Definition active_term_order_superset_dec: forall (t t': active_term), {active_term_order_superset t t'} + {~ active_term_order_superset t t'}.

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).