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

Goal state before:
pp: 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).retractClosure = (P.extensionProduct Q).retractClosure

Tactic:
by

Goal state after:
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
x✝¹ x✝ : C₃ ⥤ E
h : (L₁ ⋙ L₂) ⋙ x✝¹ = (L₁ ⋙ L₂) ⋙ x✝
⊢ L₁ ⋙ L₂ ⋙ x✝¹ = L₁ ⋙ L₂ ⋙ x✝

Tactic:
by

Goal state after:
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 : ObjectProperty C
inst✝ : IsTriangulated C
n : ℕ
⊢ P.extensionProductIter (n + 1) = (P.extensionProductIter n).extensionProduct P

Tactic:
by

Goal state after:
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
hK : K.ExactAt i
⊢ QuasiIsoAt f i ↔ L.ExactAt i

Tactic:
by

Goal state after:
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
hL : L.ExactAt i
⊢ QuasiIsoAt f i ↔ K.ExactAt i

Tactic:
by

Goal state after:
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 M K' L' : HomologicalComplex C c
f : K ⟶ L
i : ι
inst✝¹ : K.HasHomology i
inst✝ : L.HasHomology i
hf : QuasiIsoAt f i
⊢ IsIso (homologyMap f i)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : HasZeroMorphisms C
K L : CochainComplex C ℕ
f : K ⟶ L
inst✝³ : HasHomology K 0
inst✝² : HasHomology L 0
inst✝¹ : (sc' K 0 0 1).HasHomology
inst✝ : (sc' L 0 0 1).HasHomology
⊢ (ComplexShape.up ℕ).prev 0 = 0

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : HasZeroMorphisms C
K L : CochainComplex C ℕ
f : K ⟶ L
inst✝³ : HasHomology K 0
inst✝² : HasHomology L 0
inst✝¹ : (sc' K 0 0 1).HasHomology
inst✝ : (sc' L 0 0 1).HasHomology
⊢ (ComplexShape.up ℕ).next 0 = 1

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : HasZeroMorphisms C
K L : ChainComplex C ℕ
f : K ⟶ L
inst✝³ : HasHomology K 0
inst✝² : HasHomology L 0
inst✝¹ : (sc' K 1 0 0).HasHomology
inst✝ : (sc' L 1 0 0).HasHomology
⊢ (ComplexShape.down ℕ).prev 0 = 1

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : HasZeroMorphisms C
K L : ChainComplex C ℕ
f : K ⟶ L
inst✝³ : HasHomology K 0
inst✝² : HasHomology L 0
inst✝¹ : (sc' K 1 0 0).HasHomology
inst✝ : (sc' L 1 0 0).HasHomology
⊢ (ComplexShape.down ℕ).next 0 = 0

Tactic:
by

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

Goal state before:
pp: 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
⊢ (P.extensionProductIter n).Nonempty

Tactic:
by

Goal state after:
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
⊢ W₂ ≤ W₂'.inverseImage E₂.functor

Tactic:
by

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 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 : QuasiIsoAt φ i
hφ' : QuasiIsoAt φ' i
⊢ QuasiIsoAt (φ ≫ φ') i

Tactic:
by

Goal state after:
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
Φ : LocalizerMorphism W₂ W₂' := { functor := E₂.functor, map := ⋯ }
⊢ W₂' ≤ W₂.map Φ.functor

Tactic:
by

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 : QuasiIsoAt φ i
hφφ' : QuasiIsoAt (φ ≫ φ') i
⊢ QuasiIsoAt φ' i

Tactic:
by

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 : QuasiIsoAt φ i
⊢ QuasiIsoAt (φ ≫ φ') i ↔ QuasiIsoAt φ' i

Tactic:
by

Goal state after:
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
Φ : LocalizerMorphism W₂ W₂' := { functor := E₂.functor, map := ⋯ }
this : Φ.IsLocalizedEquivalence
E₃ : C₃ ≌ W₂'.Localization := (Φ.localizedFunctor L₂ W₂'.Q).asEquivalence
⊢ (W₁.Q ⋙ W₂'.Q) ⋙ E₃.inverse ≅ L₁ ⋙ L₂

Tactic:
by

Goal state after:
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 : ObjectProperty C
inst✝ : IsTriangulated C
n m n' : ℕ
h : n = n' + 1
⊢ P.extensionProductIter (n + m) = (P.extensionProductIter n').extensionProduct (P.extensionProductIter m)

Tactic:
by

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
⊢ QuasiIso (φ ≫ φ') ↔ QuasiIso φ'

Tactic:
by

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'

Tactic:
by

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' : QuasiIsoAt φ' i
hφφ' : QuasiIsoAt (φ ≫ φ') i
⊢ QuasiIsoAt φ i

Tactic:
by

Goal state after:
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
Φ : LocalizerMorphism W₂ W₂' := { functor := E₂.functor, map := ⋯ }
this : Φ.IsLocalizedEquivalence
E₃ : C₃ ≌ W₂'.Localization := (Φ.localizedFunctor L₂ W₂'.Q).asEquivalence
iso : (W₁.Q ⋙ W₂'.Q) ⋙ E₃.inverse ≅ L₁ ⋙ L₂ :=
  Trans.trans
    (Trans.trans
      (W₁.Q.associator W₂'.Q E₃.inverse ≪≫
        isoWhiskerRight (compUniqFunctor L₁ W₁.Q W₁).symm (W₂'.Q ⋙ E₃.inverse) ≪≫
          L₁.associator (uniq L₁ W₁.Q W₁).functor (W₂'.Q ⋙ E₃.inverse))
      (L₁.isoWhiskerLeft
        ((E₂.functor.associator W₂'.Q E₃.inverse).symm ≪≫
          isoWhiskerRight (CatCommSq.iso Φ.functor L₂ W₂'.Q (Φ.localizedFunctor L₂ W₂'.Q)) E₃.inverse ≪≫
            L₂.associator (Φ.localizedFunctor L₂ W₂'.Q) E₃.inverse)))
    (L₁.isoWhiskerLeft (L₂.isoWhiskerLeft E₃.unitIso.symm ≪≫ L₂.rightUnitor))
⊢ W₃.IsInvertedBy (W₁.Q ⋙ W₂'.Q)

Tactic:
by

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' : QuasiIsoAt φ' i
⊢ QuasiIsoAt (φ ≫ φ') i ↔ QuasiIsoAt φ i

Tactic:
by

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'
⊢ QuasiIso (φ ≫ φ') ↔ QuasiIso φ

Tactic:
by

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

Tactic:
by

Goal state after:
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 : ObjectProperty C
inst✝ : IsTriangulated C
n m m' : ℕ
h : m = m' + 1
⊢ P.extensionProductIter (n + m) = (P.extensionProductIter n).extensionProduct (P.extensionProductIter m')

Tactic:
by

Goal state after:
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
φ : K ⟶ L
φ' : K' ⟶ L'
e : Arrow.mk φ ≅ Arrow.mk φ'
inst✝³ : ∀ (i : ι), K.HasHomology i
inst✝² : ∀ (i : ι), L.HasHomology i
inst✝¹ : ∀ (i : ι), K'.HasHomology i
inst✝ : ∀ (i : ι), L'.HasHomology i
⊢ QuasiIso φ ↔ QuasiIso φ'

Tactic:
by

Goal state after:
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
Φ : LocalizerMorphism W₂ W₂' := { functor := E₂.functor, map := ⋯ }
this : Φ.IsLocalizedEquivalence
E₃ : C₃ ≌ W₂'.Localization := (Φ.localizedFunctor L₂ W₂'.Q).asEquivalence
iso : (W₁.Q ⋙ W₂'.Q) ⋙ E₃.inverse ≅ L₁ ⋙ L₂ :=
  Trans.trans
    (Trans.trans
      (W₁.Q.associator W₂'.Q E₃.inverse ≪≫
        isoWhiskerRight (compUniqFunctor L₁ W₁.Q W₁).symm (W₂'.Q ⋙ E₃.inverse) ≪≫
          L₁.associator (uniq L₁ W₁.Q W₁).functor (W₂'.Q ⋙ E₃.inverse))
      (L₁.isoWhiskerLeft
        ((E₂.functor.associator W₂'.Q E₃.inverse).symm ≪≫
          isoWhiskerRight (CatCommSq.iso Φ.functor L₂ W₂'.Q (Φ.localizedFunctor L₂ W₂'.Q)) E₃.inverse ≪≫
            L₂.associator (Φ.localizedFunctor L₂ W₂'.Q) E₃.inverse)))
    (L₁.isoWhiskerLeft (L₂.isoWhiskerLeft E₃.unitIso.symm ≪≫ L₂.rightUnitor))
hW₃' : W₃.IsInvertedBy (W₁.Q ⋙ W₂'.Q)
⊢ (fun x ↦ x.map E₂.functor) (W₃.map L₁) ≤ W₃.map W₁.Q

Tactic:
by

Goal state after:
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
φ : K ⟶ L
φ' : K' ⟶ L'
e : Arrow.mk φ ≅ Arrow.mk φ'
inst✝³ : ∀ (i : ι), K.HasHomology i
inst✝² : ∀ (i : ι), L.HasHomology i
inst✝¹ : ∀ (i : ι), K'.HasHomology i
inst✝ : ∀ (i : ι), L'.HasHomology i
hφ : QuasiIso φ
⊢ QuasiIso φ'

Tactic:
by

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

Goal state before:
pp: 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
hPQ : P ≤ Q
n : ℕ
⊢ P.extensionProductIter n ≤ Q.extensionProductIter n

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
C : Type u
inst✝¹¹ : Category.{v, u} C
inst✝¹⁰ : HasZeroMorphisms C
c : ComplexShape ι
K✝ L✝ M K' L' : HomologicalComplex C c
C₁ : Type u_2
C₂ : Type u_3
inst✝⁹ : Category.{v_1, u_2} C₁
inst✝⁸ : Category.{v_2, u_3} C₂
inst✝⁷ : Preadditive C₁
inst✝⁶ : Preadditive C₂
K L : HomologicalComplex C₁ c
φ : K ⟶ L
F : C₁ ⥤ C₂
inst✝⁵ : F.Additive
inst✝⁴ : F.PreservesHomology
i : ι
inst✝³ : K.HasHomology i
inst✝² : L.HasHomology i
inst✝¹ : ((F.mapHomologicalComplex c).obj K).HasHomology i
inst✝ : ((F.mapHomologicalComplex c).obj L).HasHomology i
hφ : QuasiIsoAt φ i
⊢ QuasiIsoAt ((F.mapHomologicalComplex c).map φ) i

Tactic:
rw [quasiIsoAt_iff] at hφ ⊢

Goal state after:
ι : Type u_1
C : Type u
inst✝¹¹ : Category.{v, u} C
inst✝¹⁰ : HasZeroMorphisms C
c : ComplexShape ι
K✝ L✝ M K' L' : HomologicalComplex C c
C₁ : Type u_2
C₂ : Type u_3
inst✝⁹ : Category.{v_1, u_2} C₁
inst✝⁸ : Category.{v_2, u_3} C₂
inst✝⁷ : Preadditive C₁
inst✝⁶ : Preadditive C₂
K L : HomologicalComplex C₁ c
φ : K ⟶ L
F : C₁ ⥤ C₂
inst✝⁵ : F.Additive
inst✝⁴ : F.PreservesHomology
i : ι
inst✝³ : K.HasHomology i
inst✝² : L.HasHomology i
inst✝¹ : ((F.mapHomologicalComplex c).obj K).HasHomology i
inst✝ : ((F.mapHomologicalComplex c).obj L).HasHomology i
hφ : ShortComplex.QuasiIso ((shortComplexFunctor C₁ c i).map φ)
⊢ ShortComplex.QuasiIso ((shortComplexFunctor C₂ c i).map ((F.mapHomologicalComplex c).map φ))
Module: Mathlib.Algebra.Homology.QuasiIso

Goal state before:
pp: ι : Type u_1
C : Type u
inst✝¹¹ : Category.{v, u} C
inst✝¹⁰ : HasZeroMorphisms C
c : ComplexShape ι
K✝ L✝ M K' L' : HomologicalComplex C c
C₁ : Type u_2
C₂ : Type u_3
inst✝⁹ : Category.{v_1, u_2} C₁
inst✝⁸ : Category.{v_2, u_3} C₂
inst✝⁷ : Preadditive C₁
inst✝⁶ : Preadditive C₂
K L : HomologicalComplex C₁ c
φ : K ⟶ L
F : C₁ ⥤ C₂
inst✝⁵ : F.Additive
inst✝⁴ : F.PreservesHomology
i : ι
inst✝³ : K.HasHomology i
inst✝² : L.HasHomology i
inst✝¹ : ((F.mapHomologicalComplex c).obj K).HasHomology i
inst✝ : ((F.mapHomologicalComplex c).obj L).HasHomology i
hφ : QuasiIsoAt φ i
⊢ QuasiIsoAt ((F.mapHomologicalComplex c).map φ) i

Tactic:
by

Goal state after:
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 : ObjectProperty C
inst✝ : P.ContainsZero
n m : ℕ
h : n ≤ m
⊢ P.extensionProductIter n ≤ P.extensionProductIter m

Tactic:
by

Goal state after:
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
Φ : LocalizerMorphism W₂ W₂' := { functor := E₂.functor, map := ⋯ }
this : Φ.IsLocalizedEquivalence
E₃ : C₃ ≌ W₂'.Localization := (Φ.localizedFunctor L₂ W₂'.Q).asEquivalence
iso : (W₁.Q ⋙ W₂'.Q) ⋙ E₃.inverse ≅ L₁ ⋙ L₂ :=
  Trans.trans
    (Trans.trans
      (W₁.Q.associator W₂'.Q E₃.inverse ≪≫
        isoWhiskerRight (compUniqFunctor L₁ W₁.Q W₁).symm (W₂'.Q ⋙ E₃.inverse) ≪≫
          L₁.associator (uniq L₁ W₁.Q W₁).functor (W₂'.Q ⋙ E₃.inverse))
      (L₁.isoWhiskerLeft
        ((E₂.functor.associator W₂'.Q E₃.inverse).symm ≪≫
          isoWhiskerRight (CatCommSq.iso Φ.functor L₂ W₂'.Q (Φ.localizedFunctor L₂ W₂'.Q)) E₃.inverse ≪≫
            L₂.associator (Φ.localizedFunctor L₂ W₂'.Q) E₃.inverse)))
    (L₁.isoWhiskerLeft (L₂.isoWhiskerLeft E₃.unitIso.symm ≪≫ L₂.rightUnitor))
hW₃' : W₃.IsInvertedBy (W₁.Q ⋙ W₂'.Q)
hW₂₃' : W₂' ≤ W₃.map W₁.Q
⊢ (W₁.Q ⋙ W₂'.Q).IsLocalization W₃

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
c : ComplexShape ι
C₁ : Type u_2
C₂ : Type u_3
inst✝¹⁰ : Category.{v_1, u_2} C₁
inst✝⁹ : Category.{v_2, u_3} C₂
inst✝⁸ : Preadditive C₁
inst✝⁷ : Preadditive C₂
K L : HomologicalComplex C₁ c
φ : K ⟶ L
F : C₁ ⥤ C₂
inst✝⁶ : F.Additive
inst✝⁵ : F.PreservesHomology
i : ι
inst✝⁴ : K.HasHomology i
inst✝³ : L.HasHomology i
inst✝² : ((F.mapHomologicalComplex c).obj K).HasHomology i
inst✝¹ : ((F.mapHomologicalComplex c).obj L).HasHomology i
inst✝ : F.ReflectsIsomorphisms
⊢ QuasiIsoAt ((F.mapHomologicalComplex c).map φ) i ↔ QuasiIsoAt φ i

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
c : ComplexShape ι
C₁ : Type u_2
C₂ : Type u_3
inst✝¹⁰ : Category.{v_1, u_2} C₁
inst✝⁹ : Category.{v_2, u_3} C₂
inst✝⁸ : Preadditive C₁
inst✝⁷ : Preadditive C₂
K L : HomologicalComplex C₁ c
φ : K ⟶ L
F : C₁ ⥤ C₂
inst✝⁶ : F.Additive
inst✝⁵ : F.PreservesHomology
inst✝⁴ : ∀ (i : ι), K.HasHomology i
inst✝³ : ∀ (i : ι), L.HasHomology i
inst✝² : ∀ (i : ι), ((F.mapHomologicalComplex c).obj K).HasHomology i
inst✝¹ : ∀ (i : ι), ((F.mapHomologicalComplex c).obj L).HasHomology i
inst✝ : F.ReflectsIsomorphisms
⊢ QuasiIso ((F.mapHomologicalComplex c).map φ) ↔ QuasiIso φ

Tactic:
by

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

Goal state before:
pp: 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
n : ℕ
⊢ ⊥.extensionProductIter n = ⊥

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
c : ComplexShape ι
K L : HomologicalComplex C c
inst✝ : CategoryWithHomology C
f : K ⟶ L
⊢ quasiIso C c f ↔ QuasiIso f

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
c : ComplexShape ι
K L M K' L' : HomologicalComplex C c
inst✝ : CategoryWithHomology C
x✝ : HomologicalComplex C c
⊢ quasiIso C c (𝟙 x✝)

Tactic:
by

Goal state after:
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
⊢ (L₁ ⋙ L₂).IsLocalization W₃

Tactic:
by

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

Goal state before:
pp: 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.IsStableUnderShift ℤ
n : ℕ
⊢ (P.extensionProductIter n).IsStableUnderShift ℤ

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
c : ComplexShape ι
K L M K' L' : HomologicalComplex C c
inst✝ : CategoryWithHomology C
X✝ Y✝ Z✝ : HomologicalComplex C c
x✝¹ : X✝ ⟶ Y✝
x✝ : Y✝ ⟶ Z✝
hf : quasiIso C c x✝¹
hg : quasiIso C c x✝
⊢ quasiIso C c (x✝¹ ≫ x✝)

Tactic:
by

Goal state after:
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✝ : Q.IsTriangulatedClosed₂
h : P ≤ Q
n : ℕ
⊢ P.extensionProductIter n ≤ Q.isoClosure

Tactic:
by

Goal state after:
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
W₃ : MorphismProperty C₁
inst✝¹ : L₁.IsLocalization W₁
inst✝ : (L₁ ⋙ L₂).IsLocalization W₃
hW₁₃ : W₁ ≤ W₃
hW₂₃ : W₂ = W₃.map L₁
X Y : C₁
f : X ⟶ Y
hf : W₃ f
⊢ W₂ (L₁.map f)

Tactic:
by

Goal state after: