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.

Lemma lex_trans a1 a2 a3: lex a1 a2 -> lex a2 a3 -> lex a1 a3.