awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
Goal:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
inst_2 : Preorder ι
X : SpectralObject.{u_1, u_2, v_1, u_2} C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r r' : ℤ
hrr' : r + 1 = r'
hr : r₀ ≤ r
pq pq' : κ
i₀ i₁ i₂ i₃ i₃' : ι
hi₀ : i₀ = data.i₀ r pq' ⋯
hi₁ : i₁ = data.i₁ pq'
hi₂ : i₂ = data.i₂ pq'
hi₃ : i₃ = data.i₃ r pq' ⋯
hi₃' : i₃' = data.i₃ r' pq' ⋯
n₀ n₁ n₂ : ℤ
hn₁' : n₁ = data.deg pq'
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
pos :
  ∀ (h : (c r).Rel pq pq'),
    Eq (α := (page X data r hr).X pq ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)
      ((page X data r hr).d pq pq' ≫
          Iso.hom (pageXIso X data r hr pq' i₀ i₁ i₂ i₃ hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' ⋯ ⋯) ≫
            X.mapFourδ₄Toδ₃' i₀ i₁ i₂ i₃ i₃' ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ ⋯ ⋯ :
        (page X data r hr).X pq ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)
      (0 : (page X data r hr).X pq ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)
neg :
  ∀ (h : ¬(c r).Rel pq pq'),
    Eq (α := (page X data r hr).X pq ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)
      ((page X data r hr).d pq pq' ≫
          Iso.hom (pageXIso X data r hr pq' i₀ i₁ i₂ i₃ hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' ⋯ ⋯) ≫
            X.mapFourδ₄Toδ₃' i₀ i₁ i₂ i₃ i₃' ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ ⋯ ⋯ :
        (page X data r hr).X pq ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)
      (0 : (page X data r hr).X pq ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)
⊢ Eq (α := (page X data r hr).X pq ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)
    ((page X data r hr).d pq pq' ≫
        Iso.hom (pageXIso X data r hr pq' i₀ i₁ i₂ i₃ hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' ⋯ ⋯) ≫
          X.mapFourδ₄Toδ₃' i₀ i₁ i₂ i₃ i₃' ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ ⋯ ⋯ :
      (page X data r hr).X pq ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)
    (0 : (page X data r hr).X pq ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)

Tactic:
by_cases h : (c r).Rel pq pq'

Premises used: ComplexShape.Rel
-- Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
Goal:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
inst_2 : Preorder ι
X : SpectralObject.{u_1, u_2, v_1, u_2} C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r r' : ℤ
hrr' : r + 1 = r'
hr : r₀ ≤ r
pq pq' : κ
i₀ i₁ i₂ i₃ i₃' : ι
hi₀ : i₀ = data.i₀ r pq' ⋯
hi₁ : i₁ = data.i₁ pq'
hi₂ : i₂ = data.i₂ pq'
hi₃ : i₃ = data.i₃ r pq' ⋯
hi₃' : i₃' = data.i₃ r' pq' ⋯
n₀ n₁ n₂ : ℤ
hn₁' : n₁ = data.deg pq'
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
h : (c r).Rel pq pq'
pos :
  Eq (α := pageX X data r pq ⋯ ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)
    (pageD X data r pq pq' ⋯ ≫
        Iso.hom (pageXIso X data r hr pq' i₀ i₁ i₂ i₃ hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' ⋯ ⋯) ≫
          X.mapFourδ₄Toδ₃' i₀ i₁ i₂ i₃ i₃' ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ ⋯ ⋯ :
      pageX X data r pq ⋯ ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)
    (0 : (page X data r hr).X pq ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)
⊢ Eq (α := (page X data r hr).X pq ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)
    ((page X data r hr).d pq pq' ≫
        Iso.hom (pageXIso X data r hr pq' i₀ i₁ i₂ i₃ hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' ⋯ ⋯) ≫
          X.mapFourδ₄Toδ₃' i₀ i₁ i₂ i₃ i₃' ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ ⋯ ⋯ :
      (page X data r hr).X pq ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)
    (0 : (page X data r hr).X pq ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)

Tactic:
dsimp
-- Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
Goal:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
inst_2 : Preorder ι
X : SpectralObject.{u_1, u_2, v_1, u_2} C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r r' : ℤ
hrr' : r + 1 = r'
hr : r₀ ≤ r
pq pq' : κ
i₀ i₁ i₂ i₃ i₃' : ι
hi₀ : i₀ = data.i₀ r pq' ⋯
hi₁ : i₁ = data.i₁ pq'
hi₂ : i₂ = data.i₂ pq'
hi₃ : i₃ = data.i₃ r pq' ⋯
hi₃' : i₃' = data.i₃ r' pq' ⋯
n₀ n₁ n₂ : ℤ
hn₁' : n₁ = data.deg pq'
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
h : (c r).Rel pq pq'
⊢ i₃' ≤ data.i₃ r pq ⋯

Tactic:
simpa only [hi₃', data.i₃_next r r' _ _ h] using data.le₂₃ r pq

Premises used: CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore.le₂₃, CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore.i₃_next
-- Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
Goal:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
inst_2 : Preorder ι
X : SpectralObject.{u_1, u_2, v_1, u_2} C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r r' : ℤ
hrr' : r + 1 = r'
hr : r₀ ≤ r
pq pq' : κ
i₀ i₁ i₂ i₃ i₃' : ι
hi₀ : i₀ = data.i₀ r pq' ⋯
hi₁ : i₁ = data.i₁ pq'
hi₂ : i₂ = data.i₂ pq'
hi₃ : i₃ = data.i₃ r pq' ⋯
hi₃' : i₃' = data.i₃ r' pq' ⋯
n₀ n₁ n₂ : ℤ
hn₁' : n₁ = data.deg pq'
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
h : (c r).Rel pq pq'
rw : data.i₂ pq' = data.i₀ r pq ⋯
rw₁ rw₂ : data.i₂ pq' = data.i₂ pq'
⊢ i₂ = data.i₀ r pq ⋯

Tactic:
rw [hi₂, data.hc₀₂ r _ _ h]

Premises used: CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore.hc₀₂
-- Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
Goal:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
inst_2 : Preorder ι
X : SpectralObject.{u_1, u_2, v_1, u_2} C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r r' : ℤ
hrr' : r + 1 = r'
hr : r₀ ≤ r
pq pq' : κ
i₀ i₁ i₂ i₃ i₃' : ι
hi₀ : i₀ = data.i₀ r pq' ⋯
hi₁ : i₁ = data.i₁ pq'
hi₂ : i₂ = data.i₂ pq'
hi₃ : i₃ = data.i₃ r pq' ⋯
hi₃' : i₃' = data.i₃ r' pq' ⋯
n₀ n₁ n₂ : ℤ
hn₁' : n₁ = data.deg pq'
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
h : (c r).Rel pq pq'
rw : data.i₃ r pq' ⋯ = data.i₁ pq
rw₁ rw₂ : data.i₃ r pq' ⋯ = data.i₃ r pq' ⋯
⊢ i₃ = data.i₁ pq

Tactic:
rw [hi₃, data.hc₁₃ r _ _ h]

Premises used: CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore.hc₁₃
-- Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
Goal:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
inst_2 : Preorder ι
X : SpectralObject.{u_1, u_2, v_1, u_2} C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r r' : ℤ
hrr' : r + 1 = r'
hr : r₀ ≤ r
pq pq' : κ
i₀ i₁ i₂ i₃ i₃' : ι
hi₀ : i₀ = data.i₀ r pq' ⋯
hi₁ : i₁ = data.i₁ pq'
hi₂ : i₂ = data.i₂ pq'
hi₃ : i₃ = data.i₃ r pq' ⋯
hi₃' : i₃' = data.i₃ r' pq' ⋯
n₀ n₁ n₂ : ℤ
hn₁' : n₁ = data.deg pq'
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
h : (c r).Rel pq pq'
rw : data.i₃ r' pq' ⋯ = data.i₂ pq
rw₁ rw₂ : data.i₂ pq = data.i₂ pq
⊢ i₃' = data.i₂ pq

Tactic:
rw [hi₃', data.i₃_next r r' _ _ h]

Premises used: CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore.i₃_next
-- Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
Goal:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
inst_2 : Preorder ι
X : SpectralObject.{u_1, u_2, v_1, u_2} C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r r' : ℤ
hrr' : r + 1 = r'
hr : r₀ ≤ r
pq pq' : κ
i₀ i₁ i₂ i₃ i₃' : ι
hi₀ : i₀ = data.i₀ r pq' ⋯
hi₁ : i₁ = data.i₁ pq'
hi₂ : i₂ = data.i₂ pq'
hi₃ : i₃ = data.i₃ r pq' ⋯
hi₃' : i₃' = data.i₃ r' pq' ⋯
n₀ n₁ n₂ : ℤ
hn₁' : n₁ = data.deg pq'
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
h : (c r).Rel pq pq'
⊢ data.deg pq + 1 = data.deg pq'

Tactic:
have := data.hc r pq pq' h
-- Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
Goal:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
inst_2 : Preorder ι
X : SpectralObject.{u_1, u_2, v_1, u_2} C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r r' : ℤ
hrr' : r + 1 = r'
hr : r₀ ≤ r
pq pq' : κ
i₀ i₁ i₂ i₃ i₃' : ι
hi₀ : i₀ = data.i₀ r pq' ⋯
hi₁ : i₁ = data.i₁ pq'
hi₂ : i₂ = data.i₂ pq'
hi₃ : i₃ = data.i₃ r pq' ⋯
hi₃' : i₃' = data.i₃ r' pq' ⋯
n₀ n₁ n₂ : ℤ
hn₁' : n₁ = data.deg pq'
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
h : (c r).Rel pq pq'
this : data.deg pq + 1 = data.deg pq'
⊢ n₀ = data.deg pq

Tactic:
lia
-- Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
Goal:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
inst_2 : Preorder ι
X : SpectralObject.{u_1, u_2, v_1, u_2} C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r r' : ℤ
hrr' : r + 1 = r'
hr : r₀ ≤ r
pq pq' : κ
i₀ i₁ i₂ i₃ i₃' : ι
hi₀ : i₀ = data.i₀ r pq' ⋯
hi₁ : i₁ = data.i₁ pq'
hi₂ : i₂ = data.i₂ pq'
hi₃ : i₃ = data.i₃ r pq' ⋯
hi₃' : i₃' = data.i₃ r' pq' ⋯
n₀ n₁ n₂ : ℤ
hn₁' : n₁ = data.deg pq'
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
h : (c r).Rel pq pq'
⊢ n₀ - 1 + 1 = n₀

Tactic:
simp
-- Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
Goal:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
inst_2 : Preorder ι
X : SpectralObject.{u_1, u_2, v_1, u_2} C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r r' : ℤ
hrr' : r + 1 = r'
hr : r₀ ≤ r
pq pq' : κ
i₀ i₁ i₂ i₃ i₃' : ι
hi₀ : i₀ = data.i₀ r pq' ⋯
hi₁ : i₁ = data.i₁ pq'
hi₂ : i₂ = data.i₂ pq'
hi₃ : i₃ = data.i₃ r pq' ⋯
hi₃' : i₃' = data.i₃ r' pq' ⋯
n₀ n₁ n₂ : ℤ
hn₁' : n₁ = data.deg pq'
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
h : (c r).Rel pq pq'
pos : Eq.{v_1 + 1} (α := pageX X data r pq ⋯ ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯) 0 0
⊢ Eq (α := pageX X data r pq ⋯ ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)
    (pageD X data r pq pq' ⋯ ≫
        Iso.hom (pageXIso X data r hr pq' i₀ i₁ i₂ i₃ hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' ⋯ ⋯) ≫
          X.mapFourδ₄Toδ₃' i₀ i₁ i₂ i₃ i₃' ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ ⋯ ⋯ :
      pageX X data r pq ⋯ ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)
    (0 : (page X data r hr).X pq ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)

Tactic:
rw [pageD_eq X data r hr pq pq' h (homOfLE (data.le₀₁' r hr pq' hi₀ hi₁))
      (homOfLE (data.le₁₂' pq' hi₁ hi₂)) (homOfLE (data.le₂₃' r hr pq' hi₂ hi₃))
      (homOfLE (data.le₃₃' hrr' hr pq' hi₃ hi₃'))
      (homOfLE (by simpa only [hi₃', data.i₃_next r r' _ _ h] using data.le₂₃ r pq))
      hi₀ hi₁ (by rw [hi₂, data.hc₀₂ r _ _ h])
      (by rw [hi₃, data.hc₁₃ r _ _ h]) (by rw [hi₃', data.i₃_next r r' _ _ h]) rfl
      (n₀ - 1) n₀ n₁ n₂ (by have := data.hc r pq pq' h; lia) (by simp) hn₁ hn₂,
      Category.assoc, Category.assoc, Iso.inv_hom_id_assoc,
      d_map_fourδ₄Toδ₃ .., comp_zero]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore.le₀₁', HSub.hSub, CategoryTheory.Limits.comp_zero, CategoryTheory.Iso.inv_hom_id_assoc, CategoryTheory.Abelian.SpectralObject.d_map_fourδ₄Toδ₃, CategoryTheory.homOfLE, CategoryTheory.Abelian.SpectralObject.SpectralSequence.pageD_eq, CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore.le₂₃', CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore.le₃₃', OfNat.ofNat, CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore.le₁₂', rfl
-- Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
Goal:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
inst_2 : Preorder ι
X : SpectralObject.{u_1, u_2, v_1, u_2} C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r r' : ℤ
hrr' : r + 1 = r'
hr : r₀ ≤ r
pq pq' : κ
i₀ i₁ i₂ i₃ i₃' : ι
hi₀ : i₀ = data.i₀ r pq' ⋯
hi₁ : i₁ = data.i₁ pq'
hi₂ : i₂ = data.i₂ pq'
hi₃ : i₃ = data.i₃ r pq' ⋯
hi₃' : i₃' = data.i₃ r' pq' ⋯
n₀ n₁ n₂ : ℤ
hn₁' : n₁ = data.deg pq'
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
h : (c r).Rel pq pq'
⊢ Eq.{v_1 + 1} (α := pageX X data r pq ⋯ ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯) 0 0

Tactic:
rfl
-- Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
Goal:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
inst_2 : Preorder ι
X : SpectralObject.{u_1, u_2, v_1, u_2} C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r r' : ℤ
hrr' : r + 1 = r'
hr : r₀ ≤ r
pq pq' : κ
i₀ i₁ i₂ i₃ i₃' : ι
hi₀ : i₀ = data.i₀ r pq' ⋯
hi₁ : i₁ = data.i₁ pq'
hi₂ : i₂ = data.i₂ pq'
hi₃ : i₃ = data.i₃ r pq' ⋯
hi₃' : i₃' = data.i₃ r' pq' ⋯
n₀ n₁ n₂ : ℤ
hn₁' : n₁ = data.deg pq'
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
h : ¬(c r).Rel pq pq'
neg :
  Eq (α := (page X data r hr).X pq ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)
    (0 ≫
        Iso.hom (pageXIso X data r hr pq' i₀ i₁ i₂ i₃ hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' ⋯ ⋯) ≫
          X.mapFourδ₄Toδ₃' i₀ i₁ i₂ i₃ i₃' ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ ⋯ ⋯ :
      (page X data r hr).X pq ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)
    (0 : (page X data r hr).X pq ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)
neg₁ neg₂ : Eq.{v_1 + 1} (α := (page X data r hr).X pq ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯) 0 0
⊢ Eq (α := (page X data r hr).X pq ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)
    ((page X data r hr).d pq pq' ≫
        Iso.hom (pageXIso X data r hr pq' i₀ i₁ i₂ i₃ hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' ⋯ ⋯) ≫
          X.mapFourδ₄Toδ₃' i₀ i₁ i₂ i₃ i₃' ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ ⋯ ⋯ :
      (page X data r hr).X pq ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)
    (0 : (page X data r hr).X pq ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)

Tactic:
rw [HomologicalComplex.shape _ _ _ h, zero_comp]

Premises used: CategoryTheory.Limits.zero_comp, HomologicalComplex.shape
-- Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
Goal:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
inst_2 : Preorder ι
X : SpectralObject.{u_1, u_2, v_1, u_2} C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r r' : ℤ
hrr' : r + 1 = r'
hr : r₀ ≤ r
pq pq' pq'' : κ
hpq : (c r).prev pq' = pq
hpq' : (c r).next pq' = pq''
i₀' i₀ i₁ i₂ i₃ i₃' : ι
hi₀' : i₀' = data.i₀ r' pq' ⋯
hi₀ : i₀ = data.i₀ r pq' ⋯
hi₁ : i₁ = data.i₁ pq'
hi₂ : i₂ = data.i₂ pq'
hi₃ : i₃ = data.i₃ r pq' ⋯
hi₃' : i₃' = data.i₃ r' pq' ⋯
n₀ n₁ n₂ : ℤ
hn₁' : n₁ = data.deg pq'
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
dsimp :
  Epi (Y := X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)
    (Iso.hom (pageXIso X data r hr pq' i₀ i₁ i₂ i₃ hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' ⋯ ⋯) ≫
        X.mapFourδ₄Toδ₃' i₀ i₁ i₂ i₃ i₃' ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ ⋯ ⋯ :
      pageX X data r pq' ⋯ ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)
⊢ Epi (ccSc X data r r' hrr' hr pq pq' i₀ i₁ i₂ i₃ i₃' hi₀ hi₁ hi₂ hi₃ hi₃' n₀ n₁ n₂ hn₁' hn₁ hn₂).g

Tactic:
dsimp
-- Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
Goal:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
inst_2 : Preorder ι
X : SpectralObject.{u_1, u_2, v_1, u_2} C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r r' : ℤ
hrr' : r + 1 = r'
hr : r₀ ≤ r
pq pq' pq'' : κ
hpq : (c r).prev pq' = pq
hpq' : (c r).next pq' = pq''
i₀' i₀ i₁ i₂ i₃ i₃' : ι
hi₀' : i₀' = data.i₀ r' pq' ⋯
hi₀ : i₀ = data.i₀ r pq' ⋯
hi₁ : i₁ = data.i₁ pq'
hi₂ : i₂ = data.i₂ pq'
hi₃ : i₃ = data.i₃ r pq' ⋯
hi₃' : i₃' = data.i₃ r' pq' ⋯
n₀ n₁ n₂ : ℤ
hn₁' : n₁ = data.deg pq'
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ Epi (Y := X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)
    (Iso.hom (pageXIso X data r hr pq' i₀ i₁ i₂ i₃ hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' ⋯ ⋯) ≫
        X.mapFourδ₄Toδ₃' i₀ i₁ i₂ i₃ i₃' ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ ⋯ ⋯ :
      pageX X data r pq' ⋯ ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)

Tactic:
infer_instance
-- Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
Goal:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
inst_2 : Preorder ι
X : SpectralObject.{u_1, u_2, v_1, u_2} C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r r' : ℤ
hrr' : r + 1 = r'
hr : r₀ ≤ r
pq pq' : κ
hpq : (c r).prev pq' = pq
i₀ i₁ i₂ i₃ i₃' : ι
hi₀ : i₀ = data.i₀ r pq' ⋯
hi₁ : i₁ = data.i₁ pq'
hi₂ : i₂ = data.i₂ pq'
hi₃ : i₃ = data.i₃ r pq' ⋯
hi₃' : i₃' = data.i₃ r' pq' ⋯
n₀ n₁ n₂ : ℤ
hn₁' : n₁ = data.deg pq'
inst_3 : X.HasSpectralSequence data
h : ¬(c r).Rel pq pq'
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
apply : IsZero.{v_1, u_1} ((X.H n₀).obj (mk₁ (homOfLE ⋯)))
⊢ IsIso (X := X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯) (Y :=
    X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)
    (X.mapFourδ₄Toδ₃' i₀ i₁ i₂ i₃ i₃' ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ ⋯ ⋯ :
      X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯ ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)

Tactic:
apply X.isIso_map_fourδ₄Toδ₃_of_isZero _ _ _ _ _ _ _ _ _ _

Premises used: CategoryTheory.Abelian.SpectralObject.isIso_map_fourδ₄Toδ₃_of_isZero
-- Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
Goal:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
inst_2 : Preorder ι
X : SpectralObject.{u_1, u_2, v_1, u_2} C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r r' : ℤ
hrr' : r + 1 = r'
hr : r₀ ≤ r
pq pq' : κ
hpq : (c r).prev pq' = pq
i₀ i₁ i₂ i₃ i₃' : ι
hi₀ : i₀ = data.i₀ r pq' ⋯
hi₁ : i₁ = data.i₁ pq'
hi₂ : i₂ = data.i₂ pq'
hi₃ : i₃ = data.i₃ r pq' ⋯
hi₃' : i₃' = data.i₃ r' pq' ⋯
n₀ n₁ n₂ : ℤ
hn₁' : n₁ = data.deg pq'
inst_3 : X.HasSpectralSequence data
h : ¬(c r).Rel pq pq'
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ n₀ = data.deg pq' - 1

Tactic:
lia
-- Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
Goal:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
inst_2 : Preorder ι
X : SpectralObject.{u_1, u_2, v_1, u_2} C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r r' : ℤ
hrr' : r + 1 = r'
hr : r₀ ≤ r
pq pq' : κ
hpq : (c r).prev pq' = pq
i₀ i₁ i₂ i₃ i₃' : ι
hi₀ : i₀ = data.i₀ r pq' ⋯
hi₁ : i₁ = data.i₁ pq'
hi₂ : i₂ = data.i₂ pq'
hi₃ : i₃ = data.i₃ r pq' ⋯
hi₃' : i₃' = data.i₃ r' pq' ⋯
n₀ n₁ n₂ : ℤ
hn₁' : n₁ = data.deg pq'
inst_3 : X.HasSpectralSequence data
h : ¬(c r).Rel pq pq'
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
refine : ∀ (x : κ) (hk : (c r).Rel x pq'), False
⊢ IsZero.{v_1, u_1} ((X.H n₀).obj (mk₁ (homOfLE ⋯)))

Tactic:
refine X.isZero_H_obj_mk₁_i₃_le' data r r' hrr' hr pq' (fun _ hk ↦ ?_) _ (by lia) _ _ hi₃ hi₃'

Premises used: CategoryTheory.Abelian.SpectralObject.isZero_H_obj_mk₁_i₃_le'
-- Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
Goal:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
inst_2 : Preorder ι
X : SpectralObject.{u_1, u_2, v_1, u_2} C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r r' : ℤ
hrr' : r + 1 = r'
hr : r₀ ≤ r
pq pq' : κ
hpq : (c r).prev pq' = pq
i₀ i₁ i₂ i₃ i₃' : ι
hi₀ : i₀ = data.i₀ r pq' ⋯
hi₁ : i₁ = data.i₁ pq'
hi₂ : i₂ = data.i₂ pq'
hi₃ : i₃ = data.i₃ r pq' ⋯
hi₃' : i₃' = data.i₃ r' pq' ⋯
n₀ n₁ n₂ : ℤ
hn₁' : n₁ = data.deg pq'
inst_3 : X.HasSpectralSequence data
h : ¬(c r).Rel pq pq'
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
x : κ
hk : (c r).Rel x pq'
⊢ (c r).Rel ((c r).prev pq') pq'

Tactic:
obtain rfl := (c r).prev_eq' hk

Premises used: ComplexShape.prev_eq', rfl
-- Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
Goal:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
inst_2 : Preorder ι
X : SpectralObject.{u_1, u_2, v_1, u_2} C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r r' : ℤ
hrr' : r + 1 = r'
hr : r₀ ≤ r
pq pq' : κ
hpq : (c r).prev pq' = pq
i₀ i₁ i₂ i₃ i₃' : ι
hi₀ : i₀ = data.i₀ r pq' ⋯
hi₁ : i₁ = data.i₁ pq'
hi₂ : i₂ = data.i₂ pq'
hi₃ : i₃ = data.i₃ r pq' ⋯
hi₃' : i₃' = data.i₃ r' pq' ⋯
n₀ n₁ n₂ : ℤ
hn₁' : n₁ = data.deg pq'
inst_3 : X.HasSpectralSequence data
h : ¬(c r).Rel pq pq'
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hk : (c r).Rel ((c r).prev pq') pq'
⊢ ¬(c r).Rel ((c r).prev pq') pq'

Tactic:
subst hpq
-- Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
Goal:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
inst_2 : Preorder ι
X : SpectralObject.{u_1, u_2, v_1, u_2} C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r r' : ℤ
hrr' : r + 1 = r'
hr : r₀ ≤ r
pq' : κ
i₀ i₁ i₂ i₃ i₃' : ι
hi₀ : i₀ = data.i₀ r pq' ⋯
hi₁ : i₁ = data.i₁ pq'
hi₂ : i₂ = data.i₂ pq'
hi₃ : i₃ = data.i₃ r pq' ⋯
hi₃' : i₃' = data.i₃ r' pq' ⋯
n₀ n₁ n₂ : ℤ
hn₁' : n₁ = data.deg pq'
inst_3 : X.HasSpectralSequence data
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hk : (c r).Rel ((c r).prev pq') pq'
h : ¬(c r).Rel ((c r).prev pq') pq'
⊢ False

Tactic:
h hk
-- Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
Goal:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
inst_2 : Preorder ι
X : SpectralObject.{u_1, u_2, v_1, u_2} C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r r' : ℤ
hrr' : r + 1 = r'
hr : r₀ ≤ r
pq' : κ
i₀ i₁ i₂ i₃ i₃' : ι
hi₀ : i₀ = data.i₀ r pq' ⋯
hi₁ : i₁ = data.i₁ pq'
hi₂ : i₂ = data.i₂ pq'
hi₃ : i₃ = data.i₃ r pq' ⋯
hi₃' : i₃' = data.i₃ r' pq' ⋯
n₀ n₁ n₂ : ℤ
hn₁' : n₁ = data.deg pq'
inst_3 : X.HasSpectralSequence data
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hk : (c r).Rel ((c r).prev pq') pq'
h : ¬(c r).Rel ((c r).prev pq') pq'
⊢ False

Tactic:
exact h hk
-- Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
Goal:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
inst_2 : Preorder ι
X : SpectralObject.{u_1, u_2, v_1, u_2} C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r r' : ℤ
hrr' : r + 1 = r'
hr : r₀ ≤ r
pq pq' : κ
hpq : (c r).prev pq' = pq
i₀ i₁ i₂ i₃ i₃' : ι
hi₀ : i₀ = data.i₀ r pq' ⋯
hi₁ : i₁ = data.i₁ pq'
hi₂ : i₂ = data.i₂ pq'
hi₃ : i₃ = data.i₃ r pq' ⋯
hi₃' : i₃' = data.i₃ r' pq' ⋯
n₀ n₁ n₂ : ℤ
hn₁' : n₁ = data.deg pq'
inst_3 : X.HasSpectralSequence data
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
pos :
  ∀ (h : (c r).Rel pq pq'),
    (ccSc X data r r' hrr' hr pq pq' i₀ i₁ i₂ i₃ i₃' hi₀ hi₁ hi₂ hi₃ hi₃' n₀ n₁ n₂ hn₁' ⋯ ⋯).Exact
neg :
  ∀ (h : ¬(c r).Rel pq pq'),
    (ccSc X data r r' hrr' hr pq pq' i₀ i₁ i₂ i₃ i₃' hi₀ hi₁ hi₂ hi₃ hi₃' n₀ n₁ n₂ hn₁' ⋯ ⋯).Exact
⊢ (ccSc X data r r' hrr' hr pq pq' i₀ i₁ i₂ i₃ i₃' hi₀ hi₁ hi₂ hi₃ hi₃' n₀ n₁ n₂ hn₁' ⋯ ⋯).Exact

Tactic:
by_cases h : (c r).Rel pq pq'

Premises used: ComplexShape.Rel
-- Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
Goal:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
inst_2 : Preorder ι
X : SpectralObject.{u_1, u_2, v_1, u_2} C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r r' : ℤ
hrr' : r + 1 = r'
hr : r₀ ≤ r
pq pq' : κ
hpq : (c r).prev pq' = pq
i₀ i₁ i₂ i₃ i₃' : ι
hi₀ : i₀ = data.i₀ r pq' ⋯
hi₁ : i₁ = data.i₁ pq'
hi₂ : i₂ = data.i₂ pq'
hi₃ : i₃ = data.i₃ r pq' ⋯
hi₃' : i₃' = data.i₃ r' pq' ⋯
n₀ n₁ n₂ : ℤ
hn₁' : n₁ = data.deg pq'
inst_3 : X.HasSpectralSequence data
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
h : (c r).Rel pq pq'
⊢ i₃' ≤ data.i₃ r pq ⋯

Tactic:
simpa only [hi₃', data.i₃_next r r' _ _ h] using data.le₂₃ r pq

Premises used: CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore.le₂₃, CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore.i₃_next
-- Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
Goal:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
inst_2 : Preorder ι
X : SpectralObject.{u_1, u_2, v_1, u_2} C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r r' : ℤ
hrr' : r + 1 = r'
hr : r₀ ≤ r
pq pq' : κ
hpq : (c r).prev pq' = pq
i₀ i₁ i₂ i₃ i₃' : ι
hi₀ : i₀ = data.i₀ r pq' ⋯
hi₁ : i₁ = data.i₁ pq'
hi₂ : i₂ = data.i₂ pq'
hi₃ : i₃ = data.i₃ r pq' ⋯
hi₃' : i₃' = data.i₃ r' pq' ⋯
n₀ n₁ n₂ : ℤ
hn₁' : n₁ = data.deg pq'
inst_3 : X.HasSpectralSequence data
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
h : (c r).Rel pq pq'
⊢ n₀ - 1 + 1 = n₀

Tactic:
simp