mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: n : ℕ
x✝¹ : SimplexCategoryᵒᵖ
x : Δ[n].obj x✝¹
x✝ : x ∈ ⊤.obj x✝¹
⊢ x ∈ (Subcomplex.ofSimplex (objEquiv.symm (𝟙 ⦋n⦌))).obj x✝¹

Tactic:
by
    obtain ⟨f, rfl⟩ := objEquiv.symm.surjective x
    simp only [Subcomplex.mem_ofSimplex_obj_iff, op_unop]
    exact ⟨f, by simp [map_objEquiv_symm.{u}]⟩

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

Goal state before:
pp: I₁ : Type u_1
I₂ : Type u_2
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
c₁₂ : ComplexShape I₁₂
inst✝⁴ : TotalComplexShape c₁ c₂ c₁₂
inst✝³ : TotalComplexShape c₂ c₁ c₁₂
inst✝² : TotalComplexShapeSymmetry c₁ c₂ c₁₂
inst✝¹ : TotalComplexShapeSymmetry c₂ c₁ c₁₂
inst✝ : TotalComplexShapeSymmetrySymmetry c₁ c₂ c₁₂
i₁ : I₁
i₂ : I₂
⊢ c₂.σ c₁ c₁₂ i₂ i₁ = c₁.σ c₂ c₁₂ i₁ i₂

Tactic:
apply TotalComplexShapeSymmetrySymmetry.σ_symm

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

Goal state before:
pp: case mp
C : Type u_1
I₁ : Type u_2
I₂ : Type u_3
J : Type u_4
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Preadditive C
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
c : ComplexShape J
inst✝² : TotalComplexShape c₁ c₂ c
inst✝¹ : TotalComplexShape c₂ c₁ c
inst✝ : TotalComplexShapeSymmetry c₁ c₂ c
a✝ : K.flip.HasTotal c
⊢ K.flip.flip.HasTotal c

Tactic:
have := TotalComplexShapeSymmetry.symmetry c₁ c₂ c

Goal state after:
case mp
C : Type u_1
I₁ : Type u_2
I₂ : Type u_3
J : Type u_4
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Preadditive C
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
c : ComplexShape J
inst✝² : TotalComplexShape c₁ c₂ c
inst✝¹ : TotalComplexShape c₂ c₁ c
inst✝ : TotalComplexShapeSymmetry c₁ c₂ c
a✝ : K.flip.HasTotal c
this : TotalComplexShapeSymmetry c₂ c₁ c
⊢ K.flip.flip.HasTotal c
Module: Mathlib.Algebra.Homology.CochainComplexOpposite

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
f g : K ⟶ L
h : Homotopy f g
p q : ℤ
hpq : ¬(ComplexShape.up ℤ).Rel q p
⊢ (h.hom (-q) (-p)).op = 0

Tactic:
by
    rw [h.zero, op_zero]
    dsimp at hpq ⊢
    lia

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

Goal state before:
pp: case refine_1
C : Type u
inst✝⁶ : Category.{v, u} C
P : ObjectProperty C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : P.IsTriangulated
Y : C
hY : P.trW.isLocal Y
X : C
f : X ⟶ Y
hX : P X
⊢ f = 0

Tactic:
exact (hY _ (trW.mk P (contractible_distinguished₁ X) hX)).injective (by simp)

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

Goal state before:
pp: case refine_1
C : Type u
inst✝⁹ : Category.{v, u} C
D : Type u'
inst✝⁸ : Category.{v', u'} D
P : ObjectProperty C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝³ : Pretriangulated C
inst✝² : P.IsTriangulated
inst✝¹ : IsTriangulated C
Y : C
hY : P.trW.isLocal Y
L : C ⥤ D
inst✝ : L.IsLocalization P.trW
X : C
f₁ f₂ : X ⟶ Y
hf : ∃ X_1 s, ∃ (_ : P.trW s), s ≫ f₁ = s ≫ f₂
⊢ f₁ = f₂

Tactic:
obtain ⟨Z, s, hs, eq⟩ := hf

Goal state after:
case refine_1
C : Type u
inst✝⁹ : Category.{v, u} C
D : Type u'
inst✝⁸ : Category.{v', u'} D
P : ObjectProperty C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝³ : Pretriangulated C
inst✝² : P.IsTriangulated
inst✝¹ : IsTriangulated C
Y : C
hY : P.trW.isLocal Y
L : C ⥤ D
inst✝ : L.IsLocalization P.trW
X : C
f₁ f₂ : X ⟶ Y
Z : C
s : Z ⟶ X
hs : P.trW s
eq : s ≫ f₁ = s ≫ f₂
⊢ f₁ = f₂
Module: Mathlib.CategoryTheory.Triangulated.Orthogonal

Goal state before:
pp: case refine_1
C : Type u
inst✝⁹ : Category.{v, u} C
D : Type u'
inst✝⁸ : Category.{v', u'} D
P : ObjectProperty C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝³ : Pretriangulated C
inst✝² : P.IsTriangulated
inst✝¹ : IsTriangulated C
X : C
hX : P.trW.isColocal X
L : C ⥤ D
inst✝ : L.IsLocalization P.trW
Y : C
f₁ f₂ : X ⟶ Y
hf : ∃ Z s, ∃ (_ : P.trW s), f₁ ≫ s = f₂ ≫ s
⊢ f₁ = f₂

Tactic:
obtain ⟨Z, s, hs, eq⟩ := hf

Goal state after:
case refine_1
C : Type u
inst✝⁹ : Category.{v, u} C
D : Type u'
inst✝⁸ : Category.{v', u'} D
P : ObjectProperty C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝³ : Pretriangulated C
inst✝² : P.IsTriangulated
inst✝¹ : IsTriangulated C
X : C
hX : P.trW.isColocal X
L : C ⥤ D
inst✝ : L.IsLocalization P.trW
Y : C
f₁ f₂ : X ⟶ Y
Z : C
s : Y ⟶ Z
hs : P.trW s
eq : f₁ ≫ s = f₂ ≫ s
⊢ f₁ = f₂
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
⊢ ∀ (n a n' : ℤ), a + n' = n → ∀ (X : DerivedCategory C), (∃ K x, K.IsStrictlyGE n) → ∃ K x, K.IsStrictlyGE n'

Tactic:
by
    rintro n a n' h X ⟨K, e, _⟩
    exact ⟨(shiftFunctor (CochainComplex C ℤ) a).obj K,
      (shiftFunctor (DerivedCategory C) a).mapIso e ≪≫ (Q.commShiftIso a).symm.app K,
      K.isStrictlyGE_shift n a n' h⟩

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

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
X✝ Y✝ Z W : C
f✝ : X✝ ⟶ Y✝
s : X✝ ⟶ Z
g : Z ⟶ W
t✝ : Y✝ ⟶ W
J : Type u_2
A B : J → C
inst✝² : HasProduct A
inst✝¹ : HasProduct B
f : (j : J) → A j ⟶ B j
X Y : C
p : X ⟶ Y
inst✝ : ∀ (j : J), HasLiftingProperty p (f j)
t : X ⟶ ∏ᶜ A
b : Y ⟶ ∏ᶜ B
sq : CommSq t p (Limits.Pi.map f) b
j : J
⊢ (t ≫ Pi.π A j) ≫ f j = (p ≫ b) ≫ Pi.π B j

Tactic:
← sq.w

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
X✝ Y✝ Z W : C
f✝ : X✝ ⟶ Y✝
s : X✝ ⟶ Z
g : Z ⟶ W
t✝ : Y✝ ⟶ W
J : Type u_2
A B : J → C
inst✝² : HasProduct A
inst✝¹ : HasProduct B
f : (j : J) → A j ⟶ B j
X Y : C
p : X ⟶ Y
inst✝ : ∀ (j : J), HasLiftingProperty p (f j)
t : X ⟶ ∏ᶜ A
b : Y ⟶ ∏ᶜ B
sq : CommSq t p (Limits.Pi.map f) b
j : J
⊢ (t ≫ Pi.π A j) ≫ f j = (t ≫ Limits.Pi.map f) ≫ Pi.π B j
Module: Mathlib.CategoryTheory.MorphismProperty.LiftingProperty

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

Tactic:
rw [← le_llp_iff_le_rlp]
    exact T.coproducts_le_llp_rlp

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ModelCategory C
A : C
P✝ : PathObject A
inst✝² : IsFibrant A
P P' : PathObject A
inst✝¹ : P.IsGood
inst✝ : P'.IsGood
ψ : (P.trans P').P ⟶ P.P ⨯ A := prod.lift (pullback.fst P.p₁ P'.p₀) (pullback.snd P.p₁ P'.p₀ ≫ P'.p₁)
fac : ψ ≫ prod.map P.p₁ (𝟙 A) = pullback.snd P.p₁ P'.p₀ ≫ P'.p
sq : IsPullback (ψ ≫ prod.fst) (pullback.snd P.p₁ P'.p₀) P.p₁ (P'.p ≫ prod.fst)
⊢ fibrations C P'.p

Tactic:
rw [← fibration_iff]; infer_instance

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

Goal state before:
pp: case neg
C₁ : Type u_1
C₂ : Type u_2
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝²³ : Category.{v_1, u_1} C₁
inst✝²² : Category.{v_2, u_2} C₂
inst✝²¹ : Category.{v_3, u_5} C₃
inst✝²⁰ : Category.{v_4, u_6} C₄
inst✝¹⁹ : Category.{v_6, u_4} C₂₃
inst✝¹⁸ : HasZeroMorphisms C₁
inst✝¹⁷ : HasZeroMorphisms C₂
inst✝¹⁶ : HasZeroMorphisms C₃
inst✝¹⁵ : Preadditive C₂₃
inst✝¹⁴ : Preadditive C₄
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
inst✝¹³ : G₂₃.PreservesZeroMorphisms
inst✝¹² : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst✝¹¹ : F.PreservesZeroMorphisms
inst✝¹⁰ : ∀ (X₁ : C₁), (F.obj X₁).Additive
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₂₃ : Type u_11
ι₄ : Type u_12
inst✝⁹ : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
K₃ : HomologicalComplex C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₂₃ : ComplexShape ι₂₃
c₄ : ComplexShape ι₄
inst✝⁸ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁷ : TotalComplexShape c₁₂ c₃ c₄
inst✝⁶ : TotalComplexShape c₂ c₃ c₂₃
inst✝⁵ : TotalComplexShape c₁ c₂₃ c₄
inst✝⁴ : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst✝³ : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst✝² : DecidableEq ι₂₃
inst✝¹ : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄
inst✝ : HasGoodTrifunctor₂₃Obj F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄
j j' : ι₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
i₂₃ : ι₂₃ := c₂.π c₃ c₂₃ (i₂, i₃)
h₁ : c₂₃.Rel i₂₃ (c₂₃.next i₂₃)
h₂ : ¬c₁.π c₂₃ c₄ (i₁, c₂₃.next i₂₃) = j'
⊢ (F.obj (K₁.X i₁)).map (K₂.ιMapBifunctor K₃ G₂₃ c₂₃ i₂ i₃ (c₂.π c₃ c₂₃ (i₂, i₃)) ⋯) ≫
      mapBifunctor.d₂ K₁ (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄ i₁ (c₂.π c₃ c₂₃ (i₂, i₃)) j' =
    d₂ F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ i₃ j' + d₃ F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ i₃ j'

Tactic:
rw [mapBifunctor.d₂_eq_zero' _ _ _ _ _ h₁ _ h₂, comp_zero]
      trans 0 + 0
      · simp
      · congr 1
        · by_cases h₃ : c₂.Rel i₂ (c₂.next i₂)
          · rw [d₂_eq _ _ _ _ _ _ _ _ _ h₃, ιOrZero_eq_zero, comp_zero, smul_zero]
            intro h₄
            apply h₂
            rw [← h₄]
            dsimp [ComplexShape.r]
            rw [ComplexShape.assoc c₁ c₂ c₃ c₁₂ c₂₃ c₄, ComplexShape.next_π₁ c₃ c₂₃ h₃ i₃]
          · rw [d₂_eq_zero _ _ _ _ _ _ _ _ _ _ _ _ h₃]
        · by_cases h₃ : c₃.Rel i₃ (c₃.next i₃)
          · rw [d₃_eq _ _ _ _ _ _ _ _ _ _ h₃, ιOrZero_eq_zero, comp_zero, smul_zero]
            intro h₄
            apply h₂
            rw [← h₄]
            dsimp [ComplexShape.r]
            rw [ComplexShape.assoc c₁ c₂ c₃ c₁₂ c₂₃ c₄, ComplexShape.next_π₂ c₂ c₂₃ i₂ h₃]
          · rw [d₃_eq_zero _ _ _ _ _ _ _ _ _ _ _ _ h₃]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
r : HomRel C
inst✝ : Congruence r
W : MorphismProperty C
hW : W.IsInvertedBy (functor r)
hr : ∀ ⦃X Y : C⦄ (f₀ f₁ : X ⟶ Y), r f₀ f₁ → ∃ P x, W P.ι
X Y : C
f₀ f₁ : X ⟶ Y
hf : (functor r).map f₀ = (functor r).map f₁
⊢ ∃ P x, W P.ι

Tactic:
functor_map_eq_iff

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
r : HomRel C
inst✝ : Congruence r
W : MorphismProperty C
hW : W.IsInvertedBy (functor r)
hr : ∀ ⦃X Y : C⦄ (f₀ f₁ : X ⟶ Y), r f₀ f₁ → ∃ P x, W P.ι
X Y : C
f₀ f₁ : X ⟶ Y
hf : r f₀ f₁
⊢ ∃ P x, W P.ι
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✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : Category.{v_3, u_3} D₁
inst✝¹² : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : HasShift C ℤ
inst✝⁹ : Preadditive C
inst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁷ : Pretriangulated C
inst✝⁶ : IsTriangulated C
inst✝⁵ : A.IsTriangulated
inst✝⁴ : B.IsTriangulated
inst✝³ : B.IsClosedUnderIsomorphisms
inst✝² : A.IsVerdierRightLocalizing B
L₁ : A.FullSubcategory ⥤ D₁
L₂ : C ⥤ D₂
inst✝¹ : L₁.IsLocalization (B.inverseImage A.ι).trW
inst✝ : L₂.IsLocalization B.trW
this✝² : Preadditive D₁ := Localization.preadditive L₁ (B.inverseImage A.ι).trW
this✝¹ : Preadditive D₂ := Localization.preadditive L₂ B.trW
this✝ : L₁.Additive
this : L₂.Additive
F : D₁ ⥤ D₂ := (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂
e : A.ι ⋙ L₂ ≅ L₁ ⋙ F := CatCommSq.iso (A.triangulatedLocalizerMorphism B).functor L₁ L₂ F
X₁ X₂ : A.FullSubcategory
f : X₁ ⟶ X₂
X₃ : C
s : X₂.obj ⟶ X₃
hs : B.trW s
fac : f.hom ≫ s = 0 ≫ s
X₄ : C
t : X₂.obj ⟶ X₄
a : X₃ ⟶ X₄
hX₄ : A X₄
ht : (A ⊓ B).trW t
fac' : s ≫ a = t
t' : X₂ ⟶ { obj := X₄, property := hX₄ } := homMk t
⊢ L₁.map f = 0

Tactic:
have := Localization.inverts L₁ (B.inverseImage A.ι).trW t'
    (by rwa [trW_inverseImage_ι_iff])

Goal state after:
C : Type u_1
D : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : Category.{v_3, u_3} D₁
inst✝¹² : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : HasShift C ℤ
inst✝⁹ : Preadditive C
inst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁷ : Pretriangulated C
inst✝⁶ : IsTriangulated C
inst✝⁵ : A.IsTriangulated
inst✝⁴ : B.IsTriangulated
inst✝³ : B.IsClosedUnderIsomorphisms
inst✝² : A.IsVerdierRightLocalizing B
L₁ : A.FullSubcategory ⥤ D₁
L₂ : C ⥤ D₂
inst✝¹ : L₁.IsLocalization (B.inverseImage A.ι).trW
inst✝ : L₂.IsLocalization B.trW
this✝³ : Preadditive D₁ := Localization.preadditive L₁ (B.inverseImage A.ι).trW
this✝² : Preadditive D₂ := Localization.preadditive L₂ B.trW
this✝¹ : L₁.Additive
this✝ : L₂.Additive
F : D₁ ⥤ D₂ := (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂
e : A.ι ⋙ L₂ ≅ L₁ ⋙ F := CatCommSq.iso (A.triangulatedLocalizerMorphism B).functor L₁ L₂ F
X₁ X₂ : A.FullSubcategory
f : X₁ ⟶ X₂
X₃ : C
s : X₂.obj ⟶ X₃
hs : B.trW s
fac : f.hom ≫ s = 0 ≫ s
X₄ : C
t : X₂.obj ⟶ X₄
a : X₃ ⟶ X₄
hX₄ : A X₄
ht : (A ⊓ B).trW t
fac' : s ≫ a = t
t' : X₂ ⟶ { obj := X₄, property := hX₄ } := homMk t
this : IsIso (L₁.map t')
⊢ L₁.map f = 0
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: case refine_2
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
X Y : C
f : X ⟶ Y
n : ℤ
x✝ : ∃ Z g h, ∃ (_ : Triangle.mk ((t.truncLTι n).app X ≫ f) g h ∈ distinguishedTriangles), t.IsGE Z n
Z : C
g : Y ⟶ Z
h : Z ⟶ (shiftFunctor C 1).obj ((t.truncLT n).obj X)
mem : Triangle.mk ((t.truncLTι n).app X ≫ f) g h ∈ distinguishedTriangles
h✝ : t.IsGE Z n
⊢ IsIso ((t.truncLT n).map f)

Tactic:
obtain ⟨e, he⟩ := t.triangle_iso_exists
      mem (t.triangleLTGE_distinguished n Y) (Iso.refl _) (n - 1) n
      (by dsimp; infer_instance) (by dsimp; infer_instance)
      (by dsimp; infer_instance) (by dsimp; infer_instance)
    suffices ((t.truncLT n).map f) = e.hom.hom₁ by rw [this]; infer_instance
    exact t.to_truncLT_obj_ext (Eq.trans (by cat_disch) e.hom.comm₁)

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

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : ModelCategory C
X Y : C
f : X ⟶ Y
h✝ : FibrantBrownFactorization f
inst✝¹ : IsFibrant X
inst✝ : IsFibrant Y
h : (trivialCofibrations C).MapFactorizationData (fibrations C) (prod.lift f (𝟙 X))
⊢ fibrations C (h.p ≫ prod.fst)

Tactic:
rw [← fibration_iff]; infer_instance

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

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
W : MorphismProperty C
homRel : HomRel C
inst✝² : HomRel.IsStableUnderPrecomp homRel
inst✝¹ : HomRel.IsStableUnderPostcomp homRel
inst✝ : W.HasQuotient homRel
X Y : C
f : X ⟶ Y
⊢ W.quotient homRel ((Quotient.functor homRel).map f) ↔ W f

Tactic:
refine ⟨fun ⟨f', hf', h⟩ ↦ ?_, fun hf ↦ ⟨f, hf, rfl⟩⟩
  rw [← Functor.homRel_iff, Quotient.functor_homRel_eq_compClosure_eqvGen,
    HomRel.compClosure_eq_self homRel] at h
  rwa [HasQuotient.iff_of_eqvGen W h]

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
inst✝ : EnoughInjectives C
n₁ q : ℤ
⊢ epiWithInjectiveKernel ((π K L n₁).f q)

Tactic:
rw [Abelian.epiWithInjectiveKernel_iff]

Goal state after:
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 q : ℤ
⊢ ∃ I,
    ∃ (_ : Injective I),
      ∃ f,
        ∃ (w : f ≫ (π K L n₁).f q = 0),
          Nonempty { X₁ := I, X₂ := (mid K L n₁).X q, X₃ := L.X q, f := f, g := (π K L n₁).f q, zero := w }.Splitting
Module: Mathlib.CategoryTheory.GuitartExact.Opposite

Goal state before:
pp: case mp
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

pp: case mpr
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.GuitartExact → w.op.GuitartExact

Tactic:
· 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

Goal state after:
case mpr
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.GuitartExact → w.op.GuitartExact
Module: Mathlib.CategoryTheory.GuitartExact.Opposite

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
C₄ : Type u₄
inst✝⁶ : Category.{v₁, u₁} C₁
inst✝⁵ : Category.{v₂, u₂} C₂
inst✝⁴ : Category.{v₃, u₃} C₃
inst✝³ : Category.{v₄, u₄} C₄
T : C₁ ⥤ C₂
L : C₁ ⥤ C₃
R : C₂ ⥤ C₄
B : C₃ ⥤ C₄
w : TwoSquare T L R B
inst✝² : T.IsEquivalence
inst✝¹ : B.IsEquivalence
inst✝ : IsIso w.natTrans
⊢ w.GuitartExact

Tactic:
rw [← guitartExact_op_iff]
  infer_instance

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

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝² : EnoughInjectives C
n : ℤ
inst✝¹ : K.IsStrictlyGE (n + 1)
inst✝ : L.IsStrictlyGE n
⊢ ∃ K', ∃ (_ : K'.IsStrictlyGE n), ∃ ι π, Mono ι ∧ QuasiIso ι ∧ degreewiseEpiWithInjectiveKernel π ∧ ι ≫ π = f

Tactic:
by
  have : K.IsStrictlyGE n := K.isStrictlyGE_of_ge n (n + 1) (by lia)
  obtain ⟨L', _, i, p, _, hp, _, rfl⟩ := cm5b f n
  obtain ⟨K', _, ι, π, _, _, hπ, rfl⟩ := cm5a_cof i n
  exact ⟨K', inferInstance, ι, π ≫ p, inferInstance, inferInstance,
    MorphismProperty.comp_mem _ _ _ hπ hp, by simp⟩

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

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
K : CochainComplex C ℤ
inst✝¹ : EnoughInjectives C
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
inst✝ : K.IsStrictlyGE n₁
⊢ ∃ L i, ∃ (_ : Mono i) (_ : QuasiIso i) (_ : ∀ (n : ℤ), Injective (L.X n)), L.IsStrictlyGE n₀

Tactic:
have : K.IsStrictlyGE (n₀ + 1) := by rw [h]; infer_instance

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
K : CochainComplex C ℤ
inst✝¹ : EnoughInjectives C
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
inst✝ : K.IsStrictlyGE n₁
this : K.IsStrictlyGE (n₀ + 1)
⊢ ∃ L i, ∃ (_ : Mono i) (_ : QuasiIso i) (_ : ∀ (n : ℤ), Injective (L.X n)), L.IsStrictlyGE n₀
Module: Mathlib.CategoryTheory.GuitartExact.KanExtension

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
C₄ : Type u₄
D : Type u₅
inst✝⁶ : Category.{v₁, u₁} C₁
inst✝⁵ : Category.{v₂, u₂} C₂
inst✝⁴ : Category.{v₃, u₃} C₃
inst✝³ : Category.{v₄, u₄} C₄
inst✝² : Category.{v₅, u₅} D
T : C₁ ⥤ C₂
L : C₁ ⥤ C₃
R : C₂ ⥤ C₄
B : C₃ ⥤ C₄
w : TwoSquare T L R B
inst✝¹ : ∀ (F : C₁ ⥤ D), L.HasLeftKanExtension F
inst✝ : ∀ (F : C₂ ⥤ D), R.HasLeftKanExtension F
F₁ F₂ : C₂ ⥤ D
τ : F₁ ⟶ F₂
⊢ L.lan.map (T.whiskerLeft τ) ≫
      ((L.lanAdjunction D).homEquiv (T ⋙ F₂) (B ⋙ R.lan.obj F₂)).symm
        (T.whiskerLeft (R.lanUnit.app F₂) ≫
          (T.associator R (R.lan.obj F₂)).inv ≫
            whiskerRight w.natTrans (R.lan.obj F₂) ≫ (L.associator B (R.lan.obj F₂)).hom) =
    ((L.lanAdjunction D).homEquiv (T ⋙ F₁) (B ⋙ R.lan.obj F₁)).symm
        (T.whiskerLeft (R.lanUnit.app F₁) ≫
          (T.associator R (R.lan.obj F₁)).inv ≫
            whiskerRight w.natTrans (R.lan.obj F₁) ≫ (L.associator B (R.lan.obj F₁)).hom) ≫
      B.whiskerLeft (R.lan.map τ)

Tactic:
refine (Adjunction.homEquiv_naturality_left_symm ..).symm.trans
      (Eq.trans ?_ (Adjunction.homEquiv_naturality_right_symm ..))

Goal state after:
C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
C₄ : Type u₄
D : Type u₅
inst✝⁶ : Category.{v₁, u₁} C₁
inst✝⁵ : Category.{v₂, u₂} C₂
inst✝⁴ : Category.{v₃, u₃} C₃
inst✝³ : Category.{v₄, u₄} C₄
inst✝² : Category.{v₅, u₅} D
T : C₁ ⥤ C₂
L : C₁ ⥤ C₃
R : C₂ ⥤ C₄
B : C₃ ⥤ C₄
w : TwoSquare T L R B
inst✝¹ : ∀ (F : C₁ ⥤ D), L.HasLeftKanExtension F
inst✝ : ∀ (F : C₂ ⥤ D), R.HasLeftKanExtension F
F₁ F₂ : C₂ ⥤ D
τ : F₁ ⟶ F₂
⊢ ((L.lanAdjunction D).homEquiv (T ⋙ F₁) (B ⋙ R.lan.obj F₂)).symm
      (T.whiskerLeft τ ≫
        T.whiskerLeft (R.lanUnit.app F₂) ≫
          (T.associator R (R.lan.obj F₂)).inv ≫
            whiskerRight w.natTrans (R.lan.obj F₂) ≫ (L.associator B (R.lan.obj F₂)).hom) =
    ((L.lanAdjunction D).homEquiv (T ⋙ F₁) (B ⋙ R.lan.obj F₂)).symm
      ((T.whiskerLeft (R.lanUnit.app F₁) ≫
          (T.associator R (R.lan.obj F₁)).inv ≫
            whiskerRight w.natTrans (R.lan.obj F₁) ≫ (L.associator B (R.lan.obj F₁)).hom) ≫
        ((whiskeringLeft C₁ C₃ D).obj L).map (B.whiskerLeft (R.lan.map τ)))
Module: Mathlib.CategoryTheory.GuitartExact.KanExtension

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
C₄ : Type u₄
D : Type u₅
inst✝⁶ : Category.{v₁, u₁} C₁
inst✝⁵ : Category.{v₂, u₂} C₂
inst✝⁴ : Category.{v₃, u₃} C₃
inst✝³ : Category.{v₄, u₄} C₄
inst✝² : Category.{v₅, u₅} D
T : C₁ ⥤ C₂
L : C₁ ⥤ C₃
R : C₂ ⥤ C₄
B : C₃ ⥤ C₄
w : TwoSquare T L R B
inst✝¹ : ∀ (F : C₁ ⥤ D), L.HasLeftKanExtension F
inst✝ : ∀ (F : C₂ ⥤ D), R.HasLeftKanExtension F
F : C₂ ⥤ D
⊢ IsIso (w.lanBaseChange.app F) ↔
    (StructuredArrow.right ((LeftExtension.mk (R.lan.obj F) (R.lanUnit.app F)).compTwoSquare w)).IsLeftKanExtension
      (StructuredArrow.hom ((LeftExtension.mk (R.lan.obj F) (R.lanUnit.app F)).compTwoSquare w))

Tactic:
rw [lanBaseChange_app, isIso_lanAdjunction_homEquiv_symm_iff]
  simp

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

Goal state before:
pp: C : Type u_1
H : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Category.{v_2, u_2} H
this :
  ((L C).functor ⋙ 𝟭 (Plus (InjectiveObject C))).IsLocalization
    ((CochainComplex.Plus.quasiIso C).inverseImage (InjectiveObject.ι C).mapCochainComplexPlus)
⊢ (L C).IsLocalizedEquivalence

Tactic:
exact LocalizerMorphism.IsLocalizedEquivalence.of_isLocalization_of_isLocalization (L C) (𝟭 _)

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

Goal state before:
pp: case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁴ : Category.{v_1, u_3} C
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroObject C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝¹ : e.IsRelIff
ι'' : Type u_4
c'' : ComplexShape ι''
e' : c''.Embedding c'
inst✝ : K.IsStrictlySupported e'
i' : ι'
hi' : ∀ (i : ι''), e'.f i ≠ i'
hi'' : ¬∃ i, e.f i = i'
⊢ IsZero ((K.stupidTrunc e).X i')

Tactic:
apply isZero_stupidTrunc_X
      simpa using hi''

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

Goal state before:
pp: case mpr
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝ : e.IsRelIff
⊢ K.IsStrictlySupportedOutside e → IsZero (K.stupidTrunc e)

Tactic:
· intro h
    rw [isZero_iff_isStrictlySupported_and_isStrictlySupportedOutside _ e]
    constructor
    · infer_instance
    · exact ⟨fun i ↦ (h.isZero i).of_iso (K.stupidTruncXIso e rfl)⟩

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

Goal state before:
pp: case neg
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
i₁ i j k : ι
hij : c.Rel i j
hjk : c.Rel j k
hi₀₁ : c.Rel i i₁
hj : ¬j = i₁
⊢ ((if hk : i = i ∧ j = i₁ ∧ ¬i = i₁ then eqToHom ⋯ ≫ f ≫ eqToHom ⋯ else 0) ≫
      if hk : j = i ∧ k = i₁ ∧ ¬i = i₁ then eqToHom ⋯ ≫ f ≫ eqToHom ⋯ else 0) =
    0

Tactic:
rw [dif_neg (by tauto), zero_comp]

Goal state after: