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

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u'
inst✝⁵ : Category.{v', u'} D
P : ObjectProperty C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
h₁ : P.leftOrthogonal T.obj₁
h₃ : P.leftOrthogonal T.obj₃
Y : C
f : T.obj₂ ⟶ Y
hY : P Y
⊢ f = 0

Tactic:
obtain ⟨g, rfl⟩ := Pretriangulated.Triangle.yoneda_exact₂ T hT f (h₁ _ hY)

Goal state after:
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u'
inst✝⁵ : Category.{v', u'} D
P : ObjectProperty C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
h₁ : P.leftOrthogonal T.obj₁
h₃ : P.leftOrthogonal T.obj₃
Y : C
hY : P Y
g : T.obj₃ ⟶ Y
⊢ T.mor₂ ≫ g = 0
Module: Mathlib.CategoryTheory.Triangulated.TStructure.Basic

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} 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
⊢ Monotone t.le

Tactic:
let H := fun (a : ℕ) => ∀ (n : ℤ), t.le n ≤ t.le (n + a)

Goal state after:
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} 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
H : ℕ → Prop := fun a ↦ ∀ (n : ℤ), t.le n ≤ t.le (n + ↑a)
⊢ Monotone t.le
Module: Mathlib.Algebra.Homology.HomotopyCategory.KInjective

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
L : CochainComplex C ℤ
d : ℤ
inst✝¹ : L.IsStrictlyGE d
inst✝ : ∀ (n : ℤ), Injective (L.X n)
K : CochainComplex C ℤ
f : K ⟶ L
hK : HomologicalComplex.Acyclic K
X : ℕ → Set (Cochain K L (-1)) := fun n ↦ {α | (δ (-1) 0 α).EqUpTo (Cochain.ofHom f) (↑n + d - 1)}
x₀ : ↑(X 0) := ⟨0, ⋯⟩
⊢ Nonempty (Homotopy f 0)

Tactic:
let φ (n : ℕ) (α : X n) : X (n + 1) :=
      ⟨_, (isKInjective_of_injective_aux f α.1 (n + d - 1) ((n + 1 : ℕ) + d - 1)
        (by lia) (hK _) α.2).choose_spec⟩

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
L : CochainComplex C ℤ
d : ℤ
inst✝¹ : L.IsStrictlyGE d
inst✝ : ∀ (n : ℤ), Injective (L.X n)
K : CochainComplex C ℤ
f : K ⟶ L
hK : HomologicalComplex.Acyclic K
X : ℕ → Set (Cochain K L (-1)) := fun n ↦ {α | (δ (-1) 0 α).EqUpTo (Cochain.ofHom f) (↑n + d - 1)}
x₀ : ↑(X 0) := ⟨0, ⋯⟩
φ : (n : ℕ) → ↑(X n) → ↑(X (n + 1)) := fun n α ↦ ⟨↑α + Cochain.single ⋯.choose (-1), ⋯⟩
⊢ Nonempty (Homotopy f 0)
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
K : CochainComplex C ℤ
n : ℤ
⊢ (Q.obj K).IsLE n ↔ K.IsLE n

Tactic:
by
  have eq := fun i ↦ ((homologyFunctorFactors C i).app K).isZero_iff
  simp only [Functor.comp_obj, HomologicalComplex.homologyFunctor_obj] at eq
  simp only [isLE_iff, CochainComplex.isLE_iff,
    HomologicalComplex.exactAt_iff_isZero_homology, eq]

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
X Y : C
f : X ⟶ Y
inst✝¹ : CategoryWithCofibrations C
inst✝ : CategoryWithWeakEquivalences C
⊢ trivialCofibrations C f ↔ Cofibration f ∧ WeakEquivalence f

Tactic:
rw [cofibration_iff, weakEquivalence_iff]

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
X Y : C
f : X ⟶ Y
inst✝¹ : CategoryWithCofibrations C
inst✝ : CategoryWithWeakEquivalences C
⊢ trivialCofibrations C f ↔ cofibrations C f ∧ weakEquivalences C f
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'
h₃ : ¬c₂.Rel i₂ (c₂.next i₂)
⊢ 0 = d₂ F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ i₃ j'

Tactic:
rw [d₂_eq_zero _ _ _ _ _ _ _ _ _ _ _ _ h₃]

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
⊢ (Triangle.mk u₂₃ v₂₃ w₂₃).mor₁.unop ≫ (Triangle.mk u₁₂ v₁₂ w₁₂).mor₁.unop = (Triangle.mk u₁₃ v₁₃ w₁₃).mor₁.unop

Tactic:
by aesop

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

Goal state before:
pp: case neg
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
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
h₁ : ¬c₁.Rel i₁ (c₁.next i₁)
⊢ ((associator.hom.app (K₁.X i₁)).app (K₂.X i₂)).app (K₃.X i₃) ≫
      mapBifunctor₂₃.d₁ F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ i₃ j' =
    mapBifunctor₁₂.d₁ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j' ≫
      (mapBifunctorAssociatorX associator K₁ K₂ K₃ c₁₂ c₂₃ c₄ j').hom

Tactic:
· rw [mapBifunctor₁₂.d₁_eq_zero _ _ _ _ _ _ _ _ _ _ _ h₁,
      mapBifunctor₂₃.d₁_eq_zero _ _ _ _ _ _ _ _ _ _ _ _ h₁, comp_zero, zero_comp]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X Y Z : C
g₁ : X ⟶ Y
g₂ : Y ⟶ Z
I₁ : C
w✝¹ : Injective I₁
f₁ : I₁ ⟶ X
w₁ : f₁ ≫ g₁ = 0
σ₁ : { X₁ := I₁, X₂ := X, X₃ := Y, f := f₁, g := g₁, zero := w₁ }.Splitting
I₂ : C
w✝ : Injective I₂
f₂ : I₂ ⟶ Y
w₂ : f₂ ≫ g₂ = 0
σ₂ : { X₁ := I₂, X₂ := Y, X₃ := Z, f := f₂, g := g₂, zero := w₂ }.Splitting
⊢ { X₁ := I₁ ⊞ I₂, X₂ := X, X₃ := Z, f := biprod.fst ≫ f₁ + biprod.snd ≫ f₂ ≫ σ₁.s, g := g₁ ≫ g₂, zero := ⋯ }.f ≫
      (σ₁.r ≫ biprod.inl + g₁ ≫ σ₂.r ≫ biprod.inr) =
    𝟙 { X₁ := I₁ ⊞ I₂, X₂ := X, X₃ := Z, f := biprod.fst ≫ f₁ + biprod.snd ≫ f₂ ≫ σ₁.s, g := g₁ ≫ g₂, zero := ⋯ }.X₁

Tactic:
by
            ext
            · simp [σ₁.f_r]
            · simp [reassoc_of% w₁]
            · simp
            · simp [reassoc_of% σ₁.s_g, σ₂.f_r]

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

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
inst✝⁶ : Category.{v_1, u_1} C₁
inst✝⁵ : Category.{v_2, u_2} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
inst✝⁴ : ∀ (X₂ : C₂), IsConnected (Φ.RightResolution X₂)
inst✝³ : Φ.arrow.HasRightResolutions
inst✝² : W₂.ContainsIdentities
D : Type u_3
inst✝¹ : Category.{v_3, u_3} D
L : C₂ ⥤ D
inst✝ : L.IsLocalization W₂
X₂ : C₂
X₃ : D
y : L.obj X₂ ⟶ X₃
⊢ IsConnected
    ((TwoSquare.mk Φ.functor (Φ.functor ⋙ L) L (𝟭 D) (Φ.functor ⋙ L).rightUnitor.inv).CostructuredArrowDownwards y)

Tactic:
by
  let w := (TwoSquare.mk Φ.functor (Φ.functor ⋙ L) L (𝟭 _) (Functor.rightUnitor _).inv)
  have : Nonempty (w.CostructuredArrowDownwards y) :=
    ⟨(fromRightResolution Φ L y).obj (Classical.arbitrary _)⟩
  suffices ∀ (X : w.CostructuredArrowDownwards y),
      ∃ Y, Zigzag X ((fromRightResolution Φ L y).obj Y) by
    refine zigzag_isConnected (fun X X' => ?_)
    obtain ⟨Y, hX⟩ := this X
    obtain ⟨Y', hX'⟩ := this X'
    exact hX.trans ((zigzag_obj_of_zigzag _ (isPreconnected_zigzag Y Y')).trans hX'.symm)
  intro X
  obtain ⟨c, g, x, fac, rfl⟩ := TwoSquare.CostructuredArrowDownwards.mk_surjective X
  dsimp [w] at x fac
  rw [id_comp] at fac
  let ρ : Φ.arrow.RightResolution (Arrow.mk g) := Classical.arbitrary _
  refine ⟨RightResolution.mk ρ.w.left ρ.hw.1, ?_⟩
  have := zigzag_obj_of_zigzag
    (fromRightResolution Φ L x ⋙ w.costructuredArrowDownwardsPrecomp x y g fac)
      (isPreconnected_zigzag (RightResolution.mk (𝟙 _) (W₂.id_mem _))
        (RightResolution.mk ρ.w.right ρ.hw.2))
  refine Zigzag.trans ?_ (Zigzag.trans this ?_)
  · exact Zigzag.of_hom (eqToHom (by simp))
  · apply Zigzag.of_inv
    refine CostructuredArrow.homMk (StructuredArrow.homMk ρ.X₁.hom (by simp)) ?_
    ext
    simp [← cancel_epi (isoOfHom L W₂ ρ.w.left ρ.hw.1).hom, ← L.map_comp_assoc, fac]

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

Goal state before:
pp: case refine_1
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)
j : CostructuredArrow L (L.obj Y)
X₁ X₂ : C
f : X₁ ⟶ X₂
φ : L.obj X₂ ⟶ L.obj Y
hφ :
  e.hom.app X₂ ≫ G.map φ ≫ e.inv.app Y ≫ s.ι.app (CostructuredArrow.mk (𝟙 (L.obj Y))) = s.ι.app (CostructuredArrow.mk φ)
eq : F.map f ≫ s.ι.app (CostructuredArrow.mk φ) = s.ι.app (CostructuredArrow.mk (L.map f ≫ φ))
⊢ (e.hom.app X₁ ≫ G.map (L.map f ≫ φ)) ≫ e.inv.app Y ≫ s.ι.app (CostructuredArrow.mk (𝟙 (L.obj Y))) =
    s.ι.app (CostructuredArrow.mk (L.map f ≫ φ))

Tactic:
rw [assoc, map_comp_assoc, ← eq, ← hφ, NatTrans.naturality_assoc, comp_map]

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

Goal state before:
pp: case refine_2
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)
j : CostructuredArrow L (L.obj Y)
X₁ X₂ : C
w : X₁ ⟶ X₂
hw : W w
φ : L.obj X₁ ⟶ L.obj Y
hφ :
  ((LeftExtension.mk G e.hom).coconeAt (L.obj Y)).ι.app (CostructuredArrow.mk φ) ≫
      e.inv.app Y ≫ s.ι.app (CostructuredArrow.mk (𝟙 (L.obj Y))) =
    s.ι.app (CostructuredArrow.mk φ)
this : IsIso (F.map w)
⊢ ((LeftExtension.mk G e.hom).coconeAt (L.obj Y)).ι.app
        (CostructuredArrow.mk ((Localization.isoOfHom L W w hw).inv ≫ φ)) ≫
      e.inv.app Y ≫ s.ι.app (CostructuredArrow.mk (𝟙 (L.obj Y))) =
    s.ι.app (CostructuredArrow.mk ((Localization.isoOfHom L W w hw).inv ≫ φ))

Tactic:
have eq := s.ι.naturality
        (CostructuredArrow.homMk w : CostructuredArrow.mk φ ⟶ CostructuredArrow.mk
          ((Localization.isoOfHom L W w hw).inv ≫ φ))

Goal state after:
case refine_2
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)
j : CostructuredArrow L (L.obj Y)
X₁ X₂ : C
w : X₁ ⟶ X₂
hw : W w
φ : L.obj X₁ ⟶ L.obj Y
hφ :
  ((LeftExtension.mk G e.hom).coconeAt (L.obj Y)).ι.app (CostructuredArrow.mk φ) ≫
      e.inv.app Y ≫ s.ι.app (CostructuredArrow.mk (𝟙 (L.obj Y))) =
    s.ι.app (CostructuredArrow.mk φ)
this : IsIso (F.map w)
eq :
  (CostructuredArrow.proj L (L.obj Y) ⋙ F).map (CostructuredArrow.homMk w ⋯) ≫
      s.ι.app (CostructuredArrow.mk ((Localization.isoOfHom L W w hw).inv ≫ φ)) =
    s.ι.app (CostructuredArrow.mk φ) ≫
      ((const (CostructuredArrow L (L.obj Y))).obj s.pt).map (CostructuredArrow.homMk w ⋯)
⊢ ((LeftExtension.mk G e.hom).coconeAt (L.obj Y)).ι.app
        (CostructuredArrow.mk ((Localization.isoOfHom L W w hw).inv ≫ φ)) ≫
      e.inv.app Y ≫ s.ι.app (CostructuredArrow.mk (𝟙 (L.obj Y))) =
    s.ι.app (CostructuredArrow.mk ((Localization.isoOfHom L W w hw).inv ≫ φ))
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✝¹ : B.HasLeftResolutions
F : W₁'.Localization ⥤ W₂'.Localization := B.localizedFunctor W₁'.Q W₂'.Q
e' : B.functor ⋙ W₂'.Q ≅ W₁'.Q ⋙ F := CatCommSq.iso B.functor W₁'.Q W₂'.Q F
iso' : CatCommSq T.functor L.functor R.functor B.functor := { iso := iso }
this✝ : CatCommSq T.functor (L.functor ⋙ W₁'.Q) (R.functor ⋙ W₂'.Q) F :=
  CatCommSq.vComp L.functor W₁'.Q T.functor B.functor F R.functor W₂'.Q
this : (TwoSquare.hComp iso.inv e'.inv).GuitartExact
⊢ B.IsLeftDerivabilityStructure

Tactic:
B.isLeftDerivabilityStructure_iff W₁'.Q W₂'.Q F e'

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✝¹ : B.HasLeftResolutions
F : W₁'.Localization ⥤ W₂'.Localization := B.localizedFunctor W₁'.Q W₂'.Q
e' : B.functor ⋙ W₂'.Q ≅ W₁'.Q ⋙ F := CatCommSq.iso B.functor W₁'.Q W₂'.Q F
iso' : CatCommSq T.functor L.functor R.functor B.functor := { iso := iso }
this✝ : CatCommSq T.functor (L.functor ⋙ W₁'.Q) (R.functor ⋙ W₂'.Q) F :=
  CatCommSq.vComp L.functor W₁'.Q T.functor B.functor F R.functor W₂'.Q
this : (TwoSquare.hComp iso.inv e'.inv).GuitartExact
⊢ TwoSquare.GuitartExact e'.inv
Module: Mathlib.CategoryTheory.Functor.Derived.PointwiseRightDerived

Goal state before:
pp: C : Type u₁
D : Type u₂
H : Type u₃
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : Category.{v₂, u₂} D
inst✝² : Category.{v₃, u₃} H
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
F : C ⥤ H
RF : D ⥤ H
α : F ⟶ L ⋙ RF
hF : W.IsInvertedBy F
inst✝ : RF.IsRightDerivedFunctor α W
⊢ IsIso α

Tactic:
by
  have : α = (Localization.fac F hF L).inv ≫
    whiskerLeft _ (rightDerivedUnique _ _ (Localization.fac F hF L).inv α W).hom := by simp
  rw [this]
  infer_instance

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✝¹⁰ : Category.{v_1, u_1} C₁
inst✝⁹ : Category.{v_2, u_2} C₂
inst✝⁸ : Category.{v_3, u_3} D₁
inst✝⁷ : 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✝⁶ : W₁'.RespectsIso
inst✝⁵ : W₂'.RespectsIso
inst✝⁴ : L.IsInduced
inst✝³ : L.functor.IsEquivalence
inst✝² : R.IsInduced
inst✝¹ : R.functor.IsEquivalence
inst✝ : T.IsLeftDerivabilityStructure
iso : T.functor ⋙ R.functor ≅ L.functor ⋙ B.functor
⊢ B.IsLeftDerivabilityStructure

Tactic:
have := L.isLocalizedEquivalence_of_isInduced

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹⁰ : Category.{v_1, u_1} C₁
inst✝⁹ : Category.{v_2, u_2} C₂
inst✝⁸ : Category.{v_3, u_3} D₁
inst✝⁷ : 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✝⁶ : W₁'.RespectsIso
inst✝⁵ : W₂'.RespectsIso
inst✝⁴ : L.IsInduced
inst✝³ : L.functor.IsEquivalence
inst✝² : R.IsInduced
inst✝¹ : R.functor.IsEquivalence
inst✝ : T.IsLeftDerivabilityStructure
iso : T.functor ⋙ R.functor ≅ L.functor ⋙ B.functor
this : L.IsLocalizedEquivalence
⊢ B.IsLeftDerivabilityStructure
Module: Mathlib.Algebra.Homology.DerivedCategory.KProjective

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
K L : CochainComplex C ℤ
n : ℤ
inst✝¹ : HasSmallLocalizedShiftedHom (HomologicalComplex.quasiIso C (ComplexShape.up ℤ)) ℤ K L
inst✝ : K.IsKProjective
this : HasDerivedCategory C := HasDerivedCategory.standard C
⊢ Function.Bijective
    (⇑(((quotientCompQhIso C).symm.app K).homCongr
          ((Functor.commShiftIso Q n).symm.app L ≪≫
            (quotientCompQhIso C).symm.app ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj L))) ∘
      ⇑(SmallShiftedHom.equiv (HomologicalComplex.quasiIso C (ComplexShape.up ℤ)) Q) ∘ toSmallShiftedHom)

Tactic:
convert! (CochainComplex.IsKProjective.Qh_map_bijective _ _).comp (toHom_bijective K L n)

Goal state after:
case e'_3
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
K L : CochainComplex C ℤ
n : ℤ
inst✝¹ : HasSmallLocalizedShiftedHom (HomologicalComplex.quasiIso C (ComplexShape.up ℤ)) ℤ K L
inst✝ : K.IsKProjective
this : HasDerivedCategory C := ⋯
e_2✝ :
  ((HomotopyCategory.quotient C (ComplexShape.up ℤ) ⋙ Qh).obj K ⟶
      (HomotopyCategory.quotient C (ComplexShape.up ℤ) ⋙ Qh).obj
        ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj L)) =
    (Qh.obj ((HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj K) ⟶
      Qh.obj
        ((HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj
          ((CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) n).obj L)))
⊢ ⇑(((quotientCompQhIso C).symm.app K).homCongr
          ((Functor.commShiftIso Q n).symm.app L ≪≫
            (quotientCompQhIso C).symm.app ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj L))) ∘
      ⇑(SmallShiftedHom.equiv (HomologicalComplex.quasiIso C (ComplexShape.up ℤ)) Q) ∘ toSmallShiftedHom =
    Qh.map ∘ ⇑toHom
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.Map

Goal state before:
pp: C : Type u
inst✝¹⁰ : Category.{v, u} C
inst✝⁹ : Abelian C
D : Type u'
inst✝⁸ : Category.{v', u'} D
inst✝⁷ : Abelian D
F : C ⥤ D
inst✝⁶ : F.Additive
inst✝⁵ : PreservesFiniteLimits F
inst✝⁴ : PreservesFiniteColimits F
inst✝³ : HasDerivedCategory C
inst✝² : HasDerivedCategory D
inst✝¹ : HasExt C
inst✝ : HasExt D
X Y : C
n : ℕ
e : Ext X Y n
⊢ (mapExactFunctor F e).hom =
    (F.mapDerivedCategorySingleFunctor 0).inv.app X ≫
      e.hom.map F.mapDerivedCategory ≫
        (shiftFunctor (DerivedCategory D) ↑n).map ((F.mapDerivedCategorySingleFunctor 0).hom.app Y)

Tactic:
by
  have : (e.mapExactFunctor F).hom = _ :=
    ((F.mapHomologicalComplexUpToQuasiIsoLocalizerMorphism
      (ComplexShape.up ℤ)).equiv_smallShiftedHomMap DerivedCategory.Q DerivedCategory.Q
        ((F.mapCochainComplexSingleFunctor 0).app X) ((F.mapCochainComplexSingleFunctor 0).app Y)
          F.mapDerivedCategory F.mapDerivedCategoryFactors.symm e)
  rw [this, ← ShiftedHom.comp_mk₀ _ 0 rfl, ← ShiftedHom.mk₀_comp 0 rfl]
  congr 2
  · simp [← F.mapDerivedCategorySingleFunctor_inv_app_mapDerivedCategoryFactors_hom_app_assoc,
      CochainComplex.singleFunctor, CochainComplex.singleFunctors]
  · simp [CochainComplex.singleFunctor, CochainComplex.singleFunctors,
      ← Functor.mapDerivedCategoryFactors_inv_app_mapDerivedCategorySingleFunctor_hom_app]

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
i j : ι
hij : ¬c.Rel i j
⊢ ¬(i = i₀ ∧ j = i₁ ∧ i₀ ≠ i₁)

Tactic:
by aesop

Goal state after:
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 p : ℤ
⊢ h.d n m ≫ h.d m p = 0

Tactic:
by
  by_cases hnm : n + 1 = m; swap
  · rw [h.shape n m hnm, zero_comp]
  by_cases hmp : m + 1 = p; swap
  · rw [h.shape m p hmp, comp_zero]
  obtain n | (_ | _ | n) := n
  · obtain rfl : m = .ofNat (n + 1) := by simp [← hnm]
    obtain rfl : p = .ofNat (n + 2) := by simp [← hmp]; lia
    simp only [Int.ofNat_eq_natCast, X_ofNat, d_ofNat, HomologicalComplex.d_comp_d]
  · obtain rfl : m = 0 := by lia
    obtain rfl : p = 1 := by lia
    simp
  · obtain rfl : m = -1 := by lia
    obtain rfl : p = 0 := by lia
    simp
  · obtain rfl : m = .negSucc (n + 1) := by lia
    obtain rfl : p = .negSucc n := by lia
    simp

Goal state after: