mathlib-initiative/mathlib-tactics huggingface dataset:
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✝⁵ : Category.{v_1, u_3} C
inst✝⁴ : HasZeroMorphisms C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝³ : e.IsTruncGE
inst✝² : ∀ (i' : ι'), K.HasHomology i'
inst✝¹ : ∀ (i' : ι'), L.HasHomology i'
inst✝ : ∀ (i' : ι'), M.HasHomology i'
i : ι
⊢ (truncGE'Map (φ ≫ φ') e).f i = (truncGE'Map φ e ≫ truncGE'Map φ' e).f i

Tactic:
by_cases hi : e.BoundaryGE i

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

case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁵ : Category.{v_1, u_3} C
inst✝⁴ : HasZeroMorphisms C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝³ : e.IsTruncGE
inst✝² : ∀ (i' : ι'), K.HasHomology i'
inst✝¹ : ∀ (i' : ι'), L.HasHomology i'
inst✝ : ∀ (i' : ι'), M.HasHomology i'
i : ι
hi : ¬e.BoundaryGE i
⊢ (truncGE'Map (φ ≫ φ') e).f i = (truncGE'Map φ e ≫ truncGE'Map φ' e).f i
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 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✝ j k : ι
hi : c.prev j = i✝
hk : c.next j = k
i : ι
⊢ (K.truncGE' e).HasHomology i

Tactic:
by_cases hi : e.BoundaryGE i

Goal state after:
case pos
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✝ j k : ι
hi✝ : c.prev j = i✝
hk : c.next j = k
i : ι
hi : e.BoundaryGE i
⊢ (K.truncGE' e).HasHomology i

case neg
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✝ j k : ι
hi✝ : c.prev j = i✝
hk : c.next j = k
i : ι
hi : ¬e.BoundaryGE i
⊢ (K.truncGE' e).HasHomology i
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✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝¹ : e.IsTruncGE
inst✝ : ∀ (i' : ι'), K.HasHomology i'
i j : ι
hij : c.Rel i j
⊢ f K e i ≫ (K.truncGE' e).d i j = (K.restriction e).d i j ≫ f K e j

Tactic:
by_cases hi : e.BoundaryGE i

Goal state after:
case pos
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 : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝¹ : e.IsTruncGE
inst✝ : ∀ (i' : ι'), K.HasHomology i'
i j : ι
hij : c.Rel i j
hi : e.BoundaryGE i
⊢ f K e i ≫ (K.truncGE' e).d i j = (K.restriction e).d i j ≫ f K e j

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 : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝¹ : e.IsTruncGE
inst✝ : ∀ (i' : ι'), K.HasHomology i'
i j : ι
hij : c.Rel i j
hi : ¬e.BoundaryGE i
⊢ f K e i ≫ (K.truncGE' e).d i j = (K.restriction e).d i j ≫ f K e j
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✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝¹ : e.IsTruncGE
inst✝ : ∀ (i' : ι'), K.HasHomology i'
i j : ι
hij : c.Rel i j
hi : e.BoundaryGE i
⊢ ((K.restrictionXIso e ⋯).hom ≫ K.pOpcycles (e.f i) ≫ (K.truncGE'XIsoOpcycles e ⋯ hi).inv) ≫ (K.truncGE' e).d i j =
    (K.restriction e).d i j ≫ f K e j

Tactic:
f_eq_iso_hom_iso_inv K e rfl (e.not_boundaryGE_next hij),

Goal state after:
case pos
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 : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝¹ : e.IsTruncGE
inst✝ : ∀ (i' : ι'), K.HasHomology i'
i j : ι
hij : c.Rel i j
hi : e.BoundaryGE i
⊢ ((K.restrictionXIso e ⋯).hom ≫ K.pOpcycles (e.f i) ≫ (K.truncGE'XIsoOpcycles e ⋯ hi).inv) ≫ (K.truncGE' e).d i j =
    (K.restriction e).d i j ≫ (K.restrictionXIso e ⋯).hom ≫ (K.truncGE'XIso e ⋯ ⋯).inv
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✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝¹ : e.IsTruncGE
inst✝ : ∀ (i' : ι'), K.HasHomology i'
i j : ι
hij : c.Rel i j
hi : ¬e.BoundaryGE i
⊢ ((K.restrictionXIso e ⋯).hom ≫ (K.truncGE'XIso e ⋯ hi).inv) ≫ (K.truncGE' e).d i j = (K.restriction e).d i j ≫ f K e j

Tactic:
f_eq_iso_hom_iso_inv K e rfl (e.not_boundaryGE_next hij),

Goal state after:
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 : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝¹ : e.IsTruncGE
inst✝ : ∀ (i' : ι'), K.HasHomology i'
i j : ι
hij : c.Rel i j
hi : ¬e.BoundaryGE i
⊢ ((K.restrictionXIso e ⋯).hom ≫ (K.truncGE'XIso e ⋯ hi).inv) ≫ (K.truncGE' e).d i j =
    (K.restriction e).d i j ≫ (K.restrictionXIso e ⋯).hom ≫ (K.truncGE'XIso e ⋯ ⋯).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
⊢ (K.restrictionToTruncGE' e ≫ truncGE'Map φ e).f i = (restrictionMap φ e ≫ L.restrictionToTruncGE' e).f i

Tactic:
by_cases hi : e.BoundaryGE i

Goal state after:
case pos
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
⊢ (K.restrictionToTruncGE' e ≫ truncGE'Map φ e).f i = (restrictionMap φ e ≫ L.restrictionToTruncGE' e).f i

case neg
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
⊢ (K.restrictionToTruncGE' e ≫ truncGE'Map φ e).f i = (restrictionMap φ e ≫ L.restrictionToTruncGE' e).f i
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✝⁵ : Category.{v_1, u_3} C
inst✝⁴ : HasZeroMorphisms C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝³ : e.IsTruncGE
inst✝² : ∀ (i' : ι'), K.HasHomology i'
inst✝¹ : ∀ (i' : ι'), L.HasHomology i'
inst✝ : ∀ (i' : ι'), M.HasHomology i'
i : ι
⊢ Epi ((K.restrictionToTruncGE' e).f i)

Tactic:
by_cases hi : e.BoundaryGE i

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

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

Tactic:
by_cases hi : e.BoundaryGE i

Goal state after:
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)

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✝ : K.IsStrictlySupported e
i : ι
hi : ¬e.BoundaryGE i
⊢ IsIso ((K.restrictionToTruncGE' e).f i)
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✝⁷ : Category.{v_1, u_3} C
inst✝⁶ : HasZeroMorphisms C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝⁵ : e.IsTruncGE
inst✝⁴ : ∀ (i' : ι'), K.HasHomology i'
inst✝³ : ∀ (i' : ι'), L.HasHomology i'
inst✝² : ∀ (i' : ι'), M.HasHomology i'
inst✝¹ : HasZeroObject C
inst✝ : K.IsStrictlySupported e
i' : ι'
hn : ∃ i, e.f i = i'
⊢ IsIso ((K.πTruncGE e).f i')

pp: case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁷ : Category.{v_1, u_3} C
inst✝⁶ : HasZeroMorphisms C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝⁵ : e.IsTruncGE
inst✝⁴ : ∀ (i' : ι'), K.HasHomology i'
inst✝³ : ∀ (i' : ι'), L.HasHomology i'
inst✝² : ∀ (i' : ι'), M.HasHomology i'
inst✝¹ : HasZeroObject C
inst✝ : K.IsStrictlySupported e
i' : ι'
hn : ∀ (i : ι), e.f i ≠ i'
⊢ IsIso ((K.πTruncGE e).f i')

Tactic:
· obtain ⟨i, hi⟩ := hn
    dsimp [πTruncGE]
    rw [e.isIso_liftExtend_f_iff _ _ hi]
    infer_instance

Goal state after:
case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁷ : Category.{v_1, u_3} C
inst✝⁶ : HasZeroMorphisms C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝⁵ : e.IsTruncGE
inst✝⁴ : ∀ (i' : ι'), K.HasHomology i'
inst✝³ : ∀ (i' : ι'), L.HasHomology i'
inst✝² : ∀ (i' : ι'), M.HasHomology i'
inst✝¹ : HasZeroObject C
inst✝ : K.IsStrictlySupported e
i' : ι'
hn : ∀ (i : ι), e.f i ≠ i'
⊢ IsIso ((K.πTruncGE e).f i')
Module: Mathlib.Algebra.Homology.Embedding.AreComplementary

Goal state before:
pp: case inr
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✝³ : Category.{v_1, u_4} C
inst✝² : Abelian C
K : HomologicalComplex C c
e₁ : c₁.Embedding c
e₂ : c₂.Embedding c
inst✝¹ : e₁.IsTruncLE
inst✝ : e₂.IsTruncGE
ac : e₁.AreComplementary e₂
i₂ : ι₂
⊢ QuasiIsoAt (K.shortComplexTruncLEX₃ToTruncGE ac) (e₂.f i₂)

Tactic:
have := quasiIsoAt_shortComplexTruncLE_g K e₁ (e₂.f i₂) (fun _ => ac.disjoint _ _)

Goal state after:
case inr
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✝³ : Category.{v_1, u_4} C
inst✝² : Abelian C
K : HomologicalComplex C c
e₁ : c₁.Embedding c
e₂ : c₂.Embedding c
inst✝¹ : e₁.IsTruncLE
inst✝ : e₂.IsTruncGE
ac : e₁.AreComplementary e₂
i₂ : ι₂
this : QuasiIsoAt (K.shortComplexTruncLE e₁).g (e₂.f i₂)
⊢ QuasiIsoAt (K.shortComplexTruncLEX₃ToTruncGE ac) (e₂.f i₂)
Module: Mathlib.Algebra.Homology.DerivedCategory.FullyFaithful

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
n : ℤ
A B : C
X : CochainComplex C ℤ
w✝² : X.IsStrictlyGE n
w✝¹ : X.IsStrictlyLE n
s : X ⟶ (CochainComplex.singleFunctor C n).obj A
w✝ : IsIso (Q.map s)
g : X ⟶ (CochainComplex.singleFunctor C n).obj B
A₀ : C
e : X ≅ (HomologicalComplex.single C (ComplexShape.up ℤ) n).obj A₀
φ : A₀ ⟶ A
hφ : (CochainComplex.singleFunctor C n).map φ = e.inv ≫ s
this✝ : IsIso ((singleFunctor C n).map φ)
this : IsIso φ
⊢ IsIso (e.inv ≫ s)

Tactic:
rw [← hφ]

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
n : ℤ
A B : C
X : CochainComplex C ℤ
w✝² : X.IsStrictlyGE n
w✝¹ : X.IsStrictlyLE n
s : X ⟶ (CochainComplex.singleFunctor C n).obj A
w✝ : IsIso (Q.map s)
g : X ⟶ (CochainComplex.singleFunctor C n).obj B
A₀ : C
e : X ≅ (HomologicalComplex.single C (ComplexShape.up ℤ) n).obj A₀
φ : A₀ ⟶ A
hφ : (CochainComplex.singleFunctor C n).map φ = e.inv ≫ s
this✝ : IsIso ((singleFunctor C n).map φ)
this : IsIso φ
⊢ IsIso ((CochainComplex.singleFunctor C n).map φ)
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
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
⊢ (homologyFunctorFactors C (up ℤ) n₀).inv.app (mappingCone S.f) ≫
      (homologyFunctor C (up ℤ) 0).homologySequenceδ
          (Triangle.mk ((quotient C (up ℤ)).map S.f) ((quotient C (up ℤ)).map (inr S.f))
            ((quotient C (up ℤ)).map (triangle S.f).mor₃ ≫ (Functor.commShiftIso (quotient C (up ℤ)) 1).hom.app S.X₁))
          n₀ n₁ h ≫
        (homologyFunctorFactors C (up ℤ) n₁).hom.app S.X₁ =
    HomologicalComplex.homologyMap (descShortComplex S) n₀ ≫
      hS.δ n₀ n₁ h ≫ 𝟙 ((HomologicalComplex.homologyFunctor C (up ℤ) n₁).obj S.X₁)

Tactic:
apply yoneda.map_injective

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
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
⊢ yoneda.map
      ((homologyFunctorFactors C (up ℤ) n₀).inv.app (mappingCone S.f) ≫
        (homologyFunctor C (up ℤ) 0).homologySequenceδ
            (Triangle.mk ((quotient C (up ℤ)).map S.f) ((quotient C (up ℤ)).map (inr S.f))
              ((quotient C (up ℤ)).map (triangle S.f).mor₃ ≫ (Functor.commShiftIso (quotient C (up ℤ)) 1).hom.app S.X₁))
            n₀ n₁ h ≫
          (homologyFunctorFactors C (up ℤ) n₁).hom.app S.X₁) =
    yoneda.map
      (HomologicalComplex.homologyMap (descShortComplex S) n₀ ≫
        hS.δ n₀ n₁ h ≫ 𝟙 ((HomologicalComplex.homologyFunctor C (up ℤ) n₁).obj S.X₁))
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
n : ℕ
R₁ R₂ : ComposableArrows C n
hR₁ : R₁.Exact
hR₂ : R₂.Exact
φ : R₁ ⟶ R₂
k₀ : ℕ
h : k₀ + 4 ≤ n
h₁ : IsIso (app' φ (k₀ + 1) ⋯)
h₃ : IsIso (app' φ (k₀ + 3) ⋯)
h₄ : Mono (app' φ (k₀ + 4) ⋯)
h₀ : Epi (app' φ (k₀ + 0) ⋯)
⊢ IsIso (app' φ (k₀ + 2) ⋯)

Tactic:
rw [← natAddLEFunctor_app' h] at h₀ h₁ h₃ h₄ ⊢

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
n : ℕ
R₁ R₂ : ComposableArrows C n
hR₁ : R₁.Exact
hR₂ : R₂.Exact
φ : R₁ ⟶ R₂
k₀ : ℕ
h : k₀ + 4 ≤ n
h₁ : IsIso (app' ((natAddLEFunctor h).map φ) 1 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_4)
h₃ : IsIso (app' ((natAddLEFunctor h).map φ) 3 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_6)
h₄ : Mono (app' ((natAddLEFunctor h).map φ) 4 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_7)
h₀ : Epi (app' ((natAddLEFunctor h).map φ) 0 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_2)
⊢ IsIso (app' ((natAddLEFunctor h).map φ) 2 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_9)
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.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_epi_mono'._proof_1 = 0
hR₁' : Epi (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_epi_mono'._proof_1)
hR₂ : R₂.Exact
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_2)
h₁ : Mono (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_6)
ψ : mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0 ⟶ mk₃ (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯) 0 :=
  homMk₃ (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) (𝟙 ((mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0).obj' 3 _proof_300✝)) ⋯ ⋯
    ⋯
⊢ Mono (app' ψ 3 mono_of_epi_of_mono_of_mono'._proof_7)

Tactic:
by dsimp [ψ]; infer_instance

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.ExtClass

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
S₁ S₂ : ShortComplex C
h₁ : S₁.ShortExact
h₂ : S₂.ShortExact
f : S₁ ⟶ S₂
⊢ h₁.extClass.comp (Ext.mk₀ f.τ₁) ⋯ = (Ext.mk₀ f.τ₃).comp h₂.extClass ⋯

Tactic:
by
  let := HasDerivedCategory.standard C
  ext
  simpa [ShiftedHom.comp_mk₀, ShiftedHom.mk₀_comp] using! (singleTriangle.map h₁ h₂ f).comm₃

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 : 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 ⋯) ⋯ ⋯
    ⋯
⊢ Epi (app' ψ 0 mono_of_epi_of_mono_of_mono'._proof_13)

Tactic:
by dsimp [ψ]; infer_instance

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 ⋯) ⋯ ⋯
    ⋯
⊢ Epi (app' ψ 0 mono_of_epi_of_mono_of_mono'._proof_13)

Tactic:
by dsimp [ψ]; infer_instance

Goal state after:
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
this :
  (shiftFunctor C a).map (adj.unit.app (G.obj Y)) ≫ e₂.inv.app (F.obj (G.obj Y)) =
    adj.unit.app ((shiftFunctor C a).obj (G.obj Y)) ≫
      G.map (e₁.hom.app (G.obj Y)) ≫ 𝟙 ((shiftFunctor D a ⋙ G).obj (F.obj (G.obj Y)))
⊢ e₂.inv.app Y =
    adj.unit.app ((shiftFunctor C a).obj (G.obj Y)) ≫
      G.map (e₁.hom.app (G.obj Y)) ≫ G.map ((shiftFunctor D a).map (adj.counit.app Y))

Tactic:
erw [comp_id] at this

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
this :
  (shiftFunctor C a).map (adj.unit.app (G.obj Y)) ≫ e₂.inv.app (F.obj (G.obj Y)) =
    adj.unit.app ((shiftFunctor C a).obj (G.obj Y)) ≫ G.map (e₁.hom.app (G.obj Y))
⊢ e₂.inv.app Y =
    adj.unit.app ((shiftFunctor C a).obj (G.obj Y)) ≫
      G.map (e₁.hom.app (G.obj Y)) ≫ G.map ((shiftFunctor D a).map (adj.counit.app Y))
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
hR₂ : R₂.Exact
hR₁' : Epi (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_epi_mono'._proof_1)
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_2)
h₁ : Epi (app' φ 2 mono_of_epi_of_epi_mono'._proof_1)
ψ : mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0 ⟶ mk₃ (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯) 0 :=
  homMk₃ (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) (𝟙 ((mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0).obj' 3 _proof_300✝)) ⋯ ⋯
    ⋯
⊢ Mono (app' ψ 3 mono_of_epi_of_mono_of_mono'._proof_7)

Tactic:
by dsimp [ψ]; infer_instance

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

Goal state before:
pp: case mp
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
T : Triangle Cᵒᵖ
hT : Opposite.unop ((triangleOpEquivalence C).inverse.obj T) ∈ Pretriangulated.distinguishedTriangles
⊢ ∃ T',
    ∃ (_ : T' ∈ Pretriangulated.distinguishedTriangles),
      Nonempty (T ≅ (triangleOpEquivalence C).functor.obj (Opposite.op T'))

Tactic:
exact ⟨_, hT, ⟨(triangleOpEquivalence C).counitIso.symm.app T⟩⟩

Goal state after: