Imported Upstream version 8.4~betaupstream/8.4_beta

12 files changed, 249 insertions, 70 deletions

diff --git a/theories/QArith/QArith.v b/theories/QArith/QArith.v
index 2255cd41..fe8d639c 100644
--- a/theories/QArith/QArith.v
+++ b/theories/QArith/QArith.v
@@ -1,13 +1,11 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011     *)
+(* <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        *)
 (************************************************************************)
 
-(*i $Id: QArith.v 14641 2011-11-06 11:59:10Z herbelin $ i*)
-
 Require Export QArith_base.
 Require Export Qring.
 Require Export Qreduction.
diff --git a/theories/QArith/QArith_base.v b/theories/QArith/QArith_base.v
index 18b8823d..94ea4906 100644
--- a/theories/QArith/QArith_base.v
+++ b/theories/QArith/QArith_base.v
@@ -1,13 +1,11 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011     *)
+(* <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        *)
 (************************************************************************)
 
-(*i $Id: QArith_base.v 14641 2011-11-06 11:59:10Z herbelin $ i*)
-
 Require Export ZArith.
 Require Export ZArithRing.
 Require Export Morphisms Setoid Bool.
@@ -20,7 +18,7 @@ Record Q : Set := Qmake {Qnum : Z; Qden : positive}.
 
 Delimit Scope Q_scope with Q.
 Bind Scope Q_scope with Q.
-Arguments Scope Qmake [Z_scope positive_scope].
+Arguments Qmake _%Z _%positive.
 Open Scope Q_scope.
 Ltac simpl_mult := repeat rewrite Zpos_mult_morphism.
 
@@ -29,7 +27,7 @@ Ltac simpl_mult := repeat rewrite Zpos_mult_morphism.
 Notation "a # b" := (Qmake a b) (at level 55, no associativity) : Q_scope.
 
 Definition inject_Z (x : Z) := Qmake x 1.
-Arguments Scope inject_Z [Z_scope].
+Arguments inject_Z x%Z.
 
 Notation QDen p := (Zpos (Qden p)).
 Notation " 0 " := (0#1) : Q_scope.
@@ -48,6 +46,13 @@ Notation "x > y" := (Qlt y x)(only parsing) : Q_scope.
 Notation "x >= y" := (Qle y x)(only parsing) : Q_scope.
 Notation "x <= y <= z" := (x<=y/\y<=z) : Q_scope.
 
+(** injection from Z is injective. *)
+
+Lemma inject_Z_injective (a b: Z): inject_Z a == inject_Z b <-> a = b.
+Proof.
+ unfold Qeq. simpl. omega.
+Qed.
+
 (** Another approach : using Qcompare for defining order relations. *)
 
 Definition Qcompare (p q : Q) := (Qnum p * QDen q ?= Qnum q * QDen p)%Z.
@@ -92,7 +97,7 @@ Proof.
  unfold "?=". intros. apply Zcompare_antisym.
 Qed.
 
-Lemma Qcompare_spec : forall x y, CompSpec Qeq Qlt x y (x ?= y).
+Lemma Qcompare_spec : forall x y, CompareSpec (x==y) (x<y) (y<x) (x ?= y).
 Proof.
  intros.
  destruct (x ?= y) as [ ]_eqn:H; constructor; auto.
@@ -387,6 +392,26 @@ Proof.
   red; simpl; intro; ring.
 Qed.
 
+(** Injectivity of addition (uses theory about Qopp above): *)
+
+Lemma Qplus_inj_r (x y z: Q):
+  x + z == y + z <-> x == y.
+Proof.
+ split; intro E.
+  rewrite <- (Qplus_0_r x), <- (Qplus_0_r y).
+  rewrite <- (Qplus_opp_r z); auto.
+  do 2 rewrite Qplus_assoc.
+  rewrite E. reflexivity.
+ rewrite E. reflexivity.
+Qed.
+
+Lemma Qplus_inj_l (x y z: Q):
+  z + x == z + y <-> x == y.
+Proof.
+ rewrite (Qplus_comm z x), (Qplus_comm z y).
+ apply Qplus_inj_r.
+Qed.
+
 
 (** * Properties of [Qmult] *)
 
@@ -462,6 +487,21 @@ Proof.
   rewrite <- H0; ring.
 Qed.
 
+
+(** * inject_Z is a ring homomorphism: *)
+
+Lemma inject_Z_plus (x y: Z): inject_Z (x + y) = inject_Z x + inject_Z y.
+Proof.
+ unfold Qplus, inject_Z. simpl. f_equal. ring.
+Qed.
+
+Lemma inject_Z_mult (x y: Z): inject_Z (x * y) = inject_Z x * inject_Z y.
+Proof. reflexivity. Qed.
+
+Lemma inject_Z_opp (x: Z): inject_Z (- x) = - inject_Z x.
+Proof. reflexivity. Qed.
+
+
 (** * Inverse and division. *)
 
 Lemma Qinv_involutive : forall q, (/ / q) == q.
@@ -500,6 +540,25 @@ Proof.
   apply Qdiv_mult_l; auto.
 Qed.
 
+(** Injectivity of Qmult (requires theory about Qinv above): *)
+
+Lemma Qmult_inj_r (x y z: Q): ~ z == 0 -> (x * z == y * z <-> x == y).
+Proof.
+ intro z_ne_0.
+ split; intro E.
+  rewrite <- (Qmult_1_r x), <- (Qmult_1_r y).
+  rewrite <- (Qmult_inv_r z); auto.
+  do 2 rewrite Qmult_assoc.
+  rewrite E. reflexivity.
+ rewrite E. reflexivity.
+Qed.
+
+Lemma Qmult_inj_l (x y z: Q): ~ z == 0 -> (z * x == z * y <-> x == y).
+Proof.
+ rewrite (Qmult_comm z x), (Qmult_comm z y).
+ apply Qmult_inj_r.
+Qed.
+
 (** * Properties of order upon Q. *)
 
 Lemma Qle_refl : forall x, x<=x.
@@ -539,6 +598,19 @@ Proof.
   unfold Qlt, Qeq; auto with zarith.
 Qed.
 
+Lemma Zle_Qle (x y: Z): (x <= y)%Z = (inject_Z x <= inject_Z y).
+Proof.
+ unfold Qle. intros. simpl.
+ do 2 rewrite Zmult_1_r. reflexivity.
+Qed.
+
+Lemma Zlt_Qlt (x y: Z): (x < y)%Z = (inject_Z x < inject_Z y).
+Proof.
+ unfold Qlt. intros. simpl.
+ do 2 rewrite Zmult_1_r. reflexivity.
+Qed.
+
+
 (** Large = strict or equal *)
 
 Lemma Qle_lteq : forall x y, x<=y <-> x<y \/ x==y.
@@ -677,6 +749,54 @@ Proof.
   Close Scope Z_scope.
 Qed.
 
+Lemma Qplus_lt_le_compat :
+  forall x y z t, x<y -> z<=t -> x+z < y+t.
+Proof.
+  unfold Qplus, Qle, Qlt; intros (x1, x2) (y1, y2) (z1, z2) (t1, t2);
+    simpl; simpl_mult.
+  Open Scope Z_scope.
+  intros.
+  match goal with |- ?a < ?b => ring_simplify a b end.
+  rewrite Zplus_comm.
+  apply Zplus_le_lt_compat.
+  match goal with |- ?a <= ?b => ring_simplify z1 t1 ('z2) ('t2) a b end.
+  auto with zarith.
+  match goal with |- ?a < ?b => ring_simplify x1 y1 ('x2) ('y2) a b end.
+  assert (forall p, 0 < ' p) by reflexivity.
+  repeat (apply Zmult_lt_compat_r; auto).
+  Close Scope Z_scope.
+Qed.
+
+Lemma Qplus_le_l (x y z: Q): x + z <= y + z <-> x <= y.
+Proof.
+ split; intros.
+  rewrite <- (Qplus_0_r x), <- (Qplus_0_r y), <- (Qplus_opp_r z).
+  do 2 rewrite Qplus_assoc.
+  apply Qplus_le_compat; auto with *.
+ apply Qplus_le_compat; auto with *.
+Qed.
+
+Lemma Qplus_le_r (x y z: Q): z + x <= z + y <-> x <= y.
+Proof.
+ rewrite (Qplus_comm z x), (Qplus_comm z y).
+ apply Qplus_le_l.
+Qed.
+
+Lemma Qplus_lt_l (x y z: Q): x + z < y + z <-> x < y.
+Proof.
+ split; intros.
+  rewrite <- (Qplus_0_r x), <- (Qplus_0_r y), <- (Qplus_opp_r z).
+  do 2 rewrite Qplus_assoc.
+  apply Qplus_lt_le_compat; auto with *.
+ apply Qplus_lt_le_compat; auto with *.
+Qed.
+
+Lemma Qplus_lt_r (x y z: Q): z + x < z + y <-> x < y.
+Proof.
+ rewrite (Qplus_comm z x), (Qplus_comm z y).
+ apply Qplus_lt_l.
+Qed.
+
 Lemma Qmult_le_compat_r : forall x y z, x <= y -> 0 <= z -> x*z <= y*z.
 Proof.
   intros (a1,a2) (b1,b2) (c1,c2); unfold Qle, Qlt; simpl.
@@ -699,6 +819,19 @@ Proof.
   Close Scope Z_scope.
 Qed.
 
+Lemma Qmult_le_r (x y z: Q): 0 < z -> (x*z <= y*z <-> x <= y).
+Proof.
+ split; intro.
+  now apply Qmult_lt_0_le_reg_r with z.
+ apply Qmult_le_compat_r; auto with qarith.
+Qed.
+
+Lemma Qmult_le_l (x y z: Q): 0 < z -> (z*x <= z*y <-> x <= y).
+Proof.
+ rewrite (Qmult_comm z x), (Qmult_comm z y).
+ apply Qmult_le_r.
+Qed.
+
 Lemma Qmult_lt_compat_r : forall x y z, 0 < z  -> x < y -> x*z < y*z.
 Proof.
   intros (a1,a2) (b1,b2) (c1,c2); unfold Qle, Qlt; simpl.
@@ -713,6 +846,30 @@ Proof.
   Close Scope Z_scope.
 Qed.
 
+Lemma Qmult_lt_r: forall x y z, 0 < z -> (x*z < y*z <-> x < y).
+Proof.
+ Open Scope Z_scope.
+ intros (a1,a2) (b1,b2) (c1,c2).
+ unfold Qle, Qlt; simpl.
+ simpl_mult.
+ replace (a1*c1*('b2*'c2)) with ((a1*'b2)*(c1*'c2)) by ring.
+ replace (b1*c1*('a2*'c2)) with ((b1*'a2)*(c1*'c2)) by ring.
+ assert (forall p, 0 < ' p) by reflexivity.
+ split; intros.
+  apply Zmult_lt_reg_r with (c1*'c2); auto with zarith.
+  apply Zmult_lt_0_compat; auto with zarith.
+ apply Zmult_lt_compat_r; auto with zarith.
+ apply Zmult_lt_0_compat. omega.
+ compute; auto.
+ Close Scope Z_scope.
+Qed.
+
+Lemma Qmult_lt_l (x y z: Q): 0 < z -> (z*x < z*y <-> x < y).
+Proof.
+ rewrite (Qmult_comm z x), (Qmult_comm z y).
+ apply Qmult_lt_r.
+Qed.
+
 Lemma Qmult_le_0_compat : forall a b, 0 <= a -> 0 <= b -> 0 <= a*b.
 Proof.
 intros a b Ha Hb.
diff --git a/theories/QArith/QOrderedType.v b/theories/QArith/QOrderedType.v
index be894419..238de6fa 100644
--- a/theories/QArith/QOrderedType.v
+++ b/theories/QArith/QOrderedType.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011     *)
+(* <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        *)
diff --git a/theories/QArith/Qabs.v b/theories/QArith/Qabs.v
index 747c2c3c..557fabc8 100644
--- a/theories/QArith/Qabs.v
+++ b/theories/QArith/Qabs.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011     *)
+(* <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        *)
@@ -100,6 +100,13 @@ rewrite Zabs_Zmult.
 reflexivity.
 Qed.
 
+Lemma Qabs_Qminus x y: Qabs (x - y) = Qabs (y - x).
+Proof.
+ unfold Qminus, Qopp. simpl.
+ rewrite Pmult_comm, <- Zabs_Zopp.
+ do 2 f_equal. ring.
+Qed.
+
 Lemma Qle_Qabs : forall a, a <= Qabs a.
 Proof.
 intros a.
@@ -122,3 +129,31 @@ apply Qabs_triangle.
 apply Qabs_wd.
 ring.
 Qed.
+
+Lemma Qabs_Qle_condition x y: Qabs x <= y <-> -y <= x <= y.
+Proof.
+ split.
+  split.
+   rewrite <- (Qopp_opp x).
+   apply Qopp_le_compat.
+   apply Qle_trans with (Qabs (-x)).
+   apply Qle_Qabs.
+   now rewrite Qabs_opp.
+  apply Qle_trans with (Qabs x); auto using Qle_Qabs.
+ intros (H,H').
+ apply Qabs_case; trivial.
+ intros. rewrite <- (Qopp_opp y). now apply Qopp_le_compat.
+Qed.
+
+Lemma Qabs_diff_Qle_condition x y r: Qabs (x - y) <= r <-> x - r <= y <= x + r.
+Proof.
+ intros. unfold Qminus.
+ rewrite Qabs_Qle_condition.
+ rewrite <- (Qplus_le_l (-r) (x+-y) (y+r)).
+ rewrite <- (Qplus_le_l (x+-y) r (y-r)).
+ setoid_replace (-r + (y + r)) with y by ring.
+ setoid_replace (r + (y - r)) with y by ring.
+ setoid_replace (x + - y + (y + r)) with (x + r) by ring.
+ setoid_replace (x + - y + (y - r)) with (x - r) by ring.
+ intuition.
+Qed.
diff --git a/theories/QArith/Qcanon.v b/theories/QArith/Qcanon.v
index 71a3b474..fea2ba39 100644
--- a/theories/QArith/Qcanon.v
+++ b/theories/QArith/Qcanon.v
@@ -1,13 +1,11 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011     *)
+(* <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        *)
 (************************************************************************)
 
-(*i $Id: Qcanon.v 14641 2011-11-06 11:59:10Z herbelin $ i*)
-
 Require Import Field.
 Require Import QArith.
 Require Import Znumtheory.
@@ -20,7 +18,7 @@ Record Qc : Set := Qcmake { this :> Q ; canon : Qred this = this }.
 
 Delimit Scope Qc_scope with Qc.
 Bind Scope Qc_scope with Qc.
-Arguments Scope Qcmake [Q_scope].
+Arguments Qcmake this%Q _.
 Open Scope Qc_scope.
 
 Lemma Qred_identity :
@@ -71,7 +69,7 @@ Proof.
 Qed.
 
 Definition Q2Qc (q:Q) : Qc := Qcmake (Qred q) (Qred_involutive q).
-Arguments Scope Q2Qc [Q_scope].
+Arguments Q2Qc q%Q.
 Notation " !! " := Q2Qc : Qc_scope.
 
 Lemma Qc_is_canon : forall q q' : Qc, q == q' -> q = q'.
@@ -468,18 +466,16 @@ Proof.
   destruct n; simpl.
   destruct 1; auto.
   intros.
-  apply Qc_is_canon.
-  simpl.
-  compute; auto.
+  now apply Qc_is_canon.
 Qed.
 
 Lemma Qcpower_pos : forall p n, 0 <= p -> 0 <= p^n.
 Proof.
   induction n; simpl; auto with qarith.
-  intros; compute; intro; discriminate.
+  easy.
   intros.
   apply Qcle_trans with (0*(p^n)).
-  compute; intro; discriminate.
+  easy.
   apply Qcmult_le_compat_r; auto.
 Qed.
 
diff --git a/theories/QArith/Qfield.v b/theories/QArith/Qfield.v
index 81d59714..5e27f381 100644
--- a/theories/QArith/Qfield.v
+++ b/theories/QArith/Qfield.v
@@ -1,13 +1,11 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011     *)
+(* <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        *)
 (************************************************************************)
 
-(*i $Id: Qfield.v 14641 2011-11-06 11:59:10Z herbelin $ i*)
-
 Require Export Field.
 Require Export QArith_base.
 Require Import NArithRing.
diff --git a/theories/QArith/Qminmax.v b/theories/QArith/Qminmax.v
index a458fc6e..2da24ee6 100644
--- a/theories/QArith/Qminmax.v
+++ b/theories/QArith/Qminmax.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011     *)
+(* <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        *)
@@ -64,4 +64,4 @@ Proof.
  apply plus_min_distr_l.
 Qed.
 
-End Q.
\ No newline at end of file
+End Q.
diff --git a/theories/QArith/Qpower.v b/theories/QArith/Qpower.v
index 9568c796..b05ee649 100644
--- a/theories/QArith/Qpower.v
+++ b/theories/QArith/Qpower.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011     *)
+(* <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        *)
@@ -136,9 +136,9 @@ Proof.
 intros a [|n|n] [|m|m] H; simpl; try ring;
 try rewrite Qpower_plus_positive;
 try apply Qinv_mult_distr; try reflexivity;
-case_eq ((n ?= m)%positive Eq); intros H0; simpl;
+rewrite ?Z.pos_sub_spec;
+case Pos.compare_spec; intros H0; simpl; subst;
  try rewrite Qpower_minus_positive;
- try rewrite (Pcompare_Eq_eq _ _ H0);
  try (field; try split; apply Qpower_not_0_positive);
  try assumption;
  apply ZC2;
diff --git a/theories/QArith/Qreals.v b/theories/QArith/Qreals.v
index 8a0ebcff..24f6d720 100644
--- a/theories/QArith/Qreals.v
+++ b/theories/QArith/Qreals.v
@@ -1,13 +1,11 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011     *)
+(* <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        *)
 (************************************************************************)
 
-(*i $Id: Qreals.v 14641 2011-11-06 11:59:10Z herbelin $ i*)
-
 Require Export Rbase.
 Require Export QArith_base.
 
@@ -210,4 +208,4 @@ intro; apply H; apply eqR_Qeq.
 rewrite H0; unfold Q2R in |- *; simpl in |- *; field; auto with real.
 Qed.
 
-End LegacyQField.
\ No newline at end of file
+End LegacyQField.
diff --git a/theories/QArith/Qreduction.v b/theories/QArith/Qreduction.v
index eb8c1164..e39eca0c 100644
--- a/theories/QArith/Qreduction.v
+++ b/theories/QArith/Qreduction.v
@@ -1,13 +1,11 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011     *)
+(* <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        *)
 (************************************************************************)
 
-(*i $Id: Qreduction.v 14641 2011-11-06 11:59:10Z herbelin $ i*)
-
 (** Normalisation functions for rational numbers. *)
 
 Require Export QArith_base.
@@ -43,23 +41,16 @@ Definition Qred (q:Q) :=
 Lemma Qred_correct : forall q, (Qred q) == q.
 Proof.
   unfold Qred, Qeq; intros (n,d); simpl.
-  generalize (Zggcd_gcd n ('d)) (Zgcd_is_pos n ('d))
-    (Zgcd_is_gcd n ('d)) (Zggcd_correct_divisors n ('d)).
+  generalize (Zggcd_gcd n ('d)) (Zgcd_nonneg n ('d))
+    (Zggcd_correct_divisors n ('d)).
   destruct (Zggcd n (Zpos d)) as (g,(nn,dd)); simpl.
   Open Scope Z_scope.
-  intuition.
-  rewrite <- H in H0,H1; clear H.
-  rewrite H3; rewrite H4.
-  assert (0 <> g).
-  intro; subst g; discriminate.
-
-  assert (0 < dd).
-  apply Zmult_gt_0_lt_0_reg_r with g.
-  omega.
-  rewrite Zmult_comm.
-  rewrite <- H4; compute; auto.
-  rewrite Z2P_correct; auto.
-  ring.
+  intros Hg LE (Hn,Hd). rewrite Hd, Hn.
+  rewrite <- Hg in LE; clear Hg.
+  assert (0 <> g) by (intro; subst; discriminate).
+  rewrite Z2P_correct. ring.
+  apply Zmult_gt_0_lt_0_reg_r with g; auto with zarith.
+  now rewrite Zmult_comm, <- Hd.
   Close Scope Z_scope.
 Qed.
 
@@ -69,10 +60,10 @@ Proof.
   unfold Qred, Qeq in *; simpl in *.
   Open Scope Z_scope.
   generalize (Zggcd_gcd a ('b)) (Zgcd_is_gcd a ('b))
-    (Zgcd_is_pos a ('b)) (Zggcd_correct_divisors a ('b)).
+    (Zgcd_nonneg a ('b)) (Zggcd_correct_divisors a ('b)).
   destruct (Zggcd a (Zpos b)) as (g,(aa,bb)).
   generalize (Zggcd_gcd c ('d)) (Zgcd_is_gcd c ('d))
-    (Zgcd_is_pos c ('d)) (Zggcd_correct_divisors c ('d)).
+    (Zgcd_nonneg c ('d)) (Zggcd_correct_divisors c ('d)).
   destruct (Zggcd c (Zpos d)) as (g',(cc,dd)).
   simpl.
   intro H; rewrite <- H; clear H.
@@ -80,10 +71,9 @@ Proof.
   intro H; rewrite <- H; clear H.
   intros Hg1 Hg2 (Hg3,Hg4).
   intros.
-  assert (g <> 0).
-  intro; subst g; discriminate.
-  assert (g' <> 0).
-  intro; subst g'; discriminate.
+  assert (g <> 0) by (intro; subst g; discriminate).
+  assert (g' <> 0) by (intro; subst g'; discriminate).
+
   elim (rel_prime_cross_prod aa bb cc dd).
   congruence.
   unfold rel_prime in |- *.
@@ -93,14 +83,13 @@ Proof.
   exists bb; auto with zarith.
   intros.
   inversion Hg1.
-  destruct (H6 (g*x)).
+  destruct (H6 (g*x)) as (x',Hx).
   rewrite Hg3.
   destruct H2 as (xa,Hxa); exists xa; rewrite Hxa; ring.
   rewrite Hg4.
   destruct H3 as (xb,Hxb); exists xb; rewrite Hxb; ring.
-  exists q.
-  apply Zmult_reg_l with g; auto.
-  pattern g at 1; rewrite H7; ring.
+  exists x'.
+  apply Zmult_reg_l with g; auto. rewrite Hx at 1; ring.
   (* /rel_prime *)
   unfold rel_prime in |- *.
   (* rel_prime *)
@@ -109,14 +98,13 @@ Proof.
   exists dd; auto with zarith.
   intros.
   inversion Hg'1.
-  destruct (H6 (g'*x)).
+  destruct (H6 (g'*x)) as (x',Hx).
   rewrite Hg'3.
   destruct H2 as (xc,Hxc); exists xc; rewrite Hxc; ring.
   rewrite Hg'4.
   destruct H3 as (xd,Hxd); exists xd; rewrite Hxd; ring.
-  exists q.
-  apply Zmult_reg_l with g'; auto.
-  pattern g' at 1; rewrite H7; ring.
+  exists x'.
+  apply Zmult_reg_l with g'; auto. rewrite Hx at 1; ring.
     (* /rel_prime *)
   assert (0<bb); [|auto with zarith].
   apply Zmult_gt_0_lt_0_reg_r with g.
diff --git a/theories/QArith/Qring.v b/theories/QArith/Qring.v
index 173136b8..c88a8141 100644
--- a/theories/QArith/Qring.v
+++ b/theories/QArith/Qring.v
@@ -1,11 +1,9 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011     *)
+(* <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        *)
 (************************************************************************)
 
-(*i $Id: Qring.v 14641 2011-11-06 11:59:10Z herbelin $ i*)
-
 Require Export Qfield.
diff --git a/theories/QArith/Qround.v b/theories/QArith/Qround.v
index 01a97870..ce363a33 100644
--- a/theories/QArith/Qround.v
+++ b/theories/QArith/Qround.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011     *)
+(* <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        *)
@@ -136,4 +136,15 @@ intros x y H.
 apply Zle_antisym.
  auto with *.
 symmetry in H; auto with *.
-Qed.
\ No newline at end of file
+Qed.
+
+Lemma Zdiv_Qdiv (n m: Z): (n / m)%Z = Qfloor (n / m).
+Proof.
+ unfold Qfloor. intros. simpl.
+ destruct m as [?|?|p]; simpl.
+  now rewrite Zdiv_0_r, Zmult_0_r.
+  now rewrite Zmult_1_r.
+ rewrite <- Zopp_eq_mult_neg_1.
+ rewrite <- (Zopp_involutive (Zpos p)).
+ now rewrite Zdiv_opp_opp.
+Qed.