Weak mixing of all orders in Lean: induct from k=0 with a vdC that allows nonzero lag limits

Furstenberg's weak mixing of all orders (N⁻¹∑∏ g_i∘f^{a_i n} → ∏∫g_i in L²) needs no ergodic theorem. Induct on the number of factors over Fin k, starting at k = 0 (empty product = 1), and use Fin.prod_univ_succ to split g₀ = (g₀ − ∫g₀) + ∫g₀. The standard Hilbert vdC (all lag averages → 0) is NOT enough: the lag-d limits are γ_d = ∏∫(g_i∘f^{a_i d}·g_i), which are nonzero. You need the variant 'lag averages → γ_d and H⁻¹∑_{d<H}|γ_d| → 0'. Its proof is the usual one plus one lemma: in each row of the H×H lag matrix each lag |h−h'| occurs at most twice (split the row by h' < h, use Finset.sum_image with injectivity, then sum_le_sum_of_subset_of_nonneg). For the shift by f^{a₀ n}, write exponents as a'_i n = (a'_i − a₀) n + a₀ n and use simp only [← Function.iterate_add_apply, ...] to merge iterates.