awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGE (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'
j : ι
hj : e.BoundaryGE j
i' : ι'
h : c'.Rel i' (e.f j)
⊢ Eq (α := K.X i' ⟶ (K.truncGE' e).X j) (K.d i' (e.f j) ≫ restrictionToTruncGE'.f K e j) 0
have rw : Eq (α := K.X i' ⟶ (K.truncGE' e).X j)
  (K.d i' (e.f j) ≫ Iso.hom (K.restrictionXIso e sorry) ≫ K.pOpcycles (e.f j) ≫ Iso.inv (K.truncGE'XIsoOpcycles e sorry hj)) 0
Tactic: rw [restrictionToTruncGE'.f_eq_iso_hom_pOpcycles_iso_inv K e rfl hj]
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGE (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'
i : ι
hi : ¬e.BoundaryGE i
⊢ IsIso (X := (K.restriction e).X i) (Y := (K.truncGE' e).X i) (Hom.f (K.restrictionToTruncGE' e) i)
have rw : IsIso (X := (K.restriction e).X i) (Y := (K.truncGE' e).X i)
  (Iso.hom (K.restrictionXIso e sorry) ≫ Iso.inv (K.truncGE'XIso e sorry hi))
Tactic: rw [K.restrictionToTruncGE'_f_eq_iso_hom_iso_inv e rfl hi]
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGE (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'
⊢ Eq (α := K.restriction e ⟶ L.truncGE' e)
    (CategoryStruct.comp (Y := K.truncGE' e) (K.restrictionToTruncGE' e) (truncGE'Map φ e))
    (restrictionMap φ e ≫ L.restrictionToTruncGE' e)
have h : ∀ (i : ι),
  Eq (α := (K.restriction e).X i ⟶ (L.truncGE' e).X i)
    (Hom.f (CategoryStruct.comp (Y := K.truncGE' e) (K.restrictionToTruncGE' e) (truncGE'Map φ e)) i)
    (Hom.f (restrictionMap φ e ≫ L.restrictionToTruncGE' e) i)
Tactic: ext i
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGE (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'
i : ι
⊢ Eq (α := (K.restriction e).X i ⟶ (L.truncGE' e).X i)
    (Hom.f (CategoryStruct.comp (Y := K.truncGE' e) (K.restrictionToTruncGE' e) (truncGE'Map φ e)) i)
    (Hom.f (restrictionMap φ e ≫ L.restrictionToTruncGE' e) i)
have pos : ∀ (hi : e.BoundaryGE i),
  Eq (α := (K.restriction e).X i ⟶ (L.truncGE' e).X i)
    (Hom.f (CategoryStruct.comp (Y := K.truncGE' e) (K.restrictionToTruncGE' e) (truncGE'Map φ e)) i)
    (Hom.f (restrictionMap φ e ≫ L.restrictionToTruncGE' e) i)
Tactic: by_cases hi : e.BoundaryGE i
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGE (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'
i : ι
pos :
  ∀ (hi : e.BoundaryGE i),
    Eq (α := (K.restriction e).X i ⟶ (L.truncGE' e).X i)
      (Hom.f (CategoryStruct.comp (Y := K.truncGE' e) (K.restrictionToTruncGE' e) (truncGE'Map φ e)) i)
      (Hom.f (restrictionMap φ e ≫ L.restrictionToTruncGE' e) i)
⊢ Eq (α := (K.restriction e).X i ⟶ (L.truncGE' e).X i)
    (Hom.f (CategoryStruct.comp (Y := K.truncGE' e) (K.restrictionToTruncGE' e) (truncGE'Map φ e)) i)
    (Hom.f (restrictionMap φ e ≫ L.restrictionToTruncGE' e) i)
have neg : ∀ (hi : ¬e.BoundaryGE i),
  Eq (α := (K.restriction e).X i ⟶ (L.truncGE' e).X i)
    (Hom.f (CategoryStruct.comp (Y := K.truncGE' e) (K.restrictionToTruncGE' e) (truncGE'Map φ e)) i)
    (Hom.f (restrictionMap φ e ≫ L.restrictionToTruncGE' e) i)
Tactic: by_cases hi : e.BoundaryGE i
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGE (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 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'
i : ι
⊢ Epi (X := (K.restriction e).X i) (Y := (K.truncGE' e).X i) (Hom.f (K.restrictionToTruncGE' e) i)
have pos : ∀ (hi : e.BoundaryGE i),
  Epi (X := (K.restriction e).X i) (Y := (K.truncGE' e).X i) (Hom.f (K.restrictionToTruncGE' e) i)
Tactic: by_cases hi : e.BoundaryGE i
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGE (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 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'
i : ι
pos :
  ∀ (hi : e.BoundaryGE i),
    Epi (X := (K.restriction e).X i) (Y := (K.truncGE' e).X i) (Hom.f (K.restrictionToTruncGE' e) i)
⊢ Epi (X := (K.restriction e).X i) (Y := (K.truncGE' e).X i) (Hom.f (K.restrictionToTruncGE' e) i)
have neg : ∀ (hi : ¬e.BoundaryGE i),
  Epi (X := (K.restriction e).X i) (Y := (K.truncGE' e).X i) (Hom.f (K.restrictionToTruncGE' e) i)
Tactic: by_cases hi : e.BoundaryGE i
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGE (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 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'
i : ι
hi : e.BoundaryGE i
⊢ Epi (X := (K.restriction e).X i) (Y := (K.truncGE' e).X i) (Hom.f (K.restrictionToTruncGE' e) i)
have pos : Epi (X := (K.restriction e).X i) (Y := (K.truncGE' e).X i)
  (Iso.hom (K.restrictionXIso e sorry) ≫ K.pOpcycles (e.f i) ≫ Iso.inv (K.truncGE'XIsoOpcycles e sorry hi))
Tactic: rw [K.restrictionToTruncGE'_f_eq_iso_hom_pOpcycles_iso_inv e rfl hi]
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGE (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 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'
i : ι
hi : ¬e.BoundaryGE i
⊢ Epi (X := (K.restriction e).X i) (Y := (K.truncGE' e).X i) (Hom.f (K.restrictionToTruncGE' e) i)
have this : IsIso (X := (K.restriction e).X i) (Y := (K.truncGE' e).X i) (Hom.f (K.restrictionToTruncGE' e) i)
Tactic: have := K.isIso_restrictionToTruncGE' e i hi
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGE (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 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 : K.IsStrictlySupported e
i : ι
⊢ IsIso (X := (K.restriction e).X i) (Y := (K.truncGE' e).X i) (Hom.f (K.restrictionToTruncGE' e) i)
have pos : ∀ (hi : e.BoundaryGE i),
  IsIso (X := (K.restriction e).X i) (Y := (K.truncGE' e).X i) (Hom.f (K.restrictionToTruncGE' e) i)
Tactic: by_cases hi : e.BoundaryGE i
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGE (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 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 : K.IsStrictlySupported e
i : ι
pos :
  ∀ (hi : e.BoundaryGE i),
    IsIso (X := (K.restriction e).X i) (Y := (K.truncGE' e).X i) (Hom.f (K.restrictionToTruncGE' e) i)
⊢ IsIso (X := (K.restriction e).X i) (Y := (K.truncGE' e).X i) (Hom.f (K.restrictionToTruncGE' e) i)
have neg : ∀ (hi : ¬e.BoundaryGE i),
  IsIso (X := (K.restriction e).X i) (Y := (K.truncGE' e).X i) (Hom.f (K.restrictionToTruncGE' e) i)
Tactic: by_cases hi : e.BoundaryGE i
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGE (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 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 : K.IsStrictlySupported e
i : ι
hi : e.BoundaryGE i
⊢ IsIso (X := (K.restriction e).X i) (Y := (K.truncGE' e).X i) (Hom.f (K.restrictionToTruncGE' e) i)
have pos : IsIso (X := (K.restriction e).X i) (Y := (K.truncGE' e).X i)
  (Iso.hom (K.restrictionXIso e sorry) ≫ K.pOpcycles (e.f i) ≫ Iso.inv (K.truncGE'XIsoOpcycles e sorry hi))
Tactic: rw [K.restrictionToTruncGE'_f_eq_iso_hom_pOpcycles_iso_inv e rfl hi]
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGE (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 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 : K.IsStrictlySupported e
i : ι
hi : e.BoundaryGE i
⊢ Eq (α := K.X (c'.prev (e.f i)) ⟶ K.X (e.f i)) (K.d (c'.prev (e.f i)) (e.f i)) 0
have hi₁ : c'.Rel (c'.prev (e.f i)) (e.f i)
Tactic: obtain ⟨hi₁, hi₂⟩ := hi
Cleared hypotheses: hi
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGE (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 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 : K.IsStrictlySupported e
i : ι
hi : e.BoundaryGE i
hi₁ : c'.Rel (c'.prev (e.f i)) (e.f i)
⊢ Eq (α := K.X (c'.prev (e.f i)) ⟶ K.X (e.f i)) (K.d (c'.prev (e.f i)) (e.f i)) 0
have hi₂ : ∀ (i_1 : ι), ¬c'.Rel (e.f i_1) (e.f i)
Tactic: obtain ⟨hi₁, hi₂⟩ := hi
Cleared hypotheses: hi
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGE (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 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 : K.IsStrictlySupported e
i : ι
hi : e.BoundaryGE i
⊢ IsIso (X := (K.restriction e).X i) (Y := (K.truncGE' e).X i)
    (Iso.hom (K.restrictionXIso e ⋯) ≫ K.pOpcycles (e.f i) ≫ Iso.inv (K.truncGE'XIsoOpcycles e ⋯ hi))
have this : IsIso (X := K.X (e.f i)) (Y := K.opcycles (e.f i)) (K.pOpcycles (e.f i))
Tactic: have : IsIso (K.pOpcycles (e.f i)) := K.isIso_pOpcycles _ _ rfl (by
      obtain ⟨hi₁, hi₂⟩ := hi
      apply IsZero.eq_of_src (K.isZero_X_of_isStrictlySupported e _
        (fun j hj ↦ hi₂ j (by simpa only [hj] using hi₁))))
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGE (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 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 : K.IsStrictlySupported e
i : ι
hi : ¬e.BoundaryGE i
⊢ IsIso (X := (K.restriction e).X i) (Y := (K.truncGE' e).X i) (Hom.f (K.restrictionToTruncGE' e) i)
have neg : IsIso (X := (K.restriction e).X i) (Y := (K.truncGE' e).X i)
  (Iso.hom (K.restrictionXIso e sorry) ≫ Iso.inv (K.truncGE'XIso e sorry hi))
Tactic: rw [K.restrictionToTruncGE'_f_eq_iso_hom_iso_inv e rfl hi]
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGE (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 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
i' : ι'
⊢ Epi (Y := (K.truncGE e).X i') (Hom.f (K.πTruncGE e) i')
have pos : ∀ (hi' : ∃ (i : ι), e.f i = i'), Epi (Y := (K.truncGE e).X i') (Hom.f (K.πTruncGE e) i')
Tactic: by_cases hi' : ∃ i, e.f i = i'
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGE (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 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
i' : ι'
pos : ∀ (hi' : ∃ (i : ι), e.f i = i'), Epi (Y := (K.truncGE e).X i') (Hom.f (K.πTruncGE e) i')
⊢ Epi (Y := (K.truncGE e).X i') (Hom.f (K.πTruncGE e) i')
have neg : ∀ (hi' : ¬∃ (i : ι), e.f i = i'), Epi (Y := (K.truncGE e).X i') (Hom.f (K.πTruncGE e) i')
Tactic: by_cases hi' : ∃ i, e.f i = i'
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGE (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 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
i' : ι'
hi' : ∃ (i : ι), e.f i = i'
⊢ Epi (Y := (K.truncGE e).X i') (Hom.f (K.πTruncGE e) i')
have pos : ∀ (i : ι) (hi : e.f i = i'), Epi (Y := (K.truncGE e).X i') (Hom.f (K.πTruncGE e) i')
Tactic: obtain ⟨i, hi⟩ := hi'
Cleared hypotheses: hi'
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGE (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 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
i' : ι'
i : ι
hi : e.f i = i'
⊢ Epi (Y := (K.truncGE e).X i') (Hom.f (e.liftExtend (K.restrictionToTruncGE' e) ⋯) i')
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.TruncGE (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
⊢ Eq (α := K ⟶ L.truncGE e) (CategoryStruct.comp (Y := K.truncGE e) (K.πTruncGE e) (truncGEMap φ e)) (φ ≫ L.πTruncGE e)
have a : Eq (α := { φ : K.restriction e ⟶ L.truncGE' e // e.HasLift φ })
  ((e.homEquiv K (L.truncGE' e) :
        (a : K ⟶ (L.truncGE' e).extend e) → { φ : K.restriction e ⟶ L.truncGE' e // e.HasLift φ })
      (CategoryStruct.comp (Y := K.truncGE e) (K.πTruncGE e) (truncGEMap φ e)) :
    { φ : K.restriction e ⟶ L.truncGE' e // e.HasLift φ })
  ((e.homEquiv K (L.truncGE' e) :
        (a : K ⟶ (L.truncGE' e).extend e) → { φ : K.restriction e ⟶ L.truncGE' e // e.HasLift φ })
      (φ ≫ L.πTruncGE e) :
    { φ : K.restriction e ⟶ L.truncGE' e // e.HasLift φ })
Tactic: apply (e.homEquiv _ _).injective
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGE (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
⊢ Eq (α := { φ : K.restriction e ⟶ L.truncGE' e // e.HasLift φ })
    ((e.homEquiv K (L.truncGE' e) :
          (a : K ⟶ (L.truncGE' e).extend e) → { φ : K.restriction e ⟶ L.truncGE' e // e.HasLift φ })
        (CategoryStruct.comp (Y := K.truncGE e) (K.πTruncGE e) (truncGEMap φ e)) :
      { φ : K.restriction e ⟶ L.truncGE' e // e.HasLift φ })
    ((e.homEquiv K (L.truncGE' e) :
          (a : K ⟶ (L.truncGE' e).extend e) → { φ : K.restriction e ⟶ L.truncGE' e // e.HasLift φ })
        (φ ≫ L.πTruncGE e) :
      { φ : K.restriction e ⟶ L.truncGE' e // e.HasLift φ })
have a : Eq (α := K.restriction e ⟶ L.truncGE' e)
  (↑((e.homEquiv K (L.truncGE' e) :
          (a : K ⟶ (L.truncGE' e).extend e) → { φ : K.restriction e ⟶ L.truncGE' e // e.HasLift φ })
        (CategoryStruct.comp (Y := K.truncGE e) (K.πTruncGE e) (truncGEMap φ e))) :
    K.restriction e ⟶ L.truncGE' e)
  (↑((e.homEquiv K (L.truncGE' e) :
          (a : K ⟶ (L.truncGE' e).extend e) → { φ : K.restriction e ⟶ L.truncGE' e // e.HasLift φ })
        (φ ≫ L.πTruncGE e)) :
    K.restriction e ⟶ L.truncGE' e)
Tactic: ext1
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGE (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
⊢ Eq (α := K.restriction e ⟶ L.truncGE' e)
    (e.homRestrict
      (CategoryStruct.comp (Y := K.truncGE e) (e.liftExtend (K.restrictionToTruncGE' e) ⋯)
        (extendMap (truncGE'Map φ e) e)))
    (e.homRestrict (φ ≫ e.liftExtend (L.restrictionToTruncGE' e) ⋯))
have a : Eq (α := K.restriction e ⟶ L.truncGE' e)
  (CategoryStruct.comp (Y := K.truncGE' e) (e.homRestrict (e.liftExtend (K.restrictionToTruncGE' e) sorry))
    (truncGE'Map φ e))
  (e.homRestrict (φ ≫ e.liftExtend (L.restrictionToTruncGE' e) sorry))
Tactic: rw [e.homRestrict_comp_extendMap, e.homRestrict_liftExtend, e.homRestrict_precomp,
    e.homRestrict_liftExtend, restrictionToTruncGE'_naturality]
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGE (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
a :
  Eq (α := K.restriction e ⟶ L.truncGE' e)
    (CategoryStruct.comp (Y := K.truncGE' e) (e.homRestrict (e.liftExtend (K.restrictionToTruncGE' e) ⋯))
      (truncGE'Map φ e))
    (e.homRestrict (φ ≫ e.liftExtend (L.restrictionToTruncGE' e) ⋯))
⊢ Eq (α := K.restriction e ⟶ L.truncGE' e)
    (e.homRestrict
      (CategoryStruct.comp (Y := K.truncGE e) (e.liftExtend (K.restrictionToTruncGE' e) ⋯)
        (extendMap (truncGE'Map φ e) e)))
    (e.homRestrict (φ ≫ e.liftExtend (L.restrictionToTruncGE' e) ⋯))
have a₁ : Eq (α := K.restriction e ⟶ L.truncGE' e)
  (CategoryStruct.comp (Y := K.truncGE' e) (K.restrictionToTruncGE' e) (truncGE'Map φ e))
  (e.homRestrict (φ ≫ e.liftExtend (L.restrictionToTruncGE' e) sorry))
Tactic: rw [e.homRestrict_comp_extendMap, e.homRestrict_liftExtend, e.homRestrict_precomp,
    e.homRestrict_liftExtend, restrictionToTruncGE'_naturality]
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.TruncGE (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
a :
  Eq (α := K.restriction e ⟶ L.truncGE' e)
    (CategoryStruct.comp (Y := K.truncGE' e) (e.homRestrict (e.liftExtend (K.restrictionToTruncGE' e) ⋯))
      (truncGE'Map φ e))
    (e.homRestrict (φ ≫ e.liftExtend (L.restrictionToTruncGE' e) ⋯))
a₁ :
  Eq (α := K.restriction e ⟶ L.truncGE' e)
    (CategoryStruct.comp (Y := K.truncGE' e) (K.restrictionToTruncGE' e) (truncGE'Map φ e))
    (e.homRestrict (φ ≫ e.liftExtend (L.restrictionToTruncGE' e) ⋯))
⊢ Eq (α := K.restriction e ⟶ L.truncGE' e)
    (e.homRestrict
      (CategoryStruct.comp (Y := K.truncGE e) (e.liftExtend (K.restrictionToTruncGE' e) ⋯)
        (extendMap (truncGE'Map φ e) e)))
    (e.homRestrict (φ ≫ e.liftExtend (L.restrictionToTruncGE' e) ⋯))
have a₂ : Eq (α := K.restriction e ⟶ L.truncGE' e)
  (CategoryStruct.comp (Y := K.truncGE' e) (K.restrictionToTruncGE' e) (truncGE'Map φ e))
  (restrictionMap φ e ≫ e.homRestrict (e.liftExtend (L.restrictionToTruncGE' e) sorry))
Tactic: rw [e.homRestrict_comp_extendMap, e.homRestrict_liftExtend, e.homRestrict_precomp,
    e.homRestrict_liftExtend, restrictionToTruncGE'_naturality]