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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 Relations_1.
Local Ltac Tauto.intuition_solver ::= auto with sets.
Definition Complement (U:Type) (R:Relation U) : Relation U :=
fun x y:U => ~ R x y.

(* Specifications: *)
Theorem Rsym_imp_notRsym :
forall (U:Type) (R:Relation U),
Symmetric U R -> Symmetric U (Complement U R).
Theorem Equiv_from_preorder :
forall (U:Type) (R:Relation U),
Preorder U R -> Equivalence U (fun x y:U => R x y /\ R y x).