awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.RestrictionHomology (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.IsRelIff
i j k : ι
hi : c.prev j = i
hk : c.next j = k
i' j' k' : ι'
hi' : e.f i = i'
hj' : e.f j = j'
hk' : e.f k = k'
hi'' : c'.prev j' = i'
hk'' : c'.next j' = k'
inst_3 : K.HasHomology j'
inst_4 : (K.restriction e).HasHomology j
this_1 : ((K.restriction e).sc' i j k).HasHomology
this : (K.sc' i' j' k').HasHomology
⊢ Eq (α := (K.sc j').cycles ⟶ ((K.restriction e).sc j).opcycles)
    ((K.sc j').iCycles ≫
        CategoryStruct.comp (Y := K.X (e.f j)) (Iso.inv (K.restrictionXIso e hj') : K.X j' ⟶ K.X (e.f j))
          (((K.restriction e).sc j).pOpcycles : ((K.restriction e).sc j).X₂ ⟶ ((K.restriction e).sc j).opcycles) :
      (K.sc j').cycles ⟶ ((K.restriction e).sc j).opcycles)
    (CategoryStruct.comp (Y := (K.sc j').homology) (K.sc j').homologyπ
        (K.homologyι j' ≫ Iso.inv (K.restrictionOpcyclesIso e i j hi hi' hj' hi'')) :
      (K.sc j').cycles ⟶ (K.restriction e).opcycles j)
have refine : Eq (α := (K.sc j').cycles ⟶ ((K.restriction e).sc j).opcycles)
  ((K.sc j').iCycles ≫
      CategoryStruct.comp (Y := (K.sc j').opcycles) ((K.sc j').pOpcycles : (K.sc j').X₂ ⟶ (K.sc j').opcycles)
        (Iso.inv (K.restrictionOpcyclesIso e i j hi hi' hj' hi'') : K.opcycles j' ⟶ (K.restriction e).opcycles j) :
    (K.sc j').cycles ⟶ ((K.restriction e).sc j).opcycles)
  ((K.sc j').iCycles ≫
      CategoryStruct.comp (Y := K.X (e.f j)) (Iso.inv (K.restrictionXIso e hj') : K.X j' ⟶ K.X (e.f j))
        (((K.restriction e).sc j).pOpcycles : ((K.restriction e).sc j).X₂ ⟶ ((K.restriction e).sc j).opcycles) :
    (K.sc j').cycles ⟶ ((K.restriction e).sc j).opcycles)
Tactic: refine ((ShortComplex.homology_π_ι_assoc _ _).trans ?_).symm
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.RestrictionHomology (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.IsRelIff
i j k : ι
hi : c.prev j = i
hk : c.next j = k
i' j' k' : ι'
hi' : e.f i = i'
hj' : e.f j = j'
hk' : e.f k = k'
hi'' : c'.prev j' = i'
hk'' : c'.next j' = k'
inst_3 : K.HasHomology j'
inst_4 : (K.restriction e).HasHomology j
this_1 : ((K.restriction e).sc' i j k).HasHomology
this : (K.sc' i' j' k').HasHomology
⊢ Eq (α := (K.sc j').cycles ⟶ ((K.restriction e).sc j).opcycles)
    ((K.sc j').iCycles ≫
        CategoryStruct.comp (Y := (K.sc j').opcycles) ((K.sc j').pOpcycles : (K.sc j').X₂ ⟶ (K.sc j').opcycles)
          (Iso.inv (K.restrictionOpcyclesIso e i j hi hi' hj' hi'') : K.opcycles j' ⟶ (K.restriction e).opcycles j) :
      (K.sc j').cycles ⟶ ((K.restriction e).sc j).opcycles)
    ((K.sc j').iCycles ≫
        CategoryStruct.comp (Y := K.X (e.f j)) (Iso.inv (K.restrictionXIso e hj') : K.X j' ⟶ K.X (e.f j))
          (((K.restriction e).sc j).pOpcycles : ((K.restriction e).sc j).X₂ ⟶ ((K.restriction e).sc j).opcycles) :
      (K.sc j').cycles ⟶ ((K.restriction e).sc j).opcycles)
have e_a : Eq (α := (K.sc j').X₂ ⟶ ((K.restriction e).sc j).opcycles)
  (CategoryStruct.comp (Y := (K.sc j').opcycles) ((K.sc j').pOpcycles : (K.sc j').X₂ ⟶ (K.sc j').opcycles)
      (Iso.inv (K.restrictionOpcyclesIso e i j hi hi' hj' hi'') : K.opcycles j' ⟶ (K.restriction e).opcycles j) :
    (K.sc j').X₂ ⟶ ((K.restriction e).sc j).opcycles)
  (CategoryStruct.comp (Y := K.X (e.f j)) (Iso.inv (K.restrictionXIso e hj') : K.X j' ⟶ K.X (e.f j))
      (((K.restriction e).sc j).pOpcycles : ((K.restriction e).sc j).X₂ ⟶ ((K.restriction e).sc j).opcycles) :
    K.X j' ⟶ ((K.restriction e).sc j).opcycles)
Tactic: congr 1
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.RestrictionHomology (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.IsRelIff
i j k : ι
hi : c.prev j = i
hk : c.next j = k
i' j' k' : ι'
hi' : e.f i = i'
hj' : e.f j = j'
hk' : e.f k = k'
hi'' : c'.prev j' = i'
hk'' : c'.next j' = k'
inst_3 : K.HasHomology j'
inst_4 : (K.restriction e).HasHomology j
⊢ Eq (α := (K.restriction e).homology j ⟶ K.opcycles j')
    (Iso.hom (K.restrictionHomologyIso e i j k hi hk hi' hj' hk' hi'' hk'') ≫ K.homologyι j')
    ((K.restriction e).homologyι j ≫ Iso.hom (K.restrictionOpcyclesIso e i j hi hi' hj' hi''))
have rw : Eq (α := K.homology j' ⟶ K.opcycles j')
  (CategoryStruct.comp (Y := (K.restriction e).homology j)
    (Iso.inv (K.restrictionHomologyIso e i j k hi hk hi' hj' hk' hi'' hk''))
    (Iso.hom (K.restrictionHomologyIso e i j k hi hk hi' hj' hk' hi'' hk'') ≫ K.homologyι j'))
  (CategoryStruct.comp (Y := (K.restriction e).homology j)
    (Iso.inv (K.restrictionHomologyIso e i j k hi hk hi' hj' hk' hi'' hk''))
    ((K.restriction e).homologyι j ≫ Iso.hom (K.restrictionOpcyclesIso e i j hi hi' hj' hi'')))
Tactic: rw [← cancel_epi (K.restrictionHomologyIso e i j k hi hk hi' hj' hk' hi'' hk'').inv,
    Iso.inv_hom_id_assoc, restrictionHomologyIso_inv_homologyι_assoc,
      Iso.inv_hom_id, comp_id]
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.RestrictionHomology (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.IsRelIff
i j k : ι
hi : c.prev j = i
hk : c.next j = k
i' j' k' : ι'
hi' : e.f i = i'
hj' : e.f j = j'
hk' : e.f k = k'
hi'' : c'.prev j' = i'
hk'' : c'.next j' = k'
inst_3 : K.HasHomology j'
inst_4 : (K.restriction e).HasHomology j
rw :
  Eq (α := K.homology j' ⟶ K.opcycles j')
    (CategoryStruct.comp (Y := (K.restriction e).homology j)
      (Iso.inv (K.restrictionHomologyIso e i j k hi hk hi' hj' hk' hi'' hk''))
      (Iso.hom (K.restrictionHomologyIso e i j k hi hk hi' hj' hk' hi'' hk'') ≫ K.homologyι j'))
    (CategoryStruct.comp (Y := (K.restriction e).homology j)
      (Iso.inv (K.restrictionHomologyIso e i j k hi hk hi' hj' hk' hi'' hk''))
      ((K.restriction e).homologyι j ≫ Iso.hom (K.restrictionOpcyclesIso e i j hi hi' hj' hi'')))
⊢ Eq (α := (K.restriction e).homology j ⟶ K.opcycles j')
    (Iso.hom (K.restrictionHomologyIso e i j k hi hk hi' hj' hk' hi'' hk'') ≫ K.homologyι j')
    ((K.restriction e).homologyι j ≫ Iso.hom (K.restrictionOpcyclesIso e i j hi hi' hj' hi''))
have rw₁ : Eq (α := K.homology j' ⟶ K.opcycles j') (K.homologyι j')
  (CategoryStruct.comp (Y := (K.restriction e).homology j)
    (Iso.inv (K.restrictionHomologyIso e i j k hi hk hi' hj' hk' hi'' hk''))
    ((K.restriction e).homologyι j ≫ Iso.hom (K.restrictionOpcyclesIso e i j hi hi' hj' hi'')))
Tactic: rw [← cancel_epi (K.restrictionHomologyIso e i j k hi hk hi' hj' hk' hi'' hk'').inv,
    Iso.inv_hom_id_assoc, restrictionHomologyIso_inv_homologyι_assoc,
      Iso.inv_hom_id, comp_id]
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.RestrictionHomology (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.IsRelIff
i j k : ι
hi : c.prev j = i
hk : c.next j = k
i' j' k' : ι'
hi' : e.f i = i'
hj' : e.f j = j'
hk' : e.f k = k'
hi'' : c'.prev j' = i'
hk'' : c'.next j' = k'
inst_3 : K.HasHomology j'
inst_4 : (K.restriction e).HasHomology j
rw :
  Eq (α := K.homology j' ⟶ K.opcycles j')
    (CategoryStruct.comp (Y := (K.restriction e).homology j)
      (Iso.inv (K.restrictionHomologyIso e i j k hi hk hi' hj' hk' hi'' hk''))
      (Iso.hom (K.restrictionHomologyIso e i j k hi hk hi' hj' hk' hi'' hk'') ≫ K.homologyι j'))
    (CategoryStruct.comp (Y := (K.restriction e).homology j)
      (Iso.inv (K.restrictionHomologyIso e i j k hi hk hi' hj' hk' hi'' hk''))
      ((K.restriction e).homologyι j ≫ Iso.hom (K.restrictionOpcyclesIso e i j hi hi' hj' hi'')))
rw₁ :
  Eq (α := K.homology j' ⟶ K.opcycles j') (K.homologyι j')
    (CategoryStruct.comp (Y := (K.restriction e).homology j)
      (Iso.inv (K.restrictionHomologyIso e i j k hi hk hi' hj' hk' hi'' hk''))
      ((K.restriction e).homologyι j ≫ Iso.hom (K.restrictionOpcyclesIso e i j hi hi' hj' hi'')))
⊢ Eq (α := (K.restriction e).homology j ⟶ K.opcycles j')
    (Iso.hom (K.restrictionHomologyIso e i j k hi hk hi' hj' hk' hi'' hk'') ≫ K.homologyι j')
    ((K.restriction e).homologyι j ≫ Iso.hom (K.restrictionOpcyclesIso e i j hi hi' hj' hi''))
have rw₂ : Eq (α := K.homology j' ⟶ K.opcycles j') (K.homologyι j')
  (K.homologyι j' ≫
    CategoryStruct.comp (Y := (K.restriction e).opcycles j) (Iso.inv (K.restrictionOpcyclesIso e i j hi hi' hj' hi''))
      (Iso.hom (K.restrictionOpcyclesIso e i j hi hi' hj' hi'')))
Tactic: rw [← cancel_epi (K.restrictionHomologyIso e i j k hi hk hi' hj' hk' hi'' hk'').inv,
    Iso.inv_hom_id_assoc, restrictionHomologyIso_inv_homologyι_assoc,
      Iso.inv_hom_id, comp_id]
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.RestrictionHomology (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.IsRelIff
i j k : ι
hi : c.prev j = i
hk : c.next j = k
i' j' k' : ι'
hi' : e.f i = i'
hj' : e.f j = j'
hk' : e.f k = k'
hi'' : c'.prev j' = i'
hk'' : c'.next j' = k'
inst_3 : K.HasHomology j'
inst_4 : (K.restriction e).HasHomology j
rw :
  Eq (α := K.homology j' ⟶ K.opcycles j')
    (CategoryStruct.comp (Y := (K.restriction e).homology j)
      (Iso.inv (K.restrictionHomologyIso e i j k hi hk hi' hj' hk' hi'' hk''))
      (Iso.hom (K.restrictionHomologyIso e i j k hi hk hi' hj' hk' hi'' hk'') ≫ K.homologyι j'))
    (CategoryStruct.comp (Y := (K.restriction e).homology j)
      (Iso.inv (K.restrictionHomologyIso e i j k hi hk hi' hj' hk' hi'' hk''))
      ((K.restriction e).homologyι j ≫ Iso.hom (K.restrictionOpcyclesIso e i j hi hi' hj' hi'')))
rw₁ :
  Eq (α := K.homology j' ⟶ K.opcycles j') (K.homologyι j')
    (CategoryStruct.comp (Y := (K.restriction e).homology j)
      (Iso.inv (K.restrictionHomologyIso e i j k hi hk hi' hj' hk' hi'' hk''))
      ((K.restriction e).homologyι j ≫ Iso.hom (K.restrictionOpcyclesIso e i j hi hi' hj' hi'')))
rw₂ :
  Eq (α := K.homology j' ⟶ K.opcycles j') (K.homologyι j')
    (K.homologyι j' ≫
      CategoryStruct.comp (Y := (K.restriction e).opcycles j) (Iso.inv (K.restrictionOpcyclesIso e i j hi hi' hj' hi''))
        (Iso.hom (K.restrictionOpcyclesIso e i j hi hi' hj' hi'')))
⊢ Eq (α := (K.restriction e).homology j ⟶ K.opcycles j')
    (Iso.hom (K.restrictionHomologyIso e i j k hi hk hi' hj' hk' hi'' hk'') ≫ K.homologyι j')
    ((K.restriction e).homologyι j ≫ Iso.hom (K.restrictionOpcyclesIso e i j hi hi' hj' hi''))
have rw₃ : K.homologyι j' = K.homologyι j' ≫ 𝟙 (K.opcycles j')
Tactic: rw [← cancel_epi (K.restrictionHomologyIso e i j k hi hk hi' hj' hk' hi'' hk'').inv,
    Iso.inv_hom_id_assoc, restrictionHomologyIso_inv_homologyι_assoc,
      Iso.inv_hom_id, comp_id]
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.RestrictionHomology (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.IsRelIff
i j k : ι
hi : c.prev j = i
hk : c.next j = k
i' j' k' : ι'
hi' : e.f i = i'
hj' : e.f j = j'
hk' : e.f k = k'
hi'' : c'.prev j' = i'
hk'' : c'.next j' = k'
inst_3 : K.HasHomology j'
inst_4 : (K.restriction e).HasHomology j
rw :
  Eq (α := K.homology j' ⟶ K.opcycles j')
    (CategoryStruct.comp (Y := (K.restriction e).homology j)
      (Iso.inv (K.restrictionHomologyIso e i j k hi hk hi' hj' hk' hi'' hk''))
      (Iso.hom (K.restrictionHomologyIso e i j k hi hk hi' hj' hk' hi'' hk'') ≫ K.homologyι j'))
    (CategoryStruct.comp (Y := (K.restriction e).homology j)
      (Iso.inv (K.restrictionHomologyIso e i j k hi hk hi' hj' hk' hi'' hk''))
      ((K.restriction e).homologyι j ≫ Iso.hom (K.restrictionOpcyclesIso e i j hi hi' hj' hi'')))
rw₁ :
  Eq (α := K.homology j' ⟶ K.opcycles j') (K.homologyι j')
    (CategoryStruct.comp (Y := (K.restriction e).homology j)
      (Iso.inv (K.restrictionHomologyIso e i j k hi hk hi' hj' hk' hi'' hk''))
      ((K.restriction e).homologyι j ≫ Iso.hom (K.restrictionOpcyclesIso e i j hi hi' hj' hi'')))
rw₂ :
  Eq (α := K.homology j' ⟶ K.opcycles j') (K.homologyι j')
    (K.homologyι j' ≫
      CategoryStruct.comp (Y := (K.restriction e).opcycles j) (Iso.inv (K.restrictionOpcyclesIso e i j hi hi' hj' hi''))
        (Iso.hom (K.restrictionOpcyclesIso e i j hi hi' hj' hi'')))
rw₃ : K.homologyι j' = K.homologyι j' ≫ 𝟙 (K.opcycles j')
⊢ Eq (α := (K.restriction e).homology j ⟶ K.opcycles j')
    (Iso.hom (K.restrictionHomologyIso e i j k hi hk hi' hj' hk' hi'' hk'') ≫ K.homologyι j')
    ((K.restriction e).homologyι j ≫ Iso.hom (K.restrictionOpcyclesIso e i j hi hi' hj' hi''))
have rw₄ : Eq (α := K.homology j' ⟶ K.opcycles j') (K.homologyι j') (K.homologyι j')
Tactic: rw [← cancel_epi (K.restrictionHomologyIso e i j k hi hk hi' hj' hk' hi'' hk'').inv,
    Iso.inv_hom_id_assoc, restrictionHomologyIso_inv_homologyι_assoc,
      Iso.inv_hom_id, comp_id]
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.StupidTrunc (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
inst_2 : HasZeroObject.{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_3 : e.IsRelIff
i : ι
i' : ι'
hi' : e.f i = i'
⊢ (K.restriction e).X i = K.X i'
have subst : (K.restriction e).X i = K.X (e.f i)
Tactic: subst hi'
Cleared hypotheses: i', hi'
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.StupidTrunc (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
inst_2 : HasZeroObject.{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_3 : e.IsRelIff
ι'' : Type u_4
c'' : ComplexShape ι''
e' : c''.Embedding c'
inst_4 : K.IsStrictlySupported e'
i' : ι'
hi' : ∀ (i : ι''), e'.f i ≠ i'
⊢ IsZero.{v_1, u_3} ((K.stupidTrunc e).X i')
have pos : ∀ (hi'' : ∃ (i : ι), e.f i = i'), IsZero.{v_1, u_3} ((K.stupidTrunc e).X i')
Tactic: by_cases hi'' : ∃ i, e.f i = i'
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.StupidTrunc (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
inst_2 : HasZeroObject.{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_3 : e.IsRelIff
ι'' : Type u_4
c'' : ComplexShape ι''
e' : c''.Embedding c'
inst_4 : K.IsStrictlySupported e'
i' : ι'
hi' : ∀ (i : ι''), e'.f i ≠ i'
pos : ∀ (hi'' : ∃ (i : ι), e.f i = i'), IsZero.{v_1, u_3} ((K.stupidTrunc e).X i')
⊢ IsZero.{v_1, u_3} ((K.stupidTrunc e).X i')
have neg : ∀ (hi'' : ¬∃ (i : ι), e.f i = i'), IsZero.{v_1, u_3} ((K.stupidTrunc e).X i')
Tactic: by_cases hi'' : ∃ i, e.f i = i'
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.StupidTrunc (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
inst_2 : HasZeroObject.{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_3 : e.IsRelIff
ι'' : Type u_4
c'' : ComplexShape ι''
e' : c''.Embedding c'
inst_4 : 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.stupidTrunc e).X i')
have pos : ∀ (i : ι) (hi : e.f i = i'), IsZero.{v_1, u_3} ((K.stupidTrunc e).X i')
Tactic: obtain ⟨i, hi⟩ := hi''
Cleared hypotheses: hi''
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.StupidTrunc (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
inst_2 : HasZeroObject.{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_3 : e.IsRelIff
ι'' : Type u_4
c'' : ComplexShape ι''
e' : c''.Embedding c'
inst_4 : 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.stupidTrunc e).X i')
have neg : ∀ (i : ι), e.f i ≠ i'
Tactic: apply isZero_stupidTrunc_X
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.StupidTrunc (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
inst_2 : HasZeroObject.{v_1, u_3} C
K : HomologicalComplex.{v_1, u_3, u_2} C c'
e : c.Embedding c'
inst_3 : e.IsRelIff
⊢ IsZero (K.stupidTrunc e) ↔ K.IsStrictlySupportedOutside e
have mp : IsZero (K.stupidTrunc e) → K.IsStrictlySupportedOutside e
Tactic: constructor
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.StupidTrunc (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
inst_2 : HasZeroObject.{v_1, u_3} C
K : HomologicalComplex.{v_1, u_3, u_2} C c'
e : c.Embedding c'
inst_3 : e.IsRelIff
mp : IsZero (K.stupidTrunc e) → K.IsStrictlySupportedOutside e
⊢ IsZero (K.stupidTrunc e) ↔ K.IsStrictlySupportedOutside e
have mpr : K.IsStrictlySupportedOutside e → IsZero (K.stupidTrunc e)
Tactic: constructor
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.StupidTrunc (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
inst_2 : HasZeroObject.{v_1, u_3} C
K : HomologicalComplex.{v_1, u_3, u_2} C c'
e : c.Embedding c'
inst_3 : e.IsRelIff
h : K.IsStrictlySupportedOutside e
⊢ IsZero (K.stupidTrunc e)
have mpr : (K.stupidTrunc e).IsStrictlySupported e ∧ (K.stupidTrunc e).IsStrictlySupportedOutside e
Tactic: rw [isZero_iff_isStrictlySupported_and_isStrictlySupportedOutside _ e]
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.StupidTrunc (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
inst_2 : HasZeroObject.{v_1, u_3} C
K : HomologicalComplex.{v_1, u_3, u_2} C c'
e : c.Embedding c'
inst_3 : e.IsRelIff
h : K.IsStrictlySupportedOutside e
⊢ (K.stupidTrunc e).IsStrictlySupported e ∧ (K.stupidTrunc e).IsStrictlySupportedOutside e
have mpr : (K.stupidTrunc e).IsStrictlySupported e
Tactic: constructor
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.StupidTrunc (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
inst_2 : HasZeroObject.{v_1, u_3} C
K : HomologicalComplex.{v_1, u_3, u_2} C c'
e : c.Embedding c'
inst_3 : e.IsRelIff
h : K.IsStrictlySupportedOutside e
mpr : (K.stupidTrunc e).IsStrictlySupported e
⊢ (K.stupidTrunc e).IsStrictlySupported e ∧ (K.stupidTrunc e).IsStrictlySupportedOutside e
have mpr₁ : (K.stupidTrunc e).IsStrictlySupportedOutside e
Tactic: constructor
-- Lean 4 proof step from Mathlib.Algebra.Homology.Embedding.StupidTrunc (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
inst_2 : HasZeroObject.{v_1, u_3} C
K L : HomologicalComplex.{v_1, u_3, u_2} C c'
φ : K ⟶ L
e : c.Embedding c'
inst_3 : e.IsRelIff
i : ι
i' : ι'
hi : e.f i = i'
⊢ Eq (α := (K.stupidTrunc e).X i' ⟶ L.X i') (Hom.f (stupidTruncMap φ e) i' ≫ Iso.hom (L.stupidTruncXIso e hi))
    (Iso.hom (K.stupidTruncXIso e hi) ≫ φ.f i')
have subst : Eq (α := (K.stupidTrunc e).X (e.f i) ⟶ L.X (e.f i))
  (Hom.f (stupidTruncMap φ e) (e.f i) ≫ Iso.hom (L.stupidTruncXIso e sorry)) (Iso.hom (K.stupidTruncXIso e sorry) ≫ φ.f (e.f i))
Tactic: subst hi
Cleared hypotheses: 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 : 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
⊢ Eq (α := X K e i ⟶ X K e k) (d K e i j ≫ d K e j k) 0
have dsimp : Eq.{v_1 + 1} (α := X K e i ⟶ X K e k)
  (CategoryStruct.comp (Y := X K e j)
    (if hij : c.Rel i j then
      if hi : e.BoundaryGE i then Iso.hom (XIsoOpcycles K e hi) ≫ K.fromOpcycles (e.f i) (e.f j) ≫ Iso.inv (XIso K e sorry)
      else Iso.hom (XIso K e hi) ≫ K.d (e.f i) (e.f j) ≫ Iso.inv (XIso K e sorry)
    else 0)
    (if hij : c.Rel j k then
      if hi : e.BoundaryGE j then Iso.hom (XIsoOpcycles K e hi) ≫ K.fromOpcycles (e.f j) (e.f k) ≫ Iso.inv (XIso K e sorry)
      else Iso.hom (XIso K e hi) ≫ K.d (e.f j) (e.f k) ≫ Iso.inv (XIso K e sorry)
    else 0))
  0
Tactic: dsimp [d]
-- 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 j k : ι
⊢ Eq.{v_1 + 1} (α := X K e i ⟶ X K e k)
    (CategoryStruct.comp (Y := X K e j)
      (if hij : c.Rel i j then
        if hi : e.BoundaryGE i then
          Iso.hom (XIsoOpcycles K e hi) ≫ K.fromOpcycles (e.f i) (e.f j) ≫ Iso.inv (XIso K e ⋯)
        else Iso.hom (XIso K e hi) ≫ K.d (e.f i) (e.f j) ≫ Iso.inv (XIso K e ⋯)
      else 0)
      (if hij : c.Rel j k then
        if hi : e.BoundaryGE j then
          Iso.hom (XIsoOpcycles K e hi) ≫ K.fromOpcycles (e.f j) (e.f k) ≫ Iso.inv (XIso K e ⋯)
        else Iso.hom (XIso K e hi) ≫ K.d (e.f j) (e.f k) ≫ Iso.inv (XIso K e ⋯)
      else 0))
    0
have pos : ∀ (hij : c.Rel i j),
  Eq.{v_1 + 1} (α := X K e i ⟶ X K e k)
    (CategoryStruct.comp (Y := X K e j)
      (if hij : c.Rel i j then
        if hi : e.BoundaryGE i then
          Iso.hom (XIsoOpcycles K e hi) ≫ K.fromOpcycles (e.f i) (e.f j) ≫ Iso.inv (XIso K e sorry)
        else Iso.hom (XIso K e hi) ≫ K.d (e.f i) (e.f j) ≫ Iso.inv (XIso K e sorry)
      else 0)
      (if hij : c.Rel j k then
        if hi : e.BoundaryGE j then
          Iso.hom (XIsoOpcycles K e hi) ≫ K.fromOpcycles (e.f j) (e.f k) ≫ Iso.inv (XIso K e sorry)
        else Iso.hom (XIso K e hi) ≫ K.d (e.f j) (e.f k) ≫ Iso.inv (XIso K e sorry)
      else 0))
    0
Tactic: by_cases hij : c.Rel i j