-- Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean
lemma binaryBiproductTriangle_distinguished (X₁ X₂ : C) :
    binaryBiproductTriangle X₁ X₂ ∈ distTriang C

Proof state:
case intro.intro.intro.intro.intro
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
X₁ X₂ Y : C
g : X₁ ⟶ Y
h : Y ⟶ X₂
mem : Triangle.mk g h 0 ∈ distinguishedTriangles
e : (Triangle.mk g h 0).obj₂ ≅ (Triangle.mk g h 0).obj₁ ⊞ (Triangle.mk g h 0).obj₃
he₁ : (Triangle.mk g h 0).mor₁ ≫ e.hom = biprod.inl
he₂ : (Triangle.mk g h 0).mor₂ = e.hom ≫ biprod.snd
⊢ binaryBiproductTriangle X₁ X₂ ∈ distinguishedTriangles

Tactic:
dsimp at he₁ he₂

Result:
case intro.intro.intro.intro.intro
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
X₁ X₂ Y : C
g : X₁ ⟶ Y
h : Y ⟶ X₂
mem : Triangle.mk g h 0 ∈ distinguishedTriangles
e : (Triangle.mk g h 0).obj₂ ≅ (Triangle.mk g h 0).obj₁ ⊞ (Triangle.mk g h 0).obj₃
he₁ : g ≫ e.hom = biprod.inl
he₂ : h = e.hom ≫ biprod.snd
⊢ binaryBiproductTriangle X₁ X₂ ∈ distinguishedTriangles

Proof state:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
X₁ X₂ Y : C
g : X₁ ⟶ Y
h : Y ⟶ X₂
mem : Triangle.mk g h 0 ∈ distinguishedTriangles
e : (Triangle.mk g h 0).obj₂ ≅ (Triangle.mk g h 0).obj₁ ⊞ (Triangle.mk g h 0).obj₃
he₁ : g ≫ e.hom = biprod.inl
he₂ : h = e.hom ≫ biprod.snd
⊢ (Triangle.mk g h 0).mor₁ ≫ e.hom = (Iso.refl (Triangle.mk g h 0).obj₁).hom ≫ (binaryBiproductTriangle X₁ X₂).mor₁

Tactic:
aesop_cat

Result:
no goals

Proof state:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
X₁ X₂ Y : C
g : X₁ ⟶ Y
h : Y ⟶ X₂
mem : Triangle.mk g h 0 ∈ distinguishedTriangles
e : (Triangle.mk g h 0).obj₂ ≅ (Triangle.mk g h 0).obj₁ ⊞ (Triangle.mk g h 0).obj₃
he₁ : g ≫ e.hom = biprod.inl
he₂ : h = e.hom ≫ biprod.snd
⊢ (Triangle.mk g h 0).mor₃ ≫ (shiftFunctor C 1).map (Iso.refl (Triangle.mk g h 0).obj₁).hom =
    (Iso.refl (Triangle.mk g h 0).obj₃).hom ≫ (binaryBiproductTriangle X₁ X₂).mor₃

Tactic:
simp

Result:
no goals

Proof state:
case intro.intro.intro.intro.intro
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
X₁ X₂ Y : C
g : X₁ ⟶ Y
h : Y ⟶ X₂
mem : Triangle.mk g h 0 ∈ distinguishedTriangles
e : (Triangle.mk g h 0).obj₂ ≅ (Triangle.mk g h 0).obj₁ ⊞ (Triangle.mk g h 0).obj₃
he₁ : g ≫ e.hom = biprod.inl
he₂ : h = e.hom ≫ biprod.snd
⊢ binaryBiproductTriangle X₁ X₂ ∈ distinguishedTriangles

Tactic:
refine isomorphic_distinguished _ mem _ (Iso.symm ?_)

Result:
case intro.intro.intro.intro.intro
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
X₁ X₂ Y : C
g : X₁ ⟶ Y
h : Y ⟶ X₂
mem : Triangle.mk g h 0 ∈ distinguishedTriangles
e : (Triangle.mk g h 0).obj₂ ≅ (Triangle.mk g h 0).obj₁ ⊞ (Triangle.mk g h 0).obj₃
he₁ : g ≫ e.hom = biprod.inl
he₂ : h = e.hom ≫ biprod.snd
⊢ Triangle.mk g h 0 ≅ binaryBiproductTriangle X₁ X₂

Proof state:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
X₁ X₂ : C
⊢ binaryBiproductTriangle X₁ X₂ ∈ distinguishedTriangles

Tactic:
obtain ⟨Y, g, h, mem⟩ := distinguished_cocone_triangle₂ (0 : X₂ ⟶ X₁⟦(1 : ℤ)⟧)

Result:
case intro.intro.intro
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
X₁ X₂ Y : C
g : X₁ ⟶ Y
h : Y ⟶ X₂
mem : Triangle.mk g h 0 ∈ distinguishedTriangles
⊢ binaryBiproductTriangle X₁ X₂ ∈ distinguishedTriangles

Proof state:
case intro.intro.intro.intro.intro
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
X₁ X₂ Y : C
g : X₁ ⟶ Y
h : Y ⟶ X₂
mem : Triangle.mk g h 0 ∈ distinguishedTriangles
e : (Triangle.mk g h 0).obj₂ ≅ (Triangle.mk g h 0).obj₁ ⊞ (Triangle.mk g h 0).obj₃
he₁ : g ≫ e.hom = biprod.inl
he₂ : h = e.hom ≫ biprod.snd
⊢ Triangle.mk g h 0 ≅ binaryBiproductTriangle X₁ X₂

Tactic:
refine Triangle.isoMk _ _ (Iso.refl _) e (Iso.refl _)
  (by aesop_cat) (by aesop_cat) (by simp)

Result:
no goals

Proof state:
case intro.intro.intro
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
X₁ X₂ Y : C
g : X₁ ⟶ Y
h : Y ⟶ X₂
mem : Triangle.mk g h 0 ∈ distinguishedTriangles
⊢ binaryBiproductTriangle X₁ X₂ ∈ distinguishedTriangles

Tactic:
obtain ⟨e, ⟨he₁, he₂⟩⟩ := exists_iso_binaryBiproduct_of_distTriang _ mem rfl

Result:
case intro.intro.intro.intro.intro
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
X₁ X₂ Y : C
g : X₁ ⟶ Y
h : Y ⟶ X₂
mem : Triangle.mk g h 0 ∈ distinguishedTriangles
e : (Triangle.mk g h 0).obj₂ ≅ (Triangle.mk g h 0).obj₁ ⊞ (Triangle.mk g h 0).obj₃
he₁ : (Triangle.mk g h 0).mor₁ ≫ e.hom = biprod.inl
he₂ : (Triangle.mk g h 0).mor₂ = e.hom ≫ biprod.snd
⊢ binaryBiproductTriangle X₁ X₂ ∈ distinguishedTriangles

Proof state:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
X₁ X₂ Y : C
g : X₁ ⟶ Y
h : Y ⟶ X₂
mem : Triangle.mk g h 0 ∈ distinguishedTriangles
e : (Triangle.mk g h 0).obj₂ ≅ (Triangle.mk g h 0).obj₁ ⊞ (Triangle.mk g h 0).obj₃
he₁ : g ≫ e.hom = biprod.inl
he₂ : h = e.hom ≫ biprod.snd
⊢ (Triangle.mk g h 0).mor₂ ≫ (Iso.refl (Triangle.mk g h 0).obj₃).hom = e.hom ≫ (binaryBiproductTriangle X₁ X₂).mor₂

Tactic:
aesop_cat

Result:
no goals