mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y Z : C
z₁ : W.LeftFraction X Y
z₂ : W.LeftFraction Y Z
z₃ z₃' : W.LeftFraction z₁.Y' z₂.Y'
h₃ : z₂.f ≫ z₃.s = z₁.s ≫ z₃.f
h₃' : z₂.f ≫ z₃'.s = z₁.s ≫ z₃'.f
z₄ : W.LeftFraction z₃.Y' z₃'.Y'
fac : z₃'.s ≫ z₄.s = z₃.s ≫ z₄.f
⊢ z₂.f ≫ z₃.s ≫ z₄.f = z₂.f ≫ z₃'.s ≫ z₄.s

Tactic:
fac

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y Z : C
z₁ : W.LeftFraction X Y
z₂ : W.LeftFraction Y Z
z₃ z₃' : W.LeftFraction z₁.Y' z₂.Y'
h₃ : z₂.f ≫ z₃.s = z₁.s ≫ z₃.f
h₃' : z₂.f ≫ z₃'.s = z₁.s ≫ z₃'.f
z₄ : W.LeftFraction z₃.Y' z₃'.Y'
fac : z₃'.s ≫ z₄.s = z₃.s ≫ z₄.f
⊢ z₂.f ≫ z₃.s ≫ z₄.f = z₂.f ≫ z₃.s ≫ z₄.f
Module: Mathlib.Order.TransfiniteIteration

Goal state before:
pp: I : Type u
inst✝³ : SupSet I
φ : I → I
J : Type w
inst✝² : LinearOrder J
inst✝¹ : SuccOrder J
inst✝ : WellFoundedLT J
i₀ : I
j : J
hj : Order.IsSuccLimit j
⊢ transfiniteIterate φ j i₀ = ⨆ x, transfiniteIterate φ (↑x) i₀

Tactic:
dsimp [transfiniteIterate]
  rw [SuccOrder.limitRecOn_of_isSuccLimit _ _ _ hj]
  simp only [iSup_apply]

Goal state after:
Module: Mathlib.Order.TransfiniteIteration

Goal state before:
pp: case isMin
I : Type u
inst✝⁴ : CompleteLattice I
φ : I → I
i₀ : I
J : Type w
inst✝³ : LinearOrder J
inst✝² : OrderBot J
inst✝¹ : SuccOrder J
inst✝ : WellFoundedLT J
hφ : ∀ (i : I), i ≤ φ i
k✝ k : J
hk : IsMin k
hkj : k✝ ≤ k
⊢ (fun j ↦ transfiniteIterate φ j i₀) k✝ ≤ (fun j ↦ transfiniteIterate φ j i₀) k

Tactic:
obtain rfl := hk.eq_bot

Goal state after:
case isMin
I : Type u
inst✝⁴ : CompleteLattice I
φ : I → I
i₀ : I
J : Type w
inst✝³ : LinearOrder J
inst✝² : OrderBot J
inst✝¹ : SuccOrder J
inst✝ : WellFoundedLT J
hφ : ∀ (i : I), i ≤ φ i
k : J
hk : IsMin ⊥
hkj : k ≤ ⊥
⊢ (fun j ↦ transfiniteIterate φ j i₀) k ≤ (fun j ↦ transfiniteIterate φ j i₀) ⊥
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X₁ X₂ X₃ X₄ : C
z₁ : W.LeftFraction X₁ X₂
z₂ : W.LeftFraction X₂ X₃
z₃ : W.LeftFraction X₃ X₄
z₁₂ : W.LeftFraction z₁.Y' z₂.Y'
fac₁₂ : z₂.f ≫ z₁₂.s = z₁.s ≫ z₁₂.f
z₂₃ : W.LeftFraction z₂.Y' z₃.Y'
fac₂₃ : z₃.f ≫ z₂₃.s = z₂.s ≫ z₂₃.f
z' : W.LeftFraction z₁₂.Y' z₂₃.Y'
fac : z₂₃.f ≫ z'.s = z₁₂.s ≫ z'.f
⊢ z₂.s ≫ z₂₃.f ≫ z'.s = z₂.s ≫ z₁₂.s ≫ z'.f

Tactic:
fac

Goal state after:
C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X₁ X₂ X₃ X₄ : C
z₁ : W.LeftFraction X₁ X₂
z₂ : W.LeftFraction X₂ X₃
z₃ : W.LeftFraction X₃ X₄
z₁₂ : W.LeftFraction z₁.Y' z₂.Y'
fac₁₂ : z₂.f ≫ z₁₂.s = z₁.s ≫ z₁₂.f
z₂₃ : W.LeftFraction z₂.Y' z₃.Y'
fac₂₃ : z₃.f ≫ z₂₃.s = z₂.s ≫ z₂₃.f
z' : W.LeftFraction z₁₂.Y' z₂₃.Y'
fac : z₂₃.f ≫ z'.s = z₁₂.s ≫ z'.f
⊢ z₂.s ≫ z₁₂.s ≫ z'.f = z₂.s ≫ z₁₂.s ≫ z'.f
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X₁ X₂ X₃ X₄ : C
z₁ : W.LeftFraction X₁ X₂
z₂ : W.LeftFraction X₂ X₃
z₃ : W.LeftFraction X₃ X₄
z₁₂ : W.LeftFraction z₁.Y' z₂.Y'
fac₁₂ : z₂.f ≫ z₁₂.s = z₁.s ≫ z₁₂.f
z₂₃ : W.LeftFraction z₂.Y' z₃.Y'
fac₂₃ : z₃.f ≫ z₂₃.s = z₂.s ≫ z₂₃.f
z' : W.LeftFraction z₁₂.Y' z₂₃.Y'
fac : z₂₃.f ≫ z'.s = z₁₂.s ≫ z'.f
⊢ z₂.f ≫ z₂₃.f ≫ z'.s = z₂.f ≫ z₁₂.s ≫ z'.f

Tactic:
fac

Goal state after:
C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X₁ X₂ X₃ X₄ : C
z₁ : W.LeftFraction X₁ X₂
z₂ : W.LeftFraction X₂ X₃
z₃ : W.LeftFraction X₃ X₄
z₁₂ : W.LeftFraction z₁.Y' z₂.Y'
fac₁₂ : z₂.f ≫ z₁₂.s = z₁.s ≫ z₁₂.f
z₂₃ : W.LeftFraction z₂.Y' z₃.Y'
fac₂₃ : z₃.f ≫ z₂₃.s = z₂.s ≫ z₂₃.f
z' : W.LeftFraction z₁₂.Y' z₂₃.Y'
fac : z₂₃.f ≫ z'.s = z₁₂.s ≫ z'.f
⊢ z₂.f ≫ z₁₂.s ≫ z'.f = z₂.f ≫ z₁₂.s ≫ z'.f
Module: Mathlib.CategoryTheory.Localization.Triangulated

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹⁰ : Category.{v_1, u_1} C
inst✝⁹ : Category.{v_2, u_2} D
L : C ⥤ D
inst✝⁸ : HasShift C ℤ
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁴ : Pretriangulated C
inst✝³ : HasShift D ℤ
inst✝² : L.CommShift ℤ
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : D
f : X ⟶ Y
⊢ ∃ Z g h, Triangle.mk f g h ∈ L.essImageDistTriang

Tactic:
have := essSurj_mapArrow L W
  obtain ⟨φ, ⟨e⟩⟩ : ∃ (φ : Arrow C), Nonempty (L.mapArrow.obj φ ≅ Arrow.mk f) :=
    ⟨_, ⟨Functor.objObjPreimageIso _ _⟩⟩
  obtain ⟨Z, g, h, H⟩ := Pretriangulated.distinguished_cocone_triangle φ.hom
  refine ⟨L.obj Z, e.inv.right ≫ L.map g,
    L.map h ≫ (L.commShiftIso (1 : ℤ)).hom.app _ ≫ e.hom.left⟦(1 : ℤ)⟧', _, ?_, H⟩
  refine Triangle.isoMk _ _ (Arrow.leftFunc.mapIso e.symm) (Arrow.rightFunc.mapIso e.symm)
    (Iso.refl _) e.inv.w.symm (by simp) ?_
  dsimp
  simp only [assoc, id_comp, ← Functor.map_comp, ← Arrow.comp_left, e.hom_inv_id, Arrow.id_left,
    Functor.mapArrow_obj, Arrow.mk_left, Functor.map_id, comp_id]

Goal state after:
Module: Mathlib.CategoryTheory.Localization.Triangulated

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : Category.{v_2, u_2} D
L : C ⥤ D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : Preadditive C
inst✝⁷ : HasZeroObject C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : HasShift D ℤ
inst✝³ : L.CommShift ℤ
W : MorphismProperty C
inst✝² : L.IsLocalization W
inst✝¹ : W.HasLeftCalculusOfFractions
inst✝ : W.IsCompatibleWithTriangulation
T₁ T₂ : Triangle C
hT₁ : T₁ ∈ distinguishedTriangles
hT₂ : T₂ ∈ distinguishedTriangles
a : L.obj T₁.obj₁ ⟶ L.obj T₂.obj₁
b : L.obj T₁.obj₂ ⟶ L.obj T₂.obj₂
⊢ L.map T₁.mor₁ ≫ b = a ≫ L.map T₂.mor₁ → ∃ φ, φ.hom₁ = a ∧ φ.hom₂ = b

Tactic:
fac

Goal state after:
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : Category.{v_2, u_2} D
L : C ⥤ D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : Preadditive C
inst✝⁷ : HasZeroObject C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : HasShift D ℤ
inst✝³ : L.CommShift ℤ
W : MorphismProperty C
inst✝² : L.IsLocalization W
inst✝¹ : W.HasLeftCalculusOfFractions
inst✝ : W.IsCompatibleWithTriangulation
T₁ T₂ : Triangle C
hT₁ : T₁ ∈ distinguishedTriangles
hT₂ : T₂ ∈ distinguishedTriangles
a : L.obj T₁.obj₁ ⟶ L.obj T₂.obj₁
b : L.obj T₁.obj₂ ⟶ L.obj T₂.obj₂
fac : L.map T₁.mor₁ ≫ b = a ≫ L.map T₂.mor₁
⊢ ∃ φ, φ.hom₁ = a ∧ φ.hom₂ = b
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝¹ : W.HasLeftCalculusOfFractions
E : Type u_3
inst✝ : Category.{v_3, u_3} E
F : C ⥤ E
hF : W.IsInvertedBy F
X Y Z : C
f : W.LeftFraction X Y
g : W.LeftFraction Y Z
z : W.LeftFraction f.Y' g.Y'
fac : g.f ≫ z.s = f.s ≫ z.f
this✝ : IsIso (F.map g.s)
this : IsIso (F.map z.s)
⊢ F.map { Y' := z.Y', f := f.f ≫ z.f, s := g.s ≫ z.s, hs := ⋯ }.f = f.map F hF ≫ F.map (g.f ≫ z.s)

Tactic:
fac

Goal state after:
C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝¹ : W.HasLeftCalculusOfFractions
E : Type u_3
inst✝ : Category.{v_3, u_3} E
F : C ⥤ E
hF : W.IsInvertedBy F
X Y Z : C
f : W.LeftFraction X Y
g : W.LeftFraction Y Z
z : W.LeftFraction f.Y' g.Y'
fac : g.f ≫ z.s = f.s ≫ z.f
this✝ : IsIso (F.map g.s)
this : IsIso (F.map z.s)
⊢ F.map { Y' := z.Y', f := f.f ≫ z.f, s := g.s ≫ z.s, hs := ⋯ }.f = f.map F hF ≫ F.map (f.s ≫ z.f)
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝¹ : W.HasLeftCalculusOfFractions
X Y Z : C
z₁ : W.LeftFraction X Y
z₂ : W.LeftFraction Y Z
z₃ : W.LeftFraction z₁.Y' z₂.Y'
h₃ : z₂.f ≫ z₃.s = z₁.s ≫ z₃.f
L : C ⥤ D
inst✝ : L.IsLocalization W
⊢ z₁.map L ⋯ ≫ z₂.map L ⋯ = (z₁.comp₀ z₂ z₃).map L ⋯

Tactic:
have : IsIso (L.map (z₂.s ≫ z₃.s)) := by
    rw [L.map_comp]
    infer_instance

Goal state after:
C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝¹ : W.HasLeftCalculusOfFractions
X Y Z : C
z₁ : W.LeftFraction X Y
z₂ : W.LeftFraction Y Z
z₃ : W.LeftFraction z₁.Y' z₂.Y'
h₃ : z₂.f ≫ z₃.s = z₁.s ≫ z₃.f
L : C ⥤ D
inst✝ : L.IsLocalization W
this : IsIso (L.map (z₂.s ≫ z₃.s))
⊢ z₁.map L ⋯ ≫ z₂.map L ⋯ = (z₁.comp₀ z₂ z₃).map L ⋯
Module: Mathlib.CategoryTheory.Triangulated.HomologicalFunctor

Goal state before:
pp: C : Type u_1
A : Type u_3
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : Category.{v_3, u_3} A
F : C ⥤ A
inst✝⁶ : HasZeroObject C
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝³ : Pretriangulated C
inst✝² : Abelian A
inst✝¹ : F.IsHomological
inst✝ : F.ShiftSequence ℤ
T : Triangle C
hT : T ∈ distinguishedTriangles
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
⊢ (F.shift n₀).map T.mor₂ ≫ F.homologySequenceδ T n₀ n₁ h = 0

Tactic:
by
  dsimp only [homologySequenceδ]
  rw [← F.shiftMap_comp', comp_distTriang_mor_zero₂₃ _ hT, shiftMap_zero]

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.HomologicalFunctor

Goal state before:
pp: case mpr
C : Type u_1
A : Type u_3
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : Category.{v_3, u_3} A
F : C ⥤ A
inst✝⁶ : HasZeroObject C
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝³ : Pretriangulated C
inst✝² : Abelian A
inst✝¹ : F.IsHomological
inst✝ : F.ShiftSequence ℤ
X Y : C
f : X ⟶ Y
Z : C
g : Y ⟶ Z
h : Z ⟶ (shiftFunctor C 1).obj X
hT : Triangle.mk f g h ∈ distinguishedTriangles
h₁ :
  ∀ (n : ℤ),
    IsZero ((F.shift n).obj Z) ↔ (F.shift n).map g = 0 ∧ F.homologySequenceδ (Triangle.mk f g h) n (n + 1) ⋯ = 0
h₂ : ∀ (n : ℤ), Mono ((F.shift (n + 1)).map f) ↔ F.homologySequenceδ (Triangle.mk f g h) n (n + 1) ⋯ = 0
h₃ : ∀ (n : ℤ), Epi ((F.shift n).map f) ↔ (F.shift n).map g = 0
⊢ (∀ (n : ℤ), IsIso ((F.shift n).map f)) → ∀ (n : ℤ), Epi ((F.shift n).map f) ∧ Mono ((F.shift (n + 1)).map f)

Tactic:
intros

Goal state after:
case mpr
C : Type u_1
A : Type u_3
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : Category.{v_3, u_3} A
F : C ⥤ A
inst✝⁶ : HasZeroObject C
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝³ : Pretriangulated C
inst✝² : Abelian A
inst✝¹ : F.IsHomological
inst✝ : F.ShiftSequence ℤ
X Y : C
f : X ⟶ Y
Z : C
g : Y ⟶ Z
h : Z ⟶ (shiftFunctor C 1).obj X
hT : Triangle.mk f g h ∈ distinguishedTriangles
h₁ :
  ∀ (n : ℤ),
    IsZero ((F.shift n).obj Z) ↔ (F.shift n).map g = 0 ∧ F.homologySequenceδ (Triangle.mk f g h) n (n + 1) ⋯ = 0
h₂ : ∀ (n : ℤ), Mono ((F.shift (n + 1)).map f) ↔ F.homologySequenceδ (Triangle.mk f g h) n (n + 1) ⋯ = 0
h₃ : ∀ (n : ℤ), Epi ((F.shift n).map f) ↔ (F.shift n).map g = 0
a✝ : ∀ (n : ℤ), IsIso ((F.shift n).map f)
n✝ : ℤ
⊢ Epi ((F.shift n✝).map f) ∧ Mono ((F.shift (n✝ + 1)).map f)
Module: Mathlib.CategoryTheory.Localization.Composition

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
inst✝⁴ : Category.{v₁, u₁} C₁
inst✝³ : Category.{v₂, u₂} C₂
inst✝² : Category.{v₃, u₃} C₃
L₁ : C₁ ⥤ C₂
L₂ : C₂ ⥤ C₃
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
inst✝¹ : L₁.IsLocalization W₁
inst✝ : L₂.IsLocalization W₂
W₃ : MorphismProperty C₁
hW₃ : W₃.IsInvertedBy (L₁ ⋙ L₂)
hW₁₃ : W₁ ≤ W₃
hW₂₃ : W₂ ≤ W₃.map L₁
E₂ : C₂ ≌ W₁.Localization := uniq L₁ W₁.Q W₁
W₂' : MorphismProperty W₁.Localization := W₂.map E₂.functor
⊢ W₂ ≤ W₂'.inverseImage E₂.functor

Tactic:
have eq := W₂.isoClosure.inverseImage_map_eq_of_isEquivalence E₂.functor

Goal state after:
C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
inst✝⁴ : Category.{v₁, u₁} C₁
inst✝³ : Category.{v₂, u₂} C₂
inst✝² : Category.{v₃, u₃} C₃
L₁ : C₁ ⥤ C₂
L₂ : C₂ ⥤ C₃
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
inst✝¹ : L₁.IsLocalization W₁
inst✝ : L₂.IsLocalization W₂
W₃ : MorphismProperty C₁
hW₃ : W₃.IsInvertedBy (L₁ ⋙ L₂)
hW₁₃ : W₁ ≤ W₃
hW₂₃ : W₂ ≤ W₃.map L₁
E₂ : C₂ ≌ W₁.Localization := uniq L₁ W₁.Q W₁
W₂' : MorphismProperty W₁.Localization := W₂.map E₂.functor
eq : (W₂.isoClosure.map E₂.functor).inverseImage E₂.functor = W₂.isoClosure
⊢ W₂ ≤ W₂'.inverseImage E₂.functor
Module: Mathlib.CategoryTheory.Triangulated.Subcategory

Goal state before:
pp: C : Type u_1
inst✝¹⁴ : Category.{v_1, u_1} C
inst✝¹³ : HasZeroObject C
inst✝¹² : HasShift C ℤ
inst✝¹¹ : Preadditive C
inst✝¹⁰ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁹ : Pretriangulated C
D : Type u_2
inst✝⁸ : Category.{v_2, u_2} D
inst✝⁷ : Preadditive D
inst✝⁶ : HasZeroObject D
inst✝⁵ : HasShift D ℤ
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝³ : Pretriangulated D
E : Type u_3
inst✝² : Category.{v_3, u_3} E
inst✝¹ : HasShift E ℤ
P Q R : ObjectProperty C
inst✝ : P.IsStableUnderShift ℤ
n : ℕ
⊢ (P.extensionProductIter n).IsStableUnderShift ℤ

Tactic:
by
  induction n with
  | zero => assumption
  | succ n h =>
    rw [extensionProductIter_succ]
    infer_instance

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.ShiftSequence

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
n m mn : ℤ
hmn : m + n = mn
a a' a'' : ℤ
ha' : n + a = a'
ha'' : m + a' = a''
K : CochainComplex C ℤ
⊢ mn + a = a''

Tactic:
by rw [← ha'', ← ha', ← add_assoc, hmn]

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Subcategory

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
P : ObjectProperty C
inst✝ : P.IsStableUnderShift ℤ
T : Triangle C
hT : T ∈ distinguishedTriangles
h : P T.obj₁
⊢ P.trW T.mor₂

Tactic:
rw [trW_iff']
  exact ⟨_, _, _, hT, h⟩

Goal state after:
Module: Mathlib.Algebra.Homology.Localization

Goal state before:
pp: case mp
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
ι : Type u_2
c : ComplexShape ι
inst✝² : HasZeroMorphisms C
inst✝¹ : CategoryWithHomology C
inst✝ : (HomologicalComplex.quasiIso C c).HasLocalization
K L : HomologicalComplex C c
f : K ⟶ L
h : IsIso (Q.map f)
⊢ HomologicalComplex.quasiIso C c f

Tactic:
HomologicalComplex.mem_quasiIso_iff,

Goal state after:
case mp
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
ι : Type u_2
c : ComplexShape ι
inst✝² : HasZeroMorphisms C
inst✝¹ : CategoryWithHomology C
inst✝ : (HomologicalComplex.quasiIso C c).HasLocalization
K L : HomologicalComplex C c
f : K ⟶ L
h : IsIso (Q.map f)
⊢ QuasiIso f
Module: Mathlib.Algebra.Homology.Embedding.Basic

Goal state before:
pp: case mpr
p n : ℤ
⊢ p < n → ∀ (i : ℕ), (embeddingUpIntLE p).f i ≠ n

Tactic:
intros

Goal state after:
case mpr
p n : ℤ
a✝ : p < n
i✝ : ℕ
⊢ (embeddingUpIntLE p).f i✝ ≠ n
Module: Mathlib.Algebra.Homology.Embedding.Basic

Goal state before:
pp: case mpr
p n : ℤ
⊢ n < p → ∀ (i : ℕ), (embeddingUpIntGE p).f i ≠ n

Tactic:
intros

Goal state after:
case mpr
p n : ℤ
a✝ : n < p
i✝ : ℕ
⊢ (embeddingUpIntGE p).f i✝ ≠ n
Module: Mathlib.Algebra.Homology.Embedding.IsSupported

Goal state before:
pp: case mp.left
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝¹ : Category.{v_1, u_3} C
inst✝ : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
hK : IsZero K
⊢ ∀ (i' : ι'), (∀ (i : ι), e.f i ≠ i') → IsZero (K.X i')

Tactic:
intros

Goal state after:
case mp.left
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝¹ : Category.{v_1, u_3} C
inst✝ : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
hK : IsZero K
i'✝ : ι'
hi'✝ : ∀ (i : ι), e.f i ≠ i'✝
⊢ IsZero (K.X i'✝)
Module: Mathlib.Algebra.Homology.Embedding.IsSupported

Goal state before:
pp: case mp.right
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝¹ : Category.{v_1, u_3} C
inst✝ : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
hK : IsZero K
⊢ ∀ (i : ι), IsZero (K.X (e.f i))

Tactic:
intros

Goal state after:
case mp.right
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝¹ : Category.{v_1, u_3} C
inst✝ : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
hK : IsZero K
i✝ : ι
⊢ IsZero (K.X (e.f i✝))
Module: Mathlib.Algebra.Homology.Embedding.ExtendHomology

Goal state before:
pp: case pos
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
K L M : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝ : ∀ (j : ι), K.HasHomology j
j' : ι'
h : ∃ j, e.f j = j'
⊢ (K.extend e).HasHomology j'

Tactic:
obtain ⟨j, rfl⟩ := h
    infer_instance

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.ExtendHomology

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁴ : Category.{v_1, u_3} C
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroObject C
K L : HomologicalComplex C c
φ : K ⟶ L
e : c.Embedding c'
inst✝¹ : ∀ (j : ι), K.HasHomology j
inst✝ : ∀ (j : ι), L.HasHomology j
⊢ QuasiIso (extendMap φ e) ↔ QuasiIso φ

Tactic:
simp only [quasiIso_iff, ← fun j ↦ quasiIsoAt_extendMap_iff φ e (j := j) (hj' := rfl)]

Goal state after:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁴ : Category.{v_1, u_3} C
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroObject C
K L : HomologicalComplex C c
φ : K ⟶ L
e : c.Embedding c'
inst✝¹ : ∀ (j : ι), K.HasHomology j
inst✝ : ∀ (j : ι), L.HasHomology j
⊢ (∀ (i : ι'), QuasiIsoAt (extendMap φ e) i) ↔ ∀ (i : ι), QuasiIsoAt (extendMap φ e) (e.f i)
Module: Mathlib.Algebra.Homology.Embedding.AreComplementary

Goal state before:
pp: ι : Type u_1
ι₁ : Type u_2
ι₂ : Type u_3
c : ComplexShape ι
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
e₁ : c₁.Embedding c
e₂ : c₂.Embedding c
ac : e₁.AreComplementary e₂
i : ι
hi : ∀ (i₂ : ι₂), e₂.f i₂ ≠ i
⊢ ∃ i₁, i = e₁.f i₁

Tactic:
obtain ⟨i₁, rfl⟩ | ⟨i₂, rfl⟩ := ac.union i

Goal state after:
case inl
ι : Type u_1
ι₁ : Type u_2
ι₂ : Type u_3
c : ComplexShape ι
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
e₁ : c₁.Embedding c
e₂ : c₂.Embedding c
ac : e₁.AreComplementary e₂
i₁ : ι₁
hi : ∀ (i₂ : ι₂), e₂.f i₂ ≠ e₁.f i₁
⊢ ∃ i₁_1, e₁.f i₁ = e₁.f i₁_1

case inr
ι : Type u_1
ι₁ : Type u_2
ι₂ : Type u_3
c : ComplexShape ι
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
e₁ : c₁.Embedding c
e₂ : c₂.Embedding c
ac : e₁.AreComplementary e₂
i₂ : ι₂
hi : ∀ (i₂_1 : ι₂), e₂.f i₂_1 ≠ e₂.f i₂
⊢ ∃ i₁, e₂.f i₂ = e₁.f i₁
Module: Mathlib.Algebra.Homology.Embedding.AreComplementary

Goal state before:
pp: ι : Type u_1
ι₁ : Type u_2
ι₂ : Type u_3
c : ComplexShape ι
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
C : Type u_4
inst✝¹ : Category.{v_1, u_4} C
inst✝ : HasZeroMorphisms C
e₁ : c₁.Embedding c
e₂ : c₂.Embedding c
ac : e₁.AreComplementary e₂
K L : HomologicalComplex C c
φ : K ⟶ L
hK : K.IsStrictlySupportedOutside e₂
hL : L.IsStrictlySupportedOutside e₁
i : ι
⊢ φ.f i = HomologicalComplex.Hom.f 0 i

Tactic:
obtain ⟨i₁, rfl⟩ | ⟨i₂, rfl⟩ := ac.union i

Goal state after:
case inl
ι : Type u_1
ι₁ : Type u_2
ι₂ : Type u_3
c : ComplexShape ι
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
C : Type u_4
inst✝¹ : Category.{v_1, u_4} C
inst✝ : HasZeroMorphisms C
e₁ : c₁.Embedding c
e₂ : c₂.Embedding c
ac : e₁.AreComplementary e₂
K L : HomologicalComplex C c
φ : K ⟶ L
hK : K.IsStrictlySupportedOutside e₂
hL : L.IsStrictlySupportedOutside e₁
i₁ : ι₁
⊢ φ.f (e₁.f i₁) = HomologicalComplex.Hom.f 0 (e₁.f i₁)

case inr
ι : Type u_1
ι₁ : Type u_2
ι₂ : Type u_3
c : ComplexShape ι
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
C : Type u_4
inst✝¹ : Category.{v_1, u_4} C
inst✝ : HasZeroMorphisms C
e₁ : c₁.Embedding c
e₂ : c₂.Embedding c
ac : e₁.AreComplementary e₂
K L : HomologicalComplex C c
φ : K ⟶ L
hK : K.IsStrictlySupportedOutside e₂
hL : L.IsStrictlySupportedOutside e₁
i₂ : ι₂
⊢ φ.f (e₂.f i₂) = HomologicalComplex.Hom.f 0 (e₂.f i₂)
Module: Mathlib.Algebra.Homology.Embedding.AreComplementary

Goal state before:
pp: ι : Type u_1
ι₁ : Type u_2
ι₂ : Type u_3
c : ComplexShape ι
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
C : Type u_4
inst✝³ : Category.{v_1, u_4} C
inst✝² : Abelian C
K : HomologicalComplex C c
e₁ : c₁.Embedding c
e₂ : c₂.Embedding c
inst✝¹ : e₁.IsTruncLE
inst✝ : e₂.IsTruncGE
ac : e₁.AreComplementary e₂
i : ι
⊢ QuasiIsoAt (K.shortComplexTruncLEX₃ToTruncGE ac) i

Tactic:
obtain ⟨i₁, rfl⟩ | ⟨i₂, rfl⟩ := ac.union i

Goal state after:
case inl
ι : Type u_1
ι₁ : Type u_2
ι₂ : Type u_3
c : ComplexShape ι
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
C : Type u_4
inst✝³ : Category.{v_1, u_4} C
inst✝² : Abelian C
K : HomologicalComplex C c
e₁ : c₁.Embedding c
e₂ : c₂.Embedding c
inst✝¹ : e₁.IsTruncLE
inst✝ : e₂.IsTruncGE
ac : e₁.AreComplementary e₂
i₁ : ι₁
⊢ QuasiIsoAt (K.shortComplexTruncLEX₃ToTruncGE ac) (e₁.f i₁)

case inr
ι : Type u_1
ι₁ : Type u_2
ι₂ : Type u_3
c : ComplexShape ι
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
C : Type u_4
inst✝³ : Category.{v_1, u_4} C
inst✝² : Abelian C
K : HomologicalComplex C c
e₁ : c₁.Embedding c
e₂ : c₂.Embedding c
inst✝¹ : e₁.IsTruncLE
inst✝ : e₂.IsTruncGE
ac : e₁.AreComplementary e₂
i₂ : ι₂
⊢ QuasiIsoAt (K.shortComplexTruncLEX₃ToTruncGE ac) (e₂.f i₂)
Module: Mathlib.Algebra.Homology.SingleHomology

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A B : C
f : A ⟶ B
⊢ (singleObjCyclesSelfIso c j A).inv ≫ cyclesMap ((single C c j).map f) j = f ≫ (singleObjCyclesSelfIso c j B).inv

Tactic:
by
  rw [← cancel_epi (singleObjCyclesSelfIso c j A).hom, Iso.hom_inv_id_assoc,
    ← singleObjCyclesSelfIso_hom_naturality_assoc, Iso.hom_inv_id, comp_id]

Goal state after:
Module: Mathlib.CategoryTheory.Localization.HomEquiv

Goal state before:
pp: C : Type u_1
D₁ : Type u_5
D₂ : Type u_6
D₃ : Type u_7
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Category.{v_5, u_5} D₁
inst✝⁴ : Category.{v_6, u_6} D₂
inst✝³ : Category.{v_7, u_7} D₃
W : MorphismProperty C
L₁ : C ⥤ D₁
inst✝² : L₁.IsLocalization W
L₂ : C ⥤ D₂
inst✝¹ : L₂.IsLocalization W
L₃ : C ⥤ D₃
inst✝ : L₃.IsLocalization W
X Y : C
f : L₁.obj X ⟶ L₁.obj Y
⊢ (homEquiv W L₂ L₃) ((homEquiv W L₁ L₂) f) = (homEquiv W L₁ L₃) f

Tactic:
dsimp only [homEquiv_apply]

Goal state after:
C : Type u_1
D₁ : Type u_5
D₂ : Type u_6
D₃ : Type u_7
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Category.{v_5, u_5} D₁
inst✝⁴ : Category.{v_6, u_6} D₂
inst✝³ : Category.{v_7, u_7} D₃
W : MorphismProperty C
L₁ : C ⥤ D₁
inst✝² : L₁.IsLocalization W
L₂ : C ⥤ D₂
inst✝¹ : L₂.IsLocalization W
L₃ : C ⥤ D₃
inst✝ : L₃.IsLocalization W
X Y : C
f : L₁.obj X ⟶ L₁.obj Y
⊢ (LocalizerMorphism.id W).homMap L₂ L₃ ((LocalizerMorphism.id W).homMap L₁ L₂ f) =
    (LocalizerMorphism.id W).homMap L₁ L₃ f
Module: Mathlib.Algebra.Homology.SingleHomology

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A B : C
f : A ⟶ B
⊢ opcyclesMap ((single C c j).map f) j = (singleObjOpcyclesSelfIso c j A).inv ≫ f ≫ (singleObjOpcyclesSelfIso c j B).hom

Tactic:
← singleObjOpcyclesSelfIso_hom_naturality,

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A B : C
f : A ⟶ B
⊢ opcyclesMap ((single C c j).map f) j =
    (singleObjOpcyclesSelfIso c j A).inv ≫ (singleObjOpcyclesSelfIso c j A).hom ≫ opcyclesMap ((single C c j).map f) j