mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: case refine_2
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
f₁₂ : i ⟶ k
h₁₂ : f₁ ≫ f₂ = f₁₂
f₂₃ : j ⟶ l
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hp : IsColimit (CokernelCofork.ofπ (X.cokernelSequenceOpcycles f₂ f₃ n₀ n₁ hn₁).g ⋯) := ⋯.gIsCokernel
this : hp.desc (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯) = X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ hn₂
⊢ (Cone.postcompose
          (parallelPair.ext
              (Iso.refl
                ((parallelPair (hp.desc (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯)) 0).obj
                  WalkingParallelPair.zero))
              (Iso.refl
                ((parallelPair (hp.desc (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯)) 0).obj
                  WalkingParallelPair.one))
              ⋯ ⋯).hom).obj
      (KernelFork.ofι (kernel.ι (X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ ⋯)) ⋯) ≅
    Fork.ofι (kernel.ι (X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ ⋯)) ⋯

Tactic:
·

Goal state after:
case refine_2
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
f₁₂ : i ⟶ k
h₁₂ : f₁ ≫ f₂ = f₁₂
f₂₃ : j ⟶ l
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hp : IsColimit (CokernelCofork.ofπ (X.cokernelSequenceOpcycles f₂ f₃ n₀ n₁ hn₁).g ⋯) := ⋯.gIsCokernel
this : hp.desc (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯) = X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ hn₂
⊢ (Cone.postcompose
          (parallelPair.ext
              (Iso.refl
                ((parallelPair (hp.desc (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯)) 0).obj
                  WalkingParallelPair.zero))
              (Iso.refl
                ((parallelPair (hp.desc (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯)) 0).obj
                  WalkingParallelPair.one))
              ⋯ ⋯).hom).obj
      (KernelFork.ofι (kernel.ι (X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ ⋯)) ⋯) ≅
    Fork.ofι (kernel.ι (X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ ⋯)) ⋯
Module: Mathlib.Algebra.Homology.SpectralObject.FirstPage

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁴ : Category.{u_4, u_1} C
inst✝³ : Abelian C
inst✝² : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
inst✝¹ : data.HasFirstPageComputation
inst✝ : X.HasSpectralSequence data
pq : κ
i₁ i₂ : ι
hi₁ : i₁ = data.i₁ pq
hi₂ : i₂ = data.i₂ pq
n₀ n₁ n₂ : ℤ
hn₁' : n₁ = data.deg pq
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ n₀ = n₁ - 1

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
l : ℕ
x✝³ : ℤ
x✝² x✝¹ : ℤ × Fin l
x✝ : (ComplexShape.spectralSequenceFin l (x✝³, 1 - x✝³)).Rel x✝² x✝¹
h₁ : x✝².1 + (x✝³, 1 - x✝³).1 = x✝¹.1
h₂ : ↑↑x✝².2 + (x✝³, 1 - x✝³).2 = ↑↑x✝¹.2
⊢ autoParam (2 ≤ x✝³) SpectralSequenceDataCore._auto_14 → x✝².1 + ↑↑x✝².2 + 1 = x✝¹.1 + ↑↑x✝¹.2

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
l : ℕ
r : ℤ
⊢ ∀ (pq pq' : ℤ × Fin l),
    (ComplexShape.spectralSequenceFin l (r, 1 - r)).Rel pq pq' →
      autoParam (2 ≤ r) SpectralSequenceDataCore._auto_20 → pq.2.castSucc = Fin.clamp (↑↑pq'.2 + (r - 1)).toNat l

Tactic:
by
    rintro ⟨a₁, ⟨a₂, _⟩⟩ ⟨b₁, ⟨b₂, _⟩⟩ ⟨h₁, h₂⟩ hr
    rw [Fin.ext_iff]
    dsimp
    grind

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i₀ i₁ i₂ i₃ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
i₀' i₁' i₂' i₃' : ι
f₁' : i₀' ⟶ i₁'
f₂' : i₁' ⟶ i₂'
f₃' : i₂' ⟶ i₃'
α : mk₃ f₁ f₂ f₃ ⟶ mk₃ f₁' f₂' f₃'
β : mk₂ f₁ f₂ ⟶ mk₂ f₁' f₂'
⊢ 0 ≤ 1

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i₀ i₁ i₂ i₃ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
i₀' i₁' i₂' i₃' : ι
f₁' : i₀' ⟶ i₁'
f₂' : i₁' ⟶ i₂'
f₃' : i₂' ⟶ i₃'
α : mk₃ f₁ f₂ f₃ ⟶ mk₃ f₁' f₂' f₃'
β : mk₂ f₁ f₂ ⟶ mk₂ f₁' f₂'
⊢ 1 ≤ 3

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
l : ℕ
r r' a : ℤ
a' : ℕ
isLt✝ : a' < l
hr : autoParam (2 ≤ r) SpectralSequenceDataCore._auto_32
hrr' : autoParam (r ≤ r') SpectralSequenceDataCore._auto_34
⊢ (↑↑(a, ⟨a', isLt✝⟩).2 + (r - 1)).toNat ≤ (↑↑(a, ⟨a', isLt✝⟩).2 + (r' - 1)).toNat

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i₀ i₁ i₂ i₃ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
i₀' i₁' i₂' i₃' : ι
f₁' : i₀' ⟶ i₁'
f₂' : i₁' ⟶ i₂'
f₃' : i₂' ⟶ i₃'
α : mk₃ f₁ f₂ f₃ ⟶ mk₃ f₁' f₂' f₃'
β : mk₂ f₁ f₂ ⟶ mk₂ f₁' f₂'
⊢ 1 ≤ 2

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i₀ i₁ i₂ i₃ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
i₀' i₁' i₂' i₃' : ι
f₁' : i₀' ⟶ i₁'
f₂' : i₁' ⟶ i₂'
f₃' : i₂' ⟶ i₃'
α : mk₃ f₁ f₂ f₃ ⟶ mk₃ f₁' f₂' f₃'
β : mk₂ f₁ f₂ ⟶ mk₂ f₁' f₂'
⊢ 2 ≤ 3

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i₀ i₁ i₂ i₃ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
i₀' i₁' i₂' i₃' : ι
f₁' : i₀' ⟶ i₁'
f₂' : i₁' ⟶ i₂'
f₃' : i₂' ⟶ i₃'
α : mk₃ f₁ f₂ f₃ ⟶ mk₃ f₁' f₂' f₃'
β : mk₂ f₁ f₂ ⟶ mk₂ f₁' f₂'
n₀ n₁ n₂ : ℤ
⊢ 0 ≤ 1

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i₀ i₁ i₂ i₃ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
i₀' i₁' i₂' i₃' : ι
f₁' : i₀' ⟶ i₁'
f₂' : i₁' ⟶ i₂'
f₃' : i₂' ⟶ i₃'
α : mk₃ f₁ f₂ f₃ ⟶ mk₃ f₁' f₂' f₃'
β : mk₂ f₁ f₂ ⟶ mk₂ f₁' f₂'
n₀ n₁ n₂ : ℤ
⊢ 1 ≤ 3

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i₀ i₁ i₂ i₃ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
i₀' i₁' i₂' i₃' : ι
f₁' : i₀' ⟶ i₁'
f₂' : i₁' ⟶ i₂'
f₃' : i₂' ⟶ i₃'
α : mk₃ f₁ f₂ f₃ ⟶ mk₃ f₁' f₂' f₃'
β : mk₂ f₁ f₂ ⟶ mk₂ f₁' f₂'
n₀ n₁ n₂ : ℤ
⊢ 1 ≤ 2

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i₀ i₁ i₂ i₃ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
i₀' i₁' i₂' i₃' : ι
f₁' : i₀' ⟶ i₁'
f₂' : i₁' ⟶ i₂'
f₃' : i₂' ⟶ i₃'
α : mk₃ f₁ f₂ f₃ ⟶ mk₃ f₁' f₂' f₃'
β : mk₂ f₁ f₂ ⟶ mk₂ f₁' f₂'
n₀ n₁ n₂ : ℤ
⊢ 2 ≤ 3

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
inst✝¹ : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
X : SpectralObject C ι
data : SpectralSequenceDataCore ι c r₀
inst✝ : X.HasSpectralSequence data
l : ℕ
E : SpectralObject C (Fin (l + 1))
n : ℤ
k : ℕ
hr : 2 ≤ 2 + ↑k
p : ℤ
t : ℕ
hq : k + 1 + t < l
hn : n = (coreE₂CohomologicalFin l).deg (p, ⟨k + 1 + t, hq⟩) + 1
this : k < k + 1 + t
hpq : ∀ (a : ℤ) (b : Fin l), p + (2 + ↑k) = a → ¬↑(k + 1 + t) + (1 - (2 + ↑k)) = ↑↑b
⊢ t < l

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
inst✝¹ : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
X : SpectralObject C ι
data : SpectralSequenceDataCore ι c r₀
inst✝ : X.HasSpectralSequence data
l : ℕ
E : SpectralObject C (Fin (l + 1))
r n : ℤ
hr : 2 ≤ r
p : ℤ
q : ℕ
hq : q < l
hn : n = (coreE₂CohomologicalFin l).deg (p, ⟨q, hq⟩) - 1
this : ↑q + r ≤ ↑l
t : ℕ
ht : ↑q + r + ↑t = ↑l
hpq : ∀ (a : ℤ) (b : Fin l), a + r = p → ¬↑↑b + (1 - r) = ↑q
⊢ l - 1 - t < l

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
inst✝¹ : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
X : SpectralObject C ι
data : SpectralSequenceDataCore ι c r₀
inst✝ : X.HasSpectralSequence data
l : ℕ
E : SpectralObject C (Fin (l + 1))
r n : ℤ
hr : 2 ≤ r
p : ℤ
q : ℕ
hq : q < l
hn : n = (coreE₂CohomologicalFin l).deg (p, ⟨q, hq⟩) - 1
this : ↑q + r ≤ ↑l
t : ℕ
ht : ↑q + r + ↑t = ↑l
hpq : ∀ (a : ℤ) (b : Fin l), a + r = p → ¬↑↑b + (1 - r) = ↑q
⊢ p - r + r = p

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
inst✝¹ : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
X : SpectralObject C ι
data : SpectralSequenceDataCore ι c r₀
inst✝ : X.HasSpectralSequence data
l : ℕ
E : SpectralObject C (Fin (l + 1))
r n : ℤ
hr : 2 ≤ r
p : ℤ
q : ℕ
hq : q < l
hn : n = (coreE₂CohomologicalFin l).deg (p, ⟨q, hq⟩) - 1
this : ↑q + r ≤ ↑l
t : ℕ
ht : ↑q + r + ↑t = ↑l
hpq : ∀ (a : ℤ) (b : Fin l), a + r = p → ¬↑↑b + (1 - r) = ↑q
⊢ ↑↑⟨l - 1 - t, ⋯⟩ + (1 - r) = ↑q

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r : ℤ
hr : r₀ ≤ r
pq : κ
i₀ i₁ i₂ i₃ : ι
h₀ : i₀ = data.i₀ r pq ⋯
h₁ : i₁ = data.i₁ pq
h₂ : i₂ = data.i₂ pq
h₃ : i₃ = data.i₃ r pq ⋯
n₀ n₁ n₂ : ℤ
h : n₁ = data.deg pq
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ n₀ = n₁ - 1

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
inst✝¹ : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
X : SpectralObject C ι
data : SpectralSequenceDataCore ι c r₀
inst✝ : X.HasSpectralSequence data
l : ℕ
E : SpectralObject C (Fin (l + 1))
r n : ℤ
hr : 2 ≤ r
p : ℤ
q : ℕ
hq : q < l
hpq : ∀ (pq' : ℤ × Fin l), ¬(ComplexShape.spectralSequenceFin l (r, 1 - r)).Rel pq' (p, ⟨q, hq⟩)
hn : n = (coreE₂CohomologicalFin l).deg (p, ⟨q, hq⟩) - 1
h : ↑l < ↑q + r
⊢ l ≤ (↑q + (r - 1)).toNat

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
inst✝¹ : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
X : SpectralObject C ι
data : SpectralSequenceDataCore ι c r₀
inst✝ : X.HasSpectralSequence data
l : ℕ
E : SpectralObject C (Fin (l + 1))
r n : ℤ
hr : 2 ≤ r
p : ℤ
q : ℕ
hq : q < l
hpq : ∀ (pq' : ℤ × Fin l), ¬(ComplexShape.spectralSequenceFin l (r, 1 - r)).Rel pq' (p, ⟨q, hq⟩)
hn : n = (coreE₂CohomologicalFin l).deg (p, ⟨q, hq⟩) - 1
h : ↑l < ↑q + r
⊢ l ≤ (↑q + r).toNat

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r : ℤ
hr : r₀ ≤ r
pq pq' : κ
hpq : (c r).Rel pq pq'
n₀ n₂ n₃ : ℤ
hn₃ : n₂ + 1 = n₃
hn₁ : n₀ + 1 = data.deg pq
hn₂ : data.deg pq + 1 = n₂
f₁ : data.i₀ r pq' ⋯ ⟶ data.i₁ pq'
f₂ : data.i₁ pq' ⟶ data.i₀ r pq ⋯
f₃ : data.i₀ r pq ⋯ ⟶ data.i₁ pq
f₄ : data.i₁ pq ⟶ data.i₂ pq
f₅ : data.i₂ pq ⟶ data.i₃ r pq ⋯
⊢ n₀ = data.deg pq - 1

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r : ℤ
hr : r₀ ≤ r
pq pq' : κ
hpq : (c r).Rel pq pq'
n₂ n₃ : ℤ
hn₃ : n₂ + 1 = n₃
hn₂ : data.deg pq + 1 = n₂
f₁ : data.i₀ r pq' ⋯ ⟶ data.i₁ pq'
f₂ : data.i₁ pq' ⟶ data.i₀ r pq ⋯
f₃ : data.i₀ r pq ⋯ ⟶ data.i₁ pq
f₄ : data.i₁ pq ⟶ data.i₂ pq
f₅ : data.i₂ pq ⟶ data.i₃ r pq ⋯
hn₁ : data.deg pq - 1 + 1 = data.deg pq
⊢ n₂ = data.deg pq + 1

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r : ℤ
hr : r₀ ≤ r
pq pq' : κ
hpq : (c r).Rel pq pq'
n₃ : ℤ
f₁ : data.i₀ r pq' ⋯ ⟶ data.i₁ pq'
f₂ : data.i₁ pq' ⟶ data.i₀ r pq ⋯
f₃ : data.i₀ r pq ⋯ ⟶ data.i₁ pq
f₄ : data.i₁ pq ⟶ data.i₂ pq
f₅ : data.i₂ pq ⟶ data.i₃ r pq ⋯
hn₁ : data.deg pq - 1 + 1 = data.deg pq
hn₃ : data.deg pq + 1 + 1 = n₃
hn₂ : data.deg pq + 1 = data.deg pq + 1
⊢ n₃ = data.deg pq + 2

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence

Goal state before:
pp: case pos
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r : ℤ
hr : r₀ ≤ r
pq pq' pq'' : κ
hpq : (c r).Rel pq pq'
⊢ pageD X data r pq pq' hr ≫ pageD X data r pq' pq'' hr = 0

pp: case neg
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r : ℤ
hr : r₀ ≤ r
pq pq' pq'' : κ
hpq : ¬(c r).Rel pq pq'
⊢ pageD X data r pq pq' hr ≫ pageD X data r pq' pq'' hr = 0

Tactic:
·

Goal state after:
case pos
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r : ℤ
hr : r₀ ≤ r
pq pq' pq'' : κ
hpq : (c r).Rel pq pq'
⊢ pageD X data r pq pq' hr ≫ pageD X data r pq' pq'' hr = 0
Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence

Goal state before:
pp: case pos
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r : ℤ
hr : r₀ ≤ r
pq pq' pq'' : κ
hpq : (c r).Rel pq pq'
hpq' : (c r).Rel pq' pq''
⊢ pageD X data r pq pq' hr ≫ pageD X data r pq' pq'' hr = 0

pp: case neg
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r : ℤ
hr : r₀ ≤ r
pq pq' pq'' : κ
hpq : (c r).Rel pq pq'
hpq' : ¬(c r).Rel pq' pq''
⊢ pageD X data r pq pq' hr ≫ pageD X data r pq' pq'' hr = 0

Tactic:
·

Goal state after:
case pos
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r : ℤ
hr : r₀ ≤ r
pq pq' pq'' : κ
hpq : (c r).Rel pq pq'
hpq' : (c r).Rel pq' pq''
⊢ pageD X data r pq pq' hr ≫ pageD X data r pq' pq'' hr = 0
Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r : ℤ
hr : r₀ ≤ r
pq pq' pq'' : κ
hpq : (c r).Rel pq pq'
hpq' : (c r).Rel pq' pq''
⊢ data.deg pq + 1 + 1 = data.deg pq + 2

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i₀ i₁ : ι
f : i₀ ⟶ i₁
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
⊢ n₀ - 1 + 1 = n₀

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence

Goal state before:
pp: case neg
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r : ℤ
hr : r₀ ≤ r
pq pq' pq'' : κ
hpq : (c r).Rel pq pq'
hpq' : ¬(c r).Rel pq' pq''
⊢ pageD X data r pq pq' hr ≫ pageD X data r pq' pq'' hr = 0

Tactic:
·

Goal state after:
case neg
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r : ℤ
hr : r₀ ≤ r
pq pq' pq'' : κ
hpq : (c r).Rel pq pq'
hpq' : ¬(c r).Rel pq' pq''
⊢ pageD X data r pq pq' hr ≫ pageD X data r pq' pq'' hr = 0
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i₀ i₁ : ι
f : i₀ ⟶ i₁
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
⊢ n₁ + 1 = n₁ + 1

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence

Goal state before:
pp: case neg
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r : ℤ
hr : r₀ ≤ r
pq pq' pq'' : κ
hpq : ¬(c r).Rel pq pq'
⊢ pageD X data r pq pq' hr ≫ pageD X data r pq' pq'' hr = 0

Tactic:
·

Goal state after:
case neg
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r : ℤ
hr : r₀ ≤ r
pq pq' pq'' : κ
hpq : ¬(c r).Rel pq pq'
⊢ pageD X data r pq pq' hr ≫ pageD X data r pq' pq'' hr = 0
Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence

Goal state before:
pp: case refine_1
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r : ℤ
hr : r₀ ≤ r
pq pq' pq'' : κ
hpq : (c r).Rel pq pq'
hpq' : (c r).Rel pq' pq''
n₀ n₁ n₂ n₃ n₄ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
hn₄ : n₃ + 1 = n₄
hn₂' : n₂ = data.deg pq'
⊢ (pageXIso X data r hr pq (data.i₀ r pq ⋯) (data.i₁ pq) (data.i₂ pq) (data.i₃ r pq ⋯) ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ ⋯ ⋯ ⋯).hom ≫
      (X.dShortComplex (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ n₃
          n₄ hn₁ hn₂ hn₃ hn₄).f =
    ((page X data r hr).sc' pq pq' pq'').f ≫
      (pageXIso X data r hr pq' (data.i₂ pq'') (data.i₃ r pq'' ⋯) (data.i₀ r pq ⋯) (data.i₁ pq) ⋯ ⋯ ⋯ ⋯ n₁ n₂ n₃ hn₂' ⋯
          ⋯).hom

pp: case refine_2
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r : ℤ
hr : r₀ ≤ r
pq pq' pq'' : κ
hpq : (c r).Rel pq pq'
hpq' : (c r).Rel pq' pq''
n₀ n₁ n₂ n₃ n₄ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
hn₄ : n₃ + 1 = n₄
hn₂' : n₂ = data.deg pq'
⊢ (pageXIso X data r hr pq' (data.i₂ pq'') (data.i₃ r pq'' ⋯) (data.i₀ r pq ⋯) (data.i₁ pq) ⋯ ⋯ ⋯ ⋯ n₁ n₂ n₃ hn₂' ⋯
          ⋯).hom ≫
      (X.dShortComplex (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ n₃
          n₄ hn₁ hn₂ hn₃ hn₄).g =
    ((page X data r hr).sc' pq pq' pq'').g ≫
      (pageXIso X data r hr pq'' (data.i₀ r pq'' ⋯) (data.i₁ pq'') (data.i₂ pq'') (data.i₃ r pq'' ⋯) ⋯ ⋯ ⋯ ⋯ n₂ n₃ n₄ ⋯
          ⋯ ⋯).hom

Tactic:
·

Goal state after:
case refine_1
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r : ℤ
hr : r₀ ≤ r
pq pq' pq'' : κ
hpq : (c r).Rel pq pq'
hpq' : (c r).Rel pq' pq''
n₀ n₁ n₂ n₃ n₄ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
hn₄ : n₃ + 1 = n₄
hn₂' : n₂ = data.deg pq'
⊢ (pageXIso X data r hr pq (data.i₀ r pq ⋯) (data.i₁ pq) (data.i₂ pq) (data.i₃ r pq ⋯) ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ ⋯ ⋯ ⋯).hom ≫
      (X.dShortComplex (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ n₃
          n₄ hn₁ hn₂ hn₃ hn₄).f =
    ((page X data r hr).sc' pq pq' pq'').f ≫
      (pageXIso X data r hr pq' (data.i₂ pq'') (data.i₃ r pq'' ⋯) (data.i₀ r pq ⋯) (data.i₁ pq) ⋯ ⋯ ⋯ ⋯ n₁ n₂ n₃ hn₂' ⋯
          ⋯).hom
Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence

Goal state before:
pp: case refine_2
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r : ℤ
hr : r₀ ≤ r
pq pq' pq'' : κ
hpq : (c r).Rel pq pq'
hpq' : (c r).Rel pq' pq''
n₀ n₁ n₂ n₃ n₄ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
hn₄ : n₃ + 1 = n₄
hn₂' : n₂ = data.deg pq'
⊢ (pageXIso X data r hr pq' (data.i₂ pq'') (data.i₃ r pq'' ⋯) (data.i₀ r pq ⋯) (data.i₁ pq) ⋯ ⋯ ⋯ ⋯ n₁ n₂ n₃ hn₂' ⋯
          ⋯).hom ≫
      (X.dShortComplex (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ n₃
          n₄ hn₁ hn₂ hn₃ hn₄).g =
    ((page X data r hr).sc' pq pq' pq'').g ≫
      (pageXIso X data r hr pq'' (data.i₀ r pq'' ⋯) (data.i₁ pq'') (data.i₂ pq'') (data.i₃ r pq'' ⋯) ⋯ ⋯ ⋯ ⋯ n₂ n₃ n₄ ⋯
          ⋯ ⋯).hom

Tactic:
·

Goal state after:
case refine_2
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r : ℤ
hr : r₀ ≤ r
pq pq' pq'' : κ
hpq : (c r).Rel pq pq'
hpq' : (c r).Rel pq' pq''
n₀ n₁ n₂ n₃ n₄ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
hn₄ : n₃ + 1 = n₄
hn₂' : n₂ = data.deg pq'
⊢ (pageXIso X data r hr pq' (data.i₂ pq'') (data.i₃ r pq'' ⋯) (data.i₀ r pq ⋯) (data.i₁ pq) ⋯ ⋯ ⋯ ⋯ n₁ n₂ n₃ hn₂' ⋯
          ⋯).hom ≫
      (X.dShortComplex (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ n₃
          n₄ hn₁ hn₂ hn₃ hn₄).g =
    ((page X data r hr).sc' pq pq' pq'').g ≫
      (pageXIso X data r hr pq'' (data.i₀ r pq'' ⋯) (data.i₁ pq'') (data.i₂ pq'') (data.i₃ r pq'' ⋯) ⋯ ⋯ ⋯ ⋯ n₂ n₃ n₄ ⋯
          ⋯ ⋯).hom
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
i j : ι
f : i ⟶ j
h : (X.shortComplex (𝟙 i) f (𝟙 j) n₀ n₁ n₂ hn₁ hn₂).LeftHomologyData := (X.homologyDataIdId f n₀ n₁ n₂ hn₁ hn₂).left
this : h.cyclesIso.inv = X.toCycles (𝟙 i) f f ⋯ n₁ ≫ (X.cyclesIso (𝟙 i) f (𝟙 j) n₀ n₁ n₂ hn₁ hn₂).inv
⊢ n₀ = n₁ - 1

Tactic:
by lia

Goal state after: