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_comp n s h gN gS f: max_arity_prog (rules h) ≤ maxar h → (max_arity_prog (concat (map rules gN))) ≤ maxl (map maxar gN) → (max_arity_prog (concat (map rules gS))) ≤ maxl (map maxar gS) → max_arity_prog (rules (make_comp n s h gN gS f)) ≤ maxar (make_comp n s h gN gS f).