Definition min_elt m : option (key*elt) := match elements m with
   | nil => None
   | (x,e)::_ => Some (x,e)
  end.

Lemma fold_Add_Below : forall m1 m2 x e (A:Type)(eqA:A->A->Prop)(st:Equivalence eqA) (f:key->elt->A->A)(i:A) (P:Proper (E.eq==>eq==>eqA==>eqA) f), Below x m1 -> Add x e m1 m2 -> eqA (fold f m2 i) (fold f m1 (f x e i)).