mathlib-initiative/mathlib-tactics huggingface dataset:
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 := ⋯ }
⊢ S'.Exact

Tactic:
]

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 := ⋯ }
⊢ S'.Exact
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 := ⋯ }
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 := ⋯ }
π : T' ⟶ T := { τ₁ := K.pOpcycles i, τ₂ := 𝟙 T'.X₂, τ₃ := 𝟙 T'.X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }
⊢ T'.Exact

Tactic:
]

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 := ⋯ }
π : T' ⟶ T := { τ₁ := K.pOpcycles i, τ₂ := 𝟙 T'.X₂, τ₃ := 𝟙 T'.X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }
⊢ T'.Exact
Module: Mathlib.CategoryTheory.Shift.ShiftSequence

Goal state before:
pp: C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
inst✝⁵ : Category.{v_3, u_3} A
F : C ⥤ A
π : C ⥤ D
H : D ⥤ A
e : π ⋙ H ≅ F
M : Type u_4
inst✝⁴ : AddMonoid M
inst✝³ : HasShift C M
inst✝² : HasShift D M
G : Type u_5
inst✝¹ : AddGroup G
inst✝ : HasShift C G
sequence : M → C ⥤ A
isoZero : sequence 0 ≅ F
shiftIso : (n a a' : M) → n + a = a' → (shiftFunctor C n ⋙ sequence a ≅ sequence a')
shiftIso_zero :
  ∀ (a : M), shiftIso 0 a a ⋯ = isoWhiskerRight (shiftFunctorZero C M) (sequence a) ≪≫ (sequence a).leftUnitor
n m a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
⊢ a'' = a''

Tactic:
]

Goal state after:
Module: Mathlib.CategoryTheory.Shift.ShiftSequence

Goal state before:
pp: C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
inst✝⁵ : Category.{v_3, u_3} A
F : C ⥤ A
π : C ⥤ D
H : D ⥤ A
e : π ⋙ H ≅ F
M : Type u_4
inst✝⁴ : AddMonoid M
inst✝³ : HasShift C M
inst✝² : HasShift D M
G : Type u_5
inst✝¹ : AddGroup G
inst✝ : HasShift C G
a : M
⊢ ((shiftFunctor C 0).associator (shiftFunctor C a) F).symm ≪≫
      isoWhiskerRight
        ((shiftFunctor C a).leftUnitor.symm ≪≫ isoWhiskerRight (shiftFunctorZero C M).symm (shiftFunctor C a)).symm F =
    isoWhiskerRight (shiftFunctorZero C M) (shiftFunctor C a ⋙ F) ≪≫ (shiftFunctor C a ⋙ F).leftUnitor

Tactic:
]

Goal state after:
C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
inst✝⁵ : Category.{v_3, u_3} A
F : C ⥤ A
π : C ⥤ D
H : D ⥤ A
e : π ⋙ H ≅ F
M : Type u_4
inst✝⁴ : AddMonoid M
inst✝³ : HasShift C M
inst✝² : HasShift D M
G : Type u_5
inst✝¹ : AddGroup G
inst✝ : HasShift C G
a : M
⊢ ((shiftFunctor C 0).associator (shiftFunctor C a) F).symm ≪≫
      isoWhiskerRight
        ((shiftFunctor C a).leftUnitor.symm ≪≫ isoWhiskerRight (shiftFunctorZero C M).symm (shiftFunctor C a)).symm F =
    isoWhiskerRight (shiftFunctorZero C M) (shiftFunctor C a ⋙ F) ≪≫ (shiftFunctor C a ⋙ F).leftUnitor
Module: Mathlib.CategoryTheory.Shift.ShiftSequence

Goal state before:
pp: C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
inst✝⁵ : Category.{v_3, u_3} A
F : C ⥤ A
π : C ⥤ D
H : D ⥤ A
e : π ⋙ H ≅ F
M : Type u_4
inst✝⁴ : AddMonoid M
inst✝³ : HasShift C M
inst✝² : HasShift D M
G : Type u_5
inst✝¹ : AddGroup G
inst✝ : HasShift C G
n m a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
X : C
⊢ m + (n + a) = m + (n + a)

Tactic:
]

Goal state after:
Module: Mathlib.CategoryTheory.Shift.ShiftSequence

Goal state before:
pp: C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
inst✝⁶ : Category.{v_3, u_3} A
F : C ⥤ A
π : C ⥤ D
H : D ⥤ A
e : π ⋙ H ≅ F
M : Type u_4
inst✝⁵ : AddMonoid M
inst✝⁴ : HasShift C M
inst✝³ : HasShift D M
G : Type u_5
inst✝² : AddGroup G
inst✝¹ : HasShift C G
inst✝ : F.ShiftSequence M
n m a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
⊢ a'' = a''

Tactic:
]

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

Goal state before:
pp: case refine_1
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : IsTriangulated 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
o : Octahedron comm ⋯ ⋯ ⋯ := someOctahedron comm ⋯ ⋯ ⋯
m₁ : Z₁₂ ⟶ Z₁₃ := (shiftShiftNeg Z₁₂ 1).inv ≫ (shiftFunctor C (-1)).map o.m₁ ≫ (shiftShiftNeg Z₁₃ 1).hom
m₃ : Z₁₃ ⟶ Z₂₃ := (shiftShiftNeg Z₁₃ 1).inv ≫ (shiftFunctor C (-1)).map o.m₃ ≫ (shiftShiftNeg Z₂₃ 1).hom
eq₁ : w₁₂ ≫ o.m₁ = u₂₃ ≫ w₁₃
eq₂ : o.m₁ ≫ (shiftFunctor C 1).map v₁₃ = (shiftFunctor C 1).map v₁₂
eq₃ : w₁₃ ≫ o.m₃ = w₂₃
eq₄ : (shiftFunctor C 1).map v₁₃ ≫ (shiftFunctor C 1).map u₁₂ = o.m₃ ≫ (shiftFunctor C 1).map v₂₃
⊢ m₁ ≫
      (shiftFunctorCompIsoId C 1 (-1) ⋯).inv.app Z₁₃ ≫
        (shiftFunctor C (-1)).map ((shiftFunctor C 1).map v₁₃) ≫ (shiftFunctorCompIsoId C 1 (-1) ⋯).hom.app X₁ =
    v₁₂

Tactic:
]

Goal state after:
case refine_1
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : IsTriangulated 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
o : Octahedron comm ⋯ ⋯ ⋯ := someOctahedron comm ⋯ ⋯ ⋯
m₁ : Z₁₂ ⟶ Z₁₃ := (shiftShiftNeg Z₁₂ 1).inv ≫ (shiftFunctor C (-1)).map o.m₁ ≫ (shiftShiftNeg Z₁₃ 1).hom
m₃ : Z₁₃ ⟶ Z₂₃ := (shiftShiftNeg Z₁₃ 1).inv ≫ (shiftFunctor C (-1)).map o.m₃ ≫ (shiftShiftNeg Z₂₃ 1).hom
eq₁ : w₁₂ ≫ o.m₁ = u₂₃ ≫ w₁₃
eq₂ : o.m₁ ≫ (shiftFunctor C 1).map v₁₃ = (shiftFunctor C 1).map v₁₂
eq₃ : w₁₃ ≫ o.m₃ = w₂₃
eq₄ : (shiftFunctor C 1).map v₁₃ ≫ (shiftFunctor C 1).map u₁₂ = o.m₃ ≫ (shiftFunctor C 1).map v₂₃
⊢ m₁ ≫
      (shiftFunctorCompIsoId C 1 (-1) ⋯).inv.app Z₁₃ ≫
        (shiftFunctor C (-1)).map ((shiftFunctor C 1).map v₁₃) ≫ (shiftFunctorCompIsoId C 1 (-1) ⋯).hom.app X₁ =
    v₁₂
Module: Mathlib.CategoryTheory.Shift.ShiftSequence

Goal state before:
pp: C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
inst✝⁶ : Category.{v_3, u_3} A
F : C ⥤ A
π : C ⥤ D
H : D ⥤ A
e : π ⋙ H ≅ F
M : Type u_4
inst✝⁵ : AddMonoid M
inst✝⁴ : HasShift C M
inst✝³ : HasShift D M
G : Type u_5
inst✝² : AddGroup G
inst✝¹ : HasShift C G
inst✝ : F.ShiftSequence M
n m a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
X : C
⊢ a'' = a''

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Homology.HomologySequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
c : ComplexShape ι
S : ShortComplex (HomologicalComplex C c)
hS : S.Exact
inst✝³ : Epi S.g
i : ι
inst✝² : S.X₁.HasHomology i
inst✝¹ : S.X₂.HasHomology i
inst✝ : S.X₃.HasHomology i
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.CategoryTheory.Shift.ShiftSequence

Goal state before:
pp: C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
inst✝⁶ : Category.{v_3, u_3} A
F : C ⥤ A
π : C ⥤ D
H : D ⥤ A
e : π ⋙ H ≅ F
M : Type u_4
inst✝⁵ : AddMonoid M
inst✝⁴ : HasShift C M
inst✝³ : HasShift D M
G : Type u_5
inst✝² : AddGroup G
inst✝¹ : HasShift C G
inst✝ : F.ShiftSequence M
n m mn : M
hnm : m + n = mn
a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
⊢ m + (n + a) = m + (n + a)

Tactic:
]

Goal state after:
Module: Mathlib.CategoryTheory.PathCategory.Basic

Goal state before:
pp: case nil
V : Type u₁
inst✝¹ : Quiver V
C : Type ?u.6
inst✝ : Category.{v_1, ?u.6} C
φ : V ⥤q C
X✝ Y✝ Z✝ : Paths V
f : X✝ ⟶ Y✝
⊢ Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) (f ≫ Quiver.Path.nil) =
    Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) f

Tactic:
]

Goal state after:
case nil
V : Type u₁
inst✝¹ : Quiver V
C : Type ?u.6
inst✝ : Category.{v_1, ?u.6} C
φ : V ⥤q C
X✝ Y✝ Z✝ : Paths V
f : X✝ ⟶ Y✝
⊢ Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) (f ≫ Quiver.Path.nil) =
    Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) f
Module: Mathlib.CategoryTheory.Shift.ShiftSequence

Goal state before:
pp: C : Type u_1
A : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_3, u_3} A
F : C ⥤ A
M : Type u_4
inst✝² : AddMonoid M
inst✝¹ : HasShift C M
inst✝ : F.ShiftSequence M
n m a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
⊢ isoWhiskerRight (shiftFunctorAdd C m n) (F.shift a) ≪≫
      (shiftFunctor C m).associator (shiftFunctor C n) (F.shift a) ≪≫
        (shiftFunctor C m).isoWhiskerLeft (F.shiftIso n a ?a' ?ha') ≪≫ F.shiftIso m ?a' a'' ?ha'' =
    isoWhiskerRight (shiftFunctorAdd C m n) (F.shift a) ≪≫
      (shiftFunctor C m).associator (shiftFunctor C n) (F.shift a) ≪≫
        (shiftFunctor C m).isoWhiskerLeft (F.shiftIso n a a' ha') ≪≫ F.shiftIso m a' a'' ha''

pp: case a'
C : Type u_1
A : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_3, u_3} A
F : C ⥤ A
M : Type u_4
inst✝² : AddMonoid M
inst✝¹ : HasShift C M
inst✝ : F.ShiftSequence M
n m a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
⊢ M

pp: case ha'
C : Type u_1
A : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_3, u_3} A
F : C ⥤ A
M : Type u_4
inst✝² : AddMonoid M
inst✝¹ : HasShift C M
inst✝ : F.ShiftSequence M
n m a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
⊢ n + a = ?a'

pp: case ha''
C : Type u_1
A : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_3, u_3} A
F : C ⥤ A
M : Type u_4
inst✝² : AddMonoid M
inst✝¹ : HasShift C M
inst✝ : F.ShiftSequence M
n m a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
⊢ m + ?a' = a''

Tactic:
]

Goal state after:
Module: Mathlib.CategoryTheory.Shift.ShiftSequence

Goal state before:
pp: C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
inst✝⁶ : Category.{v_3, u_3} A
F : C ⥤ A
π : C ⥤ D
H : D ⥤ A
e : π ⋙ H ≅ F
M : Type u_4
inst✝⁵ : AddMonoid M
inst✝⁴ : HasShift C M
inst✝³ : HasShift D M
G : Type u_5
inst✝² : AddGroup G
inst✝¹ : HasShift C G
inst✝ : F.ShiftSequence M
n m mn : M
hnm : m + n = mn
a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
X : C
⊢ m + (n + a) = m + (n + a)

Tactic:
]

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

Goal state before:
pp: case refine_1
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : IsTriangulated 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
o : Octahedron comm ⋯ ⋯ ⋯ := someOctahedron comm ⋯ ⋯ ⋯
m₁ : Z₁₂ ⟶ Z₁₃ := (shiftShiftNeg Z₁₂ 1).inv ≫ (shiftFunctor C (-1)).map o.m₁ ≫ (shiftShiftNeg Z₁₃ 1).hom
m₃ : Z₁₃ ⟶ Z₂₃ := (shiftShiftNeg Z₁₃ 1).inv ≫ (shiftFunctor C (-1)).map o.m₃ ≫ (shiftShiftNeg Z₂₃ 1).hom
eq₁ : w₁₂ ≫ o.m₁ = u₂₃ ≫ w₁₃
eq₂ : o.m₁ ≫ (shiftFunctor C 1).map v₁₃ = (shiftFunctor C 1).map v₁₂
eq₃ : w₁₃ ≫ o.m₃ = w₂₃
eq₄ : (shiftFunctor C 1).map v₁₃ ≫ (shiftFunctor C 1).map u₁₂ = o.m₃ ≫ (shiftFunctor C 1).map v₂₃
⊢ (shiftFunctorCompIsoId C 1 (-1) ⋯).inv.app Z₁₂ ≫
      (shiftFunctor C (-1)).map o.m₁ ≫
        (shiftFunctor C (-1)).map ((shiftFunctor C 1).map v₁₃) ≫ (shiftFunctorCompIsoId C 1 (-1) ⋯).hom.app X₁ =
    v₁₂

Tactic:
]

Goal state after:
case refine_1
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : IsTriangulated 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
o : Octahedron comm ⋯ ⋯ ⋯ := someOctahedron comm ⋯ ⋯ ⋯
m₁ : Z₁₂ ⟶ Z₁₃ := (shiftShiftNeg Z₁₂ 1).inv ≫ (shiftFunctor C (-1)).map o.m₁ ≫ (shiftShiftNeg Z₁₃ 1).hom
m₃ : Z₁₃ ⟶ Z₂₃ := (shiftShiftNeg Z₁₃ 1).inv ≫ (shiftFunctor C (-1)).map o.m₃ ≫ (shiftShiftNeg Z₂₃ 1).hom
eq₁ : w₁₂ ≫ o.m₁ = u₂₃ ≫ w₁₃
eq₂ : o.m₁ ≫ (shiftFunctor C 1).map v₁₃ = (shiftFunctor C 1).map v₁₂
eq₃ : w₁₃ ≫ o.m₃ = w₂₃
eq₄ : (shiftFunctor C 1).map v₁₃ ≫ (shiftFunctor C 1).map u₁₂ = o.m₃ ≫ (shiftFunctor C 1).map v₂₃
⊢ (shiftFunctorCompIsoId C 1 (-1) ⋯).inv.app Z₁₂ ≫
      (shiftFunctor C (-1)).map o.m₁ ≫
        (shiftFunctor C (-1)).map ((shiftFunctor C 1).map v₁₃) ≫ (shiftFunctorCompIsoId C 1 (-1) ⋯).hom.app X₁ =
    v₁₂
Module: Mathlib.CategoryTheory.Shift.ShiftSequence

Goal state before:
pp: C : Type u_1
A : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_3, u_3} A
F : C ⥤ A
M : Type u_4
inst✝² : AddMonoid M
inst✝¹ : HasShift C M
inst✝ : F.ShiftSequence M
n m mn : M
hnm : m + n = mn
a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
X : C
⊢ (F.shiftIso mn a a'' ⋯).hom.app X =
    (F.shift a).map ((shiftFunctorAdd' C m n mn hnm).hom.app X) ≫
      (F.shiftIso n a a' ha').hom.app ((shiftFunctor C m).obj X) ≫ (F.shiftIso m a' a'' ha'').hom.app X

Tactic:
simp [F.shiftIso_add' n m mn hnm a a' a'' ha' ha'']

Goal state after:
Module: Mathlib.Algebra.Homology.HomologySequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
c : ComplexShape ι
S : ShortComplex (HomologicalComplex C c)
hS : S.Exact
inst✝³ : Epi S.g
i : ι
inst✝² : S.X₁.HasHomology i
inst✝¹ : S.X₂.HasHomology i
inst✝ : S.X₃.HasHomology i
this : Epi (S.map (eval C c i)).g
hj : IsColimit (CokernelCofork.ofπ (S.map (eval C c i)).g ⋯)
A : C
k : S.X₂.opcycles i ⟶ A
hk : opcyclesMap S.f i ≫ k = 0
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.CategoryTheory.PathCategory.Basic

Goal state before:
pp: case cons
V : Type u₁
inst✝¹ : Quiver V
C : Type ?u.6
inst✝ : Category.{v_1, ?u.6} C
φ : V ⥤q C
X✝ Y✝ Z✝ : Paths V
f : X✝ ⟶ Y✝
b✝ c✝ : Paths V
g' : Quiver.Path Y✝ b✝
p : b✝ ⟶ c✝
ih :
  Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) (f ≫ g') =
    Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) f ≫
      Quiver.Path.rec (𝟙 (φ.obj Y✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) g'
this : f ≫ g'.cons p = Quiver.Path.cons (f ≫ g') p
⊢ Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) (Quiver.Path.cons (f ≫ g') p) =
    Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) f ≫
      Quiver.Path.rec (𝟙 (φ.obj Y✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) (g'.cons p)

Tactic:
]

Goal state after:
case cons
V : Type u₁
inst✝¹ : Quiver V
C : Type ?u.6
inst✝ : Category.{v_1, ?u.6} C
φ : V ⥤q C
X✝ Y✝ Z✝ : Paths V
f : X✝ ⟶ Y✝
b✝ c✝ : Paths V
g' : Quiver.Path Y✝ b✝
p : b✝ ⟶ c✝
ih :
  Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) (f ≫ g') =
    Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) f ≫
      Quiver.Path.rec (𝟙 (φ.obj Y✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) g'
this : f ≫ g'.cons p = Quiver.Path.cons (f ≫ g') p
⊢ Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) (Quiver.Path.cons (f ≫ g') p) =
    Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) f ≫
      Quiver.Path.rec (𝟙 (φ.obj Y✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) (g'.cons p)
Module: Mathlib.CategoryTheory.Shift.ShiftSequence

Goal state before:
pp: C : Type u_1
A : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_3, u_3} A
F : C ⥤ A
M : Type u_4
inst✝² : AddMonoid M
inst✝¹ : HasShift C M
inst✝ : F.ShiftSequence M
n m mn : M
hnm : m + n = mn
a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
X : C
⊢ (F.shiftIso mn a a'' ⋯).inv.app X =
    (F.shiftIso m a' a'' ha'').inv.app X ≫
      (F.shiftIso n a a' ha').inv.app ((shiftFunctor C m).obj X) ≫
        (F.shift a).map ((shiftFunctorAdd' C m n mn hnm).inv.app X)

Tactic:
simp [F.shiftIso_add' n m mn hnm a a' a'' ha' ha'']

Goal state after:
Module: Mathlib.CategoryTheory.Shift.ShiftSequence

Goal state before:
pp: C : Type u_1
A : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_3, u_3} A
F : C ⥤ A
M : Type u_4
inst✝² : AddMonoid M
inst✝¹ : HasShift C M
inst✝ : F.ShiftSequence M
n m mn : M
hnm : m + n = mn
a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
X : C
⊢ (F.shiftIso n a a' ha').hom.app ((shiftFunctor C m).obj X) ≫ (F.shiftIso m a' a'' ha'').hom.app X =
    (F.shiftIso n a a' ha').hom.app ((shiftFunctor C m).obj X) ≫ (F.shiftIso m a' a'' ha'').hom.app X

Tactic:
]

Goal state after:
Module: Mathlib.CategoryTheory.PathCategory.Basic

Goal state before:
pp: case cons
V : Type u₁
inst✝¹ : Quiver V
C : Type ?u.6
inst✝ : Category.{v_1, ?u.6} C
φ : V ⥤q C
X✝ Y✝ Z✝ : Paths V
f : X✝ ⟶ Y✝
b✝ c✝ : Paths V
g' : Quiver.Path Y✝ b✝
p : b✝ ⟶ c✝
ih :
  Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) (f ≫ g') =
    Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) f ≫
      Quiver.Path.rec (𝟙 (φ.obj Y✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) g'
this : f ≫ g'.cons p = Quiver.Path.cons (f ≫ g') p
⊢ Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) f ≫
      Quiver.Path.rec (𝟙 (φ.obj Y✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) g' ≫ φ.map p =
    Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) f ≫
      Quiver.Path.rec (𝟙 (φ.obj Y✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) g' ≫ φ.map p

Tactic:
]

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

Goal state before:
pp: case refine_1
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : IsTriangulated 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
o : Octahedron comm ⋯ ⋯ ⋯ := someOctahedron comm ⋯ ⋯ ⋯
m₁ : Z₁₂ ⟶ Z₁₃ := (shiftShiftNeg Z₁₂ 1).inv ≫ (shiftFunctor C (-1)).map o.m₁ ≫ (shiftShiftNeg Z₁₃ 1).hom
m₃ : Z₁₃ ⟶ Z₂₃ := (shiftShiftNeg Z₁₃ 1).inv ≫ (shiftFunctor C (-1)).map o.m₃ ≫ (shiftShiftNeg Z₂₃ 1).hom
eq₁ : w₁₂ ≫ o.m₁ = u₂₃ ≫ w₁₃
eq₂ : o.m₁ ≫ (shiftFunctor C 1).map v₁₃ = (shiftFunctor C 1).map v₁₂
eq₃ : w₁₃ ≫ o.m₃ = w₂₃
eq₄ : (shiftFunctor C 1).map v₁₃ ≫ (shiftFunctor C 1).map u₁₂ = o.m₃ ≫ (shiftFunctor C 1).map v₂₃
⊢ (shiftFunctorCompIsoId C 1 (-1) ⋯).inv.app Z₁₂ ≫
      ((shiftFunctor C (-1)).map o.m₁ ≫ (shiftFunctor C (-1)).map ((shiftFunctor C 1).map v₁₃)) ≫
        (shiftFunctorCompIsoId C 1 (-1) ⋯).hom.app X₁ =
    v₁₂

Tactic:
]

Goal state after:
case refine_1
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : IsTriangulated 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
o : Octahedron comm ⋯ ⋯ ⋯ := someOctahedron comm ⋯ ⋯ ⋯
m₁ : Z₁₂ ⟶ Z₁₃ := (shiftShiftNeg Z₁₂ 1).inv ≫ (shiftFunctor C (-1)).map o.m₁ ≫ (shiftShiftNeg Z₁₃ 1).hom
m₃ : Z₁₃ ⟶ Z₂₃ := (shiftShiftNeg Z₁₃ 1).inv ≫ (shiftFunctor C (-1)).map o.m₃ ≫ (shiftShiftNeg Z₂₃ 1).hom
eq₁ : w₁₂ ≫ o.m₁ = u₂₃ ≫ w₁₃
eq₂ : o.m₁ ≫ (shiftFunctor C 1).map v₁₃ = (shiftFunctor C 1).map v₁₂
eq₃ : w₁₃ ≫ o.m₃ = w₂₃
eq₄ : (shiftFunctor C 1).map v₁₃ ≫ (shiftFunctor C 1).map u₁₂ = o.m₃ ≫ (shiftFunctor C 1).map v₂₃
⊢ (shiftFunctorCompIsoId C 1 (-1) ⋯).inv.app Z₁₂ ≫
      ((shiftFunctor C (-1)).map o.m₁ ≫ (shiftFunctor C (-1)).map ((shiftFunctor C 1).map v₁₃)) ≫
        (shiftFunctorCompIsoId C 1 (-1) ⋯).hom.app X₁ =
    v₁₂
Module: Mathlib.CategoryTheory.Shift.ShiftSequence

Goal state before:
pp: C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
inst✝⁶ : Category.{v_3, u_3} A
F : C ⥤ A
π : C ⥤ D
H : D ⥤ A
e : π ⋙ H ≅ F
M : Type u_4
inst✝⁵ : AddMonoid M
inst✝⁴ : HasShift C M
inst✝³ : HasShift D M
G : Type u_5
inst✝² : AddGroup G
inst✝¹ : HasShift C G
inst✝ : F.ShiftSequence M
X Y : C
m : M
f : X ⟶ (shiftFunctor C m).obj Y
n mn : M
hnm : m + n = mn
a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
⊢ m + (n + a) = m + (n + a)

Tactic:
]

Goal state after: