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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 *)
-(************************************************************************)
-
-(* $Id: NArithRing.v,v 1.1.2.1 2004/07/16 19:30:18 herbelin Exp $ *)
-
-(* Instantiation of the Ring tactic for the binary natural numbers *)
-
-Require Export Ring.
-Require Export ZArith_base.
-Require NArith.
-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.
- Apply Nplus_assoc.
- Apply Nmult_comm.
- Apply Nmult_assoc.
- Apply Nplus_0_l.
- Apply Nmult_1_l.
- Apply Nmult_0_l.
- Apply Nmult_plus_distr_r.
- Apply Nplus_reg_l.
- Apply Neq_prop.
-Save.
-
-Add Semi Ring entier Nplus Nmult (Pos xH) Nul Neq NTheory [Pos Nul xO xI xH].