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

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
K : HomologicalComplex C c'
L : HomologicalComplex C c
inst✝ : e.IsRelIff
ψ : K ⟶ L.extend e
i : ι
i' : ι'
h : e.f i = i'
⊢ f ψ i = (K.restrictionXIso e h).hom ≫ ψ.f i' ≫ (L.extendXIso e h).hom

Tactic:
subst h

Goal state after:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
K : HomologicalComplex C c'
L : HomologicalComplex C c
inst✝ : e.IsRelIff
ψ : K ⟶ L.extend e
i : ι
⊢ f ψ i = (K.restrictionXIso e ⋯).hom ≫ ψ.f (e.f i) ≫ (L.extendXIso e ⋯).hom
Module: Mathlib.Algebra.Homology.Embedding.TruncGE

Goal state before:
pp: 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
K L : HomologicalComplex C c'
φ : K ⟶ L
e : c.Embedding c'
inst✝² : e.IsTruncGE
inst✝¹ : ∀ (i' : ι'), K.HasHomology i'
inst✝ : ∀ (i' : ι'), L.HasHomology i'
i : ι
hi : e.BoundaryGE i
i' : ι'
h : e.f i = i'
⊢ (truncGE'Map φ e).f i = (K.truncGE'XIsoOpcycles e h hi).hom ≫ opcyclesMap φ i' ≫ (L.truncGE'XIsoOpcycles e h hi).inv

Tactic:
subst h

Goal state after:
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
K L : HomologicalComplex C c'
φ : K ⟶ L
e : c.Embedding c'
inst✝² : e.IsTruncGE
inst✝¹ : ∀ (i' : ι'), K.HasHomology i'
inst✝ : ∀ (i' : ι'), L.HasHomology i'
i : ι
hi : e.BoundaryGE i
⊢ (truncGE'Map φ e).f i =
    (K.truncGE'XIsoOpcycles e ⋯ hi).hom ≫ opcyclesMap φ (e.f i) ≫ (L.truncGE'XIsoOpcycles e ⋯ hi).inv
Module: Mathlib.Algebra.Homology.Embedding.TruncGE

Goal state before:
pp: 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
K L : HomologicalComplex C c'
φ : K ⟶ L
e : c.Embedding c'
inst✝² : e.IsTruncGE
inst✝¹ : ∀ (i' : ι'), K.HasHomology i'
inst✝ : ∀ (i' : ι'), L.HasHomology i'
i : ι
hi : ¬e.BoundaryGE i
i' : ι'
h : e.f i = i'
⊢ (truncGE'Map φ e).f i = (K.truncGE'XIso e h hi).hom ≫ φ.f i' ≫ (L.truncGE'XIso e h hi).inv

Tactic:
subst h

Goal state after:
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
K L : HomologicalComplex C c'
φ : K ⟶ L
e : c.Embedding c'
inst✝² : e.IsTruncGE
inst✝¹ : ∀ (i' : ι'), K.HasHomology i'
inst✝ : ∀ (i' : ι'), L.HasHomology i'
i : ι
hi : ¬e.BoundaryGE i
⊢ (truncGE'Map φ e).f i = (K.truncGE'XIso e ⋯ hi).hom ≫ φ.f (e.f i) ≫ (L.truncGE'XIso e ⋯ hi).inv
Module: Mathlib.Algebra.Homology.Embedding.ExtendHomology

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝² : Category.{v_1, u_3} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
K L M : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
i j k : ι
i' j' k' : ι'
hj' : e.f j = j'
hi : c.prev j = i
hi' : c'.prev j' = i'
hk : c.next j = k
hk' : c'.next j' = k'
h : (K.sc' i j k).RightHomologyData
⊢ h.ι ≫
      (isColimitCokernelCofork K e hj' hi hi' (CokernelCofork.ofπ h.p ⋯) h.hp).desc
        (CokernelCofork.ofπ ((K.extend e).sc' i' j' k').g ⋯) =
    0

Tactic:
dsimp
    rw [← d_comp_desc_eq_zero_iff K e hj' hi hi' hk hk' _ h.hp]
    exact h.wι

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

Goal state before:
pp: case pos
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁶ : Category.{v_1, u_3} C
inst✝⁵ : HasZeroMorphisms C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝⁴ : e.IsTruncGE
inst✝³ : ∀ (i' : ι'), K.HasHomology i'
inst✝² : ∀ (i' : ι'), L.HasHomology i'
inst✝¹ : ∀ (i' : ι'), M.HasHomology i'
inst✝ : K.IsStrictlySupported e
i : ι
hi : e.BoundaryGE i
⊢ IsIso ((K.restrictionToTruncGE' e).f i)

Tactic:
rw [K.restrictionToTruncGE'_f_eq_iso_hom_pOpcycles_iso_inv e rfl hi]
    have : IsIso (K.pOpcycles (e.f i)) := K.isIso_pOpcycles _ _ rfl (by
      obtain ⟨hi₁, hi₂⟩ := hi
      apply IsZero.eq_of_src (K.isZero_X_of_isStrictlySupported e _
        (fun j hj ↦ hi₂ j (by simpa only [hj] using hi₁))))
    infer_instance

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

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✝⁶ : Category.{v_1, u_3} C
inst✝⁵ : HasZeroMorphisms C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝⁴ : e.IsTruncGE
inst✝³ : ∀ (i' : ι'), K.HasHomology i'
inst✝² : ∀ (i' : ι'), L.HasHomology i'
inst✝¹ : ∀ (i' : ι'), M.HasHomology i'
inst✝ : HasZeroObject C
i' : ι'
hi' : ¬∃ i, e.f i = i'
⊢ Epi ((K.πTruncGE e).f i')

Tactic:
apply (isZero_extend_X _ _ _ (by simpa using hi')).epi

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

Goal state before:
pp: 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
K : HomologicalComplex C c'
inst✝² : ∀ (i' : ι'), K.HasHomology i'
inst✝¹ : HasZeroObject C
hK : K.Acyclic
e : c.Embedding c'
inst✝ : e.IsTruncLE
⊢ K.IsSupportedOutside e

Tactic:
exact ⟨fun _ ↦ hK _⟩

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

Goal state before:
pp: ι : Type u_1
ι₁ : Type u_2
ι₂ : Type u_3
c : ComplexShape ι
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
C : Type u_4
inst✝¹ : Category.{v_1, u_4} C
inst✝ : HasZeroMorphisms C
e₁ : c₁.Embedding c
e₂ : c₂.Embedding c
ac : e₁.AreComplementary e₂
X : ι → Type u_5
x₁ : (i₁ : ι₁) → X (e₁.f i₁)
x₂ : (i₂ : ι₂) → X (e₂.f i₂)
i : ι₁ ⊕ ι₂
i₁ : ι₁
h : Sum.inl i₁ = i
⊢ e₁.f i₁ = ac.equiv i

Tactic:
subst h

Goal state after:
ι : Type u_1
ι₁ : Type u_2
ι₂ : Type u_3
c : ComplexShape ι
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
C : Type u_4
inst✝¹ : Category.{v_1, u_4} C
inst✝ : HasZeroMorphisms C
e₁ : c₁.Embedding c
e₂ : c₂.Embedding c
ac : e₁.AreComplementary e₂
X : ι → Type u_5
x₁ : (i₁ : ι₁) → X (e₁.f i₁)
x₂ : (i₂ : ι₂) → X (e₂.f i₂)
i₁ : ι₁
⊢ e₁.f i₁ = ac.equiv (Sum.inl i₁)
Module: Mathlib.Algebra.Homology.Embedding.TruncGEHomology

Goal state before:
pp: 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
K : HomologicalComplex C c'
inst✝² : ∀ (i' : ι'), K.HasHomology i'
inst✝¹ : HasZeroObject C
hK : K.Acyclic
e : c.Embedding c'
inst✝ : e.IsTruncGE
⊢ K.IsSupportedOutside e

Tactic:
exact ⟨fun _ ↦ hK _⟩

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

Goal state before:
pp: ι : Type u_1
ι₁ : Type u_2
ι₂ : Type u_3
c : ComplexShape ι
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
e₁ : c₁.Embedding c
e₂ : c₂.Embedding c
ac : e₁.AreComplementary e₂
X : ι → Type u_5
x₁ : (i₁ : ι₁) → X (e₁.f i₁)
x₂ : (i₂ : ι₂) → X (e₂.f i₂)
i : ι₁ ⊕ ι₂
i₁ : ι₁
h : Sum.inl i₁ = i
⊢ ac.desc' x₁ x₂ i = (desc.aux X (e₁.f i₁) (ac.equiv i) ⋯) (x₁ i₁)

Tactic:
subst h

Goal state after:
ι : Type u_1
ι₁ : Type u_2
ι₂ : Type u_3
c : ComplexShape ι
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
e₁ : c₁.Embedding c
e₂ : c₂.Embedding c
ac : e₁.AreComplementary e₂
X : ι → Type u_5
x₁ : (i₁ : ι₁) → X (e₁.f i₁)
x₂ : (i₂ : ι₂) → X (e₂.f i₂)
i₁ : ι₁
⊢ ac.desc' x₁ x₂ (Sum.inl i₁) = (desc.aux X (e₁.f i₁) (ac.equiv (Sum.inl i₁)) ⋯) (x₁ i₁)
Module: Mathlib.Algebra.Homology.Embedding.AreComplementary

Goal state before:
pp: ι : Type u_1
ι₁ : Type u_2
ι₂ : Type u_3
c : ComplexShape ι
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
C : Type u_4
inst✝¹ : Category.{v_1, u_4} C
inst✝ : HasZeroMorphisms C
e₁ : c₁.Embedding c
e₂ : c₂.Embedding c
ac : e₁.AreComplementary e₂
X : ι → Type u_5
x₁ : (i₁ : ι₁) → X (e₁.f i₁)
x₂ : (i₂ : ι₂) → X (e₂.f i₂)
i : ι₁ ⊕ ι₂
i₂ : ι₂
h : Sum.inr i₂ = i
⊢ e₂.f i₂ = ac.equiv i

Tactic:
subst h

Goal state after:
ι : Type u_1
ι₁ : Type u_2
ι₂ : Type u_3
c : ComplexShape ι
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
C : Type u_4
inst✝¹ : Category.{v_1, u_4} C
inst✝ : HasZeroMorphisms C
e₁ : c₁.Embedding c
e₂ : c₂.Embedding c
ac : e₁.AreComplementary e₂
X : ι → Type u_5
x₁ : (i₁ : ι₁) → X (e₁.f i₁)
x₂ : (i₂ : ι₂) → X (e₂.f i₂)
i₂ : ι₂
⊢ e₂.f i₂ = ac.equiv (Sum.inr i₂)
Module: Mathlib.Algebra.Homology.Embedding.AreComplementary

Goal state before:
pp: ι : Type u_1
ι₁ : Type u_2
ι₂ : Type u_3
c : ComplexShape ι
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
e₁ : c₁.Embedding c
e₂ : c₂.Embedding c
ac : e₁.AreComplementary e₂
X : ι → Type u_5
x₁ : (i₁ : ι₁) → X (e₁.f i₁)
x₂ : (i₂ : ι₂) → X (e₂.f i₂)
i : ι₁ ⊕ ι₂
i₂ : ι₂
h : Sum.inr i₂ = i
⊢ ac.desc' x₁ x₂ i = (desc.aux X (e₂.f i₂) (ac.equiv i) ⋯) (x₂ i₂)

Tactic:
subst h

Goal state after:
ι : Type u_1
ι₁ : Type u_2
ι₂ : Type u_3
c : ComplexShape ι
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
e₁ : c₁.Embedding c
e₂ : c₂.Embedding c
ac : e₁.AreComplementary e₂
X : ι → Type u_5
x₁ : (i₁ : ι₁) → X (e₁.f i₁)
x₂ : (i₂ : ι₂) → X (e₂.f i₂)
i₂ : ι₂
⊢ ac.desc' x₁ x₂ (Sum.inr i₂) = (desc.aux X (e₂.f i₂) (ac.equiv (Sum.inr i₂)) ⋯) (x₂ i₂)
Module: Mathlib.Algebra.Homology.Embedding.TruncLEHomology

Goal state before:
pp: case pos.hf
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝² : Category.{v_1, u_3} C
inst✝¹ : Abelian C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝ : e.IsTruncLE
i : ι
j' : ι'
hj' : c'.Rel (e.f i) j'
⊢ { X₁ := (K.shortComplexTruncLE e).X₂.homology (e.f i), X₂ := (K.shortComplexTruncLE e).X₃.homology (e.f i),
        X₃ := (K.shortComplexTruncLE e).X₁.homology j', f := homologyMap (K.shortComplexTruncLE e).g (e.f i),
        g := ⋯.δ (e.f i) j' hj', zero := ⋯ }.f =
    0

pp: case pos.hg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝² : Category.{v_1, u_3} C
inst✝¹ : Abelian C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝ : e.IsTruncLE
i : ι
j' : ι'
hj' : c'.Rel (e.f i) j'
⊢ { X₁ := (K.shortComplexTruncLE e).X₂.homology (e.f i), X₂ := (K.shortComplexTruncLE e).X₃.homology (e.f i),
        X₃ := (K.shortComplexTruncLE e).X₁.homology j', f := homologyMap (K.shortComplexTruncLE e).g (e.f i),
        g := ⋯.δ (e.f i) j' hj', zero := ⋯ }.g =
    0

Tactic:
· rw [← cancel_epi (homologyMap (K.ιTruncLE e) (e.f i)), comp_zero]
        dsimp [shortComplexTruncLE]
        rw [← homologyMap_comp, cokernel.condition, homologyMap_zero]

Goal state after:
case pos.hg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝² : Category.{v_1, u_3} C
inst✝¹ : Abelian C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝ : e.IsTruncLE
i : ι
j' : ι'
hj' : c'.Rel (e.f i) j'
⊢ { X₁ := (K.shortComplexTruncLE e).X₂.homology (e.f i), X₂ := (K.shortComplexTruncLE e).X₃.homology (e.f i),
        X₃ := (K.shortComplexTruncLE e).X₁.homology j', f := homologyMap (K.shortComplexTruncLE e).g (e.f i),
        g := ⋯.δ (e.f i) j' hj', zero := ⋯ }.g =
    0
Module: Mathlib.Algebra.Homology.SingleHomology

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A : C
⊢ (singleObjOpcyclesSelfIso c j A).hom = (singleObjHomologySelfIso c j A).inv ≫ ((single C c j).obj A).homologyι j

Tactic:
singleObjHomologySelfIso_inv_homologyι

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A : C
⊢ (singleObjOpcyclesSelfIso c j A).hom = (singleObjOpcyclesSelfIso c j A).hom
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic

Goal state before:
pp: case mp
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
h :
  ∀ (X Y : C) (a b : ℤ),
    Small.{w, w'}
      ((shiftFunctor (DerivedCategory C) a).obj (Q.obj ((CochainComplex.singleFunctor C 0).obj X)) ⟶
        (shiftFunctor (DerivedCategory C) b).obj (Q.obj ((CochainComplex.singleFunctor C 0).obj Y)))
X : C
⊢ ∀ (Y : C) (n : ℤ),
    0 ≤ n →
      Small.{w, w'} ((singleFunctor C 0).obj X ⟶ (shiftFunctor (DerivedCategory C) n).obj ((singleFunctor C 0).obj Y))

Tactic:
Y

Goal state after:
case mp
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
h :
  ∀ (X Y : C) (a b : ℤ),
    Small.{w, w'}
      ((shiftFunctor (DerivedCategory C) a).obj (Q.obj ((CochainComplex.singleFunctor C 0).obj X)) ⟶
        (shiftFunctor (DerivedCategory C) b).obj (Q.obj ((CochainComplex.singleFunctor C 0).obj Y)))
X Y : C
⊢ ∀ (n : ℤ),
    0 ≤ n →
      Small.{w, w'} ((singleFunctor C 0).obj X ⟶ (shiftFunctor (DerivedCategory C) n).obj ((singleFunctor C 0).obj Y))
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic

Goal state before:
pp: case mpr
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
h :
  ∀ (X Y : C) (n : ℤ),
    0 ≤ n →
      Small.{w, w'} ((singleFunctor C 0).obj X ⟶ (shiftFunctor (DerivedCategory C) n).obj ((singleFunctor C 0).obj Y))
X : C
⊢ ∀ (Y : C) (a b : ℤ),
    Small.{w, w'}
      ((shiftFunctor (DerivedCategory C) a).obj (Q.obj ((CochainComplex.singleFunctor C 0).obj X)) ⟶
        (shiftFunctor (DerivedCategory C) b).obj (Q.obj ((CochainComplex.singleFunctor C 0).obj Y)))

Tactic:
Y

Goal state after:
case mpr
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
h :
  ∀ (X Y : C) (n : ℤ),
    0 ≤ n →
      Small.{w, w'} ((singleFunctor C 0).obj X ⟶ (shiftFunctor (DerivedCategory C) n).obj ((singleFunctor C 0).obj Y))
X Y : C
⊢ ∀ (a b : ℤ),
    Small.{w, w'}
      ((shiftFunctor (DerivedCategory C) a).obj (Q.obj ((CochainComplex.singleFunctor C 0).obj X)) ⟶
        (shiftFunctor (DerivedCategory C) b).obj (Q.obj ((CochainComplex.singleFunctor C 0).obj Y)))
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic

Goal state before:
pp: case e_a
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasExt C
X Y : C
n : ℕ
inst✝ : HasDerivedCategory C
α β : Ext X Y n
α' : Ext (X ⊞ X) Y n := (mk₀ biprod.fst).comp α ⋯
β' : Ext (X ⊞ X) Y n := (mk₀ biprod.snd).comp β ⋯
eq₁ : α + β = (mk₀ (biprod.lift (𝟙 X) (𝟙 X))).comp (α' + β') ⋯
eq₂ : α' + β' = homEquiv.symm (α'.hom + β'.hom)
⊢ (mk₀ (biprod.lift (𝟙 X) (𝟙 X))).hom.comp α'.hom ⋯ = α.hom

pp: case e_a
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasExt C
X Y : C
n : ℕ
inst✝ : HasDerivedCategory C
α β : Ext X Y n
α' : Ext (X ⊞ X) Y n := (mk₀ biprod.fst).comp α ⋯
β' : Ext (X ⊞ X) Y n := (mk₀ biprod.snd).comp β ⋯
eq₁ : α + β = (mk₀ (biprod.lift (𝟙 X) (𝟙 X))).comp (α' + β') ⋯
eq₂ : α' + β' = homEquiv.symm (α'.hom + β'.hom)
⊢ (mk₀ (biprod.lift (𝟙 X) (𝟙 X))).hom.comp β'.hom ⋯ = β.hom

Tactic:
· dsimp [α']
    rw [comp_hom, mk₀_hom, mk₀_hom]
    dsimp
    rw [ShiftedHom.mk₀_comp_mk₀_assoc, ← Functor.map_comp,
      biprod.lift_fst, Functor.map_id, ShiftedHom.mk₀_id_comp]

Goal state after:
case e_a
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasExt C
X Y : C
n : ℕ
inst✝ : HasDerivedCategory C
α β : Ext X Y n
α' : Ext (X ⊞ X) Y n := (mk₀ biprod.fst).comp α ⋯
β' : Ext (X ⊞ X) Y n := (mk₀ biprod.snd).comp β ⋯
eq₁ : α + β = (mk₀ (biprod.lift (𝟙 X) (𝟙 X))).comp (α' + β') ⋯
eq₂ : α' + β' = homEquiv.symm (α'.hom + β'.hom)
⊢ (mk₀ (biprod.lift (𝟙 X) (𝟙 X))).hom.comp β'.hom ⋯ = β.hom
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasExt C
J : Type u_1
inst✝ : Fintype J
X : J → C
c : Bicone X
hc : c.IsBilimit
Y : C
n : ℕ
x : Ext c.pt Y n
⊢ (∑ i, mk₀ (c.π i ≫ c.ι i)).comp x ⋯ = x

Tactic:
← Ext.mk₀_sum,

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasExt C
J : Type u_1
inst✝ : Fintype J
X : J → C
c : Bicone X
hc : c.IsBilimit
Y : C
n : ℕ
x : Ext c.pt Y n
⊢ (mk₀ (∑ i, c.π i ≫ c.ι i)).comp x ⋯ = x
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic

Goal state before:
pp: case mp
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
this : HasDerivedCategory C := HasDerivedCategory.standard C
h :
  ∀ (X Y : C) (n : ℤ),
    0 ≤ n →
      Small.{w', max u v}
        ((singleFunctor C 0).obj X ⟶ (shiftFunctor (DerivedCategory C) n).obj ((singleFunctor C 0).obj Y))
X : C
⊢ ∀ (Y : C) (n : ℕ), Small.{w', max u v} (ShiftedHom ((singleFunctor C 0).obj X) ((singleFunctor C 0).obj Y) ↑n)

Tactic:
Y

Goal state after:
case mp
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
this : HasDerivedCategory C := HasDerivedCategory.standard C
h :
  ∀ (X Y : C) (n : ℤ),
    0 ≤ n →
      Small.{w', max u v}
        ((singleFunctor C 0).obj X ⟶ (shiftFunctor (DerivedCategory C) n).obj ((singleFunctor C 0).obj Y))
X Y : C
⊢ ∀ (n : ℕ), Small.{w', max u v} (ShiftedHom ((singleFunctor C 0).obj X) ((singleFunctor C 0).obj Y) ↑n)
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic

Goal state before:
pp: case mpr
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
this : HasDerivedCategory C := HasDerivedCategory.standard C
h : ∀ (X Y : C) (n : ℕ), Small.{w', max u v} (ShiftedHom ((singleFunctor C 0).obj X) ((singleFunctor C 0).obj Y) ↑n)
X : C
⊢ ∀ (Y : C) (n : ℤ),
    0 ≤ n →
      Small.{w', max u v}
        ((singleFunctor C 0).obj X ⟶ (shiftFunctor (DerivedCategory C) n).obj ((singleFunctor C 0).obj Y))

Tactic:
Y

Goal state after:
case mpr
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
this : HasDerivedCategory C := HasDerivedCategory.standard C
h : ∀ (X Y : C) (n : ℕ), Small.{w', max u v} (ShiftedHom ((singleFunctor C 0).obj X) ((singleFunctor C 0).obj Y) ↑n)
X Y : C
⊢ ∀ (n : ℤ),
    0 ≤ n →
      Small.{w', max u v}
        ((singleFunctor C 0).obj X ⟶ (shiftFunctor (DerivedCategory C) n).obj ((singleFunctor C 0).obj Y))
Module: Mathlib.CategoryTheory.Shift.Adjunction

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝² : AddMonoid A
inst✝¹ : HasShift C A
inst✝ : HasShift D A
a : A
e₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a
e₂ : shiftFunctor D a ⋙ G ≅ G ⋙ shiftFunctor C a
h : CompatibilityUnit adj e₁ e₂
Y : D
⊢ adj.counit.app ((shiftFunctor D a).obj Y) =
    F.map (e₂.hom.app Y) ≫ e₁.hom.app (G.obj Y) ≫ (shiftFunctor D a).map (adj.counit.app Y)

Tactic:
have eq := h (G.obj Y)

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝² : AddMonoid A
inst✝¹ : HasShift C A
inst✝ : HasShift D A
a : A
e₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a
e₂ : shiftFunctor D a ⋙ G ≅ G ⋙ shiftFunctor C a
h : CompatibilityUnit adj e₁ e₂
Y : D
eq :
  (shiftFunctor C a).map (adj.unit.app (G.obj Y)) =
    adj.unit.app ((shiftFunctor C a).obj (G.obj Y)) ≫ G.map (e₁.hom.app (G.obj Y)) ≫ e₂.hom.app (F.obj (G.obj Y))
⊢ adj.counit.app ((shiftFunctor D a).obj Y) =
    F.map (e₂.hom.app Y) ≫ e₁.hom.app (G.obj Y) ≫ (shiftFunctor D a).map (adj.counit.app Y)
Module: Mathlib.CategoryTheory.Shift.Opposite

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
A : Type u_2
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
X : OppositeShift C A
a b c : A
h : a + b = c
⊢ (shiftFunctorAdd' (OppositeShift C A) a b c h).inv.app X = ((shiftFunctorAdd' C a b c h).hom.app (Opposite.unop X)).op

Tactic:
subst h

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
A : Type u_2
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
X : OppositeShift C A
a b : A
⊢ (shiftFunctorAdd' (OppositeShift C A) a b (a + b) ⋯).inv.app X =
    ((shiftFunctorAdd' C a b (a + b) ⋯).hom.app (Opposite.unop X)).op
Module: Mathlib.CategoryTheory.Shift.Opposite

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
A : Type u_2
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
X : OppositeShift C A
a b c : A
h : a + b = c
⊢ (shiftFunctorAdd' (OppositeShift C A) a b c h).hom.app X = ((shiftFunctorAdd' C a b c h).inv.app (Opposite.unop X)).op

Tactic:
subst h

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
A : Type u_2
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
X : OppositeShift C A
a b : A
⊢ (shiftFunctorAdd' (OppositeShift C A) a b (a + b) ⋯).hom.app X =
    ((shiftFunctorAdd' C a b (a + b) ⋯).inv.app (Opposite.unop X)).op
Module: Mathlib.CategoryTheory.Shift.Pullback

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
A : Type u_2
B : Type u_3
inst✝² : AddMonoid A
inst✝¹ : AddMonoid B
inst✝ : HasShift C B
φ : A →+ B
a : A
b : B
h : b = φ a
⊢ shiftFunctor (PullbackShift C φ) a = shiftFunctor C b

Tactic:
subst h

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
A : Type u_2
B : Type u_3
inst✝² : AddMonoid A
inst✝¹ : AddMonoid B
inst✝ : HasShift C B
φ : A →+ B
a : A
⊢ shiftFunctor (PullbackShift C φ) a = shiftFunctor C (φ a)
Module: Mathlib.CategoryTheory.Shift.Adjunction

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝³ : AddGroup A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
inst✝ : G.CommShift A
a : A
X✝ : C
⊢ (shiftFunctor C a).map (adj.unit.app X✝) =
    (shiftFunctor C a).map (adj.unit.app X✝) ≫
      (shiftFunctorCompIsoId C (-a) a ⋯).inv.app ((shiftFunctor C a).obj (G.obj (F.obj X✝))) ≫
        (shiftFunctor C (-a) ⋙ shiftFunctor C a).map ((Functor.commShiftIso G a).inv.app (F.obj X✝)) ≫
          (shiftFunctorCompIsoId C (-a) a ⋯).hom.app (G.obj ((shiftFunctor D a).obj (F.obj X✝))) ≫
            (Functor.commShiftIso G a).hom.app (F.obj X✝)

Tactic:
(shiftFunctorCompIsoId C _ _ (neg_add_cancel a)).hom.naturality_assoc

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝³ : AddGroup A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
inst✝ : G.CommShift A
a : A
X✝ : C
⊢ (shiftFunctor C a).map (adj.unit.app X✝) =
    (shiftFunctor C a).map (adj.unit.app X✝) ≫
      (shiftFunctorCompIsoId C (-a) a ⋯).inv.app ((shiftFunctor C a).obj (G.obj (F.obj X✝))) ≫
        (shiftFunctorCompIsoId C (-a) a ⋯).hom.app ((shiftFunctor C a).obj (G.obj (F.obj X✝))) ≫
          (𝟭 C).map ((Functor.commShiftIso G a).inv.app (F.obj X✝)) ≫ (Functor.commShiftIso G a).hom.app (F.obj X✝)
Module: Mathlib.CategoryTheory.Localization.Linear

Goal state before:
pp: R : Type w
inst✝⁷ : Ring R
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : Preadditive C
inst✝³ : Preadditive D
L : C ⥤ D
W : MorphismProperty C
inst✝² : L.IsLocalization W
inst✝¹ : L.Additive
inst✝ : Linear R C
this : Linear R D := linear R L W
X : C
⊢ ∀ {Y : C} (f : X ⟶ Y) (r : R), L.map (r • f) = r • L.map f

Tactic:
Y

Goal state after:
R : Type w
inst✝⁷ : Ring R
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : Preadditive C
inst✝³ : Preadditive D
L : C ⥤ D
W : MorphismProperty C
inst✝² : L.IsLocalization W
inst✝¹ : L.Additive
inst✝ : Linear R C
this : Linear R D := linear R L W
X Y : C
⊢ ∀ (f : X ⟶ Y) (r : R), L.map (r • f) = r • L.map f
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.EnoughProjectives

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : LocallySmall.{w, v, u} C
inst✝ : EnoughProjectives C
this✝ : HasDerivedCategory C := HasDerivedCategory.standard C
this : HasExt C
X : C
⊢ ∀ (Y : C) (n : ℕ), Small.{w, max u v} (Ext X Y n)

Tactic:
Y

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : LocallySmall.{w, v, u} C
inst✝ : EnoughProjectives C
this✝ : HasDerivedCategory C := HasDerivedCategory.standard C
this : HasExt C
X Y : C
⊢ ∀ (n : ℕ), Small.{w, max u v} (Ext X Y n)