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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.