Section Elt.
Variable elt elt' elt'':Type.
Implicit Types m : t elt.
Implicit Types x y : key.
Implicit Types e : elt.
Definition empty : t elt := Build_slist (Raw.empty_NoDup elt).
Definition is_empty m : bool := Raw.is_empty (this m).
Definition add x e m : t elt := Build_slist (Raw.add_NoDup (NoDup m) x e).
Definition find x m : option elt := Raw.find x (this m).
Definition remove x m : t elt := Build_slist (Raw.remove_NoDup (NoDup m) x).
Definition mem x m : bool := Raw.mem x (this m).
Definition map f m : t elt' := Build_slist (Raw.map_NoDup (NoDup m) f).
Definition mapi (f:key->elt->elt') m : t elt' := Build_slist (Raw.mapi_NoDup (NoDup m) f).
Definition map2 f m (m':t elt') : t elt'' :=
Build_slist (Raw.map2_NoDup f (NoDup m) (NoDup m')).
Definition elements m : list (key*elt) := @Raw.elements elt (this m).
Definition cardinal m := length (this m).
Definition fold (A:Type)(f:key->elt->A->A) m (i:A) : A := @Raw.fold elt A f (this m) i.
Definition equal cmp m m' : bool := @Raw.equal elt cmp (this m) (this m').
Definition MapsTo x e m : Prop := Raw.PX.MapsTo x e (this m).
Definition In x m : Prop := Raw.PX.In x (this m).
Definition Empty m : Prop := Raw.Empty (this m).
Definition Equal m m' := forall y, find y m = find y m'.
Definition Equiv (eq_elt:elt->elt->Prop) m m' :=
(forall k, In k m <-> In k m') /\
(forall k e e', MapsTo k e m -> MapsTo k e' m' -> eq_elt e e').
Definition Equivb cmp m m' : Prop := @Raw.Equivb elt cmp (this m) (this m').
Definition eq_key : (key*elt) -> (key*elt) -> Prop := @Raw.PX.eqk elt.
Definition eq_key_elt : (key*elt) -> (key*elt) -> Prop:= @Raw.PX.eqke elt.

Lemma MapsTo_1 : forall m x y e, E.eq x y -> MapsTo x e m -> MapsTo y e m.
Lemma map2_2 : forall (elt elt' elt'':Type)(m: t elt)(m': t elt')
(x:key)(f:option elt->option elt'->option elt''),
In x (map2 f m m') -> In x m \/ In x m'.