mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.MorphismProperty.LiftingProperty

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
T : MorphismProperty C
J : Type u_1
A B : J → C
inst✝¹ : Limits.HasProduct A
inst✝ : Limits.HasProduct B
f : (j : J) → A j ⟶ B j
hf : ∀ (j : J), T.rlp (f j)
⊢ T.rlp (Limits.Pi.map f)

Tactic:
X

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
T : MorphismProperty C
J : Type u_1
A B : J → C
inst✝¹ : Limits.HasProduct A
inst✝ : Limits.HasProduct B
f : (j : J) → A j ⟶ B j
hf : ∀ (j : J), T.rlp (f j)
X : C
⊢ ∀ ⦃Y : C⦄ (g : X ⟶ Y), T g → HasLiftingProperty g (Limits.Pi.map f)
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
⊢ ∃ Y, Nonempty (X ≅ (singleFunctor C n).obj Y)

Tactic:
obtain ⟨K, _, _, ⟨e⟩⟩ := exists_iso_Q_obj_of_isGE_of_isLE X n n

Goal state after:
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, Nonempty (X ≅ (singleFunctor C n).obj Y)
Module: Mathlib.AlgebraicTopology.ModelCategory.Instances

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : CategoryWithWeakEquivalences C
inst✝⁴ : CategoryWithCofibrations C
inst✝³ : CategoryWithFibrations C
X₁ X₂ Y₁ Y₂ : C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₂
inst✝² : (cofibrations C).IsWeakFactorizationSystem (trivialFibrations C)
h₁ : cofibrations C f₁
h₂ : cofibrations C f₂
inst✝¹ : HasBinaryCoproduct X₁ X₂
inst✝ : HasBinaryCoproduct Y₁ Y₂
⊢ cofibrations C (coprod.map f₁ f₂)

Tactic:
apply MorphismProperty.colimMap

Goal state after:
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : CategoryWithWeakEquivalences C
inst✝⁴ : CategoryWithCofibrations C
inst✝³ : CategoryWithFibrations C
X₁ X₂ Y₁ Y₂ : C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₂
inst✝² : (cofibrations C).IsWeakFactorizationSystem (trivialFibrations C)
h₁ : cofibrations C f₁
h₂ : cofibrations C f₂
inst✝¹ : HasBinaryCoproduct X₁ X₂
inst✝ : HasBinaryCoproduct Y₁ Y₂
⊢ (cofibrations C).functorCategory (Discrete WalkingPair) (mapPair f₁ f₂)
Module: Mathlib.AlgebraicTopology.ModelCategory.Basic

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ModelCategory C
A B X Y : C
i : A ⟶ B
p : X ⟶ Y
hi : trivialCofibrations C i
hp : fibrations C p
⊢ HasLiftingProperty i p

Tactic:
obtain ⟨_, _⟩ := mem_trivialCofibrations_iff i |>.mp hi
    rw [← fibration_iff] at hp
    infer_instance

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ModelCategory C
A : C
P : PathObject A
h : (trivialCofibrations C).MapFactorizationData (fibrations C) (diag A)
⊢ Fibration (ofFactorizationData h).p

Tactic:
simpa using inferInstanceAs (Fibration h.p)

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ModelCategory C
A : C
P : PathObject A
h : (trivialCofibrations C).MapFactorizationData (fibrations C) (diag A)
⊢ Fibration (ofFactorizationData h).p

Tactic:
by simpa using inferInstanceAs (Fibration h.p)

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
X Y : C
inst✝ : ModelCategory C
P : Cylinder X
f g : X ⟶ Y
h : P.LeftHomotopy f g
d : (cofibrations C).MapFactorizationData (trivialFibrations C) P.i :=
  (cofibrations C).factorizationData (trivialFibrations C) P.i
⊢ Cofibration
    { I := d.Z, i₀ := coprod.inl ≫ d.i, i₁ := coprod.inr ≫ d.i, π := d.p ≫ P.π, i₀_π := ⋯, i₁_π := ⋯,
        weakEquivalence_π := ⋯ }.i

Tactic:
rw [cofibration_iff]
        convert! d.hi
        aesop

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
X Y : C
inst✝ : CategoryWithWeakEquivalences C
f g : X ⟶ Y
h : RightHomotopyRel f g
Z : C
i : Z ⟶ X
⊢ RightHomotopyRel (i ≫ f) (i ≫ g)

Tactic:
obtain ⟨P, ⟨h⟩⟩ := h
  exact (h.precomp i).rightHomotopyRel

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

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
inst✝ : IsTriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : Cᵒᵖ
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : X₂ ⟶ Z₁₂
w₁₂ : Z₁₂ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁
h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ Pretriangulated.distinguishedTriangles
⊢ ∀ {v₂₃ : X₃ ⟶ Z₂₃} {w₂₃ : Z₂₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₂}
    (h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ Pretriangulated.distinguishedTriangles) {v₁₃ : X₃ ⟶ Z₁₃}
    {w₁₃ : Z₁₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁} (h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ Pretriangulated.distinguishedTriangles),
    Nonempty (Triangulated.Octahedron comm h₁₂ h₂₃ h₁₃)

Tactic:
v₂₃

Goal state after:
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
inst✝ : IsTriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : Cᵒᵖ
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : X₂ ⟶ Z₁₂
w₁₂ : Z₁₂ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁
h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ Pretriangulated.distinguishedTriangles
v₂₃ : X₃ ⟶ Z₂₃
⊢ ∀ {w₂₃ : Z₂₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₂} (h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ Pretriangulated.distinguishedTriangles)
    {v₁₃ : X₃ ⟶ Z₁₃} {w₁₃ : Z₁₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁}
    (h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ Pretriangulated.distinguishedTriangles),
    Nonempty (Triangulated.Octahedron comm h₁₂ h₂₃ h₁₃)
Module: Mathlib.Algebra.Homology.HomotopyCategory.Plus

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
K : Plus C
L : HomologicalComplex C (ComplexShape.up ℤ)
hL : (HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj L = K.obj
⊢ ((quotient C).obj { obj := L, property := ?m.38 }).obj = K.obj

Tactic:
exact hL

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

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝³³ : Category.{v_1, u_1} C₁
inst✝³² : Category.{v_2, u_2} C₂
inst✝³¹ : Category.{v_3, u_5} C₃
inst✝³⁰ : Category.{v_4, u_6} C₄
inst✝²⁹ : Category.{v_5, u_3} C₁₂
inst✝²⁸ : Category.{v_6, u_4} C₂₃
inst✝²⁷ : HasZeroMorphisms C₁
inst✝²⁶ : HasZeroMorphisms C₂
inst✝²⁵ : HasZeroMorphisms C₃
inst✝²⁴ : Preadditive C₁₂
inst✝²³ : Preadditive C₂₃
inst✝²² : Preadditive C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
inst✝²¹ : F₁₂.PreservesZeroMorphisms
inst✝²⁰ : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst✝¹⁹ : G.Additive
inst✝¹⁸ : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
inst✝¹⁷ : G₂₃.PreservesZeroMorphisms
inst✝¹⁶ : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst✝¹⁵ : F.PreservesZeroMorphisms
inst✝¹⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
associator : bifunctorComp₁₂ F₁₂ G ≅ bifunctorComp₂₃ F G₂₃
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✝¹³ : 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✝¹² : TotalComplexShape c₁ c₂ c₁₂
inst✝¹¹ : TotalComplexShape c₁₂ c₃ c₄
inst✝¹⁰ : TotalComplexShape c₂ c₃ c₂₃
inst✝⁹ : TotalComplexShape c₁ c₂₃ c₄
inst✝⁸ : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst✝⁷ : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst✝⁶ : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst✝⁵ : DecidableEq ι₁₂
inst✝⁴ : DecidableEq ι₂₃
inst✝³ : (K₁.mapBifunctor K₂ F₁₂ c₁₂).HasMapBifunctor K₃ G c₄
inst✝² : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄
inst✝¹ : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
inst✝ : HasGoodTrifunctor₂₃Obj F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄
j j' : ι₄
⊢ (mapBifunctorAssociatorX associator K₁ K₂ K₃ c₁₂ c₂₃ c₄ j).hom ≫ mapBifunctor₂₃.D₁ F G₂₃ K₁ K₂ K₃ c₂₃ c₄ j j' =
    mapBifunctor₁₂.D₁ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ j j' ≫ (mapBifunctorAssociatorX associator K₁ K₂ K₃ c₁₂ c₂₃ c₄ j').hom

Tactic:
by
  ext i₁ i₂ i₃ h
  rw [mapBifunctor₁₂.ι_D₁_assoc, ι_mapBifunctorAssociatorX_hom_assoc, mapBifunctor₂₃.ι_D₁]
  by_cases h₁ : c₁.Rel i₁ (c₁.next i₁)
  · have := NatTrans.naturality_app_app associator.hom
      (K₁.d i₁ (c₁.next i₁)) (K₂.X i₂) (K₃.X i₃)
    dsimp at this
    rw [mapBifunctor₁₂.d₁_eq _ _ _ _ _ _ _ h₁, mapBifunctor₂₃.d₁_eq _ _ _ _ _ _ _ _ h₁,
      Linear.comp_units_smul, Linear.units_smul_comp, assoc,
        ComplexShape.associative_ε₁_eq_mul c₁ c₂ c₃ c₁₂ c₂₃ c₄,
      ιOrZero_mapBifunctorAssociatorX_hom, smul_left_cancel_iff,
      reassoc_of% this]
  · rw [mapBifunctor₁₂.d₁_eq_zero _ _ _ _ _ _ _ _ _ _ _ h₁,
      mapBifunctor₂₃.d₁_eq_zero _ _ _ _ _ _ _ _ _ _ _ _ h₁, comp_zero, zero_comp]

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
K : HomotopyCategory C (ComplexShape.up ℤ)
hK : HomotopyCategory.plus C K
⊢ t.plus ((HomotopyCategory.Plus.ι C ⋙ DerivedCategory.Qh).obj { obj := K, property := hK })

Tactic:
obtain ⟨K, rfl⟩ := HomotopyCategory.quotient_obj_surjective K

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
K : HomologicalComplex C (ComplexShape.up ℤ)
hK : HomotopyCategory.plus C ((HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj K)
⊢ t.plus
    ((HomotopyCategory.Plus.ι C ⋙ DerivedCategory.Qh).obj
      { obj := (HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj K, property := hK })
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
n : ℤ
X Y : C
φ : X ⟶ Y
H : ∃ f, f.hom₂ = φ
⊢ (triangle t n X).mor₁ ≫ φ = H.choose.hom₁ ≫ (triangle t n Y).mor₁

Tactic:
← H.choose.comm₁,

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
n : ℤ
X Y : C
φ : X ⟶ Y
H : ∃ f, f.hom₂ = φ
⊢ (triangle t n X).mor₁ ≫ φ = (triangle t n X).mor₁ ≫ H.choose.hom₂
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
n : ℤ
X Y : C
φ : X ⟶ Y
A : C
a b : ℤ
h : a ≤ b
H : ∃ f, f.hom₂ = 𝟙 (triangle t a A).obj₂
⊢ (triangle t a A).mor₁ ≫ 𝟙 (triangle t a A).obj₂ = H.choose.hom₁ ≫ (triangle t b A).mor₁

Tactic:
← H.choose.comm₁,

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
n : ℤ
X Y : C
φ : X ⟶ Y
A : C
a b : ℤ
h : a ≤ b
H : ∃ f, f.hom₂ = 𝟙 (triangle t a A).obj₂
⊢ (triangle t a A).mor₁ ≫ 𝟙 (triangle t a A).obj₂ = (triangle t a A).mor₁ ≫ H.choose.hom₂
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
X✝ : A.FullSubcategoryᵒᵖ
⊢ ∀ ⦃Y : A.FullSubcategoryᵒᵖ⦄ (f : X✝ ⟶ Y),
    (B.op.inverseImage A.op.ι).trW.inverseImage A.opEquivalence.inverse f →
      (B.op.inverseImage A.op.ι).trW.isoClosure.inverseImage A.opEquivalence.symm.functor f

Tactic:
_

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
X✝ Y✝ : A.FullSubcategoryᵒᵖ
⊢ ∀ (f : X✝ ⟶ Y✝),
    (B.op.inverseImage A.op.ι).trW.inverseImage A.opEquivalence.inverse f →
      (B.op.inverseImage A.op.ι).trW.isoClosure.inverseImage A.opEquivalence.symm.functor f
Module: Mathlib.Algebra.Homology.HomologicalComplexLimitsEventuallyConstant

Goal state before:
pp: C : Type u_1
J : Type u_2
ι : Type u_3
inst✝⁵ : Category.{u_4, u_1} C
inst✝⁴ : Category.{u_5, u_2} J
c : ComplexShape ι
inst✝³ : IsCofiltered J
inst✝² : HasZeroMorphisms C
F : J ⥤ HomologicalComplex C c
inst✝¹ : ∀ (j : ι), HasLimit (F ⋙ eval C c j)
cF : Cone F
hcF : IsLimit cF
inst✝ : CategoryWithHomology C
q₀ q₁ q₂ : ι
h₀ : c.prev q₁ = q₀
h₂ : c.next q₁ = q₂
j : J
hq₀ : (F ⋙ eval C c q₀).IsEventuallyConstantTo j
hq₁ : (F ⋙ eval C c q₁).IsEventuallyConstantTo j
hq₂ : (F ⋙ eval C c q₂).IsEventuallyConstantTo j
⊢ QuasiIsoAt (cF.π.app j) q₁

Tactic:
quasiIsoAt_iff' _ q₀ q₁ q₂ h₀ h₂

Goal state after:
C : Type u_1
J : Type u_2
ι : Type u_3
inst✝⁵ : Category.{u_4, u_1} C
inst✝⁴ : Category.{u_5, u_2} J
c : ComplexShape ι
inst✝³ : IsCofiltered J
inst✝² : HasZeroMorphisms C
F : J ⥤ HomologicalComplex C c
inst✝¹ : ∀ (j : ι), HasLimit (F ⋙ eval C c j)
cF : Cone F
hcF : IsLimit cF
inst✝ : CategoryWithHomology C
q₀ q₁ q₂ : ι
h₀ : c.prev q₁ = q₀
h₂ : c.next q₁ = q₂
j : J
hq₀ : (F ⋙ eval C c q₀).IsEventuallyConstantTo j
hq₁ : (F ⋙ eval C c q₁).IsEventuallyConstantTo j
hq₂ : (F ⋙ eval C c q₂).IsEventuallyConstantTo j
⊢ ShortComplex.QuasiIso ((shortComplexFunctor' C c q₀ q₁ q₂).map (cF.π.app j))
Module: Mathlib.CategoryTheory.Functor.OfSequence

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
F G : ℕᵒᵖ ⥤ C
app : (n : ℕ) → F.obj (Opposite.op n) ⟶ G.obj (Opposite.op n)
naturality : ∀ (n : ℕ), F.map (homOfLE ⋯).op ≫ app n = app (n + 1) ≫ G.map (homOfLE ⋯).op
x✝¹ x✝ : ℕᵒᵖ
f : x✝¹ ⟶ x✝
⊢ F.map f ≫ app (Opposite.unop x✝) = app (Opposite.unop x✝¹) ≫ G.map f

Tactic:
by
    let φ : G.rightOp ⟶ F.rightOp := ofSequence (fun n ↦ (app n).op)
      (fun n ↦ Quiver.Hom.unop_inj (naturality n).symm)
    exact Quiver.Hom.op_inj (φ.naturality f.unop).symm

Goal state after:
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
Y Z : C
g : Y ⟶ Z
n₀ n₁ : ℤ
hn : n₀ + 1 = n₁
x✝ : ∃ X f h, ∃ (_ : Triangle.mk f (g ≫ (t.truncGEπ n₁).app Z) h ∈ distinguishedTriangles), t.IsLE X n₀
X : C
f : X ⟶ Y
h : (t.truncGE n₁).obj Z ⟶ (shiftFunctor C 1).obj X
mem : Triangle.mk f (g ≫ (t.truncGEπ n₁).app Z) h ∈ distinguishedTriangles
h✝ : t.IsLE X n₀
⊢ t.IsLE ((t.truncLT n₁).obj Y) n₀

Tactic:
rw [← hn]

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
Y Z : C
g : Y ⟶ Z
n₀ n₁ : ℤ
hn : n₀ + 1 = n₁
x✝ : ∃ X f h, ∃ (_ : Triangle.mk f (g ≫ (t.truncGEπ n₁).app Z) h ∈ distinguishedTriangles), t.IsLE X n₀
X : C
f : X ⟶ Y
h : (t.truncGE n₁).obj Z ⟶ (shiftFunctor C 1).obj X
mem : Triangle.mk f (g ≫ (t.truncGEπ n₁).app Z) h ∈ distinguishedTriangles
h✝ : t.IsLE X n₀
⊢ t.IsLE ((t.truncLT (n₀ + 1)).obj Y) n₀
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: case pos
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
X : C
a b : ℤ
inst✝ : t.IsLE X b
h : a ≤ b + 1
⊢ t.IsLE ((t.truncLT a).obj X) b

Tactic:
exact t.isLE_truncLT_obj ..

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
n m : ℤ
hnm : n + 1 = m
fac_left : ∀ (k : ℤ), i.f k ≫ (hsq k).l = t.f k
⊢ i.f n ≫ (↑(cocycle₁' sq hsq)).v n m hnm = 0

Tactic:
simp [cocycle₁', fac_left, reassoc_of% fac_left]

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
f : K ⟶ L
inst✝ : EnoughInjectives C
n₀✝ n₁ : ℤ
hn₁ : n₀✝ + 1 = n₁
hf : ∀ i ≤ n₀✝, QuasiIsoAt f i
n₀ : ℤ := n₁ - 1
A : C
x₁ : A ⟶ K.X n₁
⊢ x₁ ≫ K.d n₁ (n₁ + 1) = 0 →
    ∀ (y₁ : A ⟶ (mid K L n₁).X n₀),
      x₁ ≫ (ι f n₁).f n₁ = y₁ ≫ (mid K L n₁).d n₀ n₁ → ∃ A' π, ∃ (_ : Epi π), ∃ x₁_1, π ≫ x₁ = x₁_1 ≫ K.d n₀ n₁

Tactic:
_

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝ : EnoughInjectives C
n₀✝ n₁ : ℤ
hn₁ : n₀✝ + 1 = n₁
hf : ∀ i ≤ n₀✝, QuasiIsoAt f i
n₀ : ℤ := n₁ - 1
A : C
x₁ : A ⟶ K.X n₁
x✝ : x₁ ≫ K.d n₁ (n₁ + 1) = 0
⊢ ∀ (y₁ : A ⟶ (mid K L n₁).X n₀),
    x₁ ≫ (ι f n₁).f n₁ = y₁ ≫ (mid K L n₁).d n₀ n₁ → ∃ A' π, ∃ (_ : Epi π), ∃ x₁_1, π ≫ x₁ = x₁_1 ≫ K.d n₀ n₁
Module: Mathlib.AlgebraicTopology.ModelCategory.FibrantObjectHomotopy

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : ModelCategory C
X Y : C
inst✝ : IsFibrant Y
f g : resolutionObj X ⟶ Y
h : RightHomotopyRel (iResolutionObj X ≫ f) (iResolutionObj X ≫ g)
⊢ homRel C (homMk f) (homMk g)

Tactic:
homRel_iff_leftHomotopyRel

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : ModelCategory C
X Y : C
inst✝ : IsFibrant Y
f g : resolutionObj X ⟶ Y
h : RightHomotopyRel (iResolutionObj X ≫ f) (iResolutionObj X ≫ g)
⊢ LeftHomotopyRel (homMk f).hom (homMk g).hom
Module: Mathlib.CategoryTheory.GuitartExact.Quotient

Goal state before:
pp: C₀ : Type u_1
C : Type u_2
H₀ : Type u_3
H : Type u_4
inst✝⁶ : Category.{v_1, u_1} C₀
inst✝⁵ : Category.{v_2, u_2} C
inst✝⁴ : Category.{v_3, u_3} H₀
inst✝³ : Category.{v_4, u_4} H
T : C₀ ⥤ H₀
L : C₀ ⥤ C
R : H₀ ⥤ H
B : C ⥤ H
inst✝² : T.EssSurj
inst✝¹ : T.Full
inst✝ : B.Full
e : T ⋙ R ≅ L ⋙ B
he :
  ∀ ⦃X₀ : C₀⦄ ⦃Y : C⦄ (f₀ f₁ : L.obj X₀ ⟶ Y),
    B.map f₀ = B.map f₁ → ∃ P, T.map P.i₀ = T.map P.i₁ ∧ Nonempty ((P.map L).LeftHomotopy f₀ f₁)
Y₀ : H₀
⊢ ∀ {X₃ : C} (g : R.obj Y₀ ⟶ B.obj X₃), IsConnected (CostructuredArrowDownwards e.hom g)

Tactic:
X

Goal state after:
C₀ : Type u_1
C : Type u_2
H₀ : Type u_3
H : Type u_4
inst✝⁶ : Category.{v_1, u_1} C₀
inst✝⁵ : Category.{v_2, u_2} C
inst✝⁴ : Category.{v_3, u_3} H₀
inst✝³ : Category.{v_4, u_4} H
T : C₀ ⥤ H₀
L : C₀ ⥤ C
R : H₀ ⥤ H
B : C ⥤ H
inst✝² : T.EssSurj
inst✝¹ : T.Full
inst✝ : B.Full
e : T ⋙ R ≅ L ⋙ B
he :
  ∀ ⦃X₀ : C₀⦄ ⦃Y : C⦄ (f₀ f₁ : L.obj X₀ ⟶ Y),
    B.map f₀ = B.map f₁ → ∃ P, T.map P.i₀ = T.map P.i₁ ∧ Nonempty ((P.map L).LeftHomotopy f₀ f₁)
Y₀ : H₀
X : C
⊢ ∀ (g : R.obj Y₀ ⟶ B.obj X), IsConnected (CostructuredArrowDownwards e.hom g)
Module: Mathlib.Algebra.Homology.DerivedCategory.DerivabilityStructureInjectives

Goal state before:
pp: case refine_1
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)
X✝ : CochainComplex.Plus C
⊢ ∀ ⦃Y : CochainComplex.Plus C⦄ (f : X✝ ⟶ Y), CochainComplex.Plus.quasiIso C f → IsIso ((quotient C ⋙ L).map f)

Tactic:
_

Goal state after:
case refine_1
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)
X✝ Y✝ : CochainComplex.Plus C
⊢ ∀ (f : X✝ ⟶ Y✝), CochainComplex.Plus.quasiIso C f → IsIso ((quotient C ⋙ L).map f)
Module: Mathlib.Algebra.Homology.DerivedCategory.DerivabilityStructureInjectives

Goal state before:
pp: case refine_2.refine_1
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
K₁ : CochainComplex C ℤ
n₁ : ℤ
hn₁ : K₁.IsStrictlyGE n₁
K₂ : CochainComplex (InjectiveObject C) ℤ
n₂ : ℤ
hn₂ : K₂.IsStrictlyGE n₂
f₀ f₁ : K₁ ⟶ ((InjectiveObject.ι C).mapHomologicalComplex (ComplexShape.up ℤ)).obj K₂
hf : (R C).functor.map (ObjectProperty.homMk f₀) = (R C).functor.map (ObjectProperty.homMk f₁)
⊢ (pathObject.lift f₀ f₁ (homotopyOfEq f₀ f₁ ⋯) ≫
        (pathObject.mapHomologicalComplexObjIso K₂ (InjectiveObject.ι C) ⋯).inv) ≫
      (((Plus.prepathObject { obj := K₂, property := ⋯ }).map (CochainComplex.Plus.localizerMorphism C).functor).map
          (CochainComplex.plus C).ι).p₀ =
    (ObjectProperty.homMk f₀).hom

Tactic:
dsimp [Functor.mapCochainComplexPlus]
        cat_disch

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

Goal state before:
pp: case refine_2.refine_2
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
K₁ : CochainComplex C ℤ
n₁ : ℤ
hn₁ : K₁.IsStrictlyGE n₁
K₂ : CochainComplex (InjectiveObject C) ℤ
n₂ : ℤ
hn₂ : K₂.IsStrictlyGE n₂
f₀ f₁ : K₁ ⟶ ((InjectiveObject.ι C).mapHomologicalComplex (ComplexShape.up ℤ)).obj K₂
hf : (R C).functor.map (ObjectProperty.homMk f₀) = (R C).functor.map (ObjectProperty.homMk f₁)
⊢ (pathObject.lift f₀ f₁ (homotopyOfEq f₀ f₁ ⋯) ≫
        (pathObject.mapHomologicalComplexObjIso K₂ (InjectiveObject.ι C) ⋯).inv) ≫
      (((Plus.prepathObject { obj := K₂, property := ⋯ }).map (CochainComplex.Plus.localizerMorphism C).functor).map
          (CochainComplex.plus C).ι).p₁ =
    (ObjectProperty.homMk f₁).hom

Tactic:
dsimp [Functor.mapCochainComplexPlus]
        cat_disch

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

Goal state before:
pp: 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 i₁ : ι
hi₀₁ : c.Rel i₀ i₁
k k' : ι
hk : k = i₀ ∧ k' = i₁ ∧ i₀ ≠ i₁
⊢ X₁ = if k' = i₀ then X₀ else if k' = i₁ then X₁ else 0

Tactic:
rw [if_neg, if_pos hk.2.1]

Goal state after:
case hnc
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 i₁ : ι
hi₀₁ : c.Rel i₀ i₁
k k' : ι
hk : k = i₀ ∧ k' = i₁ ∧ i₀ ≠ i₁
⊢ ¬k' = i₀