mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Triangulated.Functor

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
inst✝¹⁰ : Category.{v_1, u_1} C
inst✝⁹ : Category.{v_2, u_2} D
inst✝⁸ : Category.{v_3, u_3} E
inst✝⁷ : HasShift C ℤ
inst✝⁶ : HasShift D ℤ
inst✝⁵ : HasShift E ℤ
F : C ⥤ D
inst✝⁴ : F.CommShift ℤ
G : D ⥤ E
inst✝³ : G.CommShift ℤ
inst✝² : Preadditive C
inst✝¹ : Preadditive D
inst✝ : F.Additive
n : ℤ
T : Triangle C
⊢ ((Triangle.shiftFunctor C n ⋙ F.mapTriangle).obj T).mor₃ ≫
      (shiftFunctor D 1).map ((commShiftIso F n).app T.obj₁).hom =
    ((commShiftIso F n).app T.obj₃).hom ≫ ((F.mapTriangle ⋙ Triangle.shiftFunctor D n).obj T).mor₃

Tactic:
dsimp

Goal state after:
C : Type u_1
D : Type u_2
E : Type u_3
inst✝¹⁰ : Category.{v_1, u_1} C
inst✝⁹ : Category.{v_2, u_2} D
inst✝⁸ : Category.{v_3, u_3} E
inst✝⁷ : HasShift C ℤ
inst✝⁶ : HasShift D ℤ
inst✝⁵ : HasShift E ℤ
F : C ⥤ D
inst✝⁴ : F.CommShift ℤ
G : D ⥤ E
inst✝³ : G.CommShift ℤ
inst✝² : Preadditive C
inst✝¹ : Preadditive D
inst✝ : F.Additive
n : ℤ
T : Triangle C
⊢ (F.map (n.negOnePow • (shiftFunctor C n).map T.mor₃ ≫ (shiftFunctorComm C 1 n).hom.app T.obj₁) ≫
        (commShiftIso F 1).hom.app ((shiftFunctor C n).obj T.obj₁)) ≫
      (shiftFunctor D 1).map ((commShiftIso F n).hom.app T.obj₁) =
    (commShiftIso F n).hom.app T.obj₃ ≫
      (n.negOnePow •
        (shiftFunctor D n).map (F.map T.mor₃ ≫ (commShiftIso F 1).hom.app T.obj₁) ≫
          (shiftFunctorComm D 1 n).hom.app (F.obj T.obj₁))
Module: Mathlib.CategoryTheory.Triangulated.Functor

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
inst✝¹³ : Category.{v_1, u_1} C
inst✝¹² : Category.{v_2, u_2} D
inst✝¹¹ : Category.{v_3, u_3} E
inst✝¹⁰ : HasShift C ℤ
inst✝⁹ : HasShift D ℤ
inst✝⁸ : HasShift E ℤ
F : C ⥤ D
inst✝⁷ : F.CommShift ℤ
G : D ⥤ E
inst✝⁶ : G.CommShift ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : Preadditive D
inst✝³ : F.Additive
F₁ F₂ : C ⥤ D
e : F₁ ≅ F₂
inst✝² : F₁.CommShift ℤ
inst✝¹ : F₂.CommShift ℤ
inst✝ : NatTrans.CommShift e.hom ℤ
T : Triangle C
⊢ (F₁.mapTriangle.obj T).mor₃ ≫ (shiftFunctor D 1).map (e.app T.obj₁).hom =
    (e.app T.obj₃).hom ≫ (F₂.mapTriangle.obj T).mor₃

Tactic:
dsimp

Goal state after:
C : Type u_1
D : Type u_2
E : Type u_3
inst✝¹³ : Category.{v_1, u_1} C
inst✝¹² : Category.{v_2, u_2} D
inst✝¹¹ : Category.{v_3, u_3} E
inst✝¹⁰ : HasShift C ℤ
inst✝⁹ : HasShift D ℤ
inst✝⁸ : HasShift E ℤ
F : C ⥤ D
inst✝⁷ : F.CommShift ℤ
G : D ⥤ E
inst✝⁶ : G.CommShift ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : Preadditive D
inst✝³ : F.Additive
F₁ F₂ : C ⥤ D
e : F₁ ≅ F₂
inst✝² : F₁.CommShift ℤ
inst✝¹ : F₂.CommShift ℤ
inst✝ : NatTrans.CommShift e.hom ℤ
T : Triangle C
⊢ (F₁.map T.mor₃ ≫ (commShiftIso F₁ 1).hom.app T.obj₁) ≫ (shiftFunctor D 1).map (e.hom.app T.obj₁) =
    e.hom.app T.obj₃ ≫ F₂.map T.mor₃ ≫ (commShiftIso F₂ 1).hom.app T.obj₁
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.mapTriangle.obj (contractibleTriangle 0)).mor₁

Tactic:
dsimp

Goal state after:
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))
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
⊢ (F.mapTriangle.obj (binaryProductTriangle X₁ X₃)).mor₁ ≫ prodComparison F X₁ X₃ =
    𝟙 (F.mapTriangle.obj (binaryProductTriangle X₁ X₃)).obj₁ ≫ (binaryProductTriangle (F.obj X₁) (F.obj X₃)).mor₁

Tactic:
dsimp

Goal state after:
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
⊢ F.map (prod.lift (𝟙 X₁) 0) ≫ prodComparison F X₁ X₃ = 𝟙 (F.obj X₁) ≫ prod.lift (𝟙 (F.obj X₁)) 0
Module: Mathlib.CategoryTheory.Triangulated.Functor

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹³ : Category.{v_1, u_1} C
inst✝¹² : Category.{v_2, u_2} D
inst✝¹¹ : HasShift C ℤ
inst✝¹⁰ : HasShift D 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
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : X₂ ⟶ Z₁₂
w₁₂ : Z₁₂ ⟶ (shiftFunctor C 1).obj X₁
h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ distinguishedTriangles
v₂₃ : X₃ ⟶ Z₂₃
w₂₃ : Z₂₃ ⟶ (shiftFunctor C 1).obj X₂
h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ distinguishedTriangles
v₁₃ : X₃ ⟶ Z₁₃
w₁₃ : Z₁₃ ⟶ (shiftFunctor C 1).obj X₁
h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ distinguishedTriangles
h : Octahedron comm h₁₂ h₂₃ h₁₃
F : C ⥤ D
inst✝¹ : F.CommShift ℤ
inst✝ : F.IsTriangulated
⊢ F.map (Triangle.mk u₁₂ v₁₂ w₁₂).mor₁ ≫ F.map (Triangle.mk u₂₃ v₂₃ w₂₃).mor₁ = F.map (Triangle.mk u₁₃ v₁₃ w₁₃).mor₁

Tactic:
dsimp

Goal state after:
C : Type u_1
D : Type u_2
inst✝¹³ : Category.{v_1, u_1} C
inst✝¹² : Category.{v_2, u_2} D
inst✝¹¹ : HasShift C ℤ
inst✝¹⁰ : HasShift D 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
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : X₂ ⟶ Z₁₂
w₁₂ : Z₁₂ ⟶ (shiftFunctor C 1).obj X₁
h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ distinguishedTriangles
v₂₃ : X₃ ⟶ Z₂₃
w₂₃ : Z₂₃ ⟶ (shiftFunctor C 1).obj X₂
h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ distinguishedTriangles
v₁₃ : X₃ ⟶ Z₁₃
w₁₃ : Z₁₃ ⟶ (shiftFunctor C 1).obj X₁
h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ distinguishedTriangles
h : Octahedron comm h₁₂ h₂₃ h₁₃
F : C ⥤ D
inst✝¹ : F.CommShift ℤ
inst✝ : F.IsTriangulated
⊢ F.map u₁₂ ≫ F.map u₂₃ = F.map u₁₃
Module: Mathlib.CategoryTheory.Triangulated.Triangulated

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Z₁₂ ⟶ X₁
w₁₂ : X₂ ⟶ (shiftFunctor C 1).obj Z₁₂
h₁₂ : Triangle.mk v₁₂ u₁₂ w₁₂ ∈ distinguishedTriangles
v₂₃ : Z₂₃ ⟶ X₂
w₂₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₂₃
h₂₃ : Triangle.mk v₂₃ u₂₃ w₂₃ ∈ distinguishedTriangles
v₁₃ : Z₁₃ ⟶ X₁
w₁₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₁₃
h₁₃ : Triangle.mk v₁₃ u₁₃ w₁₃ ∈ distinguishedTriangles
h : Octahedron' comm h₁₂ h₂₃ h₁₃
⊢ (Triangle.mk v₁₂ u₁₂ w₁₂).mor₁ ≫ 𝟙 X₁ = h.m₁ ≫ (Triangle.mk v₁₃ u₁₃ w₁₃).mor₁

Tactic:
dsimp

Goal state after:
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Z₁₂ ⟶ X₁
w₁₂ : X₂ ⟶ (shiftFunctor C 1).obj Z₁₂
h₁₂ : Triangle.mk v₁₂ u₁₂ w₁₂ ∈ distinguishedTriangles
v₂₃ : Z₂₃ ⟶ X₂
w₂₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₂₃
h₂₃ : Triangle.mk v₂₃ u₂₃ w₂₃ ∈ distinguishedTriangles
v₁₃ : Z₁₃ ⟶ X₁
w₁₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₁₃
h₁₃ : Triangle.mk v₁₃ u₁₃ w₁₃ ∈ distinguishedTriangles
h : Octahedron' comm h₁₂ h₂₃ h₁₃
⊢ v₁₂ ≫ 𝟙 X₁ = h.m₁ ≫ v₁₃
Module: Mathlib.CategoryTheory.Triangulated.Triangulated

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Z₁₂ ⟶ X₁
w₁₂ : X₂ ⟶ (shiftFunctor C 1).obj Z₁₂
h₁₂ : Triangle.mk v₁₂ u₁₂ w₁₂ ∈ distinguishedTriangles
v₂₃ : Z₂₃ ⟶ X₂
w₂₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₂₃
h₂₃ : Triangle.mk v₂₃ u₂₃ w₂₃ ∈ distinguishedTriangles
v₁₃ : Z₁₃ ⟶ X₁
w₁₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₁₃
h₁₃ : Triangle.mk v₁₃ u₁₃ w₁₃ ∈ distinguishedTriangles
h : Octahedron' comm h₁₂ h₂₃ h₁₃
⊢ (Triangle.mk v₁₂ u₁₂ w₁₂).mor₂ ≫ u₂₃ = 𝟙 X₁ ≫ (Triangle.mk v₁₃ u₁₃ w₁₃).mor₂

Tactic:
dsimp

Goal state after:
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Z₁₂ ⟶ X₁
w₁₂ : X₂ ⟶ (shiftFunctor C 1).obj Z₁₂
h₁₂ : Triangle.mk v₁₂ u₁₂ w₁₂ ∈ distinguishedTriangles
v₂₃ : Z₂₃ ⟶ X₂
w₂₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₂₃
h₂₃ : Triangle.mk v₂₃ u₂₃ w₂₃ ∈ distinguishedTriangles
v₁₃ : Z₁₃ ⟶ X₁
w₁₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₁₃
h₁₃ : Triangle.mk v₁₃ u₁₃ w₁₃ ∈ distinguishedTriangles
h : Octahedron' comm h₁₂ h₂₃ h₁₃
⊢ u₁₂ ≫ u₂₃ = 𝟙 X₁ ≫ u₁₃
Module: Mathlib.CategoryTheory.Triangulated.Triangulated

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Z₁₂ ⟶ X₁
w₁₂ : X₂ ⟶ (shiftFunctor C 1).obj Z₁₂
h₁₂ : Triangle.mk v₁₂ u₁₂ w₁₂ ∈ distinguishedTriangles
v₂₃ : Z₂₃ ⟶ X₂
w₂₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₂₃
h₂₃ : Triangle.mk v₂₃ u₂₃ w₂₃ ∈ distinguishedTriangles
v₁₃ : Z₁₃ ⟶ X₁
w₁₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₁₃
h₁₃ : Triangle.mk v₁₃ u₁₃ w₁₃ ∈ distinguishedTriangles
h : Octahedron' comm h₁₂ h₂₃ h₁₃
⊢ (Triangle.mk v₁₂ u₁₂ w₁₂).mor₃ ≫ (shiftFunctor C 1).map h.m₁ = u₂₃ ≫ (Triangle.mk v₁₃ u₁₃ w₁₃).mor₃

Tactic:
dsimp

Goal state after:
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Z₁₂ ⟶ X₁
w₁₂ : X₂ ⟶ (shiftFunctor C 1).obj Z₁₂
h₁₂ : Triangle.mk v₁₂ u₁₂ w₁₂ ∈ distinguishedTriangles
v₂₃ : Z₂₃ ⟶ X₂
w₂₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₂₃
h₂₃ : Triangle.mk v₂₃ u₂₃ w₂₃ ∈ distinguishedTriangles
v₁₃ : Z₁₃ ⟶ X₁
w₁₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₁₃
h₁₃ : Triangle.mk v₁₃ u₁₃ w₁₃ ∈ distinguishedTriangles
h : Octahedron' comm h₁₂ h₂₃ h₁₃
⊢ w₁₂ ≫ (shiftFunctor C 1).map h.m₁ = u₂₃ ≫ w₁₃
Module: Mathlib.CategoryTheory.Triangulated.Triangulated

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Z₁₂ ⟶ X₁
w₁₂ : X₂ ⟶ (shiftFunctor C 1).obj Z₁₂
h₁₂ : Triangle.mk v₁₂ u₁₂ w₁₂ ∈ distinguishedTriangles
v₂₃ : Z₂₃ ⟶ X₂
w₂₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₂₃
h₂₃ : Triangle.mk v₂₃ u₂₃ w₂₃ ∈ distinguishedTriangles
v₁₃ : Z₁₃ ⟶ X₁
w₁₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₁₃
h₁₃ : Triangle.mk v₁₃ u₁₃ w₁₃ ∈ distinguishedTriangles
h : Octahedron' comm h₁₂ h₂₃ h₁₃
⊢ (Triangle.mk v₁₃ u₁₃ w₁₃).mor₁ ≫ u₁₂ = h.m₃ ≫ (Triangle.mk v₂₃ u₂₃ w₂₃).mor₁

Tactic:
dsimp

Goal state after:
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Z₁₂ ⟶ X₁
w₁₂ : X₂ ⟶ (shiftFunctor C 1).obj Z₁₂
h₁₂ : Triangle.mk v₁₂ u₁₂ w₁₂ ∈ distinguishedTriangles
v₂₃ : Z₂₃ ⟶ X₂
w₂₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₂₃
h₂₃ : Triangle.mk v₂₃ u₂₃ w₂₃ ∈ distinguishedTriangles
v₁₃ : Z₁₃ ⟶ X₁
w₁₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₁₃
h₁₃ : Triangle.mk v₁₃ u₁₃ w₁₃ ∈ distinguishedTriangles
h : Octahedron' comm h₁₂ h₂₃ h₁₃
⊢ v₁₃ ≫ u₁₂ = h.m₃ ≫ v₂₃
Module: Mathlib.CategoryTheory.Triangulated.Triangulated

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Z₁₂ ⟶ X₁
w₁₂ : X₂ ⟶ (shiftFunctor C 1).obj Z₁₂
h₁₂ : Triangle.mk v₁₂ u₁₂ w₁₂ ∈ distinguishedTriangles
v₂₃ : Z₂₃ ⟶ X₂
w₂₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₂₃
h₂₃ : Triangle.mk v₂₃ u₂₃ w₂₃ ∈ distinguishedTriangles
v₁₃ : Z₁₃ ⟶ X₁
w₁₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₁₃
h₁₃ : Triangle.mk v₁₃ u₁₃ w₁₃ ∈ distinguishedTriangles
h : Octahedron' comm h₁₂ h₂₃ h₁₃
⊢ (Triangle.mk v₁₃ u₁₃ w₁₃).mor₂ ≫ 𝟙 X₃ = u₁₂ ≫ (Triangle.mk v₂₃ u₂₃ w₂₃).mor₂

Tactic:
dsimp

Goal state after:
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Z₁₂ ⟶ X₁
w₁₂ : X₂ ⟶ (shiftFunctor C 1).obj Z₁₂
h₁₂ : Triangle.mk v₁₂ u₁₂ w₁₂ ∈ distinguishedTriangles
v₂₃ : Z₂₃ ⟶ X₂
w₂₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₂₃
h₂₃ : Triangle.mk v₂₃ u₂₃ w₂₃ ∈ distinguishedTriangles
v₁₃ : Z₁₃ ⟶ X₁
w₁₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₁₃
h₁₃ : Triangle.mk v₁₃ u₁₃ w₁₃ ∈ distinguishedTriangles
h : Octahedron' comm h₁₂ h₂₃ h₁₃
⊢ u₁₃ ≫ 𝟙 X₃ = u₁₂ ≫ u₂₃
Module: Mathlib.CategoryTheory.Triangulated.Triangulated

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Z₁₂ ⟶ X₁
w₁₂ : X₂ ⟶ (shiftFunctor C 1).obj Z₁₂
h₁₂ : Triangle.mk v₁₂ u₁₂ w₁₂ ∈ distinguishedTriangles
v₂₃ : Z₂₃ ⟶ X₂
w₂₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₂₃
h₂₃ : Triangle.mk v₂₃ u₂₃ w₂₃ ∈ distinguishedTriangles
v₁₃ : Z₁₃ ⟶ X₁
w₁₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₁₃
h₁₃ : Triangle.mk v₁₃ u₁₃ w₁₃ ∈ distinguishedTriangles
h : Octahedron' comm h₁₂ h₂₃ h₁₃
⊢ (Triangle.mk v₁₃ u₁₃ w₁₃).mor₃ ≫ (shiftFunctor C 1).map h.m₃ = 𝟙 X₃ ≫ (Triangle.mk v₂₃ u₂₃ w₂₃).mor₃

Tactic:
dsimp

Goal state after:
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Z₁₂ ⟶ X₁
w₁₂ : X₂ ⟶ (shiftFunctor C 1).obj Z₁₂
h₁₂ : Triangle.mk v₁₂ u₁₂ w₁₂ ∈ distinguishedTriangles
v₂₃ : Z₂₃ ⟶ X₂
w₂₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₂₃
h₂₃ : Triangle.mk v₂₃ u₂₃ w₂₃ ∈ distinguishedTriangles
v₁₃ : Z₁₃ ⟶ X₁
w₁₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₁₃
h₁₃ : Triangle.mk v₁₃ u₁₃ w₁₃ ∈ distinguishedTriangles
h : Octahedron' comm h₁₂ h₂₃ h₁₃
⊢ w₁₃ ≫ (shiftFunctor C 1).map h.m₃ = 𝟙 X₃ ≫ w₂₃
Module: Mathlib.CategoryTheory.Triangulated.Triangulated

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Z₁₂ ⟶ X₁
w₁₂ : X₂ ⟶ (shiftFunctor C 1).obj Z₁₂
h₁₂ : Triangle.mk v₁₂ u₁₂ w₁₂ ∈ distinguishedTriangles
v₂₃ : Z₂₃ ⟶ X₂
w₂₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₂₃
h₂₃ : Triangle.mk v₂₃ u₂₃ w₂₃ ∈ distinguishedTriangles
v₁₃ : Z₁₃ ⟶ X₁
w₁₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₁₃
h₁₃ : Triangle.mk v₁₃ u₁₃ w₁₃ ∈ distinguishedTriangles
h : Octahedron' comm h₁₂ h₂₃ h₁₃
⊢ w₁₃ ≫ (shiftFunctor C 1).map h.m₃ = 𝟙 X₃ ≫ w₂₃

Tactic:
rw [id_comp, h.comm₃]

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
S : ShortComplex (CochainComplex C ℤ)
σ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting
inst✝ : HasBinaryBiproducts C
⊢ (triangleOfDegreewiseSplit S σ).rotate.rotate.mor₁ ≫
      (Iso.refl (triangleOfDegreewiseSplit S σ).rotate.rotate.obj₂).hom =
    (Iso.refl (triangleOfDegreewiseSplit S σ).rotate.rotate.obj₁).hom ≫
      (mappingCone.triangle (homOfDegreewiseSplit S σ)).mor₁

Tactic:
dsimp

Goal state after:
C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
S : ShortComplex (CochainComplex C ℤ)
σ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting
inst✝ : HasBinaryBiproducts C
⊢ homOfDegreewiseSplit S σ ≫ 𝟙 ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) 1).obj S.X₁) =
    𝟙 S.X₃ ≫ homOfDegreewiseSplit S σ
Module: Mathlib.Algebra.Homology.HomotopyCategory.DegreewiseSplit

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
S : ShortComplex (CochainComplex C ℤ)
σ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting
inst✝ : HasBinaryBiproducts C
⊢ (triangleOfDegreewiseSplit S σ).rotate.rotate.mor₂ ≫ (mappingConeHomOfDegreewiseSplitIso S σ).symm.hom =
    (Iso.refl (triangleOfDegreewiseSplit S σ).rotate.rotate.obj₂).hom ≫
      (mappingCone.triangle (homOfDegreewiseSplit S σ)).mor₂

Tactic:
dsimp

Goal state after:
C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
S : ShortComplex (CochainComplex C ℤ)
σ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting
inst✝ : HasBinaryBiproducts C
⊢ (-(CategoryTheory.shiftFunctor (CochainComplex C ℤ) 1).map S.f) ≫ (mappingConeHomOfDegreewiseSplitIso S σ).inv =
    𝟙 ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) 1).obj S.X₁) ≫ mappingCone.inr (homOfDegreewiseSplit S σ)
Module: Mathlib.Algebra.Homology.HomotopyCategory.DegreewiseSplit

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
S : ShortComplex (CochainComplex C ℤ)
σ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting
inst✝ : HasBinaryBiproducts C
⊢ (triangleOfDegreewiseSplit S σ).rotate.rotate.mor₃ ≫
      (CategoryTheory.shiftFunctor (CochainComplex C ℤ) 1).map
        (Iso.refl (triangleOfDegreewiseSplit S σ).rotate.rotate.obj₁).hom =
    (mappingConeHomOfDegreewiseSplitIso S σ).symm.hom ≫ (mappingCone.triangle (homOfDegreewiseSplit S σ)).mor₃

Tactic:
dsimp

Goal state after:
C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
S : ShortComplex (CochainComplex C ℤ)
σ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting
inst✝ : HasBinaryBiproducts C
⊢ (-(CategoryTheory.shiftFunctor (CochainComplex C ℤ) 1).map S.g) ≫
      (CategoryTheory.shiftFunctor (CochainComplex C ℤ) 1).map (𝟙 S.X₃) =
    (mappingConeHomOfDegreewiseSplitIso S σ).inv ≫ (mappingCone.triangle (homOfDegreewiseSplit S σ)).mor₃
Module: Mathlib.Algebra.Homology.HomologySequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
c : ComplexShape ι
K : HomologicalComplex C c
i j : ι
hij : c.Rel i j
inst✝¹ : K.HasHomology i
inst✝ : K.HasHomology j
⊢ Mono
    ((composableArrows₃ K i j).map' 0 1 instMonoMap'ComposableArrows₃OfNatNat._proof_1
      instMonoMap'ComposableArrows₃OfNatNat._proof_3)

Tactic:
dsimp

Goal state after:
C : Type u_1
ι : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
c : ComplexShape ι
K : HomologicalComplex C c
i j : ι
hij : c.Rel i j
inst✝¹ : K.HasHomology i
inst✝ : K.HasHomology j
⊢ Mono (K.homologyι i)
Module: Mathlib.Algebra.Homology.HomologySequence

Goal state before:
pp: C : Type u_1
ι✝ : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
c : ComplexShape ι✝
K : HomologicalComplex C c
i j : ι✝
hij : c.Rel i j
inst✝ : CategoryWithHomology C
S : ShortComplex C :=
  { X₁ := K.homology i, X₂ := K.opcycles i, X₃ := K.cycles j, f := K.homologyι i, g := K.opcyclesToCycles i j,
    zero := ⋯ }
S' : ShortComplex C :=
  { X₁ := K.homology i, X₂ := K.opcycles i, X₃ := K.X j, f := K.homologyι i, g := K.fromOpcycles i j, zero := ⋯ }
ι : S ⟶ S' := { τ₁ := 𝟙 S.X₁, τ₂ := 𝟙 S.X₂, τ₃ := K.iCycles j, comm₁₂ := ⋯, comm₂₃ := ⋯ }
hS : S.Exact
T : ShortComplex C :=
  { X₁ := K.opcycles i, X₂ := K.cycles j, X₃ := K.homology j, f := K.opcyclesToCycles i j, g := K.homologyπ j,
    zero := ⋯ }
⊢ (composableArrows₃ K i j).Exact

Tactic:
let T' := ShortComplex.mk (K.toCycles i j) (K.homologyπ j) (by simp)

Goal state after:
C : Type u_1
ι✝ : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
c : ComplexShape ι✝
K : HomologicalComplex C c
i j : ι✝
hij : c.Rel i j
inst✝ : CategoryWithHomology C
S : ShortComplex C :=
  { X₁ := K.homology i, X₂ := K.opcycles i, X₃ := K.cycles j, f := K.homologyι i, g := K.opcyclesToCycles i j,
    zero := ⋯ }
S' : ShortComplex C :=
  { X₁ := K.homology i, X₂ := K.opcycles i, X₃ := K.X j, f := K.homologyι i, g := K.fromOpcycles i j, zero := ⋯ }
ι : S ⟶ S' := { τ₁ := 𝟙 S.X₁, τ₂ := 𝟙 S.X₂, τ₃ := K.iCycles j, comm₁₂ := ⋯, comm₂₃ := ⋯ }
hS : S.Exact
T : ShortComplex C :=
  { X₁ := K.opcycles i, X₂ := K.cycles j, X₃ := K.homology j, f := K.opcyclesToCycles i j, g := K.homologyπ j,
    zero := ⋯ }
T' : ShortComplex C :=
  { X₁ := K.X i, X₂ := K.cycles j, X₃ := K.homology j, f := K.toCycles i j, g := K.homologyπ j, zero := ⋯ }
⊢ (composableArrows₃ K i j).Exact