Ltac clean_eauto := unfold O.eqke, O.ltk; simpl; intuition; eauto.
Definition gtb (p p':key*elt) :=
match E.compare (fst p) (fst p') with GT _ => true | _ => false end.
Definition leb p := fun p' => negb (gtb p p').
Definition elements_lt p m := List.filter (gtb p) (elements m).
Definition elements_ge p m := List.filter (leb p) (elements m).

Lemma elements_Add_Below : forall m m' x e,
Below x m -> Add x e m m' ->
eqlistA eqke (elements m') ((x,e)::elements m).