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

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Abelian C
inst✝³ : HasDerivedCategory C
X Y : CochainComplex C ℤ
a b : ℤ
inst✝² : X.IsStrictlyGE a
inst✝¹ : X.IsStrictlyLE b
inst✝ : Y.IsStrictlyGE a
f : Q.obj X ⟶ Q.obj Y
X' : CochainComplex C ℤ
hX' : X'.IsStrictlyLE b
s : X' ⟶ X
hs✝ : IsIso (Q.map s)
hs : QuasiIso s
g : X' ⟶ Y
fac : f = inv (Q.map s) ≫ Q.map g
⊢ IsIso (Q.map (CochainComplex.truncGEMap s a))

Tactic:
rw [isIso_Q_map_iff_quasiIso, CochainComplex.quasiIso_truncGEMap_iff]

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Abelian C
inst✝³ : HasDerivedCategory C
X Y : CochainComplex C ℤ
a b : ℤ
inst✝² : X.IsStrictlyGE a
inst✝¹ : X.IsStrictlyLE b
inst✝ : Y.IsStrictlyGE a
f : Q.obj X ⟶ Q.obj Y
X' : CochainComplex C ℤ
hX' : X'.IsStrictlyLE b
s : X' ⟶ X
hs✝ : IsIso (Q.map s)
hs : QuasiIso s
g : X' ⟶ Y
fac : f = inv (Q.map s) ≫ Q.map g
⊢ ∀ (i : ℤ), a ≤ i → QuasiIsoAt s i
Module: Mathlib.Algebra.Homology.DerivedCategory.ShortExact

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
S₁ S₂ : ShortComplex (CochainComplex C ℤ)
hS₁ : S₁.ShortExact
hS₂ : S₂.ShortExact
f : S₁ ⟶ S₂
⊢ Q.map (CochainComplex.mappingCone.triangle S₁.f).mor₃ ≫ (shiftFunctor (CochainComplex C ℤ) 1 ⋙ Q).map f.τ₁ =
    ((Q.map (CochainComplex.mappingCone.descShortComplex S₁) ≫ Q.map f.τ₃) ≫
        inv (Q.map (CochainComplex.mappingCone.descShortComplex S₂))) ≫
      Q.map (CochainComplex.mappingCone.triangle S₂.f).mor₃

Tactic:
← Q.map_comp

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
S₁ S₂ : ShortComplex (CochainComplex C ℤ)
hS₁ : S₁.ShortExact
hS₂ : S₂.ShortExact
f : S₁ ⟶ S₂
⊢ Q.map (CochainComplex.mappingCone.triangle S₁.f).mor₃ ≫ (shiftFunctor (CochainComplex C ℤ) 1 ⋙ Q).map f.τ₁ =
    (Q.map (CochainComplex.mappingCone.descShortComplex S₁ ≫ f.τ₃) ≫
        inv (Q.map (CochainComplex.mappingCone.descShortComplex S₂))) ≫
      Q.map (CochainComplex.mappingCone.triangle S₂.f).mor₃
Module: Mathlib.Algebra.Homology.DerivedCategory.ShortExact

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
S₁ S₂ : ShortComplex (CochainComplex C ℤ)
hS₁ : S₁.ShortExact
hS₂ : S₂.ShortExact
f : S₁ ⟶ S₂
⊢ Q.map ((CochainComplex.mappingCone.triangle S₁.f).mor₃ ≫ (shiftFunctor (CochainComplex C ℤ) 1).map f.τ₁) =
    Q.map (CochainComplex.mappingCone.map S₁.f S₂.f f.τ₁ f.τ₂ ⋯) ≫ Q.map (CochainComplex.mappingCone.triangle S₂.f).mor₃

Tactic:
← Q.map_comp

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
S₁ S₂ : ShortComplex (CochainComplex C ℤ)
hS₁ : S₁.ShortExact
hS₂ : S₂.ShortExact
f : S₁ ⟶ S₂
⊢ Q.map ((CochainComplex.mappingCone.triangle S₁.f).mor₃ ≫ (shiftFunctor (CochainComplex C ℤ) 1).map f.τ₁) =
    Q.map (CochainComplex.mappingCone.map S₁.f S₂.f f.τ₁ f.τ₂ ⋯ ≫ (CochainComplex.mappingCone.triangle S₂.f).mor₃)
Module: Mathlib.Algebra.Homology.DerivedCategory.ShortExact

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
S₁ S₂ : ShortComplex (CochainComplex C ℤ)
h₁ : S₁.ShortExact
h₂ : S₂.ShortExact
f : S₁ ⟶ S₂
⊢ Q.map (CochainComplex.mappingCone.triangle S₁.f).mor₃ ≫
      (Functor.commShiftIso Q 1).hom.app S₁.X₁ ≫ (shiftFunctor (DerivedCategory C) 1).map (Q.map f.τ₁) =
    Q.map (CochainComplex.mappingCone.map S₁.f S₂.f f.τ₁ f.τ₂ ⋯ ≫ CochainComplex.mappingCone.descShortComplex S₂) ≫
      inv (Q.map (CochainComplex.mappingCone.descShortComplex S₂)) ≫
        Q.map (CochainComplex.mappingCone.triangle S₂.f).mor₃ ≫ (Functor.commShiftIso Q 1).hom.app S₂.X₁

Tactic:
Functor.map_comp_assoc,

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
S₁ S₂ : ShortComplex (CochainComplex C ℤ)
h₁ : S₁.ShortExact
h₂ : S₂.ShortExact
f : S₁ ⟶ S₂
⊢ Q.map (CochainComplex.mappingCone.triangle S₁.f).mor₃ ≫
      (Functor.commShiftIso Q 1).hom.app S₁.X₁ ≫ (shiftFunctor (DerivedCategory C) 1).map (Q.map f.τ₁) =
    Q.map (CochainComplex.mappingCone.map S₁.f S₂.f f.τ₁ f.τ₂ ⋯) ≫
      Q.map (CochainComplex.mappingCone.descShortComplex S₂) ≫
        inv (Q.map (CochainComplex.mappingCone.descShortComplex S₂)) ≫
          Q.map (CochainComplex.mappingCone.triangle S₂.f).mor₃ ≫ (Functor.commShiftIso Q 1).hom.app S₂.X₁
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✝ : (i : J) → Ext (X i) Y n
i : J
⊢ ∀ b ∈ Finset.univ, b ≠ i → (mk₀ (c.ι i ≫ c.π b)).comp (x✝ b) ⋯ = 0

Tactic:
intro _

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✝ : (i : J) → Ext (X i) Y n
i b✝ : J
⊢ b✝ ∈ Finset.univ → b✝ ≠ i → (mk₀ (c.ι i ≫ c.π b✝)).comp (x✝ b✝) ⋯ = 0
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
X : C
J : Type u_1
inst✝ : Fintype J
Y : J → C
c : Bicone Y
hc : c.IsBilimit
n : ℕ
x✝ : (i : J) → Ext X (Y i) n
i : J
⊢ ∀ b ∈ Finset.univ, b ≠ i → (x✝ b).comp (mk₀ (c.ι b ≫ c.π i)) ⋯ = 0

Tactic:
intro _

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasExt C
X : C
J : Type u_1
inst✝ : Fintype J
Y : J → C
c : Bicone Y
hc : c.IsBilimit
n : ℕ
x✝ : (i : J) → Ext X (Y i) n
i b✝ : J
⊢ b✝ ∈ Finset.univ → b✝ ≠ i → (x✝ b✝).comp (mk₀ (c.ι b✝ ≫ c.π i)) ⋯ = 0
Module: Mathlib.Algebra.Homology.HomotopyCategory.ShortExact

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
⊢ (homologyFunctorFactors C (up ℤ) n).hom.app S.X₂ ≫
      HomologicalComplex.homologyMap (inr S.f) n ≫ HomologicalComplex.homologyMap (descShortComplex S) n =
    (homologyFunctorFactors C (up ℤ) n).hom.app S.X₂ ≫ HomologicalComplex.homologyMap S.g n

Tactic:
← HomologicalComplex.homologyMap_comp,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
⊢ (homologyFunctorFactors C (up ℤ) n).hom.app S.X₂ ≫ HomologicalComplex.homologyMap (inr S.f ≫ descShortComplex S) n =
    (homologyFunctorFactors C (up ℤ) n).hom.app S.X₂ ≫ HomologicalComplex.homologyMap S.g n
Module: Mathlib.CategoryTheory.Shift.Opposite

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
A : Type u_3
inst✝³ : AddMonoid A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
F : C ⥤ D
inst✝ : (OppositeShift.functor A F).CommShift A
x✝ : C
⊢ ((OppositeShift.functor A F).map ((shiftFunctorZero Cᵒᵖ A).inv.app (Opposite.op x✝))).unop ≫
      ((shiftFunctorZero D A).inv.app (Opposite.unop ((OppositeShift.functor A F).obj (Opposite.op x✝)))).op.unop =
    F.map ((shiftFunctorZero C A).hom.app x✝) ≫ (shiftFunctorZero D A).inv.app (F.obj x✝)

Tactic:
oppositeShiftFunctorZero_inv_app

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
A : Type u_3
inst✝³ : AddMonoid A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
F : C ⥤ D
inst✝ : (OppositeShift.functor A F).CommShift A
x✝ : C
⊢ ((OppositeShift.functor A F).map ((shiftFunctorZero C A).hom.app (Opposite.unop (Opposite.op x✝))).op).unop ≫
      ((shiftFunctorZero D A).inv.app (Opposite.unop ((OppositeShift.functor A F).obj (Opposite.op x✝)))).op.unop =
    F.map ((shiftFunctorZero C A).hom.app x✝) ≫ (shiftFunctorZero D A).inv.app (F.obj x✝)
Module: Mathlib.CategoryTheory.Shift.Opposite

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
A : Type u_3
inst✝³ : AddMonoid A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
F : C ⥤ D
inst✝ : (OppositeShift.functor A F).CommShift A
x✝ : C
⊢ ((OppositeShift.functor A F).map ((shiftFunctorZero Cᵒᵖ A).inv.app (Opposite.op x✝))).unop ≫
      ((shiftFunctorZero Dᵒᵖ A).hom.app ((OppositeShift.functor A F).obj (Opposite.op x✝))).unop =
    F.map ((shiftFunctorZero C A).hom.app x✝) ≫ (shiftFunctorZero D A).inv.app (F.obj x✝)

Tactic:
erw [oppositeShiftFunctorZero_hom_app, oppositeShiftFunctorZero_inv_app]

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
A : Type u_3
inst✝³ : AddMonoid A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
F : C ⥤ D
inst✝ : (OppositeShift.functor A F).CommShift A
x✝ : C
⊢ ((OppositeShift.functor A F).map ((shiftFunctorZero C A).hom.app (Opposite.unop (Opposite.op x✝))).op).unop ≫
      ((shiftFunctorZero D A).inv.app (Opposite.unop ((OppositeShift.functor A F).obj (Opposite.op x✝)))).op.unop =
    F.map ((shiftFunctorZero C A).hom.app x✝) ≫ (shiftFunctorZero D A).inv.app (F.obj x✝)
Module: Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four

Goal state before:
pp: case refine_1
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.Exact
hR₂ : R₂.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_epi_mono'._proof_1 = 0
hR₂' : Mono (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
h₀ : Epi (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_6)
h₁ : Mono (app' φ 2 mono_of_epi_of_epi_mono'._proof_1)
ψ : mk₃ 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) ⟶ mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯) :=
  homMk₃ (𝟙 ((mk₃ 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯)).obj' 0 _proof_297✝)) (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) ⋯ ⋯
    ⋯
⊢ Mono
    {
        X₁ :=
          (mk₂
                ((mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9
                  mono_of_epi_of_mono_of_mono'._proof_11)
                ((mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6
                  mono_of_epi_of_mono_of_mono'._proof_4)).obj
            ⟨0, ⋯⟩,
        X₂ :=
          (mk₂
                ((mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9
                  mono_of_epi_of_mono_of_mono'._proof_11)
                ((mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6
                  mono_of_epi_of_mono_of_mono'._proof_4)).obj
            ⟨1, ⋯⟩,
        X₃ :=
          (mk₂
                ((mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9
                  mono_of_epi_of_mono_of_mono'._proof_11)
                ((mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6
                  mono_of_epi_of_mono_of_mono'._proof_4)).obj
            ⟨2, ⋯⟩,
        f :=
          (mk₂
                ((mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9
                  mono_of_epi_of_mono_of_mono'._proof_11)
                ((mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6
                  mono_of_epi_of_mono_of_mono'._proof_4)).map'
            0 1 isComplex₂_iff._proof_2 isComplex₂_iff._proof_4,
        g :=
          (mk₂
                ((mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9
                  mono_of_epi_of_mono_of_mono'._proof_11)
                ((mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6
                  mono_of_epi_of_mono_of_mono'._proof_4)).map'
            1 2 isComplex₂_iff._proof_4 isComplex₂_iff._proof_5,
        zero := ⋯ }.g

Tactic:
exact hR₂'

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.Exact
hR₂' : Mono (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
h₀ : Mono (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_2)
h₁ : Mono (app' φ 2 mono_of_epi_of_epi_mono'._proof_1)
ψ : mk₃ 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) ⟶ mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯) :=
  homMk₃ (𝟙 ((mk₃ 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯)).obj' 0 _proof_297✝)) (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) ⋯ ⋯
    ⋯
⊢ Mono
    {
        X₁ :=
          (mk₂
                ((mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9
                  mono_of_epi_of_mono_of_mono'._proof_11)
                ((mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6
                  mono_of_epi_of_mono_of_mono'._proof_4)).obj
            ⟨0, ⋯⟩,
        X₂ :=
          (mk₂
                ((mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9
                  mono_of_epi_of_mono_of_mono'._proof_11)
                ((mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6
                  mono_of_epi_of_mono_of_mono'._proof_4)).obj
            ⟨1, ⋯⟩,
        X₃ :=
          (mk₂
                ((mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9
                  mono_of_epi_of_mono_of_mono'._proof_11)
                ((mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6
                  mono_of_epi_of_mono_of_mono'._proof_4)).obj
            ⟨2, ⋯⟩,
        f :=
          (mk₂
                ((mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9
                  mono_of_epi_of_mono_of_mono'._proof_11)
                ((mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6
                  mono_of_epi_of_mono_of_mono'._proof_4)).map'
            0 1 isComplex₂_iff._proof_2 isComplex₂_iff._proof_4,
        g :=
          (mk₂
                ((mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9
                  mono_of_epi_of_mono_of_mono'._proof_11)
                ((mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6
                  mono_of_epi_of_mono_of_mono'._proof_4)).map'
            1 2 isComplex₂_iff._proof_4 isComplex₂_iff._proof_5,
        zero := ⋯ }.g

Tactic:
exact hR₂'

Goal state after:
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
X : PullbackShift C φ
a₁ a₂ a₃ : A
h : a₁ + a₂ = a₃
b₁ b₂ b₃ : B
h₁ : b₁ = φ a₁
h₂ : b₂ = φ a₂
h₃ : b₃ = φ a₃
⊢ (shiftFunctorAdd' (PullbackShift C φ) a₁ a₂ a₃ h).inv.app X =
    (shiftFunctor (PullbackShift C φ) a₂).map ((pullbackShiftIso C φ a₁ b₁ h₁).hom.app X) ≫
      (pullbackShiftIso C φ a₂ b₂ h₂).hom.app ((shiftFunctor C b₁).obj X) ≫
        (shiftFunctorAdd' C b₁ b₂ b₃ ⋯).inv.app X ≫ (pullbackShiftIso C φ a₃ b₃ h₃).inv.app X

Tactic:
subst h₁ h₂ h
  obtain rfl : b₃ = φ a₁ + φ a₂ := by rw [h₃, φ.map_add]
  simp only [NatTrans.naturality_assoc]
  erw [Functor.map_id, id_comp, id_comp, shiftFunctorAdd'_eq_shiftFunctorAdd,
    shiftFunctorAdd'_eq_shiftFunctorAdd]
  change _ ≫ _ = _
  congr 1
  rw [Discrete.addMonoidalFunctor_μ]
  dsimp [Discrete.eqToHom]
  congr 2
  apply eqToHom_map

Goal state after:
Module: Mathlib.CategoryTheory.Shift.Pullback

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
A : Type u_2
B : Type u_3
inst✝⁵ : AddMonoid A
inst✝⁴ : AddMonoid B
inst✝³ : HasShift C B
φ : A →+ B
X : PullbackShift C φ
a₁ a₂ a₃ : A
h : a₁ + a₂ = a₃
b₁ b₂ b₃ : B
h₁ : b₁ = φ a₁
h₂ : b₂ = φ a₂
h₃ : b₃ = φ a₃
D : Type u_4
inst✝² : Category.{v_2, u_4} D
inst✝¹ : HasShift D B
F : C ⥤ D
inst✝ : F.CommShift B
x✝² x✝¹ : A
x✝ : PullbackShift C φ
⊢ F.map ((pullbackShiftIso C φ (x✝² + x✝¹) (φ (x✝² + x✝¹)) ⋯).hom.app x✝) ≫
      (CommShift.isoAdd' ⋯ (commShiftIso F (φ x✝²)) (commShiftIso F (φ x✝¹))).hom.app x✝ ≫
        (pullbackShiftIso D φ (x✝² + x✝¹) (φ (x✝² + x✝¹)) ⋯).inv.app (F.obj x✝) =
    F.map ((shiftFunctorAdd (PullbackShift C φ) x✝² x✝¹).hom.app x✝) ≫
      F.map ((pullbackShiftIso C φ x✝¹ (φ x✝¹) ⋯).hom.app ((shiftFunctor (PullbackShift C φ) x✝²).obj x✝)) ≫
        (commShiftIso F (φ x✝¹)).hom.app ((shiftFunctor (PullbackShift C φ) x✝²).obj x✝) ≫
          (pullbackShiftIso D φ x✝¹ (φ x✝¹) ⋯).inv.app (F.obj ((shiftFunctor (PullbackShift C φ) x✝²).obj x✝)) ≫
            (shiftFunctor (PullbackShift D φ) x✝¹).map (F.map ((pullbackShiftIso C φ x✝² (φ x✝²) ⋯).hom.app x✝)) ≫
              (shiftFunctor (PullbackShift D φ) x✝¹).map ((commShiftIso F (φ x✝²)).hom.app x✝) ≫
                (shiftFunctor (PullbackShift D φ) x✝¹).map ((pullbackShiftIso D φ x✝² (φ x✝²) ⋯).inv.app (F.obj x✝)) ≫
                  (shiftFunctorAdd (PullbackShift D φ) x✝² x✝¹).inv.app (F.obj x✝)

Tactic:
← shiftFunctorAdd'_eq_shiftFunctorAdd,

Goal state after:
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
A : Type u_2
B : Type u_3
inst✝⁵ : AddMonoid A
inst✝⁴ : AddMonoid B
inst✝³ : HasShift C B
φ : A →+ B
X : PullbackShift C φ
a₁ a₂ a₃ : A
h : a₁ + a₂ = a₃
b₁ b₂ b₃ : B
h₁ : b₁ = φ a₁
h₂ : b₂ = φ a₂
h₃ : b₃ = φ a₃
D : Type u_4
inst✝² : Category.{v_2, u_4} D
inst✝¹ : HasShift D B
F : C ⥤ D
inst✝ : F.CommShift B
x✝² x✝¹ : A
x✝ : PullbackShift C φ
⊢ F.map ((pullbackShiftIso C φ (x✝² + x✝¹) (φ (x✝² + x✝¹)) ⋯).hom.app x✝) ≫
      (CommShift.isoAdd' ⋯ (commShiftIso F (φ x✝²)) (commShiftIso F (φ x✝¹))).hom.app x✝ ≫
        (pullbackShiftIso D φ (x✝² + x✝¹) (φ (x✝² + x✝¹)) ⋯).inv.app (F.obj x✝) =
    F.map ((shiftFunctorAdd' (PullbackShift C φ) x✝² x✝¹ (x✝² + x✝¹) ⋯).hom.app x✝) ≫
      F.map ((pullbackShiftIso C φ x✝¹ (φ x✝¹) ⋯).hom.app ((shiftFunctor (PullbackShift C φ) x✝²).obj x✝)) ≫
        (commShiftIso F (φ x✝¹)).hom.app ((shiftFunctor (PullbackShift C φ) x✝²).obj x✝) ≫
          (pullbackShiftIso D φ x✝¹ (φ x✝¹) ⋯).inv.app (F.obj ((shiftFunctor (PullbackShift C φ) x✝²).obj x✝)) ≫
            (shiftFunctor (PullbackShift D φ) x✝¹).map (F.map ((pullbackShiftIso C φ x✝² (φ x✝²) ⋯).hom.app x✝)) ≫
              (shiftFunctor (PullbackShift D φ) x✝¹).map ((commShiftIso F (φ x✝²)).hom.app x✝) ≫
                (shiftFunctor (PullbackShift D φ) x✝¹).map ((pullbackShiftIso D φ x✝² (φ x✝²) ⋯).inv.app (F.obj x✝)) ≫
                  (shiftFunctorAdd (PullbackShift D φ) x✝² x✝¹).inv.app (F.obj x✝)
Module: Mathlib.CategoryTheory.Shift.Pullback

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
A : Type u_2
B : Type u_3
inst✝⁵ : AddMonoid A
inst✝⁴ : AddMonoid B
inst✝³ : HasShift C B
φ : A →+ B
X : PullbackShift C φ
a₁ a₂ a₃ : A
h : a₁ + a₂ = a₃
b₁ b₂ b₃ : B
h₁ : b₁ = φ a₁
h₂ : b₂ = φ a₂
h₃ : b₃ = φ a₃
D : Type u_4
inst✝² : Category.{v_2, u_4} D
inst✝¹ : HasShift D B
F : C ⥤ D
inst✝ : F.CommShift B
x✝² x✝¹ : A
x✝ : PullbackShift C φ
⊢ F.map ((pullbackShiftIso C φ (x✝² + x✝¹) (φ (x✝² + x✝¹)) ⋯).hom.app x✝) ≫
      (CommShift.isoAdd' ⋯ (commShiftIso F (φ x✝²)) (commShiftIso F (φ x✝¹))).hom.app x✝ ≫
        (pullbackShiftIso D φ (x✝² + x✝¹) (φ (x✝² + x✝¹)) ⋯).inv.app (F.obj x✝) =
    F.map ((shiftFunctorAdd' (PullbackShift C φ) x✝² x✝¹ (x✝² + x✝¹) ⋯).hom.app x✝) ≫
      F.map ((pullbackShiftIso C φ x✝¹ (φ x✝¹) ⋯).hom.app ((shiftFunctor (PullbackShift C φ) x✝²).obj x✝)) ≫
        (commShiftIso F (φ x✝¹)).hom.app ((shiftFunctor (PullbackShift C φ) x✝²).obj x✝) ≫
          (pullbackShiftIso D φ x✝¹ (φ x✝¹) ⋯).inv.app (F.obj ((shiftFunctor (PullbackShift C φ) x✝²).obj x✝)) ≫
            (shiftFunctor (PullbackShift D φ) x✝¹).map (F.map ((pullbackShiftIso C φ x✝² (φ x✝²) ⋯).hom.app x✝)) ≫
              (shiftFunctor (PullbackShift D φ) x✝¹).map ((commShiftIso F (φ x✝²)).hom.app x✝) ≫
                (shiftFunctor (PullbackShift D φ) x✝¹).map ((pullbackShiftIso D φ x✝² (φ x✝²) ⋯).inv.app (F.obj x✝)) ≫
                  (shiftFunctorAdd (PullbackShift D φ) x✝² x✝¹).inv.app (F.obj x✝)

Tactic:
← shiftFunctorAdd'_eq_shiftFunctorAdd,

Goal state after:
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
A : Type u_2
B : Type u_3
inst✝⁵ : AddMonoid A
inst✝⁴ : AddMonoid B
inst✝³ : HasShift C B
φ : A →+ B
X : PullbackShift C φ
a₁ a₂ a₃ : A
h : a₁ + a₂ = a₃
b₁ b₂ b₃ : B
h₁ : b₁ = φ a₁
h₂ : b₂ = φ a₂
h₃ : b₃ = φ a₃
D : Type u_4
inst✝² : Category.{v_2, u_4} D
inst✝¹ : HasShift D B
F : C ⥤ D
inst✝ : F.CommShift B
x✝² x✝¹ : A
x✝ : PullbackShift C φ
⊢ F.map ((pullbackShiftIso C φ (x✝² + x✝¹) (φ (x✝² + x✝¹)) ⋯).hom.app x✝) ≫
      (CommShift.isoAdd' ⋯ (commShiftIso F (φ x✝²)) (commShiftIso F (φ x✝¹))).hom.app x✝ ≫
        (pullbackShiftIso D φ (x✝² + x✝¹) (φ (x✝² + x✝¹)) ⋯).inv.app (F.obj x✝) =
    F.map ((shiftFunctorAdd' (PullbackShift C φ) x✝² x✝¹ (x✝² + x✝¹) ⋯).hom.app x✝) ≫
      F.map ((pullbackShiftIso C φ x✝¹ (φ x✝¹) ⋯).hom.app ((shiftFunctor (PullbackShift C φ) x✝²).obj x✝)) ≫
        (commShiftIso F (φ x✝¹)).hom.app ((shiftFunctor (PullbackShift C φ) x✝²).obj x✝) ≫
          (pullbackShiftIso D φ x✝¹ (φ x✝¹) ⋯).inv.app (F.obj ((shiftFunctor (PullbackShift C φ) x✝²).obj x✝)) ≫
            (shiftFunctor (PullbackShift D φ) x✝¹).map (F.map ((pullbackShiftIso C φ x✝² (φ x✝²) ⋯).hom.app x✝)) ≫
              (shiftFunctor (PullbackShift D φ) x✝¹).map ((commShiftIso F (φ x✝²)).hom.app x✝) ≫
                (shiftFunctor (PullbackShift D φ) x✝¹).map ((pullbackShiftIso D φ x✝² (φ x✝²) ⋯).inv.app (F.obj x✝)) ≫
                  (shiftFunctorAdd' (PullbackShift D φ) x✝² x✝¹ (x✝² + x✝¹) ⋯).inv.app (F.obj x✝)
Module: Mathlib.CategoryTheory.Triangulated.Opposite.Pretriangulated

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasShift C ℤ
inst✝³ : HasZeroObject C
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
X : Cᵒᵖ
⊢ contractibleTriangle X ∈ distinguishedTriangles C

Tactic:
mem_distinguishedTriangles_iff'

Goal state after:
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasShift C ℤ
inst✝³ : HasZeroObject C
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
X : Cᵒᵖ
⊢ ∃ T',
    ∃ (_ : T' ∈ Pretriangulated.distinguishedTriangles),
      Nonempty (contractibleTriangle X ≅ (triangleOpEquivalence C).functor.obj (Opposite.op T'))
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
E : C ≌ D
A : Type u_3
inst✝³ : AddGroup A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
inst✝ : E.functor.CommShift A
⊢ E.CommShift A

Tactic:
let := E.commShiftInverse A

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
E : C ≌ D
A : Type u_3
inst✝³ : AddGroup A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
inst✝ : E.functor.CommShift A
this : E.inverse.CommShift A := E.commShiftInverse A
⊢ E.CommShift A
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.EnoughProjectives

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Abelian C
P : C
inst✝² : Projective P
K : CochainComplex C ℤ
i : ℤ
π : K ⟶ (singleFunctor C i).obj P
inst✝¹ : K.IsStrictlyLE i
inst✝ : QuasiIsoAt π i
e : cycles K i ≅ K.X i := iCyclesIso K i (i + 1) ⋯ ⋯
α : K.X i ⟶ P := e.inv ≫ homologyπ K i ≫ homologyMap π i ≫ (singleObjHomologySelfIso (ComplexShape.up ℤ) i P).hom
⊢ e.hom ≫ π.f i = e.hom ≫ α ≫ (singleObjXSelf (ComplexShape.up ℤ) i P).inv

Tactic:
dsimp [α, e]

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Abelian C
P : C
inst✝² : Projective P
K : CochainComplex C ℤ
i : ℤ
π : K ⟶ (singleFunctor C i).obj P
inst✝¹ : K.IsStrictlyLE i
inst✝ : QuasiIsoAt π i
e : cycles K i ≅ K.X i := iCyclesIso K i (i + 1) ⋯ ⋯
α : K.X i ⟶ P := e.inv ≫ homologyπ K i ≫ homologyMap π i ≫ (singleObjHomologySelfIso (ComplexShape.up ℤ) i P).hom
⊢ iCycles K i ≫ π.f i =
    iCycles K i ≫
      ((iCyclesIso K i (i + 1) ⋯ ⋯).inv ≫
          homologyπ K i ≫ homologyMap π i ≫ (singleObjHomologySelfIso (ComplexShape.up ℤ) i P).hom) ≫
        (singleObjXSelf (ComplexShape.up ℤ) i P).inv