mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j ↦ (T j).obj₁
inst✝² : HasProduct fun j ↦ (T j).obj₂
inst✝¹ : HasProduct fun j ↦ (T j).obj₃
inst✝ : HasProduct fun j ↦ (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j ↦ (T j).obj₁) ⟶ ∏ᶜ fun j ↦ (T j).obj₂ := Limits.Pi.map fun j ↦ (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j ↦ (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j ↦ (T j).obj₁) j) (Pi.π (fun j ↦ (T j).obj₂) j) ⋯
φ' : T' ⟶ productTriangle T := productTriangle.lift T φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
this✝ : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A : C
a : A ⟶ (productTriangle T).obj₃
a' : A ⟶ Z
ha' : a' ≫ T'.mor₃ = a ≫ (productTriangle T).mor₃
ha'' : ∀ (j : J), ∃ g, (a - a' ≫ φ'.hom₃) ≫ Pi.π (fun j ↦ (T j).obj₃) j = g ≫ (T j).mor₂
b : (j : J) → A ⟶ (T j).obj₂ := fun j ↦ ⋯.choose
hb : ∀ (j : J), (a - a' ≫ φ'.hom₃) ≫ Pi.π (fun j ↦ (T j).obj₃) j = b j ≫ (T j).mor₂
hb' : a - a' ≫ φ'.hom₃ = Pi.lift b ≫ (productTriangle T).mor₂
⊢ a' ≫ φ'.hom₃ + Pi.lift b ≫ (productTriangle T).mor₂ = a

Tactic:
← hb',

Goal state after:
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j ↦ (T j).obj₁
inst✝² : HasProduct fun j ↦ (T j).obj₂
inst✝¹ : HasProduct fun j ↦ (T j).obj₃
inst✝ : HasProduct fun j ↦ (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j ↦ (T j).obj₁) ⟶ ∏ᶜ fun j ↦ (T j).obj₂ := Limits.Pi.map fun j ↦ (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j ↦ (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j ↦ (T j).obj₁) j) (Pi.π (fun j ↦ (T j).obj₂) j) ⋯
φ' : T' ⟶ productTriangle T := productTriangle.lift T φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
this✝ : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A : C
a : A ⟶ (productTriangle T).obj₃
a' : A ⟶ Z
ha' : a' ≫ T'.mor₃ = a ≫ (productTriangle T).mor₃
ha'' : ∀ (j : J), ∃ g, (a - a' ≫ φ'.hom₃) ≫ Pi.π (fun j ↦ (T j).obj₃) j = g ≫ (T j).mor₂
b : (j : J) → A ⟶ (T j).obj₂ := fun j ↦ ⋯.choose
hb : ∀ (j : J), (a - a' ≫ φ'.hom₃) ≫ Pi.π (fun j ↦ (T j).obj₃) j = b j ≫ (T j).mor₂
hb' : a - a' ≫ φ'.hom₃ = Pi.lift b ≫ (productTriangle T).mor₂
⊢ a' ≫ φ'.hom₃ + (a - a' ≫ φ'.hom₃) = a
Module: Mathlib.CategoryTheory.PathCategory.Basic

Goal state before:
pp: V : Type u₁
inst✝¹ : Quiver V
C : Type ?u.6
inst✝ : Category.{v_1, ?u.6} C
φ : V ⥤q C
X✝ Y✝ Z✝ : Paths V
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) (f ≫ g) =
    Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) f ≫
      Quiver.Path.rec (𝟙 (φ.obj Y✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) g

Tactic:
induction g with
    | nil =>
      rw [Category.comp_id]
      rfl
    | cons g' p ih =>
      have : f ≫ Quiver.Path.cons g' p = (f ≫ g').cons p := by apply Quiver.Path.comp_cons
      rw [this]
      simp only at ih ⊢
      rw [ih, Category.assoc]

Goal state after:
Module: Mathlib.CategoryTheory.PathCategory.Basic

Goal state before:
pp: case h_map.nil
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{v_1, u_1} C
F G : Paths V ⥤ C
h_obj : F.obj = G.obj
h : ∀ (a b : V) (e : a ⟶ b), F.map e.toPath = eqToHom ⋯ ≫ G.map e.toPath ≫ eqToHom ⋯
X Y : Paths V
⊢ 𝟙 (F.obj X) = eqToHom ⋯ ≫ G.map Quiver.Path.nil ≫ eqToHom ⋯

Tactic:
G.map_id,

Goal state after:
case h_map.nil
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{v_1, u_1} C
F G : Paths V ⥤ C
h_obj : F.obj = G.obj
h : ∀ (a b : V) (e : a ⟶ b), F.map e.toPath = eqToHom ⋯ ≫ G.map e.toPath ≫ eqToHom ⋯
X Y : Paths V
⊢ 𝟙 (F.obj X) = eqToHom ⋯ ≫ 𝟙 (G.obj X) ≫ eqToHom ⋯
Module: Mathlib.CategoryTheory.Localization.Construction

Goal state before:
pp: case refine_2
C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
G₁ G₂ : W.Localization ⥤ D
h : W.Q ⋙ G₁ = W.Q ⋙ G₂
⊢ ∀ (a b : LocQuiver W) (e : a ⟶ b),
    (Quotient.functor (relations W) ⋙ G₁).map e.toPath =
      eqToHom ⋯ ≫ (Quotient.functor (relations W) ⋙ G₂).map e.toPath ≫ eqToHom ⋯

Tactic:
rintro ⟨X⟩ ⟨Y⟩ (f | ⟨w, hw⟩)

Goal state after:
case refine_2.inl
C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
G₁ G₂ : W.Localization ⥤ D
h : W.Q ⋙ G₁ = W.Q ⋙ G₂
X Y : C
f : { obj := X }.obj ⟶ { obj := Y }.obj
⊢ (Quotient.functor (relations W) ⋙ G₁).map (Quiver.Hom.toPath (Sum.inl f)) =
    eqToHom ⋯ ≫ (Quotient.functor (relations W) ⋙ G₂).map (Quiver.Hom.toPath (Sum.inl f)) ≫ eqToHom ⋯

case refine_2.inr
C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
G₁ G₂ : W.Localization ⥤ D
h : W.Q ⋙ G₁ = W.Q ⋙ G₂
X Y : C
w : { obj := Y }.obj ⟶ { obj := X }.obj
hw : W w
⊢ (Quotient.functor (relations W) ⋙ G₁).map (Quiver.Hom.toPath (Sum.inr ⟨w, hw⟩)) =
    eqToHom ⋯ ≫ (Quotient.functor (relations W) ⋙ G₂).map (Quiver.Hom.toPath (Sum.inr ⟨w, hw⟩)) ≫ eqToHom ⋯
Module: Mathlib.CategoryTheory.Localization.Predicate

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
W : MorphismProperty C
X Y : C
f : X ⟶ Y
hf : W f
⊢ wIso f hf = isoOfHom W.Q W f hf

Tactic:
ext; rfl

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

Goal state before:
pp: case refine_5
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : IsTriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Z₁₂ ⟶ X₁
w₁₂ : X₂ ⟶ (shiftFunctor C 1).obj Z₁₂
h₁₂ : Triangle.mk v₁₂ u₁₂ w₁₂ ∈ distinguishedTriangles
v₂₃ : Z₂₃ ⟶ X₂
w₂₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₂₃
h₂₃ : Triangle.mk v₂₃ u₂₃ w₂₃ ∈ distinguishedTriangles
v₁₃ : Z₁₃ ⟶ X₁
w₁₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₁₃
h₁₃ : Triangle.mk v₁₃ u₁₃ w₁₃ ∈ distinguishedTriangles
o : Octahedron comm ⋯ ⋯ ⋯ := someOctahedron comm ⋯ ⋯ ⋯
m₁ : Z₁₂ ⟶ Z₁₃ := (shiftShiftNeg Z₁₂ 1).inv ≫ (shiftFunctor C (-1)).map o.m₁ ≫ (shiftShiftNeg Z₁₃ 1).hom
m₃ : Z₁₃ ⟶ Z₂₃ := (shiftShiftNeg Z₁₃ 1).inv ≫ (shiftFunctor C (-1)).map o.m₃ ≫ (shiftShiftNeg Z₂₃ 1).hom
eq₁ : w₁₂ ≫ o.m₁ = u₂₃ ≫ w₁₃
eq₂ : o.m₁ ≫ (shiftFunctor C 1).map v₁₃ = (shiftFunctor C 1).map v₁₂
eq₃ : w₁₃ ≫ o.m₃ = w₂₃
eq₄ : (shiftFunctor C 1).map v₁₃ ≫ (shiftFunctor C 1).map u₁₂ = o.m₃ ≫ (shiftFunctor C 1).map v₂₃
⊢ Triangle.mk m₁ m₃ (v₂₃ ≫ w₁₂) ∈ distinguishedTriangles

Tactic:
· apply isomorphic_distinguished _ ((Triangle.shift_distinguished_iff _ (-1 : ℤ)).mpr o.mem)
    refine Triangle.isoMk _ _ (shiftShiftNeg Z₁₂ (1 : ℤ)).symm
      (-(shiftShiftNeg Z₁₃ (1 : ℤ)).symm) (shiftShiftNeg Z₂₃ (1 : ℤ)).symm (comm₃ := ?_)
    dsimp
    simp only [Int.reduceNeg, assoc, Int.negOnePow_neg, Int.negOnePow_one, neg_comp,
      Functor.map_neg, Functor.map_comp, smul_neg, Units.neg_smul, one_smul, neg_neg]
    rw [shift_shift_neg', shift_shift_neg', shift_shiftFunctorCompIsoId_inv_app,
      shiftFunctorComm_hom_app_of_add_eq_zero _ _ (Int.add_right_neg 1)]
    simp

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

Goal state before:
pp: C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
F G : W.Localization ⥤ D
τ₁ τ₂ : F ⟶ G
h : 𝟙 W.Q ◫ τ₁ = 𝟙 W.Q ◫ τ₂
X : W.Localization
eq : W.Q.obj X.as.obj = X
⊢ τ₁.app X = τ₂.app X

Tactic:
rw [← eq, ← NatTrans.id_hcomp_app, ← NatTrans.id_hcomp_app, h]

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

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
L : C ⥤ D
W : MorphismProperty C
inst✝ : L.IsLocalization W
X Y : Cᵒᵖ
w : X ⟶ Y
hw : W.op w
⊢ (isoOfHom L.op W.op w hw).unop = isoOfHom L W w.unop hw

Tactic:
ext; rfl

Goal state after:
Module: Mathlib.SetTheory.Cardinal.HasCardinalLT

Goal state before:
pp: X : Type v
κ : Cardinal.{w}
⊢ HasCardinalLT X (Cardinal.lift.{u, w} κ) ↔ HasCardinalLT X κ

Tactic:
by
  simp [HasCardinalLT, ← (Cardinal.lift_strictMono.{max v w, max u}).lt_iff_lt]

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

Goal state before:
pp: case e_a
C : Type u_1
inst✝² : Category.{v_1, u_1} C
W : MorphismProperty C
inst✝¹ : W.HasLeftCalculusOfFractions
E : Type u_3
inst✝ : Category.{v_3, u_3} E
F₁ F₂ : Localization W ⥤ E
h : Q W ⋙ F₁ = Q W ⋙ F₂
X Y : C
f : W.LeftFraction X Y
this : IsIso ((Q W).map f.s)
⊢ eqToHom ⋯ ≫ F₁.map (Qinv f.s ⋯) = F₂.map (Qinv f.s ⋯) ≫ eqToHom ⋯

Tactic:
rw [← cancel_epi (F₂.map ((Q W).map f.s)), ← F₂.map_comp_assoc,
      Qiso_hom_inv_id, Functor.map_id, id_comp]

Goal state after:
case e_a
C : Type u_1
inst✝² : Category.{v_1, u_1} C
W : MorphismProperty C
inst✝¹ : W.HasLeftCalculusOfFractions
E : Type u_3
inst✝ : Category.{v_3, u_3} E
F₁ F₂ : Localization W ⥤ E
h : Q W ⋙ F₁ = Q W ⋙ F₂
X Y : C
f : W.LeftFraction X Y
this : IsIso ((Q W).map f.s)
⊢ F₂.map ((Q W).map f.s) ≫ eqToHom ⋯ ≫ F₁.map (Qinv f.s ⋯) = eqToHom ⋯
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 : C
f g : W.LeftFraction X Y
⊢ f.map (Q W) ⋯ = g.map (Q W) ⋯ ↔ LeftFractionRel f g

Tactic:
by
  simp only [← Hom.map_mk _ (Q W)]
  constructor
  · intro h
    rw [← homMk_eq_iff_leftFractionRel, homMk_eq, homMk_eq]
    exact h
  · intro h
    congr 1
    exact Quot.sound h

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
fac✝ : L.map T₁.mor₁ ≫ b = a ≫ L.map T₂.mor₁
α : W.LeftFraction T₁.obj₁ T₂.obj₁
hα : a = α.map L ⋯
β : W.LeftFraction α.Y' T₂.obj₂
hβ : T₂.mor₁ ≫ β.s = α.s ≫ β.f
γ : W.LeftFraction T₁.obj₂ β.Y'
hγ : b ≫ L.map β.s = γ.map L ⋯
Z₂ : C
σ : γ.Y' ⟶ Z₂
hσ : W σ
fac : α.f ≫ β.f ≫ γ.s ≫ σ = T₁.mor₁ ≫ γ.f ≫ σ
Y₃ : C
g : Z₂ ⟶ Y₃
h : Y₃ ⟶ (shiftFunctor C 1).obj α.Y'
hT₃ : Triangle.mk (β.f ≫ γ.s ≫ σ) g h ∈ distinguishedTriangles
T₃ : Triangle C := Triangle.mk (β.f ≫ γ.s ≫ σ) g h
⊢ ∃ φ, φ.hom₁ = a ∧ φ.hom₂ = b

Tactic:
change T₃ ∈ distTriang C at hT₃

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
α : W.LeftFraction T₁.obj₁ T₂.obj₁
hα : a = α.map L ⋯
β : W.LeftFraction α.Y' T₂.obj₂
hβ : T₂.mor₁ ≫ β.s = α.s ≫ β.f
γ : W.LeftFraction T₁.obj₂ β.Y'
hγ : b ≫ L.map β.s = γ.map L ⋯
Z₂ : C
σ : γ.Y' ⟶ Z₂
hσ : W σ
fac : α.f ≫ β.f ≫ γ.s ≫ σ = T₁.mor₁ ≫ γ.f ≫ σ
Y₃ : C
g : Z₂ ⟶ Y₃
h : Y₃ ⟶ (shiftFunctor C 1).obj α.Y'
T₃ : Triangle C := Triangle.mk (β.f ≫ γ.s ≫ σ) g h
hT₃ : T₃ ∈ distinguishedTriangles
⊢ ∃ 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
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
E : MorphismProperty.LeftFraction.Localization W ≌ D := uniq (MorphismProperty.LeftFraction.Localization.Q W) L W
e : MorphismProperty.LeftFraction.Localization.Q W ⋙ E.functor ≅ L :=
  compUniqFunctor (MorphismProperty.LeftFraction.Localization.Q W) L W
f' : E.functor.obj X ⟶ E.functor.obj Y
⊢ ∃ φ, e.inv.app X ≫ f' ≫ e.hom.app Y = φ.map L ⋯

Tactic:
obtain ⟨g, rfl⟩ := E.functor.map_surjective f'

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
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
E : MorphismProperty.LeftFraction.Localization W ≌ D := uniq (MorphismProperty.LeftFraction.Localization.Q W) L W
e : MorphismProperty.LeftFraction.Localization.Q W ⋙ E.functor ≅ L :=
  compUniqFunctor (MorphismProperty.LeftFraction.Localization.Q W) L W
g : X ⟶ Y
⊢ ∃ φ, e.inv.app X ≫ E.functor.map g ≫ e.hom.app Y = φ.map L ⋯
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
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasRightCalculusOfFractions
X Y : C
f : L.obj X ⟶ L.obj Y
⊢ ∃ φ, f = φ.map L ⋯

Tactic:
obtain ⟨φ, eq⟩ := Localization.exists_leftFraction L.op W.op f.op
  refine ⟨φ.unop, Quiver.Hom.op_inj ?_⟩
  rw [eq, MorphismProperty.RightFraction.op_map]
  rfl

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.IsTriangulatedClosed₁
Y Z : C
b : Y ⟶ Z
hY : P Y
hZ : P Z
⊢ ∃ X, ∃ (_ : P X), ∃ a c, Triangle.mk a b c ∈ distinguishedTriangles

Tactic:
by
  obtain ⟨X, a, c, h⟩ := distinguished_cocone_triangle₁ b
  obtain ⟨X', hX', ⟨e⟩⟩ := P.ext_of_isTriangulatedClosed₁' _ h hY hZ
  exact ⟨X', hX', e.inv ≫ a, c ≫ e.hom⟦1⟧', isomorphic_distinguished _ h _
    (Triangle.isoMk _ _ e.symm (Iso.refl _) (Iso.refl _))⟩

Goal state after:
Module: Mathlib.CategoryTheory.Shift.InducedShiftSequence

Goal state before:
pp: C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : Category.{v_2, u_2} D
inst✝⁷ : Category.{v_3, u_3} A
L : C ⥤ D
F : D ⥤ A
G : C ⥤ A
e : L ⋙ F ≅ G
M : Type u_4
inst✝⁶ : AddMonoid M
inst✝⁵ : HasShift C M
inst✝⁴ : G.ShiftSequence M
F' : M → D ⥤ A
e' : (m : M) → L ⋙ F' m ≅ G.shift m
inst✝³ : ((whiskeringLeft C D A).obj L).Full
inst✝² : ((whiskeringLeft C D A).obj L).Faithful
inst✝¹ : HasShift D M
inst✝ : L.CommShift M
n : M
X Y : C
f : X ⟶ (shiftFunctor C n).obj Y
a a' : M
h : n + a = a'
⊢ F.shiftMap (L.map f ≫ (commShiftIso L n).hom.app Y) a a' h =
    (e' a).hom.app X ≫ G.shiftMap f a a' h ≫ (e' a').inv.app Y

Tactic:
by
  dsimp [shiftMap]
  rw [Functor.map_comp, induced_shiftIso_hom_app_obj, assoc, assoc]
  nth_rw 2 [← Functor.map_comp_assoc]
  simp only [comp_obj, Iso.hom_inv_id_app, map_id, id_comp]
  rw [← NatTrans.naturality_assoc]
  rfl

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Retract

Goal state before:
pp: case refine_2
C : Type u
inst✝ : Category.{v, u} C
X Y Z W : C
f : X ⟶ Y
g : Z ⟶ W
h : RetractArrow f g
x✝ : IsIso g
⊢ (Arrow.Hom.right h.i ≫ inv g ≫ Arrow.Hom.left h.r) ≫ f = 𝟙 Y

Tactic:
rw [Category.assoc, Category.assoc, h.r_w, IsIso.inv_hom_id_assoc, h.retract_right]

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

Goal state before:
pp: C : Type u_1
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : HasZeroObject C
inst✝¹³ : HasShift C ℤ
inst✝¹² : Preadditive C
inst✝¹¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹⁰ : Pretriangulated C
D : Type u_2
inst✝⁹ : Category.{v_2, u_2} D
inst✝⁸ : Preadditive D
inst✝⁷ : HasZeroObject D
inst✝⁶ : HasShift D ℤ
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁴ : Pretriangulated D
E : Type u_3
inst✝³ : Category.{v_3, u_3} E
inst✝² : HasShift E ℤ
P Q R : ObjectProperty C
inst✝¹ : P.IsStableUnderShift ℤ
inst✝ : Q.IsStableUnderShift ℤ
a : ℤ
⊢ P.extensionProduct Q ≤ (P.extensionProduct Q).shift a

Tactic:
intro X ⟨Y, Z, f, g, h, hT, hP, hQ⟩
    refine ⟨_, _, _, _, _, Triangle.shift_distinguished _ hT a, ?_, ?_⟩
    all_goals apply IsStableUnderShiftBy.le_shift; assumption

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

Goal state before:
pp: ι : Type u_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
c : ComplexShape ι
K L M K' L' : HomologicalComplex C c
inst✝ : CategoryWithHomology C
X✝ Y✝ Z✝ W✝ : HomologicalComplex C c
f✝ : X✝ ⟶ Y✝
g✝ : Z✝ ⟶ W✝
h : RetractArrow f✝ g✝
hg : quasiIso C c g✝
⊢ quasiIso C c f✝

Tactic:
by
    rw [mem_quasiIso_iff] at hg ⊢
    exact quasiIso_of_retractArrow h

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.HomologySequence

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
K L : CochainComplex C ℤ
n : ℤ
f : K ⟶ (shiftFunctor (CochainComplex C ℤ) n).obj L
a a' : ℤ
h : n + a = a'
⊢ ((homologyFunctor C 0).shift a).map ((quotientCompQhIso C).inv.app K) ≫
      (((homologyFunctor C a).map ((quotientCompQhIso C).hom.app K) ≫
            (homologyFunctorFactors C a).hom.app K ≫
              (HomotopyCategory.homologyFunctorFactors C (ComplexShape.up ℤ) a).inv.app K) ≫
          ((HomotopyCategory.homologyFunctorFactors C (ComplexShape.up ℤ) a).hom.app K ≫
              (HomologicalComplex.homologyFunctor C (ComplexShape.up ℤ) 0).shiftMap f a a' ⋯ ≫
                (HomotopyCategory.homologyFunctorFactors C (ComplexShape.up ℤ) a').inv.app L) ≫
            (HomotopyCategory.homologyFunctorFactors C (ComplexShape.up ℤ) a').hom.app L ≫
              (homologyFunctorFactors C a').inv.app L ≫ (homologyFunctor C a').map ((quotientCompQhIso C).inv.app L)) ≫
        ((homologyFunctor C 0).shift a').map ((quotientCompQhIso C).hom.app L) =
    (homologyFunctorFactors C a).hom.app K ≫
      (HomologicalComplex.homologyFunctor C (ComplexShape.up ℤ) 0).shiftMap f a a' h ≫
        (homologyFunctorFactors C a').inv.app L

Tactic:
simp [shift_homologyFunctor, ← Functor.map_comp, ← Functor.map_comp_assoc]

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

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

Tactic:
by_contra

Goal state after:
case mp
p n : ℤ
h : ∀ (i : ℕ), (embeddingUpIntLE p).f i ≠ n
this : ¬p < n
⊢ False
Module: Mathlib.Algebra.Homology.Embedding.Basic

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

Tactic:
by_contra

Goal state after:
case mp
p n : ℤ
h : ∀ (i : ℕ), (embeddingUpIntGE p).f i ≠ n
this : ¬n < p
⊢ False
Module: Mathlib.Algebra.Homology.Opposite

Goal state before:
pp: ι : Type u_1
V : Type u_2
inst✝³ : Category.{v_1, u_2} V
c : ComplexShape ι
inst✝² : HasZeroMorphisms V
K L : HomologicalComplex Vᵒᵖ c
φ : K ⟶ L
i : ι
inst✝¹ : K.HasHomology i
inst✝ : L.HasHomology i
⊢ QuasiIsoAt ((unopFunctor V c).map φ.op) i ↔ QuasiIsoAt φ i

Tactic:
by
  rw [← quasiIsoAt_opFunctor_map_iff]
  rfl

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

Goal state before:
pp: case inr.inr.a
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝² : Category.{v_1, u_3} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
K L M : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
i' j' : ι'
h : ¬c'.Rel i' j'
i : ι
hi : e.r i' = some i
j : ι
hj : e.r j' = some j
⊢ ¬c.Rel i j

Tactic:
obtain rfl := e.f_eq_of_r_eq_some hi

Goal state after:
case inr.inr.a
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝² : Category.{v_1, u_3} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
K L M : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
j' : ι'
i j : ι
hj : e.r j' = some j
h : ¬c'.Rel (e.f i) j'
hi : e.r (e.f i) = some i
⊢ ¬c.Rel i j
Module: Mathlib.Algebra.Homology.HomotopyCategory.Triangulated

Goal state before:
pp: case refine_3
C : Type u_1
inst✝³ : Category.{v, u_1} C
inst✝² : Preadditive C
inst✝¹ : HasBinaryBiproducts C
X₁✝ X₂✝ X₃✝ : CochainComplex C ℤ
f : X₁✝ ⟶ X₂✝
g : X₂✝ ⟶ X₃✝
inst✝ : HasZeroObject C
X₁ X₂ X₃ : CochainComplex C ℤ
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
α : mappingCone.triangle u₁₂ ⟶ mappingCone.triangle (u₁₂ ≫ u₂₃) := mappingCone.triangleMap u₁₂ (u₁₂ ≫ u₂₃) (𝟙 X₁) u₂₃ ⋯
β : mappingCone.triangle (u₁₂ ≫ u₂₃) ⟶ mappingCone.triangle u₂₃ := mappingCone.triangleMap (u₁₂ ≫ u₂₃) u₂₃ u₁₂ (𝟙 X₃) ⋯
⊢ (quotient C (ComplexShape.up ℤ)).map (mappingCone.triangle (u₁₂ ≫ u₂₃)).mor₂ ≫
      (quotient C (ComplexShape.up ℤ)).map β.hom₃ =
    (quotient C (ComplexShape.up ℤ)).map (mappingCone.triangle u₂₃).mor₂

Tactic:
exact ((quotient _ _).mapTriangle.map β).comm₂.trans (by dsimp [β]; simp)

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

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✝² : Category.{v_1, u_3} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
K L M : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
i' j' : ι'
x✝ : c'.Rel i' j'
i : ι
hi : e.f i = i'
j : ι
hj : e.f j = j'
⊢ ((extend.XIso K ⋯).hom ≫ φ.f i ≫ (extend.XIso L ⋯).inv) ≫
      (L.extendXIso e hi).hom ≫ L.d i j ≫ (L.extendXIso e hj).inv =
    ((K.extendXIso e hi).hom ≫ K.d i j ≫ (K.extendXIso e hj).inv) ≫
      (extend.XIso K ⋯).hom ≫ φ.f j ≫ (extend.XIso L ⋯).inv

Tactic:
simp only [extendXIso, assoc, Iso.inv_hom_id_assoc, Hom.comm_assoc]

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

Goal state before:
pp: case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁵ : Category.{v_1, u_3} C
inst✝⁴ : HasZeroMorphisms C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝³ : e.IsTruncGE
inst✝² : ∀ (i' : ι'), K.HasHomology i'
inst✝¹ : ∀ (i' : ι'), L.HasHomology i'
inst✝ : ∀ (i' : ι'), M.HasHomology i'
i : ι
hi : ¬e.BoundaryGE i
⊢ Epi ((K.restrictionToTruncGE' e).f i)

Tactic:
· have := K.isIso_restrictionToTruncGE' e i hi
    infer_instance

Goal state after:
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
⊢ ((single C c j).obj A).iCycles j ≫ ((single C c j).map f).f j ≫ ((single C c j).obj B).pOpcycles j =
    (singleObjCyclesSelfIso c j A).hom ≫ f ≫ (singleObjOpcyclesSelfIso c j B).hom

Tactic:
single_map_f_self,

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
⊢ ((single C c j).obj A).iCycles j ≫
      ((singleObjXSelf c j A).hom ≫ f ≫ (singleObjXSelf c j B).inv) ≫ ((single C c j).obj B).pOpcycles j =
    (singleObjCyclesSelfIso c j A).hom ≫ f ≫ (singleObjOpcyclesSelfIso c j B).hom