(* $Id$ *)

(* Instantiation of the Ring tactic for the binary integers of ZArith *)

Require Export ArithRing.
Require Export ZArith.
Require Eqdep_dec.

Definition Zeq := [x,y:Z]
  Cases `x ?= y ` of
    EGAL => true
  | _ => false
  end.

Lemma Zeq_prop : (x,y:Z)(Is_true (Zeq x y)) -> x==y.
  Intros x y; Unfold Zeq.
  Generalize (let (H1,H2)=(Zcompare_EGAL x y) in H1).
  Elim (Zcompare x y); [Intro; Rewrite H; Trivial | Contradiction | 
	Contradiction ].
Save.

Definition ZTheory : (Ring_Theory Zplus Zmult `1` `0` Zopp Zeq).
  Split; Intros; Apply eq2eqT; EAuto with zarith.
  Apply eqT2eq; Apply Zeq_prop; Assumption.
Save.

(* NatConstants and NatTheory are defined in Ring_theory.v *)
Add Ring Z Zplus Zmult `1` `0` Zopp Zeq ZTheory [POS NEG ZERO xO xI xH].