awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Algebra.Homology.Embedding.TruncGE
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroMorphisms.{v_1, u_3} C
K L M : HomologicalComplex.{v_1, u_3, u_2} C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
inst_4 : ∀ (i' : ι'), L.HasHomology i'
inst_5 : ∀ (i' : ι'), M.HasHomology i'
inst_6 : HasZeroObject.{v_1, u_3} C
ι'' : Type u_4
c'' : ComplexShape ι''
e' : c''.Embedding c'
inst_7 : K.IsStrictlySupported e'
i' : ι'
hi' : ∀ (i : ι''), e'.f i ≠ i'
hi'' : ¬∃ (i : ι), e.f i = i'
⊢ IsZero.{v_1, u_3} ((K.truncGE e).X i')

Tactic:
exact (K.truncGE e).isZero_X_of_isStrictlySupported e _ (by simpa using hi'')

Premises used: HomologicalComplex.truncGE, HomologicalComplex.isZero_X_of_isStrictlySupported
-- Module: Mathlib.Algebra.Homology.Embedding.TruncGE
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroMorphisms.{v_1, u_3} C
K L M : HomologicalComplex.{v_1, u_3, u_2} C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
inst_4 : ∀ (i' : ι'), L.HasHomology i'
inst_5 : ∀ (i' : ι'), M.HasHomology i'
inst_6 : HasZeroObject.{v_1, u_3} C
inst_7 : K.IsStrictlySupported e
this : ∀ (i' : ι'), IsIso (Y := (K.truncGE e).X i') (Hom.f (K.πTruncGE e) i')
⊢ IsIso (Y := K.truncGE e) (K.πTruncGE e)

Tactic:
apply Hom.isIso_of_components

Premises used: HomologicalComplex.Hom.isIso_of_components
-- Module: Mathlib.Algebra.Homology.Embedding.TruncGE
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroMorphisms.{v_1, u_3} C
K L M : HomologicalComplex.{v_1, u_3, u_2} C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
inst_4 : ∀ (i' : ι'), L.HasHomology i'
inst_5 : ∀ (i' : ι'), M.HasHomology i'
inst_6 : HasZeroObject.{v_1, u_3} C
inst_7 : K.IsStrictlySupported e
suffices : ∀ (i' : ι'), IsIso (Y := (K.truncGE e).X i') (Hom.f (K.πTruncGE e) i')
⊢ IsIso (Y := K.truncGE e) (K.πTruncGE e)

Tactic:
suffices ∀ (i' : ι'), IsIso ((K.πTruncGE e).f i') by
    apply Hom.isIso_of_components
-- Module: Mathlib.Algebra.Homology.Embedding.TruncGE
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroMorphisms.{v_1, u_3} C
K L M : HomologicalComplex.{v_1, u_3, u_2} C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
inst_4 : ∀ (i' : ι'), L.HasHomology i'
inst_5 : ∀ (i' : ι'), M.HasHomology i'
inst_6 : HasZeroObject.{v_1, u_3} C
inst_7 : K.IsStrictlySupported e
i' : ι'
pos : ∀ (hn : ∃ (i : ι), e.f i = i'), IsIso (Y := (K.truncGE e).X i') (Hom.f (K.πTruncGE e) i')
neg : ∀ (hn : ∀ (i : ι), e.f i ≠ i'), IsIso (Y := (K.truncGE e).X i') (Hom.f (K.πTruncGE e) i')
⊢ IsIso (Y := (K.truncGE e).X i') (Hom.f (K.πTruncGE e) i')

Tactic:
by_cases! hn : ∃ i, e.f i = i'
-- Module: Mathlib.Algebra.Homology.Embedding.TruncGE
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroMorphisms.{v_1, u_3} C
K L M : HomologicalComplex.{v_1, u_3, u_2} C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
inst_4 : ∀ (i' : ι'), L.HasHomology i'
inst_5 : ∀ (i' : ι'), M.HasHomology i'
inst_6 : HasZeroObject.{v_1, u_3} C
inst_7 : K.IsStrictlySupported e
i' : ι'
hn : ∃ (i : ι), e.f i = i'
pos : ∀ (i : ι) (hi : e.f i = i'), IsIso (Y := (K.truncGE e).X i') (Hom.f (K.πTruncGE e) i')
⊢ IsIso (Y := (K.truncGE e).X i') (Hom.f (K.πTruncGE e) i')

Tactic:
obtain ⟨i, hi⟩ := hn
-- Module: Mathlib.Algebra.Homology.Embedding.TruncGE
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroMorphisms.{v_1, u_3} C
K L M : HomologicalComplex.{v_1, u_3, u_2} C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
inst_4 : ∀ (i' : ι'), L.HasHomology i'
inst_5 : ∀ (i' : ι'), M.HasHomology i'
inst_6 : HasZeroObject.{v_1, u_3} C
inst_7 : K.IsStrictlySupported e
i' : ι'
i : ι
hi : e.f i = i'
pos : IsIso (X := (K.restriction e).X i) (Y := (K.truncGE' e).X i) (Hom.f (K.restrictionToTruncGE' e) i)
⊢ IsIso (Y := (K.truncGE e).X i') (Hom.f (e.liftExtend (K.restrictionToTruncGE' e) ⋯) i')

Tactic:
rw [e.isIso_liftExtend_f_iff _ _ hi]

Premises used: ComplexShape.Embedding.isIso_liftExtend_f_iff
-- Module: Mathlib.Algebra.Homology.Embedding.TruncGE
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroMorphisms.{v_1, u_3} C
K L M : HomologicalComplex.{v_1, u_3, u_2} C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
inst_4 : ∀ (i' : ι'), L.HasHomology i'
inst_5 : ∀ (i' : ι'), M.HasHomology i'
inst_6 : HasZeroObject.{v_1, u_3} C
inst_7 : K.IsStrictlySupported e
i' : ι'
i : ι
hi : e.f i = i'
⊢ IsIso (X := (K.restriction e).X i) (Y := (K.truncGE' e).X i) (Hom.f (K.restrictionToTruncGE' e) i)

Tactic:
infer_instance
-- Module: Mathlib.Algebra.Homology.Embedding.TruncGE
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroMorphisms.{v_1, u_3} C
K L M : HomologicalComplex.{v_1, u_3, u_2} C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
inst_4 : ∀ (i' : ι'), L.HasHomology i'
inst_5 : ∀ (i' : ι'), M.HasHomology i'
inst_6 : HasZeroObject.{v_1, u_3} C
inst_7 : K.IsStrictlySupported e
i' : ι'
hn : ∀ (i : ι), e.f i ≠ i'
⊢ IsIso (Y := (K.truncGE e).X i') (Hom.f (K.πTruncGE e) i')

Tactic:
⟨0, ?_, ?_⟩

Premises used: And.intro, Exists.intro, OfNat.ofNat, CategoryTheory.IsIso.mk
-- Module: Mathlib.Algebra.Homology.Embedding.TruncGE
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroMorphisms.{v_1, u_3} C
K L M : HomologicalComplex.{v_1, u_3, u_2} C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
inst_4 : ∀ (i' : ι'), L.HasHomology i'
inst_5 : ∀ (i' : ι'), M.HasHomology i'
inst_6 : HasZeroObject.{v_1, u_3} C
inst_7 : K.IsStrictlySupported e
i' : ι'
hn : ∀ (i : ι), e.f i ≠ i'
neg : CategoryStruct.comp (Y := (K.truncGE e).X i') (Hom.f (K.πTruncGE e) i') 0 = 𝟙 (K.X i')
neg₁ : 0 ≫ Hom.f (K.πTruncGE e) i' = 𝟙 ((K.truncGE e).X i')
⊢ IsIso (Y := (K.truncGE e).X i') (Hom.f (K.πTruncGE e) i')

Tactic:
refine ⟨0, ?_, ?_⟩
-- Module: Mathlib.Algebra.Homology.Embedding.TruncGE
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroMorphisms.{v_1, u_3} C
K L M : HomologicalComplex.{v_1, u_3, u_2} C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
inst_4 : ∀ (i' : ι'), L.HasHomology i'
inst_5 : ∀ (i' : ι'), M.HasHomology i'
inst_6 : HasZeroObject.{v_1, u_3} C
inst_7 : K.IsStrictlySupported e
i' : ι'
hn : ∀ (i : ι), e.f i ≠ i'
⊢ CategoryStruct.comp (Y := (K.truncGE e).X i') (Hom.f (K.πTruncGE e) i') 0 = 𝟙 (K.X i')

Tactic:
apply (isZero_X_of_isStrictlySupported _ e i' hn).eq_of_src

Premises used: CategoryTheory.Limits.IsZero.eq_of_src, HomologicalComplex.isZero_X_of_isStrictlySupported
-- Module: Mathlib.Algebra.Homology.Embedding.TruncGE
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroMorphisms.{v_1, u_3} C
K L M : HomologicalComplex.{v_1, u_3, u_2} C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
inst_4 : ∀ (i' : ι'), L.HasHomology i'
inst_5 : ∀ (i' : ι'), M.HasHomology i'
inst_6 : HasZeroObject.{v_1, u_3} C
inst_7 : K.IsStrictlySupported e
i' : ι'
hn : ∀ (i : ι), e.f i ≠ i'
⊢ 0 ≫ Hom.f (K.πTruncGE e) i' = 𝟙 ((K.truncGE e).X i')

Tactic:
apply (isZero_X_of_isStrictlySupported _ e i' hn).eq_of_src

Premises used: CategoryTheory.Limits.IsZero.eq_of_src, HomologicalComplex.isZero_X_of_isStrictlySupported
-- Module: Mathlib.Algebra.Homology.Embedding.TruncGE
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroMorphisms.{v_1, u_3} C
K : HomologicalComplex.{v_1, u_3, u_2} C c'
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
inst_4 : HasZeroObject.{v_1, u_3} C
⊢ IsIso (Y := K.truncGE e) (K.πTruncGE e) ↔ K.IsStrictlySupported e

Tactic:
⟨fun _ ↦ isStrictlySupported_of_iso (asIso (K.πTruncGE e)).symm e,
    fun _ ↦ inferInstance⟩

Premises used: inferInstance, Iff.intro, CategoryTheory.Iso.symm, HomologicalComplex.isStrictlySupported_of_iso, CategoryTheory.asIso, HomologicalComplex.πTruncGE
-- Module: Mathlib.Algebra.Homology.Embedding.TruncGEHomology
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroMorphisms.{v_1, u_3} C
K : HomologicalComplex.{v_1, u_3, u_2} C c'
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
i j k : ι
hi : c.prev j = i
hk : c.next j = k
hj : ¬e.BoundaryGE j
⊢ (K.restriction e).HasHomology j

Tactic:
have : (K.restriction e).HasHomology j :=
    restriction.hasHomology K e i j k hi hk rfl rfl rfl
      (e.prev_f_of_not_boundaryGE hi hj) (e.next_f hk)
-- Module: Mathlib.Algebra.Homology.Embedding.TruncGEHomology
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroMorphisms.{v_1, u_3} C
K : HomologicalComplex.{v_1, u_3, u_2} C c'
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
i j k : ι
hi : c.prev j = i
hk : c.next j = k
hj : ¬e.BoundaryGE j
this : (K.restriction e).HasHomology j
⊢ ((K.restriction e).sc' i j k).HasHomology

Tactic:
have := ShortComplex.hasHomology_of_iso ((K.restriction e).isoSc' i j k hi hk)
-- Module: Mathlib.Algebra.Homology.Embedding.TruncGEHomology
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroMorphisms.{v_1, u_3} C
K : HomologicalComplex.{v_1, u_3, u_2} C c'
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
i j k : ι
hi : c.prev j = i
hk : c.next j = k
hj : ¬e.BoundaryGE j
this_1 : (K.restriction e).HasHomology j
this : ((K.restriction e).sc' i j k).HasHomology
φ : Functor.obj (shortComplexFunctor' C c i j k) (K.restriction e) ⟶
  Functor.obj (shortComplexFunctor' C c i j k) (K.truncGE' e) :=
  CategoryTheory.Functor.map (shortComplexFunctor' C c i j k) (K.restrictionToTruncGE' e)
dsimp : Epi (X := K.X (e.f i)) (Y := (K.truncGE' e).X i) (Hom.f (K.restrictionToTruncGE' e) i)
⊢ Epi φ.τ₁

Tactic:
dsimp [φ]
-- Module: Mathlib.Algebra.Homology.Embedding.TruncGEHomology
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroMorphisms.{v_1, u_3} C
K : HomologicalComplex.{v_1, u_3, u_2} C c'
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
i j k : ι
hi : c.prev j = i
hk : c.next j = k
hj : ¬e.BoundaryGE j
this_1 : (K.restriction e).HasHomology j
this : ((K.restriction e).sc' i j k).HasHomology
φ : Functor.obj (shortComplexFunctor' C c i j k) (K.restriction e) ⟶
  Functor.obj (shortComplexFunctor' C c i j k) (K.truncGE' e) :=
  CategoryTheory.Functor.map (shortComplexFunctor' C c i j k) (K.restrictionToTruncGE' e)
⊢ Epi (X := K.X (e.f i)) (Y := (K.truncGE' e).X i) (Hom.f (K.restrictionToTruncGE' e) i)

Tactic:
infer_instance
-- Module: Mathlib.Algebra.Homology.Embedding.TruncGEHomology
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroMorphisms.{v_1, u_3} C
K : HomologicalComplex.{v_1, u_3, u_2} C c'
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
i j k : ι
hi : c.prev j = i
hk : c.next j = k
hj : ¬e.BoundaryGE j
this_1 : (K.restriction e).HasHomology j
this : ((K.restriction e).sc' i j k).HasHomology
φ : Functor.obj (shortComplexFunctor' C c i j k) (K.restriction e) ⟶
  Functor.obj (shortComplexFunctor' C c i j k) (K.truncGE' e) :=
  CategoryTheory.Functor.map (shortComplexFunctor' C c i j k) (K.restrictionToTruncGE' e)
⊢ Epi φ.τ₁

Tactic:
have : Epi φ.τ₁ := by dsimp [φ]; infer_instance
-- Module: Mathlib.Algebra.Homology.Embedding.TruncGEHomology
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroMorphisms.{v_1, u_3} C
K : HomologicalComplex.{v_1, u_3, u_2} C c'
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
i j k : ι
hi : c.prev j = i
hk : c.next j = k
hj : ¬e.BoundaryGE j
this_1 : (K.restriction e).HasHomology j
this_2 : ((K.restriction e).sc' i j k).HasHomology
φ : Functor.obj (shortComplexFunctor' C c i j k) (K.restriction e) ⟶
  Functor.obj (shortComplexFunctor' C c i j k) (K.truncGE' e) :=
  CategoryTheory.Functor.map (shortComplexFunctor' C c i j k) (K.restrictionToTruncGE' e)
this : Epi φ.τ₁
⊢ IsIso φ.τ₂

Tactic:
have : IsIso φ.τ₂ := K.isIso_restrictionToTruncGE' e j hj
-- Module: Mathlib.Algebra.Homology.Embedding.TruncGEHomology
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroMorphisms.{v_1, u_3} C
K : HomologicalComplex.{v_1, u_3, u_2} C c'
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
i j k : ι
hi : c.prev j = i
hk : c.next j = k
hj : ¬e.BoundaryGE j
this_1 : (K.restriction e).HasHomology j
this_2 : ((K.restriction e).sc' i j k).HasHomology
φ : Functor.obj (shortComplexFunctor' C c i j k) (K.restriction e) ⟶
  Functor.obj (shortComplexFunctor' C c i j k) (K.truncGE' e) :=
  CategoryTheory.Functor.map (shortComplexFunctor' C c i j k) (K.restrictionToTruncGE' e)
this_3 : Epi φ.τ₁
this : IsIso φ.τ₂
⊢ IsIso φ.τ₃

Tactic:
have : IsIso φ.τ₃ := K.isIso_restrictionToTruncGE' e k (e.not_boundaryGE_next' hj hk)
-- Module: Mathlib.Algebra.Homology.Embedding.TruncGEHomology
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroMorphisms.{v_1, u_3} C
K : HomologicalComplex.{v_1, u_3, u_2} C c'
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
i j k : ι
hi : c.prev j = i
hk : c.next j = k
hj : ¬e.BoundaryGE j
this_1 : (K.restriction e).HasHomology j
this_2 : ((K.restriction e).sc' i j k).HasHomology
φ : Functor.obj (shortComplexFunctor' C c i j k) (K.restriction e) ⟶
  Functor.obj (shortComplexFunctor' C c i j k) (K.truncGE' e) :=
  CategoryTheory.Functor.map (shortComplexFunctor' C c i j k) (K.restrictionToTruncGE' e)
this_3 : Epi φ.τ₁
this_4 : IsIso φ.τ₂
this : IsIso φ.τ₃
⊢ ((K.truncGE' e).sc' i j k).HasHomology

Tactic:
ShortComplex.hasHomology_of_epi_of_isIso_of_mono φ

Premises used: CategoryTheory.ShortComplex.hasHomology_of_epi_of_isIso_of_mono
-- Module: Mathlib.Algebra.Homology.Embedding.TruncGEHomology
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroMorphisms.{v_1, u_3} C
K : HomologicalComplex.{v_1, u_3, u_2} C c'
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
i j k : ι
hi : c.prev j = i
hk : c.next j = k
hj : ¬e.BoundaryGE j
this_1 : (K.restriction e).HasHomology j
this_2 : ((K.restriction e).sc' i j k).HasHomology
φ : Functor.obj (shortComplexFunctor' C c i j k) (K.restriction e) ⟶
  Functor.obj (shortComplexFunctor' C c i j k) (K.truncGE' e) :=
  CategoryTheory.Functor.map (shortComplexFunctor' C c i j k) (K.restrictionToTruncGE' e)
this_3 : Epi φ.τ₁
this_4 : IsIso φ.τ₂
this : IsIso φ.τ₃
⊢ ((K.truncGE' e).sc' i j k).HasHomology

Tactic:
exact ShortComplex.hasHomology_of_epi_of_isIso_of_mono φ

Premises used: CategoryTheory.ShortComplex.hasHomology_of_epi_of_isIso_of_mono
-- Module: Mathlib.Algebra.Homology.Embedding.TruncGEHomology
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroMorphisms.{v_1, u_3} C
K : HomologicalComplex.{v_1, u_3, u_2} C c'
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
j : ι
hj : ¬e.BoundaryGE j
⊢ (K.truncGE' e).HasHomology j

Tactic:
hasHomology_sc'_of_not_mem_boundary K e _ j _ rfl rfl hj

Premises used: HomologicalComplex.truncGE'.hasHomology_sc'_of_not_mem_boundary, rfl
-- Module: Mathlib.Algebra.Homology.Embedding.TruncGEHomology
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroMorphisms.{v_1, u_3} C
K : HomologicalComplex.{v_1, u_3, u_2} C c'
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
j : ι
hj : ¬e.BoundaryGE j
inst_4 : (K.restriction e).HasHomology j
inst_5 : (K.truncGE' e).HasHomology j
rw :
  ShortComplex.QuasiIso (S₁ := Functor.obj (shortComplexFunctor C c j) (K.restriction e)) (S₂ :=
    Functor.obj (shortComplexFunctor C c j) (K.truncGE' e))
    (CategoryTheory.Functor.map (shortComplexFunctor C c j) (K.restrictionToTruncGE' e))
⊢ QuasiIsoAt (K := K.restriction e) (L := K.truncGE' e) (K.restrictionToTruncGE' e) j

Tactic:
rw [quasiIsoAt_iff]

Premises used: quasiIsoAt_iff
-- Module: Mathlib.Algebra.Homology.Embedding.TruncGEHomology
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroMorphisms.{v_1, u_3} C
K : HomologicalComplex.{v_1, u_3, u_2} C c'
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
j : ι
hj : ¬e.BoundaryGE j
inst_4 : (K.restriction e).HasHomology j
inst_5 : (K.truncGE' e).HasHomology j
φ : Functor.obj (shortComplexFunctor C c j) (K.restriction e) ⟶
  Functor.obj (shortComplexFunctor C c j) (K.truncGE' e) :=
  CategoryTheory.Functor.map (shortComplexFunctor C c j) (K.restrictionToTruncGE' e)
dsimp :
  Epi (X := K.X (e.f (c.prev j))) (Y := (K.truncGE' e).X (c.prev j)) (Hom.f (K.restrictionToTruncGE' e) (c.prev j))
⊢ Epi φ.τ₁

Tactic:
dsimp [φ]
-- Module: Mathlib.Algebra.Homology.Embedding.TruncGEHomology
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroMorphisms.{v_1, u_3} C
K : HomologicalComplex.{v_1, u_3, u_2} C c'
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
j : ι
hj : ¬e.BoundaryGE j
inst_4 : (K.restriction e).HasHomology j
inst_5 : (K.truncGE' e).HasHomology j
φ : Functor.obj (shortComplexFunctor C c j) (K.restriction e) ⟶
  Functor.obj (shortComplexFunctor C c j) (K.truncGE' e) :=
  CategoryTheory.Functor.map (shortComplexFunctor C c j) (K.restrictionToTruncGE' e)
⊢ Epi (X := K.X (e.f (c.prev j))) (Y := (K.truncGE' e).X (c.prev j)) (Hom.f (K.restrictionToTruncGE' e) (c.prev j))

Tactic:
infer_instance
-- Module: Mathlib.Algebra.Homology.Embedding.TruncGEHomology
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroMorphisms.{v_1, u_3} C
K : HomologicalComplex.{v_1, u_3, u_2} C c'
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
j : ι
hj : ¬e.BoundaryGE j
inst_4 : (K.restriction e).HasHomology j
inst_5 : (K.truncGE' e).HasHomology j
φ : Functor.obj (shortComplexFunctor C c j) (K.restriction e) ⟶
  Functor.obj (shortComplexFunctor C c j) (K.truncGE' e) :=
  CategoryTheory.Functor.map (shortComplexFunctor C c j) (K.restrictionToTruncGE' e)
⊢ Epi φ.τ₁

Tactic:
have : Epi φ.τ₁ := by dsimp [φ]; infer_instance
-- Module: Mathlib.Algebra.Homology.Embedding.TruncGEHomology
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroMorphisms.{v_1, u_3} C
K : HomologicalComplex.{v_1, u_3, u_2} C c'
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
j : ι
hj : ¬e.BoundaryGE j
inst_4 : (K.restriction e).HasHomology j
inst_5 : (K.truncGE' e).HasHomology j
φ : Functor.obj (shortComplexFunctor C c j) (K.restriction e) ⟶
  Functor.obj (shortComplexFunctor C c j) (K.truncGE' e) :=
  CategoryTheory.Functor.map (shortComplexFunctor C c j) (K.restrictionToTruncGE' e)
this : Epi φ.τ₁
⊢ IsIso φ.τ₂

Tactic:
have : IsIso φ.τ₂ := K.isIso_restrictionToTruncGE' e j hj
-- Module: Mathlib.Algebra.Homology.Embedding.TruncGEHomology
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroMorphisms.{v_1, u_3} C
K : HomologicalComplex.{v_1, u_3, u_2} C c'
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
j : ι
hj : ¬e.BoundaryGE j
inst_4 : (K.restriction e).HasHomology j
inst_5 : (K.truncGE' e).HasHomology j
φ : Functor.obj (shortComplexFunctor C c j) (K.restriction e) ⟶
  Functor.obj (shortComplexFunctor C c j) (K.truncGE' e) :=
  CategoryTheory.Functor.map (shortComplexFunctor C c j) (K.restrictionToTruncGE' e)
this_1 : Epi φ.τ₁
this : IsIso φ.τ₂
⊢ IsIso φ.τ₃

Tactic:
have : IsIso φ.τ₃ := K.isIso_restrictionToTruncGE' e _ (e.not_boundaryGE_next' hj rfl)
-- Module: Mathlib.Algebra.Homology.Embedding.TruncGEHomology
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroMorphisms.{v_1, u_3} C
K : HomologicalComplex.{v_1, u_3, u_2} C c'
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
j : ι
hj : ¬e.BoundaryGE j
inst_4 : (K.restriction e).HasHomology j
inst_5 : (K.truncGE' e).HasHomology j
φ : Functor.obj (shortComplexFunctor C c j) (K.restriction e) ⟶
  Functor.obj (shortComplexFunctor C c j) (K.truncGE' e) :=
  CategoryTheory.Functor.map (shortComplexFunctor C c j) (K.restrictionToTruncGE' e)
this_1 : Epi φ.τ₁
this_2 : IsIso φ.τ₂
this : IsIso φ.τ₃
⊢ ShortComplex.QuasiIso (S₁ := Functor.obj (shortComplexFunctor C c j) (K.restriction e)) (S₂ :=
    Functor.obj (shortComplexFunctor C c j) (K.truncGE' e))
    (CategoryTheory.Functor.map (shortComplexFunctor C c j) (K.restrictionToTruncGE' e))

Tactic:
ShortComplex.quasiIso_of_epi_of_isIso_of_mono φ

Premises used: CategoryTheory.ShortComplex.quasiIso_of_epi_of_isIso_of_mono
-- Module: Mathlib.Algebra.Homology.Embedding.TruncGEHomology
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroMorphisms.{v_1, u_3} C
K : HomologicalComplex.{v_1, u_3, u_2} C c'
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
j : ι
hj : ¬e.BoundaryGE j
inst_4 : (K.restriction e).HasHomology j
inst_5 : (K.truncGE' e).HasHomology j
φ : Functor.obj (shortComplexFunctor C c j) (K.restriction e) ⟶
  Functor.obj (shortComplexFunctor C c j) (K.truncGE' e) :=
  CategoryTheory.Functor.map (shortComplexFunctor C c j) (K.restrictionToTruncGE' e)
this_1 : Epi φ.τ₁
this_2 : IsIso φ.τ₂
this : IsIso φ.τ₃
⊢ ShortComplex.QuasiIso (S₁ := Functor.obj (shortComplexFunctor C c j) (K.restriction e)) (S₂ :=
    Functor.obj (shortComplexFunctor C c j) (K.truncGE' e))
    (CategoryTheory.Functor.map (shortComplexFunctor C c j) (K.restrictionToTruncGE' e))

Tactic:
exact ShortComplex.quasiIso_of_epi_of_isIso_of_mono φ

Premises used: CategoryTheory.ShortComplex.quasiIso_of_epi_of_isIso_of_mono