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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 1.5.2.1 2004/07/16 19:30:13 herbelin Exp $ *)
(* Instantiation of the Ring tactic for the binary natural numbers *)
Require Export Ring.
Require Export ZArith_base.
Require Import NArith.
Require Import Eqdep_dec.
Definition Neq (n m:N) :=
match (n ?= m)%N with
| Datatypes.Eq => true
| _ => false
end.
Lemma Neq_prop : forall n m:N, Is_true (Neq n m) -> n = m.
intros n m H; unfold Neq in H.
apply Ncompare_Eq_eq.
destruct (n ?= m)%N; [ reflexivity | contradiction | contradiction ].
Qed.
Definition NTheory : Semi_Ring_Theory Nplus Nmult 1%N 0%N 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.
Qed.
Add Semi Ring N Nplus Nmult 1%N 0%N Neq NTheory [ Npos 0%N xO xI 1%positive ].
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