Ajout NArithRing

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@4703 85f007b7-540e-0410-9357-904b9bb8a0f7

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+(***********************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team    *)
+(* <O___,, *        INRIA-Rocquencourt  &  LRI-CNRS-Orsay              *)
+(*   \VV/  *************************************************************)
+(*    //   *      This file is distributed under the terms of the      *)
+(*         *       GNU Lesser General Public License Version 2.1       *)
+(***********************************************************************)
+
+(* $Id$ *)
+
+(* Instantiation of the Ring tactic for the binary natural numbers *)
+
+Require Export ArithRing.
+Require Export ZArith_base.
+Require Eqdep_dec.
+
+Definition Neq := [n,m:entier]
+  Cases (Ncompare n m) of
+    EGAL => true
+  | _ => false
+  end.
+
+Lemma Neq_prop : (n,m:entier)(Is_true (Neq n m)) -> n=m.
+  Intros n m H; Unfold Neq in H.
+  Apply Ncompare_Eq_eq.
+  NewDestruct (Ncompare n m); [Reflexivity | Contradiction | Contradiction ].
+Save.
+
+Definition NTheory : (Semi_Ring_Theory Nplus Nmult (Pos xH) Nul Neq).
+  Split.
+    Apply Nplus_comm.
+    Symmetry; Apply Nplus_assoc_l.
+    Apply Nmult_comm.
+    Symmetry; Apply Nmult_assoc_l.
+    Apply Nplus_0_l.
+    Apply Nmult_1_l.
+    Apply Nmult_0_l.
+    Apply Nmult_Nplus_distr_l.
+    Apply Nplus_reg_l.
+    Apply Neq_prop.
+Save.
+
+Add Semi Ring entier Nplus Nmult (Pos xH) Nul Neq NTheory [Pos Nul xO xI xH].