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
hpq' : (c r).next pq' = pq''
i₀' i₀ i₁ i₂ 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' ⋯
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 : ∀ (k : κ) (hk : (c r).Rel pq' k), False
⊢ IsZero.{v_1, u_1} ((X.H n₂).obj (mk₁ (homOfLE ⋯)))

Tactic:
refine X.isZero_H_obj_mk₁_i₀_le' data r r' hrr' hr pq' (fun k hk ↦ ?_) _ (by lia) _ _ hi₀' hi₀

Premises used: CategoryTheory.Abelian.SpectralObject.isZero_H_obj_mk₁_i₀_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).next pq' = pq''
i₀' i₀ i₁ i₂ 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' ⋯
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
k : κ
hk : (c r).Rel pq' k
⊢ (c r).Rel pq' ((c r).next pq')

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

Premises used: rfl, ComplexShape.next_eq'
-- 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).next pq' = pq''
i₀' i₀ i₁ i₂ 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' ⋯
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 pq' ((c r).next pq')
⊢ ¬(c r).Rel pq' ((c r).next 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' i₀ i₁ i₂ 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' ⋯
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 pq' ((c r).next pq')
h : ¬(c r).Rel pq' ((c r).next 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' i₀ i₁ i₂ 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' ⋯
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 pq' ((c r).next pq')
h : ¬(c r).Rel pq' ((c r).next 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).next pq' = pq''
i₀' i₀ i₁ i₂ 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' ⋯
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''),
    (kfSc X data r r' hrr' hr pq' pq'' i₀' i₀ i₁ i₂ i₃ hi₀' hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' hn₁ hn₂).Exact
neg :
  ∀ (h : ¬(c r).Rel pq' pq''),
    (kfSc X data r r' hrr' hr pq' pq'' i₀' i₀ i₁ i₂ i₃ hi₀' hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' hn₁ hn₂).Exact
⊢ (kfSc X data r r' hrr' hr pq' pq'' i₀' i₀ i₁ i₂ i₃ hi₀' hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' hn₁ hn₂).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).next pq' = pq''
i₀' i₀ i₁ i₂ 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' ⋯
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''
⊢ data.i₀ r pq'' ⋯ ≤ i₀'

Tactic:
simpa only [hi₀', data.i₀_prev r r' _ _ h] using data.le₀₁ r pq''

Premises used: CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore.le₀₁, CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore.i₀_prev
-- 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).next pq' = pq''
i₀' i₀ i₁ i₂ 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' ⋯
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''
pos :
  kfSc X data r r' hrr' hr pq' pq'' i₀' i₀ i₁ i₂ i₃ hi₀' hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' hn₁ hn₂ ≅
    X.dKernelSequence (i₀ := data.i₀ r pq'' ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯)
      (homOfLE ⋯ ≫ homOfLE ⋯) ⋯ n₀ n₁ n₂ (n₂ + 1) ⋯ ⋯ ⋯
⊢ (kfSc X data r r' hrr' hr pq' pq'' i₀' i₀ i₁ i₂ i₃ hi₀' hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' hn₁ hn₂).Exact

Tactic:
refine ShortComplex.exact_of_iso (Iso.symm ?_)
      (X.dKernelSequence_exact
        (homOfLE (show data.i₀ r pq'' ≤ i₀' by
          simpa only [hi₀', data.i₀_prev r r' _ _ h] using data.le₀₁ r pq''))
        (homOfLE (data.i₀_le' hrr' hr pq' hi₀' hi₀)) (homOfLE (data.le₀₁' r hr pq' hi₀ hi₁))
        (homOfLE (data.le₁₂' pq' hi₁ hi₂)) (homOfLE (data.le₂₃' r hr pq' hi₂ hi₃)) _ rfl
        n₀ n₁ n₂ (n₂ + 1) hn₁ hn₂ rfl)

Premises used: CategoryTheory.Abelian.SpectralObject.dKernelSequence_exact, CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore.le₀₁', CategoryTheory.ShortComplex.exact_of_iso, HAdd.hAdd, CategoryTheory.homOfLE, CategoryTheory.Iso.symm, CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore.le₂₃', OfNat.ofNat, CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore.le₁₂', CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore.i₀_le', 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).next pq' = pq''
i₀' i₀ i₁ i₂ 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' ⋯
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''
rw : data.i₀ r' pq' ⋯ = data.i₁ pq''
rw₁ rw₂ : data.i₁ pq'' = data.i₁ pq''
⊢ i₀' = data.i₁ pq''

Tactic:
rw [hi₀', data.i₀_prev r r' _ _ h]

Premises used: CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore.i₀_prev
-- 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).next pq' = pq''
i₀' i₀ i₁ i₂ 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' ⋯
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''
rw : data.i₀ r pq' ⋯ = data.i₂ pq''
rw₁ rw₂ : data.i₂ pq'' = data.i₂ pq''
⊢ i₀ = data.i₂ pq''

Tactic:
rw [hi₀, 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
hpq' : (c r).next pq' = pq''
i₀' i₀ i₁ i₂ 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' ⋯
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''
rw : data.i₁ pq' = data.i₃ r pq'' ⋯
rw₁ rw₂ : data.i₃ r pq'' ⋯ = data.i₃ r pq'' ⋯
⊢ i₁ = data.i₃ r pq'' ⋯

Tactic:
rw [hi₁, 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
hpq' : (c r).next pq' = pq''
i₀' i₀ i₁ i₂ 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' ⋯
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''
⊢ data.deg pq' + 1 = data.deg pq''

Tactic:
have := data.hc r _ _ 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
hpq' : (c r).next pq' = pq''
i₀' i₀ i₁ i₂ 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' ⋯
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''
this : data.deg pq' + 1 = data.deg pq''
⊢ n₂ = 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
hpq' : (c r).next pq' = pq''
i₀' i₀ i₁ i₂ 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' ⋯
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''
pos :
  Iso.hom (Iso.refl (kfSc X data r r' hrr' hr pq' pq'' i₀' i₀ i₁ i₂ i₃ hi₀' hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' hn₁ hn₂).X₁) ≫
      (X.dKernelSequence (i₀ := data.i₀ r pq'' ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯)
          (homOfLE ⋯ ≫ homOfLE ⋯) ⋯ n₀ n₁ n₂ (n₂ + 1) ⋯ ⋯ ⋯).f =
    (kfSc X data r r' hrr' hr pq' pq'' i₀' i₀ i₁ i₂ i₃ hi₀' hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' hn₁ hn₂).f ≫
      Iso.hom (pageXIso X data r hr pq' i₀ i₁ i₂ i₃ hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' ⋯ ⋯)
pos₁ :
  Iso.hom (pageXIso X data r hr pq' i₀ i₁ i₂ i₃ hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' ⋯ ⋯) ≫
      (X.dKernelSequence (i₀ := data.i₀ r pq'' ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯)
          (homOfLE ⋯ ≫ homOfLE ⋯) ⋯ n₀ n₁ n₂ (n₂ + 1) ⋯ ⋯ ⋯).g =
    (kfSc X data r r' hrr' hr pq' pq'' i₀' i₀ i₁ i₂ i₃ hi₀' hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' hn₁ hn₂).g ≫
      Iso.hom (pageXIso X data r hr pq'' (data.i₀ r pq'' ⋯) i₀' i₀ i₁ ⋯ ⋯ ⋯ ⋯ n₁ n₂ (n₂ + 1) ⋯ ⋯ ⋯)
⊢ kfSc X data r r' hrr' hr pq' pq'' i₀' i₀ i₁ i₂ i₃ hi₀' hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' hn₁ hn₂ ≅
    X.dKernelSequence (i₀ := data.i₀ r pq'' ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯)
      (homOfLE ⋯ ≫ homOfLE ⋯) ⋯ n₀ n₁ n₂ (n₂ + 1) ⋯ ⋯ ⋯

Tactic:
refine ShortComplex.isoMk (Iso.refl _)
      (pageXIso X data _ hr _ _ _ _ _ hi₀ hi₁ hi₂ hi₃ _ _ _ hn₁')
      (pageXIso X data _ hr _ _ _ _ _ rfl (by rw [hi₀', data.i₀_prev r r' _ _ h])
      (by rw [hi₀, data.hc₀₂ r _ _ h]) (by rw [hi₁, data.hc₁₃ r _ _ h]) _ _ _
      (by have := data.hc r _ _ h; lia)) ?_ ?_

Premises used: CategoryTheory.Iso.refl, CategoryTheory.ShortComplex.isoMk
-- 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).next pq' = pq''
i₀' i₀ i₁ i₂ 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' ⋯
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''
⊢ Iso.hom (Iso.refl (kfSc X data r r' hrr' hr pq' pq'' i₀' i₀ i₁ i₂ i₃ hi₀' hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' hn₁ hn₂).X₁) ≫
      (X.dKernelSequence (i₀ := data.i₀ r pq'' ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯)
          (homOfLE ⋯ ≫ homOfLE ⋯) ⋯ n₀ n₁ n₂ (n₂ + 1) ⋯ ⋯ ⋯).f =
    (kfSc X data r r' hrr' hr pq' pq'' i₀' i₀ i₁ i₂ i₃ hi₀' hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' hn₁ hn₂).f ≫
      Iso.hom (pageXIso X data r hr pq' i₀ i₁ i₂ i₃ hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' ⋯ ⋯)

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
hpq' : (c r).next pq' = pq''
i₀' i₀ i₁ i₂ 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' ⋯
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''
pos :
  Eq (α := pageX X data r pq' hr ⟶ X.E (i := data.i₀ r pq'' ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₁ n₂ (n₂ + 1) ⋯ ⋯)
    (CategoryStruct.comp (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.d (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ (n₂ + 1) ⋯ ⋯ ⋯) :
      pageX X data r pq' hr ⟶ X.E (i := data.i₀ r pq'' ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₁ n₂ (n₂ + 1) ⋯ ⋯)
    (pageD X data r pq' pq'' ⋯ ≫
        Iso.hom (pageXIso X data r hr pq'' (data.i₀ r pq'' ⋯) i₀' i₀ i₁ ⋯ ⋯ ⋯ ⋯ n₁ n₂ (n₂ + 1) ⋯ ⋯ ⋯) :
      pageX X data r pq' hr ⟶ X.E (i := data.i₀ r pq'' ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₁ n₂ (n₂ + 1) ⋯ ⋯)
⊢ Iso.hom (pageXIso X data r hr pq' i₀ i₁ i₂ i₃ hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' ⋯ ⋯) ≫
      (X.dKernelSequence (i₀ := data.i₀ r pq'' ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯)
          (homOfLE ⋯ ≫ homOfLE ⋯) ⋯ n₀ n₁ n₂ (n₂ + 1) ⋯ ⋯ ⋯).g =
    (kfSc X data r r' hrr' hr pq' pq'' i₀' i₀ i₁ i₂ i₃ hi₀' hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' hn₁ hn₂).g ≫
      Iso.hom (pageXIso X data r hr pq'' (data.i₀ r pq'' ⋯) i₀' i₀ i₁ ⋯ ⋯ ⋯ ⋯ n₁ n₂ (n₂ + 1) ⋯ ⋯ ⋯)

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
hpq' : (c r).next pq' = pq''
i₀' i₀ i₁ i₂ 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' ⋯
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''
⊢ i₀' = data.i₁ pq''

Tactic:
simpa [← data.i₀_prev r r' _ _ h]

Premises used: CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore.i₀_prev
-- 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).next pq' = pq''
i₀' i₀ i₁ i₂ 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' ⋯
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''
neg : Epi (kfSc X data r r' hrr' hr pq' pq'' i₀' i₀ i₁ i₂ i₃ hi₀' hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' hn₁ hn₂).f
⊢ (kfSc X data r r' hrr' hr pq' pq'' i₀' i₀ i₁ i₂ i₃ hi₀' hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' hn₁ hn₂).Exact

Tactic:
rw [ShortComplex.exact_iff_epi _ ((page X data r hr).shape _ _ h)]

Premises used: CategoryTheory.ShortComplex.exact_iff_epi
-- 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).next pq' = pq''
i₀' i₀ i₁ i₂ 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' ⋯
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''
⊢ IsIso (X := X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯) (Y :=
    X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)
    (X.mapFourδ₁Toδ₀' i₀' i₀ i₁ i₂ i₃ ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ ⋯ ⋯ :
      X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯ ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)

Tactic:
have := isIso_mapFourδ₁Toδ₀' X data r r' hrr' hr pq' pq'' hpq'
      i₀' i₀ i₁ i₂ i₃ hi₀' hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' 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
hpq' : (c r).next pq' = pq''
i₀' i₀ i₁ i₂ 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' ⋯
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''
this :
  IsIso (X := X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯) (Y :=
    X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)
    (X.mapFourδ₁Toδ₀' i₀' i₀ i₁ i₂ i₃ ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ ⋯ ⋯ :
      X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯ ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)
neg :
  Epi (X := X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ hn₁ ⋯)
    (X.mapFourδ₁Toδ₀' i₀' i₀ i₁ i₂ i₃ ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ hn₁ ⋯ ≫
        Iso.inv (pageXIso X data r hr pq' i₀ i₁ i₂ i₃ hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' hn₁ ⋯) :
      X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ hn₁ ⋯ ⟶ pageX X data r pq' hr)
⊢ Epi (kfSc X data r r' hrr' hr pq' pq'' i₀' i₀ i₁ i₂ i₃ hi₀' hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' hn₁ hn₂).f

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
hpq' : (c r).next pq' = pq''
i₀' i₀ i₁ i₂ 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' ⋯
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''
this :
  IsIso (X := X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯) (Y :=
    X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)
    (X.mapFourδ₁Toδ₀' i₀' i₀ i₁ i₂ i₃ ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ ⋯ ⋯ :
      X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯ ⟶ X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯)
⊢ Epi (X := X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ hn₁ ⋯)
    (X.mapFourδ₁Toδ₀' i₀' i₀ i₁ i₂ i₃ ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ hn₁ ⋯ ≫
        Iso.inv (pageXIso X data r hr pq' i₀ i₁ i₂ i₃ hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' hn₁ ⋯) :
      X.E (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ hn₁ ⋯ ⟶ pageX X data r pq' hr)

Tactic:
infer_instance