Definition Add x s s' := forall y, In y s' <-> O.eq x y \/ In y s.

(* Specifications: *)
Lemma lt_empty_l : forall x s1 s2 s2', Empty s1 -> Above x s2 -> Add x s2 s2' -> lt s1 s2'.
Lemma lt_add_lt : forall x1 x2 s1 s1' s2 s2', Above x1 s1 -> Above x2 s2 -> Add x1 s1 s1' -> Add x2 s2 s2' -> O.lt x1 x2 -> lt s1' s2'.
Lemma lt_add_eq : forall x1 x2 s1 s1' s2 s2', Above x1 s1 -> Above x2 s2 -> Add x1 s1 s1' -> Add x2 s2 s2' -> O.eq x1 x2 -> lt s1 s2 -> lt s1' s2'.