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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 integers of ZArith *)
Require Export LegacyArithRing.
Require Export ZArith_base.
Require Import Eqdep_dec.
Require Import LegacyRing.
Definition Zeq (x y:Z) :=
match (x ?= y)%Z with
| Datatypes.Eq => true
| _ => false
end.
Lemma Zeq_prop : forall x y:Z, Is_true (Zeq x y) -> x = y.
intros x y H; unfold Zeq in H.
apply Z.compare_eq.
destruct (x ?= y)%Z; [ reflexivity | contradiction | contradiction ].
Qed.
Definition ZTheory : Ring_Theory Z.add Z.mul 1%Z 0%Z Z.opp Zeq.
split; intros; eauto with zarith.
apply Zeq_prop; assumption.
Qed.
(* NatConstants and NatTheory are defined in Ring_theory.v *)
Add Legacy Ring Z Z.add Z.mul 1%Z 0%Z Z.opp Zeq ZTheory
[ Zpos Zneg 0%Z xO xI 1%positive ].
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