mathlib-initiative/mathlib-tactics huggingface dataset:
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
f₁₂ : i₀ ⟶ i₂
f₂₃ : i₁ ⟶ i₃
h₁₂ : f₁ ≫ f₂ = f₁₂
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ Mono (X.kernelSequenceCyclesE f₁ f₂ f₃ f₂₃ h₂₃ n₀ n₁ n₂ hn₁ hn₂).f

Tactic:
by

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✝⁴ : Category.{v_1, u_1} C
inst✝³ : Abelian C
inst✝² : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
X : SpectralObject C ι
data : SpectralSequenceDataCore ι c r₀
inst✝¹ : X.HasSpectralSequence data
Y : SpectralObject C EInt
inst✝ : Y.IsFirstQuadrant
r : ℤ
p q : ℕ
hpq : ∀ (pq' : ℕ × ℕ), ¬(ComplexShape.spectralSequenceNat (r, 1 - r)).Rel (p, q) pq'
hr : autoParam (2 ≤ r) HasSpectralSequence._auto_3
⊢ IsZero ((Y.H (coreE₂CohomologicalNat.deg (p, q) + 1)).obj (mk₁ (homOfLE ⋯)))

Tactic:
apply isZero₁_of_isFirstQuadrant

Goal state after:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Abelian C
inst✝² : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
X : SpectralObject C ι
data : SpectralSequenceDataCore ι c r₀
inst✝¹ : X.HasSpectralSequence data
Y : SpectralObject C EInt
inst✝ : Y.IsFirstQuadrant
r : ℤ
p q : ℕ
hpq : ∀ (pq' : ℕ × ℕ), ¬(ComplexShape.spectralSequenceNat (r, 1 - r)).Rel (p, q) pq'
hr : autoParam (2 ≤ r) HasSpectralSequence._auto_3
⊢ coreE₂CohomologicalNat.i₀ r (p, q) ⋯ ≤ WithBotTop.coe 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' : κ
hpq : (c r).Rel pq pq'
i₀ i₁ i₂ i₃ i₄ i₅ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₄ : i₃ ⟶ i₄
f₅ : i₄ ⟶ i₅
h₀ : i₀ = data.i₀ r pq' ⋯
h₁ : i₁ = data.i₁ pq'
h₂ : i₂ = data.i₀ r pq ⋯
h₃ : i₃ = data.i₁ pq
h₄ : i₄ = data.i₂ pq
h₅ : i₅ = data.i₃ r pq ⋯
n₀ n₁ n₂ n₃ : ℤ
hn₁' : n₁ = data.deg pq
hn₁ : autoParam (n₀ + 1 = n₁) pageD_eq._auto_1
hn₂ : autoParam (n₁ + 1 = n₂) pageD_eq._auto_3
hn₃ : autoParam (n₂ + 1 = n₃) pageD_eq._auto_5
⊢ i₃ = data.i₃ r pq' ⋯

Tactic:
by

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'
i₀ i₁ i₂ i₃ i₄ i₅ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₄ : i₃ ⟶ i₄
f₅ : i₄ ⟶ i₅
h₀ : i₀ = data.i₀ r pq' ⋯
h₁ : i₁ = data.i₁ pq'
h₂ : i₂ = data.i₀ r pq ⋯
h₃ : i₃ = data.i₁ pq
h₄ : i₄ = data.i₂ pq
h₅ : i₅ = data.i₃ r pq ⋯
n₀ n₁ n₂ n₃ : ℤ
hn₁' : n₁ = data.deg pq
hn₁ : autoParam (n₀ + 1 = n₁) pageD_eq._auto_1
hn₂ : autoParam (n₁ + 1 = n₂) pageD_eq._auto_3
hn₃ : autoParam (n₂ + 1 = n₃) pageD_eq._auto_5
⊢ n₂ = data.deg pq'

Tactic:
by

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✝⁴ : Category.{v_1, u_1} C
inst✝³ : Abelian C
inst✝² : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
X : SpectralObject C ι
data : SpectralSequenceDataCore ι c r₀
inst✝¹ : X.HasSpectralSequence data
Y : SpectralObject C EInt
inst✝ : Y.IsFirstQuadrant
r : ℤ
p q : ℕ
hpq : ∀ (pq' : ℕ × ℕ), ¬(ComplexShape.spectralSequenceNat (r, 1 - r)).Rel (p, q) pq'
hr : autoParam (2 ≤ r) HasSpectralSequence._auto_3
this : 0 < ↑q - r + 2
p' : ℕ
hp' : ↑p + r = ↑p'
q' : ℕ
hq' : ↑q + 1 - r = ↑q'
⊢ (ComplexShape.spectralSequenceNat (r, 1 - r)).Rel (p, q) (p', q')

Tactic:
by

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✝⁴ : Category.{v_1, u_1} C
inst✝³ : Abelian C
inst✝² : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
X : SpectralObject C ι
data : SpectralSequenceDataCore ι c r₀
inst✝¹ : X.HasSpectralSequence data
Y : SpectralObject C EInt
inst✝ : Y.IsFirstQuadrant
⊢ ∀ (r r' : ℤ) (pq : ℕ × ℕ),
    (∀ (pq' : ℕ × ℕ), ¬(ComplexShape.spectralSequenceNat (r, 1 - r)).Rel pq pq') →
      ∀ (n : ℤ),
        n = coreE₂CohomologicalNat.deg pq + 1 →
          ∀ (hrr' : autoParam (r + 1 = r') HasSpectralSequence._auto_1)
            (hr : autoParam (2 ≤ r) HasSpectralSequence._auto_3), IsZero ((Y.H n).obj (mk₁ (homOfLE ⋯)))

Tactic:
by

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

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

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
f₂₃ : i₁ ⟶ i₃
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
A : C
x₂ : A ⟶ X.cycles f₁ f₂₃ n₁
hx₂ : x₂ ≫ X.iCycles f₁ f₂₃ n₁ ≫ (X.H n₁).map (twoδ₁Toδ₀ f₂ f₃ f₂₃ h₂₃) = 0
⊢ (x₂ ≫ X.iCycles f₁ f₂₃ n₁) ≫ (X.kernelSequenceE f₁ f₂ f₃ f₂₃ h₂₃ n₀ n₁ n₂ ⋯ ⋯).g = 0

Tactic:
by

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

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
f₂₃ : i₁ ⟶ i₃
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
A : C
x₂ : A ⟶ X.cycles f₁ f₂₃ n₁
hx₂ : x₂ ≫ X.iCycles f₁ f₂₃ n₁ ≫ (X.H n₁).map (twoδ₁Toδ₀ f₂ f₃ f₂₃ h₂₃) = 0
A₁ : C
π₁ : A₁ ⟶ A
w✝ : Epi π₁
x₁ : A₁ ⟶ (X.kernelSequenceE f₁ f₂ f₃ f₂₃ h₂₃ n₀ n₁ n₂ ⋯ ⋯).X₁
hx₁ : π₁ ≫ x₂ ≫ X.iCycles f₁ f₂₃ n₁ = x₁ ≫ (X.kernelSequenceE f₁ f₂ f₃ f₂₃ h₂₃ n₀ n₁ n₂ ⋯ ⋯).f
⊢ π₁ ≫ x₂ = x₁ ≫ (X.kernelSequenceCyclesE f₁ f₂ f₃ f₂₃ h₂₃ n₀ n₁ n₂ hn₁ hn₂).f

Tactic:
by

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'
i₀ i₁ i₂ i₃ i₄ i₅ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₄ : i₃ ⟶ i₄
f₅ : i₄ ⟶ i₅
h₀ : i₀ = data.i₀ r pq' ⋯
h₁ : i₁ = data.i₁ pq'
h₂ : i₂ = data.i₀ r pq ⋯
h₃ : i₃ = data.i₁ pq
h₄ : i₄ = data.i₂ pq
h₅ : i₅ = data.i₃ r pq ⋯
n₀ n₁ n₂ n₃ : ℤ
hn₁' : n₁ = data.deg pq
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
⊢ pageD X data r pq pq' ⋯ =
    (pageXIso X data r ⋯ pq i₂ i₃ i₄ i₅ h₂ h₃ h₄ h₅ n₀ n₁ n₂ hn₁' hn₁ hn₂).hom ≫
      X.d f₁ f₂ f₃ f₄ f₅ n₀ n₁ n₂ n₃ hn₁ hn₂ hn₃ ≫
        (pageXIso X data r ⋯ pq' i₀ i₁ i₂ i₃ h₀ h₁ ⋯ ⋯ n₁ n₂ n₃ ⋯ hn₂ hn₃).inv

Tactic:
by

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
f₂₃ : i₁ ⟶ i₃
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ (X.kernelSequenceCyclesE f₁ f₂ f₃ f₂₃ h₂₃ n₀ n₁ n₂ hn₁ hn₂).Exact

Tactic:
by

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
i j : ι
f : i ⟶ j
i' j' : ι
f' : i' ⟶ j'
α : mk₁ f ⟶ mk₁ f'
β : mk₃ (𝟙 i) f (𝟙 j) ⟶ mk₃ (𝟙 i') f' (𝟙 j')
n₀ n₁ n₂ : ℤ
⊢ (mk₃ (𝟙 i) f (𝟙 j)).map' 0 1 homMk₁._proof_4 _proof_298✝ ≫ α.app 0 =
    α.app 0 ≫ (mk₃ (𝟙 i') f' (𝟙 j')).map' 0 1 homMk₁._proof_4 _proof_298✝

Tactic:
by

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✝⁴ : Category.{v_1, u_1} C
inst✝³ : Abelian C
inst✝² : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
X : SpectralObject C ι
data : SpectralSequenceDataCore ι c r₀
inst✝¹ : X.HasSpectralSequence data
Y : SpectralObject C EInt
inst✝ : Y.IsFirstQuadrant
r : ℤ
p q : ℕ
hpq : ∀ (pq' : ℕ × ℕ), ¬(ComplexShape.spectralSequenceNat (r, 1 - r)).Rel pq' (p, q)
hr : autoParam (2 ≤ r) HasSpectralSequence._auto_11
this : ↑q + r - 1 ≤ ↑p + ↑q - 1
p' : ℕ
hp' : ↑p - r = ↑p'
q' : ℕ
hq' : ↑q - 1 + r = ↑q'
⊢ (ComplexShape.spectralSequenceNat (r, 1 - r)).Rel (p', q') (p, q)

Tactic:
by

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
i j : ι
f : i ⟶ j
i' j' : ι
f' : i' ⟶ j'
α : mk₁ f ⟶ mk₁ f'
β : mk₃ (𝟙 i) f (𝟙 j) ⟶ mk₃ (𝟙 i') f' (𝟙 j')
n₀ n₁ n₂ : ℤ
⊢ (mk₃ (𝟙 i) f (𝟙 j)).map' 2 3 _proof_299✝ _proof_300✝ ≫ α.app 1 =
    α.app 1 ≫ (mk₃ (𝟙 i') f' (𝟙 j')).map' 2 3 _proof_299✝ _proof_300✝

Tactic:
by

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✝⁴ : Category.{v_1, u_1} C
inst✝³ : Abelian C
inst✝² : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
X : SpectralObject C ι
data : SpectralSequenceDataCore ι c r₀
inst✝¹ : X.HasSpectralSequence data
Y : SpectralObject C EInt
inst✝ : Y.IsFirstQuadrant
⊢ ∀ (r r' : ℤ) (pq : ℕ × ℕ),
    (∀ (pq' : ℕ × ℕ), ¬(ComplexShape.spectralSequenceNat (r, 1 - r)).Rel pq' pq) →
      ∀ (n : ℤ),
        n = coreE₂CohomologicalNat.deg pq - 1 →
          ∀ (hrr' : autoParam (r + 1 = r') HasSpectralSequence._auto_9)
            (hr : autoParam (2 ≤ r) HasSpectralSequence._auto_11), IsZero ((Y.H n).obj (mk₁ (homOfLE ⋯)))

Tactic:
by

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' 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

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
i j : ι
f : i ⟶ j
i' j' : ι
f' : i' ⟶ j'
α : mk₁ f ⟶ mk₁ f'
β : mk₃ (𝟙 i) f (𝟙 j) ⟶ mk₃ (𝟙 i') f' (𝟙 j')
n₀ n₁ n₂ : ℤ
hβ : β = homMk₃ (α.app 0) (α.app 0) (α.app 1) (α.app 1) ⋯ ⋯ ⋯
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ α = homMk₁ (β.app 1) (β.app 2) ⋯

Tactic:
by

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
i j : ι
f : i ⟶ j
i' j' : ι
f' : i' ⟶ j'
α : mk₁ f ⟶ mk₁ f'
β : mk₃ (𝟙 i) f (𝟙 j) ⟶ mk₃ (𝟙 i') f' (𝟙 j')
n₀ n₁ n₂ : ℤ
hβ : β = homMk₃ (α.app 0) (α.app 0) (α.app 1) (α.app 1) ⋯ ⋯ ⋯
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ X.map (𝟙 i) f (𝟙 j) (𝟙 i') f' (𝟙 j') β n₀ n₁ n₂ hn₁ hn₂ ≫ (X.EIsoH f' n₀ n₁ n₂ hn₁ hn₂).hom =
    (X.EIsoH f n₀ n₁ n₂ hn₁ hn₂).hom ≫ (X.H n₁).map α

Tactic:
by

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

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 : autoParam (n₀ + 1 = n₁) cyclesIsoH_inv._auto_1
⊢ 𝟙 i₀ ≫ f = f

Tactic:
by

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✝⁴ : Category.{v_1, u_1} C
inst✝³ : Abelian C
inst✝² : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
X : SpectralObject C ι
data : SpectralSequenceDataCore ι c r₀
inst✝¹ : X.HasSpectralSequence data
Y : SpectralObject C EInt
inst✝ : Y.IsThirdQuadrant
r : ℤ
p q : ℕ
hpq : ∀ (pq' : ℕ × ℕ), ¬(ComplexShape.spectralSequenceNat (-r, r - 1)).Rel (p, q) pq'
hr : autoParam (2 ≤ r) HasSpectralSequence._auto_3
this : -↑p - ↑q + 1 < -↑q - r + 2
p' : ℕ
hp' : ↑p - r = ↑p'
q' : ℕ
hq' : ↑q + r - 1 = ↑q'
⊢ (ComplexShape.spectralSequenceNat (-r, r - 1)).Rel (p, q) (p', q')

Tactic:
by

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
⊢ (X.cyclesIsoH f n₀ n₁ hn₁).inv = X.toCycles (𝟙 i₀) f f ⋯ n₀

Tactic:
by

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 : autoParam (n₀ + 1 = n₁) cyclesIsoH_hom_inv_id._auto_1
⊢ 𝟙 i₀ ≫ f = f

Tactic:
by

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✝⁴ : Category.{v_1, u_1} C
inst✝³ : Abelian C
inst✝² : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
X : SpectralObject C ι
data : SpectralSequenceDataCore ι c r₀
inst✝¹ : X.HasSpectralSequence data
Y : SpectralObject C EInt
inst✝ : Y.IsThirdQuadrant
⊢ ∀ (r r' : ℤ) (pq : ℕ × ℕ),
    (∀ (pq' : ℕ × ℕ), ¬(ComplexShape.spectralSequenceNat (-r, r - 1)).Rel pq pq') →
      ∀ (n : ℤ),
        n = coreE₂HomologicalNat.deg pq + 1 →
          ∀ (hrr' : autoParam (r + 1 = r') HasSpectralSequence._auto_1)
            (hr : autoParam (2 ≤ r) HasSpectralSequence._auto_3), IsZero ((Y.H n).obj (mk₁ (homOfLE ⋯)))

Tactic:
by

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
⊢ (X.cyclesIsoH f n₀ n₁ hn₁).hom ≫ X.toCycles (𝟙 i₀) f f ⋯ n₀ = 𝟙 (X.cycles (𝟙 i₀) f n₀)

Tactic:
by

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 : autoParam (n₀ + 1 = n₁) cyclesIsoH_inv_hom_id._auto_1
⊢ 𝟙 i₀ ≫ f = f

Tactic:
by

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
⊢ X.toCycles (𝟙 i₀) f f ⋯ n₀ ≫ (X.cyclesIsoH f n₀ n₁ hn₁).hom = 𝟙 ((X.H n₀).obj (mk₁ f))

Tactic:
by

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
⊢ n₁ + 1 = n₁ + 1

Tactic:
by

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 : autoParam (n₀ + 1 = n₁) opcyclesIsoH_hom._auto_1
⊢ f ≫ 𝟙 i₁ = f

Tactic:
by

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' pq'' : κ
⊢ pageD X data r pq pq' hr ≫ pageD X data r pq' pq'' hr = 0

Tactic:
by

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' 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'
⊢ data.i₃ r pq'' ⋯ ≤ data.i₀ r pq ⋯

Tactic:
by

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' 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'
⊢ n₁ = data.deg pq

Tactic:
by

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' 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'
⊢ data.i₂ pq'' = data.i₀ r pq' ⋯

Tactic:
by

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
⊢ (X.opcyclesIsoH f n₀ n₁ hn₁).hom = X.fromOpcycles f (𝟙 i₁) f ⋯ n₁

Tactic:
by

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' 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'
⊢ data.i₃ r pq'' ⋯ = data.i₁ pq'

Tactic:
by

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 : autoParam (n₀ + 1 = n₁) opcyclesIsoH_hom_inv_id._auto_1
⊢ f ≫ 𝟙 i₁ = f

Tactic:
by

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' 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'
⊢ data.i₀ r pq ⋯ = data.i₂ pq'

Tactic:
by

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
⊢ X.fromOpcycles f (𝟙 i₁) f ⋯ n₁ ≫ (X.opcyclesIsoH f n₀ n₁ hn₁).inv = 𝟙 (X.opcycles f (𝟙 i₁) n₁)

Tactic:
by

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 : autoParam (n₀ + 1 = n₁) opcyclesIsoH_inv_hom_id._auto_1
⊢ f ≫ 𝟙 i₁ = f

Tactic:
by

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' 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'
⊢ data.i₁ pq = data.i₃ r pq' ⋯

Tactic:
by

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
⊢ (X.opcyclesIsoH f n₀ n₁ hn₁).inv ≫ X.fromOpcycles f (𝟙 i₁) f ⋯ n₁ = 𝟙 ((X.H n₁).obj (mk₁ f))

Tactic:
by

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' 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'
⊢ n₃ = data.deg pq''

Tactic:
by

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
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
⊢ 𝟙 i ≫ f = f

Tactic:
by

Goal state after: