Section Elt.
Variable elt:Type.
Definition Add x (e:elt) m m' := forall y, find y m' = find y (add x e m).
Notation eqke := (@eq_key_elt elt).
Notation eqk := (@eq_key elt).
Instance eqk_equiv : Equivalence eqk.
(** Complements about InA, NoDupA and findA *)
(** * Elements *)
(** * Conversions between maps and association lists. *)
Definition uncurry {U V W : Type} (f : U -> V -> W) : U*V -> W :=
fun p => f (fst p) (snd p).
Definition of_list :=
List.fold_right (uncurry (@add _)) (empty elt).
Definition to_list := elements.
(** * Fold *)
(** Alternative specification via [fold_right] *)
(** ** 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. *)
(** Same, with [empty] and [add] instead of [Empty] and [Add]. In this case, [P] must be compatible with equality of sets *)
(** [fold_rec_weak] is a weaker principle than [fold_rec_bis] : the step hypothesis must here be applicable anywhere. At the same time, it looks more like an induction principle, and hence can be easier to use. *)
(** From the induction principle on [fold], we can deduce some general induction principles on maps. *)

Lemma map_induction_bis :
forall P : t elt -> Type,
(forall m m', Equal m m' -> P m -> P m') ->
P (empty _) ->
(forall x e m, ~In x m -> P m -> P (add x e m)) ->
forall m, P m.