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+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+
+Require Import ZArith_base.
+Require Import Zpow_def.
+Require Import BinInt.
+Require Import BinNat.
+Require Import Setoid.
+Require Import BinList.
+Require Import BinPos.
+Require Import BinNat.
+Require Import BinInt.
+Require Import Setoid.
+Require Export Ncring.
+Require Export Ncring_polynom.
+Import List.
+
+Set Implicit Arguments.
+
+(* An object to return when an expression is not recognized as a constant *)
+Definition NotConstant := false.
+
+(** Z is a ring and a setoid*)
+
+Lemma Zsth : Setoid_Theory Z (@eq Z).
+constructor;red;intros;subst;trivial.
+Qed.
+
+Instance Zops:@Ring_ops Z 0%Z 1%Z Zplus Zmult Zminus Zopp (@eq Z).
+
+Instance Zr: (@Ring _ _ _ _ _ _ _ _ Zops).
+constructor;
+try (try apply Zsth;
+   try (unfold respectful, Proper; unfold equality; unfold eq_notation in *;
+  intros; try rewrite H; try rewrite H0; reflexivity)).
+ exact Zplus_comm. exact Zplus_assoc.
+ exact Zmult_1_l.  exact Zmult_1_r. exact Zmult_assoc.
+ exact Zmult_plus_distr_l.  intros; apply Zmult_plus_distr_r.  exact Zminus_diag.
+Defined.
+
+(*Instance ZEquality: @Equality Z:= (@eq Z).*)
+
+(** Two generic morphisms from Z to (abrbitrary) rings, *)
+(**second one is more convenient for proofs but they are ext. equal*)
+Section ZMORPHISM.
+Context {R:Type}`{Ring R}.
+
+ Ltac rrefl := reflexivity.
+
+ Fixpoint gen_phiPOS1 (p:positive) : R :=
+  match p with
+  | xH => 1
+  | xO p => (1 + 1) * (gen_phiPOS1 p)
+  | xI p => 1 + ((1 + 1) * (gen_phiPOS1 p))
+  end.
+
+ Fixpoint gen_phiPOS (p:positive) : R :=
+  match p with
+  | xH => 1
+  | xO xH => (1 + 1)
+  | xO p => (1 + 1) * (gen_phiPOS p)
+  | xI xH => 1 + (1 +1)
+  | xI p => 1 + ((1 + 1) * (gen_phiPOS p))
+  end.
+
+ Definition gen_phiZ1 z :=
+  match z with
+  | Zpos p => gen_phiPOS1 p
+  | Z0 => 0
+  | Zneg p => -(gen_phiPOS1 p)
+  end.
+
+ Definition gen_phiZ z :=
+  match z with
+  | Zpos p => gen_phiPOS p
+  | Z0 => 0
+  | Zneg p => -(gen_phiPOS p)
+  end.
+ Notation "[ x ]" := (gen_phiZ x).
+
+ Definition get_signZ z :=
+  match z with
+  | Zneg p => Some (Zpos p)
+  | _ => None
+  end.
+
+   Ltac norm := gen_rewrite.
+   Ltac add_push :=  Ncring.gen_add_push.
+Ltac rsimpl := simpl.
+
+ Lemma same_gen : forall x, gen_phiPOS1 x == gen_phiPOS x.
+ Proof.
+  induction x;rsimpl.
+  rewrite IHx. destruct x;simpl;norm.
+  rewrite IHx;destruct x;simpl;norm.
+  reflexivity.
+ Qed.
+
+ Lemma ARgen_phiPOS_Psucc : forall x,
+   gen_phiPOS1 (Psucc x) == 1 + (gen_phiPOS1 x).
+ Proof.
+  induction x;rsimpl;norm.
+ rewrite IHx. gen_rewrite. add_push 1. reflexivity.
+ Qed.
+
+ Lemma ARgen_phiPOS_add : forall x y,
+   gen_phiPOS1 (x + y) == (gen_phiPOS1 x) + (gen_phiPOS1 y).
+ Proof.
+  induction x;destruct y;simpl;norm.
+  rewrite Pplus_carry_spec.
+  rewrite ARgen_phiPOS_Psucc.
+  rewrite IHx;norm.
+  add_push (gen_phiPOS1 y);add_push 1;reflexivity.
+  rewrite IHx;norm;add_push (gen_phiPOS1 y);reflexivity.
+  rewrite ARgen_phiPOS_Psucc;norm;add_push 1;reflexivity.
+  rewrite IHx;norm;add_push(gen_phiPOS1 y); add_push 1;reflexivity.
+  rewrite IHx;norm;add_push(gen_phiPOS1 y);reflexivity.
+  add_push 1;reflexivity.
+  rewrite ARgen_phiPOS_Psucc;norm;add_push 1;reflexivity.
+ Qed.
+
+ Lemma ARgen_phiPOS_mult :
+   forall x y, gen_phiPOS1 (x * y) == gen_phiPOS1 x * gen_phiPOS1 y.
+ Proof.
+  induction x;intros;simpl;norm.
+  rewrite ARgen_phiPOS_add;simpl;rewrite IHx;norm.
+  rewrite IHx;reflexivity.
+ Qed.
+
+
+(*morphisms are extensionaly equal*)
+ Lemma same_genZ : forall x, [x] == gen_phiZ1 x.
+ Proof.
+  destruct x;rsimpl; try rewrite same_gen; reflexivity.
+ Qed.
+
+ Lemma gen_Zeqb_ok : forall x y,
+   Zeq_bool x y = true -> [x] == [y].
+ Proof.
+  intros x y H7.
+  assert (H10 := Zeq_bool_eq x y H7);unfold IDphi in H10.
+  rewrite H10;reflexivity.
+ Qed.
+
+ Lemma gen_phiZ1_add_pos_neg : forall x y,
+ gen_phiZ1 (Z.pos_sub x y)
+ == gen_phiPOS1 x + -gen_phiPOS1 y.
+ Proof.
+  intros x y.
+  rewrite Z.pos_sub_spec.
+  assert (HH0 := Pminus_mask_Gt x y). unfold Pos.gt in HH0.
+  assert (HH1 := Pminus_mask_Gt y x). unfold Pos.gt in HH1.
+  rewrite Pos.compare_antisym in HH1.
+  destruct (Pos.compare_spec x y) as [HH|HH|HH].
+  subst. rewrite ring_opp_def;reflexivity.
+  destruct HH1 as [h [HHeq1 [HHeq2 HHor]]];trivial.
+  unfold Pminus; rewrite HHeq1;rewrite <- HHeq2.
+  rewrite ARgen_phiPOS_add;simpl;norm.
+  rewrite ring_opp_def;norm.
+  destruct HH0 as [h [HHeq1 [HHeq2 HHor]]];trivial.
+  unfold Pminus; rewrite HHeq1;rewrite <- HHeq2.
+  rewrite ARgen_phiPOS_add;simpl;norm.
+  add_push (gen_phiPOS1 h). rewrite ring_opp_def ; norm.
+ Qed.
+
+ Lemma match_compOpp : forall x (B:Type) (be bl bg:B),
+  match CompOpp x with Eq => be | Lt => bl | Gt => bg end
+  = match x with Eq => be | Lt => bg | Gt => bl end.
+ Proof. destruct x;simpl;intros;trivial. Qed.
+
+ Lemma gen_phiZ_add : forall x y, [x + y] == [x] + [y].
+ Proof.
+  intros x y; repeat rewrite same_genZ; generalize x y;clear x y.
+  induction x;destruct y;simpl;norm.
+  apply ARgen_phiPOS_add.
+  apply gen_phiZ1_add_pos_neg. 
+   rewrite gen_phiZ1_add_pos_neg. rewrite ring_add_comm.
+reflexivity.
+ rewrite ARgen_phiPOS_add. rewrite ring_opp_add. reflexivity.
+Qed.
+
+Lemma gen_phiZ_opp : forall x, [- x] == - [x].
+ Proof.
+  intros x. repeat rewrite same_genZ. generalize x ;clear x.
+  induction x;simpl;norm.
+  rewrite ring_opp_opp.  reflexivity.
+ Qed.
+
+ Lemma gen_phiZ_mul : forall x y, [x * y] == [x] * [y].
+ Proof.
+  intros x y;repeat rewrite same_genZ.
+  destruct x;destruct y;simpl;norm;
+  rewrite  ARgen_phiPOS_mult;try (norm;fail).
+  rewrite ring_opp_opp ;reflexivity.
+ Qed.
+
+ Lemma gen_phiZ_ext : forall x y : Z, x = y -> [x] == [y].
+ Proof. intros;subst;reflexivity. Qed.
+
+(*proof that [.] satisfies morphism specifications*)
+Global Instance gen_phiZ_morph :
+(@Ring_morphism (Z:Type) R _ _ _ _ _ _ _ Zops Zr _ _ _ _ _ _ _ _ _ gen_phiZ) . (* beurk!*)
+ apply Build_Ring_morphism; simpl;try reflexivity.
+   apply gen_phiZ_add. intros. rewrite ring_sub_def. 
+replace (Zminus x y) with (x + (-y))%Z. rewrite gen_phiZ_add. 
+rewrite gen_phiZ_opp.  rewrite ring_sub_def. reflexivity.
+reflexivity.
+ apply gen_phiZ_mul. apply gen_phiZ_opp. apply gen_phiZ_ext. 
+ Defined.
+
+End ZMORPHISM.
+
+Instance multiplication_phi_ring{R:Type}`{Ring R} : Multiplication  :=
+  {multiplication x y := (gen_phiZ x) * y}.
+
+