Module: Mathlib.CategoryTheory.Triangulated.Triangulated

Goal state before:
pp: case refine_4
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 : 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
X₁' X₂' X₃' Z₁₂' Z₂₃' Z₁₃' : C
u₁₂' : X₁' ⟶ X₂'
u₂₃' : X₂' ⟶ X₃'
u₁₃' : X₁' ⟶ X₃'
comm' : u₁₂' ≫ u₂₃' = u₁₃'
e₁ : X₁ ≅ X₁'
e₂ : X₂ ≅ X₂'
e₃ : X₃ ≅ X₃'
comm₁₂ : u₁₂ ≫ e₂.hom = e₁.hom ≫ u₁₂'
comm₂₃ : u₂₃ ≫ e₃.hom = e₂.hom ≫ u₂₃'
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'
iso₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ≅ Triangle.mk u₁₂' v₁₂' w₁₂' :=
  isoTriangleOfIso₁₂ (Triangle.mk u₁₂ v₁₂ w₁₂) (Triangle.mk u₁₂' v₁₂' w₁₂') h₁₂ h₁₂' e₁ e₂ comm₁₂
iso₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ≅ Triangle.mk u₂₃' v₂₃' w₂₃' :=
  isoTriangleOfIso₁₂ (Triangle.mk u₂₃ v₂₃ w₂₃) (Triangle.mk u₂₃' v₂₃' w₂₃') h₂₃ h₂₃' e₂ e₃ comm₂₃
iso₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ≅ Triangle.mk u₁₃' v₁₃' w₁₃' :=
  isoTriangleOfIso₁₂ (Triangle.mk u₁₃ v₁₃ w₁₃) (Triangle.mk u₁₃' v₁₃' w₁₃') h₁₃ h₁₃' e₁ e₃ ⋯
eq₁₂ :
  v₁₂ ≫ (isoTriangleOfIso₁₂ (Triangle.mk u₁₂ v₁₂ w₁₂) (Triangle.mk u₁₂' v₁₂' w₁₂') h₁₂ h₁₂' e₁ e₂ comm₁₂).hom.hom₃ =
    e₂.hom ≫ v₁₂'
eq₁₂' :
  w₁₂ ≫ (shiftFunctor C 1).map e₁.hom =
    (isoTriangleOfIso₁₂ (Triangle.mk u₁₂ v₁₂ w₁₂) (Triangle.mk u₁₂' v₁₂' w₁₂') h₁₂ h₁₂' e₁ e₂ comm₁₂).hom.hom₃ ≫ w₁₂'
eq₁₃ :
  v₁₃ ≫ (isoTriangleOfIso₁₂ (Triangle.mk u₁₃ v₁₃ w₁₃) (Triangle.mk u₁₃' v₁₃' w₁₃') h₁₃ h₁₃' e₁ e₃ ⋯).hom.hom₃ =
    e₃.hom ≫ v₁₃'
eq₁₃' :
  w₁₃ ≫ (shiftFunctor C 1).map e₁.hom =
    (isoTriangleOfIso₁₂ (Triangle.mk u₁₃ v₁₃ w₁₃) (Triangle.mk u₁₃' v₁₃' w₁₃') h₁₃ h₁₃' e₁ e₃ ⋯).hom.hom₃ ≫ w₁₃'
eq₂₃ :
  v₂₃ ≫ (isoTriangleOfIso₁₂ (Triangle.mk u₂₃ v₂₃ w₂₃) (Triangle.mk u₂₃' v₂₃' w₂₃') h₂₃ h₂₃' e₂ e₃ comm₂₃).hom.hom₃ =
    e₃.hom ≫ v₂₃'
eq₂₃' :
  w₂₃ ≫ (shiftFunctor C 1).map e₂.hom =
    (isoTriangleOfIso₁₂ (Triangle.mk u₂₃ v₂₃ w₂₃) (Triangle.mk u₂₃' v₂₃' w₂₃') h₂₃ h₂₃' e₂ e₃ comm₂₃).hom.hom₃ ≫ w₂₃'
rel₁₂ : v₁₂' ≫ H.m₁ = u₂₃' ≫ v₁₃'
rel₁₃ : w₁₂' = H.m₁ ≫ w₁₃'
rel₂₂ : v₁₃' ≫ H.m₃ = v₂₃'
rel₂₃ : w₁₃' ≫ (shiftFunctor C 1).map u₁₂' = H.m₃ ≫ w₂₃'
⊢ w₁₃ ≫ (shiftFunctor C 1).map u₁₂ = (iso₁₃.hom.hom₃ ≫ H.m₃ ≫ iso₂₃.inv.hom₃) ≫ w₂₃

pp: case refine_5
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 : 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
X₁' X₂' X₃' Z₁₂' Z₂₃' Z₁₃' : C
u₁₂' : X₁' ⟶ X₂'
u₂₃' : X₂' ⟶ X₃'
u₁₃' : X₁' ⟶ X₃'
comm' : u₁₂' ≫ u₂₃' = u₁₃'
e₁ : X₁ ≅ X₁'
e₂ : X₂ ≅ X₂'
e₃ : X₃ ≅ X₃'
comm₁₂ : u₁₂ ≫ e₂.hom = e₁.hom ≫ u₁₂'
comm₂₃ : u₂₃ ≫ e₃.hom = e₂.hom ≫ u₂₃'
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'
iso₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ≅ Triangle.mk u₁₂' v₁₂' w₁₂' :=
  isoTriangleOfIso₁₂ (Triangle.mk u₁₂ v₁₂ w₁₂) (Triangle.mk u₁₂' v₁₂' w₁₂') h₁₂ h₁₂' e₁ e₂ comm₁₂
iso₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ≅ Triangle.mk u₂₃' v₂₃' w₂₃' :=
  isoTriangleOfIso₁₂ (Triangle.mk u₂₃ v₂₃ w₂₃) (Triangle.mk u₂₃' v₂₃' w₂₃') h₂₃ h₂₃' e₂ e₃ comm₂₃
iso₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ≅ Triangle.mk u₁₃' v₁₃' w₁₃' :=
  isoTriangleOfIso₁₂ (Triangle.mk u₁₃ v₁₃ w₁₃) (Triangle.mk u₁₃' v₁₃' w₁₃') h₁₃ h₁₃' e₁ e₃ ⋯
eq₁₂ :
  v₁₂ ≫ (isoTriangleOfIso₁₂ (Triangle.mk u₁₂ v₁₂ w₁₂) (Triangle.mk u₁₂' v₁₂' w₁₂') h₁₂ h₁₂' e₁ e₂ comm₁₂).hom.hom₃ =
    e₂.hom ≫ v₁₂'
eq₁₂' :
  w₁₂ ≫ (shiftFunctor C 1).map e₁.hom =
    (isoTriangleOfIso₁₂ (Triangle.mk u₁₂ v₁₂ w₁₂) (Triangle.mk u₁₂' v₁₂' w₁₂') h₁₂ h₁₂' e₁ e₂ comm₁₂).hom.hom₃ ≫ w₁₂'
eq₁₃ :
  v₁₃ ≫ (isoTriangleOfIso₁₂ (Triangle.mk u₁₃ v₁₃ w₁₃) (Triangle.mk u₁₃' v₁₃' w₁₃') h₁₃ h₁₃' e₁ e₃ ⋯).hom.hom₃ =
    e₃.hom ≫ v₁₃'
eq₁₃' :
  w₁₃ ≫ (shiftFunctor C 1).map e₁.hom =
    (isoTriangleOfIso₁₂ (Triangle.mk u₁₃ v₁₃ w₁₃) (Triangle.mk u₁₃' v₁₃' w₁₃') h₁₃ h₁₃' e₁ e₃ ⋯).hom.hom₃ ≫ w₁₃'
eq₂₃ :
  v₂₃ ≫ (isoTriangleOfIso₁₂ (Triangle.mk u₂₃ v₂₃ w₂₃) (Triangle.mk u₂₃' v₂₃' w₂₃') h₂₃ h₂₃' e₂ e₃ comm₂₃).hom.hom₃ =
    e₃.hom ≫ v₂₃'
eq₂₃' :
  w₂₃ ≫ (shiftFunctor C 1).map e₂.hom =
    (isoTriangleOfIso₁₂ (Triangle.mk u₂₃ v₂₃ w₂₃) (Triangle.mk u₂₃' v₂₃' w₂₃') h₂₃ h₂₃' e₂ e₃ comm₂₃).hom.hom₃ ≫ w₂₃'
rel₁₂ : v₁₂' ≫ H.m₁ = u₂₃' ≫ v₁₃'
rel₁₃ : w₁₂' = H.m₁ ≫ w₁₃'
rel₂₂ : v₁₃' ≫ H.m₃ = v₂₃'
rel₂₃ : w₁₃' ≫ (shiftFunctor C 1).map u₁₂' = H.m₃ ≫ w₂₃'
⊢ Triangle.mk (iso₁₂.hom.hom₃ ≫ H.m₁ ≫ iso₁₃.inv.hom₃) (iso₁₃.hom.hom₃ ≫ H.m₃ ≫ iso₂₃.inv.hom₃)
      (w₂₃ ≫ (shiftFunctor C 1).map v₁₂) ∈
    distinguishedTriangles

Tactic:
·

Goal state after:
case refine_4
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 : 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
X₁' X₂' X₃' Z₁₂' Z₂₃' Z₁₃' : C
u₁₂' : X₁' ⟶ X₂'
u₂₃' : X₂' ⟶ X₃'
u₁₃' : X₁' ⟶ X₃'
comm' : u₁₂' ≫ u₂₃' = u₁₃'
e₁ : X₁ ≅ X₁'
e₂ : X₂ ≅ X₂'
e₃ : X₃ ≅ X₃'
comm₁₂ : u₁₂ ≫ e₂.hom = e₁.hom ≫ u₁₂'
comm₂₃ : u₂₃ ≫ e₃.hom = e₂.hom ≫ u₂₃'
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'
iso₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ≅ Triangle.mk u₁₂' v₁₂' w₁₂' :=
  isoTriangleOfIso₁₂ (Triangle.mk u₁₂ v₁₂ w₁₂) (Triangle.mk u₁₂' v₁₂' w₁₂') h₁₂ h₁₂' e₁ e₂ comm₁₂
iso₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ≅ Triangle.mk u₂₃' v₂₃' w₂₃' :=
  isoTriangleOfIso₁₂ (Triangle.mk u₂₃ v₂₃ w₂₃) (Triangle.mk u₂₃' v₂₃' w₂₃') h₂₃ h₂₃' e₂ e₃ comm₂₃
iso₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ≅ Triangle.mk u₁₃' v₁₃' w₁₃' :=
  isoTriangleOfIso₁₂ (Triangle.mk u₁₃ v₁₃ w₁₃) (Triangle.mk u₁₃' v₁₃' w₁₃') h₁₃ h₁₃' e₁ e₃ ⋯
eq₁₂ :
  v₁₂ ≫ (isoTriangleOfIso₁₂ (Triangle.mk u₁₂ v₁₂ w₁₂) (Triangle.mk u₁₂' v₁₂' w₁₂') h₁₂ h₁₂' e₁ e₂ comm₁₂).hom.hom₃ =
    e₂.hom ≫ v₁₂'
eq₁₂' :
  w₁₂ ≫ (shiftFunctor C 1).map e₁.hom =
    (isoTriangleOfIso₁₂ (Triangle.mk u₁₂ v₁₂ w₁₂) (Triangle.mk u₁₂' v₁₂' w₁₂') h₁₂ h₁₂' e₁ e₂ comm₁₂).hom.hom₃ ≫ w₁₂'
eq₁₃ :
  v₁₃ ≫ (isoTriangleOfIso₁₂ (Triangle.mk u₁₃ v₁₃ w₁₃) (Triangle.mk u₁₃' v₁₃' w₁₃') h₁₃ h₁₃' e₁ e₃ ⋯).hom.hom₃ =
    e₃.hom ≫ v₁₃'
eq₁₃' :
  w₁₃ ≫ (shiftFunctor C 1).map e₁.hom =
    (isoTriangleOfIso₁₂ (Triangle.mk u₁₃ v₁₃ w₁₃) (Triangle.mk u₁₃' v₁₃' w₁₃') h₁₃ h₁₃' e₁ e₃ ⋯).hom.hom₃ ≫ w₁₃'
eq₂₃ :
  v₂₃ ≫ (isoTriangleOfIso₁₂ (Triangle.mk u₂₃ v₂₃ w₂₃) (Triangle.mk u₂₃' v₂₃' w₂₃') h₂₃ h₂₃' e₂ e₃ comm₂₃).hom.hom₃ =
    e₃.hom ≫ v₂₃'
eq₂₃' :
  w₂₃ ≫ (shiftFunctor C 1).map e₂.hom =
    (isoTriangleOfIso₁₂ (Triangle.mk u₂₃ v₂₃ w₂₃) (Triangle.mk u₂₃' v₂₃' w₂₃') h₂₃ h₂₃' e₂ e₃ comm₂₃).hom.hom₃ ≫ w₂₃'
rel₁₂ : v₁₂' ≫ H.m₁ = u₂₃' ≫ v₁₃'
rel₁₃ : w₁₂' = H.m₁ ≫ w₁₃'
rel₂₂ : v₁₃' ≫ H.m₃ = v₂₃'
rel₂₃ : w₁₃' ≫ (shiftFunctor C 1).map u₁₂' = H.m₃ ≫ w₂₃'
⊢ w₁₃ ≫ (shiftFunctor C 1).map u₁₂ = (iso₁₃.hom.hom₃ ≫ H.m₃ ≫ iso₂₃.inv.hom₃) ≫ w₂₃