Definition Subset' s s' := forall a, interp s a -> interp s' a.
Definition Empty' s := forall a, ~ interp s a.
Definition For_all' (P : key -> Prop) s := forall x, interp s x -> P x.
Definition Exists' (P : key -> Prop) s := exists x, interp s x /\ P x.
(* Now, some definitions to match the Module Type *)
Definition t : Type := sig valid.
Definition elt := key.
Definition empty : t := exist valid EE empty_valid.
Definition is_empty (x: t) : bool :=
match proj1_sig x with EE => true | _ => false end.
Definition mem (k: elt) (x: t) : bool := member k (proj1_sig x).
Definition add (k: elt) (x: t) : t :=
exist valid (insert k (proj1_sig x)) (insert_valid _ _ (proj2_sig x)).
Definition singleton (k: elt) : t := exist valid _ (singleton_valid k).
Definition remove (k: elt) (x: t) : t :=
exist valid _ (remove_valid k _ (proj2_sig x)).
Definition fold (A: Type) (f: elt -> A -> A) (x: t) (base: A) : A :=
fold' A f (proj1_sig x) base.
Definition for_all (f: elt -> bool) (x: t) : bool :=
fold bool (fun k y => andb (f k) y) x true.
Definition exists_ (f: elt -> bool) (x: t) : bool :=
fold bool (fun k y => orb (f k) y) x false.
Definition cardinal (x: t) := fold nat (fun _ => S) x O.
Definition filter (f: elt -> bool) (s: t) : t := exist valid _ (filter_valid f _ (proj2_sig s)).
Definition partition (f: elt -> bool) (s: t) : t*t :=
let s' := partition' f (proj1_sig s)
in (exist valid _ (partition_valid1 f _ (proj2_sig s)),
exist valid _ (partition_valid2 f _ (proj2_sig s))).
Definition union (t1 t2: t) : t := exist valid _ (union_valid _ _ (proj2_sig t1) (proj2_sig t2)).
Definition diff (t1 t2: t) : t := exist valid _ (diff_valid _ _ (proj2_sig t1) (proj2_sig t2)).
Definition inter (t1 t2: t) : t := exist valid _ (inter_valid _ _ (proj2_sig t1) (proj2_sig t2)).
Definition elements (x: t) : list elt := elements' (proj1_sig x) nil.
Definition choose (x: t) : option elt := choose' (proj1_sig x).
Definition max_elt (x: t) : option elt := max_elt' (proj1_sig x).
Definition equal' (s1 s2 : tree) : bool :=
match compare' s1 s2 with Eq => true | _ => false end.
Definition equal (t1 t2 : t) : bool := equal' (proj1_sig t1) (proj1_sig t2).
Definition subset (t1 t2: t) : bool := is_empty (diff t1 t2).
Definition In (k: key) (x: t) := interp (proj1_sig x) k.

(* Specifications: *)
Lemma In_compat : Proper (E.eq==>eq==>iff) In.
Definition Equal (s s': t)  := Equal' (proj1_sig s) (proj1_sig s').