Definition gtb x y := match E.compare x y with GT _ => true | _ => false end.
Definition leb x := fun y => negb (gtb x y).
Definition elements_lt x s := List.filter (gtb x) (elements s).
Definition elements_ge x s := List.filter (leb x) (elements s).

Lemma gtb_1 : forall x y, gtb x y = true <-> E.lt y x.