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

Goal state before:
pp: C : Type u_1
D : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹⁹ : Category.{v_1, u_1} C
inst✝¹⁸ : Category.{v_2, u_2} D
inst✝¹⁷ : Category.{v_3, u_3} D₁
inst✝¹⁶ : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝¹⁵ : HasZeroObject C
inst✝¹⁴ : HasShift C ℤ
inst✝¹³ : Preadditive C
inst✝¹² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹¹ : Pretriangulated C
inst✝¹⁰ : IsTriangulated C
inst✝⁹ : A.IsTriangulated
inst✝⁸ : B.IsTriangulated
inst✝⁷ : B.IsClosedUnderIsomorphisms
inst✝⁶ : A.IsVerdierRightLocalizing B
L₁ : A.FullSubcategory ⥤ D₁
L₂ : C ⥤ D₂
inst✝⁵ : L₁.IsLocalization (B.inverseImage A.ι).trW
inst✝⁴ : L₂.IsLocalization B.trW
inst✝³ : Preadditive D₁
inst✝² : Preadditive D₂
inst✝¹ : L₁.Additive
inst✝ : L₂.Additive
F : D₁ ⥤ D₂ := (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂
⊢ ((A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂).Additive

Tactic:
rw [Localization.functor_additive_iff L₁ (B.inverseImage A.ι).trW]

Goal state after:
C : Type u_1
D : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹⁹ : Category.{v_1, u_1} C
inst✝¹⁸ : Category.{v_2, u_2} D
inst✝¹⁷ : Category.{v_3, u_3} D₁
inst✝¹⁶ : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝¹⁵ : HasZeroObject C
inst✝¹⁴ : HasShift C ℤ
inst✝¹³ : Preadditive C
inst✝¹² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹¹ : Pretriangulated C
inst✝¹⁰ : IsTriangulated C
inst✝⁹ : A.IsTriangulated
inst✝⁸ : B.IsTriangulated
inst✝⁷ : B.IsClosedUnderIsomorphisms
inst✝⁶ : A.IsVerdierRightLocalizing B
L₁ : A.FullSubcategory ⥤ D₁
L₂ : C ⥤ D₂
inst✝⁵ : L₁.IsLocalization (B.inverseImage A.ι).trW
inst✝⁴ : L₂.IsLocalization B.trW
inst✝³ : Preadditive D₁
inst✝² : Preadditive D₂
inst✝¹ : L₁.Additive
inst✝ : L₂.Additive
F : D₁ ⥤ D₂ := (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂
⊢ (L₁ ⋙ (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂).Additive
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} 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
n₀ : ℤ
X : C
x✝ : t.IsLE X n₀
e : contractibleTriangle X ≅ (t.triangleLTGE (n₀ + 1)).obj X
he : e.hom.hom₂ = (Iso.refl X).hom
⊢ (e.inv ≫ e.hom).hom₂ = 𝟙 X ≫ e.hom.hom₂

Tactic:
e.inv_hom_id,

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} 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
n₀ : ℤ
X : C
x✝ : t.IsLE X n₀
e : contractibleTriangle X ≅ (t.triangleLTGE (n₀ + 1)).obj X
he : e.hom.hom₂ = (Iso.refl X).hom
⊢ (𝟙 ((t.triangleLTGE (n₀ + 1)).obj X)).hom₂ = 𝟙 X ≫ e.hom.hom₂
Module: Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory

Goal state before:
pp: C : Type u_1
D : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹⁹ : Category.{v_1, u_1} C
inst✝¹⁸ : Category.{v_2, u_2} D
inst✝¹⁷ : Category.{v_3, u_3} D₁
inst✝¹⁶ : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝¹⁵ : HasZeroObject C
inst✝¹⁴ : HasShift C ℤ
inst✝¹³ : Preadditive C
inst✝¹² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹¹ : Pretriangulated C
inst✝¹⁰ : IsTriangulated C
inst✝⁹ : A.IsTriangulated
inst✝⁸ : B.IsTriangulated
inst✝⁷ : B.IsClosedUnderIsomorphisms
inst✝⁶ : A.IsVerdierLeftLocalizing B
L₁ : A.FullSubcategory ⥤ D₁
L₂ : C ⥤ D₂
inst✝⁵ : L₁.IsLocalization (B.inverseImage A.ι).trW
inst✝⁴ : L₂.IsLocalization B.trW
inst✝³ : Preadditive D₁
inst✝² : Preadditive D₂
inst✝¹ : L₁.Additive
inst✝ : L₂.Additive
F : D₁ ⥤ D₂ := (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂
⊢ ((A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂).Additive

Tactic:
rw [Localization.functor_additive_iff L₁ (B.inverseImage A.ι).trW]

Goal state after:
C : Type u_1
D : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹⁹ : Category.{v_1, u_1} C
inst✝¹⁸ : Category.{v_2, u_2} D
inst✝¹⁷ : Category.{v_3, u_3} D₁
inst✝¹⁶ : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝¹⁵ : HasZeroObject C
inst✝¹⁴ : HasShift C ℤ
inst✝¹³ : Preadditive C
inst✝¹² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹¹ : Pretriangulated C
inst✝¹⁰ : IsTriangulated C
inst✝⁹ : A.IsTriangulated
inst✝⁸ : B.IsTriangulated
inst✝⁷ : B.IsClosedUnderIsomorphisms
inst✝⁶ : A.IsVerdierLeftLocalizing B
L₁ : A.FullSubcategory ⥤ D₁
L₂ : C ⥤ D₂
inst✝⁵ : L₁.IsLocalization (B.inverseImage A.ι).trW
inst✝⁴ : L₂.IsLocalization B.trW
inst✝³ : Preadditive D₁
inst✝² : Preadditive D₂
inst✝¹ : L₁.Additive
inst✝ : L₂.Additive
F : D₁ ⥤ D₂ := (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂
⊢ (L₁ ⋙ (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂).Additive
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} 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
T : Triangle C
hT : T ∈ distinguishedTriangles
n : ℤ
h₁ : t.IsLE T.obj₁ n
h₃ : t.IsLE T.obj₃ n
⊢ ∀ (Y : C) (f : T.obj₂ ⟶ Y), t.IsGE Y (n + 1) → f = 0

Tactic:
intro Y f hY

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} 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
T : Triangle C
hT : T ∈ distinguishedTriangles
n : ℤ
h₁ : t.IsLE T.obj₁ n
h₃ : t.IsLE T.obj₃ n
Y : C
f : T.obj₂ ⟶ Y
hY : t.IsGE Y (n + 1)
⊢ f = 0
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} 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
T : Triangle C
hT : T ∈ distinguishedTriangles
n : ℤ
h₁ : t.IsGE T.obj₁ n
h₃ : t.IsGE T.obj₃ n
⊢ ∀ (Y : C) (f : Y ⟶ T.obj₂), t.IsLE Y (n - 1) → f = 0

Tactic:
intro Y f hY

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} 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
T : Triangle C
hT : T ∈ distinguishedTriangles
n : ℤ
h₁ : t.IsGE T.obj₁ n
h₃ : t.IsGE T.obj₃ n
Y : C
f : Y ⟶ T.obj₂
hY : t.IsLE Y (n - 1)
⊢ f = 0
Module: Mathlib.Algebra.Homology.ModelCategory.Lifting

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
A B X Y : CochainComplex C ℤ
t : A ⟶ X
i : A ⟶ B
p : X ⟶ Y
b : B ⟶ Y
sq : CommSq t i p b
hsq : (n : ℤ) → ⋯.LiftStruct
Q : CochainComplex C ℤ
π : B ⟶ Q
hπ : i ≫ π = 0
hQ : IsColimit (CokernelCofork.ofπ π hπ)
K : CochainComplex C ℤ
ι : K ⟶ X
hι : ι ≫ p = 0
hK : IsLimit (KernelFork.ofι ι hι)
this✝¹ : Epi π
this✝ : Mono ι
n : ℤ
this :
  (↑(cocycle₁' sq hsq)).v n (n + 1) ⋯ ≫ X.d (n + 1) (n + 2) +
      B.d n (n + 1) ≫ (↑(cocycle₁' sq hsq)).v (n + 1) (n + 2) ⋯ =
    0
⊢ π.f n ≫ (cochain₁ sq hsq hQ hK).v n (n + 1) ⋯ ≫ K.d (n + 1) (n + 2) ≫ ι.f (n + 2) +
      B.d n (n + 1) ≫ (↑(cocycle₁' sq hsq)).v (n + 1) (n + 2) ⋯ =
    π.f n ≫ 0

Tactic:
comp_zero,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
A B X Y : CochainComplex C ℤ
t : A ⟶ X
i : A ⟶ B
p : X ⟶ Y
b : B ⟶ Y
sq : CommSq t i p b
hsq : (n : ℤ) → ⋯.LiftStruct
Q : CochainComplex C ℤ
π : B ⟶ Q
hπ : i ≫ π = 0
hQ : IsColimit (CokernelCofork.ofπ π hπ)
K : CochainComplex C ℤ
ι : K ⟶ X
hι : ι ≫ p = 0
hK : IsLimit (KernelFork.ofι ι hι)
this✝¹ : Epi π
this✝ : Mono ι
n : ℤ
this :
  (↑(cocycle₁' sq hsq)).v n (n + 1) ⋯ ≫ X.d (n + 1) (n + 2) +
      B.d n (n + 1) ≫ (↑(cocycle₁' sq hsq)).v (n + 1) (n + 2) ⋯ =
    0
⊢ π.f n ≫ (cochain₁ sq hsq hQ hK).v n (n + 1) ⋯ ≫ K.d (n + 1) (n + 2) ≫ ι.f (n + 2) +
      B.d n (n + 1) ≫ (↑(cocycle₁' sq hsq)).v (n + 1) (n + 2) ⋯ =
    0
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} 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
t : TStructure C
inst✝ : IsTriangulated C
a b : ℤ
X : C
⊢ IsIso ((t.truncLT b).map ((t.truncGE a).map ((t.truncLTι b).app X)))

Tactic:
rw [← t.truncLT_map_truncGE_map_truncLTι_app_fac a b X]
  infer_instance

Goal state after:
Module: Mathlib.AlgebraicTopology.ModelCategory.FibrantObjectHomotopy

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : ModelCategory C
Hfib : Type u_1 := (weakEquivalences (HoCat C)).Localization
Lfibπ : HoCat C ⥤ Hfib := (weakEquivalences (HoCat C)).Q
Lfib : FibrantObject C ⥤ Hfib := toHoCat ⋙ Lfibπ
H : Type u_1 := (weakEquivalences C).Localization
L : C ⥤ H := (weakEquivalences C).Q
F : Hfib ⥤ H := (localizerMorphism C).localizedFunctor Lfib L
eF : ι ⋙ L ≅ Lfib ⋙ F := CatCommSq.iso (localizerMorphism C).functor Lfib L F
eF' : HoCat.toLocalization L ≅ Lfibπ ⋙ F :=
  Quotient.natIsoLift (homRel C) (HoCat.toHoCatCompToLocalizationIso L ≪≫ eF ≪≫ toHoCat.associator Lfibπ F)
G : H ⥤ Hfib := (HoCat.localizerMorphismResolution C).localizedFunctor L Lfibπ
eG : HoCat.resolution ⋙ Lfibπ ≅ L ⋙ G := CatCommSq.iso (HoCat.localizerMorphismResolution C).functor L Lfibπ G
⊢ (localizerMorphism C).IsLocalizedEquivalence

Tactic:
have : Localization.Lifting L (weakEquivalences C)
      (HoCat.resolution ⋙ HoCat.toLocalization L) (G ⋙ F) :=
    ⟨(associator _ _ _).symm ≪≫ isoWhiskerRight eG.symm _ ≪≫
      associator _ _ _ ≪≫ isoWhiskerLeft _ eF'.symm⟩

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : ModelCategory C
Hfib : Type u_1 := (weakEquivalences (HoCat C)).Localization
Lfibπ : HoCat C ⥤ Hfib := (weakEquivalences (HoCat C)).Q
Lfib : FibrantObject C ⥤ Hfib := toHoCat ⋙ Lfibπ
H : Type u_1 := (weakEquivalences C).Localization
L : C ⥤ H := (weakEquivalences C).Q
F : Hfib ⥤ H := (localizerMorphism C).localizedFunctor Lfib L
eF : ι ⋙ L ≅ Lfib ⋙ F := CatCommSq.iso (localizerMorphism C).functor Lfib L F
eF' : HoCat.toLocalization L ≅ Lfibπ ⋙ F :=
  Quotient.natIsoLift (homRel C) (HoCat.toHoCatCompToLocalizationIso L ≪≫ eF ≪≫ toHoCat.associator Lfibπ F)
G : H ⥤ Hfib := (HoCat.localizerMorphismResolution C).localizedFunctor L Lfibπ
eG : HoCat.resolution ⋙ Lfibπ ≅ L ⋙ G := CatCommSq.iso (HoCat.localizerMorphismResolution C).functor L Lfibπ G
this : Localization.Lifting L (weakEquivalences C) (HoCat.resolution ⋙ HoCat.toLocalization L) (G ⋙ F)
⊢ (localizerMorphism C).IsLocalizedEquivalence
Module: Mathlib.CategoryTheory.GuitartExact.Opposite

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
C₄ : Type u₄
inst✝³ : Category.{v₁, u₁} C₁
inst✝² : Category.{v₂, u₂} C₂
inst✝¹ : Category.{v₃, u₃} C₃
inst✝ : Category.{v₄, u₄} C₄
T : C₁ ⥤ C₂
L : C₁ ⥤ C₃
R : C₂ ⥤ C₄
B : C₃ ⥤ C₄
w : TwoSquare T L R B
F : C₁ ⥤ C₂
⊢ (mk F (𝟭 C₁) (𝟭 C₂) F (𝟙 F)).op.GuitartExact

Tactic:
apply guitartExact_id

Goal state after:
Module: Mathlib.AlgebraicTopology.ModelCategory.DerivabilityStructureFibrant

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : ModelCategory C
X : C
R₀ : (localizerMorphism C).RightResolution X :=
  { X₁ := mk (HoCat.resolutionObj X), w := HoCat.iResolutionObj X, hw := ⋯ }
R : (localizerMorphism C).RightResolution X
sq :
  CommSq R.w R₀.w (terminal.from ((localizerMorphism C).functor.obj R.X₁))
    (terminal.from ((localizerMorphism C).functor.obj R₀.X₁))
⊢ Nonempty (Zigzag R R₀)

Tactic:
exact ⟨Zigzag.of_inv { f := homMk sq.lift }⟩

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: case inr.succ
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝³ : EnoughInjectives C
inst✝² : Mono f
n₀ : ℤ
inst✝¹ : K.IsStrictlyGE (n₀ + 1)
inst✝ : L.IsStrictlyGE (n₀ + 1)
i : ℤ
this : ∀ {q₁ q₂ : ℕ} (hq : q₁ ≤ q₂), i ≤ n₀ + ↑q₁ → q₁ + 1 = q₂ → IsIso (((functor f n₀).map (homOfLE hq).op).hom.h.f i)
k : ℕ
h : ∀ {q₁ q₂ : ℕ} (hq : q₁ ≤ q₂), i ≤ n₀ + ↑q₁ → q₁ + k = q₂ → IsIso (((functor f n₀).map (homOfLE hq).op).hom.h.f i)
q₁ q₂ : ℕ
hq : q₁ ≤ q₂
hi : i ≤ n₀ + ↑q₁
hk : q₁ + (k + 1) = q₂
⊢ IsIso (((functor f n₀).map (homOfLE hq).op).hom.h.f i)

Tactic:
rw [← homOfLE_comp (show q₁ ≤ q₁ + k by lia) (show q₁ + k ≤ q₂ by lia),
        op_comp, Functor.map_comp]
      exact IsIso.comp_isIso' (this _ (by lia) (by lia)) (h _ (by lia) rfl)

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝³ : EnoughInjectives C
inst✝² : Mono f
n₀ : ℤ
inst✝¹ : K.IsStrictlyGE (n₀ + 1)
inst✝ : L.IsStrictlyGE (n₀ + 1)
i : ℤ
⊢ Mono ((ι f n₀).f i)

Tactic:
obtain ⟨q, _⟩ : ∃ (q : ℕ), IsIso ((midπ f n₀ q).f i) :=
      ⟨(i - n₀).natAbs, isIso_midπ_f f n₀ _ i⟩

Goal state after:
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝³ : EnoughInjectives C
inst✝² : Mono f
n₀ : ℤ
inst✝¹ : K.IsStrictlyGE (n₀ + 1)
inst✝ : L.IsStrictlyGE (n₀ + 1)
i : ℤ
q : ℕ
h✝ : IsIso ((midπ f n₀ q).f i)
⊢ Mono ((ι f n₀).f i)
Module: Mathlib.CategoryTheory.Localization.StructuredArrow

Goal state before:
pp: C : Type u_1
D : Type u_2
D' : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Category.{v_3, u_3} D'
W : MorphismProperty C
L : C ⥤ D
L' : C ⥤ D'
inst✝¹ : L.IsLocalization W
inst✝ : L'.IsLocalization W
X : C
f : StructuredArrow (L.obj X) L
⊢ (fun f ↦ StructuredArrow.mk ((homEquiv W L' L) f.hom)) ((fun f ↦ StructuredArrow.mk ((homEquiv W L L') f.hom)) f) = f

Tactic:
obtain ⟨Y, f, rfl⟩ := f.mk_surjective

Goal state after:
C : Type u_1
D : Type u_2
D' : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Category.{v_3, u_3} D'
W : MorphismProperty C
L : C ⥤ D
L' : C ⥤ D'
inst✝¹ : L.IsLocalization W
inst✝ : L'.IsLocalization W
X Y : C
f : L.obj X ⟶ L.obj Y
⊢ (fun f ↦ StructuredArrow.mk ((homEquiv W L' L) f.hom))
      ((fun f ↦ StructuredArrow.mk ((homEquiv W L L') f.hom)) (StructuredArrow.mk f)) =
    StructuredArrow.mk f
Module: Mathlib.CategoryTheory.Localization.StructuredArrow

Goal state before:
pp: C : Type u_1
D : Type u_2
D' : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Category.{v_3, u_3} D'
W : MorphismProperty C
L : C ⥤ D
L' : C ⥤ D'
inst✝¹ : L.IsLocalization W
inst✝ : L'.IsLocalization W
X : C
f : StructuredArrow (L'.obj X) L'
⊢ (fun f ↦ StructuredArrow.mk ((homEquiv W L L') f.hom)) ((fun f ↦ StructuredArrow.mk ((homEquiv W L' L) f.hom)) f) = f

Tactic:
obtain ⟨Y, f, rfl⟩ := f.mk_surjective

Goal state after:
C : Type u_1
D : Type u_2
D' : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Category.{v_3, u_3} D'
W : MorphismProperty C
L : C ⥤ D
L' : C ⥤ D'
inst✝¹ : L.IsLocalization W
inst✝ : L'.IsLocalization W
X Y : C
f : L'.obj X ⟶ L'.obj Y
⊢ (fun f ↦ StructuredArrow.mk ((homEquiv W L L') f.hom))
      ((fun f ↦ StructuredArrow.mk ((homEquiv W L' L) f.hom)) (StructuredArrow.mk f)) =
    StructuredArrow.mk f
Module: Mathlib.CategoryTheory.Localization.DerivabilityStructure.PointwiseRightDerived

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
H : Type u₃
inst✝⁷ : Category.{v₁, u₁} C₁
inst✝⁶ : Category.{v₂, u₂} C₂
inst✝⁵ : Category.{v₃, u₃} H
D₁ : Type u₄
D₂ : Type u₅
inst✝⁴ : Category.{v₄, u₄} D₁
inst✝³ : Category.{v₅, u₅} D₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
L₁ : C₁ ⥤ D₁
L₂ : C₂ ⥤ D₂
inst✝² : L₁.IsLocalization W₁
inst✝¹ : L₂.IsLocalization W₂
F : C₂ ⥤ H
F₁ : D₁ ⥤ H
α₁ : Φ.functor ⋙ F ⟶ L₁ ⋙ F₁
F₂ : D₂ ⥤ H
α₂ : F ⟶ L₂ ⋙ F₂
inst✝ : F₁.IsRightDerivedFunctor α₁ W₁
⊢ α₁ ≫ L₁.whiskerLeft (Φ.rightDerivedFunctorComparison L₁ L₂ F F₁ α₁ F₂ α₂) =
    Φ.functor.whiskerLeft α₂ ≫
      (Φ.functor.associator L₂ F₂).inv ≫
        whiskerRight (CatCommSq.iso Φ.functor L₁ L₂ (Φ.localizedFunctor L₁ L₂)).hom F₂ ≫
          (L₁.associator (Φ.localizedFunctor L₁ L₂) F₂).hom

Tactic:
dsimp only [rightDerivedFunctorComparison]
  rw [Functor.rightDerived_fac]

Goal state after:
Module: Mathlib.CategoryTheory.Localization.DerivabilityStructure.OfLocalizedEquivalences

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝⁹ : Category.{v_1, u_1} C₁
inst✝⁸ : Category.{v_2, u_2} C₂
inst✝⁷ : Category.{v_3, u_3} D₁
inst✝⁶ : Category.{v_4, u_4} D₂
W₁ : MorphismProperty C₁
W₁' : MorphismProperty D₁
W₂ : MorphismProperty C₂
W₂' : MorphismProperty D₂
T : LocalizerMorphism W₁ W₂
L : LocalizerMorphism W₁ W₁'
R : LocalizerMorphism W₂ W₂'
B : LocalizerMorphism W₁' W₂'
inst✝⁵ : W₂'.RespectsIso
inst✝⁴ : L.IsLocalizedEquivalence
inst✝³ : R.IsLocalizedEquivalence
inst✝² : R.functor.EssSurj
inst✝¹ : T.IsLeftDerivabilityStructure
iso : T.functor ⋙ R.functor ≅ L.functor ⋙ B.functor
inst✝ : TwoSquare.GuitartExact iso.inv
this : B.HasLeftResolutions
⊢ B.IsLeftDerivabilityStructure

Tactic:
let F := B.localizedFunctor W₁'.Q W₂'.Q

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝⁹ : Category.{v_1, u_1} C₁
inst✝⁸ : Category.{v_2, u_2} C₂
inst✝⁷ : Category.{v_3, u_3} D₁
inst✝⁶ : Category.{v_4, u_4} D₂
W₁ : MorphismProperty C₁
W₁' : MorphismProperty D₁
W₂ : MorphismProperty C₂
W₂' : MorphismProperty D₂
T : LocalizerMorphism W₁ W₂
L : LocalizerMorphism W₁ W₁'
R : LocalizerMorphism W₂ W₂'
B : LocalizerMorphism W₁' W₂'
inst✝⁵ : W₂'.RespectsIso
inst✝⁴ : L.IsLocalizedEquivalence
inst✝³ : R.IsLocalizedEquivalence
inst✝² : R.functor.EssSurj
inst✝¹ : T.IsLeftDerivabilityStructure
iso : T.functor ⋙ R.functor ≅ L.functor ⋙ B.functor
inst✝ : TwoSquare.GuitartExact iso.inv
this : B.HasLeftResolutions
F : W₁'.Localization ⥤ W₂'.Localization := B.localizedFunctor W₁'.Q W₂'.Q
⊢ B.IsLeftDerivabilityStructure
Module: Mathlib.CategoryTheory.Localization.DerivabilityStructure.OfLocalizedEquivalences

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D₁
inst✝⁸ : Category.{v_4, u_4} D₂
W₁ : MorphismProperty C₁
W₁' : MorphismProperty D₁
W₂ : MorphismProperty C₂
W₂' : MorphismProperty D₂
T : LocalizerMorphism W₁ W₂
L : LocalizerMorphism W₁ W₁'
R : LocalizerMorphism W₂ W₂'
B : LocalizerMorphism W₁' W₂'
inst✝⁷ : W₂'.RespectsIso
inst✝⁶ : L.IsLocalizedEquivalence
inst✝⁵ : R.IsLocalizedEquivalence
inst✝⁴ : R.functor.EssSurj
inst✝³ : L.functor.EssSurj
inst✝² : R.functor.Full
inst✝¹ : R.IsInduced
iso : T.functor ⋙ R.functor ≅ L.functor ⋙ B.functor
inst✝ : TwoSquare.GuitartExact iso.inv
x✝ : B.IsLeftDerivabilityStructure
this : T.HasLeftResolutions
⊢ T.IsLeftDerivabilityStructure

Tactic:
let F := B.localizedFunctor W₁'.Q W₂'.Q

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D₁
inst✝⁸ : Category.{v_4, u_4} D₂
W₁ : MorphismProperty C₁
W₁' : MorphismProperty D₁
W₂ : MorphismProperty C₂
W₂' : MorphismProperty D₂
T : LocalizerMorphism W₁ W₂
L : LocalizerMorphism W₁ W₁'
R : LocalizerMorphism W₂ W₂'
B : LocalizerMorphism W₁' W₂'
inst✝⁷ : W₂'.RespectsIso
inst✝⁶ : L.IsLocalizedEquivalence
inst✝⁵ : R.IsLocalizedEquivalence
inst✝⁴ : R.functor.EssSurj
inst✝³ : L.functor.EssSurj
inst✝² : R.functor.Full
inst✝¹ : R.IsInduced
iso : T.functor ⋙ R.functor ≅ L.functor ⋙ B.functor
inst✝ : TwoSquare.GuitartExact iso.inv
x✝ : B.IsLeftDerivabilityStructure
this : T.HasLeftResolutions
F : W₁'.Localization ⥤ W₂'.Localization := B.localizedFunctor W₁'.Q W₂'.Q
⊢ T.IsLeftDerivabilityStructure
Module: Mathlib.Algebra.Homology.DerivedCategory.DerivabilityStructureInjectives

Goal state before:
pp: case refine_3
C : Type u_1
H : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
inst✝¹ : Category.{v_2, u_2} H
L✝ : Plus C ⥤ H
inst✝ : L✝.IsLocalization (quasiIso C)
K L : CochainComplex.Plus C
f : K ⟶ L
hf : quasiIso C ((quotient C).map f)
⊢ (CochainComplex.Plus.quasiIso C).map (quotient C) ((quotient C).map f)

Tactic:
apply MorphismProperty.map_mem_map

Goal state after:
case refine_3
C : Type u_1
H : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
inst✝¹ : Category.{v_2, u_2} H
L✝ : Plus C ⥤ H
inst✝ : L✝.IsLocalization (quasiIso C)
K L : CochainComplex.Plus C
f : K ⟶ L
hf : quasiIso C ((quotient C).map f)
⊢ CochainComplex.Plus.quasiIso C f
Module: Mathlib.Algebra.Homology.Embedding.Connect

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
K : ChainComplex C ℕ
L : CochainComplex C ℕ
h : ConnectData K L
n m : ℕ
⊢ h.d (Int.negSucc n) (Int.negSucc m) = K.d n m

Tactic:
by simp [d]

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

Goal state before:
pp: case pos
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
X₀ X₁ : C
f : X₀ ⟶ X₁
ι : Type u_2
c : ComplexShape ι
i₀ : ι
K : HomologicalComplex C c
φ₀ : X₀ ⟶ K.X i₀
k₀ k₁ : ι
hk : c.Rel k₀ k₁
h₀ : ¬k₀ = i₀
hi₀₁ : c.Rel i₀ k₀
h : i₀ ≠ k₀
φ₁ : X₁ ⟶ K.X k₀
comm : φ₀ ≫ K.d i₀ k₀ = f ≫ φ₁
hφ : ∀ (k : ι), c.Rel k₀ k → φ₁ ≫ K.d k₀ k = 0
⊢ (doubleXIso₁ f hi₀₁ h).hom ≫ 0 =
    (double f hi₀₁).d k₀ k₁ ≫
      if hk₀ : k₁ = i₀ then eqToHom ⋯ ≫ (doubleXIso₀ f hi₀₁).hom ≫ φ₀ ≫ eqToHom ⋯
      else if hk₁ : k₁ = k₀ then eqToHom ⋯ ≫ (doubleXIso₁ f hi₀₁ h).hom ≫ φ₁ ≫ eqToHom ⋯ else 0

Tactic:
comp_zero,

Goal state after:
case pos
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
X₀ X₁ : C
f : X₀ ⟶ X₁
ι : Type u_2
c : ComplexShape ι
i₀ : ι
K : HomologicalComplex C c
φ₀ : X₀ ⟶ K.X i₀
k₀ k₁ : ι
hk : c.Rel k₀ k₁
h₀ : ¬k₀ = i₀
hi₀₁ : c.Rel i₀ k₀
h : i₀ ≠ k₀
φ₁ : X₁ ⟶ K.X k₀
comm : φ₀ ≫ K.d i₀ k₀ = f ≫ φ₁
hφ : ∀ (k : ι), c.Rel k₀ k → φ₁ ≫ K.d k₀ k = 0
⊢ 0 =
    (double f hi₀₁).d k₀ k₁ ≫
      if hk₀ : k₁ = i₀ then eqToHom ⋯ ≫ (doubleXIso₀ f hi₀₁).hom ≫ φ₀ ≫ eqToHom ⋯
      else if hk₁ : k₁ = k₀ then eqToHom ⋯ ≫ (doubleXIso₁ f hi₀₁ h).hom ≫ φ₁ ≫ eqToHom ⋯ else 0
Module: Mathlib.Algebra.Homology.Embedding.Connect

Goal state before:
pp: case neg
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
K : ChainComplex C ℕ
L : CochainComplex C ℕ
h : ConnectData K L
n m p : ℤ
hnm : n + 1 = m
hmp : ¬m + 1 = p
⊢ h.d n m ≫ h.d m p = 0

pp: case pos
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
K : ChainComplex C ℕ
L : CochainComplex C ℕ
h : ConnectData K L
n m p : ℤ
hnm : n + 1 = m
hmp : m + 1 = p
⊢ h.d n m ≫ h.d m p = 0

Tactic:
· rw [h.shape m p hmp, comp_zero]

Goal state after:
case pos
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
K : ChainComplex C ℕ
L : CochainComplex C ℕ
h : ConnectData K L
n m p : ℤ
hnm : n + 1 = m
hmp : m + 1 = p
⊢ h.d n m ≫ h.d m p = 0
Module: Mathlib.CategoryTheory.Preadditive.Projective.Resolution

Goal state before:
pp: case zero
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
Z : C
P : ProjectiveResolution Z
⊢ Epi (P.π.f 0)

Tactic:
exact epi_of_isColimit_cofork P.isColimitCokernelCofork

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.Ext

Goal state before:
pp: R : Type u_1
inst✝⁵ : Ring R
C : Type u_2
inst✝⁴ : Category.{v_1, u_2} C
inst✝³ : Abelian C
inst✝² : Linear R C
inst✝¹ : EnoughProjectives C
X Y : C
inst✝ : Projective X
n : ℕ
⊢ IsZero (HomologicalComplex.homology ((ProjectiveResolution.self X).complex.linearYonedaObj R Y) (n + 1))

Tactic:
rw [← HomologicalComplex.exactAt_iff_isZero_homology, HomologicalComplex.exactAt_iff]

Goal state after:
R : Type u_1
inst✝⁵ : Ring R
C : Type u_2
inst✝⁴ : Category.{v_1, u_2} C
inst✝³ : Abelian C
inst✝² : Linear R C
inst✝¹ : EnoughProjectives C
X Y : C
inst✝ : Projective X
n : ℕ
⊢ (HomologicalComplex.sc ((ProjectiveResolution.self X).complex.linearYonedaObj R Y) (n + 1)).Exact