(************************************************************************)
(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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(** * Finite sets library : conversion to old [Finite_sets] *)
(** * Going from [FSets] with usual Leibniz equality to the good old [Ensembles] and [Finite_sets] theory. *)
Module WS_to_Finite_set (U:UsualDecidableType)(M: WSfun U).
Module MP:= WProperties_fun U M.
Definition mkEns : M.t -> Ensemble M.elt :=
fun s x => M.In x s.
Notation " !! " := mkEns.

(** we can even build a function from Finite Ensemble to FSet ... at least in Prop. *)
Lemma Ens_to_FSet : forall e : Ensemble M.elt, Finite _ e ->
exists s:M.t, !!s === e.