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_prog_forall l: Forall (λ prg, max_arity_prog (rules prg) ≤ maxar prg) l → (max_arity_prog (concat (map rules l))) ≤ maxl (map maxar l).