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

Lemma map_induction_min : forall P : t elt -> Type, (forall m, Empty m -> P m) -> (forall m m', P m -> forall x e, Below x m -> Add x e m m' -> P m') -> forall m, P m.