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

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} 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
P Q : ObjectProperty C
inst✝ : P.ContainsZero
X : C
hX : Q X
⊢ P.isoClosure.extensionProduct Q X

Tactic:
]

Goal state after:
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} 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
P Q : ObjectProperty C
inst✝ : P.ContainsZero
X : C
hX : Q X
⊢ P.isoClosure.extensionProduct Q X
Module: Mathlib.CategoryTheory.Localization.Triangulated

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
L : C ⥤ D
inst✝¹³ : HasShift C ℤ
inst✝¹² : Preadditive C
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁹ : Pretriangulated C
inst✝⁸ : HasShift D ℤ
inst✝⁷ : L.CommShift ℤ
W : MorphismProperty C
inst✝⁶ : L.IsLocalization W
inst✝⁵ : W.HasLeftCalculusOfFractions
inst✝⁴ : HasZeroObject D
inst✝³ : Preadditive D
inst✝² : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝¹ : W.IsCompatibleWithTriangulation
inst✝ : W.HasLocalization
n : ℤ
⊢ (W.Q' ⋙ shiftFunctor W.Localization' n).Additive

Tactic:
]

Goal state after:
C : Type u_1
D : Type u_2
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
L : C ⥤ D
inst✝¹³ : HasShift C ℤ
inst✝¹² : Preadditive C
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁹ : Pretriangulated C
inst✝⁸ : HasShift D ℤ
inst✝⁷ : L.CommShift ℤ
W : MorphismProperty C
inst✝⁶ : L.IsLocalization W
inst✝⁵ : W.HasLeftCalculusOfFractions
inst✝⁴ : HasZeroObject D
inst✝³ : Preadditive D
inst✝² : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝¹ : W.IsCompatibleWithTriangulation
inst✝ : W.HasLocalization
n : ℤ
⊢ (W.Q' ⋙ shiftFunctor W.Localization' n).Additive
Module: Mathlib.CategoryTheory.Triangulated.Subcategory

Goal state before:
pp: case mpr
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} 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
P Q R : ObjectProperty C
inst✝ : IsTriangulated C
X : C
⊢ P.extensionProduct (Q.extensionProduct R) X → (P.extensionProduct Q).extensionProduct R X

Tactic:
intro ⟨A, Z, f, g, h, hT, hP, ⟨B, C, f', g', h', hT', hQ, hR⟩⟩
    obtain ⟨Y, f'', h'', hT''⟩ := distinguished_cocone_triangle₁ (g ≫ g')
    let o := someOctahedron' rfl hT hT' hT''
    exact ⟨_, _, _, _, _, hT'', ⟨_, _, _, _, _, o.mem, hP, hQ⟩, hR⟩

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.Retract

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasZeroMorphisms C
S₁ T₁ S₂ T₂ : ShortComplex C
inst✝³ : S₁.HasHomology
inst✝² : T₁.HasHomology
inst✝¹ : S₂.HasHomology
inst✝ : T₂.HasHomology
f₁ : S₁ ⟶ T₁
f₂ : S₂ ⟶ T₂
h : RetractArrow f₁ f₂
hf₂ : IsIso (homologyMap f₂)
⊢ IsIso (homologyMap f₁)

Tactic:
]

Goal state after:
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasZeroMorphisms C
S₁ T₁ S₂ T₂ : ShortComplex C
inst✝³ : S₁.HasHomology
inst✝² : T₁.HasHomology
inst✝¹ : S₂.HasHomology
inst✝ : T₂.HasHomology
f₁ : S₁ ⟶ T₁
f₂ : S₂ ⟶ T₂
h : RetractArrow f₁ f₂
hf₂ : IsIso (homologyMap f₂)
⊢ IsIso (homologyMap f₁)
Module: Mathlib.Algebra.Homology.QuasiIso

Goal state before:
pp: ι : Type u_1
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
c : ComplexShape ι
K L M K' L' : HomologicalComplex C c
f : K ⟶ L
inst✝² : IsIso f
i : ι
inst✝¹ : K.HasHomology i
inst✝ : L.HasHomology i
⊢ ShortComplex.QuasiIso ((shortComplexFunctor C c i).map f)

Tactic:
]

Goal state after:
ι : Type u_1
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
c : ComplexShape ι
K L M K' L' : HomologicalComplex C c
f : K ⟶ L
inst✝² : IsIso f
i : ι
inst✝¹ : K.HasHomology i
inst✝ : L.HasHomology i
⊢ ShortComplex.QuasiIso ((shortComplexFunctor C c i).map f)
Module: Mathlib.Algebra.Homology.QuasiIso

Goal state before:
pp: ι : Type u_1
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : HasZeroMorphisms C
c : ComplexShape ι
K L : HomologicalComplex C c
f : K ⟶ L
i j k : ι
hi : c.prev j = i
hk : c.next j = k
inst✝³ : K.HasHomology j
inst✝² : L.HasHomology j
inst✝¹ : (K.sc' i j k).HasHomology
inst✝ : (L.sc' i j k).HasHomology
⊢ ShortComplex.QuasiIso ((shortComplexFunctor C c j).map f) ↔
    ShortComplex.QuasiIso ((shortComplexFunctor' C c i j k).map f)

Tactic:
]

Goal state after:
ι : Type u_1
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : HasZeroMorphisms C
c : ComplexShape ι
K L : HomologicalComplex C c
f : K ⟶ L
i j k : ι
hi : c.prev j = i
hk : c.next j = k
inst✝³ : K.HasHomology j
inst✝² : L.HasHomology j
inst✝¹ : (K.sc' i j k).HasHomology
inst✝ : (L.sc' i j k).HasHomology
⊢ ShortComplex.QuasiIso ((shortComplexFunctor C c j).map f) ↔
    ShortComplex.QuasiIso ((shortComplexFunctor' C c i j k).map f)
Module: Mathlib.Algebra.Homology.QuasiIso

Goal state before:
pp: ι : Type u_1
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : HasZeroMorphisms C
c : ComplexShape ι
K L K' L' : HomologicalComplex C c
f : K ⟶ L
f' : K' ⟶ L'
h : RetractArrow f f'
i : ι
inst✝³ : K.HasHomology i
inst✝² : L.HasHomology i
inst✝¹ : K'.HasHomology i
inst✝ : L'.HasHomology i
hf' : ShortComplex.QuasiIso ((shortComplexFunctor C c i).map f')
⊢ ShortComplex.QuasiIso ((shortComplexFunctor C c i).map f)

Tactic:
]

Goal state after:
ι : Type u_1
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : HasZeroMorphisms C
c : ComplexShape ι
K L K' L' : HomologicalComplex C c
f : K ⟶ L
f' : K' ⟶ L'
h : RetractArrow f f'
i : ι
inst✝³ : K.HasHomology i
inst✝² : L.HasHomology i
inst✝¹ : K'.HasHomology i
inst✝ : L'.HasHomology i
hf' : ShortComplex.QuasiIso ((shortComplexFunctor C c i).map f')
⊢ ShortComplex.QuasiIso ((shortComplexFunctor C c i).map f)
Module: Mathlib.Algebra.Homology.QuasiIso

Goal state before:
pp: ι : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
c : ComplexShape ι
K L : HomologicalComplex C c
f : K ⟶ L
i : ι
inst✝¹ : K.HasHomology i
inst✝ : L.HasHomology i
⊢ IsIso (ShortComplex.homologyMap ((shortComplexFunctor C c i).map f)) ↔ IsIso (homologyMap f i)

Tactic:
]

Goal state after:
ι : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
c : ComplexShape ι
K L : HomologicalComplex C c
f : K ⟶ L
i : ι
inst✝¹ : K.HasHomology i
inst✝ : L.HasHomology i
⊢ IsIso (ShortComplex.homologyMap ((shortComplexFunctor C c i).map f)) ↔ IsIso (homologyMap f i)
Module: Mathlib.CategoryTheory.Localization.Composition

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
E : Type u₄
inst✝³ : Category.{v₁, u₁} C₁
inst✝² : Category.{v₂, u₂} C₂
inst✝¹ : Category.{v₃, u₃} C₃
inst✝ : Category.{v₄, u₄} E
L₁ : C₁ ⥤ C₂
L₂ : C₂ ⥤ C₃
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
h₁ : StrictUniversalPropertyFixedTarget L₁ W₁ E
h₂ : StrictUniversalPropertyFixedTarget L₂ W₂ E
W₃ : MorphismProperty C₁
hW₃ : W₃.IsInvertedBy (L₁ ⋙ L₂)
hW₁₃ : W₁ ≤ W₃
hW₂₃ : W₂ ≤ W₃.map L₁
F : C₁ ⥤ E
hF : W₃.IsInvertedBy F
⊢ F = F

Tactic:
]

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

Goal state before:
pp: case a
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
P Q : ObjectProperty C
⊢ P.retractClosure.extensionProduct Q.retractClosure ≤ (P.extensionProduct Q).retractClosure

Tactic:
]

Goal state after:
case a
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
P Q : ObjectProperty C
⊢ P.retractClosure.extensionProduct Q.retractClosure ≤ (P.extensionProduct Q).retractClosure
Module: Mathlib.CategoryTheory.Triangulated.Subcategory

Goal state before:
pp: case a
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
P Q : ObjectProperty C
⊢ P.retractClosure.extensionProduct Q.retractClosure ≤ P.retractClosure.extensionProduct Q.retractClosure

Tactic:
]

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

Goal state before:
pp: case mp
ι : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
c : ComplexShape ι
K L : HomologicalComplex C c
f : K ⟶ L
i : ι
inst✝¹ : K.HasHomology i
inst✝ : L.HasHomology i
hK : IsZero (K.sc i).homology
⊢ IsIso (ShortComplex.homologyMap ((shortComplexFunctor C c i).map f)) → IsZero (L.sc i).homology

pp: case mpr
ι : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
c : ComplexShape ι
K L : HomologicalComplex C c
f : K ⟶ L
i : ι
inst✝¹ : K.HasHomology i
inst✝ : L.HasHomology i
hK : IsZero (K.sc i).homology
⊢ IsZero (L.sc i).homology → IsIso (ShortComplex.homologyMap ((shortComplexFunctor C c i).map f))

Tactic:
· intro h
    exact IsZero.of_iso hK (@asIso _ _ _ _ _ h).symm

Goal state after:
case mpr
ι : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
c : ComplexShape ι
K L : HomologicalComplex C c
f : K ⟶ L
i : ι
inst✝¹ : K.HasHomology i
inst✝ : L.HasHomology i
hK : IsZero (K.sc i).homology
⊢ IsZero (L.sc i).homology → IsIso (ShortComplex.homologyMap ((shortComplexFunctor C c i).map f))
Module: Mathlib.CategoryTheory.Triangulated.Subcategory

Goal state before:
pp: case succ
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} 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
P : ObjectProperty C
inst✝ : IsTriangulated C
n : ℕ
h : P.extensionProductIter (n + 1) = (P.extensionProductIter n).extensionProduct P
⊢ (P.extensionProductIter (n + 1)).extensionProduct P = (P.extensionProductIter (n + 1)).extensionProduct P

Tactic:
]

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

Goal state before:
pp: case zero
C : Type u_1
inst✝¹⁴ : Category.{v_1, u_1} C
inst✝¹³ : HasZeroObject C
inst✝¹² : HasShift C ℤ
inst✝¹¹ : Preadditive C
inst✝¹⁰ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁹ : Pretriangulated C
D : Type u_2
inst✝⁸ : Category.{v_2, u_2} D
inst✝⁷ : Preadditive D
inst✝⁶ : HasZeroObject D
inst✝⁵ : HasShift D ℤ
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝³ : Pretriangulated D
E : Type u_3
inst✝² : Category.{v_3, u_3} E
inst✝¹ : HasShift E ℤ
P Q R : ObjectProperty C
inst✝ : P.Nonempty
⊢ P.Nonempty

Tactic:
]

Goal state after:
case zero
C : Type u_1
inst✝¹⁴ : Category.{v_1, u_1} C
inst✝¹³ : HasZeroObject C
inst✝¹² : HasShift C ℤ
inst✝¹¹ : Preadditive C
inst✝¹⁰ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁹ : Pretriangulated C
D : Type u_2
inst✝⁸ : Category.{v_2, u_2} D
inst✝⁷ : Preadditive D
inst✝⁶ : HasZeroObject D
inst✝⁵ : HasShift D ℤ
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝³ : Pretriangulated D
E : Type u_3
inst✝² : Category.{v_3, u_3} E
inst✝¹ : HasShift E ℤ
P Q R : ObjectProperty C
inst✝ : P.Nonempty
⊢ P.Nonempty
Module: Mathlib.CategoryTheory.Localization.Composition

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
inst✝⁴ : Category.{v₁, u₁} C₁
inst✝³ : Category.{v₂, u₂} C₂
inst✝² : Category.{v₃, u₃} C₃
L₁ : C₁ ⥤ C₂
L₂ : C₂ ⥤ C₃
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
inst✝¹ : L₁.IsLocalization W₁
inst✝ : L₂.IsLocalization W₂
W₃ : MorphismProperty C₁
hW₃ : W₃.IsInvertedBy (L₁ ⋙ L₂)
hW₁₃ : W₁ ≤ W₃
hW₂₃ : W₂ ≤ W₃.map L₁
E₂ : C₂ ≌ W₁.Localization := uniq L₁ W₁.Q W₁
W₂' : MorphismProperty W₁.Localization := W₂.map E₂.functor
eq : (W₂.map E₂.functor).inverseImage E₂.functor = W₂.isoClosure
⊢ W₂ ≤ W₂'.inverseImage E₂.functor

Tactic:
]

Goal state after:
C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
inst✝⁴ : Category.{v₁, u₁} C₁
inst✝³ : Category.{v₂, u₂} C₂
inst✝² : Category.{v₃, u₃} C₃
L₁ : C₁ ⥤ C₂
L₂ : C₂ ⥤ C₃
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
inst✝¹ : L₁.IsLocalization W₁
inst✝ : L₂.IsLocalization W₂
W₃ : MorphismProperty C₁
hW₃ : W₃.IsInvertedBy (L₁ ⋙ L₂)
hW₁₃ : W₁ ≤ W₃
hW₂₃ : W₂ ≤ W₃.map L₁
E₂ : C₂ ≌ W₁.Localization := uniq L₁ W₁.Q W₁
W₂' : MorphismProperty W₁.Localization := W₂.map E₂.functor
eq : (W₂.map E₂.functor).inverseImage E₂.functor = W₂.isoClosure
⊢ W₂ ≤ W₂'.inverseImage E₂.functor
Module: Mathlib.CategoryTheory.Triangulated.Subcategory

Goal state before:
pp: case succ
C : Type u_1
inst✝¹⁴ : Category.{v_1, u_1} C
inst✝¹³ : HasZeroObject C
inst✝¹² : HasShift C ℤ
inst✝¹¹ : Preadditive C
inst✝¹⁰ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁹ : Pretriangulated C
D : Type u_2
inst✝⁸ : Category.{v_2, u_2} D
inst✝⁷ : Preadditive D
inst✝⁶ : HasZeroObject D
inst✝⁵ : HasShift D ℤ
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝³ : Pretriangulated D
E : Type u_3
inst✝² : Category.{v_3, u_3} E
inst✝¹ : HasShift E ℤ
P Q R : ObjectProperty C
inst✝ : P.Nonempty
n : ℕ
h : (P.extensionProductIter n).Nonempty
⊢ (P.extensionProduct (P.extensionProductIter n)).Nonempty

Tactic:
]

Goal state after:
case succ
C : Type u_1
inst✝¹⁴ : Category.{v_1, u_1} C
inst✝¹³ : HasZeroObject C
inst✝¹² : HasShift C ℤ
inst✝¹¹ : Preadditive C
inst✝¹⁰ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁹ : Pretriangulated C
D : Type u_2
inst✝⁸ : Category.{v_2, u_2} D
inst✝⁷ : Preadditive D
inst✝⁶ : HasZeroObject D
inst✝⁵ : HasShift D ℤ
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝³ : Pretriangulated D
E : Type u_3
inst✝² : Category.{v_3, u_3} E
inst✝¹ : HasShift E ℤ
P Q R : ObjectProperty C
inst✝ : P.Nonempty
n : ℕ
h : (P.extensionProductIter n).Nonempty
⊢ (P.extensionProduct (P.extensionProductIter n)).Nonempty
Module: Mathlib.CategoryTheory.Localization.Composition

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
inst✝⁴ : Category.{v₁, u₁} C₁
inst✝³ : Category.{v₂, u₂} C₂
inst✝² : Category.{v₃, u₃} C₃
L₁ : C₁ ⥤ C₂
L₂ : C₂ ⥤ C₃
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
inst✝¹ : L₁.IsLocalization W₁
inst✝ : L₂.IsLocalization W₂
W₃ : MorphismProperty C₁
hW₃ : W₃.IsInvertedBy (L₁ ⋙ L₂)
hW₁₃ : W₁ ≤ W₃
hW₂₃ : W₂ ≤ W₃.map L₁
E₂ : C₂ ≌ W₁.Localization := uniq L₁ W₁.Q W₁
W₂' : MorphismProperty W₁.Localization := W₂.map E₂.functor
eq : (W₂.map E₂.functor).inverseImage E₂.functor = W₂.isoClosure
⊢ W₂ ≤ W₂.isoClosure

Tactic:
]

Goal state after:
C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
inst✝⁴ : Category.{v₁, u₁} C₁
inst✝³ : Category.{v₂, u₂} C₂
inst✝² : Category.{v₃, u₃} C₃
L₁ : C₁ ⥤ C₂
L₂ : C₂ ⥤ C₃
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
inst✝¹ : L₁.IsLocalization W₁
inst✝ : L₂.IsLocalization W₂
W₃ : MorphismProperty C₁
hW₃ : W₃.IsInvertedBy (L₁ ⋙ L₂)
hW₁₃ : W₁ ≤ W₃
hW₂₃ : W₂ ≤ W₃.map L₁
E₂ : C₂ ≌ W₁.Localization := uniq L₁ W₁.Q W₁
W₂' : MorphismProperty W₁.Localization := W₂.map E₂.functor
eq : (W₂.map E₂.functor).inverseImage E₂.functor = W₂.isoClosure
⊢ W₂ ≤ W₂.isoClosure
Module: Mathlib.Algebra.Homology.QuasiIso

Goal state before:
pp: ι : Type u_1
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
c : ComplexShape ι
K L M K' L' : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
i : ι
inst✝² : K.HasHomology i
inst✝¹ : L.HasHomology i
inst✝ : M.HasHomology i
hφ : ShortComplex.QuasiIso ((shortComplexFunctor C c i).map φ)
hφ' : ShortComplex.QuasiIso ((shortComplexFunctor C c i).map φ')
⊢ ShortComplex.QuasiIso ((shortComplexFunctor C c i).map (φ ≫ φ'))

Tactic:
]

Goal state after:
ι : Type u_1
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
c : ComplexShape ι
K L M K' L' : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
i : ι
inst✝² : K.HasHomology i
inst✝¹ : L.HasHomology i
inst✝ : M.HasHomology i
hφ : ShortComplex.QuasiIso ((shortComplexFunctor C c i).map φ)
hφ' : ShortComplex.QuasiIso ((shortComplexFunctor C c i).map φ')
⊢ ShortComplex.QuasiIso ((shortComplexFunctor C c i).map (φ ≫ φ'))
Module: Mathlib.Algebra.Homology.QuasiIso

Goal state before:
pp: ι : Type u_1
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
c : ComplexShape ι
K L M K' L' : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
i : ι
inst✝² : K.HasHomology i
inst✝¹ : L.HasHomology i
inst✝ : M.HasHomology i
hφ : ShortComplex.QuasiIso ((shortComplexFunctor C c i).map φ)
hφ' : ShortComplex.QuasiIso ((shortComplexFunctor C c i).map φ')
⊢ ShortComplex.QuasiIso ((shortComplexFunctor C c i).map φ ≫ (shortComplexFunctor C c i).map φ')

Tactic:
]

Goal state after:
ι : Type u_1
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
c : ComplexShape ι
K L M K' L' : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
i : ι
inst✝² : K.HasHomology i
inst✝¹ : L.HasHomology i
inst✝ : M.HasHomology i
hφ : ShortComplex.QuasiIso ((shortComplexFunctor C c i).map φ)
hφ' : ShortComplex.QuasiIso ((shortComplexFunctor C c i).map φ')
⊢ ShortComplex.QuasiIso ((shortComplexFunctor C c i).map φ ≫ (shortComplexFunctor C c i).map φ')
Module: Mathlib.CategoryTheory.Triangulated.Subcategory

Goal state before:
pp: case zero
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} 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
P : ObjectProperty C
inst✝ : IsTriangulated C
n n' : ℕ
h : n = n' + 1
⊢ (P.extensionProductIter n').extensionProduct P = (P.extensionProductIter n').extensionProduct P

Tactic:
]

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

Goal state before:
pp: ι : Type u_1
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
c : ComplexShape ι
K L M : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
i : ι
inst✝² : K.HasHomology i
inst✝¹ : L.HasHomology i
inst✝ : M.HasHomology i
hφ : IsIso (homologyMap φ i)
hφφ' : IsIso (homologyMap (φ ≫ φ') i)
⊢ IsIso (homologyMap φ' i)

Tactic:
]

Goal state after:
ι : Type u_1
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
c : ComplexShape ι
K L M : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
i : ι
inst✝² : K.HasHomology i
inst✝¹ : L.HasHomology i
inst✝ : M.HasHomology i
hφ : IsIso (homologyMap φ i)
hφφ' : IsIso (homologyMap (φ ≫ φ') i)
⊢ IsIso (homologyMap φ' i)
Module: Mathlib.Algebra.Homology.QuasiIso

Goal state before:
pp: ι : Type u_1
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
c : ComplexShape ι
K L M : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
i : ι
inst✝² : K.HasHomology i
inst✝¹ : L.HasHomology i
inst✝ : M.HasHomology i
hφ : IsIso (homologyMap φ i)
hφφ' : IsIso (homologyMap φ i ≫ homologyMap φ' i)
⊢ IsIso (homologyMap φ' i)

Tactic:
]

Goal state after:
ι : Type u_1
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
c : ComplexShape ι
K L M : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
i : ι
inst✝² : K.HasHomology i
inst✝¹ : L.HasHomology i
inst✝ : M.HasHomology i
hφ : IsIso (homologyMap φ i)
hφφ' : IsIso (homologyMap φ i ≫ homologyMap φ' i)
⊢ IsIso (homologyMap φ' i)
Module: Mathlib.CategoryTheory.Triangulated.Subcategory

Goal state before:
pp: case succ
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} 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
P : ObjectProperty C
inst✝ : IsTriangulated C
n n' : ℕ
h : n = n' + 1
m : ℕ
hm : P.extensionProductIter (n + m) = (P.extensionProductIter n').extensionProduct (P.extensionProductIter m)
⊢ (P.extensionProductIter (n + m)).extensionProduct P =
    (P.extensionProductIter n').extensionProduct ((P.extensionProductIter m).extensionProduct P)

Tactic:
hm,

Goal state after:
case succ
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} 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
P : ObjectProperty C
inst✝ : IsTriangulated C
n n' : ℕ
h : n = n' + 1
m : ℕ
hm : P.extensionProductIter (n + m) = (P.extensionProductIter n').extensionProduct (P.extensionProductIter m)
⊢ ((P.extensionProductIter n').extensionProduct (P.extensionProductIter m)).extensionProduct P =
    (P.extensionProductIter n').extensionProduct ((P.extensionProductIter m).extensionProduct P)
Module: Mathlib.CategoryTheory.Triangulated.Subcategory

Goal state before:
pp: case succ
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} 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
P : ObjectProperty C
inst✝ : IsTriangulated C
n n' : ℕ
h : n = n' + 1
m : ℕ
hm : P.extensionProductIter (n + m) = (P.extensionProductIter n').extensionProduct (P.extensionProductIter m)
⊢ (P.extensionProductIter n').extensionProduct ((P.extensionProductIter m).extensionProduct P) =
    (P.extensionProductIter n').extensionProduct ((P.extensionProductIter m).extensionProduct P)

Tactic:
]

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

Goal state before:
pp: ι : Type u_1
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
c : ComplexShape ι
K L M : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
inst✝² : ∀ (i : ι), K.HasHomology i
inst✝¹ : ∀ (i : ι), L.HasHomology i
inst✝ : ∀ (i : ι), M.HasHomology i
hφ : QuasiIso φ
hφφ' : QuasiIso (φ ≫ φ')
⊢ QuasiIso (φ ≫ φ')

Tactic:
]

Goal state after:
ι : Type u_1
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
c : ComplexShape ι
K L M : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
inst✝² : ∀ (i : ι), K.HasHomology i
inst✝¹ : ∀ (i : ι), L.HasHomology i
inst✝ : ∀ (i : ι), M.HasHomology i
hφ : QuasiIso φ
hφφ' : QuasiIso (φ ≫ φ')
⊢ QuasiIso (φ ≫ φ')
Module: Mathlib.Algebra.Homology.QuasiIso

Goal state before:
pp: ι : Type u_1
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
c : ComplexShape ι
K L M : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
i : ι
inst✝² : K.HasHomology i
inst✝¹ : L.HasHomology i
inst✝ : M.HasHomology i
hφ' : IsIso (homologyMap φ' i)
hφφ' : IsIso (homologyMap (φ ≫ φ') i)
⊢ IsIso (homologyMap φ i)

Tactic:
]

Goal state after:
ι : Type u_1
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
c : ComplexShape ι
K L M : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
i : ι
inst✝² : K.HasHomology i
inst✝¹ : L.HasHomology i
inst✝ : M.HasHomology i
hφ' : IsIso (homologyMap φ' i)
hφφ' : IsIso (homologyMap (φ ≫ φ') i)
⊢ IsIso (homologyMap φ i)
Module: Mathlib.CategoryTheory.Triangulated.Subcategory

Goal state before:
pp: case zero
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} 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
P : ObjectProperty C
inst✝ : IsTriangulated C
m m' : ℕ
h : m = m' + 1
⊢ P.extensionProduct (P.extensionProductIter m') = P.extensionProduct (P.extensionProductIter m')

Tactic:
]

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

Goal state before:
pp: ι : Type u_1
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
c : ComplexShape ι
K L M : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
i : ι
inst✝² : K.HasHomology i
inst✝¹ : L.HasHomology i
inst✝ : M.HasHomology i
hφ' : IsIso (homologyMap φ' i)
hφφ' : IsIso (homologyMap φ i ≫ homologyMap φ' i)
⊢ IsIso (homologyMap φ i)

Tactic:
]

Goal state after:
ι : Type u_1
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
c : ComplexShape ι
K L M : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
i : ι
inst✝² : K.HasHomology i
inst✝¹ : L.HasHomology i
inst✝ : M.HasHomology i
hφ' : IsIso (homologyMap φ' i)
hφφ' : IsIso (homologyMap φ i ≫ homologyMap φ' i)
⊢ IsIso (homologyMap φ i)
Module: Mathlib.CategoryTheory.Triangulated.Subcategory

Goal state before:
pp: case succ
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} 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
P : ObjectProperty C
inst✝ : IsTriangulated C
m m' : ℕ
h : m = m' + 1
n : ℕ
hn : P.extensionProductIter (n + m) = (P.extensionProductIter n).extensionProduct (P.extensionProductIter m')
⊢ P.extensionProduct ((P.extensionProductIter n).extensionProduct (P.extensionProductIter m')) =
    P.extensionProduct ((P.extensionProductIter n).extensionProduct (P.extensionProductIter m'))

Tactic:
]

Goal state after: