mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift

Goal state before:
pp: case pos
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
n : ℤ
γ : Cochain K L n
a n' m' : ℤ
hn' : n + a = n'
m : ℤ
hm' : m + a = m'
hnm : n + 1 = m
⊢ δ n' m' (γ.leftShift a n' hn') = a.negOnePow • (δ n m γ).leftShift a m' hm'

Tactic:
have hnm' : n' + 1 = m' := by lia
    ext p q hpq
    dsimp
    rw [(δ n m γ).leftShift_v a m' hm' p q hpq (p + a) (by lia),
      δ_v n m hnm _ (p + a) q (by lia) (p + n') (p + 1 + a) (by lia) (by lia),
      δ_v n' m' hnm' _ p q hpq (p + n') (p + 1) (by lia) rfl,
      γ.leftShift_v a n' hn' p (p + n') rfl (p + a) (by lia),
      γ.leftShift_v a n' hn' (p + 1) q (by lia) (p + 1 + a) (by lia)]
    simp only [shiftFunctor_obj_X, shiftFunctorObjXIso, HomologicalComplex.XIsoOfEq_rfl,
      Iso.refl_hom, id_comp, Linear.units_smul_comp, shiftFunctor_obj_d',
      Linear.comp_units_smul, smul_add, smul_smul]
    congr 2
    · rw [← hnm', add_comm n', mul_add, mul_one]
      simp only [Int.negOnePow_add, ← mul_assoc, Int.units_mul_self, one_mul]
    · simp only [← Int.negOnePow_add, ← hn', ← hm', ← hnm]
      congr 1
      linarith

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
a b : ℤ
T : Triangle C
⊢ (a.negOnePow * b.negOnePow) •
      (CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).hom.app T.obj₂ ≫
        (CategoryTheory.shiftFunctor C b).map ((CategoryTheory.shiftFunctor C a).map T.mor₂) =
    (b.negOnePow * a.negOnePow) •
      (CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).hom.app T.obj₂ ≫
        (CategoryTheory.shiftFunctor C b).map ((CategoryTheory.shiftFunctor C a).map T.mor₂)

Tactic:
mul_comm

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
a b : ℤ
T : Triangle C
⊢ (b.negOnePow * a.negOnePow) •
      (CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).hom.app T.obj₂ ≫
        (CategoryTheory.shiftFunctor C b).map ((CategoryTheory.shiftFunctor C a).map T.mor₂) =
    (b.negOnePow * a.negOnePow) •
      (CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).hom.app T.obj₂ ≫
        (CategoryTheory.shiftFunctor C b).map ((CategoryTheory.shiftFunctor C a).map T.mor₂)
Module: Mathlib.CategoryTheory.Shift.CommShift

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
D₁ : Type u_4
D₂ : Type u_5
D₃ : Type u_6
inst✝²⁵ : Category.{v_1, u_1} C₁
inst✝²⁴ : Category.{v_2, u_2} C₂
inst✝²³ : Category.{v_3, u_3} C₃
inst✝²² : Category.{v_4, u_4} D₁
inst✝²¹ : Category.{v_5, u_5} D₂
inst✝²⁰ : Category.{v_6, u_6} D₃
F₁₂ : C₁ ⥤ C₂
F₂₃ : C₂ ⥤ C₃
F₁₃ : C₁ ⥤ C₃
α : F₁₃ ⟶ F₁₂ ⋙ F₂₃
G₁₂ : D₁ ⥤ D₂
G₂₃ : D₂ ⥤ D₃
G₁₃ : D₁ ⥤ D₃
β : G₁₂ ⋙ G₂₃ ⟶ G₁₃
L₁ : C₁ ⥤ D₁
L₂ : C₂ ⥤ D₂
L₃ : C₃ ⥤ D₃
e₁₂ : F₁₂ ⋙ L₂ ⟶ L₁ ⋙ G₁₂
e₂₃ : F₂₃ ⋙ L₃ ⟶ L₂ ⋙ G₂₃
A : Type u_7
inst✝¹⁹ : AddMonoid A
inst✝¹⁸ : HasShift C₁ A
inst✝¹⁷ : HasShift C₂ A
inst✝¹⁶ : HasShift C₃ A
inst✝¹⁵ : HasShift D₁ A
inst✝¹⁴ : HasShift D₂ A
inst✝¹³ : HasShift D₃ A
inst✝¹² : F₁₂.CommShift A
inst✝¹¹ : F₂₃.CommShift A
inst✝¹⁰ : F₁₃.CommShift A
inst✝⁹ : CommShift α A
inst✝⁸ : G₁₂.CommShift A
inst✝⁷ : G₂₃.CommShift A
inst✝⁶ : G₁₃.CommShift A
inst✝⁵ : CommShift β A
inst✝⁴ : L₁.CommShift A
inst✝³ : L₂.CommShift A
inst✝² : L₃.CommShift A
inst✝¹ : CommShift e₁₂ A
inst✝ : CommShift e₂₃ A
⊢ CommShift
    (Functor.whiskerRight α L₃ ≫
      (F₁₂.associator F₂₃ L₃).hom ≫
        F₁₂.whiskerLeft e₂₃ ≫
          (F₁₂.associator L₂ G₂₃).inv ≫ Functor.whiskerRight e₁₂ G₂₃ ≫ (L₁.associator G₁₂ G₂₃).hom ≫ L₁.whiskerLeft β)
    A

Tactic:
infer_instance

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

Goal state before:
pp: case mpr
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), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
⊢ Epi T.mor₂ → T.mor₃ = 0

Tactic:
intro

Goal state after:
case mpr
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), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
a✝ : Epi T.mor₂
⊢ T.mor₃ = 0
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: case mpr
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), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
⊢ Mono T.mor₂ → T.mor₁ = 0

Tactic:
intro

Goal state after:
case mpr
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), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
a✝ : Mono T.mor₂
⊢ T.mor₁ = 0
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: 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), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
⊢ IsZero T.obj₁ ↔ T.mor₁ = 0 ∧ T.mor₃ = 0

Tactic:
refine (isZero₂_iff _ (inv_rot_of_distTriang _ hT)).trans ?_

Goal state after:
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), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
⊢ T.invRotate.mor₁ = 0 ∧ T.invRotate.mor₂ = 0 ↔ T.mor₁ = 0 ∧ T.mor₃ = 0
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: case mpr
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), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
⊢ IsIso T.mor₂ → T.mor₁ = 0 ∧ T.mor₃ = 0

Tactic:
intro

Goal state after:
case mpr
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), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
a✝ : IsIso T.mor₂
⊢ T.mor₁ = 0 ∧ T.mor₃ = 0
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: case mpr.left
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), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
a✝ : IsIso T.mor₂
⊢ Mono T.mor₂

Tactic:
infer_instance

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

Goal state before:
pp: case mpr.right
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), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
a✝ : IsIso T.mor₂
⊢ Epi T.mor₂

Tactic:
infer_instance

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

Goal state before:
pp: 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
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₂ : IsIso φ.hom₂
⊢ IsIso ((shiftFunctor C 1).map φ.hom₁)

Tactic:
infer_instance

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

Goal state before:
pp: 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
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₂ : IsIso φ.hom₂
h₃ : IsIso φ.hom₃
⊢ IsIso ((shiftFunctor C (-1)).map φ.hom₃)

Tactic:
infer_instance

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

Goal state before:
pp: 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
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₂ : IsIso φ.hom₂
h₃ : IsIso φ.hom₃
⊢ IsIso φ.hom₂

Tactic:
infer_instance

Goal state after:
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
⊢ IsIso (𝟙 T'.obj₁)

Tactic:
infer_instance

Goal state after:
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
⊢ IsIso (𝟙 T'.obj₂)

Tactic:
infer_instance

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

Goal state before:
pp: case h₁
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 := ⋯
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 := ⋯
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j := ⋯
φ' : T' ⟶ productTriangle T := ⋯
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
this✝¹ : IsIso φ'.hom₁
this✝ : IsIso φ'.hom₂
this : IsIso φ'.hom₃
⊢ IsIso φ'.hom₁

Tactic:
infer_instance

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

Goal state before:
pp: case h₂
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 := ⋯
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 := ⋯
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j := ⋯
φ' : T' ⟶ productTriangle T := ⋯
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
this✝¹ : IsIso φ'.hom₁
this✝ : IsIso φ'.hom₂
this : IsIso φ'.hom₃
⊢ IsIso φ'.hom₂

Tactic:
infer_instance

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

Goal state before:
pp: case h₃
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 := ⋯
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 := ⋯
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j := ⋯
φ' : T' ⟶ productTriangle T := ⋯
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
this✝¹ : IsIso φ'.hom₁
this✝ : IsIso φ'.hom₂
this : IsIso φ'.hom₃
⊢ IsIso φ'.hom₃

Tactic:
infer_instance

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
T₁ T₂ : Triangle (HomotopyCategory C (ComplexShape.up ℤ))
a : T₁.obj₁ ⟶ T₂.obj₁
b : T₁.obj₂ ⟶ T₂.obj₂
fac : T₁.mor₁ ≫ b = a ≫ T₂.mor₁
K₁ L₁ : CochainComplex C ℤ
φ₁ : K₁ ⟶ L₁
e₁ : T₁ ≅ triangleh φ₁
K₂ L₂ : CochainComplex C ℤ
φ₂ : K₂ ⟶ L₂
e₂ : T₂ ≅ triangleh φ₂
a' : (quotient C (ComplexShape.up ℤ)).obj K₁ ⟶ (quotient C (ComplexShape.up ℤ)).obj K₂
ha' : a' = e₁.inv.hom₁ ≫ a ≫ e₂.hom.hom₁
b' : (quotient C (ComplexShape.up ℤ)).obj L₁ ⟶ (quotient C (ComplexShape.up ℤ)).obj L₂
hb' : b' = e₁.inv.hom₂ ≫ b ≫ e₂.hom.hom₂
⊢ ∃ c,
    T₁.mor₂ ≫ c = b ≫ T₂.mor₂ ∧ T₁.mor₃ ≫ (shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) 1).map a = c ≫ T₂.mor₃

Tactic:
obtain ⟨a'', rfl⟩ := (quotient _ _).map_surjective a'

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
T₁ T₂ : Triangle (HomotopyCategory C (ComplexShape.up ℤ))
a : T₁.obj₁ ⟶ T₂.obj₁
b : T₁.obj₂ ⟶ T₂.obj₂
fac : T₁.mor₁ ≫ b = a ≫ T₂.mor₁
K₁ L₁ : CochainComplex C ℤ
φ₁ : K₁ ⟶ L₁
e₁ : T₁ ≅ triangleh φ₁
K₂ L₂ : CochainComplex C ℤ
φ₂ : K₂ ⟶ L₂
e₂ : T₂ ≅ triangleh φ₂
b' : (quotient C (ComplexShape.up ℤ)).obj L₁ ⟶ (quotient C (ComplexShape.up ℤ)).obj L₂
hb' : b' = e₁.inv.hom₂ ≫ b ≫ e₂.hom.hom₂
a'' : K₁ ⟶ K₂
ha' : (quotient C (ComplexShape.up ℤ)).map a'' = e₁.inv.hom₁ ≫ a ≫ e₂.hom.hom₁
⊢ ∃ c,
    T₁.mor₂ ≫ c = b ≫ T₂.mor₂ ∧ T₁.mor₃ ≫ (shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) 1).map a = c ≫ T₂.mor₃
Module: Mathlib.CategoryTheory.Triangulated.Functor

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
inst✝²⁰ : Category.{v_1, u_1} C
inst✝¹⁹ : Category.{v_2, u_2} D
inst✝¹⁸ : Category.{v_3, u_3} E
inst✝¹⁷ : HasShift C ℤ
inst✝¹⁶ : HasShift D ℤ
inst✝¹⁵ : HasShift E ℤ
F : C ⥤ D
inst✝¹⁴ : F.CommShift ℤ
G : D ⥤ E
inst✝¹³ : G.CommShift ℤ
inst✝¹² : HasZeroObject C
inst✝¹¹ : HasZeroObject D
inst✝¹⁰ : HasZeroObject E
inst✝⁹ : Preadditive C
inst✝⁸ : Preadditive D
inst✝⁷ : Preadditive E
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor E n).Additive
inst✝³ : Pretriangulated C
inst✝² : Pretriangulated D
inst✝¹ : Pretriangulated E
inst✝ : F.IsTriangulated
X Y : C
h₁ : 0 = 0 ≫ 𝟙 0 ≫ 0
⊢ IsIso (F.map (𝟙 0))

Tactic:
infer_instance

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

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
inst✝²⁰ : Category.{v_1, u_1} C
inst✝¹⁹ : Category.{v_2, u_2} D
inst✝¹⁸ : Category.{v_3, u_3} E
inst✝¹⁷ : HasShift C ℤ
inst✝¹⁶ : HasShift D ℤ
inst✝¹⁵ : HasShift E ℤ
F : C ⥤ D
inst✝¹⁴ : F.CommShift ℤ
G : D ⥤ E
inst✝¹³ : G.CommShift ℤ
inst✝¹² : HasZeroObject C
inst✝¹¹ : HasZeroObject D
inst✝¹⁰ : HasZeroObject E
inst✝⁹ : Preadditive C
inst✝⁸ : Preadditive D
inst✝⁷ : Preadditive E
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor E n).Additive
inst✝³ : Pretriangulated C
inst✝² : Pretriangulated D
inst✝¹ : Pretriangulated E
inst✝ : F.IsTriangulated
X₁ X₃ : C
φ : F.mapTriangle.obj (binaryProductTriangle X₁ X₃) ⟶ binaryProductTriangle (F.obj X₁) (F.obj X₃) :=
  { hom₁ := 𝟙 (F.mapTriangle.obj (binaryProductTriangle X₁ X₃)).obj₁, hom₂ := prodComparison F X₁ X₃,
    hom₃ := 𝟙 (F.mapTriangle.obj (binaryProductTriangle X₁ X₃)).obj₃, comm₁ := ⋯, comm₂ := ⋯, comm₃ := ⋯ }
⊢ IsIso (𝟙 (F.obj X₁))

Tactic:
infer_instance

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

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
inst✝²⁰ : Category.{v_1, u_1} C
inst✝¹⁹ : Category.{v_2, u_2} D
inst✝¹⁸ : Category.{v_3, u_3} E
inst✝¹⁷ : HasShift C ℤ
inst✝¹⁶ : HasShift D ℤ
inst✝¹⁵ : HasShift E ℤ
F : C ⥤ D
inst✝¹⁴ : F.CommShift ℤ
G : D ⥤ E
inst✝¹³ : G.CommShift ℤ
inst✝¹² : HasZeroObject C
inst✝¹¹ : HasZeroObject D
inst✝¹⁰ : HasZeroObject E
inst✝⁹ : Preadditive C
inst✝⁸ : Preadditive D
inst✝⁷ : Preadditive E
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor E n).Additive
inst✝³ : Pretriangulated C
inst✝² : Pretriangulated D
inst✝¹ : Pretriangulated E
inst✝ : F.IsTriangulated
X₁ X₃ : C
φ : F.mapTriangle.obj (binaryProductTriangle X₁ X₃) ⟶ binaryProductTriangle (F.obj X₁) (F.obj X₃) :=
  { hom₁ := 𝟙 (F.mapTriangle.obj (binaryProductTriangle X₁ X₃)).obj₁, hom₂ := prodComparison F X₁ X₃,
    hom₃ := 𝟙 (F.mapTriangle.obj (binaryProductTriangle X₁ X₃)).obj₃, comm₁ := ⋯, comm₂ := ⋯, comm₃ := ⋯ }
⊢ IsIso (𝟙 (F.obj X₃))

Tactic:
infer_instance

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

Goal state before:
pp: case mp
C : Type u_1
D : Type u_2
inst✝¹⁴ : Category.{v_1, u_1} C
inst✝¹³ : Category.{v_2, u_2} D
inst✝¹² : HasShift C ℤ
inst✝¹¹ : HasShift D ℤ
inst✝¹⁰ : HasZeroObject C
inst✝⁹ : HasZeroObject D
inst✝⁸ : Preadditive C
inst✝⁷ : Preadditive D
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁴ : Pretriangulated C
inst✝³ : Pretriangulated D
F₁ F₂ : C ⥤ D
e : F₁ ≅ F₂
inst✝² : F₁.CommShift ℤ
inst✝¹ : F₂.CommShift ℤ
inst✝ : NatTrans.CommShift e.hom ℤ
⊢ F₁.IsTriangulated → F₂.IsTriangulated

Tactic:
intro

Goal state after:
case mp
C : Type u_1
D : Type u_2
inst✝¹⁴ : Category.{v_1, u_1} C
inst✝¹³ : Category.{v_2, u_2} D
inst✝¹² : HasShift C ℤ
inst✝¹¹ : HasShift D ℤ
inst✝¹⁰ : HasZeroObject C
inst✝⁹ : HasZeroObject D
inst✝⁸ : Preadditive C
inst✝⁷ : Preadditive D
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁴ : Pretriangulated C
inst✝³ : Pretriangulated D
F₁ F₂ : C ⥤ D
e : F₁ ≅ F₂
inst✝² : F₁.CommShift ℤ
inst✝¹ : F₂.CommShift ℤ
inst✝ : NatTrans.CommShift e.hom ℤ
a✝ : F₁.IsTriangulated
⊢ F₂.IsTriangulated
Module: Mathlib.CategoryTheory.Triangulated.Functor

Goal state before:
pp: case mpr
C : Type u_1
D : Type u_2
inst✝¹⁴ : Category.{v_1, u_1} C
inst✝¹³ : Category.{v_2, u_2} D
inst✝¹² : HasShift C ℤ
inst✝¹¹ : HasShift D ℤ
inst✝¹⁰ : HasZeroObject C
inst✝⁹ : HasZeroObject D
inst✝⁸ : Preadditive C
inst✝⁷ : Preadditive D
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁴ : Pretriangulated C
inst✝³ : Pretriangulated D
F₁ F₂ : C ⥤ D
e : F₁ ≅ F₂
inst✝² : F₁.CommShift ℤ
inst✝¹ : F₂.CommShift ℤ
inst✝ : NatTrans.CommShift e.hom ℤ
⊢ F₂.IsTriangulated → F₁.IsTriangulated

Tactic:
intro

Goal state after:
case mpr
C : Type u_1
D : Type u_2
inst✝¹⁴ : Category.{v_1, u_1} C
inst✝¹³ : Category.{v_2, u_2} D
inst✝¹² : HasShift C ℤ
inst✝¹¹ : HasShift D ℤ
inst✝¹⁰ : HasZeroObject C
inst✝⁹ : HasZeroObject D
inst✝⁸ : Preadditive C
inst✝⁷ : Preadditive D
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁴ : Pretriangulated C
inst✝³ : Pretriangulated D
F₁ F₂ : C ⥤ D
e : F₁ ≅ F₂
inst✝² : F₁.CommShift ℤ
inst✝¹ : F₂.CommShift ℤ
inst✝ : NatTrans.CommShift e.hom ℤ
a✝ : F₂.IsTriangulated
⊢ F₁.IsTriangulated
Module: Mathlib.CategoryTheory.Triangulated.Functor

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹⁴ : Category.{v_1, u_1} C
inst✝¹³ : Category.{v_2, u_2} D
inst✝¹² : HasShift C ℤ
inst✝¹¹ : HasShift D ℤ
F : C ⥤ D
inst✝¹⁰ : F.CommShift ℤ
inst✝⁹ : HasZeroObject C
inst✝⁸ : HasZeroObject D
inst✝⁷ : Preadditive C
inst✝⁶ : Preadditive D
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝³ : Pretriangulated C
inst✝² : Pretriangulated D
inst✝¹ : F.IsTriangulated
inst✝ : F.mapArrow.EssSurj
T : Triangle D
hT : T ∈ distinguishedTriangles
X Y : C
f : X ⟶ Y
e₁ : F.obj X ≅ T.obj₁
e₂ : F.obj Y ≅ T.obj₂
w : F.map f ≫ e₂.hom = e₁.hom ≫ T.mor₁
⊢ ∃ T', ∃ (_ : T' ∈ distinguishedTriangles), Nonempty (F.mapTriangle.obj T' ≅ T)

Tactic:
obtain ⟨W, g, h, H⟩ := distinguished_cocone_triangle f

Goal state after:
C : Type u_1
D : Type u_2
inst✝¹⁴ : Category.{v_1, u_1} C
inst✝¹³ : Category.{v_2, u_2} D
inst✝¹² : HasShift C ℤ
inst✝¹¹ : HasShift D ℤ
F : C ⥤ D
inst✝¹⁰ : F.CommShift ℤ
inst✝⁹ : HasZeroObject C
inst✝⁸ : HasZeroObject D
inst✝⁷ : Preadditive C
inst✝⁶ : Preadditive D
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝³ : Pretriangulated C
inst✝² : Pretriangulated D
inst✝¹ : F.IsTriangulated
inst✝ : F.mapArrow.EssSurj
T : Triangle D
hT : T ∈ distinguishedTriangles
X Y : C
f : X ⟶ Y
e₁ : F.obj X ≅ T.obj₁
e₂ : F.obj Y ≅ T.obj₂
w : F.map f ≫ e₂.hom = e₁.hom ≫ T.mor₁
W : C
g : Y ⟶ W
h : W ⟶ (shiftFunctor C 1).obj X
H : Triangle.mk f g h ∈ distinguishedTriangles
⊢ ∃ T', ∃ (_ : T' ∈ distinguishedTriangles), Nonempty (F.mapTriangle.obj T' ≅ T)
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: 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
T₁ T₂ : Triangle C
hT₁ : T₁ ∈ distinguishedTriangles
hT₂ : T₂ ∈ distinguishedTriangles
e : Arrow.mk T₁.mor₁ ≅ Arrow.mk T₂.mor₁
φ : T₁ ⟶ T₂ := completeDistinguishedTriangleMorphism T₁ T₂ hT₁ hT₂ (Arrow.Hom.left e.hom) (Arrow.Hom.right e.hom) ⋯
⊢ IsIso (Arrow.Hom.left e.hom)

Tactic:
infer_instance

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

Goal state before:
pp: 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
T₁ T₂ : Triangle C
hT₁ : T₁ ∈ distinguishedTriangles
hT₂ : T₂ ∈ distinguishedTriangles
e : Arrow.mk T₁.mor₁ ≅ Arrow.mk T₂.mor₁
φ : T₁ ⟶ T₂ := completeDistinguishedTriangleMorphism T₁ T₂ hT₁ hT₂ (Arrow.Hom.left e.hom) (Arrow.Hom.right e.hom) ⋯
this : IsIso φ.hom₁
⊢ IsIso (Arrow.Hom.right e.hom)

Tactic:
infer_instance

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

Goal state before:
pp: case 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), (shiftFunctor C n).Additive
hC : Pretriangulated C
T₁ T₂ : Triangle C
hT₁ : T₁ ∈ distinguishedTriangles
hT₂ : T₂ ∈ distinguishedTriangles
e : Arrow.mk T₁.mor₁ ≅ Arrow.mk T₂.mor₁
φ : T₁ ⟶ T₂ := ⋯
this✝¹ : IsIso φ.hom₁
this✝ : IsIso φ.hom₂
this : IsIso φ.hom₃
⊢ IsIso φ.hom₁

Tactic:
infer_instance

Goal state after: