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(*         *      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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(*                                                                          *)
(*                         Naive Set Theory in Coq                          *)
(*                                                                          *)
(*                     INRIA                        INRIA                   *)
(*              Rocquencourt                        Sophia-Antipolis        *)
(*                                                                          *)
(*                                 Coq V6.1                                 *)
(*									    *)
(*			         Gilles Kahn 				    *)
(*				 Gerard Huet				    *)
(*									    *)
(*									    *)
(*                                                                          *)
(* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks  *)
(* to the Newton Institute for providing an exceptional work environment    *)
(* in Summer 1995. Several developments by E. Ledinot were an inspiration.  *)
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Require Export Ensembles.
Require Export Constructive_sets.
Require Export Classical.
Section Ensembles_classical.
Variable U : Type.

(* Specifications: *)
Lemma not_included_empty_Inhabited :
forall A:Ensemble U, ~ Included U A (Empty_set U) -> Inhabited U A.
Lemma not_empty_Inhabited :
forall A:Ensemble U, A <> Empty_set U -> Inhabited U A.
Lemma Inhabited_Setminus :
forall X Y:Ensemble U,
Included U X Y -> ~ Included U Y X -> Inhabited U (Setminus U Y X).
Lemma Strict_super_set_contains_new_element :
forall X Y:Ensemble U,
Included U X Y -> X <> Y -> Inhabited U (Setminus U Y X).
Lemma Subtract_intro :
forall (A:Ensemble U) (x y:U), In U A y -> x <> y -> In U (Subtract U A x) y.