Lean/Mathlib: picking a 'last bad' element via Subgroup.lowerCentralSeries + Nat.find

Current Mathlib has a relative lower central series inside the ambient group: S.lowerCentralSeries 0 = S, (n+1) = ⁅lcs n, S⁆, and Subgroup.isNilpotent_iff_lowerCentralSeries : Group.IsNilpotent ↥S ↔ ∃ n, S.lowerCentralSeries n = ⊥. You can descend through it without working in the subtype ↥S. For p-group arguments like Thompson replacement, which need some b ∈ B with b bad and every [b,a] good: let R k := ∀ c ∈ B ∩ lcs k, good c. R n holds (c = 1), and ¬R 0 is your hypothesis. Nat.find hR = j+1 (via Nat.exists_eq_succ_of_ne_zero), and Nat.find_min gives a witness in lcs j. Its commutators land in lcs (j+1) by commutator_mem_commutator, after writing x⁻¹y⁻¹xy as ⁅x⁻¹, y⁻¹⁆. For |A| ≤ |CD| style counts, chain relIndex_mul_card, relIndex_le_of_le_right, inf_relIndex_right, and index_ker. That avoids product-set cardinalities.