l3lab/ntp-mathlib huggingface dataset:
lemma exists_iso_binaryBiproduct_of_distTriang (T : Triangle C) (hT : T ∈ distTriang C)
    (zero : T.mor₃ = 0) :
    ∃ (e : T.obj₂ ≅ T.obj₁ ⊞ T.obj₃), T.mor₁ ≫ e.hom = biprod.inl ∧
      T.mor₂ = e.hom ≫ biprod.snd := by
  have := T.epi₂ hT zero
  have := isSplitEpi_of_epi T.mor₂
  obtain ⟨fst, hfst⟩ := T.coyoneda_exact₂ hT (𝟙 T.obj₂ - T.mor₂ ≫ section_ T.mor₂) (by simp)
  let d := binaryBiproductData _ hT zero (section_ T.mor₂) (by simp) fst
    (by simp only [← hfst, sub_add_cancel])
  

Proof state:
case 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), Functor.Additive (shiftFunctor C n)
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
zero : T.mor₃ = 0
this✝ : Epi T.mor₂
this : IsSplitEpi T.mor₂
fst : T.obj₂ ⟶ T.obj₁
hfst : 𝟙 T.obj₂ - T.mor₂ ≫ section_ T.mor₂ = fst ≫ T.mor₁
d : BinaryBiproductData T.obj₁ T.obj₃ :=
  binaryBiproductData T hT zero (section_ T.mor₂) (_ : section_ T.mor₂ ≫ T.mor₂ = 𝟙 T.obj₃) fst
    (_ : fst ≫ T.mor₁ + T.mor₂ ≫ section_ T.mor₂ = 𝟙 T.obj₂)
⊢ ∃ e, T.mor₁ ≫ e.hom = biprod.inl ∧ T.mor₂ = e.hom ≫ biprod.snd

Next tactic:
refine' ⟨biprod.uniqueUpToIso _ _ d.isBilimit, ⟨_, by simp⟩⟩
lemma exists_iso_binaryBiproduct_of_distTriang (T : Triangle C) (hT : T ∈ distTriang C)
    (zero : T.mor₃ = 0) :
    ∃ (e : T.obj₂ ≅ T.obj₁ ⊞ T.obj₃), T.mor₁ ≫ e.hom = biprod.inl ∧
      T.mor₂ = e.hom ≫ biprod.snd := by
  have := T.epi₂ hT zero
  have := isSplitEpi_of_epi T.mor₂
  obtain ⟨fst, hfst⟩ := T.coyoneda_exact₂ hT (𝟙 T.obj₂ - T.mor₂ ≫ section_ T.mor₂) (by simp)
  let d := binaryBiproductData _ hT zero (section_ T.mor₂) (by simp) fst
    (by simp only [← hfst, sub_add_cancel])
  refine' ⟨biprod.uniqueUpToIso _ _ d.isBilimit, ⟨_, by 

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), Functor.Additive (shiftFunctor C n)
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
zero : T.mor₃ = 0
this✝ : Epi T.mor₂
this : IsSplitEpi T.mor₂
fst : T.obj₂ ⟶ T.obj₁
hfst : 𝟙 T.obj₂ - T.mor₂ ≫ section_ T.mor₂ = fst ≫ T.mor₁
d : BinaryBiproductData T.obj₁ T.obj₃ :=
  binaryBiproductData T hT zero (section_ T.mor₂) (_ : section_ T.mor₂ ≫ T.mor₂ = 𝟙 T.obj₃) fst
    (_ : fst ≫ T.mor₁ + T.mor₂ ≫ section_ T.mor₂ = 𝟙 T.obj₂)
⊢ T.mor₂ = (biprod.uniqueUpToIso T.obj₁ T.obj₃ d.isBilimit).hom ≫ biprod.snd

Next tactic:
simp
lemma exists_iso_binaryBiproduct_of_distTriang (T : Triangle C) (hT : T ∈ distTriang C)
    (zero : T.mor₃ = 0) :
    ∃ (e : T.obj₂ ≅ T.obj₁ ⊞ T.obj₃), T.mor₁ ≫ e.hom = biprod.inl ∧
      T.mor₂ = e.hom ≫ biprod.snd := by
  have := T.epi₂ hT zero
  have := isSplitEpi_of_epi T.mor₂
  obtain ⟨fst, hfst⟩ := T.coyoneda_exact₂ hT (𝟙 T.obj₂ - T.mor₂ ≫ section_ T.mor₂) (by simp)
  let d := binaryBiproductData _ hT zero (section_ T.mor₂) (by simp) fst
    (by simp only [← hfst, sub_add_cancel])
  refine' ⟨biprod.uniqueUpToIso _ _ d.isBilimit, ⟨_, by simp⟩⟩
  

Proof state:
case 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), Functor.Additive (shiftFunctor C n)
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
zero : T.mor₃ = 0
this✝ : Epi T.mor₂
this : IsSplitEpi T.mor₂
fst : T.obj₂ ⟶ T.obj₁
hfst : 𝟙 T.obj₂ - T.mor₂ ≫ section_ T.mor₂ = fst ≫ T.mor₁
d : BinaryBiproductData T.obj₁ T.obj₃ :=
  binaryBiproductData T hT zero (section_ T.mor₂) (_ : section_ T.mor₂ ≫ T.mor₂ = 𝟙 T.obj₃) fst
    (_ : fst ≫ T.mor₁ + T.mor₂ ≫ section_ T.mor₂ = 𝟙 T.obj₂)
⊢ T.mor₁ ≫ (biprod.uniqueUpToIso T.obj₁ T.obj₃ d.isBilimit).hom = biprod.inl

Next tactic:
ext
lemma exists_iso_binaryBiproduct_of_distTriang (T : Triangle C) (hT : T ∈ distTriang C)
    (zero : T.mor₃ = 0) :
    ∃ (e : T.obj₂ ≅ T.obj₁ ⊞ T.obj₃), T.mor₁ ≫ e.hom = biprod.inl ∧
      T.mor₂ = e.hom ≫ biprod.snd := by
  have := T.epi₂ hT zero
  have := isSplitEpi_of_epi T.mor₂
  obtain ⟨fst, hfst⟩ := T.coyoneda_exact₂ hT (𝟙 T.obj₂ - T.mor₂ ≫ section_ T.mor₂) (by simp)
  let d := binaryBiproductData _ hT zero (section_ T.mor₂) (by simp) fst
    (by simp only [← hfst, sub_add_cancel])
  refine' ⟨biprod.uniqueUpToIso _ _ d.isBilimit, ⟨_, by simp⟩⟩
  ext
  · 

Proof state:
case intro.h₀
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), Functor.Additive (shiftFunctor C n)
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
zero : T.mor₃ = 0
this✝ : Epi T.mor₂
this : IsSplitEpi T.mor₂
fst : T.obj₂ ⟶ T.obj₁
hfst : 𝟙 T.obj₂ - T.mor₂ ≫ section_ T.mor₂ = fst ≫ T.mor₁
d : BinaryBiproductData T.obj₁ T.obj₃ :=
  binaryBiproductData T hT zero (section_ T.mor₂) (_ : section_ T.mor₂ ≫ T.mor₂ = 𝟙 T.obj₃) fst
    (_ : fst ≫ T.mor₁ + T.mor₂ ≫ section_ T.mor₂ = 𝟙 T.obj₂)
⊢ (T.mor₁ ≫ (biprod.uniqueUpToIso T.obj₁ T.obj₃ d.isBilimit).hom) ≫ biprod.fst = biprod.inl ≫ biprod.fst

Next tactic:
simpa using d.bicone.inl_fst
lemma exists_iso_binaryBiproduct_of_distTriang (T : Triangle C) (hT : T ∈ distTriang C)
    (zero : T.mor₃ = 0) :
    ∃ (e : T.obj₂ ≅ T.obj₁ ⊞ T.obj₃), T.mor₁ ≫ e.hom = biprod.inl ∧
      T.mor₂ = e.hom ≫ biprod.snd := by
  have := T.epi₂ hT zero
  have := isSplitEpi_of_epi T.mor₂
  obtain ⟨fst, hfst⟩ := T.coyoneda_exact₂ hT (𝟙 T.obj₂ - T.mor₂ ≫ section_ T.mor₂) (by simp)
  let d := binaryBiproductData _ hT zero (section_ T.mor₂) (by simp) fst
    (by simp only [← hfst, sub_add_cancel])
  refine' ⟨biprod.uniqueUpToIso _ _ d.isBilimit, ⟨_, by simp⟩⟩
  ext
  · simpa using d.bicone.inl_fst
  · 

Proof state:
case intro.h₁
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), Functor.Additive (shiftFunctor C n)
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
zero : T.mor₃ = 0
this✝ : Epi T.mor₂
this : IsSplitEpi T.mor₂
fst : T.obj₂ ⟶ T.obj₁
hfst : 𝟙 T.obj₂ - T.mor₂ ≫ section_ T.mor₂ = fst ≫ T.mor₁
d : BinaryBiproductData T.obj₁ T.obj₃ :=
  binaryBiproductData T hT zero (section_ T.mor₂) (_ : section_ T.mor₂ ≫ T.mor₂ = 𝟙 T.obj₃) fst
    (_ : fst ≫ T.mor₁ + T.mor₂ ≫ section_ T.mor₂ = 𝟙 T.obj₂)
⊢ (T.mor₁ ≫ (biprod.uniqueUpToIso T.obj₁ T.obj₃ d.isBilimit).hom) ≫ biprod.snd = biprod.inl ≫ biprod.snd

Next tactic:
simpa using d.bicone.inl_snd
lemma binaryBiproductTriangle_distinguished (X₁ X₂ : C) :
    binaryBiproductTriangle X₁ X₂ ∈ distTriang C := by
  

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), Functor.Additive (shiftFunctor C n)
hC : Pretriangulated C
X₁ X₂ : C
⊢ binaryBiproductTriangle X₁ X₂ ∈ distinguishedTriangles

Next tactic:
obtain ⟨Y, g, h, mem⟩ := distinguished_cocone_triangle₂ (0 : X₂ ⟶ X₁⟦(1 : ℤ)⟧)
lemma binaryBiproductTriangle_distinguished (X₁ X₂ : C) :
    binaryBiproductTriangle X₁ X₂ ∈ distTriang C := by
  obtain ⟨Y, g, h, mem⟩ := distinguished_cocone_triangle₂ (0 : X₂ ⟶ X₁⟦(1 : ℤ)⟧)
  

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), Functor.Additive (shiftFunctor C n)
hC : Pretriangulated C
X₁ X₂ Y : C
g : X₁ ⟶ Y
h : Y ⟶ X₂
mem : Triangle.mk g h 0 ∈ distinguishedTriangles
⊢ binaryBiproductTriangle X₁ X₂ ∈ distinguishedTriangles

Next tactic:
obtain ⟨e, ⟨he₁, he₂⟩⟩ := exists_iso_binaryBiproduct_of_distTriang _ mem rfl
lemma binaryBiproductTriangle_distinguished (X₁ X₂ : C) :
    binaryBiproductTriangle X₁ X₂ ∈ distTriang C := by
  obtain ⟨Y, g, h, mem⟩ := distinguished_cocone_triangle₂ (0 : X₂ ⟶ X₁⟦(1 : ℤ)⟧)
  obtain ⟨e, ⟨he₁, he₂⟩⟩ := exists_iso_binaryBiproduct_of_distTriang _ mem rfl
  

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), Functor.Additive (shiftFunctor C n)
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

Next tactic:
dsimp at he₁ he₂
lemma binaryBiproductTriangle_distinguished (X₁ X₂ : C) :
    binaryBiproductTriangle X₁ X₂ ∈ distTriang C := by
  obtain ⟨Y, g, h, mem⟩ := distinguished_cocone_triangle₂ (0 : X₂ ⟶ X₁⟦(1 : ℤ)⟧)
  obtain ⟨e, ⟨he₁, he₂⟩⟩ := exists_iso_binaryBiproduct_of_distTriang _ mem rfl
  dsimp at he₁ he₂
  

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), Functor.Additive (shiftFunctor C n)
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

Next tactic:
refine' isomorphic_distinguished _ mem _ (Iso.symm _)
lemma binaryBiproductTriangle_distinguished (X₁ X₂ : C) :
    binaryBiproductTriangle X₁ X₂ ∈ distTriang C := by
  obtain ⟨Y, g, h, mem⟩ := distinguished_cocone_triangle₂ (0 : X₂ ⟶ X₁⟦(1 : ℤ)⟧)
  obtain ⟨e, ⟨he₁, he₂⟩⟩ := exists_iso_binaryBiproduct_of_distTriang _ mem rfl
  dsimp at he₁ he₂
  refine' isomorphic_distinguished _ mem _ (Iso.symm _)
  

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), Functor.Additive (shiftFunctor C n)
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

Next tactic:
refine' Triangle.isoMk _ _ (Iso.refl _) e (Iso.refl _)
    (by aesop_cat) (by aesop_cat) (by aesop_cat)
lemma binaryBiproductTriangle_distinguished (X₁ X₂ : C) :
    binaryBiproductTriangle X₁ X₂ ∈ distTriang C := by
  obtain ⟨Y, g, h, mem⟩ := distinguished_cocone_triangle₂ (0 : X₂ ⟶ X₁⟦(1 : ℤ)⟧)
  obtain ⟨e, ⟨he₁, he₂⟩⟩ := exists_iso_binaryBiproduct_of_distTriang _ mem rfl
  dsimp at he₁ he₂
  refine' isomorphic_distinguished _ mem _ (Iso.symm _)
  refine' Triangle.isoMk _ _ (Iso.refl _) e (Iso.refl _)
    (by 

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), Functor.Additive (shiftFunctor C n)
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

Next tactic:
aesop_cat
lemma binaryBiproductTriangle_distinguished (X₁ X₂ : C) :
    binaryBiproductTriangle X₁ X₂ ∈ distTriang C := by
  obtain ⟨Y, g, h, mem⟩ := distinguished_cocone_triangle₂ (0 : X₂ ⟶ X₁⟦(1 : ℤ)⟧)
  obtain ⟨e, ⟨he₁, he₂⟩⟩ := exists_iso_binaryBiproduct_of_distTriang _ mem rfl
  dsimp at he₁ he₂
  refine' isomorphic_distinguished _ mem _ (Iso.symm _)
  refine' Triangle.isoMk _ _ (Iso.refl _) e (Iso.refl _)
    (by aesop_cat) (by 

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), Functor.Additive (shiftFunctor C n)
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

Next tactic:
aesop_cat
lemma binaryBiproductTriangle_distinguished (X₁ X₂ : C) :
    binaryBiproductTriangle X₁ X₂ ∈ distTriang C := by
  obtain ⟨Y, g, h, mem⟩ := distinguished_cocone_triangle₂ (0 : X₂ ⟶ X₁⟦(1 : ℤ)⟧)
  obtain ⟨e, ⟨he₁, he₂⟩⟩ := exists_iso_binaryBiproduct_of_distTriang _ mem rfl
  dsimp at he₁ he₂
  refine' isomorphic_distinguished _ mem _ (Iso.symm _)
  refine' Triangle.isoMk _ _ (Iso.refl _) e (Iso.refl _)
    (by aesop_cat) (by aesop_cat) (by 

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), Functor.Additive (shiftFunctor C n)
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

Next tactic:
aesop_cat
/-- A product of distinguished triangles is distinguished -/
lemma productTriangle_distinguished {J : Type*} (T : J → Triangle C)
    (hT : ∀ j, T j ∈ distTriang C)
    [HasProduct (fun j => (T j).obj₁)] [HasProduct (fun j => (T j).obj₂)]
    [HasProduct (fun j => (T j).obj₃)] [HasProduct (fun j => (T j).obj₁⟦(1 : ℤ)⟧)] :
    productTriangle T ∈ distTriang C := by
  /- The proof proceeds by constructing a morphism of triangles
    `φ' : T' ⟶ productTriangle T` with `T'` distinguished, and such that
    `φ'.hom₁` and `φ'.hom₂` are identities. Then, it suffices to show that
    `φ'.hom₃` is an isomorphism, which is achieved by using Yoneda's lemma
    and diagram chases. -/
  

Proof state:
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), Functor.Additive (shiftFunctor C n)
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
⊢ productTriangle T ∈ distinguishedTriangles

Next tactic:
let f₁ := Pi.map (fun j => (T j).mor₁)
/-- A product of distinguished triangles is distinguished -/
lemma productTriangle_distinguished {J : Type*} (T : J → Triangle C)
    (hT : ∀ j, T j ∈ distTriang C)
    [HasProduct (fun j => (T j).obj₁)] [HasProduct (fun j => (T j).obj₂)]
    [HasProduct (fun j => (T j).obj₃)] [HasProduct (fun j => (T j).obj₁⟦(1 : ℤ)⟧)] :
    productTriangle T ∈ distTriang C := by
  /- The proof proceeds by constructing a morphism of triangles
    `φ' : T' ⟶ productTriangle T` with `T'` distinguished, and such that
    `φ'.hom₁` and `φ'.hom₂` are identities. Then, it suffices to show that
    `φ'.hom₃` is an isomorphism, which is achieved by using Yoneda's lemma
    and diagram chases. -/
  let f₁ := Pi.map (fun j => (T j).mor₁)
  

Proof state:
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), Functor.Additive (shiftFunctor C n)
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 : (∏ fun j => (T j).obj₁) ⟶ ∏ fun j => (T j).obj₂ := Pi.map fun j => (T j).mor₁
⊢ productTriangle T ∈ distinguishedTriangles

Next tactic:
obtain ⟨Z, f₂, f₃, hT'⟩ := distinguished_cocone_triangle f₁
/-- A product of distinguished triangles is distinguished -/
lemma productTriangle_distinguished {J : Type*} (T : J → Triangle C)
    (hT : ∀ j, T j ∈ distTriang C)
    [HasProduct (fun j => (T j).obj₁)] [HasProduct (fun j => (T j).obj₂)]
    [HasProduct (fun j => (T j).obj₃)] [HasProduct (fun j => (T j).obj₁⟦(1 : ℤ)⟧)] :
    productTriangle T ∈ distTriang C := by
  /- The proof proceeds by constructing a morphism of triangles
    `φ' : T' ⟶ productTriangle T` with `T'` distinguished, and such that
    `φ'.hom₁` and `φ'.hom₂` are identities. Then, it suffices to show that
    `φ'.hom₃` is an isomorphism, which is achieved by using Yoneda's lemma
    and diagram chases. -/
  let f₁ := Pi.map (fun j => (T j).mor₁)
  obtain ⟨Z, f₂, f₃, hT'⟩ := distinguished_cocone_triangle f₁
  

Proof state:
case intro.intro.intro
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), Functor.Additive (shiftFunctor C n)
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 : (∏ fun j => (T j).obj₁) ⟶ ∏ fun j => (T j).obj₂ := Pi.map fun j => (T j).mor₁
Z : C
f₂ : (∏ fun j => (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ fun j => (T j).obj₁)
hT' : Triangle.mk f₁ f₂ f₃ ∈ distinguishedTriangles
⊢ productTriangle T ∈ distinguishedTriangles

Next tactic:
let T' := Triangle.mk f₁ f₂ f₃
/-- A product of distinguished triangles is distinguished -/
lemma productTriangle_distinguished {J : Type*} (T : J → Triangle C)
    (hT : ∀ j, T j ∈ distTriang C)
    [HasProduct (fun j => (T j).obj₁)] [HasProduct (fun j => (T j).obj₂)]
    [HasProduct (fun j => (T j).obj₃)] [HasProduct (fun j => (T j).obj₁⟦(1 : ℤ)⟧)] :
    productTriangle T ∈ distTriang C := by
  /- The proof proceeds by constructing a morphism of triangles
    `φ' : T' ⟶ productTriangle T` with `T'` distinguished, and such that
    `φ'.hom₁` and `φ'.hom₂` are identities. Then, it suffices to show that
    `φ'.hom₃` is an isomorphism, which is achieved by using Yoneda's lemma
    and diagram chases. -/
  let f₁ := Pi.map (fun j => (T j).mor₁)
  obtain ⟨Z, f₂, f₃, hT'⟩ := distinguished_cocone_triangle f₁
  let T' := Triangle.mk f₁ f₂ f₃
  

Proof state:
case intro.intro.intro
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), Functor.Additive (shiftFunctor C n)
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 : (∏ fun j => (T j).obj₁) ⟶ ∏ fun j => (T j).obj₂ := Pi.map fun j => (T j).mor₁
Z : C
f₂ : (∏ fun j => (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ fun j => (T j).obj₁)
hT' : Triangle.mk f₁ f₂ f₃ ∈ distinguishedTriangles
T' : Triangle C := Triangle.mk f₁ f₂ f₃
⊢ productTriangle T ∈ distinguishedTriangles

Next tactic:
change T' ∈ distTriang C at hT'
/-- A product of distinguished triangles is distinguished -/
lemma productTriangle_distinguished {J : Type*} (T : J → Triangle C)
    (hT : ∀ j, T j ∈ distTriang C)
    [HasProduct (fun j => (T j).obj₁)] [HasProduct (fun j => (T j).obj₂)]
    [HasProduct (fun j => (T j).obj₃)] [HasProduct (fun j => (T j).obj₁⟦(1 : ℤ)⟧)] :
    productTriangle T ∈ distTriang C := by
  /- The proof proceeds by constructing a morphism of triangles
    `φ' : T' ⟶ productTriangle T` with `T'` distinguished, and such that
    `φ'.hom₁` and `φ'.hom₂` are identities. Then, it suffices to show that
    `φ'.hom₃` is an isomorphism, which is achieved by using Yoneda's lemma
    and diagram chases. -/
  let f₁ := Pi.map (fun j => (T j).mor₁)
  obtain ⟨Z, f₂, f₃, hT'⟩ := distinguished_cocone_triangle f₁
  let T' := Triangle.mk f₁ f₂ f₃
  change T' ∈ distTriang C at hT'
  

Proof state:
case intro.intro.intro
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), Functor.Additive (shiftFunctor C n)
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 : (∏ fun j => (T j).obj₁) ⟶ ∏ fun j => (T j).obj₂ := Pi.map fun j => (T j).mor₁
Z : C
f₂ : (∏ fun j => (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ fun j => (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
⊢ productTriangle T ∈ distinguishedTriangles

Next tactic:
let φ : ∀ j, T' ⟶ T j := fun j => completeDistinguishedTriangleMorphism _ _
    hT' (hT j) (Pi.π _ j) (Pi.π _ j) (by simp)
/-- A product of distinguished triangles is distinguished -/
lemma productTriangle_distinguished {J : Type*} (T : J → Triangle C)
    (hT : ∀ j, T j ∈ distTriang C)
    [HasProduct (fun j => (T j).obj₁)] [HasProduct (fun j => (T j).obj₂)]
    [HasProduct (fun j => (T j).obj₃)] [HasProduct (fun j => (T j).obj₁⟦(1 : ℤ)⟧)] :
    productTriangle T ∈ distTriang C := by
  /- The proof proceeds by constructing a morphism of triangles
    `φ' : T' ⟶ productTriangle T` with `T'` distinguished, and such that
    `φ'.hom₁` and `φ'.hom₂` are identities. Then, it suffices to show that
    `φ'.hom₃` is an isomorphism, which is achieved by using Yoneda's lemma
    and diagram chases. -/
  let f₁ := Pi.map (fun j => (T j).mor₁)
  obtain ⟨Z, f₂, f₃, hT'⟩ := distinguished_cocone_triangle f₁
  let T' := Triangle.mk f₁ f₂ f₃
  change T' ∈ distTriang C at hT'
  let φ : ∀ j, T' ⟶ T j := fun j => completeDistinguishedTriangleMorphism _ _
    hT' (hT j) (Pi.π _ j) (Pi.π _ j) (by 

Proof state:
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), Functor.Additive (shiftFunctor C n)
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 : (∏ fun j => (T j).obj₁) ⟶ ∏ fun j => (T j).obj₂ := Pi.map fun j => (T j).mor₁
Z : C
f₂ : (∏ fun j => (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ fun j => (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
j : J
⊢ T'.mor₁ ≫ Pi.π (fun j => (T j).obj₂) j = Pi.π (fun j => (T j).obj₁) j ≫ (T j).mor₁

Next tactic:
simp