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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* Instantiation of the Ring tactic for the binary natural numbers *)
Require Import Bool.
Require Export LegacyRing.
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 N.compare_eq.
destruct (n ?= m)%N; [ reflexivity | contradiction | contradiction ].
Qed.
Definition NTheory : Semi_Ring_Theory N.add N.mul 1%N 0%N Neq.
split.
apply N.add_comm.
apply N.add_assoc.
apply N.mul_comm.
apply N.mul_assoc.
apply N.add_0_l.
apply N.mul_1_l.
apply N.mul_0_l.
apply N.mul_add_distr_r.
apply Neq_prop.
Qed.
Add Legacy Semi Ring
N N.add N.mul 1%N 0%N Neq NTheory [ Npos 0%N xO xI 1%positive ].
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