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_fapply_app f (l1 l2: list term): (* => Syntax.v ? *) max_arity_term (fapply f (l1++l2)) = max (length (l1++l2)) (max (maxl (map (@max_arity_term _ _ _) l1)) (maxl (map (@max_arity_term _ _ _) l2))).