Definition choose_spec1 :
    forall (s : t) (x : elt), choose s = Some x -> In x s := min_elt_spec1.Definition choose_spec2 :
    forall s : t, choose s = None -> Empty s := min_elt_spec3.

Lemma choose_spec3: forall s s' x x', Ok s -> Ok s' -> choose s = Some x -> choose s' = Some x' -> Equal s s' -> X.eq x x'.