awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGEHomology (goal)
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
j : ι
j' : ι'
hj' : e.f j = j'
hj : ¬e.BoundaryGE j
φ : Functor.obj (shortComplexFunctor C c' j') K ⟶ Functor.obj (shortComplexFunctor C c' j') (K.truncGE e) :=
  CategoryTheory.Functor.map (shortComplexFunctor C c' j') (K.πTruncGE e)
⊢ Epi φ.τ₁
have pos : ∀ (hi : ∃ (i : ι), e.f i = c'.prev j'), Epi φ.τ₁
Tactic: by_cases hi : ∃ i, e.f i = c'.prev j'
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGEHomology (goal)
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
j : ι
j' : ι'
hj' : e.f j = j'
hj : ¬e.BoundaryGE j
φ : Functor.obj (shortComplexFunctor C c' j') K ⟶ Functor.obj (shortComplexFunctor C c' j') (K.truncGE e) :=
  CategoryTheory.Functor.map (shortComplexFunctor C c' j') (K.πTruncGE e)
pos : ∀ (hi : ∃ (i : ι), e.f i = c'.prev j'), Epi φ.τ₁
⊢ Epi φ.τ₁
have neg : ∀ (hi : ¬∃ (i : ι), e.f i = c'.prev j'), Epi φ.τ₁
Tactic: by_cases hi : ∃ i, e.f i = c'.prev j'
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGEHomology (goal)
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
j : ι
j' : ι'
hj' : e.f j = j'
hj : ¬e.BoundaryGE j
φ : Functor.obj (shortComplexFunctor C c' j') K ⟶ Functor.obj (shortComplexFunctor C c' j') (K.truncGE e) :=
  CategoryTheory.Functor.map (shortComplexFunctor C c' j') (K.πTruncGE e)
hi : ∃ (i : ι), e.f i = c'.prev j'
⊢ Epi φ.τ₁
have pos : ∀ (i : ι) (hi : e.f i = c'.prev j'), Epi φ.τ₁
Tactic: obtain ⟨i, hi⟩ := hi
Cleared hypotheses: hi
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGEHomology (goal)
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
j : ι
j' : ι'
hj' : e.f j = j'
hj : ¬e.BoundaryGE j
φ : Functor.obj (shortComplexFunctor C c' j') K ⟶ Functor.obj (shortComplexFunctor C c' j') (K.truncGE e) :=
  CategoryTheory.Functor.map (shortComplexFunctor C c' j') (K.πTruncGE e)
i : ι
hi : e.f i = c'.prev j'
⊢ Epi φ.τ₁
have pos : Epi (X := K.X (c'.prev j')) (Y := (K.truncGE e).X (c'.prev j'))
  (Hom.f (e.liftExtend (K.restrictionToTruncGE' e) sorry) (c'.prev j'))
Tactic: dsimp [φ, πTruncGE]
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGEHomology (goal)
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
j : ι
j' : ι'
hj' : e.f j = j'
hj : ¬e.BoundaryGE j
φ : Functor.obj (shortComplexFunctor C c' j') K ⟶ Functor.obj (shortComplexFunctor C c' j') (K.truncGE e) :=
  CategoryTheory.Functor.map (shortComplexFunctor C c' j') (K.πTruncGE e)
i : ι
hi : e.f i = c'.prev j'
⊢ Epi (X := K.X (c'.prev j')) (Y := (K.truncGE e).X (c'.prev j'))
    (Hom.f (e.liftExtend (K.restrictionToTruncGE' e) ⋯) (c'.prev j'))
have pos : Epi (X := (K.restriction e).X i) (Y := (K.truncGE' e).X i) (Hom.f (K.restrictionToTruncGE' e) i)
Tactic: rw [e.epi_liftExtend_f_iff _ _ hi]
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGEHomology (hypothesis)
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
j : ι
j' : ι'
hj' : e.f j = j'
hj : ¬e.BoundaryGE j
φ : Functor.obj (shortComplexFunctor C c' j') K ⟶ Functor.obj (shortComplexFunctor C c' j') (K.truncGE e) :=
  CategoryTheory.Functor.map (shortComplexFunctor C c' j') (K.πTruncGE e)
⊢ ShortComplex.QuasiIso (S₁ := Functor.obj (shortComplexFunctor C c' j') K) (S₂ :=
    Functor.obj (shortComplexFunctor C c' j') (K.truncGE e))
    (CategoryTheory.Functor.map (shortComplexFunctor C c' j') (K.πTruncGE e))
have this : Epi φ.τ₁
Tactic: have : Epi φ.τ₁ := by
      by_cases hi : ∃ i, e.f i = c'.prev j'
      · obtain ⟨i, hi⟩ := hi
        dsimp [φ, πTruncGE]
        rw [e.epi_liftExtend_f_iff _ _ hi]
        infer_instance
      · apply IsZero.epi (isZero_extend_X _ _ _ (by simpa using hi))
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGEHomology (goal)
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
j : ι
j' : ι'
hj' : e.f j = j'
hj : ¬e.BoundaryGE j
φ : Functor.obj (shortComplexFunctor C c' j') K ⟶ Functor.obj (shortComplexFunctor C c' j') (K.truncGE e) :=
  CategoryTheory.Functor.map (shortComplexFunctor C c' j') (K.πTruncGE e)
this : Epi φ.τ₁
⊢ IsIso φ.τ₂
have dsimp : IsIso (Y := (K.truncGE e).X j') (Hom.f (e.liftExtend (K.restrictionToTruncGE' e) sorry) j')
Tactic: dsimp [φ, πTruncGE]
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGEHomology (goal)
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
j : ι
j' : ι'
hj' : e.f j = j'
hj : ¬e.BoundaryGE j
φ : Functor.obj (shortComplexFunctor C c' j') K ⟶ Functor.obj (shortComplexFunctor C c' j') (K.truncGE e) :=
  CategoryTheory.Functor.map (shortComplexFunctor C c' j') (K.πTruncGE e)
this : Epi φ.τ₁
⊢ IsIso (Y := (K.truncGE e).X j') (Hom.f (e.liftExtend (K.restrictionToTruncGE' e) ⋯) j')
have rw : IsIso (X := (K.restriction e).X j) (Y := (K.truncGE' e).X j) (Hom.f (K.restrictionToTruncGE' e) j)
Tactic: rw [e.isIso_liftExtend_f_iff _ _ hj']
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGEHomology (hypothesis)
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
j : ι
j' : ι'
hj' : e.f j = j'
hj : ¬e.BoundaryGE j
φ : Functor.obj (shortComplexFunctor C c' j') K ⟶ Functor.obj (shortComplexFunctor C c' j') (K.truncGE e) :=
  CategoryTheory.Functor.map (shortComplexFunctor C c' j') (K.πTruncGE e)
this : Epi φ.τ₁
⊢ ShortComplex.QuasiIso (S₁ := Functor.obj (shortComplexFunctor C c' j') K) (S₂ :=
    Functor.obj (shortComplexFunctor C c' j') (K.truncGE e))
    (CategoryTheory.Functor.map (shortComplexFunctor C c' j') (K.πTruncGE e))
have this₁ : IsIso φ.τ₂
Tactic: have : IsIso φ.τ₂ := by
      dsimp [φ, πTruncGE]
      rw [e.isIso_liftExtend_f_iff _ _ hj']
      exact K.isIso_restrictionToTruncGE' e j hj
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGEHomology (goal)
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
j : ι
j' : ι'
hj' : e.f j = j'
hj : ¬e.BoundaryGE j
φ : Functor.obj (shortComplexFunctor C c' j') K ⟶ Functor.obj (shortComplexFunctor C c' j') (K.truncGE e) :=
  CategoryTheory.Functor.map (shortComplexFunctor C c' j') (K.πTruncGE e)
this_1 : Epi φ.τ₁
this : IsIso φ.τ₂
⊢ IsIso φ.τ₃
have dsimp : IsIso (X := K.X (c'.next j')) (Y := (K.truncGE e).X (c'.next j'))
  (Hom.f (e.liftExtend (K.restrictionToTruncGE' e) sorry) (c'.next j'))
Tactic: dsimp [φ, πTruncGE]
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGEHomology (hypothesis)
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
j : ι
j' : ι'
hj' : e.f j = j'
hj : ¬e.BoundaryGE j
φ : Functor.obj (shortComplexFunctor C c' j') K ⟶ Functor.obj (shortComplexFunctor C c' j') (K.truncGE e) :=
  CategoryTheory.Functor.map (shortComplexFunctor C c' j') (K.πTruncGE e)
this_1 : Epi φ.τ₁
this : IsIso φ.τ₂
⊢ IsIso (X := K.X (c'.next j')) (Y := (K.truncGE e).X (c'.next j'))
    (Hom.f (e.liftExtend (K.restrictionToTruncGE' e) ⋯) (c'.next j'))
have this₁ : c'.next j' = e.f (c.next j)
Tactic: have : c'.next j' = e.f (c.next j) := by simpa only [← hj'] using e.next_f rfl
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGEHomology (goal)
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
j : ι
j' : ι'
hj' : e.f j = j'
hj : ¬e.BoundaryGE j
φ : Functor.obj (shortComplexFunctor C c' j') K ⟶ Functor.obj (shortComplexFunctor C c' j') (K.truncGE e) :=
  CategoryTheory.Functor.map (shortComplexFunctor C c' j') (K.πTruncGE e)
this_1 : Epi φ.τ₁
this_2 : IsIso φ.τ₂
this : c'.next j' = e.f (c.next j)
⊢ IsIso (X := K.X (c'.next j')) (Y := (K.truncGE e).X (c'.next j'))
    (Hom.f (e.liftExtend (K.restrictionToTruncGE' e) ⋯) (c'.next j'))
have rw : IsIso (X := (K.restriction e).X (c.next j)) (Y := (K.truncGE' e).X (c.next j))
  (Hom.f (K.restrictionToTruncGE' e) (c.next j))
Tactic: rw [e.isIso_liftExtend_f_iff _ _ this.symm]
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGEHomology (hypothesis)
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
j : ι
j' : ι'
hj' : e.f j = j'
hj : ¬e.BoundaryGE j
φ : Functor.obj (shortComplexFunctor C c' j') K ⟶ Functor.obj (shortComplexFunctor C c' j') (K.truncGE e) :=
  CategoryTheory.Functor.map (shortComplexFunctor C c' j') (K.πTruncGE e)
this_1 : Epi φ.τ₁
this : IsIso φ.τ₂
⊢ ShortComplex.QuasiIso (S₁ := Functor.obj (shortComplexFunctor C c' j') K) (S₂ :=
    Functor.obj (shortComplexFunctor C c' j') (K.truncGE e))
    (CategoryTheory.Functor.map (shortComplexFunctor C c' j') (K.πTruncGE e))
have this₁ : IsIso φ.τ₃
Tactic: have : IsIso φ.τ₃ := by
      dsimp [φ, πTruncGE]
      have : c'.next j' = e.f (c.next j) := by simpa only [← hj'] using e.next_f rfl
      rw [e.isIso_liftExtend_f_iff _ _ this.symm]
      exact K.isIso_restrictionToTruncGE' e _ (e.not_boundaryGE_next' hj rfl)
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGEHomology (goal)
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
⊢ QuasiIso (L := K.truncGE e) (K.πTruncGE e) ↔ K.IsSupported e
have mp : QuasiIso (L := K.truncGE e) (K.πTruncGE e) → K.IsSupported e
Tactic: constructor
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGEHomology (goal)
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
mp : QuasiIso (L := K.truncGE e) (K.πTruncGE e) → K.IsSupported e
⊢ QuasiIso (L := K.truncGE e) (K.πTruncGE e) ↔ K.IsSupported e
have mpr : K.IsSupported e → QuasiIso (L := K.truncGE e) (K.πTruncGE e)
Tactic: constructor
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGEHomology (goal)
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
h : QuasiIso (L := K.truncGE e) (K.πTruncGE e)
⊢ K.IsSupported e
have mp : ∀ (i' : ι') (hi' : ∀ (i : ι), e.f i ≠ i'), K.ExactAt i'
Tactic: refine ⟨fun i' hi' => ?_⟩
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGEHomology (goal)
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
h : QuasiIso (L := K.truncGE e) (K.πTruncGE e)
i' : ι'
hi' : ∀ (i : ι), e.f i ≠ i'
⊢ K.ExactAt i'
have mp : (K.truncGE e).ExactAt i'
Tactic: rw [exactAt_iff_of_quasiIsoAt (K.πTruncGE e) i']
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGEHomology (goal)
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
h : K.IsSupported e
⊢ QuasiIso (L := K.truncGE e) (K.πTruncGE e)
have mpr : ∀ (i : ι'), QuasiIsoAt (L := K.truncGE e) (K.πTruncGE e) i
Tactic: rw [quasiIso_iff]
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGEHomology (goal)
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
h : K.IsSupported e
i' : ι'
⊢ QuasiIsoAt (L := K.truncGE e) (K.πTruncGE e) i'
have pos : ∀ (hi' : ∃ (i : ι), e.f i = i'), QuasiIsoAt (L := K.truncGE e) (K.πTruncGE e) i'
Tactic: by_cases hi' : ∃ i, e.f i = i'
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGEHomology (goal)
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
h : K.IsSupported e
i' : ι'
pos : ∀ (hi' : ∃ (i : ι), e.f i = i'), QuasiIsoAt (L := K.truncGE e) (K.πTruncGE e) i'
⊢ QuasiIsoAt (L := K.truncGE e) (K.πTruncGE e) i'
have neg : ∀ (hi' : ¬∃ (i : ι), e.f i = i'), QuasiIsoAt (L := K.truncGE e) (K.πTruncGE e) i'
Tactic: by_cases hi' : ∃ i, e.f i = i'
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGEHomology (goal)
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
h : K.IsSupported e
i' : ι'
hi' : ∃ (i : ι), e.f i = i'
⊢ QuasiIsoAt (L := K.truncGE e) (K.πTruncGE e) i'
have pos : ∀ (i : ι), QuasiIsoAt (L := K.truncGE e) (K.πTruncGE e) (e.f i)
Tactic: obtain ⟨i, rfl⟩ := hi'
Cleared hypotheses: i', hi'
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGEHomology (goal)
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
h : K.IsSupported e
i' : ι'
hi' : ¬∃ (i : ι), e.f i = i'
⊢ QuasiIsoAt (L := K.truncGE e) (K.πTruncGE e) i'
have neg : (K.truncGE e).ExactAt i'
Tactic: rw [quasiIsoAt_iff_exactAt (K.πTruncGE e) i']
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGEHomology (goal)
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
h : K.IsSupported e
i' : ι'
hi' : ¬∃ (i : ι), e.f i = i'
neg : (K.truncGE e).ExactAt i'
⊢ QuasiIsoAt (L := K.truncGE e) (K.πTruncGE e) i'
have neg₁ : K.ExactAt i'
Tactic: rw [quasiIsoAt_iff_exactAt (K.πTruncGE e) i']
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGEHomology (goal)
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
⊢ (K.truncGE e).Acyclic ↔ K.IsSupportedOutside e
have mp : (K.truncGE e).Acyclic → K.IsSupportedOutside e
Tactic: constructor
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGEHomology (goal)
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
mp : (K.truncGE e).Acyclic → K.IsSupportedOutside e
⊢ (K.truncGE e).Acyclic ↔ K.IsSupportedOutside e
have mpr : K.IsSupportedOutside e → (K.truncGE e).Acyclic
Tactic: constructor
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGEHomology (goal)
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
hK : K.IsSupportedOutside e
i' : ι'
⊢ (K.truncGE e).ExactAt i'
have pos : ∀ (hi' : ∃ (i : ι), e.f i = i'), (K.truncGE e).ExactAt i'
Tactic: by_cases hi' : ∃ i, e.f i = i'
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGEHomology (goal)
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
hK : K.IsSupportedOutside e
i' : ι'
pos : ∀ (hi' : ∃ (i : ι), e.f i = i'), (K.truncGE e).ExactAt i'
⊢ (K.truncGE e).ExactAt i'
have neg : ∀ (hi' : ¬∃ (i : ι), e.f i = i'), (K.truncGE e).ExactAt i'
Tactic: by_cases hi' : ∃ i, e.f i = i'
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGEHomology (goal)
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
hK : K.IsSupportedOutside e
i' : ι'
hi' : ∃ (i : ι), e.f i = i'
⊢ (K.truncGE e).ExactAt i'
have pos : ∀ (i : ι), (K.truncGE e).ExactAt (e.f i)
Tactic: obtain ⟨i, rfl⟩ := hi'
Cleared hypotheses: i', hi'
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGEHomology (goal)
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 : HomologicalComplex.{v_1, u_3, u_2} C c'
φ : K ⟶ L
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
inst_4 : ∀ (i' : ι'), L.HasHomology i'
inst_5 : HasZeroObject.{v_1, u_3} C
i : ι
⊢ QuasiIsoAt (K := K.truncGE e) (L := L.truncGE e) (truncGEMap φ e) (e.f i) ↔ QuasiIsoAt φ (e.f i)
have rw : QuasiIsoAt (L := L.truncGE e) (CategoryStruct.comp (Y := K.truncGE e) (K.πTruncGE e) (truncGEMap φ e)) (e.f i) ↔
  QuasiIsoAt φ (e.f i)
Tactic: rw [← quasiIsoAt_iff_comp_left (K.πTruncGE e), πTruncGE_naturality φ e,
      quasiIsoAt_iff_comp_right]
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGEHomology (goal)
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 : HomologicalComplex.{v_1, u_3, u_2} C c'
φ : K ⟶ L
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
inst_4 : ∀ (i' : ι'), L.HasHomology i'
inst_5 : HasZeroObject.{v_1, u_3} C
i : ι
rw :
  QuasiIsoAt (L := L.truncGE e) (CategoryStruct.comp (Y := K.truncGE e) (K.πTruncGE e) (truncGEMap φ e)) (e.f i) ↔
    QuasiIsoAt φ (e.f i)
⊢ QuasiIsoAt (K := K.truncGE e) (L := L.truncGE e) (truncGEMap φ e) (e.f i) ↔ QuasiIsoAt φ (e.f i)
have rw₁ : QuasiIsoAt (L := L.truncGE e) (φ ≫ L.πTruncGE e) (e.f i) ↔ QuasiIsoAt φ (e.f i)
Tactic: rw [← quasiIsoAt_iff_comp_left (K.πTruncGE e), πTruncGE_naturality φ e,
      quasiIsoAt_iff_comp_right]
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGEHomology (goal)
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 : HomologicalComplex.{v_1, u_3, u_2} C c'
φ : K ⟶ L
e : c.Embedding c'
inst_2 : e.IsTruncGE
inst_3 : ∀ (i' : ι'), K.HasHomology i'
inst_4 : ∀ (i' : ι'), L.HasHomology i'
inst_5 : HasZeroObject.{v_1, u_3} C
i : ι
rw :
  QuasiIsoAt (L := L.truncGE e) (CategoryStruct.comp (Y := K.truncGE e) (K.πTruncGE e) (truncGEMap φ e)) (e.f i) ↔
    QuasiIsoAt φ (e.f i)
rw₁ : QuasiIsoAt (L := L.truncGE e) (φ ≫ L.πTruncGE e) (e.f i) ↔ QuasiIsoAt φ (e.f i)
⊢ QuasiIsoAt (K := K.truncGE e) (L := L.truncGE e) (truncGEMap φ e) (e.f i) ↔ QuasiIsoAt φ (e.f i)
have rw₂ : QuasiIsoAt φ (e.f i) ↔ QuasiIsoAt φ (e.f i)
Tactic: rw [← quasiIsoAt_iff_comp_left (K.πTruncGE e), πTruncGE_naturality φ e,
      quasiIsoAt_iff_comp_right]