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_state_bounds bc st: let (st',trs) := BC_to_TRS bc st in st+1 = trs.(first) ∧ trs.(last) = st'.