(** * Final Encapsulation  Now, in order to really provide a functor implementing [S], we need to encapsulate everything into a type of binary search trees. They also happen to be well-balanced, but this has no influence on the correctness of operations, so we won't state this here, see [BalanceProps] if you need more than just the MSet interface. *)
Module Type MSetInterface_S_Ext := MSetInterface.S <+ MSetRemoveMin.
Module Make (X: Orders.OrderedType) <:
MSetInterface_S_Ext with Module E := X.
Module Raw. Include MakeRaw X. End Raw.
Include MSetInterface.Raw2Sets X Raw.
Definition opt_ok (x:option (elt * Raw.t)) :=
match x with Some (_,s) => Raw.Ok s | None => True end.
Definition mk_opt_t (x: option (elt * Raw.t))(P: opt_ok x) :
option (elt * t) :=
match x as o return opt_ok o -> option (elt * t) with
| Some (k,s') => fun P : Raw.Ok s' => Some (k, Mkt s')
| None => fun _ => None
end P.
Definition remove_min s : option (elt * t) :=
mk_opt_t (Raw.remove_min (this s)) (Raw.remove_min_ok s).

Lemma remove_min_spec1 s x s' :
remove_min s = Some (x,s') ->
min_elt s = Some x /\ Equal (remove x s) s'.