Definition diff (s s' : t) : t := let (s'', _) := diff s s' in s''.

(* Specifications: *)
Lemma diff_1 : forall (s s' : t) (x : elt), In x (diff s s') -> In x s.
Lemma diff_2 : forall (s s' : t) (x : elt), In x (diff s s') -> ~ In x s'.
Lemma diff_3 :
forall (s s' : t) (x : elt), In x s -> ~ In x s' -> In x (diff s s').