(** ** Induction principles about fold contributed by S. Lescuyer *)
(** In the following lemma, the step hypothesis is deliberately restricted to the precise map m we are considering. *)

Lemma fold_rec :
forall (A:Type)(P : t elt -> A -> Type)(f : key -> elt -> A -> A),
forall (i:A)(m:t elt),
(forall m, Empty m -> P m i) ->
(forall k e a m' m'', MapsTo k e m -> ~In k m' ->
Add k e m' m'' -> P m' a -> P m'' (f k e a)) ->
P m (fold f m i).