Proposition BC_to_TRS_rules_vars_defined bc n s st:
  let trs := snd (BC_to_TRS bc st) in
  arities bc = ok_arities n s →
  Forall rule_vars_defined trs.(rules).

Lemma max_arity_make_rec n s g h0 h1 f: max_arity_prog (rules g) ≤ maxar g → max_arity_prog (rules h0) ≤ maxar h0 → max_arity_prog (rules h1) ≤ maxar h1 → max_arity_prog (rules (make_rec (S n) s g h0 h1 f)) ≤ maxar (make_rec (S n) s g h0 h1 f).