rw [combine]
refine Finset.sum_biUnion fun i hi j hj hij => ?_
rw [Function.onFun, Finset.disjoint_left]
intro p hpi hpj
have hp_disj : Disjoint p p := (ha.pairwiseDisjoint hi hj hij).mono ((P i).le hpi) ((P j).le hpj)
exact (P i).ne_bot hpi (disjoint_self.mp hp_disj) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Partition.Finpartition | {
"line": 513,
"column": 4
} | {
"line": 513,
"column": 76
} | {
"line": 515,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝³ : Lattice α\ninst✝² : OrderBot α\ninst✝¹ : IsModularLattice α\ninst✝ : DecidableEq α\na b c : α\nP✝ : Finpartition a\nQ✝ : (i : α) → i ∈ P✝.parts → Finpartition i\nP : Finpartition a\nQ : (i : α) → i ∈ P.parts → Finpartition i\n⊢ P.parts.attach.sup Subtype.val = a",
"ppTerm": "... | [] | rw [Finset.sup_attach (f := fun x => x), ← Function.id_def, P.sup_parts] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Partition.Finpartition | {
"line": 513,
"column": 4
} | {
"line": 513,
"column": 76
} | {
"line": 515,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝³ : Lattice α\ninst✝² : OrderBot α\ninst✝¹ : IsModularLattice α\ninst✝ : DecidableEq α\na b c : α\nP✝ : Finpartition a\nQ✝ : (i : α) → i ∈ P✝.parts → Finpartition i\nP : Finpartition a\nQ : (i : α) → i ∈ P.parts → Finpartition i\n⊢ P.parts.attach.sup Subtype.val = a",
"ppTerm": "... | [] | rw [Finset.sup_attach (f := fun x => x), ← Function.id_def, P.sup_parts] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Partition.Finpartition | {
"line": 513,
"column": 4
} | {
"line": 513,
"column": 76
} | {
"line": 515,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝³ : Lattice α\ninst✝² : OrderBot α\ninst✝¹ : IsModularLattice α\ninst✝ : DecidableEq α\na b c : α\nP✝ : Finpartition a\nQ✝ : (i : α) → i ∈ P✝.parts → Finpartition i\nP : Finpartition a\nQ : (i : α) → i ∈ P.parts → Finpartition i\n⊢ P.parts.attach.sup Subtype.val = a",
"ppTerm": "... | [] | rw [Finset.sup_attach (f := fun x => x), ← Function.id_def, P.sup_parts] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Partition.Finpartition | {
"line": 528,
"column": 2
} | {
"line": 528,
"column": 18
} | {
"line": 529,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝³ : Lattice α\ninst✝² : OrderBot α\ninst✝¹ : IsModularLattice α\ninst✝ : DecidableEq α\na : α\nP : Finpartition a\nQ : (i : α) → i ∈ P.parts → Finpartition i\nb : α\nhb : b ∈ P.parts\nc : α\nhc : c ∈ P.parts\nhbc : ⟨b, hb⟩ ≠ ⟨c, hc⟩\n⊢ ∀ ⦃a_1 : α⦄, a_1 ∈ ((fun i ↦ Q ↑i ⋯) ⟨b, hb⟩).pa... | [
"α : Type u_1\ninst✝³ : Lattice α\ninst✝² : OrderBot α\ninst✝¹ : IsModularLattice α\ninst✝ : DecidableEq α\na : α\nP : Finpartition a\nQ : (i : α) → i ∈ P.parts → Finpartition i\nb : α\nhb : b ∈ P.parts\nc : α\nhc : c ∈ P.parts\nhbc : ⟨b, hb⟩ ≠ ⟨c, hc⟩\nd : α\nhdb : d ∈ ((fun i ↦ Q ↑i ⋯) ⟨b, hb⟩).parts\nhdc : d ∈ (... | rintro d hdb hdc | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Combinatorics.SimpleGraph.Regularity.Uniform | {
"line": 353,
"column": 32
} | {
"line": 356,
"column": 14
} | {
"line": 357,
"column": 2
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_2\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\ninst✝ : DecidableEq α\nA : Finset α\nP : Finpartition A\nε : 𝕜\nhε : 0 < ε\nhP : P.IsEquipartition\nhP' : 4 / ε ≤ ↑(#P.parts)\nhA : A.Nonempty\nthis✝ : ↑(#A) + ↑(#P.parts) ≤ 2 * ↑(#A)\nthis : 1 ≤ ... | [] | by
rw [mul_left_comm, ← sq]
convert! mul_le_mul_of_nonneg_left this (mul_nonneg zero_le_two <| sq_nonneg (#A : 𝕜)) using 1
<;> ring | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Regularity.Increment | {
"line": 145,
"column": 60
} | {
"line": 146,
"column": 66
} | {
"line": 147,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\ninst✝ : Nonempty α\nhP : P.IsEquipartition\nhP₇ : 7 ≤ #P.parts\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPG : ¬P.IsUniform G... | [] | by
rw [coe_energy, add_div, mul_div_cancel_left₀]; positivity | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Regularity.Increment | {
"line": 159,
"column": 38
} | {
"line": 159,
"column": 47
} | {
"line": 159,
"column": 48
} | [
{
"pp": "case hab\nα : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\ninst✝ : Nonempty α\nhP : P.IsEquipartition\nhP₇ : 7 ≤ #P.parts\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPG : ↑(#P... | [
"case hab\nα : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\ninst✝ : Nonempty α\nhP : P.IsEquipartition\nhP₇ : 7 ≤ #P.parts\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPG : ↑(#P.parts * #P.... | mul_tsub, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Copy | {
"line": 526,
"column": 2
} | {
"line": 526,
"column": 16
} | {
"line": 527,
"column": 2
} | [
{
"pp": "V : Type u_1\nW : Type u_2\nG : SimpleGraph V\nH : SimpleGraph W\ninst✝² : Fintype V\ninst✝¹ : Fintype { f // Injective ⇑f }\ninst✝ : DecidableEq G.Subgraph\n⊢ G.copyCount H = #(image Copy.toSubgraph univ)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fins... | [
"V : Type u_1\nW : Type u_2\nG : SimpleGraph V\nH : SimpleGraph W\ninst✝² : Fintype V\ninst✝¹ : Fintype { f // Injective ⇑f }\ninst✝ : DecidableEq G.Subgraph\n⊢ #{G' | Nonempty (H ≃g G'.coe)} = #(image Copy.toSubgraph univ)"
] | rw [copyCount] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Copy | {
"line": 545,
"column": 2
} | {
"line": 545,
"column": 16
} | {
"line": 546,
"column": 2
} | [
{
"pp": "V : Type u_1\ninst✝ : Fintype V\nG : SimpleGraph V\n⊢ G.copyCount ⊥ = 1",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.univ",
"SimpleGraph.Iso",
"congrArg",
"SimpleGraph.Subgraph",
"SimpleGraph.Adj",
"Classical.propDecida... | [
"V : Type u_1\ninst✝ : Fintype V\nG : SimpleGraph V\n⊢ #{G' | Nonempty (⊥ ≃g G'.coe)} = 1"
] | rw [copyCount] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Copy | {
"line": 673,
"column": 4
} | {
"line": 673,
"column": 45
} | {
"line": 674,
"column": 4
} | [
{
"pp": "case inr.calc_1\nV : Type u_1\nW : Type u_2\nG : SimpleGraph V\nH : SimpleGraph W\ninst✝¹ : Fintype ↑G.edgeSet\ninst✝ : Fintype V\nhH : H ≠ ⊥\nf : { G' // Nonempty (H ≃g G'.coe) } → Sym2 V := fun G' ↦ ⋯.some\n⊢ #G.edgeFinset - G.copyCount H = #G.edgeFinset - Fintype.card { G' // Nonempty (H ≃g G'.coe) ... | [
"case inr.calc_1\nV : Type u_1\nW : Type u_2\nG : SimpleGraph V\nH : SimpleGraph W\ninst✝¹ : Fintype ↑G.edgeSet\ninst✝ : Fintype V\nhH : H ≠ ⊥\nf : { G' // Nonempty (H ≃g G'.coe) } → Sym2 V := fun G' ↦ ⋯.some\n⊢ Fintype.card ↑G.edgeSet - G.copyCount H = Fintype.card ↑G.edgeSet - Fintype.card { G' // Nonempty (H ≃g ... | simp only [edgeFinset, Set.toFinset_card] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
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