Module Import P := Properties M.
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).
Hint Resolve gtb_compat leb_compat : fset.
Definition Above x s := forall y, In y s -> E.lt y x.
Definition Below x s := forall y, In y s -> E.lt x y.

Lemma elements_Add_Above : forall s s' x,
Above x s -> Add x s s' ->
eqlistA E.eq (elements s') (elements s ++ x::nil).