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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
x y : ℤ
inst✝ : K.HasTotal (up ℤ)
⊢ ((shiftFunctor₂ C y).obj K).totalShift₁Iso x ≪≫ (shiftFunctor (CochainComplex C ℤ) x).mapIso (K.totalShift₂Iso y) =
    (x * y).negOnePow •
      total.mapIso ((shiftFunctor₁₂CommIso C x y).app K) (up ℤ) ≪≫
        ((shiftFunctor₁ C x).obj K).totalShift₂Iso y ≪≫
          (shiftFunctor (HomologicalComplex C (up ℤ)) y).mapIso (K.totalShift₁Iso x) ≪≫
            (shiftFunctorComm (CochainComplex C ℤ) x y).app (K.total (up ℤ))

Tactic:
by
  ext n n₁ n₂ h
  dsimp at h ⊢
  rw [Linear.comp_units_smul, ι_totalShift₁Iso_hom_f_assoc _ x n₁ n₂ n h _ rfl _ rfl,
    ιTotal_map_assoc, ι_totalShift₂Iso_hom_f_assoc _ y n₁ n₂ n h _ rfl _ rfl,
    Linear.units_smul_comp, Linear.comp_units_smul]
  dsimp [shiftFunctor₁₂CommIso]
  rw [id_comp, id_comp, id_comp, id_comp, comp_id,
    ι_totalShift₂Iso_hom_f _ y (n₁ + x) n₂ (n + x) (by lia) _ rfl _ rfl, smul_smul,
    ← Int.negOnePow_add, add_mul, add_comm (x * y)]
  dsimp
  rw [id_comp, comp_id,
    ι_totalShift₁Iso_hom_f_assoc _ x n₁ (n₂ + y) (n + y) (by lia) _ rfl (n + x + y) (by lia),
    CochainComplex.shiftFunctorComm_hom_app_f]
  dsimp
  rw [Iso.inv_hom_id, comp_id, id_comp]

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

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u'
inst✝⁴ : Category.{v', u'} D
P : ObjectProperty C
M : Type u_1
inst✝³ : AddGroup M
inst✝² : HasShift C M
inst✝¹ : HasZeroMorphisms C
inst✝ : P.IsStableUnderShift M
n : M
X : C
hX : P.leftOrthogonal X
Y : C
f : (shiftFunctor C n).obj X ⟶ Y
hY : P Y
⊢ f = 0

Tactic:
by
    obtain ⟨g, rfl⟩ := ((shiftEquiv C n).toAdjunction.homEquiv _ _).symm.surjective f
    simp [hX g (P.le_shift (-n) _ hY), Adjunction.homEquiv_counit]

Goal state after:
Module: Mathlib.CategoryTheory.LiftingProperties.Limits

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X Y Z W : C
f : X ⟶ Y
s : X ⟶ Z
g : Z ⟶ W
t : Y ⟶ W
h : IsPullback s f g t
X' Y' : C
f' : X' ⟶ Y'
inst✝ : HasLiftingProperty f' g
u : X' ⟶ X
v : Y' ⟶ Y
sq : CommSq u f' f v
w : (u ≫ s) ≫ g = f' ≫ v ≫ t
⊢ (f' ≫ h.lift ⋯.lift v ⋯) ≫ s = u ≫ s

Tactic:
rw [Category.assoc, h.lift_fst, CommSq.fac_left]

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.LiftingProperty

Goal state before:
pp: case a
C : Type u
inst✝ : Category.{v, u} C
T : MorphismProperty C
⊢ T.retracts.rlp ≤ T.rlp

Tactic:
exact antitone_rlp T.le_retracts

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
X : DerivedCategory C
n : ℤ
hX : X.IsLE n
⊢ ∃ K, ∃ (_ : K.IsStrictlyLE n), Nonempty (X ≅ Q.obj K)

Tactic:
obtain ⟨K, e, _⟩ := hX

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
X : DerivedCategory C
n : ℤ
K : CochainComplex C ℤ
e : X ≅ Q.obj K
h✝ : K.IsStrictlyLE n
⊢ ∃ K, ∃ (_ : K.IsStrictlyLE n), Nonempty (X ≅ Q.obj K)
Module: Mathlib.Algebra.Homology.DerivedCategory.TStructure

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
X : DerivedCategory C
n : ℤ
hX : X.IsGE n
⊢ ∃ K, ∃ (_ : K.IsStrictlyGE n), Nonempty (X ≅ Q.obj K)

Tactic:
obtain ⟨K, e, _⟩ := hX

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
X : DerivedCategory C
n : ℤ
K : CochainComplex C ℤ
e : X ≅ Q.obj K
h✝ : K.IsStrictlyGE n
⊢ ∃ K, ∃ (_ : K.IsStrictlyGE n), Nonempty (X ≅ Q.obj K)
Module: Mathlib.Algebra.Homology.DerivedCategory.TStructure

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Abelian C
inst✝² : HasDerivedCategory C
X : DerivedCategory C
n : ℤ
inst✝¹ : X.IsGE n
inst✝ : X.IsLE n
K : CochainComplex C ℤ
w✝¹ : K.IsStrictlyGE n
w✝ : K.IsStrictlyLE n
e : X ≅ Q.obj K
Y : C
e' : K ≅ (HomologicalComplex.single C (ComplexShape.up ℤ) n).obj Y
⊢ ∃ Y, Nonempty (X ≅ (singleFunctor C n).obj Y)

Tactic:
exact ⟨Y, ⟨e ≪≫ Q.mapIso e'⟩⟩

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

Goal state before:
pp: C : Type u
inst✝⁷ : Category.{v, u} C
inst✝⁶ : CategoryWithWeakEquivalences C
inst✝⁵ : CategoryWithCofibrations C
inst✝⁴ : CategoryWithFibrations C
X Y Z : C
f : X ⟶ Y
g : X ⟶ Z
inst✝³ : HasPushout f g
inst✝² : (trivialCofibrations C).IsStableUnderCobaseChange
inst✝¹ : Cofibration f
inst✝ : WeakEquivalence f
⊢ WeakEquivalence (pushout.inr f g)

Tactic:
by
  rw [weakEquivalence_iff]
  exact (MorphismProperty.of_isPushout (IsPushout.of_hasPushout f g).flip
    (mem_trivialCofibrations f)).2

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
X Y : C
inst✝¹ : ModelCategory C
f g : X ⟶ Y
inst✝ : IsFibrant Y
h✝ : LeftHomotopyRel f g
Z : C
i : Z ⟶ X
P : Cylinder X
left✝ : P.IsVeryGood
h : P.LeftHomotopy f g
⊢ LeftHomotopyRel (i ≫ f) (i ≫ g)

Tactic:
obtain ⟨Q, _⟩ := Cylinder.exists_very_good Z

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
X Y : C
inst✝¹ : ModelCategory C
f g : X ⟶ Y
inst✝ : IsFibrant Y
h✝¹ : LeftHomotopyRel f g
Z : C
i : Z ⟶ X
P : Cylinder X
left✝ : P.IsVeryGood
h : P.LeftHomotopy f g
Q : Cylinder Z
h✝ : Q.IsVeryGood
⊢ LeftHomotopyRel (i ≫ f) (i ≫ g)
Module: Mathlib.CategoryTheory.Triangulated.Opposite.Subcategory

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasShift C ℤ
inst✝⁴ : HasZeroObject C
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
P : ObjectProperty C
inst✝ : P.IsTriangulated
X Y : C
f : X ⟶ Y
hf : P.op.trW f.op
⊢ P.trW f

Tactic:
by
  obtain ⟨Z, a, b, h₁, h₂⟩ := hf
  rw [ObjectProperty.trW_iff']
  exact ⟨_, _, _, Pretriangulated.unop_distinguished _ h₁, h₂⟩

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

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
X : C
a b : ℤ
inst✝ : t.IsLE X b
⊢ t.IsLE ((t.truncLE a).obj X) b

Tactic:
dsimp [truncLE]
  infer_instance

Goal state after:
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✝¹³ : Category.{v_1, u_1} C
inst✝¹² : Category.{v_2, u_2} D
inst✝¹¹ : Category.{v_3, u_3} D₁
inst✝¹⁰ : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝⁹ : HasZeroObject C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : Preadditive C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : IsTriangulated C
inst✝³ : A.IsTriangulated
inst✝² : B.IsTriangulated
inst✝¹ : B.IsClosedUnderIsomorphisms
inst✝ : A.IsVerdierLeftLocalizing B
L₁ : A.FullSubcategory ⥤ (B.inverseImage A.ι).trW.Localization := (B.inverseImage A.ι).trW.Q
⊢ (A.triangulatedLocalizerMorphism B).IsLocalizedFullyFaithful

Tactic:
let L₂ := B.trW.Q

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

Goal state before:
pp: case refine_1
C : Type u_1
D : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹³ : Category.{v_1, u_1} C
inst✝¹² : Category.{v_2, u_2} D
inst✝¹¹ : Category.{v_3, u_3} D₁
inst✝¹⁰ : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝⁹ : HasZeroObject C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : Preadditive C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : IsTriangulated C
inst✝³ : A.IsTriangulated
inst✝² : B.IsTriangulated
inst✝¹ : B.IsClosedUnderIsomorphisms
inst✝ : A.IsVerdierLeftLocalizing B
L₁ : A.FullSubcategory ⥤ (B.inverseImage A.ι).trW.Localization := (B.inverseImage A.ι).trW.Q
L₂ : C ⥤ B.trW.Localization := B.trW.Q
F : (B.inverseImage A.ι).trW.Localization ⥤ B.trW.Localization :=
  (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂
this✝ : CatCommSq (A.op.triangulatedLocalizerMorphism B.op).functor (A.opEquivalence.functor ⋙ L₁.op) L₂.op F.op :=
  {
    iso :=
      A.opEquivalence.functor.isoWhiskerLeft
        (NatIso.op (CatCommSq.iso (A.triangulatedLocalizerMorphism B).functor L₁ L₂ F).symm) }
this : L₂.op.IsLocalization B.op.trW
⊢ (B.inverseImage A.ι).op.trW ≤ (B.op.inverseImage A.op.ι).trW.isoClosure.inverseImage A.opEquivalence.symm.functor

Tactic:
← inverseImage_opEquivalence_inverse_trW_inverseImage_ι_op

Goal state after:
case refine_1
C : Type u_1
D : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹³ : Category.{v_1, u_1} C
inst✝¹² : Category.{v_2, u_2} D
inst✝¹¹ : Category.{v_3, u_3} D₁
inst✝¹⁰ : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝⁹ : HasZeroObject C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : Preadditive C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : IsTriangulated C
inst✝³ : A.IsTriangulated
inst✝² : B.IsTriangulated
inst✝¹ : B.IsClosedUnderIsomorphisms
inst✝ : A.IsVerdierLeftLocalizing B
L₁ : A.FullSubcategory ⥤ (B.inverseImage A.ι).trW.Localization := (B.inverseImage A.ι).trW.Q
L₂ : C ⥤ B.trW.Localization := B.trW.Q
F : (B.inverseImage A.ι).trW.Localization ⥤ B.trW.Localization :=
  (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂
this✝ : CatCommSq (A.op.triangulatedLocalizerMorphism B.op).functor (A.opEquivalence.functor ⋙ L₁.op) L₂.op F.op :=
  {
    iso :=
      A.opEquivalence.functor.isoWhiskerLeft
        (NatIso.op (CatCommSq.iso (A.triangulatedLocalizerMorphism B).functor L₁ L₂ F).symm) }
this : L₂.op.IsLocalization B.op.trW
⊢ (B.op.inverseImage A.op.ι).trW.inverseImage A.opEquivalence.inverse ≤
    (B.op.inverseImage A.op.ι).trW.isoClosure.inverseImage A.opEquivalence.symm.functor
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)
⊢ (Cochain.ofHom π).comp ((↑(cocycle₁ sq hsq hQ hK)).comp (Cochain.ofHom ι) ⋯) ⋯ = ↑(cocycle₁' sq hsq)

Tactic:
ext n m hnm
  simp [cocycle₁]

Goal state after:
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)
α : Cochain Q K 0
hα : (Cochain.ofHom π).comp ((δ 0 1 α).comp (Cochain.ofHom ι) ⋯) ⋯ = ↑(cocycle₁' sq hsq)
l : Cocycle B X 0 := Cocycle.mk (cochain₀ sq hsq - (Cochain.ofHom π).comp (α.comp (Cochain.ofHom ι) ⋯) ⋯) 1 ⋯ ⋯
n : ℤ
h₁ : i.f n ≫ π.f n = 0
h₂ :
  (HomologicalComplex.eval C (ComplexShape.up ℤ) n).map i ≫ (hsq n).l =
    (HomologicalComplex.eval C (ComplexShape.up ℤ) n).map t
⊢ (i ≫ l.homOf).f n = t.f n

Tactic:
dsimp at h₁ h₂

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)
α : Cochain Q K 0
hα : (Cochain.ofHom π).comp ((δ 0 1 α).comp (Cochain.ofHom ι) ⋯) ⋯ = ↑(cocycle₁' sq hsq)
l : Cocycle B X 0 := Cocycle.mk (cochain₀ sq hsq - (Cochain.ofHom π).comp (α.comp (Cochain.ofHom ι) ⋯) ⋯) 1 ⋯ ⋯
n : ℤ
h₁ : i.f n ≫ π.f n = 0
h₂ : i.f n ≫ (hsq n).l = t.f n
⊢ (i ≫ l.homOf).f n = t.f n
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: case inr
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
inst✝ : EnoughInjectives C
n₁ i : ℤ
hn₁ : i + 1 = n₁
hi : i ≤ i
this : homologyMap biprod.inl i = 0
⊢ IsIso (homologyMap (π K L n₁) i)

Tactic:
refine ⟨homologyMap (σ K L n₁) i, ?_, ?_⟩

Goal state after:
case inr.refine_1
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
inst✝ : EnoughInjectives C
n₁ i : ℤ
hn₁ : i + 1 = n₁
hi : i ≤ i
this : homologyMap biprod.inl i = 0
⊢ homologyMap (π K L n₁) i ≫ homologyMap (σ K L n₁) i = 𝟙 (homology (mid K L n₁) i)

case inr.refine_2
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
inst✝ : EnoughInjectives C
n₁ i : ℤ
hn₁ : i + 1 = n₁
hi : i ≤ i
this : homologyMap biprod.inl i = 0
⊢ homologyMap (σ K L n₁) i ≫ homologyMap (π K L n₁) i = 𝟙 (homology L i)
Module: Mathlib.CategoryTheory.Localization.Resolution

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D₂ : Type u_4
H : Type u_5
inst✝⁵ : Category.{v_1, u_1} C₁
inst✝⁴ : Category.{v_2, u_2} C₂
inst✝³ : Category.{v_4, u_4} D₂
inst✝² : Category.{v_5, u_5} H
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
L₂ : C₂ ⥤ D₂
inst✝¹ : L₂.IsLocalization W₂
inst✝ : Φ.HasRightResolutions
F G : D₂ ⥤ H
α : F ⟶ G
hα : ∀ (X₁ : C₁), IsIso (α.app (L₂.obj (Φ.functor.obj X₁)))
X₂ : D₂
⊢ IsIso (α.app X₂)

Tactic:
by
      have := Φ.essSurj_of_hasRightResolutions L₂
      rw [← NatTrans.isIso_app_iff_of_iso α ((Φ.functor ⋙ L₂).objObjPreimageIso X₂)]
      apply hα

Goal state after:
Module: Mathlib.CategoryTheory.Localization.Resolution

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝⁸ : Category.{v_1, u_1} C₁
inst✝⁷ : Category.{v_2, u_2} C₂
inst✝⁶ : Category.{v_3, u_3} D₁
inst✝⁵ : Category.{v_4, u_4} D₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
W₁' : MorphismProperty D₁
W₂' : MorphismProperty D₂
T : LocalizerMorphism W₁ W₂
L : LocalizerMorphism W₁ W₁'
R : LocalizerMorphism W₂ W₂'
B : LocalizerMorphism W₁' W₂'
inst✝⁴ : L.functor.EssSurj
inst✝³ : R.functor.Full
inst✝² : R.IsInduced
inst✝¹ : W₂'.RespectsIso
iso : T.functor ⋙ R.functor ≅ L.functor ⋙ B.functor
inst✝ : B.HasLeftResolutions
X₂ : C₂
ρ : B.LeftResolution (R.functor.obj X₂) := Classical.arbitrary (B.LeftResolution (R.functor.obj X₂))
X₁ : C₁
e : L.functor.obj X₁ ≅ ρ.X₁
⊢ W₂ (R.functor.preimage (iso.hom.app X₁ ≫ B.functor.map e.hom ≫ ρ.w))

Tactic:
by
      simp only [← R.inverseImage_eq, MorphismProperty.inverseImage_iff, Functor.map_preimage]
      refine (W₂'.arrow_mk_iso_iff ?_).2 ρ.hw
      exact Arrow.isoMk (iso.app _ ≪≫ B.functor.mapIso e) (Iso.refl _)

Goal state after:
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
⊢ w.op.GuitartExact ↔ w.GuitartExact

Tactic:
by
  constructor
  · intro
    let w₁ : TwoSquare T (opOp C₁) (opOp C₂) T.op.op := 𝟙 _
    let w₂ : TwoSquare B.op.op (unopUnop C₃) (unopUnop C₄) B := 𝟙 _
    have : w = (w₁ ≫ᵥ w.op.op) ≫ᵥ w₂ := by cat_disch
    rw [this]
    infer_instance
  · intro
    infer_instance

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)
⊢ ι f n₀ ≫ π f n₀ = f

Tactic:
by
  simp [π]

Goal state after:
Module: Mathlib.CategoryTheory.Functor.Derived.PointwiseRightDerived

Goal state before:
pp: C : Type u₁
D : Type u₂
H : Type u₃
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
inst✝¹ : Category.{v₃, u₃} H
F' : D ⥤ H
F : C ⥤ H
L : C ⥤ D
α : F ⟶ L ⋙ F'
W : MorphismProperty C
G : D ⥤ H
e : F ≅ L ⋙ G
inst✝ : L.IsLocalization W
Y : C
s : Cocone (CostructuredArrow.proj L (L.obj Y) ⋙ F)
m : G.obj (L.obj Y) ⟶ s.pt
hm : ∀ (j : CostructuredArrow L (L.obj Y)), (e.hom.app j.left ≫ G.map j.hom) ≫ m = s.ι.app j
this : (e.hom.app Y ≫ G.map (𝟙 (L.obj Y))) ≫ m = s.ι.app (CostructuredArrow.mk (𝟙 (L.obj Y)))
⊢ m = e.inv.app Y ≫ s.ι.app (CostructuredArrow.mk (𝟙 (L.obj Y)))

Tactic:
simp only [← this, map_id, comp_id, Iso.inv_hom_id_app_assoc]

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✝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
⊢ T.IsLeftDerivabilityStructure ↔ B.IsLeftDerivabilityStructure

Tactic:
refine ⟨fun _ ↦ isLeftDerivabilityStructure_of_isLocalizedEquivalence iso, fun _ ↦ ?_⟩
  have : T.HasLeftResolutions := fun X₂ ↦ by
    let ρ : B.LeftResolution (R.functor.obj X₂) := Classical.arbitrary _
    exact ⟨{
      X₁ := L.functor.objPreimage ρ.X₁
      w :=
        R.functor.preimage (iso.hom.app _ ≫
          B.functor.map (L.functor.objObjPreimageIso ρ.X₁).hom ≫ ρ.w)
      hw := by
        simp only [← R.inverseImage_eq, Functor.comp_obj,
          MorphismProperty.inverseImage_iff, Functor.map_preimage]
        refine (W₂'.arrow_mk_iso_iff ?_).2 ρ.hw
        exact Arrow.isoMk (iso.app _ ≪≫ B.functor.mapIso (L.functor.objObjPreimageIso ρ.X₁))
          (Iso.refl _) }⟩
  let F := B.localizedFunctor W₁'.Q W₂'.Q
  let e' := CatCommSq.iso B.functor W₁'.Q W₂'.Q F
  let e : T.functor ⋙ R.functor ⋙ W₂'.Q ≅ (L.functor ⋙ W₁'.Q) ⋙ F :=
    (Functor.associator _ _ _).symm ≪≫ Functor.isoWhiskerRight iso _ ≪≫
      Functor.associator _ _ _ ≪≫
      Functor.isoWhiskerLeft _ e' ≪≫ (Functor.associator _ _ _).symm
  have he' : TwoSquare.GuitartExact e'.inv := inferInstance
  have : TwoSquare.hComp iso.inv e'.inv = e.inv := by ext; simp [e]
  rw [T.isLeftDerivabilityStructure_iff (L.functor ⋙ W₁'.Q)
    (R.functor ⋙ W₂'.Q) F e, ← this]
  infer_instance

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.TStructure

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Abelian C
inst✝⁴ : HasExt C
K L : CochainComplex C ℤ
a b : ℤ
inst✝³ : K.IsGE a
inst✝² : K.IsLE a
inst✝¹ : L.IsGE b
inst✝ : L.IsLE b
⊢ HasSmallLocalizedShiftedHom (HomologicalComplex.quasiIso C (ComplexShape.up ℤ)) ℤ K L

Tactic:
by
  let := HasDerivedCategory.standard
  obtain ⟨X, ⟨eX⟩⟩ := DerivedCategory.exists_iso_singleFunctor_obj_of_isGE_of_isLE (Q.obj K) a
  obtain ⟨Y, ⟨eY⟩⟩ := DerivedCategory.exists_iso_singleFunctor_obj_of_isGE_of_isLE (Q.obj L) b
  simp only [hasSmallLocalizedShiftedHom_iff _ _ Q]
  exact fun p q ↦ small_of_injective (f := fun φ ↦
    ((singleFunctors C).shiftIso p (a - p) a (by simp)).inv.app X ≫
      eX.inv⟦p⟧' ≫ φ ≫ eY.hom⟦q⟧' ≫
        ((singleFunctors C).shiftIso q (b - q) b (by simp)).hom.app Y)
    (fun φ₁ φ₂ h ↦ by simpa [cancel_epi, cancel_mono] using h)

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

Goal state before:
pp: 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
inst✝ : EnoughInjectives C
n : ℤ
K : Plus C
hK : fibrantObjects (Plus C) K
⊢ Injective ({ obj := K, property := hK }.obj.obj.X n)

Tactic:
fibrantObjects,

Goal state after:
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
inst✝ : EnoughInjectives C
n : ℤ
K : Plus C
hK✝ : fibrantObjects (Plus C) K
hK : IsFibrant K
⊢ Injective ({ obj := K, property := hK✝ }.obj.obj.X n)
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.MapBijective

Goal state before:
pp: case succ
C : Type u
inst✝¹² : Category.{v, u} C
inst✝¹¹ : Abelian C
D : Type u'
inst✝¹⁰ : Category.{v', u'} D
inst✝⁹ : Abelian D
F : C ⥤ D
inst✝⁸ : F.Additive
inst✝⁷ : PreservesFiniteLimits F
inst✝⁶ : PreservesFiniteColimits F
inst✝⁵ : F.Full
inst✝⁴ : F.Faithful
inst✝³ : HasExt C
inst✝² : HasExt D
inst✝¹ : EnoughInjectives C
inst✝ : F.PreservesInjectiveObjects
X : C
n : ℕ
hn : ∀ (Y : C), Function.Bijective ⇑(F.mapExtAddHom X Y n)
Y : C
I : InjectivePresentation Y := Classical.arbitrary (InjectivePresentation Y)
⊢ Function.Bijective ⇑(F.mapExtAddHom X Y (n + 1))

Tactic:
let S := ShortComplex.mk _ _ (cokernel.condition I.f)

Goal state after:
case succ
C : Type u
inst✝¹² : Category.{v, u} C
inst✝¹¹ : Abelian C
D : Type u'
inst✝¹⁰ : Category.{v', u'} D
inst✝⁹ : Abelian D
F : C ⥤ D
inst✝⁸ : F.Additive
inst✝⁷ : PreservesFiniteLimits F
inst✝⁶ : PreservesFiniteColimits F
inst✝⁵ : F.Full
inst✝⁴ : F.Faithful
inst✝³ : HasExt C
inst✝² : HasExt D
inst✝¹ : EnoughInjectives C
inst✝ : F.PreservesInjectiveObjects
X : C
n : ℕ
hn : ∀ (Y : C), Function.Bijective ⇑(F.mapExtAddHom X Y n)
Y : C
I : InjectivePresentation Y := Classical.arbitrary (InjectivePresentation Y)
S : ShortComplex C := { X₁ := Y, X₂ := I.J, X₃ := cokernel I.f, f := I.f, g := cokernel.π I.f, zero := ⋯ }
⊢ Function.Bijective ⇑(F.mapExtAddHom X Y (n + 1))