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. *)
(** [fold] can be used to reconstruct the same initial set. *)

Lemma fold_identity : forall m : t elt, Equal (fold (@add _) m (empty _)) m.