-- Mathlib/CategoryTheory/Localization/Triangulated.lean
lemma complete_distinguished_triangle_morphism (T₁ T₂ : Triangle D)
    (hT₁ : T₁ ∈ L.essImageDistTriang) (hT₂ : T₂ ∈ L.essImageDistTriang)
    (a : T₁.obj₁ ⟶ T₂.obj₁) (b : T₁.obj₂ ⟶ T₂.obj₂) (fac : T₁.mor₁ ≫ b = a ≫ T₂.mor₁) :
    ∃ c, T₁.mor₂ ≫ c = b ≫ T₂.mor₂ ∧ T₁.mor₃ ≫ a⟦1⟧' = c ≫ T₂.mor₃

Proof state:
case intro.intro.intro
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{u_4, u_1} C
inst✝¹⁰ : Category.{u_3, 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 :
  (MorphismProperty.RightFraction.mk α.s ⋯ T₂.mor₁).f ≫ β.s = (MorphismProperty.RightFraction.mk α.s ⋯ T₂.mor₁).s ≫ β.f
γ : W.LeftFraction T₁.obj₂ β.Y'
hγ : b ≫ L.map β.s = γ.map L ⋯
⊢ ∃ φ, φ.hom₁ = a ∧ φ.hom₂ = b

Tactic:
have := inverts L W β.s β.hs

Result:
case intro.intro.intro
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{u_4, u_1} C
inst✝¹⁰ : Category.{u_3, 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 :
  (MorphismProperty.RightFraction.mk α.s ⋯ T₂.mor₁).f ≫ β.s = (MorphismProperty.RightFraction.mk α.s ⋯ T₂.mor₁).s ≫ β.f
γ : W.LeftFraction T₁.obj₂ β.Y'
hγ : b ≫ L.map β.s = γ.map L ⋯
this : IsIso (L.map β.s)
⊢ ∃ φ, φ.hom₁ = a ∧ φ.hom₂ = b

Proof state:
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{u_4, u_1} C
inst✝¹⁰ : Category.{u_3, 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 ⋯
this✝ : IsIso (L.map β.s)
this : IsIso (L.map γ.s)
⊢ L.map (α.f ≫ β.f ≫ γ.s) = L.map (T₁.mor₁ ≫ γ.f)

Tactic:
rw [← cancel_mono (L.map β.s), assoc, assoc, hγ, ← cancel_mono (L.map γ.s),
  assoc, assoc, assoc, hα, MorphismProperty.LeftFraction.map_comp_map_s,
  ← Functor.map_comp] at fac

Result:
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{u_4, u_1} C
inst✝¹⁰ : Category.{u_3, 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
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'
fac : L.map (T₁.mor₁ ≫ γ.f) = α.map L ⋯ ≫ L.map T₂.mor₁ ≫ L.map β.s ≫ L.map γ.s
hγ : b ≫ L.map β.s = γ.map L ⋯
this✝ : IsIso (L.map β.s)
this : IsIso (L.map γ.s)
⊢ L.map (α.f ≫ β.f ≫ γ.s) = L.map (T₁.mor₁ ≫ γ.f)

Proof state:
case intro.intro.intro
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{u_4, u_1} C
inst✝¹⁰ : Category.{u_3, 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 :
  (MorphismProperty.RightFraction.mk α.s ⋯ T₂.mor₁).f ≫ β.s = (MorphismProperty.RightFraction.mk α.s ⋯ T₂.mor₁).s ≫ β.f
γ : W.LeftFraction T₁.obj₂ β.Y'
hγ : b ≫ L.map β.s = γ.map L ⋯
this✝ : IsIso (L.map β.s)
this : IsIso (L.map γ.s)
⊢ ∃ φ, φ.hom₁ = a ∧ φ.hom₂ = b

Tactic:
dsimp at hβ

Result:
case intro.intro.intro
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{u_4, u_1} C
inst✝¹⁰ : Category.{u_3, 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 ⋯
this✝ : IsIso (L.map β.s)
this : IsIso (L.map γ.s)
⊢ ∃ φ, φ.hom₁ = a ∧ φ.hom₂ = b