Proposition BC_to_TRS_arity bc n s st:
  let trs := snd (BC_to_TRS bc st) in
  arities bc = ok_arities n s →
  max_arity_prog trs.(rules) ≤ trs.(maxar).

Lemma BC_to_TRS_first_le_last bc st: let trs := snd (BC_to_TRS bc st) in trs.(first) ≤ trs.(last).