From 6b649aba925b6f7462da07599fe67ebb12a3460e Mon Sep 17 00:00:00 2001
From: Samuel Mimram <samuel.mimram@ens-lyon.org>
Date: Wed, 28 Jul 2004 21:54:47 +0000
Subject: Imported Upstream version 8.0pl1

---
 theories/Relations/Operators_Properties.v | 96 +++++++++++++++++++++++++++++++
 1 file changed, 96 insertions(+)
 create mode 100755 theories/Relations/Operators_Properties.v

(limited to 'theories/Relations/Operators_Properties.v')

diff --git a/theories/Relations/Operators_Properties.v b/theories/Relations/Operators_Properties.v
new file mode 100755
index 00000000..5e0e9ec8
--- /dev/null
+++ b/theories/Relations/Operators_Properties.v
@@ -0,0 +1,96 @@
+(************************************************************************)
+(*  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 1.7.2.1 2004/07/16 19:31:16 herbelin Exp $ i*)
+
+(****************************************************************************)
+(*                            Bruno Barras                                  *)
+(****************************************************************************)
+
+Require Import Relation_Definitions.
+Require Import Relation_Operators.
+
+
+Section Properties.
+
+  Variable A : Set.
+  Variable R : relation A.
+
+  Let incl (R1 R2:relation A) : Prop := forall 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 in |- *.
+induction 1; auto with sets.
+intros.
+apply rt_trans with y; auto with sets.
+Qed.
+
+  Lemma clos_refl_trans_ind_left :
+   forall (A:Set) (R:A -> A -> Prop) (M:A) (P:A -> Prop),
+     P M ->
+     (forall P0 N:A, clos_refl_trans A R M P0 -> P P0 -> R P0 N -> P N) ->
+     forall 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 in |- *.
+induction 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 in |- *.
+induction 1; auto with sets.
+apply rst_trans with y; auto with sets.
+Qed.
+
+End Clos_Refl_Sym_Trans.
+
+End Properties.
\ No newline at end of file
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