l3lab/ntp-mathlib huggingface dataset:
/-- 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 simp)
  let φ' := productTriangle.lift _ φ
  have h₁ : φ'.hom₁ = 𝟙 _ := by aesop_cat
  have h₂ : φ'.hom₂ = 𝟙 _ := by aesop_cat
  have : IsIso φ'.hom₁ := by rw [h₁]; infer_instance
  have : IsIso φ'.hom₂ := by rw [h₂]; infer_instance
  suffices IsIso φ'.hom₃ by
    have : IsIso φ' := by
      apply Triangle.isIso_of_isIsos
      all_goals infer_instance
    exact isomorphic_distinguished _ hT' _ (asIso φ').symm
  refine' isIso_of_yoneda_map_bijective _ (fun A => ⟨_, _⟩)
  /- the proofs by diagram chase start here -/
  · 

Proof state:
case intro.intro.intro.refine'_1
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
φ : (j : J) → T' ⟶ T j :=
  fun j =>
    completeDistinguishedTriangleMorphism T' (T j) hT' (_ : T j ∈ distinguishedTriangles) (Pi.π (fun j => (T j).obj₁) j)
      (Pi.π (fun j => (T j).obj₂) j)
      (_ :
        limMap (Discrete.natTrans fun X => (fun j => (T j).mor₁) X.as) ≫
            limit.π (Discrete.functor fun j => (T j).obj₂) { as := j } =
          Pi.π (fun j => (T j).obj₁) j ≫ (T j).mor₁)
φ' : T' ⟶ productTriangle fun j => T j := productTriangle.lift (fun j => T j) φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
this✝ : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A : C
⊢ Function.Injective fun x => x ≫ φ'.hom₃

Next tactic:
suffices Mono φ'.hom₃ by
      intro a₁ a₂ ha
      simpa only [← cancel_mono φ'.hom₃] using ha
/-- 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 simp)
  let φ' := productTriangle.lift _ φ
  have h₁ : φ'.hom₁ = 𝟙 _ := by aesop_cat
  have h₂ : φ'.hom₂ = 𝟙 _ := by aesop_cat
  have : IsIso φ'.hom₁ := by rw [h₁]; infer_instance
  have : IsIso φ'.hom₂ := by rw [h₂]; infer_instance
  suffices IsIso φ'.hom₃ by
    have : IsIso φ' := by
      apply Triangle.isIso_of_isIsos
      all_goals infer_instance
    exact isomorphic_distinguished _ hT' _ (asIso φ').symm
  refine' isIso_of_yoneda_map_bijective _ (fun A => ⟨_, _⟩)
  /- the proofs by diagram chase start here -/
  · suffices Mono φ'.hom₃ 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
φ : (j : J) → T' ⟶ T j :=
  fun j =>
    completeDistinguishedTriangleMorphism T' (T j) hT' (_ : T j ∈ distinguishedTriangles) (Pi.π (fun j => (T j).obj₁) j)
      (Pi.π (fun j => (T j).obj₂) j)
      (_ :
        limMap (Discrete.natTrans fun X => (fun j => (T j).mor₁) X.as) ≫
            limit.π (Discrete.functor fun j => (T j).obj₂) { as := j } =
          Pi.π (fun j => (T j).obj₁) j ≫ (T j).mor₁)
φ' : T' ⟶ productTriangle fun j => T j := productTriangle.lift (fun j => T j) φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
this✝¹ : IsIso φ'.hom₁
this✝ : IsIso φ'.hom₂
A : C
this : Mono φ'.hom₃
⊢ Function.Injective fun x => x ≫ φ'.hom₃

Next tactic:
intro a₁ a₂ ha
/-- 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 simp)
  let φ' := productTriangle.lift _ φ
  have h₁ : φ'.hom₁ = 𝟙 _ := by aesop_cat
  have h₂ : φ'.hom₂ = 𝟙 _ := by aesop_cat
  have : IsIso φ'.hom₁ := by rw [h₁]; infer_instance
  have : IsIso φ'.hom₂ := by rw [h₂]; infer_instance
  suffices IsIso φ'.hom₃ by
    have : IsIso φ' := by
      apply Triangle.isIso_of_isIsos
      all_goals infer_instance
    exact isomorphic_distinguished _ hT' _ (asIso φ').symm
  refine' isIso_of_yoneda_map_bijective _ (fun A => ⟨_, _⟩)
  /- the proofs by diagram chase start here -/
  · suffices Mono φ'.hom₃ by
      intro a₁ a₂ ha
      

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
φ : (j : J) → T' ⟶ T j :=
  fun j =>
    completeDistinguishedTriangleMorphism T' (T j) hT' (_ : T j ∈ distinguishedTriangles) (Pi.π (fun j => (T j).obj₁) j)
      (Pi.π (fun j => (T j).obj₂) j)
      (_ :
        limMap (Discrete.natTrans fun X => (fun j => (T j).mor₁) X.as) ≫
            limit.π (Discrete.functor fun j => (T j).obj₂) { as := j } =
          Pi.π (fun j => (T j).obj₁) j ≫ (T j).mor₁)
φ' : T' ⟶ productTriangle fun j => T j := productTriangle.lift (fun j => T j) φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
this✝¹ : IsIso φ'.hom₁
this✝ : IsIso φ'.hom₂
A : C
this : Mono φ'.hom₃
a₁ a₂ : A ⟶ T'.obj₃
ha : (fun x => x ≫ φ'.hom₃) a₁ = (fun x => x ≫ φ'.hom₃) a₂
⊢ a₁ = a₂

Next tactic:
simpa only [← cancel_mono φ'.hom₃] using ha
/-- 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 simp)
  let φ' := productTriangle.lift _ φ
  have h₁ : φ'.hom₁ = 𝟙 _ := by aesop_cat
  have h₂ : φ'.hom₂ = 𝟙 _ := by aesop_cat
  have : IsIso φ'.hom₁ := by rw [h₁]; infer_instance
  have : IsIso φ'.hom₂ := by rw [h₂]; infer_instance
  suffices IsIso φ'.hom₃ by
    have : IsIso φ' := by
      apply Triangle.isIso_of_isIsos
      all_goals infer_instance
    exact isomorphic_distinguished _ hT' _ (asIso φ').symm
  refine' isIso_of_yoneda_map_bijective _ (fun A => ⟨_, _⟩)
  /- the proofs by diagram chase start here -/
  · suffices Mono φ'.hom₃ by
      intro a₁ a₂ ha
      simpa only [← cancel_mono φ'.hom₃] using ha
    

Proof state:
case intro.intro.intro.refine'_1
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
φ : (j : J) → T' ⟶ T j :=
  fun j =>
    completeDistinguishedTriangleMorphism T' (T j) hT' (_ : T j ∈ distinguishedTriangles) (Pi.π (fun j => (T j).obj₁) j)
      (Pi.π (fun j => (T j).obj₂) j)
      (_ :
        limMap (Discrete.natTrans fun X => (fun j => (T j).mor₁) X.as) ≫
            limit.π (Discrete.functor fun j => (T j).obj₂) { as := j } =
          Pi.π (fun j => (T j).obj₁) j ≫ (T j).mor₁)
φ' : T' ⟶ productTriangle fun j => T j := productTriangle.lift (fun j => T j) φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
this✝ : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A : C
⊢ Mono φ'.hom₃

Next tactic:
rw [mono_iff_cancel_zero]
/-- 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 simp)
  let φ' := productTriangle.lift _ φ
  have h₁ : φ'.hom₁ = 𝟙 _ := by aesop_cat
  have h₂ : φ'.hom₂ = 𝟙 _ := by aesop_cat
  have : IsIso φ'.hom₁ := by rw [h₁]; infer_instance
  have : IsIso φ'.hom₂ := by rw [h₂]; infer_instance
  suffices IsIso φ'.hom₃ by
    have : IsIso φ' := by
      apply Triangle.isIso_of_isIsos
      all_goals infer_instance
    exact isomorphic_distinguished _ hT' _ (asIso φ').symm
  refine' isIso_of_yoneda_map_bijective _ (fun A => ⟨_, _⟩)
  /- the proofs by diagram chase start here -/
  · suffices Mono φ'.hom₃ by
      intro a₁ a₂ ha
      simpa only [← cancel_mono φ'.hom₃] using ha
    rw [mono_iff_cancel_zero]
    

Proof state:
case intro.intro.intro.refine'_1
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
φ : (j : J) → T' ⟶ T j :=
  fun j =>
    completeDistinguishedTriangleMorphism T' (T j) hT' (_ : T j ∈ distinguishedTriangles) (Pi.π (fun j => (T j).obj₁) j)
      (Pi.π (fun j => (T j).obj₂) j)
      (_ :
        limMap (Discrete.natTrans fun X => (fun j => (T j).mor₁) X.as) ≫
            limit.π (Discrete.functor fun j => (T j).obj₂) { as := j } =
          Pi.π (fun j => (T j).obj₁) j ≫ (T j).mor₁)
φ' : T' ⟶ productTriangle fun j => T j := productTriangle.lift (fun j => T j) φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
this✝ : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A : C
⊢ ∀ (P : C) (g : P ⟶ T'.obj₃), g ≫ φ'.hom₃ = 0 → g = 0

Next tactic:
intro A f hf
/-- 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 simp)
  let φ' := productTriangle.lift _ φ
  have h₁ : φ'.hom₁ = 𝟙 _ := by aesop_cat
  have h₂ : φ'.hom₂ = 𝟙 _ := by aesop_cat
  have : IsIso φ'.hom₁ := by rw [h₁]; infer_instance
  have : IsIso φ'.hom₂ := by rw [h₂]; infer_instance
  suffices IsIso φ'.hom₃ by
    have : IsIso φ' := by
      apply Triangle.isIso_of_isIsos
      all_goals infer_instance
    exact isomorphic_distinguished _ hT' _ (asIso φ').symm
  refine' isIso_of_yoneda_map_bijective _ (fun A => ⟨_, _⟩)
  /- the proofs by diagram chase start here -/
  · suffices Mono φ'.hom₃ by
      intro a₁ a₂ ha
      simpa only [← cancel_mono φ'.hom₃] using ha
    rw [mono_iff_cancel_zero]
    intro A f hf
    

Proof state:
case intro.intro.intro.refine'_1
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
φ : (j : J) → T' ⟶ T j :=
  fun j =>
    completeDistinguishedTriangleMorphism T' (T j) hT' (_ : T j ∈ distinguishedTriangles) (Pi.π (fun j => (T j).obj₁) j)
      (Pi.π (fun j => (T j).obj₂) j)
      (_ :
        limMap (Discrete.natTrans fun X => (fun j => (T j).mor₁) X.as) ≫
            limit.π (Discrete.functor fun j => (T j).obj₂) { as := j } =
          Pi.π (fun j => (T j).obj₁) j ≫ (T j).mor₁)
φ' : T' ⟶ productTriangle fun j => T j := productTriangle.lift (fun j => T j) φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
this✝ : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A✝ A : C
f : A ⟶ T'.obj₃
hf : f ≫ φ'.hom₃ = 0
⊢ f = 0

Next tactic:
have hf' : f ≫ T'.mor₃ = 0 := by
      rw [← cancel_mono (φ'.hom₁⟦1⟧'), zero_comp, assoc, φ'.comm₃, reassoc_of% hf, zero_comp]
/-- 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 simp)
  let φ' := productTriangle.lift _ φ
  have h₁ : φ'.hom₁ = 𝟙 _ := by aesop_cat
  have h₂ : φ'.hom₂ = 𝟙 _ := by aesop_cat
  have : IsIso φ'.hom₁ := by rw [h₁]; infer_instance
  have : IsIso φ'.hom₂ := by rw [h₂]; infer_instance
  suffices IsIso φ'.hom₃ by
    have : IsIso φ' := by
      apply Triangle.isIso_of_isIsos
      all_goals infer_instance
    exact isomorphic_distinguished _ hT' _ (asIso φ').symm
  refine' isIso_of_yoneda_map_bijective _ (fun A => ⟨_, _⟩)
  /- the proofs by diagram chase start here -/
  · suffices Mono φ'.hom₃ by
      intro a₁ a₂ ha
      simpa only [← cancel_mono φ'.hom₃] using ha
    rw [mono_iff_cancel_zero]
    intro A f hf
    have hf' : f ≫ T'.mor₃ = 0 := 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
φ : (j : J) → T' ⟶ T j :=
  fun j =>
    completeDistinguishedTriangleMorphism T' (T j) hT' (_ : T j ∈ distinguishedTriangles) (Pi.π (fun j => (T j).obj₁) j)
      (Pi.π (fun j => (T j).obj₂) j)
      (_ :
        limMap (Discrete.natTrans fun X => (fun j => (T j).mor₁) X.as) ≫
            limit.π (Discrete.functor fun j => (T j).obj₂) { as := j } =
          Pi.π (fun j => (T j).obj₁) j ≫ (T j).mor₁)
φ' : T' ⟶ productTriangle fun j => T j := productTriangle.lift (fun j => T j) φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
this✝ : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A✝ A : C
f : A ⟶ T'.obj₃
hf : f ≫ φ'.hom₃ = 0
⊢ f ≫ T'.mor₃ = 0

Next tactic:
rw [← cancel_mono (φ'.hom₁⟦1⟧'), zero_comp, assoc, φ'.comm₃, reassoc_of% hf, zero_comp]
/-- General theorem for `Finset.induction_on_pi`-style induction principles. -/
theorem induction_on_pi_of_choice (r : ∀ i, α i → Finset (α i) → Prop)
    (H_ex : ∀ (i) (s : Finset (α i)), s.Nonempty → ∃ x ∈ s, r i x (s.erase x))
    {p : (∀ i, Finset (α i)) → Prop} (f : ∀ i, Finset (α i)) (h0 : p fun _ ↦ ∅)
    (step :
      ∀ (g : ∀ i, Finset (α i)) (i : ι) (x : α i),
        r i x (g i) → p g → p (update g i (insert x (g i)))) :
    p f := by
  

Proof state:
ι : Type u_1
α : ι → Type u_2
inst✝² : Finite ι
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (α i)
r : (i : ι) → α i → Finset (α i) → Prop
H_ex : ∀ (i : ι) (s : Finset (α i)), Finset.Nonempty s → ∃ x ∈ s, r i x (erase s x)
p : ((i : ι) → Finset (α i)) → Prop
f : (i : ι) → Finset (α i)
h0 : p fun x => ∅
step : ∀ (g : (i : ι) → Finset (α i)) (i : ι) (x : α i), r i x (g i) → p g → p (update g i (insert x (g i)))
⊢ p f

Next tactic:
cases nonempty_fintype ι
/-- General theorem for `Finset.induction_on_pi`-style induction principles. -/
theorem induction_on_pi_of_choice (r : ∀ i, α i → Finset (α i) → Prop)
    (H_ex : ∀ (i) (s : Finset (α i)), s.Nonempty → ∃ x ∈ s, r i x (s.erase x))
    {p : (∀ i, Finset (α i)) → Prop} (f : ∀ i, Finset (α i)) (h0 : p fun _ ↦ ∅)
    (step :
      ∀ (g : ∀ i, Finset (α i)) (i : ι) (x : α i),
        r i x (g i) → p g → p (update g i (insert x (g i)))) :
    p f := by
  cases nonempty_fintype ι
  

Proof state:
case intro
ι : Type u_1
α : ι → Type u_2
inst✝² : Finite ι
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (α i)
r : (i : ι) → α i → Finset (α i) → Prop
H_ex : ∀ (i : ι) (s : Finset (α i)), Finset.Nonempty s → ∃ x ∈ s, r i x (erase s x)
p : ((i : ι) → Finset (α i)) → Prop
f : (i : ι) → Finset (α i)
h0 : p fun x => ∅
step : ∀ (g : (i : ι) → Finset (α i)) (i : ι) (x : α i), r i x (g i) → p g → p (update g i (insert x (g i)))
val✝ : Fintype ι
⊢ p f

Next tactic:
induction' hs : univ.sigma f using Finset.strongInductionOn with s ihs generalizing f
/-- General theorem for `Finset.induction_on_pi`-style induction principles. -/
theorem induction_on_pi_of_choice (r : ∀ i, α i → Finset (α i) → Prop)
    (H_ex : ∀ (i) (s : Finset (α i)), s.Nonempty → ∃ x ∈ s, r i x (s.erase x))
    {p : (∀ i, Finset (α i)) → Prop} (f : ∀ i, Finset (α i)) (h0 : p fun _ ↦ ∅)
    (step :
      ∀ (g : ∀ i, Finset (α i)) (i : ι) (x : α i),
        r i x (g i) → p g → p (update g i (insert x (g i)))) :
    p f := by
  cases nonempty_fintype ι
  induction' hs : univ.sigma f using Finset.strongInductionOn with s ihs generalizing f; 

Proof state:
case intro.a
ι : Type u_1
α : ι → Type u_2
inst✝² : Finite ι
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (α i)
r : (i : ι) → α i → Finset (α i) → Prop
H_ex : ∀ (i : ι) (s : Finset (α i)), Finset.Nonempty s → ∃ x ∈ s, r i x (erase s x)
p : ((i : ι) → Finset (α i)) → Prop
h0 : p fun x => ∅
step : ∀ (g : (i : ι) → Finset (α i)) (i : ι) (x : α i), r i x (g i) → p g → p (update g i (insert x (g i)))
val✝ : Fintype ι
s : Finset ((i : ι) × α i)
ihs : ∀ t ⊂ s, ∀ (f : (i : ι) → Finset (α i)), Finset.sigma univ f = t → p f
f : (i : ι) → Finset (α i)
hs : Finset.sigma univ f = s
⊢ p f

Next tactic:
subst s