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 elements_split : forall x s, elements s = elements_lt x s ++ elements_ge x s.