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 bc st: (snd (BC_to_TRS bc st)).(first) = st+1.