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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 Relations_1.
Require Export Relations_1_facts.
Require Export Partial_Order.
Require Export Cpo.
Require Export Powerset.
Require Export Powerset_facts.
Require Export Classical.
Require Export Classical_sets.
Section Sets_as_an_algebra.
Variable U : Type.

(* Specifications: *)
Lemma sincl_add_x :
forall (A B:Ensemble U) (x:U),
~ In U A x ->
Strict_Included U (Add U A x) (Add U B x) -> Strict_Included U A B.
Lemma incl_soustr_in :
forall (X:Ensemble U) (x:U), In U X x -> Included U (Subtract U X x) X.
Lemma incl_soustr :
forall (X Y:Ensemble U) (x:U),
Included U X Y -> Included U (Subtract U X x) (Subtract U Y x).
Lemma incl_soustr_add_l :
forall (X:Ensemble U) (x:U), Included U (Subtract U (Add U X x) x) X.
Lemma incl_soustr_add_r :
forall (X:Ensemble U) (x:U),
~ In U X x -> Included U X (Subtract U (Add U X x) x).