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).

(* Specifications: *)
Lemma gtb_1 : forall x y, gtb x y = true <-> E.lt y x.
Lemma leb_1 : forall x y, leb x y = true <-> ~E.lt y x.
#[global]
Instance gtb_compat x : Proper (E.eq==>Logic.eq) (gtb x).