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+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+
+(*i $Id: Operators_Properties.v,v 1.1.2.1 2004/07/16 19:31:37 herbelin Exp $ i*)
+
+(****************************************************************************)
+(*                            Bruno Barras                                  *)
+(****************************************************************************)
+
+Require Relation_Definitions.
+Require Relation_Operators.
+
+
+Section Properties.
+
+  Variable A: Set.
+  Variable R: (relation A).
+
+  Local incl : (relation A)->(relation A)->Prop :=
+      [R1,R2: (relation A)] (x,y:A) (R1 x y) -> (R2 x y).
+
+Section Clos_Refl_Trans.
+
+  Lemma clos_rt_is_preorder: (preorder A (clos_refl_trans A R)).
+Apply Build_preorder.
+Exact (rt_refl A R).
+
+Exact (rt_trans A R).
+Qed.
+
+
+
+Lemma clos_rt_idempotent:
+       (incl (clos_refl_trans A (clos_refl_trans A R))
+                     (clos_refl_trans A R)).
+Red.
+NewInduction 1; Auto with sets.
+Intros.
+Apply rt_trans with y; Auto with sets.
+Qed.
+
+  Lemma  clos_refl_trans_ind_left: (A:Set)(R:A->A->Prop)(M:A)(P:A->Prop)
+           (P M)
+            ->((P0,N:A)
+                (clos_refl_trans A R M P0)->(P P0)->(R P0 N)->(P N))
+               ->(a:A)(clos_refl_trans A R M a)->(P a).
+Intros.
+Generalize H H0 .
+Clear  H H0.
+Elim H1; Intros; Auto with sets.
+Apply H2 with x; Auto with sets.
+
+Apply H3.
+Apply H0; Auto with sets.
+
+Intros.
+Apply H5 with P0; Auto with sets.
+Apply rt_trans with y; Auto with sets.
+Qed.
+
+
+End Clos_Refl_Trans.
+
+
+Section Clos_Refl_Sym_Trans.
+
+  Lemma clos_rt_clos_rst: (inclusion A (clos_refl_trans A R)
+				        (clos_refl_sym_trans A R)).
+Red.
+NewInduction 1; Auto with sets.
+Apply rst_trans with y; Auto with sets.
+Qed.
+
+  Lemma clos_rst_is_equiv: (equivalence A (clos_refl_sym_trans A R)).
+Apply Build_equivalence.
+Exact (rst_refl A R).
+
+Exact (rst_trans A R).
+
+Exact (rst_sym A R).
+Qed.
+
+  Lemma clos_rst_idempotent:
+       (incl (clos_refl_sym_trans A (clos_refl_sym_trans A R))
+                     (clos_refl_sym_trans A R)).
+Red.
+NewInduction 1; Auto with sets.
+Apply rst_trans with y; Auto with sets.
+Qed.
+
+End Clos_Refl_Sym_Trans.
+
+End Properties.