English
Repeated structure ensuring compatibility and higher relations; (technical restatement of nilpotency patterns).
Русский
Повторная структура совместимости и связи; (техническое повторение свойств нулеподобия).
LaTeX
$$$δ_δ \\text{ relations (structural identity).}$$$
Lean4
theorem δ_δ (n₀ n₁ n₂ : ℤ) (z : Cochain F G n₀) : δ n₁ n₂ (δ n₀ n₁ z) = 0 :=
by
by_cases h₁₂ : n₁ + 1 = n₂; swap
· rw [δ_shape _ _ h₁₂]
by_cases h₀₁ : n₀ + 1 = n₁; swap
· rw [δ_shape _ _ h₀₁, δ_zero]
ext p q hpq
dsimp
simp only [δ_v n₁ n₂ h₁₂ _ p q hpq _ _ rfl rfl, δ_v n₀ n₁ h₀₁ z p (q - 1) (by cutsat) (q - 2) _ (by cutsat) rfl,
δ_v n₀ n₁ h₀₁ z (p + 1) q (by cutsat) _ (p + 2) rfl (by cutsat), ← h₁₂, Int.negOnePow_succ, add_comp, assoc,
HomologicalComplex.d_comp_d, comp_zero, zero_add, comp_add, HomologicalComplex.d_comp_d_assoc, zero_comp, smul_zero,
add_zero, add_neg_cancel, Units.neg_smul, Linear.units_smul_comp, Linear.comp_units_smul]
