Definition active_term_order_superset_dec: forall (t t': active_term), {active_term_order_superset t t'} + {~ active_term_order_superset t t'}.Definition lex : (active_term * term) -> (active_term * term) -> Prop :=
  lexprod active_term_order_superset term_order.

(* Specifications: *)
Lemma wf_lex : well_founded lex.
Lemma sub_terms_trans t1 t2 t3: In t1 (sub_terms variable function constructor t2) -> In t2 (sub_terms _ _ _ t3) -> In t1 (sub_terms _ _ _ t3).
Lemma sub_terms_strict_trans t1 t2 t3: In t1 (sub_terms_strict variable function constructor t2) -> In t2 (sub_terms_strict _ _ _ t3) -> In t1 (sub_terms_strict _ _ _ t3).
Lemma lex_trans a1 a2 a3: lex a1 a2 -> lex a2 a3 -> lex a1 a3.