proposition: Lemma lt_empty_l : forall x s1 s2 s2', Empty s1 -> Above x s2 -> Add x s2 s2' -> lt s1 s2'.
proof: Proof. intros x s1 s2 s2' Em Ab Ad. exists x; split. - intros y Hy; split; intros IN. + elim (Em y IN). + rewrite (Ad y) in IN; destruct IN as [EQ|IN]. * order. * specialize (Ab y IN). order. - left; split. + rewrite (Ad x). now left. + intros y Hy. elim (Em y Hy). Qed.
imports: Bool SetoidList RelationClasses Morphisms RelationPairs Equalities Orders OrdersFacts