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

Goal state before:
pp: case pos
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₃ : Type u_5
C₄ : Type u_6
inst✝²⁰ : Category.{v_1, u_1} C₁
inst✝¹⁹ : Category.{v_2, u_2} C₂
inst✝¹⁸ : Category.{v_3, u_5} C₃
inst✝¹⁷ : Category.{v_4, u_6} C₄
inst✝¹⁶ : Category.{v_5, u_3} C₁₂
inst✝¹⁵ : HasZeroMorphisms C₁
inst✝¹⁴ : HasZeroMorphisms C₂
inst✝¹³ : HasZeroMorphisms C₃
inst✝¹² : Preadditive C₁₂
inst✝¹¹ : Preadditive C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
inst✝¹⁰ : F₁₂.PreservesZeroMorphisms
inst✝⁹ : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst✝⁸ : G.Additive
inst✝⁷ : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₄ : Type u_12
inst✝⁶ : 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
c₄ : ComplexShape ι₄
inst✝⁵ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁴ : TotalComplexShape c₁₂ c₃ c₄
inst✝³ : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst✝² : DecidableEq ι₁₂
inst✝¹ : (K₁.mapBifunctor K₂ F₁₂ c₁₂).HasMapBifunctor K₃ G c₄
j j' : ι₄
inst✝ : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
i₁₂ : ι₁₂ := c₁.π c₂ c₁₂ (i₁, i₂)
h₁ : c₁₂.Rel i₁₂ (c₁₂.next i₁₂)
h₂ : ¬c₁₂.π c₃ c₄ (c₁₂.next i₁₂, i₃) = j'
h₃ : c₁.Rel i₁ (c₁.next i₁)
⊢ 0 = d₁ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j'

Tactic:
rw [d₁_eq _ _ _ _ _ _ _ h₃, ιOrZero_eq_zero, comp_zero, smul_zero]

Goal state after:
case pos.h
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₃ : Type u_5
C₄ : Type u_6
inst✝²⁰ : Category.{v_1, u_1} C₁
inst✝¹⁹ : Category.{v_2, u_2} C₂
inst✝¹⁸ : Category.{v_3, u_5} C₃
inst✝¹⁷ : Category.{v_4, u_6} C₄
inst✝¹⁶ : Category.{v_5, u_3} C₁₂
inst✝¹⁵ : HasZeroMorphisms C₁
inst✝¹⁴ : HasZeroMorphisms C₂
inst✝¹³ : HasZeroMorphisms C₃
inst✝¹² : Preadditive C₁₂
inst✝¹¹ : Preadditive C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
inst✝¹⁰ : F₁₂.PreservesZeroMorphisms
inst✝⁹ : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst✝⁸ : G.Additive
inst✝⁷ : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₄ : Type u_12
inst✝⁶ : 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
c₄ : ComplexShape ι₄
inst✝⁵ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁴ : TotalComplexShape c₁₂ c₃ c₄
inst✝³ : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst✝² : DecidableEq ι₁₂
inst✝¹ : (K₁.mapBifunctor K₂ F₁₂ c₁₂).HasMapBifunctor K₃ G c₄
j j' : ι₄
inst✝ : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
i₁₂ : ι₁₂ := c₁.π c₂ c₁₂ (i₁, i₂)
h₁ : c₁₂.Rel i₁₂ (c₁₂.next i₁₂)
h₂ : ¬c₁₂.π c₃ c₄ (c₁₂.next i₁₂, i₃) = j'
h₃ : c₁.Rel i₁ (c₁.next i₁)
⊢ c₁.r c₂ c₃ c₁₂ c₄ (c₁.next i₁, i₂, i₃) ≠ j'
Module: Mathlib.Algebra.Homology.CommSq

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
X₁ X₂ X₃ X₄ : C
inst✝ : HasBinaryBiproduct X₂ X₃
fst : X₁ ⟶ X₂
snd : X₁ ⟶ X₃
f : X₂ ⟶ X₄
g : X₃ ⟶ X₄
h : IsPullback fst snd f g
⊢ Mono ⋯.shortComplex'.f

Tactic:
by
  rw [Preadditive.mono_iff_cancel_zero]
  intro _ b hb
  exact Fork.IsLimit.hom_ext h.isLimitKernelFork (by simpa using hb)

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
P : ObjectProperty C
X✝ Y✝ : C
e : X✝ ≅ Y✝
h : P.rightOrthogonal X✝
X : C
f : X ⟶ Y✝
hX : P X
⊢ f = 0

Tactic:
rw [← cancel_mono e.inv, zero_comp]
    exact h _ hX

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

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝²² : Category.{v_1, u_1} C₁
inst✝²¹ : Category.{v_2, u_2} C₂
inst✝²⁰ : Category.{v_3, u_5} C₃
inst✝¹⁹ : Category.{v_4, u_6} C₄
inst✝¹⁸ : Category.{v_6, u_4} C₂₃
inst✝¹⁷ : HasZeroMorphisms C₁
inst✝¹⁶ : HasZeroMorphisms C₂
inst✝¹⁵ : HasZeroMorphisms C₃
inst✝¹⁴ : Preadditive C₂₃
inst✝¹³ : Preadditive C₄
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
inst✝¹² : G₂₃.PreservesZeroMorphisms
inst✝¹¹ : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst✝¹⁰ : F.PreservesZeroMorphisms
inst✝⁹ : ∀ (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✝⁸ : 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✝⁷ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁶ : TotalComplexShape c₁₂ c₃ c₄
inst✝⁵ : TotalComplexShape c₂ c₃ c₂₃
inst✝⁴ : TotalComplexShape c₁ c₂₃ c₄
inst✝³ : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst✝² : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst✝¹ : DecidableEq ι₂₃
inst✝ : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
j j' : ι₄
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
⊢ c₁.π c₂₃ c₄ (i₁, c₂.π c₃ c₂₃ (i₂, i₃)) = j

Tactic:
by rw [← h, ← ComplexShape.assoc c₁ c₂ c₃ c₁₂ c₂₃ c₄]; rfl

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

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✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroObject C
inst✝¹ : Preadditive C
K L : HomologicalComplex C c
f g : K ⟶ L
h : Homotopy f g
e : c.Embedding c'
inst✝ : e.IsRelIff
i : ι
hi : ¬c.Rel i (c.next i)
hi' : ¬c'.Rel (e.f i) (c'.next (e.f i))
⊢ (K.extendXIso e ⋯).hom ≫ 0 ≫ (L.extendXIso e ⋯).inv = (dNext (e.f i)) (extend.hom e h.hom)

Tactic:
simp [dNext_eq_zero _ _ hi']

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
K L : CochainComplex C ℤ
inst✝¹ : K.IsKInjective
inst✝ : L.IsKInjective
f : K ⟶ L
x✝ : QuasiIso f
⊢ IsIso ((HomotopyCategory.quotient C (ComplexShape.up ℤ)).map f)

Tactic:
obtain ⟨g, hg⟩ := (Qh_map_bijective _ _).surjective
    ((quotientCompQhIso C).hom.app L ≫ inv (Q.map f) ≫ (quotientCompQhIso C).inv.app K)

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
K L : CochainComplex C ℤ
inst✝¹ : K.IsKInjective
inst✝ : L.IsKInjective
f : K ⟶ L
x✝ : QuasiIso f
g : (HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj L ⟶ (HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj K
hg : Qh.map g = (quotientCompQhIso C).hom.app L ≫ inv (Q.map f) ≫ (quotientCompQhIso C).inv.app K
⊢ IsIso ((HomotopyCategory.quotient C (ComplexShape.up ℤ)).map f)
Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy

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✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroObject C
inst✝¹ : Preadditive C
K L : HomologicalComplex C c
f g : K ⟶ L
e : c.Embedding c'
inst✝ : e.IsRelIff
h : Homotopy (extendMap f e) (extendMap g e)
i : ι
this :
  (K.extendXIso e ⋯).hom ≫ f.f i ≫ (L.extendXIso e ⋯).inv =
    (dNext (e.f i)) h.hom + (prevD (e.f i)) h.hom + (K.extendXIso e ⋯).hom ≫ g.f i ≫ (L.extendXIso e ⋯).inv
hi : ¬c.Rel i (c.next i)
hi' : ¬c'.Rel (e.f i) (c'.next (e.f i))
⊢ (dNext (e.f i)) h.hom = (K.extendXIso e ⋯).hom ≫ 0 ≫ (L.extendXIso e ⋯).inv

Tactic:
simp [dNext_eq_zero _ _ hi']

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

Goal state before:
pp: 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
⊢ c₁.π c₂₃ c₄ (i₁, c₂.π c₃ c₂₃ (i₂, i₃)) = j

Tactic:
by rw [← h, ← ComplexShape.assoc c₁ c₂ c₃ c₁₂ c₂₃ c₄]; rfl

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.KInjective

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
n : ℤ
z : Cocycle K L n
inst✝ : L.IsKInjective
hK : HomologicalComplex.Acyclic K
m : ℤ
hm : m + 1 = n
φ : K ⟶ (CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj L
hφ : Cochain.ofHom φ = (↑z).rightShift n 0 ⋯
⊢ ∃ α, δ m n α = ↑z

Tactic:
obtain ⟨h⟩ := IsKInjective.nonempty_homotopy_zero φ hK

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
n : ℤ
z : Cocycle K L n
inst✝ : L.IsKInjective
hK : HomologicalComplex.Acyclic K
m : ℤ
hm : m + 1 = n
φ : K ⟶ (CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj L
hφ : Cochain.ofHom φ = (↑z).rightShift n 0 ⋯
h : Homotopy φ 0
⊢ ∃ α, δ m n α = ↑z
Module: Mathlib.AlgebraicTopology.ModelCategory.Instances

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : CategoryWithWeakEquivalences C
inst✝² : CategoryWithCofibrations C
inst✝¹ : CategoryWithFibrations C
inst✝ : (cofibrations C).IsWeakFactorizationSystem (trivialFibrations C)
⊢ (trivialFibrations C).IsStableUnderBaseChange

Tactic:
by
  rw [← cofibrations_rlp]
  infer_instance

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : CategoryWithWeakEquivalences C
inst✝² : CategoryWithCofibrations C
inst✝¹ : CategoryWithFibrations C
inst✝ : (cofibrations C).IsWeakFactorizationSystem (trivialFibrations C)
⊢ (trivialFibrations C).IsMultiplicative

Tactic:
by
  rw [← cofibrations_rlp]
  infer_instance

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : CategoryWithWeakEquivalences C
inst✝¹ : CategoryWithCofibrations C
inst✝ : CategoryWithFibrations C
X Y : C
f : X ⟶ Y
h : (trivialCofibrations C).MapFactorizationData (fibrations C) f
⊢ Fibration h.p

Tactic:
by
  simpa only [fibration_iff] using h.hp

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

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
A : C
inst✝⁴ : CategoryWithWeakEquivalences C
P : PathObject A
inst✝³ : HasBinaryProducts C
inst✝² : CategoryWithFibrations C
inst✝¹ : P.IsGood
inst✝ : (fibrations C).RespectsIso
hp : fibrations C P.p
⊢ Fibration P.symm.p

Tactic:
P.symm_p,

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
A : C
inst✝⁴ : CategoryWithWeakEquivalences C
P : PathObject A
inst✝³ : HasBinaryProducts C
inst✝² : CategoryWithFibrations C
inst✝¹ : P.IsGood
inst✝ : (fibrations C).RespectsIso
hp : fibrations C P.p
⊢ Fibration (P.p ≫ (prod.braiding A A).hom)
Module: Mathlib.CategoryTheory.Localization.OfQuotient

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
L : C ⥤ D
inst✝¹ : L.EssSurj
inst✝ : L.Full
W : MorphismProperty C
hW : W.IsInvertedBy L
hr : ∀ ⦃X Y : C⦄ (f₀ f₁ : X ⟶ Y), L.map f₀ = L.map f₁ → ∃ P x, W P.π
F : Quotient L.homRel ⥤ D := Quotient.lift L.homRel L ⋯
hW' : W.IsInvertedBy (Quotient.functor L.homRel)
this : (Quotient.functor L.homRel).IsLocalization W
⊢ L.IsLocalization W

Tactic:
exact IsLocalization.of_equivalence_target (Quotient.functor L.homRel) W L
    F.asEquivalence (Iso.refl _)

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
K : CochainComplex C ℤ
n : ℤ
hn : K.IsStrictlyGE n
i : ℤ
hi : autoParam (i < n - 1) isStrictlyGE_iff._auto_1
⊢ IsZero (homotopyCofiber.X (biprod.lift (𝟙 K) (-𝟙 K)) i)

Tactic:
refine homotopyCofiber.isZero_X _ _ ?_ (fun j hj ↦ ?_)

Goal state after:
case refine_1
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
K : CochainComplex C ℤ
n : ℤ
hn : K.IsStrictlyGE n
i : ℤ
hi : autoParam (i < n - 1) isStrictlyGE_iff._auto_1
⊢ IsZero ((K ⊞ K).X i)

case refine_2
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
K : CochainComplex C ℤ
n : ℤ
hn : K.IsStrictlyGE n
i : ℤ
hi : autoParam (i < n - 1) isStrictlyGE_iff._auto_1
j : ℤ
hj : (ComplexShape.up ℤ).Rel i j
⊢ IsZero (K.X j)
Module: Mathlib.Algebra.Homology.HomotopyCategory.Plus

Goal state before:
pp: case refine_1
C : Type u_1
D : Type u_2
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
inst✝⁵ : Preadditive C
inst✝⁴ : Preadditive D
A : Type u_3
inst✝³ : Category.{v_3, u_3} A
inst✝² : Abelian A
inst✝¹ : HasZeroObject C
inst✝ : HasBinaryBiproducts C
T : Triangle (HomotopyCategory C (ComplexShape.up ℤ))
hT : T ∈ distinguishedTriangles
h₁ : plus C T.obj₁
h₂ : plus C T.obj₂
n₁ : ℤ
h✝¹ : CochainComplex.IsStrictlyGE T.obj₁.as n₁
n₂ : ℤ
h✝ : CochainComplex.IsStrictlyGE T.obj₂.as n₂
f : T.obj₁.as ⟶ T.obj₂.as
hf : (quotient C (ComplexShape.up ℤ)).map f = T.mor₁
⊢ plus C (Triangle.π₃.obj (CochainComplex.mappingCone.triangleh f))

pp: case refine_2
C : Type u_1
D : Type u_2
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
inst✝⁵ : Preadditive C
inst✝⁴ : Preadditive D
A : Type u_3
inst✝³ : Category.{v_3, u_3} A
inst✝² : Abelian A
inst✝¹ : HasZeroObject C
inst✝ : HasBinaryBiproducts C
T : Triangle (HomotopyCategory C (ComplexShape.up ℤ))
hT : T ∈ distinguishedTriangles
h₁ : plus C T.obj₁
h₂ : plus C T.obj₂
n₁ : ℤ
h✝¹ : CochainComplex.IsStrictlyGE T.obj₁.as n₁
n₂ : ℤ
h✝ : CochainComplex.IsStrictlyGE T.obj₂.as n₂
f : T.obj₁.as ⟶ T.obj₂.as
hf : (quotient C (ComplexShape.up ℤ)).map f = T.mor₁
⊢ T.mor₁ ≫ (Iso.refl T.obj₂).hom = (Iso.refl T.obj₁).hom ≫ (CochainComplex.mappingCone.triangleh f).mor₁

Tactic:
· dsimp
      simp only [plus_quotient_obj_iff]
      exact ⟨min (n₁ - 1) n₂, CochainComplex.isStrictlyGE_mappingCone f n₁ n₂ _
        (by simp) (by simp)⟩

Goal state after:
case refine_2
C : Type u_1
D : Type u_2
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
inst✝⁵ : Preadditive C
inst✝⁴ : Preadditive D
A : Type u_3
inst✝³ : Category.{v_3, u_3} A
inst✝² : Abelian A
inst✝¹ : HasZeroObject C
inst✝ : HasBinaryBiproducts C
T : Triangle (HomotopyCategory C (ComplexShape.up ℤ))
hT : T ∈ distinguishedTriangles
h₁ : plus C T.obj₁
h₂ : plus C T.obj₂
n₁ : ℤ
h✝¹ : CochainComplex.IsStrictlyGE T.obj₁.as n₁
n₂ : ℤ
h✝ : CochainComplex.IsStrictlyGE T.obj₂.as n₂
f : T.obj₁.as ⟶ T.obj₂.as
hf : (quotient C (ComplexShape.up ℤ)).map f = T.mor₁
⊢ T.mor₁ ≫ (Iso.refl T.obj₂).hom = (Iso.refl T.obj₁).hom ≫ (CochainComplex.mappingCone.triangleh f).mor₁
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 : C
⊢ t.IsGE X n ↔ IsZero ((t.truncLT n).obj X)

Tactic:
rw [t.isGE_iff_isIso_truncGEπ_app n X]

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 : C
⊢ IsIso ((t.truncGEπ n).app X) ↔ IsZero ((t.truncLT n).obj X)
Module: Mathlib.CategoryTheory.Functor.OfSequence

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

Tactic:
obtain ⟨k, hk⟩ := Nat.exists_eq_add_of_le (leOfHom φ)

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
F G : ℕ ⥤ C
app : (n : ℕ) → F.obj n ⟶ G.obj n
naturality : ∀ (n : ℕ), F.map (homOfLE ⋯) ≫ app (n + 1) = app n ≫ G.map (homOfLE ⋯)
i j : ℕ
φ : i ⟶ j
k : ℕ
hk : j = i + k
⊢ F.map φ ≫ app j = app i ≫ G.map φ
Module: Mathlib.Algebra.Homology.ModelCategory.Injective

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Abelian C
X Y : Plus C
p : X ⟶ Y
inst✝¹ : Fibration p
A : CochainComplex C ℤ
hA : CochainComplex.plus C A
t : { obj := A, property := hA } ⟶ X
B : CochainComplex C ℤ
hB : CochainComplex.plus C B
b : { obj := B, property := hB } ⟶ Y
i : { obj := A, property := hA } ⟶ { obj := B, property := hB }
inst✝ : Mono i
hip : WeakEquivalence i ∨ WeakEquivalence p
sq : CommSq t i p b
⊢ sq.HasLift

Tactic:
obtain ⟨X, hX⟩ := X

Goal state after:
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Abelian C
Y : Plus C
A : CochainComplex C ℤ
hA : CochainComplex.plus C A
B : CochainComplex C ℤ
hB : CochainComplex.plus C B
b : { obj := B, property := hB } ⟶ Y
i : { obj := A, property := hA } ⟶ { obj := B, property := hB }
inst✝¹ : Mono i
X : CochainComplex C ℤ
hX : CochainComplex.plus C X
p : { obj := X, property := hX } ⟶ Y
inst✝ : Fibration p
t : { obj := A, property := hA } ⟶ { obj := X, property := hX }
hip : WeakEquivalence i ∨ WeakEquivalence p
sq : CommSq t i p b
⊢ sq.HasLift
Module: Mathlib.CategoryTheory.GuitartExact.VerticalComposition

Goal state before:
pp: case mp
C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
D₁ : Type u_4
D₂ : Type u_5
D₃ : Type u_6
inst✝⁵ : Category.{v_1, u_1} C₁
inst✝⁴ : Category.{v_2, u_2} C₂
inst✝³ : Category.{v_3, u_3} C₃
inst✝² : Category.{v_4, u_4} D₁
inst✝¹ : Category.{v_5, u_5} D₂
inst✝ : Category.{v_6, u_6} D₃
H₁ : C₁ ⥤ D₁
L₁ : C₁ ⥤ C₂
R₁ : D₁ ⥤ D₂
H₂ : C₂ ⥤ D₂
w : TwoSquare H₁ L₁ R₁ H₂
H₃ : C₃ ⥤ D₃
eL : C₂ ≌ C₃
eR : D₂ ≌ D₃
w' : H₂ ⋙ eR.functor ≅ eL.functor ⋙ H₃
⊢ (w ≫ᵥ w'.hom).GuitartExact → w.GuitartExact

Tactic:
intro hww'

Goal state after:
case mp
C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
D₁ : Type u_4
D₂ : Type u_5
D₃ : Type u_6
inst✝⁵ : Category.{v_1, u_1} C₁
inst✝⁴ : Category.{v_2, u_2} C₂
inst✝³ : Category.{v_3, u_3} C₃
inst✝² : Category.{v_4, u_4} D₁
inst✝¹ : Category.{v_5, u_5} D₂
inst✝ : Category.{v_6, u_6} D₃
H₁ : C₁ ⥤ D₁
L₁ : C₁ ⥤ C₂
R₁ : D₁ ⥤ D₂
H₂ : C₂ ⥤ D₂
w : TwoSquare H₁ L₁ R₁ H₂
H₃ : C₃ ⥤ D₃
eL : C₂ ≌ C₃
eR : D₂ ≌ D₃
w' : H₂ ⋙ eR.functor ≅ eL.functor ⋙ H₃
hww' : (w ≫ᵥ w'.hom).GuitartExact
⊢ w.GuitartExact
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
R R' : Φ.RightResolution X₂
φ : R ⟶ R'
⊢ (L.map (Φ.functor.map φ.f) ≫ (isoOfHom L W₂ R'.w ⋯).inv) ≫ y = (isoOfHom L W₂ R.w ⋯).inv ≫ y

Tactic:
← cancel_epi (isoOfHom L W₂ R.w R.hw).hom,

Goal state after:
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
R R' : Φ.RightResolution X₂
φ : R ⟶ R'
⊢ (isoOfHom L W₂ R.w ⋯).hom ≫ (L.map (Φ.functor.map φ.f) ≫ (isoOfHom L W₂ R'.w ⋯).inv) ≫ y =
    (isoOfHom L W₂ R.w ⋯).hom ≫ (isoOfHom L W₂ R.w ⋯).inv ≫ y
Module: Mathlib.CategoryTheory.GuitartExact.Basic

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
X₂ X₂' : C₂
X₃ : C₃
g : R.obj X₂ ⟶ B.obj X₃
g' : R.obj X₂' ⟶ B.obj X₃
γ : X₂' ⟶ X₂
hγ : R.map γ ≫ g = g'
A : w.CostructuredArrowDownwards g
⊢ R.map (γ ≫ (CostructuredArrow.left A).hom) ≫
      w.app (CostructuredArrow.left A).right ≫ B.map (StructuredArrow.Hom.right (CostructuredArrow.hom A)) =
    g'

Tactic:
simpa [← hγ] using R.map γ ≫= StructuredArrow.w A.hom

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
n : ℤ
inst✝ : K.IsStrictlyGE n
this : HasDerivedCategory C
⊢ ∃ L i, ∃ (_ : QuasiIso i) (_ : ∀ (n : ℤ), Injective (L.X n)), L.IsStrictlyGE n

Tactic:
obtain ⟨L, i, _, _, hL, _⟩ := exists_mono_quasiIso_injective K (n - 1) n (by simp)

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
n : ℤ
inst✝ : K.IsStrictlyGE n
this : HasDerivedCategory C
L : CochainComplex C ℤ
i : K ⟶ L
w✝¹ : Mono i
w✝ : QuasiIso i
hL : ∀ (n : ℤ), Injective (L.X n)
h✝ : L.IsStrictlyGE (n - 1)
⊢ ∃ L i, ∃ (_ : QuasiIso i) (_ : ∀ (n : ℤ), Injective (L.X n)), L.IsStrictlyGE n
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)
j : CostructuredArrow L (L.obj Y)
⊢ ((LeftExtension.mk G e.hom).coconeAt (L.obj Y)).ι.app j ≫ e.inv.app Y ≫ s.ι.app (CostructuredArrow.mk (𝟙 (L.obj Y))) =
    s.ι.app j

Tactic:
refine Localization.induction_costructuredArrow L W _ (by simp)
      (fun X₁ X₂ f φ hφ ↦ ?_) (fun X₁ X₂ w hw φ hφ ↦ ?_) j
    · have eq := s.ι.naturality
        (CostructuredArrow.homMk f : CostructuredArrow.mk (L.map f ≫ φ) ⟶ CostructuredArrow.mk φ)
      dsimp at eq hφ ⊢
      rw [comp_id] at eq
      rw [assoc] at hφ
      rw [assoc, map_comp_assoc, ← eq, ← hφ, NatTrans.naturality_assoc, comp_map]
    · have : IsIso (F.map w) := by
        have := Localization.inverts L W w hw
        rw [← NatIso.naturality_2 e w]
        dsimp
        infer_instance
      have eq := s.ι.naturality
        (CostructuredArrow.homMk w : CostructuredArrow.mk φ ⟶ CostructuredArrow.mk
          ((Localization.isoOfHom L W w hw).inv ≫ φ))
      dsimp at eq hφ ⊢
      rw [comp_id] at eq
      rw [assoc] at hφ
      rw [map_comp, assoc, assoc, ← cancel_epi (F.map w), eq, ← hφ,
        NatTrans.naturality_assoc, comp_map, ← G.map_comp_assoc]
      simp

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
⊢ ∀ (X : FibrantObject (Plus C)),
    CochainComplex.plus (InjectiveObject C)
      ((HomologicalComplex.liftFunctorObjectProperty (isInjective C) (FibrantObject.ι ⋙ ι C) ⋯).obj X)

Tactic:
by
        rintro ⟨⟨_, n, _⟩, _⟩
        refine ⟨n, ?_⟩
        rwa [← isStrictlyGE_mapHomologicalComplex_obj_iff _ (InjectiveObject.ι _)]

Goal state after: