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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        *)
+(************************************************************************)
+
+Require Import ZArith.
+Require Import Zpow_facts.
+Require Import Wf_Z.
+Require Import Znumtheory.
+Require Import List.
+Require Export GroebnerR.
+
+
+Open Scope Z_scope.
+
+
+Lemma groebnerZhypR: forall x y:Z, x=y -> IZR x = IZR y.
+Admitted.
+
+Lemma groebnerZconclR: forall x y:Z, IZR x = IZR y -> x = y.
+Admitted.
+
+Lemma groebnerZhypnotR: forall x y:Z, x<>y -> IZR x <> IZR y.
+Admitted.
+
+Lemma groebnerZconclnotR: forall x y:Z, IZR x <> IZR y -> x <> y.
+Admitted.
+
+Ltac groebnerZversR1 :=
+ repeat 
+  (match goal with
+   | H:(@eq Z ?x ?y) |- _ =>
+       generalize (@groebnerZhypR _ _ H); clear H; intro H
+   | |- (@eq Z ?x ?y) => apply groebnerZconclR
+   | H:not (@eq Z ?x ?y) |- _ =>
+       generalize (@groebnerZhypnotR _ _ H); clear H; intro H
+   | |- not (@eq Z ?x ?y) => apply groebnerZconclnotR
+   end).
+
+Lemma groebnerZR1: forall x y:Z, IZR(x+y) = (IZR x + IZR y)%R.
+Admitted.
+Lemma groebnerZR2: forall x y:Z, IZR(x*y) = (IZR x * IZR y)%R.
+Admitted.
+Lemma groebnerZR3: forall x y:Z, IZR(x-y) = (IZR x - IZR y)%R.
+Admitted.
+Lemma groebnerZR4: forall (x:Z) p, IZR(x ^ Zpos p) = (IZR x ^ nat_of_P p)%R.
+Admitted.
+Ltac groebnerZversR2:=
+  repeat
+   (rewrite groebnerZR1 in * ||
+    rewrite groebnerZR2 in * ||
+    rewrite groebnerZR3 in * ||
+    rewrite groebnerZR4 in *).
+
+Ltac groebnerZ_begin :=
+ intros;
+ groebnerZversR1;
+ groebnerZversR2;
+ simpl in *.
+ (*cbv beta iota zeta delta [nat_of_P Pmult_nat plus mult] in *.*)
+
+Ltac groebnerZ := 
+ groebnerZ_begin; (*idtac "groebnerZ_begin;";*)
+ groebnerR.