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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.
Require Export Relations_1_facts.
Require Export Relations_2.

(* Specifications: *)
Theorem Rstar_reflexive :
forall (U:Type) (R:Relation U), Reflexive U (Rstar U R).
Theorem Rplus_contains_R :
forall (U:Type) (R:Relation U), contains U (Rplus U R) R.
Theorem Rstar_contains_R :
forall (U:Type) (R:Relation U), contains U (Rstar U R) R.
Theorem Rstar_contains_Rplus :
forall (U:Type) (R:Relation U), contains U (Rstar U R) (Rplus U R).
Theorem Rstar_transitive :
forall (U:Type) (R:Relation U), Transitive U (Rstar U R).
Theorem Rstar_cases :
forall (U:Type) (R:Relation U) (x y:U),
Rstar U R x y -> x = y \/ (exists u : _, R x u /\ Rstar U R u y).
Theorem Rstar_equiv_Rstar1 :
forall (U:Type) (R:Relation U), same_relation U (Rstar U R) (Rstar1 U R).
Theorem Rsym_imp_Rstarsym :
forall (U:Type) (R:Relation U), Symmetric U R -> Symmetric U (Rstar U R).
Theorem Sstar_contains_Rstar :
forall (U:Type) (R S:Relation U),
contains U (Rstar U S) R -> contains U (Rstar U S) (Rstar U R).
Theorem star_monotone :
forall (U:Type) (R S:Relation U),
contains U S R -> contains U (Rstar U S) (Rstar U R).
Theorem RstarRplus_RRstar :
forall (U:Type) (R:Relation U) (x y z:U),
Rstar U R x y -> Rplus U R y z ->  exists u : _, R x u /\ Rstar U R u z.
Theorem Lemma1 :
forall (U:Type) (R:Relation U),
Strongly_confluent U R ->
forall x b:U,
Rstar U R x b ->
forall a:U, R x a ->  exists z : _, Rstar U R a z /\ R b z.