mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroObject C
inst✝¹ : Preadditive C
K L : HomologicalComplex C c
f g : K ⟶ L
h : Homotopy f g
e : c.Embedding c'
inst✝ : e.IsRelIff
i j : ι
hij' : ¬c'.Rel (e.f j) (e.f i)
⊢ ¬c.Rel j i

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
P : ObjectProperty C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : P.IsTriangulated
X : C
hX : P.leftOrthogonal X
Y₂ Y₃ : C
h : Y₂ ⟶ Y₃
Y₁ : C
f : Y₁ ⟶ Y₂
g : Y₃ ⟶ (shiftFunctor C 1).obj Y₁
hT : Triangle.mk f h g ∈ distinguishedTriangles
hY₁ : P Y₁
this : ∀ (α : X ⟶ Y₂), α ≫ h = 0 → α = 0
α₁ α₂ : X ⟶ Y₂
hα : (fun f ↦ f ≫ h) α₁ = (fun f ↦ f ≫ h) α₂
⊢ α₁ = α₂

Tactic:
by

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

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝²² : Category.{v_1, u_1} C₁
inst✝²¹ : Category.{v_2, u_2} C₂
inst✝²⁰ : Category.{v_3, u_5} C₃
inst✝¹⁹ : Category.{v_4, u_6} C₄
inst✝¹⁸ : Category.{v_6, u_4} C₂₃
inst✝¹⁷ : HasZeroMorphisms C₁
inst✝¹⁶ : HasZeroMorphisms C₂
inst✝¹⁵ : HasZeroMorphisms C₃
inst✝¹⁴ : Preadditive C₂₃
inst✝¹³ : Preadditive C₄
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
inst✝¹² : G₂₃.PreservesZeroMorphisms
inst✝¹¹ : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst✝¹⁰ : F.PreservesZeroMorphisms
inst✝⁹ : ∀ (X₁ : C₁), (F.obj X₁).Additive
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₂₃ : Type u_11
ι₄ : Type u_12
inst✝⁸ : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
K₃ : HomologicalComplex C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₂₃ : ComplexShape ι₂₃
c₄ : ComplexShape ι₄
inst✝⁷ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁶ : TotalComplexShape c₁₂ c₃ c₄
inst✝⁵ : TotalComplexShape c₂ c₃ c₂₃
inst✝⁴ : TotalComplexShape c₁ c₂₃ c₄
inst✝³ : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst✝² : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst✝¹ : DecidableEq ι₂₃
inst✝ : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
j j' : ι₄
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
⊢ c₁.π c₂₃ c₄ (i₁, c₂.π c₃ c₂₃ (i₂, i₃)) = j

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroObject C
inst✝¹ : Preadditive C
K L : HomologicalComplex C c
f g : K ⟶ L
h : Homotopy f g
e : c.Embedding c'
inst✝ : e.IsRelIff
j' : ι'
i : ι
hij' : ¬c'.Rel j' (e.f i)
hj' : ¬∃ j, e.f j = j'
⊢ ∀ (j : ι), e.f j ≠ j'

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroObject C
inst✝¹ : Preadditive C
K L : HomologicalComplex C c
f g : K ⟶ L
h : Homotopy f g
e : c.Embedding c'
inst✝ : e.IsRelIff
i' j' : ι'
hij' : ¬c'.Rel j' i'
hi' : ¬∃ i, e.f i = i'
⊢ ∀ (i : ι), e.f i ≠ i'

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroObject C
inst✝¹ : Preadditive C
K L : HomologicalComplex C c
f g : K ⟶ L
h : Homotopy f g
e : c.Embedding c'
inst✝ : e.IsRelIff
i' j' : ι'
hij' : ¬c'.Rel j' i'
⊢ extend.hom e h.hom i' j' = 0

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
P : ObjectProperty C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : P.IsTriangulated
⊢ P.trW.isColocal = P.leftOrthogonal

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
L₁ L₂ : CochainComplex C ℤ
e : HomotopyEquiv L₁ L₂
inst✝ : L₁.IsKInjective
K : CochainComplex C ℤ
f : K ⟶ L₂
hK : HomologicalComplex.Acyclic K
⊢ f = f ≫ 𝟙 L₂

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
L₁ L₂ : CochainComplex C ℤ
e : HomotopyEquiv L₁ L₂
inst✝ : L₁.IsKInjective
K : CochainComplex C ℤ
f : K ⟶ L₂
hK : HomologicalComplex.Acyclic K
⊢ f ≫ e.inv ≫ e.hom = (f ≫ e.inv) ≫ e.hom

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
L₁ L₂ : CochainComplex C ℤ
e : HomotopyEquiv L₁ L₂
inst✝ : L₁.IsKInjective
K : CochainComplex C ℤ
f : K ⟶ L₂
hK : HomologicalComplex.Acyclic K
⊢ 0 ≫ e.hom = 0

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁹ : Category.{v, u} C
D : Type u'
inst✝⁸ : Category.{v', u'} D
P : ObjectProperty C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝³ : Pretriangulated C
inst✝² : P.IsTriangulated
inst✝¹ : IsTriangulated C
Y : C
hY : P.rightOrthogonal Y
L : C ⥤ D
inst✝ : L.IsLocalization P.trW
X : C
⊢ Function.Bijective L.map

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroObject C
inst✝¹ : Preadditive C
K L : HomologicalComplex C c
f g : K ⟶ L
h : Homotopy f g
e : c.Embedding c'
inst✝ : e.IsRelIff
i : ι
hi : c.Rel i (c.next i)
⊢ c'.Rel (e.f i) (e.f (c.next i))

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroObject C
inst✝¹ : Preadditive C
K L : HomologicalComplex C c
f g : K ⟶ L
h : Homotopy f g
e : c.Embedding c'
inst✝ : e.IsRelIff
i : ι
hi : c.Rel (c.prev i) i
⊢ c'.Rel (e.f (c.prev i)) (e.f i)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁹ : Category.{v, u} C
D : Type u'
inst✝⁸ : Category.{v', u'} D
P : ObjectProperty C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝³ : Pretriangulated C
inst✝² : P.IsTriangulated
inst✝¹ : IsTriangulated C
X : C
hX : P.leftOrthogonal X
L : C ⥤ D
inst✝ : L.IsLocalization P.trW
Y : C
⊢ Function.Bijective L.map

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.KInjective

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
K L : CochainComplex C ℤ
inst✝¹ : K.IsKInjective
inst✝ : L.IsKInjective
f : K ⟶ L
⊢ QuasiIso f ↔ homotopyEquivalences C (ComplexShape.up ℤ) f

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
L : CochainComplex C ℤ
⊢ L.IsKInjective ↔
    (HomotopyCategory.subcategoryAcyclic C).rightOrthogonal ((HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj L)

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
L : CochainComplex C ℤ
inst✝ : L.IsKInjective
⊢ (HomotopyCategory.subcategoryAcyclic C).rightOrthogonal ((HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj L)

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
L : CochainComplex C ℤ
hL : L.IsKInjective
n : ℤ
⊢ ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj L).IsKInjective

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
α : Cochain K L (-1)
n : ℤ
hα : (δ (-1) 0 α).EqUpTo (Cochain.ofHom f) n
hK : HomologicalComplex.ExactAt K (n + 1)
inst✝ : Injective (L.X (n + 1))
⊢ n + 1 + -1 = n

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
α : Cochain K L (-1)
n : ℤ
hα : (δ (-1) 0 α).EqUpTo (Cochain.ofHom f) n
hK : HomologicalComplex.ExactAt K (n + 1)
inst✝ : Injective (L.X (n + 1))
⊢ n + 2 + -1 = n + 1

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
α : Cochain K L (-1)
n : ℤ
hα : (δ (-1) 0 α).EqUpTo (Cochain.ofHom f) n
hK : HomologicalComplex.ExactAt K (n + 1)
inst✝ : Injective (L.X (n + 1))
u : K.X (n + 1) ⟶ L.X (n + 1) :=
  f.f (n + 1) - α.v (n + 1) n ⋯ ≫ L.d n (n + 1) - K.d (n + 1) (n + 2) ≫ α.v (n + 2) (n + 1) ⋯
⊢ n ≤ n

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroObject C
inst✝¹ : Preadditive C
K L : HomologicalComplex C c
f g : K ⟶ L
h : Homotopy f g
e : c.Embedding c'
inst✝ : e.IsRelIff
i' : ι'
hi' : ¬∃ i, e.f i = i'
⊢ ∀ (i : ι), e.f i ≠ i'

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
α : Cochain K L (-1)
n : ℤ
hα : (δ (-1) 0 α).EqUpTo (Cochain.ofHom f) n
hK : HomologicalComplex.ExactAt K (n + 1)
inst✝ : Injective (L.X (n + 1))
u : K.X (n + 1) ⟶ L.X (n + 1) :=
  f.f (n + 1) - α.v (n + 1) n ⋯ ≫ L.d n (n + 1) - K.d (n + 1) (n + 2) ≫ α.v (n + 2) (n + 1) ⋯
eq : (δ (-1) 0 α).v n n ⋯ = (Cochain.ofHom f).v n n ⋯
⊢ n - 1 = n - 1

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
α : Cochain K L (-1)
n : ℤ
hα : (δ (-1) 0 α).EqUpTo (Cochain.ofHom f) n
hK : HomologicalComplex.ExactAt K (n + 1)
inst✝ : Injective (L.X (n + 1))
u : K.X (n + 1) ⟶ L.X (n + 1) :=
  f.f (n + 1) - α.v (n + 1) n ⋯ ≫ L.d n (n + 1) - K.d (n + 1) (n + 2) ≫ α.v (n + 2) (n + 1) ⋯
eq : (δ (-1) 0 α).v n n ⋯ = (Cochain.ofHom f).v n n ⋯
⊢ n + 1 = n + 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroObject C
inst✝¹ : Preadditive C
K L : HomologicalComplex C c
f g : K ⟶ L
h : Homotopy f g
e : c.Embedding c'
inst✝ : e.IsRelIff
i' : ι'
⊢ (extendMap f e).f i' = (dNext i') (extend.hom e h.hom) + (prevD i') (extend.hom e h.hom) + (extendMap g e).f i'

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
α : Cochain K L (-1)
n : ℤ
hα : (δ (-1) 0 α).EqUpTo (Cochain.ofHom f) n
hK : HomologicalComplex.ExactAt K (n + 1)
inst✝ : Injective (L.X (n + 1))
u : K.X (n + 1) ⟶ L.X (n + 1) :=
  f.f (n + 1) - α.v (n + 1) n ⋯ ≫ L.d n (n + 1) - K.d (n + 1) (n + 2) ≫ α.v (n + 2) (n + 1) ⋯
⊢ K.d n (n + 1) ≫ u = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroObject C
inst✝¹ : Preadditive C
K L : HomologicalComplex C c
f g : K ⟶ L
e : c.Embedding c'
inst✝ : e.IsRelIff
h : Homotopy (extendMap f e) (extendMap g e)
i j : ι
hij : ¬c.Rel j i
⊢ ¬c'.Rel (e.f j) (e.f i)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroObject C
inst✝¹ : Preadditive C
K L : HomologicalComplex C c
f g : K ⟶ L
e : c.Embedding c'
inst✝ : e.IsRelIff
h : Homotopy (extendMap f e) (extendMap g e)
i j : ι
hij : ¬c.Rel j i
⊢ (K.extendXIso e ⋯).inv ≫ h.hom (e.f i) (e.f j) ≫ (L.extendXIso e ⋯).hom = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.KInjective

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
K L : CochainComplex C ℤ
n : ℤ
inst✝¹ : HasSmallLocalizedShiftedHom (HomologicalComplex.quasiIso C (ComplexShape.up ℤ)) ℤ K L
inst✝ : L.IsKInjective
⊢ Function.Bijective toSmallShiftedHom

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
α : Cochain K L (-1)
n : ℤ
hα : (δ (-1) 0 α).EqUpTo (Cochain.ofHom f) n
hK : HomologicalComplex.ExactAt K (n + 1)
inst✝ : Injective (L.X (n + 1))
u : K.X (n + 1) ⟶ L.X (n + 1) :=
  f.f (n + 1) - α.v (n + 1) n ⋯ ≫ L.d n (n + 1) - K.d (n + 1) (n + 2) ≫ α.v (n + 2) (n + 1) ⋯
hu : K.d n (n + 1) ≫ u = 0
⊢ (ComplexShape.up ℤ).prev (n + 1) = n

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.KInjective

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
K L : CochainComplex C ℕ
inst✝¹ : ∀ (n : ℕ), Injective (K.X n)
inst✝ : ∀ (n : ℕ), Injective (L.X n)
f : K ⟶ L
⊢ QuasiIso f ↔ homotopyEquivalences C (ComplexShape.up ℕ) f

Tactic:
by

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

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
t : TStructure C
A : C
n₀ n₁ : ℤ
h✝ : n₀ + 1 = n₁
X Y : C
hX : t.le 0 X
hY : t.ge 1 Y
f : X ⟶ (shiftFunctor C n₀).obj A
g : (shiftFunctor C n₀).obj A ⟶ Y
h : Y ⟶ (shiftFunctor C 1).obj X
mem : Triangle.mk f g h ∈ distinguishedTriangles
T : Triangle C := (Triangle.shiftFunctor C (-n₀)).obj (Triangle.mk f g h)
e : ((shiftEquiv C n₀).functor ⋙ (shiftEquiv C n₀).inverse).obj A ≅ (𝟭 C).obj A := (shiftEquiv C n₀).unitIso.symm.app A
⊢ Triangle.mk (T.mor₁ ≫ e.hom) (e.inv ≫ T.mor₂) T.mor₃ ∈ distinguishedTriangles

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
α : Cochain K L (-1)
n : ℤ
hα : (δ (-1) 0 α).EqUpTo (Cochain.ofHom f) n
hK : HomologicalComplex.ExactAt K (n + 1)
inst✝ : Injective (L.X (n + 1))
u : K.X (n + 1) ⟶ L.X (n + 1) :=
  f.f (n + 1) - α.v (n + 1) n ⋯ ≫ L.d n (n + 1) - K.d (n + 1) (n + 2) ≫ α.v (n + 2) (n + 1) ⋯
hu : K.d n (n + 1) ≫ u = 0
⊢ (ComplexShape.up ℤ).next (n + 1) = n + 2

Tactic:
by

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

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
t : TStructure C
A : C
n₀ n₁ : ℤ
h✝ : n₀ + 1 = n₁
X Y : C
hX : t.le 0 X
hY : t.ge 1 Y
f : X ⟶ (shiftFunctor C n₀).obj A
g : (shiftFunctor C n₀).obj A ⟶ Y
h : Y ⟶ (shiftFunctor C 1).obj X
mem : Triangle.mk f g h ∈ distinguishedTriangles
T : Triangle C := (Triangle.shiftFunctor C (-n₀)).obj (Triangle.mk f g h)
e : ((shiftEquiv C n₀).functor ⋙ (shiftEquiv C n₀).inverse).obj A ≅ (𝟭 C).obj A := (shiftEquiv C n₀).unitIso.symm.app A
hT' : Triangle.mk (T.mor₁ ≫ e.hom) (e.inv ≫ T.mor₂) T.mor₃ ∈ distinguishedTriangles
⊢ -n₀ + n₁ = 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroObject C
inst✝¹ : Preadditive C
K L : HomologicalComplex C c
f g : K ⟶ L
e : c.Embedding c'
inst✝ : e.IsRelIff
h : Homotopy (extendMap f e) (extendMap g e)
i : ι
this :
  (K.extendXIso e ⋯).hom ≫ f.f i ≫ (L.extendXIso e ⋯).inv =
    (dNext (e.f i)) h.hom + (prevD (e.f i)) h.hom + (K.extendXIso e ⋯).hom ≫ g.f i ≫ (L.extendXIso e ⋯).inv
hi : c.Rel i (c.next i)
⊢ c'.Rel (e.f i) (e.f (c.next i))

Tactic:
by

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

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
t : TStructure C
A : C
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
⊢ ∃ X Y, ∃ (_ : t.le n₀ X) (_ : t.ge n₁ Y), ∃ f g h, Triangle.mk f g h ∈ distinguishedTriangles

Tactic:
by

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

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
t : TStructure C
a n n' : ℤ
hn' : a + n = n'
X : C
hX : (t.le n).shift a X
⊢ -a + n' = n

Tactic:
by

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

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
t : TStructure C
a n n' : ℤ
hn' : a + n = n'
⊢ (t.le n).shift a = t.le n'

Tactic:
by

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

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
t : TStructure C
a n n' : ℤ
hn' : a + n = n'
X : C
hX : (t.ge n).shift a X
⊢ -a + n' = n

Tactic:
by

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

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
t : TStructure C
a n n' : ℤ
hn' : a + n = n'
⊢ (t.ge n).shift a = t.ge n'

Tactic:
by

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

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
t : TStructure C
H : ℕ → Prop := fun a ↦ ∀ (n : ℤ), t.le n ≤ t.le (n + ↑a)
this : ∀ (a : ℕ), H a
n₀ n₁ : ℤ
h : n₀ ≤ n₁
a : ℕ
ha : n₁ - n₀ = ↑a
⊢ n₁ = n₀ + ↑a

Tactic:
by

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

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
t : TStructure C
H : ℕ → Prop := fun a ↦ ∀ (n : ℤ), t.le n ≤ t.le (n + ↑a)
this : ∀ (a : ℕ), H a
⊢ Monotone t.le

Tactic:
by

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

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝²² : Category.{v_1, u_1} C₁
inst✝²¹ : Category.{v_2, u_2} C₂
inst✝²⁰ : Category.{v_3, u_5} C₃
inst✝¹⁹ : Category.{v_4, u_6} C₄
inst✝¹⁸ : Category.{v_6, u_4} C₂₃
inst✝¹⁷ : HasZeroMorphisms C₁
inst✝¹⁶ : HasZeroMorphisms C₂
inst✝¹⁵ : HasZeroMorphisms C₃
inst✝¹⁴ : Preadditive C₂₃
inst✝¹³ : Preadditive C₄
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
inst✝¹² : G₂₃.PreservesZeroMorphisms
inst✝¹¹ : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst✝¹⁰ : F.PreservesZeroMorphisms
inst✝⁹ : ∀ (X₁ : C₁), (F.obj X₁).Additive
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₂₃ : Type u_11
ι₄ : Type u_12
inst✝⁸ : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
K₃ : HomologicalComplex C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₂₃ : ComplexShape ι₂₃
c₄ : ComplexShape ι₄
inst✝⁷ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁶ : TotalComplexShape c₁₂ c₃ c₄
inst✝⁵ : TotalComplexShape c₂ c₃ c₂₃
inst✝⁴ : TotalComplexShape c₁ c₂₃ c₄
inst✝³ : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst✝² : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst✝¹ : DecidableEq ι₂₃
inst✝ : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
j j' : ι₄
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
h₁ : c₁.Rel i₁ (c₁.next i₁)
h₂ : ¬c₁.π c₂₃ c₄ (c₁.next i₁, c₂.π c₃ c₂₃ (i₂, i₃)) = j'
⊢ c₁.r c₂ c₃ c₁₂ c₄ (c₁.next i₁, i₂, i₃) ≠ j'

Tactic:
by

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

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
t : TStructure C
H : ℕ → Prop := fun a ↦ ∀ (n : ℤ), t.le n ≤ t.le (n + ↑a)
n : ℤ
⊢ t.le n ≤ t.le (n + ↑0)

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
α : Cochain K L (-1)
n : ℤ
hα : (δ (-1) 0 α).EqUpTo (Cochain.ofHom f) n
hK : (HomologicalComplex.sc' K n (n + 1) (n + 2)).Exact
inst✝ : Injective (L.X (n + 1))
u : K.X (n + 1) ⟶ L.X (n + 1) :=
  f.f (n + 1) - α.v (n + 1) n ⋯ ≫ L.d n (n + 1) - K.d (n + 1) (n + 2) ≫ α.v (n + 2) (n + 1) ⋯
hu : K.d n (n + 1) ≫ u = 0
β : K.X (n + 2) ⟶ L.X (n + 1)
hβ : K.d (n + 1) (n + 2) ≫ β = u
p q : ℤ
hpq : p + 0 = q
hp : p ≤ n + 1
⊢ p = q

Tactic:
by

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

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
t : TStructure C
H : ℕ → Prop := fun a ↦ ∀ (n : ℤ), t.le n ≤ t.le (n + ↑a)
H_zero : H 0
n : ℤ
X : C
hX : t.le n X
⊢ t.le (n + ↑1) X

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
α : Cochain K L (-1)
n : ℤ
hα : (δ (-1) 0 α).EqUpTo (Cochain.ofHom f) n
hK : (HomologicalComplex.sc' K n (n + 1) (n + 2)).Exact
inst✝ : Injective (L.X (n + 1))
u : K.X (n + 1) ⟶ L.X (n + 1) :=
  f.f (n + 1) - α.v (n + 1) n ⋯ ≫ L.d n (n + 1) - K.d (n + 1) (n + 2) ≫ α.v (n + 2) (n + 1) ⋯
hu : K.d n (n + 1) ≫ u = 0
β : K.X (n + 2) ⟶ L.X (n + 1)
hβ : K.d (n + 1) (n + 2) ≫ β = u
p : ℤ
hp✝ : p ≤ n + 1
hpq : p + 0 = p
hp : p < n + 1
⊢ p + 0 = p

Tactic:
by

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

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
t : TStructure C
H : ℕ → Prop := fun a ↦ ∀ (n : ℤ), t.le n ≤ t.le (n + ↑a)
H_zero : H 0
H_one : H 1
⊢ ∀ (a b c : ℕ), a + b = c → H a → H b → H c

Tactic:
by

Goal state after: