BigNums: remove files about BigN,BigZ,BigQ (now in an separate git repo)

 See now https://github.com/coq/bignums
 Int31 is still in the stdlib.
 Some proofs there has be adapted to avoid the need for BigNumPrelude.

14 files changed, 171 insertions, 6671 deletions

diff --git a/theories/Numbers/Cyclic/Abstract/CyclicAxioms.v b/theories/Numbers/Cyclic/Abstract/CyclicAxioms.v
index 3312161ae..857580198 100644
--- a/theories/Numbers/Cyclic/Abstract/CyclicAxioms.v
+++ b/theories/Numbers/Cyclic/Abstract/CyclicAxioms.v
@@ -17,7 +17,7 @@ Set Implicit Arguments.
 
 Require Import ZArith.
 Require Import Znumtheory.
-Require Import BigNumPrelude.
+Require Import Zpow_facts.
 Require Import DoubleType.
 
 Local Open Scope Z_scope.
diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleType.v b/theories/Numbers/Cyclic/Abstract/DoubleType.v
index abd567a85..d60c19ea5 100644
--- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleType.v
+++ b/theories/Numbers/Cyclic/Abstract/DoubleType.v
@@ -67,4 +67,3 @@ Fixpoint word (w:Type) (n:nat) : Type :=
  | O => w
  | S n => zn2z (word w n)
  end.
-
diff --git a/theories/Numbers/Cyclic/Abstract/NZCyclic.v b/theories/Numbers/Cyclic/Abstract/NZCyclic.v
index df9b83392..3f9b7b297 100644
--- a/theories/Numbers/Cyclic/Abstract/NZCyclic.v
+++ b/theories/Numbers/Cyclic/Abstract/NZCyclic.v
@@ -9,7 +9,8 @@
 (************************************************************************)
 
 Require Export NZAxioms.
-Require Import BigNumPrelude.
+Require Import ZArith.
+Require Import Zpow_facts.
 Require Import DoubleType.
 Require Import CyclicAxioms.
 
@@ -139,6 +140,26 @@ rewrite 2 ZnZ.of_Z_correct; auto with zarith.
 symmetry; apply Zmod_small; auto with zarith.
 Qed.
 
+Theorem Zbounded_induction :
+  (forall Q : Z -> Prop, forall b : Z,
+    Q 0 ->
+    (forall n, 0 <= n -> n < b - 1 -> Q n -> Q (n + 1)) ->
+      forall n, 0 <= n -> n < b -> Q n)%Z.
+Proof.
+intros Q b Q0 QS.
+set (Q' := fun n => (n < b /\ Q n) \/ (b <= n)).
+assert (H : forall n, 0 <= n -> Q' n).
+apply natlike_rec2; unfold Q'.
+destruct (Z.le_gt_cases b 0) as [H | H]. now right. left; now split.
+intros n H IH. destruct IH as [[IH1 IH2] | IH].
+destruct (Z.le_gt_cases (b - 1) n) as [H1 | H1].
+right; auto with zarith.
+left. split; [auto with zarith | now apply (QS n)].
+right; auto with zarith.
+unfold Q' in *; intros n H1 H2. destruct (H n H1) as [[H3 H4] | H3].
+assumption. now apply Z.le_ngt in H3.
+Qed.
+
 Lemma B_holds : forall n : Z, 0 <= n < wB -> B n.
 Proof.
 intros n [H1 H2].
diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleAdd.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleAdd.v
deleted file mode 100644
index 407bcca4b..000000000
--- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleAdd.v
+++ /dev/null
@@ -1,317 +0,0 @@
-(************************************************************************)
-(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016     *)
-(*   \VV/  **************************************************************)
-(*    //   *      This file is distributed under the terms of the       *)
-(*         *       GNU Lesser General Public License Version 2.1        *)
-(************************************************************************)
-(*            Benjamin Gregoire, Laurent Thery, INRIA, 2007             *)
-(************************************************************************)
-
-Set Implicit Arguments.
-
-Require Import ZArith.
-Require Import BigNumPrelude.
-Require Import DoubleType.
-Require Import DoubleBase.
-
-Local Open Scope Z_scope.
-
-Section DoubleAdd.
- Variable w : Type.
- Variable w_0 : w.
- Variable w_1 : w.
- Variable w_WW : w -> w -> zn2z w.
- Variable w_W0 : w -> zn2z w.
- Variable ww_1 : zn2z w.
- Variable w_succ_c : w -> carry w.
- Variable w_add_c : w -> w -> carry w.
- Variable w_add_carry_c : w -> w -> carry w.
- Variable w_succ : w -> w.
- Variable w_add : w -> w -> w.
- Variable w_add_carry : w -> w -> w.
-
- Definition ww_succ_c x :=
-  match x with
-  | W0 => C0 ww_1
-  | WW xh xl =>
-    match w_succ_c xl with
-    | C0 l => C0 (WW xh l)
-    | C1 l =>
-      match w_succ_c xh with
-      | C0 h => C0 (WW h w_0)
-      | C1 h => C1 W0
-      end
-    end
-  end.
-
- Definition ww_succ x :=
-  match x with
-  | W0 => ww_1
-  | WW xh xl =>
-    match w_succ_c xl with
-    | C0 l => WW xh l
-    | C1 l => w_W0 (w_succ xh)
-    end
-  end.
-
- Definition ww_add_c x y :=
-  match x, y with
-  | W0, _ => C0 y
-  | _, W0 => C0 x
-  | WW xh xl, WW yh yl =>
-    match w_add_c xl yl with
-    | C0 l =>
-      match w_add_c xh yh with
-      | C0 h => C0 (WW h l)
-      | C1 h => C1 (w_WW h l)
-      end
-    | C1 l =>
-      match w_add_carry_c xh yh with
-      | C0 h => C0 (WW h l)
-      | C1 h => C1 (w_WW h l)
-      end
-    end
-  end.
-
- Variable R : Type.
- Variable f0 f1 : zn2z w -> R.
-
- Definition ww_add_c_cont x y :=
-  match x, y with
-  | W0, _ => f0 y
-  | _, W0 => f0 x
-  | WW xh xl, WW yh yl =>
-    match w_add_c xl yl with
-    | C0 l =>
-      match w_add_c xh yh with
-      | C0 h => f0 (WW h l)
-      | C1 h => f1 (w_WW h l)
-      end
-    | C1 l =>
-      match w_add_carry_c xh yh with
-      | C0 h => f0 (WW h l)
-      | C1 h => f1 (w_WW h l)
-      end
-    end
-  end.
-
- (* ww_add et ww_add_carry conserve la forme normale s'il n'y a pas
-    de debordement *)
- Definition ww_add x y :=
-  match x, y with
-  | W0, _ => y
-  | _, W0 => x
-  | WW xh xl, WW yh yl =>
-    match w_add_c xl yl with
-    | C0 l => WW (w_add xh yh) l
-    | C1 l => WW (w_add_carry xh yh) l
-    end
-  end.
-
- Definition ww_add_carry_c x y :=
-  match x, y with
-  | W0, W0 => C0 ww_1
-  | W0, WW yh yl => ww_succ_c (WW yh yl)
-  | WW xh xl, W0 => ww_succ_c (WW xh xl)
-  | WW xh xl, WW yh yl =>
-    match w_add_carry_c xl yl with
-    | C0 l =>
-      match w_add_c xh yh with
-      | C0 h => C0 (WW h l)
-      | C1 h => C1 (WW h l)
-      end
-    | C1 l =>
-      match w_add_carry_c xh yh with
-      | C0 h => C0 (WW h l)
-      | C1 h => C1 (w_WW h l)
-      end
-    end
-  end.
-
- Definition ww_add_carry x y :=
-  match x, y with
-  | W0, W0 => ww_1
-  | W0, WW yh yl => ww_succ (WW yh yl)
-  | WW xh xl, W0 => ww_succ (WW xh xl)
-  | WW xh xl, WW yh yl =>
-    match w_add_carry_c xl yl with
-    | C0 l => WW (w_add xh yh) l
-    | C1 l => WW (w_add_carry xh yh) l
-    end
-  end.
-
- (*Section DoubleProof.*)
-  Variable w_digits : positive.
-  Variable w_to_Z : w -> Z.
-
-
-  Notation wB  := (base w_digits).
-  Notation wwB := (base (ww_digits w_digits)).
-  Notation "[| x |]" := (w_to_Z x)  (at level 0, x at level 99).
-  Notation "[+| c |]" :=
-   (interp_carry 1 wB w_to_Z c) (at level 0, c at level 99).
-  Notation "[-| c |]" :=
-   (interp_carry (-1) wB w_to_Z c) (at level 0, c at level 99).
-
-  Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99).
-  Notation "[+[ c ]]" :=
-   (interp_carry 1 wwB (ww_to_Z w_digits w_to_Z) c)
-   (at level 0, c at level 99).
-  Notation "[-[ c ]]" :=
-   (interp_carry (-1) wwB (ww_to_Z w_digits w_to_Z) c)
-   (at level 0, c at level 99).
-
-  Variable spec_w_0   : [|w_0|] = 0.
-  Variable spec_w_1   : [|w_1|] = 1.
-  Variable spec_ww_1   : [[ww_1]] = 1.
-  Variable spec_to_Z  : forall x, 0 <= [|x|] < wB.
-  Variable spec_w_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|].
-  Variable spec_w_W0 : forall h, [[w_W0 h]] = [|h|] * wB.
-  Variable spec_w_succ_c : forall x, [+|w_succ_c x|] = [|x|] + 1.
-  Variable spec_w_add_c  : forall x y, [+|w_add_c x y|] = [|x|] + [|y|].
-  Variable spec_w_add_carry_c :
-         forall x y, [+|w_add_carry_c x y|] = [|x|] + [|y|] + 1.
-  Variable spec_w_succ : forall x, [|w_succ x|] = ([|x|] + 1) mod wB.
-  Variable spec_w_add : forall x y, [|w_add x y|] = ([|x|] + [|y|]) mod wB.
-  Variable spec_w_add_carry :
-	 forall x y, [|w_add_carry x y|] = ([|x|] + [|y|] + 1) mod wB.
-
-  Lemma spec_ww_succ_c : forall x, [+[ww_succ_c x]] = [[x]] + 1.
-  Proof.
-   destruct x as [ |xh xl];simpl. apply spec_ww_1.
-   generalize (spec_w_succ_c xl);destruct (w_succ_c xl) as [l|l];
-    intro H;unfold interp_carry in H. simpl;rewrite H;ring.
-   rewrite <- Z.add_assoc;rewrite <- H;rewrite Z.mul_1_l.
-   assert ([|l|] = 0). generalize (spec_to_Z xl)(spec_to_Z l);omega.
-   rewrite H0;generalize (spec_w_succ_c xh);destruct (w_succ_c xh) as [h|h];
-    intro H1;unfold interp_carry in H1.
-   simpl;rewrite H1;rewrite spec_w_0;ring.
-   unfold interp_carry;simpl ww_to_Z;rewrite wwB_wBwB.
-   assert ([|xh|] = wB - 1). generalize (spec_to_Z xh)(spec_to_Z h);omega.
-   rewrite H2;ring.
-  Qed.
-
-  Lemma spec_ww_add_c  : forall x y, [+[ww_add_c x y]] = [[x]] + [[y]].
-  Proof.
-   destruct x as [ |xh xl];trivial.
-   destruct y as [ |yh yl]. rewrite Z.add_0_r;trivial.
-   simpl. replace ([|xh|] * wB + [|xl|] + ([|yh|] * wB + [|yl|]))
-   with (([|xh|]+[|yh|])*wB + ([|xl|]+[|yl|])). 2:ring.
-   generalize (spec_w_add_c xl yl);destruct (w_add_c xl yl) as [l|l];
-    intros H;unfold interp_carry in H;rewrite <- H.
-   generalize (spec_w_add_c xh yh);destruct (w_add_c xh yh) as [h|h];
-    intros H1;unfold interp_carry in *;rewrite <- H1. trivial.
-   repeat rewrite Z.mul_1_l;rewrite spec_w_WW;rewrite wwB_wBwB; ring.
-   rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r.
-   generalize (spec_w_add_carry_c xh yh);destruct (w_add_carry_c xh yh)
-    as [h|h]; intros H1;unfold interp_carry in *;rewrite <- H1.
-   simpl;ring.
-   repeat rewrite Z.mul_1_l;rewrite wwB_wBwB;rewrite spec_w_WW;ring.
-  Qed.
-
-  Section Cont.
-   Variable P : zn2z w -> zn2z w -> R -> Prop.
-   Variable x y : zn2z w.
-   Variable spec_f0 : forall r, [[r]] = [[x]] + [[y]] -> P x y (f0 r).
-   Variable spec_f1 : forall r, wwB + [[r]] = [[x]] + [[y]] -> P x y (f1 r).
-
-   Lemma spec_ww_add_c_cont  : P x y (ww_add_c_cont x y).
-   Proof.
-    destruct x as [ |xh xl];trivial.
-    apply spec_f0;trivial.
-    destruct y as [ |yh yl].
-    apply spec_f0;rewrite Z.add_0_r;trivial.
-    simpl.
-    generalize (spec_w_add_c xl yl);destruct (w_add_c xl yl) as [l|l];
-     intros H;unfold interp_carry in H.
-    generalize (spec_w_add_c xh yh);destruct (w_add_c xh yh) as [h|h];
-     intros H1;unfold interp_carry in *.
-    apply spec_f0. simpl;rewrite H;rewrite H1;ring.
-    apply spec_f1. simpl;rewrite spec_w_WW;rewrite H.
-    rewrite Z.add_assoc;rewrite wwB_wBwB. rewrite Z.pow_2_r; rewrite <- Z.mul_add_distr_r.
-    rewrite Z.mul_1_l in H1;rewrite H1;ring.
-    generalize (spec_w_add_carry_c xh yh);destruct (w_add_carry_c xh yh)
-     as [h|h]; intros H1;unfold interp_carry in *.
-    apply spec_f0;simpl;rewrite H1. rewrite Z.mul_add_distr_r.
-    rewrite <- Z.add_assoc;rewrite H;ring.
-    apply spec_f1. rewrite spec_w_WW;rewrite wwB_wBwB.
-    rewrite Z.add_assoc; rewrite Z.pow_2_r; rewrite <- Z.mul_add_distr_r.
-    rewrite Z.mul_1_l in H1;rewrite H1. rewrite Z.mul_add_distr_r.
-    rewrite <- Z.add_assoc;rewrite H; simpl; ring.
-   Qed.
-
-  End Cont.
-
-  Lemma spec_ww_add_carry_c :
-         forall x y, [+[ww_add_carry_c x y]] = [[x]] + [[y]] + 1.
-  Proof.
-   destruct x as [ |xh xl];intro y.
-   exact (spec_ww_succ_c y).
-   destruct y as [ |yh yl].
-   rewrite Z.add_0_r;exact (spec_ww_succ_c (WW xh xl)).
-   simpl; replace ([|xh|] * wB + [|xl|] + ([|yh|] * wB + [|yl|]) + 1)
-   with (([|xh|]+[|yh|])*wB + ([|xl|]+[|yl|]+1)). 2:ring.
-   generalize (spec_w_add_carry_c xl yl);destruct (w_add_carry_c xl yl)
-    as [l|l];intros H;unfold interp_carry in H;rewrite <- H.
-   generalize (spec_w_add_c xh yh);destruct (w_add_c xh yh) as [h|h];
-    intros H1;unfold interp_carry in H1;rewrite <- H1. trivial.
-   unfold interp_carry;repeat rewrite Z.mul_1_l;simpl;rewrite wwB_wBwB;ring.
-   rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r.
-   generalize (spec_w_add_carry_c xh yh);destruct (w_add_carry_c xh yh)
-    as [h|h];intros H1;unfold interp_carry in H1;rewrite <- H1. trivial.
-   unfold interp_carry;rewrite spec_w_WW;
-   repeat rewrite Z.mul_1_l;simpl;rewrite wwB_wBwB;ring.
-  Qed.
-
-  Lemma spec_ww_succ : forall x, [[ww_succ x]] = ([[x]] + 1) mod wwB.
-  Proof.
-   destruct x as [ |xh xl];simpl.
-   rewrite spec_ww_1;rewrite Zmod_small;trivial.
-   split;[intro;discriminate|apply wwB_pos].
-   rewrite <- Z.add_assoc;generalize (spec_w_succ_c xl);
-   destruct (w_succ_c xl) as[l|l];intro H;unfold interp_carry in H;rewrite <-H.
-   rewrite Zmod_small;trivial.
-   rewrite wwB_wBwB;apply beta_mult;apply spec_to_Z.
-   assert ([|l|] = 0). clear spec_ww_1 spec_w_1 spec_w_0.
-    assert (H1:= spec_to_Z l); assert (H2:= spec_to_Z xl); omega.
-   rewrite H0;rewrite Z.add_0_r;rewrite <- Z.mul_add_distr_r;rewrite wwB_wBwB.
-   rewrite Z.pow_2_r; rewrite Zmult_mod_distr_r;try apply lt_0_wB.
-   rewrite spec_w_W0;rewrite spec_w_succ;trivial.
-  Qed.
-
-  Lemma spec_ww_add : forall x y, [[ww_add x y]] = ([[x]] + [[y]]) mod wwB.
-  Proof.
-   destruct x as [ |xh xl];intros y.
-   rewrite Zmod_small;trivial. apply spec_ww_to_Z;trivial.
-   destruct y as [ |yh yl].
-   change [[W0]] with 0;rewrite Z.add_0_r.
-   rewrite Zmod_small;trivial.
-    exact (spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW xh xl)).
-   simpl. replace ([|xh|] * wB + [|xl|] + ([|yh|] * wB + [|yl|]))
-   with (([|xh|]+[|yh|])*wB + ([|xl|]+[|yl|])). 2:ring.
-   generalize (spec_w_add_c xl yl);destruct (w_add_c xl yl) as [l|l];
-   unfold interp_carry;intros H;simpl;rewrite <- H.
-   rewrite (mod_wwB w_digits w_to_Z spec_to_Z);rewrite spec_w_add;trivial.
-   rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r.
-   rewrite(mod_wwB w_digits w_to_Z spec_to_Z);rewrite spec_w_add_carry;trivial.
-  Qed.
-
-  Lemma spec_ww_add_carry :
-	 forall x y, [[ww_add_carry x y]] = ([[x]] + [[y]] + 1) mod wwB.
-  Proof.
-   destruct x as [ |xh xl];intros y.
-   exact (spec_ww_succ y).
-   destruct y as [ |yh yl].
-   change [[W0]] with 0;rewrite Z.add_0_r. exact (spec_ww_succ (WW xh xl)).
-   simpl;replace ([|xh|] * wB + [|xl|] + ([|yh|] * wB + [|yl|]) + 1)
-   with (([|xh|]+[|yh|])*wB + ([|xl|]+[|yl|]+1)). 2:ring.
-   generalize (spec_w_add_carry_c xl yl);destruct (w_add_carry_c xl yl)
-   as [l|l];unfold interp_carry;intros H;rewrite <- H;simpl ww_to_Z.
-   rewrite(mod_wwB w_digits w_to_Z spec_to_Z);rewrite spec_w_add;trivial.
-   rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r.
-   rewrite(mod_wwB w_digits w_to_Z spec_to_Z);rewrite spec_w_add_carry;trivial.
-  Qed.
-
-(* End DoubleProof. *)
-End DoubleAdd.
diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleBase.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleBase.v
deleted file mode 100644
index e94a891dd..000000000
--- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleBase.v
+++ /dev/null
@@ -1,437 +0,0 @@
-(************************************************************************)
-(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016     *)
-(*   \VV/  **************************************************************)
-(*    //   *      This file is distributed under the terms of the       *)
-(*         *       GNU Lesser General Public License Version 2.1        *)
-(************************************************************************)
-(*            Benjamin Gregoire, Laurent Thery, INRIA, 2007             *)
-(************************************************************************)
-
-Set Implicit Arguments.
-
-Require Import ZArith Ndigits.
-Require Import BigNumPrelude.
-Require Import DoubleType.
-
-Local Open Scope Z_scope.
-
-Local Infix "<<" := Pos.shiftl_nat (at level 30).
-
-Section DoubleBase.
- Variable w : Type.
- Variable w_0   : w.
- Variable w_1   : w.
- Variable w_Bm1 : w.
- Variable w_WW  : w -> w -> zn2z w.
- Variable w_0W  : w -> zn2z w.
- Variable w_digits : positive.
- Variable w_zdigits: w.
- Variable w_add: w -> w -> zn2z w.
- Variable w_to_Z   : w -> Z.
- Variable w_compare : w -> w -> comparison.
-
- Definition ww_digits := xO w_digits.
-
- Definition ww_zdigits := w_add w_zdigits w_zdigits.
-
- Definition ww_to_Z := zn2z_to_Z (base w_digits) w_to_Z.
-
- Definition ww_1 := WW w_0 w_1.
-
- Definition ww_Bm1 := WW w_Bm1 w_Bm1.
-
- Definition ww_WW xh xl : zn2z (zn2z w) :=
-  match xh, xl with
-  | W0, W0 => W0
-  | _, _ => WW xh xl
-  end.
-
- Definition ww_W0 h : zn2z (zn2z w) :=
-  match h with
-  | W0 => W0
-  | _ => WW h W0
-  end.
-
- Definition ww_0W l : zn2z (zn2z w) :=
-  match l with
-  | W0 => W0
-  | _ => WW W0 l
-  end.
-
- Definition double_WW (n:nat) :=
-  match n return word w n -> word w n -> word w (S n) with
-  | O => w_WW
-  | S n =>
-    fun (h l : zn2z (word w n)) =>
-     match h, l with
-     | W0, W0 => W0
-     | _, _ => WW h l
-     end
-  end.
-
- Definition double_wB n := base (w_digits << n).
-
- Fixpoint double_to_Z (n:nat) : word w n -> Z :=
-  match n return word w n -> Z with
-  | O => w_to_Z
-  | S n => zn2z_to_Z (double_wB n) (double_to_Z n)
-  end.
-
- Fixpoint extend_aux (n:nat) (x:zn2z w) {struct n}: word w (S n) :=
-  match n return word w (S n) with
-  | O => x
-  | S n1 => WW W0 (extend_aux n1 x)
-  end.
-
- Definition extend (n:nat) (x:w) : word w (S n) :=
-  let r := w_0W x in
-  match r with
-  | W0 => W0
-  | _ => extend_aux n r
-  end.
-
- Definition double_0 n : word w n :=
-   match n return word w n with
-   | O => w_0
-   | S _ => W0
-   end.
-
- Definition double_split (n:nat) (x:zn2z (word w n)) :=
-  match x with
-  | W0 =>
-    match n return word w n * word w n with
-    | O => (w_0,w_0)
-    | S _ => (W0, W0)
-    end
-  | WW h l => (h,l)
-  end.
-
- Definition ww_compare x y :=
-  match x, y with
-  | W0, W0 => Eq
-  | W0, WW yh yl =>
-    match w_compare w_0 yh with
-    | Eq => w_compare w_0 yl
-    | _ => Lt
-    end
-  | WW xh xl, W0 =>
-    match w_compare xh w_0 with
-    | Eq => w_compare xl w_0
-    | _ => Gt
-    end
-  | WW xh xl, WW yh yl =>
-    match w_compare xh yh with
-    | Eq => w_compare xl yl
-    | Lt => Lt
-    | Gt => Gt
-    end
-  end.
-
-
- (* Return the low part of the composed word*)
- Fixpoint get_low (n : nat) {struct n}:
-  word w n -> w :=
-  match n return (word w n -> w) with
-  | 0%nat => fun x => x
-  | S n1 =>
-      fun x =>
-      match x with
-      | W0 => w_0
-      | WW _ x1 => get_low n1 x1
-      end
-  end.
-
-
- Section DoubleProof.
-  Notation wB  := (base w_digits).
-  Notation wwB := (base ww_digits).
-  Notation "[| x |]" := (w_to_Z x)  (at level 0, x at level 99).
-  Notation "[[ x ]]" := (ww_to_Z x)  (at level 0, x at level 99).
-  Notation "[+[ c ]]" :=
-   (interp_carry 1 wwB ww_to_Z c) (at level 0, c at level 99).
-  Notation "[-[ c ]]" :=
-   (interp_carry (-1) wwB ww_to_Z c) (at level 0, c at level 99).
-  Notation "[! n | x !]" := (double_to_Z n x) (at level 0, x at level 99).
-
-  Variable spec_w_0   : [|w_0|] = 0.
-  Variable spec_w_1   : [|w_1|] = 1.
-  Variable spec_w_Bm1 : [|w_Bm1|] = wB - 1.
-  Variable spec_w_WW  : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|].
-  Variable spec_w_0W  : forall l, [[w_0W l]] = [|l|].
-  Variable spec_to_Z  : forall x, 0 <= [|x|] < wB.
-  Variable spec_w_compare : forall x y,
-     w_compare x y = Z.compare [|x|] [|y|].
-
-  Lemma wwB_wBwB : wwB = wB^2.
-  Proof.
-   unfold base, ww_digits;rewrite Z.pow_2_r; rewrite (Pos2Z.inj_xO w_digits).
-   replace (2 * Zpos w_digits) with (Zpos w_digits + Zpos w_digits).
-   apply Zpower_exp; unfold Z.ge;simpl;intros;discriminate.
-   ring.
-  Qed.
-
-  Lemma spec_ww_1    : [[ww_1]] = 1.
-  Proof. simpl;rewrite spec_w_0;rewrite spec_w_1;ring. Qed.
-
-  Lemma spec_ww_Bm1  : [[ww_Bm1]] = wwB - 1.
-  Proof. simpl;rewrite spec_w_Bm1;rewrite wwB_wBwB;ring. Qed.
-
-  Lemma lt_0_wB : 0 < wB.
-  Proof.
-   unfold base;apply Z.pow_pos_nonneg. unfold Z.lt;reflexivity.
-   unfold Z.le;intros H;discriminate H.
-  Qed.
-
-  Lemma lt_0_wwB : 0 < wwB.
-  Proof. rewrite wwB_wBwB; rewrite Z.pow_2_r; apply Z.mul_pos_pos;apply lt_0_wB. Qed.
-
-  Lemma wB_pos: 1 < wB.
-  Proof.
-   unfold base;apply Z.lt_le_trans with (2^1).  unfold Z.lt;reflexivity.
-   apply Zpower_le_monotone.  unfold Z.lt;reflexivity.
-   split;unfold Z.le;intros H. discriminate H.
-   clear spec_w_0W w_0W spec_w_Bm1 spec_to_Z spec_w_WW w_WW.
-   destruct w_digits; discriminate H.
-  Qed.
-
-  Lemma wwB_pos: 1 < wwB.
-  Proof.
-   assert (H:= wB_pos);rewrite wwB_wBwB;rewrite <-(Z.mul_1_r 1).
-   rewrite Z.pow_2_r.
-   apply Zmult_lt_compat2;(split;[unfold Z.lt;reflexivity|trivial]).
-   apply Z.lt_le_incl;trivial.
-  Qed.
-
-  Theorem wB_div_2:  2 * (wB / 2) = wB.
-  Proof.
-   clear spec_w_0 w_0 spec_w_1 w_1 spec_w_Bm1 w_Bm1 spec_w_WW spec_w_0W
-    spec_to_Z;unfold base.
-   assert (2 ^ Zpos w_digits = 2 * (2 ^ (Zpos w_digits - 1))).
-   pattern 2 at 2; rewrite <- Z.pow_1_r.
-   rewrite <- Zpower_exp; auto with zarith.
-   f_equal; auto with zarith.
-   case w_digits; compute; intros; discriminate.
-   rewrite H; f_equal; auto with zarith.
-   rewrite Z.mul_comm; apply Z_div_mult; auto with zarith.
-  Qed.
-
-  Theorem wwB_div_2 : wwB / 2 = wB / 2 * wB.
-  Proof.
-   clear spec_w_0 w_0 spec_w_1 w_1 spec_w_Bm1 w_Bm1 spec_w_WW spec_w_0W
-    spec_to_Z.
-   rewrite wwB_wBwB; rewrite Z.pow_2_r.
-   pattern wB at 1; rewrite <- wB_div_2; auto.
-   rewrite <- Z.mul_assoc.
-   repeat (rewrite (Z.mul_comm 2); rewrite Z_div_mult); auto with zarith.
-  Qed.
-
-  Lemma mod_wwB : forall z x,
-   (z*wB + [|x|]) mod wwB = (z mod wB)*wB + [|x|].
-  Proof.
-   intros z x.
-   rewrite Zplus_mod.
-   pattern wwB at 1;rewrite wwB_wBwB; rewrite Z.pow_2_r.
-   rewrite Zmult_mod_distr_r;try apply lt_0_wB.
-   rewrite (Zmod_small [|x|]).
-   apply Zmod_small;rewrite wwB_wBwB;apply beta_mult;try apply spec_to_Z.
-   apply Z_mod_lt;apply Z.lt_gt;apply lt_0_wB.
-   destruct (spec_to_Z x);split;trivial.
-   change [|x|] with (0*wB+[|x|]). rewrite wwB_wBwB.
-   rewrite Z.pow_2_r;rewrite <- (Z.add_0_r (wB*wB));apply beta_lex_inv.
-   apply lt_0_wB. apply spec_to_Z. split;[apply Z.le_refl | apply lt_0_wB].
- Qed.
-
-  Lemma wB_div : forall x y, ([|x|] * wB + [|y|]) / wB = [|x|].
-  Proof.
-   clear spec_w_0 spec_w_1 spec_w_Bm1 w_0 w_1 w_Bm1.
-   intros x y;unfold base;rewrite Zdiv_shift_r;auto with zarith.
-   rewrite Z_div_mult;auto with zarith.
-   destruct (spec_to_Z x);trivial.
-  Qed.
-
-  Lemma wB_div_plus : forall x y p,
-   0 <= p ->
-   ([|x|]*wB + [|y|]) / 2^(Zpos w_digits + p) = [|x|] / 2^p.
-  Proof.
-   clear spec_w_0 spec_w_1 spec_w_Bm1 w_0 w_1 w_Bm1.
-   intros x y p Hp;rewrite Zpower_exp;auto with zarith.
-   rewrite <- Zdiv_Zdiv;auto with zarith.
-   rewrite wB_div;trivial.
-  Qed.
-
-  Lemma lt_wB_wwB : wB < wwB.
-  Proof.
-   clear spec_w_0 spec_w_1 spec_w_Bm1 w_0 w_1 w_Bm1.
-   unfold base;apply Zpower_lt_monotone;auto with zarith.
-   assert (0 < Zpos w_digits). compute;reflexivity.
-   unfold ww_digits;rewrite Pos2Z.inj_xO;auto with zarith.
-  Qed.
-
-  Lemma w_to_Z_wwB : forall x, x < wB -> x < wwB.
-  Proof.
-   intros x H;apply Z.lt_trans with wB;trivial;apply lt_wB_wwB.
-  Qed.
-
-  Lemma spec_ww_to_Z : forall x, 0 <= [[x]] < wwB.
-  Proof.
-   clear spec_w_0 spec_w_1 spec_w_Bm1 w_0 w_1 w_Bm1.
-   destruct x as [ |h l];simpl.
-   split;[apply Z.le_refl|apply lt_0_wwB].
-   assert (H:=spec_to_Z h);assert (L:=spec_to_Z l);split.
-   apply Z.add_nonneg_nonneg;auto with zarith.
-   rewrite <- (Z.add_0_r wwB);rewrite wwB_wBwB; rewrite Z.pow_2_r;
-    apply beta_lex_inv;auto with zarith.
-  Qed.
-
-  Lemma double_wB_wwB : forall n, double_wB n * double_wB n = double_wB (S n).
-  Proof.
-   intros n;unfold double_wB;simpl.
-   unfold base. rewrite (Pos2Z.inj_xO (_ << _)).
-   replace  (2 * Zpos (w_digits << n)) with
-     (Zpos (w_digits << n) + Zpos (w_digits << n)) by ring.
-   symmetry; apply Zpower_exp;intro;discriminate.
-  Qed.
-
-  Lemma double_wB_pos:
-   forall n, 0 <= double_wB n.
- Proof.
- intros n; unfold double_wB, base; auto with zarith.
- Qed.
-
-  Lemma double_wB_more_digits:
-  forall n, wB <= double_wB n.
-  Proof.
-  clear spec_w_0 spec_w_1 spec_w_Bm1 w_0 w_1 w_Bm1.
-  intros n; elim n; clear n; auto.
-    unfold double_wB, "<<"; auto with zarith.
-  intros n H1; rewrite <- double_wB_wwB.
-  apply Z.le_trans with (wB * 1).
-    rewrite Z.mul_1_r; apply Z.le_refl.
-  unfold base; auto with zarith.
-  apply Z.mul_le_mono_nonneg; auto with zarith.
-  apply Z.le_trans with wB; auto with zarith.
-  unfold base.
-  rewrite <- (Z.pow_0_r 2).
-  apply Z.pow_le_mono_r; auto with zarith.
-  Qed.
-
-  Lemma spec_double_to_Z :
-   forall n (x:word w n), 0 <= [!n | x!] < double_wB n.
-  Proof.
-  clear spec_w_0 spec_w_1 spec_w_Bm1 w_0 w_1 w_Bm1.
-  induction n;intros. exact (spec_to_Z x).
-  unfold double_to_Z;fold double_to_Z.
-  destruct x;unfold zn2z_to_Z.
-  unfold double_wB,base;split;auto with zarith.
-  assert (U0:= IHn w0);assert (U1:= IHn w1).
-  split;auto with zarith.
-  apply Z.lt_le_trans with ((double_wB n - 1) * double_wB n + double_wB n).
-  assert (double_to_Z n w0*double_wB n <= (double_wB n - 1)*double_wB n).
-  apply Z.mul_le_mono_nonneg_r;auto with zarith.
-  auto with zarith.
-  rewrite <- double_wB_wwB.
-  replace ((double_wB n - 1) * double_wB n + double_wB n) with (double_wB n * double_wB n);
-   [auto with zarith | ring].
-  Qed.
-
-  Lemma spec_get_low:
-  forall n x,
-     [!n | x!] < wB ->  [|get_low n x|] = [!n | x!].
-  Proof.
-  clear spec_w_1 spec_w_Bm1.
-  intros n; elim n; auto; clear n.
-  intros n Hrec x; case x; clear x; auto.
-  intros xx yy; simpl.
-  destruct (spec_double_to_Z n xx) as [F1 _]. Z.le_elim F1.
-  - (* 0 < [!n | xx!] *)
-    intros; exfalso.
-    assert (F3 := double_wB_more_digits n).
-    destruct (spec_double_to_Z n yy) as [F4 _].
-    assert (F5: 1 * wB <=  [!n | xx!] * double_wB n);
-     auto with zarith.
-    apply Z.mul_le_mono_nonneg; auto with zarith.
-    unfold base; auto with zarith.
-  - (* 0 = [!n | xx!] *)
-    rewrite <- F1; rewrite Z.mul_0_l, Z.add_0_l.
-    intros; apply Hrec; auto.
-  Qed.
-
-  Lemma spec_double_WW : forall n (h l : word w n),
-    [!S n|double_WW n h l!] = [!n|h!] * double_wB n + [!n|l!].
-  Proof.
-   induction n;simpl;intros;trivial.
-   destruct h;auto.
-   destruct l;auto.
-  Qed.
-
-  Lemma spec_extend_aux : forall n x, [!S n|extend_aux n x!] = [[x]].
-  Proof. induction n;simpl;trivial. Qed.
-
-  Lemma spec_extend : forall n x, [!S n|extend n x!] = [|x|].
-  Proof.
-   intros n x;assert (H:= spec_w_0W x);unfold extend.
-   destruct (w_0W x);simpl;trivial.
-   rewrite <- H;exact (spec_extend_aux n (WW w0 w1)).
-  Qed.
-
-  Lemma spec_double_0 : forall n, [!n|double_0 n!] = 0.
-  Proof. destruct n;trivial. Qed.
-
-  Lemma spec_double_split : forall n x,
-   let (h,l) := double_split n x in
-   [!S n|x!] = [!n|h!] * double_wB n + [!n|l!].
-  Proof.
-   destruct x;simpl;auto.
-   destruct n;simpl;trivial.
-   rewrite spec_w_0;trivial.
-  Qed.
-
-  Lemma wB_lex_inv: forall a b c d,
-      a < c ->
-      a * wB + [|b|] < c  * wB + [|d|].
-  Proof.
-   intros a b c d H1; apply beta_lex_inv with (1 := H1); auto.
-  Qed.
-
-  Ltac comp2ord := match goal with
-   | |- Lt = (?x ?= ?y) => symmetry; change (x < y)
-   | |- Gt = (?x ?= ?y) => symmetry; change (x > y); apply Z.lt_gt
-  end.
-
-  Lemma spec_ww_compare : forall x y,
-    ww_compare x y = Z.compare [[x]] [[y]].
-  Proof.
-   destruct x as [ |xh xl];destruct y as [ |yh yl];simpl;trivial.
-   (* 1st case *)
-   rewrite 2 spec_w_compare, spec_w_0.
-   destruct (Z.compare_spec 0 [|yh|]) as [H|H|H].
-   rewrite <- H;simpl. reflexivity.
-   symmetry. change (0 < [|yh|]*wB+[|yl|]).
-   change 0 with (0*wB+0). rewrite <- spec_w_0 at 2.
-   apply wB_lex_inv;trivial.
-   absurd (0 <= [|yh|]). apply Z.lt_nge; trivial.
-   destruct (spec_to_Z yh);trivial.
-   (* 2nd case *)
-   rewrite 2 spec_w_compare, spec_w_0.
-   destruct (Z.compare_spec [|xh|] 0) as [H|H|H].
-   rewrite H;simpl;reflexivity.
-   absurd (0 <= [|xh|]). apply Z.lt_nge; trivial.
-   destruct (spec_to_Z xh);trivial.
-   comp2ord.
-   change 0 with (0*wB+0). rewrite <- spec_w_0 at 2.
-   apply wB_lex_inv;trivial.
-   (* 3rd case *)
-   rewrite 2 spec_w_compare.
-   destruct (Z.compare_spec [|xh|] [|yh|]) as [H|H|H].
-   rewrite H.
-   symmetry. apply Z.add_compare_mono_l.
-   comp2ord. apply wB_lex_inv;trivial.
-   comp2ord. apply wB_lex_inv;trivial.
-  Qed.
-
-
- End DoubleProof.
-
-End DoubleBase.
-
diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleCyclic.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleCyclic.v
deleted file mode 100644
index 4ebe8fac1..000000000
--- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleCyclic.v
+++ /dev/null
@@ -1,966 +0,0 @@
-(************************************************************************)
-(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016     *)
-(*   \VV/  **************************************************************)
-(*    //   *      This file is distributed under the terms of the       *)
-(*         *       GNU Lesser General Public License Version 2.1        *)
-(************************************************************************)
-(*            Benjamin Gregoire, Laurent Thery, INRIA, 2007             *)
-(************************************************************************)
-
-Set Implicit Arguments.
-
-Require Import ZArith.
-Require Import BigNumPrelude.
-Require Import DoubleType.
-Require Import DoubleBase.
-Require Import DoubleAdd.
-Require Import DoubleSub.
-Require Import DoubleMul.
-Require Import DoubleSqrt.
-Require Import DoubleLift.
-Require Import DoubleDivn1.
-Require Import DoubleDiv.
-Require Import CyclicAxioms.
-
-Local Open Scope Z_scope.
-
-
-Section Z_2nZ.
-
- Context {t : Type}{ops : ZnZ.Ops t}.
-
- Let w_digits      := ZnZ.digits.
- Let w_zdigits      := ZnZ.zdigits.
-
- Let w_to_Z        := ZnZ.to_Z.
- Let w_of_pos      := ZnZ.of_pos.
- Let w_head0       := ZnZ.head0.
- Let w_tail0       := ZnZ.tail0.
-
- Let w_0            := ZnZ.zero.
- Let w_1            := ZnZ.one.
- Let w_Bm1          := ZnZ.minus_one.
-
- Let w_compare     := ZnZ.compare.
- Let w_eq0         := ZnZ.eq0.
-
- Let w_opp_c       := ZnZ.opp_c.
- Let w_opp         := ZnZ.opp.
- Let w_opp_carry   := ZnZ.opp_carry.
-
- Let w_succ_c      := ZnZ.succ_c.
- Let w_add_c       := ZnZ.add_c.
- Let w_add_carry_c := ZnZ.add_carry_c.
- Let w_succ        := ZnZ.succ.
- Let w_add         := ZnZ.add.
- Let w_add_carry   := ZnZ.add_carry.
-
- Let w_pred_c      := ZnZ.pred_c.
- Let w_sub_c       := ZnZ.sub_c.
- Let w_sub_carry_c := ZnZ.sub_carry_c.
- Let w_pred        := ZnZ.pred.
- Let w_sub         := ZnZ.sub.
- Let w_sub_carry   := ZnZ.sub_carry.
-
-
- Let w_mul_c       := ZnZ.mul_c.
- Let w_mul         := ZnZ.mul.
- Let w_square_c    := ZnZ.square_c.
-
- Let w_div21       := ZnZ.div21.
- Let w_div_gt      := ZnZ.div_gt.
- Let w_div         := ZnZ.div.
-
- Let w_mod_gt      := ZnZ.modulo_gt.
- Let w_mod         := ZnZ.modulo.
-
- Let w_gcd_gt      := ZnZ.gcd_gt.
- Let w_gcd         := ZnZ.gcd.
-
- Let w_add_mul_div := ZnZ.add_mul_div.
-
- Let w_pos_mod := ZnZ.pos_mod.
-
- Let w_is_even     := ZnZ.is_even.
- Let w_sqrt2       := ZnZ.sqrt2.
- Let w_sqrt        := ZnZ.sqrt.
-
- Let _zn2z := zn2z t.
-
- Let wB := base w_digits.
-
- Let w_Bm2 := w_pred w_Bm1.
-
- Let ww_1 := ww_1 w_0 w_1.
- Let ww_Bm1 := ww_Bm1 w_Bm1.
-
- Let w_add2 a b := match w_add_c a b with C0 p => WW w_0 p | C1 p => WW w_1 p end.
-
- Let _ww_digits := xO w_digits.
-
- Let _ww_zdigits := w_add2 w_zdigits w_zdigits.
-
- Let to_Z := zn2z_to_Z wB w_to_Z.
-
- Let w_W0 := ZnZ.WO.
- Let w_0W := ZnZ.OW.
- Let w_WW := ZnZ.WW.
-
- Let ww_of_pos p :=
-  match w_of_pos p with
-  | (N0, l) => (N0, WW w_0 l)
-  | (Npos ph,l) =>
-    let (n,h) := w_of_pos ph in (n, w_WW h l)
-  end.
-
- Let head0 :=
-  Eval lazy beta delta [ww_head0] in
-  ww_head0 w_0 w_0W w_compare w_head0 w_add2 w_zdigits _ww_zdigits.
-
- Let tail0 :=
-  Eval lazy beta delta [ww_tail0] in
-  ww_tail0 w_0 w_0W w_compare w_tail0 w_add2 w_zdigits _ww_zdigits.
-
- Let ww_WW := Eval lazy beta delta [ww_WW] in (@ww_WW t).
- Let ww_0W := Eval lazy beta delta [ww_0W] in (@ww_0W t).
- Let ww_W0 := Eval lazy beta delta [ww_W0] in (@ww_W0 t).
-
- (* ** Comparison ** *)
- Let compare :=
-  Eval lazy beta delta[ww_compare] in ww_compare w_0 w_compare.
-
- Let eq0 (x:zn2z t) :=
-  match x with
-  | W0 => true
-  | _ => false
-  end.
-
- (* ** Opposites ** *)
- Let opp_c :=
-  Eval lazy beta delta [ww_opp_c] in ww_opp_c w_0 w_opp_c w_opp_carry.
-
- Let opp :=
-  Eval lazy beta delta [ww_opp] in ww_opp w_0 w_opp_c w_opp_carry w_opp.
-
- Let opp_carry :=
-  Eval lazy beta delta [ww_opp_carry] in ww_opp_carry w_WW ww_Bm1 w_opp_carry.
-
- (* ** Additions ** *)
-
- Let succ_c :=
-  Eval lazy beta delta [ww_succ_c] in ww_succ_c w_0 ww_1 w_succ_c.
-
- Let add_c :=
-  Eval lazy beta delta [ww_add_c] in ww_add_c w_WW w_add_c w_add_carry_c.
-
- Let add_carry_c :=
-  Eval lazy beta iota delta [ww_add_carry_c ww_succ_c] in
-  ww_add_carry_c w_0 w_WW ww_1 w_succ_c w_add_c w_add_carry_c.
-
- Let succ :=
-  Eval lazy beta delta [ww_succ] in ww_succ w_W0 ww_1 w_succ_c w_succ.
-
- Let add :=
-  Eval lazy beta delta [ww_add] in ww_add w_add_c w_add w_add_carry.
-
- Let add_carry :=
-  Eval lazy beta iota delta [ww_add_carry ww_succ] in
-  ww_add_carry w_W0 ww_1 w_succ_c w_add_carry_c w_succ w_add w_add_carry.
-
- (* ** Subtractions ** *)
-
- Let pred_c :=
-  Eval lazy beta delta [ww_pred_c] in ww_pred_c w_Bm1 w_WW ww_Bm1 w_pred_c.
-
- Let sub_c :=
-  Eval lazy beta iota delta [ww_sub_c ww_opp_c] in
-  ww_sub_c w_0 w_WW w_opp_c w_opp_carry w_sub_c w_sub_carry_c.
-
- Let sub_carry_c :=
-  Eval lazy beta iota delta [ww_sub_carry_c ww_pred_c ww_opp_carry] in
-  ww_sub_carry_c w_Bm1 w_WW ww_Bm1 w_opp_carry w_pred_c w_sub_c w_sub_carry_c.
-
- Let pred :=
-  Eval lazy beta delta [ww_pred] in ww_pred w_Bm1 w_WW ww_Bm1 w_pred_c w_pred.
-
- Let sub :=
-  Eval lazy beta iota delta [ww_sub ww_opp] in
-  ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c w_opp w_sub w_sub_carry.
-
- Let sub_carry :=
-  Eval lazy beta iota delta [ww_sub_carry ww_pred ww_opp_carry] in
-  ww_sub_carry w_Bm1 w_WW ww_Bm1 w_opp_carry w_pred_c w_sub_carry_c w_pred
-  w_sub w_sub_carry.
-
-
- (* ** Multiplication ** *)
-
- Let mul_c :=
-  Eval lazy beta iota delta [ww_mul_c double_mul_c] in
-  ww_mul_c w_0 w_1 w_WW w_W0 w_mul_c add_c add add_carry.
-
- Let karatsuba_c :=
-  Eval lazy beta iota delta [ww_karatsuba_c double_mul_c kara_prod] in
-  ww_karatsuba_c w_0 w_1 w_WW w_W0 w_compare w_add w_sub w_mul_c
-    add_c add add_carry sub_c sub.
-
- Let mul :=
-  Eval lazy beta delta [ww_mul] in
-  ww_mul w_W0 w_add w_mul_c w_mul add.
-
- Let square_c :=
-  Eval lazy beta delta [ww_square_c] in
-  ww_square_c w_0 w_1 w_WW w_W0 w_mul_c w_square_c add_c add add_carry.
-
- (* Division operation *)
-
- Let div32 :=
-   Eval lazy beta iota delta [w_div32] in
-   w_div32 w_0 w_Bm1 w_Bm2 w_WW w_compare w_add_c w_add_carry_c
-   w_add w_add_carry w_pred w_sub w_mul_c w_div21 sub_c.
-
- Let div21 :=
-  Eval lazy beta iota delta [ww_div21] in
-   ww_div21 w_0 w_0W div32 ww_1 compare sub.
-
- Let low (p: zn2z t) := match p with WW _ p1 => p1 | _ => w_0 end.
-
- Let add_mul_div :=
-  Eval lazy beta delta [ww_add_mul_div] in
-  ww_add_mul_div w_0 w_WW w_W0 w_0W compare w_add_mul_div sub w_zdigits low.
-
- Let div_gt :=
-  Eval lazy beta delta [ww_div_gt] in
-  ww_div_gt w_0 w_WW w_0W w_compare w_eq0 w_opp_c w_opp
-  w_opp_carry w_sub_c w_sub w_sub_carry
-  w_div_gt w_add_mul_div w_head0 w_div21 div32 _ww_zdigits ww_1 add_mul_div w_zdigits.
-
- Let div :=
-  Eval lazy beta delta [ww_div] in ww_div ww_1 compare div_gt.
-
- Let mod_gt :=
-  Eval lazy beta delta [ww_mod_gt] in
-  ww_mod_gt w_0 w_WW w_0W w_compare w_eq0 w_opp_c w_opp w_opp_carry w_sub_c w_sub w_sub_carry
-  w_mod_gt w_add_mul_div w_head0 w_div21 div32 _ww_zdigits add_mul_div w_zdigits.
-
- Let mod_ :=
-  Eval lazy beta delta [ww_mod] in ww_mod compare mod_gt.
-
- Let pos_mod :=
-  Eval lazy beta delta [ww_pos_mod] in
-    ww_pos_mod w_0 w_zdigits w_WW w_pos_mod compare w_0W low sub _ww_zdigits.
-
- Let is_even :=
-  Eval lazy beta delta [ww_is_even] in ww_is_even w_is_even.
-
- Let sqrt2 :=
-  Eval lazy beta delta [ww_sqrt2] in
-    ww_sqrt2 w_is_even w_compare w_0 w_1 w_Bm1 w_0W w_sub w_square_c
-    w_div21 w_add_mul_div w_zdigits w_add_c w_sqrt2 w_pred pred_c
-    pred add_c add sub_c add_mul_div.
-
- Let sqrt :=
-  Eval lazy beta delta [ww_sqrt] in
-    ww_sqrt w_is_even w_0 w_sub w_add_mul_div w_zdigits
-     _ww_zdigits w_sqrt2 pred add_mul_div head0 compare low.
-
- Let gcd_gt_fix :=
-  Eval cbv beta delta [ww_gcd_gt_aux ww_gcd_gt_body] in
-  ww_gcd_gt_aux w_0 w_WW w_0W w_compare w_opp_c w_opp w_opp_carry
-      w_sub_c w_sub w_sub_carry w_gcd_gt
-      w_add_mul_div w_head0 w_div21 div32 _ww_zdigits add_mul_div
-      w_zdigits.
-
- Let gcd_cont :=
-  Eval lazy beta delta [gcd_cont] in gcd_cont ww_1 w_1 w_compare.
-
- Let gcd_gt :=
-  Eval lazy beta delta [ww_gcd_gt] in
-  ww_gcd_gt w_0 w_eq0 w_gcd_gt _ww_digits gcd_gt_fix gcd_cont.
-
- Let gcd :=
-  Eval lazy beta delta [ww_gcd] in
-  ww_gcd compare w_0 w_eq0 w_gcd_gt _ww_digits gcd_gt_fix gcd_cont.
-
- Definition lor (x y : zn2z t) :=
-  match x, y with
-  | W0, _ => y
-  | _, W0 => x
-  | WW hx lx, WW hy ly => WW (ZnZ.lor hx hy) (ZnZ.lor lx ly)
-  end.
-
- Definition land (x y : zn2z t) :=
-  match x, y with
-  | W0, _ => W0
-  | _, W0 => W0
-  | WW hx lx, WW hy ly => WW (ZnZ.land hx hy) (ZnZ.land lx ly)
-  end.
-
- Definition lxor (x y : zn2z t) :=
-  match x, y with
-  | W0, _ => y
-  | _, W0 => x
-  | WW hx lx, WW hy ly => WW (ZnZ.lxor hx hy) (ZnZ.lxor lx ly)
-  end.
-
- (* ** Record of operators on 2 words *)
-
- Global Instance mk_zn2z_ops : ZnZ.Ops (zn2z t) | 1 :=
-  ZnZ.MkOps _ww_digits _ww_zdigits
-    to_Z ww_of_pos head0 tail0
-    W0 ww_1 ww_Bm1
-    compare eq0
-    opp_c opp opp_carry
-    succ_c add_c add_carry_c
-    succ add add_carry
-    pred_c sub_c sub_carry_c
-    pred sub sub_carry
-    mul_c mul square_c
-    div21 div_gt div
-    mod_gt mod_
-    gcd_gt gcd
-    add_mul_div
-    pos_mod
-    is_even
-    sqrt2
-    sqrt
-    lor
-    land
-    lxor.
-
- Global Instance mk_zn2z_ops_karatsuba : ZnZ.Ops (zn2z t) | 2 :=
-   ZnZ.MkOps _ww_digits _ww_zdigits
-    to_Z ww_of_pos head0 tail0
-    W0 ww_1 ww_Bm1
-    compare eq0
-    opp_c opp opp_carry
-    succ_c add_c add_carry_c
-    succ add add_carry
-    pred_c sub_c sub_carry_c
-    pred sub sub_carry
-    karatsuba_c mul square_c
-    div21 div_gt div
-    mod_gt mod_
-    gcd_gt gcd
-    add_mul_div
-    pos_mod
-    is_even
-    sqrt2
-    sqrt
-    lor
-    land
-    lxor.
-
- (* Proof *)
- Context {specs : ZnZ.Specs ops}.
- 
- Create HintDb ZnZ.
-
- Hint Resolve
-    ZnZ.spec_to_Z
-    ZnZ.spec_of_pos
-    ZnZ.spec_0
-    ZnZ.spec_1
-    ZnZ.spec_m1
-    ZnZ.spec_compare
-    ZnZ.spec_eq0
-    ZnZ.spec_opp_c
-    ZnZ.spec_opp
-    ZnZ.spec_opp_carry
-    ZnZ.spec_succ_c
-    ZnZ.spec_add_c
-    ZnZ.spec_add_carry_c
-    ZnZ.spec_succ
-    ZnZ.spec_add
-    ZnZ.spec_add_carry
-    ZnZ.spec_pred_c
-    ZnZ.spec_sub_c
-    ZnZ.spec_sub_carry_c
-    ZnZ.spec_pred
-    ZnZ.spec_sub
-    ZnZ.spec_sub_carry
-    ZnZ.spec_mul_c
-    ZnZ.spec_mul
-    ZnZ.spec_square_c
-    ZnZ.spec_div21
-    ZnZ.spec_div_gt
-    ZnZ.spec_div
-    ZnZ.spec_modulo_gt
-    ZnZ.spec_modulo
-    ZnZ.spec_gcd_gt
-    ZnZ.spec_gcd
-    ZnZ.spec_head0
-    ZnZ.spec_tail0
-    ZnZ.spec_add_mul_div
-    ZnZ.spec_pos_mod
-    ZnZ.spec_is_even
-    ZnZ.spec_sqrt2
-    ZnZ.spec_sqrt
-    ZnZ.spec_WO
-    ZnZ.spec_OW
-    ZnZ.spec_WW : ZnZ.
- 
- Ltac wwauto := unfold ww_to_Z; eauto with ZnZ.
-
- Let wwB := base _ww_digits.
-
- Notation "[| x |]" := (to_Z x)  (at level 0, x at level 99).
-
- Notation "[+| c |]" :=
-   (interp_carry 1 wwB to_Z c)  (at level 0, c at level 99).
-
- Notation "[-| c |]" :=
-   (interp_carry (-1) wwB to_Z c)  (at level 0, c at level 99).
-
- Notation "[[ x ]]" := (zn2z_to_Z wwB to_Z x)  (at level 0, x at level 99).
-
- Let spec_ww_to_Z   : forall x, 0 <= [| x |] < wwB.
- Proof. refine (spec_ww_to_Z w_digits w_to_Z _); wwauto. Qed.
-
- Let spec_ww_of_pos : forall p,
-     Zpos p = (Z.of_N (fst (ww_of_pos p)))*wwB + [|(snd (ww_of_pos p))|].
- Proof.
-  unfold ww_of_pos;intros.
-  rewrite (ZnZ.spec_of_pos p). unfold w_of_pos.
-  case (ZnZ.of_pos p); intros. simpl.
-  destruct n; simpl ZnZ.to_Z.
-  simpl;unfold w_to_Z,w_0; rewrite ZnZ.spec_0;trivial.
-  unfold Z.of_N.
-  rewrite (ZnZ.spec_of_pos p0).
-  case (ZnZ.of_pos p0); intros. simpl.
-  unfold fst, snd,Z.of_N, to_Z, wB, w_digits, w_to_Z, w_WW.
-  rewrite ZnZ.spec_WW.
-  replace wwB with (wB*wB).
-  unfold wB,w_to_Z,w_digits;destruct n;ring.
-  symmetry. rewrite <- Z.pow_2_r; exact (wwB_wBwB w_digits).
- Qed.
-
- Let spec_ww_0 : [|W0|] = 0.
- Proof. reflexivity. Qed.
-
- Let spec_ww_1 : [|ww_1|] = 1.
- Proof. refine (spec_ww_1 w_0 w_1 w_digits w_to_Z _ _);wwauto. Qed.
-
- Let spec_ww_Bm1 : [|ww_Bm1|] = wwB - 1.
- Proof. refine (spec_ww_Bm1 w_Bm1 w_digits w_to_Z _);wwauto. Qed.
-
- Let spec_ww_compare :
-  forall x y, compare x y = Z.compare [|x|] [|y|].
- Proof.
-  refine (spec_ww_compare w_0 w_digits w_to_Z w_compare _ _ _);wwauto.
- Qed.
-
- Let spec_ww_eq0 : forall x, eq0 x = true -> [|x|] = 0.
- Proof. destruct x;simpl;intros;trivial;discriminate. Qed.
-
- Let spec_ww_opp_c : forall x, [-|opp_c x|] = -[|x|].
- Proof.
-  refine(spec_ww_opp_c w_0 w_0 W0 w_opp_c w_opp_carry w_digits w_to_Z _ _ _ _);
-   wwauto.
- Qed.
-
- Let spec_ww_opp : forall x, [|opp x|] = (-[|x|]) mod wwB.
- Proof.
-  refine(spec_ww_opp w_0 w_0 W0 w_opp_c w_opp_carry w_opp
-   w_digits w_to_Z _ _ _ _ _);
-  wwauto.
- Qed.
-
- Let spec_ww_opp_carry : forall x, [|opp_carry x|] = wwB - [|x|] - 1.
- Proof.
-  refine (spec_ww_opp_carry w_WW ww_Bm1 w_opp_carry w_digits w_to_Z _ _ _);
-  wwauto.
- Qed.
-
- Let spec_ww_succ_c : forall x, [+|succ_c x|] = [|x|] + 1.
- Proof.
-  refine (spec_ww_succ_c w_0 w_0 ww_1 w_succ_c w_digits w_to_Z _ _ _ _);wwauto.
- Qed.
-
- Let spec_ww_add_c  : forall x y, [+|add_c x y|] = [|x|] + [|y|].
- Proof.
-  refine (spec_ww_add_c w_WW w_add_c w_add_carry_c w_digits w_to_Z _ _ _);wwauto.
- Qed.
-
- Let spec_ww_add_carry_c : forall x y, [+|add_carry_c x y|] = [|x|]+[|y|]+1.
- Proof.
-  refine (spec_ww_add_carry_c w_0 w_0 w_WW ww_1 w_succ_c w_add_c w_add_carry_c
-   w_digits w_to_Z _ _ _ _ _ _ _);wwauto.
- Qed.
-
- Let spec_ww_succ : forall x, [|succ x|] = ([|x|] + 1) mod wwB.
- Proof.
-  refine (spec_ww_succ w_W0 ww_1 w_succ_c w_succ w_digits w_to_Z _ _ _ _ _);
-  wwauto.
- Qed.
-
- Let spec_ww_add : forall x y, [|add x y|] = ([|x|] + [|y|]) mod wwB.
- Proof.
-  refine (spec_ww_add w_add_c w_add w_add_carry w_digits w_to_Z _ _ _ _);wwauto.
- Qed.
-
- Let spec_ww_add_carry : forall x y, [|add_carry x y|]=([|x|]+[|y|]+1)mod wwB.
- Proof.
-  refine (spec_ww_add_carry w_W0 ww_1 w_succ_c w_add_carry_c w_succ
-  w_add w_add_carry w_digits w_to_Z _ _ _ _ _ _ _ _);wwauto.
- Qed.
-
- Let spec_ww_pred_c : forall x, [-|pred_c x|] = [|x|] - 1.
- Proof.
-  refine (spec_ww_pred_c w_0 w_Bm1 w_WW ww_Bm1 w_pred_c w_digits w_to_Z
-   _ _ _ _ _);wwauto.
- Qed.
-
- Let spec_ww_sub_c : forall x y, [-|sub_c x y|] = [|x|] - [|y|].
- Proof.
-  refine (spec_ww_sub_c w_0 w_0 w_WW W0 w_opp_c w_opp_carry w_sub_c
-   w_sub_carry_c w_digits w_to_Z _ _ _ _ _ _ _);wwauto.
- Qed.
-
- Let spec_ww_sub_carry_c : forall x y, [-|sub_carry_c x y|] = [|x|]-[|y|]-1.
- Proof.
-  refine (spec_ww_sub_carry_c w_0 w_Bm1 w_WW ww_Bm1 w_opp_carry w_pred_c
-   w_sub_c w_sub_carry_c w_digits w_to_Z _ _ _ _ _ _ _ _);wwauto.
- Qed.
-
- Let spec_ww_pred : forall x, [|pred x|] = ([|x|] - 1) mod wwB.
- Proof.
-  refine (spec_ww_pred w_0 w_Bm1 w_WW ww_Bm1 w_pred_c w_pred w_digits w_to_Z
-   _ _ _ _ _ _);wwauto.
- Qed.
-
- Let spec_ww_sub : forall x y, [|sub x y|] = ([|x|] - [|y|]) mod wwB.
- Proof.
-  refine (spec_ww_sub w_0 w_0 w_WW W0 w_opp_c w_opp_carry w_sub_c w_opp
-   w_sub w_sub_carry w_digits w_to_Z _ _ _ _ _ _ _ _ _);wwauto.
- Qed.
-
- Let spec_ww_sub_carry : forall x y, [|sub_carry x y|]=([|x|]-[|y|]-1) mod wwB.
- Proof.
-  refine (spec_ww_sub_carry w_0 w_Bm1 w_WW ww_Bm1 w_opp_carry w_pred_c
-   w_sub_carry_c w_pred w_sub w_sub_carry w_digits w_to_Z _ _ _ _ _ _ _ _ _ _);
-  wwauto.
- Qed.
-
- Let spec_ww_mul_c : forall x y, [[mul_c x y ]] = [|x|] * [|y|].
- Proof.
-  refine (spec_ww_mul_c w_0 w_1 w_WW w_W0 w_mul_c add_c add add_carry w_digits
-   w_to_Z _ _ _ _ _ _ _ _ _);wwauto.
- Qed.
-
- Let spec_ww_karatsuba_c : forall x y, [[karatsuba_c x y ]] = [|x|] * [|y|].
- Proof.
- refine (spec_ww_karatsuba_c _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
-          _ _ _ _ _ _ _ _ _ _ _ _); wwauto.
-  unfold w_digits; apply ZnZ.spec_more_than_1_digit; auto.
- Qed.
-
- Let spec_ww_mul : forall x y, [|mul x y|] = ([|x|] * [|y|]) mod wwB.
- Proof.
-  refine (spec_ww_mul w_W0 w_add w_mul_c w_mul add w_digits w_to_Z _ _ _ _ _);
-  wwauto.
- Qed.
-
- Let spec_ww_square_c : forall x, [[square_c x]] = [|x|] * [|x|].
- Proof.
-  refine (spec_ww_square_c w_0 w_1 w_WW w_W0 w_mul_c w_square_c add_c add
-   add_carry w_digits w_to_Z _ _ _ _ _ _ _ _ _ _);wwauto.
- Qed.
-
- Let spec_w_div32 : forall a1 a2 a3 b1 b2,
-       wB / 2 <= (w_to_Z b1) ->
-       [|WW a1 a2|] < [|WW b1 b2|] ->
-       let (q, r) := div32 a1 a2 a3 b1 b2 in
-       (w_to_Z a1) * wwB + (w_to_Z a2) * wB + (w_to_Z a3) =
-       (w_to_Z q) * ((w_to_Z b1)*wB + (w_to_Z b2)) + [|r|] /\
-       0 <= [|r|] < (w_to_Z b1)*wB + w_to_Z b2.
- Proof.
-  refine (spec_w_div32 w_0 w_Bm1 w_Bm2 w_WW w_compare w_add_c w_add_carry_c
-   w_add w_add_carry w_pred w_sub w_mul_c w_div21 sub_c w_digits w_to_Z
-   _ _ _ _ _ _ _ _ _ _ _ _ _ _ _);wwauto.
-  unfold w_Bm2, w_to_Z, w_pred, w_Bm1.
-  rewrite ZnZ.spec_pred, ZnZ.spec_m1.
-  unfold w_digits;rewrite Zmod_small. ring.
-  assert (H:= wB_pos(ZnZ.digits)). omega.
-  exact ZnZ.spec_div21.
- Qed.
-
- Let spec_ww_div21 : forall a1 a2 b,
-      wwB/2 <= [|b|] ->
-      [|a1|] < [|b|] ->
-      let (q,r) := div21 a1 a2 b in
-      [|a1|] *wwB+ [|a2|] = [|q|] * [|b|] + [|r|] /\
-      0 <= [|r|] < [|b|].
- Proof.
-  refine (spec_ww_div21 w_0 w_0W div32 ww_1 compare sub w_digits w_to_Z
-   _ _ _ _ _ _ _);wwauto. 
- Qed.
-
- Let spec_add2: forall x y,
-  [|w_add2 x y|] = w_to_Z x + w_to_Z y.
-  unfold w_add2.
-  intros xh xl; generalize (ZnZ.spec_add_c xh xl).
-  unfold w_add_c; case ZnZ.add_c; unfold interp_carry; simpl ww_to_Z.
-    intros w0 Hw0; simpl; unfold w_to_Z; rewrite Hw0.
-  unfold w_0; rewrite ZnZ.spec_0; simpl; auto with zarith.
-  intros w0; rewrite Z.mul_1_l; simpl.
-  unfold w_to_Z, w_1; rewrite ZnZ.spec_1; auto with zarith.
-  rewrite Z.mul_1_l; auto.
- Qed.
-
- Let spec_low: forall x,
-  w_to_Z (low x) = [|x|] mod wB. 
-  intros x; case x; simpl low.
-    unfold ww_to_Z, w_to_Z, w_0; rewrite ZnZ.spec_0; simpl; wwauto.
-  intros xh xl; simpl.
-  rewrite Z.add_comm; rewrite Z_mod_plus; auto with zarith.
-  rewrite Zmod_small; auto with zarith.
-  unfold wB, base; eauto with ZnZ zarith.
-  unfold wB, base; eauto with ZnZ zarith.
- Qed.
-
- Let spec_ww_digits:
-  [|_ww_zdigits|] = Zpos (xO w_digits).
- Proof.
- unfold w_to_Z, _ww_zdigits.
- rewrite spec_add2.
- unfold w_to_Z, w_zdigits, w_digits.
- rewrite ZnZ.spec_zdigits; auto.
- rewrite Pos2Z.inj_xO; auto with zarith.
- Qed.
-
-
- Let spec_ww_head00  : forall x,  [|x|] = 0 -> [|head0 x|] = Zpos _ww_digits.
- Proof.
- refine (spec_ww_head00 w_0 w_0W
-                w_compare w_head0 w_add2 w_zdigits _ww_zdigits
-                w_to_Z _ _ _ (eq_refl _ww_digits) _ _ _ _); wwauto.
- exact ZnZ.spec_head00.
- exact ZnZ.spec_zdigits.
- Qed.
-
- Let spec_ww_head0  : forall x,  0 < [|x|] ->
-	 wwB/ 2 <= 2 ^ [|head0 x|] * [|x|] < wwB.
- Proof.
-  refine (spec_ww_head0 w_0 w_0W w_compare w_head0
-           w_add2 w_zdigits _ww_zdigits
-   w_to_Z _ _ _ _ _ _ _);wwauto.
-  exact ZnZ.spec_zdigits.
- Qed.
-
- Let spec_ww_tail00  : forall x,  [|x|] = 0 -> [|tail0 x|] = Zpos _ww_digits.
- Proof.
- refine (spec_ww_tail00 w_0 w_0W
-                w_compare w_tail0 w_add2 w_zdigits _ww_zdigits
-                w_to_Z _ _ _ (eq_refl _ww_digits) _ _ _ _); wwauto.
- exact ZnZ.spec_tail00.
- exact ZnZ.spec_zdigits.
- Qed.
-
-
- Let spec_ww_tail0  : forall x,  0 < [|x|] ->
-  exists y, 0 <= y /\ [|x|] = (2 * y + 1) * 2 ^ [|tail0 x|].
- Proof.
-  refine (spec_ww_tail0 (w_digits := w_digits) w_0 w_0W w_compare w_tail0
-           w_add2 w_zdigits _ww_zdigits w_to_Z _ _ _ _ _ _ _);wwauto.
-  exact ZnZ.spec_zdigits.
- Qed.
-
- Lemma spec_ww_add_mul_div : forall x y p,
-       [|p|] <= Zpos _ww_digits ->
-       [| add_mul_div p x y |] =
-         ([|x|] * (2 ^ [|p|]) +
-          [|y|] / (2 ^ ((Zpos _ww_digits) - [|p|]))) mod wwB.
- Proof.
- refine (@spec_ww_add_mul_div t w_0 w_WW w_W0 w_0W compare w_add_mul_div
-              sub w_digits w_zdigits low w_to_Z
-           _ _ _ _ _ _ _ _ _ _ _);wwauto.
-  exact ZnZ.spec_zdigits.
- Qed.
-
- Let spec_ww_div_gt : forall a b,
-      [|a|] > [|b|] -> 0 < [|b|] ->
-      let (q,r) := div_gt a b in
-      [|a|] = [|q|] * [|b|] + [|r|] /\ 0 <= [|r|] < [|b|].
- Proof.
-refine
-(@spec_ww_div_gt t w_digits w_0 w_WW w_0W w_compare w_eq0
-   w_opp_c w_opp w_opp_carry w_sub_c w_sub w_sub_carry w_div_gt
-   w_add_mul_div w_head0 w_div21 div32 _ww_zdigits ww_1 add_mul_div w_zdigits w_to_Z
- _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
-).
-  exact ZnZ.spec_0.
-  exact ZnZ.spec_to_Z.
-  wwauto.
-  wwauto.
-  exact ZnZ.spec_compare.
-  exact ZnZ.spec_eq0.
-  exact ZnZ.spec_opp_c.
-  exact ZnZ.spec_opp.
-  exact ZnZ.spec_opp_carry.
-  exact ZnZ.spec_sub_c.
-  exact ZnZ.spec_sub.
-  exact ZnZ.spec_sub_carry.
-  exact ZnZ.spec_div_gt.
-  exact ZnZ.spec_add_mul_div.
-  exact ZnZ.spec_head0.
-  exact ZnZ.spec_div21.
-  exact spec_w_div32.
-  exact ZnZ.spec_zdigits.
-  exact spec_ww_digits.
-  exact spec_ww_1.
-  exact spec_ww_add_mul_div.
- Qed.
-
- Let spec_ww_div : forall a b, 0 < [|b|] ->
-      let (q,r) := div a b in
-      [|a|] = [|q|] * [|b|] + [|r|] /\
-      0 <= [|r|] < [|b|].
- Proof.
-  refine (spec_ww_div w_digits ww_1 compare div_gt w_to_Z _ _ _ _);wwauto.
- Qed.
-
- Let spec_ww_mod_gt : forall a b,
-      [|a|] > [|b|] -> 0 < [|b|] ->
-      [|mod_gt a b|] = [|a|] mod [|b|].
- Proof.
-  refine (@spec_ww_mod_gt t w_digits w_0 w_WW w_0W w_compare w_eq0
-   w_opp_c w_opp w_opp_carry w_sub_c w_sub w_sub_carry w_div_gt w_mod_gt
-   w_add_mul_div w_head0 w_div21 div32 _ww_zdigits ww_1 add_mul_div
-   w_zdigits w_to_Z
-   _ _ _ _ _   _ _ _ _ _   _ _ _ _ _   _ _ _ _ _ _ _);wwauto.
-  exact ZnZ.spec_div_gt.
-  exact ZnZ.spec_div21.
-  exact ZnZ.spec_zdigits.
-  exact spec_ww_add_mul_div.
- Qed.
-
- Let spec_ww_mod :  forall a b, 0 < [|b|] -> [|mod_ a b|] = [|a|] mod [|b|].
- Proof.
-  refine (spec_ww_mod w_digits W0 compare mod_gt w_to_Z _ _ _);wwauto.
- Qed.
-
- Let spec_ww_gcd_gt : forall a b, [|a|] > [|b|] ->
-      Zis_gcd [|a|] [|b|] [|gcd_gt a b|].
- Proof.
-  refine (@spec_ww_gcd_gt t w_digits W0 w_to_Z _
-    w_0 w_0 w_eq0 w_gcd_gt _ww_digits
-  _ gcd_gt_fix _ _ _ _ gcd_cont _);wwauto.
-  refine (@spec_ww_gcd_gt_aux t w_digits w_0 w_WW w_0W w_compare w_opp_c w_opp
-   w_opp_carry w_sub_c w_sub w_sub_carry w_gcd_gt w_add_mul_div w_head0
-   w_div21 div32 _ww_zdigits ww_1 add_mul_div w_zdigits w_to_Z
-   _ _ _ _ _   _ _ _ _ _   _ _ _ _ _   _ _ _ _ _);wwauto.
-  exact ZnZ.spec_div21.
-  exact ZnZ.spec_zdigits.
-  exact spec_ww_add_mul_div.
-  refine (@spec_gcd_cont t w_digits ww_1 w_to_Z _ _ w_0 w_1 w_compare
-   _ _);wwauto.
- Qed.
-
- Let spec_ww_gcd : forall a b, Zis_gcd [|a|] [|b|] [|gcd a b|].
- Proof.
-  refine (@spec_ww_gcd t w_digits W0 compare w_to_Z _ _ w_0 w_0 w_eq0 w_gcd_gt
-  _ww_digits _ gcd_gt_fix _ _ _ _ gcd_cont _);wwauto.
-  refine (@spec_ww_gcd_gt_aux t w_digits w_0 w_WW w_0W w_compare w_opp_c w_opp
-   w_opp_carry w_sub_c w_sub w_sub_carry w_gcd_gt w_add_mul_div w_head0
-   w_div21 div32 _ww_zdigits ww_1 add_mul_div w_zdigits w_to_Z
-   _ _ _ _ _   _ _ _ _ _   _ _ _ _ _   _ _ _ _ _);wwauto.
-  exact ZnZ.spec_div21.
-  exact ZnZ.spec_zdigits.
-  exact spec_ww_add_mul_div.
-  refine (@spec_gcd_cont t w_digits ww_1 w_to_Z _ _ w_0 w_1 w_compare
-   _ _);wwauto.
- Qed.
-
- Let spec_ww_is_even : forall x,
-      match is_even x with
-         true => [|x|] mod 2 = 0
-      | false => [|x|] mod 2 = 1
-      end.
- Proof.
- refine (@spec_ww_is_even t w_is_even w_digits _ _ ).
- exact ZnZ.spec_is_even.
- Qed.
-
- Let spec_ww_sqrt2 : forall x y,
-       wwB/ 4 <= [|x|] ->
-       let (s,r) := sqrt2 x y in
-          [[WW x y]] = [|s|] ^ 2 + [+|r|] /\
-          [+|r|] <= 2 * [|s|].
- Proof.
- intros x y H.
- refine (@spec_ww_sqrt2 t w_is_even w_compare w_0 w_1 w_Bm1
-               w_0W w_sub w_square_c w_div21 w_add_mul_div w_digits w_zdigits
-               _ww_zdigits
-               w_add_c w_sqrt2 w_pred pred_c pred add_c add sub_c add_mul_div
-                _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _); wwauto.
- exact ZnZ.spec_zdigits.
- exact ZnZ.spec_more_than_1_digit.
- exact ZnZ.spec_is_even.
- exact ZnZ.spec_div21.
- exact spec_ww_add_mul_div.
- exact ZnZ.spec_sqrt2.
- Qed.
-
- Let spec_ww_sqrt : forall x,
-       [|sqrt x|] ^ 2 <= [|x|] < ([|sqrt x|] + 1) ^ 2.
- Proof.
- refine (@spec_ww_sqrt t w_is_even w_0 w_1 w_Bm1
-           w_sub w_add_mul_div w_digits w_zdigits _ww_zdigits
-               w_sqrt2 pred add_mul_div head0 compare
-            _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _); wwauto.
- exact ZnZ.spec_zdigits.
- exact ZnZ.spec_more_than_1_digit.
- exact ZnZ.spec_is_even.
- exact spec_ww_add_mul_div.
- exact ZnZ.spec_sqrt2.
- Qed.
-
- Let wB_pos : 0 < wB.
- Proof.
- unfold wB, base; apply Z.pow_pos_nonneg; auto with zarith.
- Qed.
- 
- Hint Transparent ww_to_Z.
-
- Let ww_testbit_high n x y : Z.pos w_digits <= n ->
-  Z.testbit [|WW x y|] n =
-  Z.testbit (ZnZ.to_Z x) (n - Z.pos w_digits).
- Proof.
-  intros Hn.
-  assert (E : ZnZ.to_Z x = [|WW x y|] / wB).
-  { simpl.
-    rewrite Z.div_add_l; eauto with ZnZ zarith.
-    now rewrite Z.div_small, Z.add_0_r; wwauto. }
-  rewrite E.
-  unfold wB, base. rewrite Z.div_pow2_bits.
-  - f_equal; auto with zarith.
-  - easy.
-  - auto with zarith.
- Qed.
-
- Let ww_testbit_low n x y : 0 <= n < Z.pos w_digits ->
-  Z.testbit [|WW x y|] n = Z.testbit (ZnZ.to_Z y) n.
- Proof.
-  intros (Hn,Hn').
-  assert (E : ZnZ.to_Z y = [|WW x y|] mod wB).
-  { simpl; symmetry.
-    rewrite Z.add_comm, Z.mod_add; auto with zarith nocore.
-    apply Z.mod_small; eauto with ZnZ zarith. }
-  rewrite E.
-  unfold wB, base. symmetry. apply Z.mod_pow2_bits_low; auto.
- Qed.
-
- Let spec_lor x y : [|lor x y|] = Z.lor [|x|] [|y|].
- Proof.
-  destruct x as [ |hx lx]. trivial.
-  destruct y as [ |hy ly]. now rewrite Z.lor_comm.
-  change ([|WW (ZnZ.lor hx hy) (ZnZ.lor lx ly)|] =
-          Z.lor [|WW hx lx|] [|WW hy ly|]).
-  apply Z.bits_inj'; intros n Hn.
-  rewrite Z.lor_spec.
-  destruct (Z.le_gt_cases (Z.pos w_digits) n) as [LE|GT].
-  - now rewrite !ww_testbit_high, ZnZ.spec_lor, Z.lor_spec.
-  - rewrite !ww_testbit_low; auto.
-    now rewrite ZnZ.spec_lor, Z.lor_spec.
- Qed.
-
- Let spec_land x y : [|land x y|] = Z.land [|x|] [|y|].
- Proof.
-  destruct x as [ |hx lx]. trivial.
-  destruct y as [ |hy ly]. now rewrite Z.land_comm.
-  change ([|WW (ZnZ.land hx hy) (ZnZ.land lx ly)|] =
-          Z.land [|WW hx lx|] [|WW hy ly|]).
-  apply Z.bits_inj'; intros n Hn.
-  rewrite Z.land_spec.
-  destruct (Z.le_gt_cases (Z.pos w_digits) n) as [LE|GT].
-  - now rewrite !ww_testbit_high, ZnZ.spec_land, Z.land_spec.
-  - rewrite !ww_testbit_low; auto.
-    now rewrite ZnZ.spec_land, Z.land_spec.
- Qed.
-
- Let spec_lxor x y : [|lxor x y|] = Z.lxor [|x|] [|y|].
- Proof.
-  destruct x as [ |hx lx]. trivial.
-  destruct y as [ |hy ly]. now rewrite Z.lxor_comm.
-  change ([|WW (ZnZ.lxor hx hy) (ZnZ.lxor lx ly)|] =
-          Z.lxor [|WW hx lx|] [|WW hy ly|]).
-  apply Z.bits_inj'; intros n Hn.
-  rewrite Z.lxor_spec.
-  destruct (Z.le_gt_cases (Z.pos w_digits) n) as [LE|GT].
-  - now rewrite !ww_testbit_high, ZnZ.spec_lxor, Z.lxor_spec.
-  - rewrite !ww_testbit_low; auto.
-    now rewrite ZnZ.spec_lxor, Z.lxor_spec.
- Qed.
-
- Global Instance mk_zn2z_specs : ZnZ.Specs mk_zn2z_ops.
- Proof.
-  apply ZnZ.MkSpecs; auto.
-  exact spec_ww_add_mul_div.
-
-  refine (@spec_ww_pos_mod t w_0 w_digits w_zdigits w_WW
-           w_pos_mod compare w_0W low sub _ww_zdigits w_to_Z
-       _ _ _ _ _ _ _ _ _ _ _ _);wwauto.
-  exact ZnZ.spec_zdigits.
-  unfold w_to_Z, w_zdigits.
-  rewrite ZnZ.spec_zdigits.
-  rewrite <- Pos2Z.inj_xO; exact spec_ww_digits.
- Qed.
-
- Global Instance mk_zn2z_specs_karatsuba : ZnZ.Specs mk_zn2z_ops_karatsuba.
- Proof.
-  apply ZnZ.MkSpecs; auto.
-  exact spec_ww_add_mul_div.
-  refine (@spec_ww_pos_mod t w_0 w_digits w_zdigits w_WW
-           w_pos_mod compare w_0W low sub _ww_zdigits w_to_Z
-       _ _ _ _ _ _ _ _ _ _ _ _);wwauto.
-  exact ZnZ.spec_zdigits.
-  unfold w_to_Z, w_zdigits.
-  rewrite ZnZ.spec_zdigits.
-  rewrite <- Pos2Z.inj_xO; exact spec_ww_digits.
- Qed.
-
-End Z_2nZ.
-
-
-Section MulAdd.
-
-  Context {t : Type}{ops : ZnZ.Ops t}{specs : ZnZ.Specs ops}.
-
-  Definition mul_add:= w_mul_add ZnZ.zero ZnZ.succ ZnZ.add_c ZnZ.mul_c.
-
-  Notation "[| x |]" := (ZnZ.to_Z x)  (at level 0, x at level 99).
-
-  Notation "[|| x ||]" :=
-   (zn2z_to_Z (base ZnZ.digits) ZnZ.to_Z x)  (at level 0, x at level 99).
-
-  Lemma spec_mul_add: forall x y z,
-     let (zh, zl) := mul_add x y z in
-  [||WW zh zl||] = [|x|] * [|y|] + [|z|].
-  Proof.
-  intros x y z.
-   refine (spec_w_mul_add _ _ _ _ _ _ _ _ _ _ _ _ x y z); auto.
-  exact ZnZ.spec_0.
-  exact ZnZ.spec_to_Z.
-  exact ZnZ.spec_succ.
-  exact ZnZ.spec_add_c.
-  exact ZnZ.spec_mul_c.
-  Qed.
-
-End MulAdd.
-
-
-(** Modular versions of DoubleCyclic *)
-
-Module DoubleCyclic (C:CyclicType) <: CyclicType.
- Definition t := zn2z C.t.
- Instance ops : ZnZ.Ops t := mk_zn2z_ops.
- Instance specs : ZnZ.Specs ops := mk_zn2z_specs.
-End DoubleCyclic.
-
-Module DoubleCyclicKaratsuba (C:CyclicType) <: CyclicType.
- Definition t := zn2z C.t.
- Definition ops : ZnZ.Ops t := mk_zn2z_ops_karatsuba.
- Definition specs : ZnZ.Specs ops := mk_zn2z_specs_karatsuba.
-End DoubleCyclicKaratsuba.
diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleDiv.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleDiv.v
deleted file mode 100644
index 09d7329b6..000000000
--- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleDiv.v
+++ /dev/null
@@ -1,1494 +0,0 @@
-(************************************************************************)
-(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016     *)
-(*   \VV/  **************************************************************)
-(*    //   *      This file is distributed under the terms of the       *)
-(*         *       GNU Lesser General Public License Version 2.1        *)
-(************************************************************************)
-(*         Benjamin Gregoire, Laurent Thery, INRIA, 2007                *)
-(************************************************************************)
-
-Set Implicit Arguments.
-
-Require Import ZArith.
-Require Import BigNumPrelude.
-Require Import DoubleType.
-Require Import DoubleBase.
-Require Import DoubleDivn1.
-Require Import DoubleAdd.
-Require Import DoubleSub.
-
-Local Open Scope Z_scope.
-
-Ltac zarith := auto with zarith.
-
-
-Section POS_MOD.
-
- Variable w:Type.
- Variable w_0 : w.
- Variable w_digits : positive.
- Variable w_zdigits : w.
- Variable w_WW : w -> w -> zn2z w.
- Variable w_pos_mod : w -> w -> w.
- Variable w_compare : w -> w -> comparison.
- Variable ww_compare : zn2z w -> zn2z w -> comparison.
- Variable w_0W : w ->  zn2z w.
- Variable low: zn2z w -> w.
- Variable ww_sub: zn2z w -> zn2z w -> zn2z w.
- Variable ww_zdigits : zn2z w.
-
-
- Definition ww_pos_mod p x :=
-  let zdigits := w_0W w_zdigits in
-  match x with
-  | W0 => W0
-  | WW xh xl =>
-    match ww_compare p zdigits with
-    | Eq => w_WW w_0 xl
-    | Lt => w_WW w_0 (w_pos_mod (low p) xl)
-    | Gt =>
-      match ww_compare p ww_zdigits with
-      | Lt =>
-        let n := low (ww_sub p zdigits) in
-        w_WW (w_pos_mod n xh) xl
-      | _ => x
-      end
-    end
-  end.
-
-
-  Variable w_to_Z : w -> Z.
-
-  Notation wB  := (base w_digits).
-  Notation wwB := (base (ww_digits w_digits)).
-  Notation "[| x |]" := (w_to_Z x)  (at level 0, x at level 99).
-
-  Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99).
-
-
-  Variable spec_w_0   : [|w_0|] = 0.
-
-  Variable spec_to_Z  : forall x, 0 <= [|x|] < wB.
-
-  Variable spec_to_w_Z  : forall x, 0 <= [[x]] < wwB.
-
-  Variable spec_w_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|].
-
-  Variable spec_pos_mod : forall w p,
-       [|w_pos_mod p w|] = [|w|] mod (2 ^ [|p|]).
-
-  Variable spec_w_0W : forall l, [[w_0W l]] = [|l|].
-  Variable spec_ww_compare : forall x y,
-    ww_compare x y = Z.compare [[x]] [[y]].
- Variable spec_ww_sub: forall x y,
-   [[ww_sub x y]] = ([[x]] - [[y]]) mod wwB.
-
- Variable spec_zdigits : [| w_zdigits |] = Zpos w_digits.
- Variable spec_low: forall x, [| low x|] = [[x]] mod wB.
- Variable spec_ww_zdigits : [[ww_zdigits]] = 2 * [|w_zdigits|].
- Variable spec_ww_digits : ww_digits w_digits = xO w_digits.
-
-
-  Hint Rewrite spec_w_0 spec_w_WW : w_rewrite.
-
- Lemma spec_ww_pos_mod : forall w p,
-       [[ww_pos_mod p w]] = [[w]] mod (2 ^ [[p]]).
- assert (HHHHH:= lt_0_wB w_digits).
- assert (F0: forall x y, x - y + y = x); auto with zarith.
- intros w1 p; case (spec_to_w_Z p); intros HH1 HH2.
- unfold ww_pos_mod; case w1. reflexivity.
- intros xh xl; rewrite spec_ww_compare.
-    case Z.compare_spec;
-    rewrite spec_w_0W; rewrite spec_zdigits; fold wB;
-    intros H1.
-   rewrite H1; simpl ww_to_Z.
-   autorewrite with w_rewrite rm10.
-   rewrite Zplus_mod; auto with zarith.
-   rewrite Z_mod_mult; auto with zarith.
-   autorewrite with rm10.
-   rewrite Zmod_mod; auto with zarith.
-   rewrite Zmod_small; auto with zarith.
-   autorewrite with w_rewrite rm10.
-   simpl ww_to_Z.
-   rewrite spec_pos_mod.
-   assert (HH0: [|low p|] = [[p]]).
-     rewrite spec_low.
-     apply Zmod_small; auto with zarith.
-     case (spec_to_w_Z p); intros HHH1 HHH2; split; auto with zarith.
-     apply Z.lt_le_trans with (1 := H1).
-     unfold base; apply Zpower2_le_lin; auto with zarith.
-   rewrite HH0.
-   rewrite Zplus_mod; auto with zarith.
-   unfold base.
-   rewrite <- (F0 (Zpos w_digits) [[p]]).
-   rewrite Zpower_exp; auto with zarith.
-   rewrite Z.mul_assoc.
-   rewrite Z_mod_mult; auto with zarith.
-   autorewrite with w_rewrite rm10.
-   rewrite Zmod_mod; auto with zarith.
-  rewrite spec_ww_compare.
-    case Z.compare_spec; rewrite spec_ww_zdigits;
-                     rewrite spec_zdigits; intros H2.
-  replace (2^[[p]]) with wwB.
-    rewrite Zmod_small; auto with zarith.
-  unfold base; rewrite H2.
-  rewrite spec_ww_digits; auto.
-  assert (HH0: [|low (ww_sub p (w_0W w_zdigits))|] =
-         [[p]] - Zpos w_digits).
-    rewrite spec_low.
-    rewrite spec_ww_sub.
-    rewrite spec_w_0W; rewrite spec_zdigits.
-    rewrite <- Zmod_div_mod; auto with zarith.
-    rewrite Zmod_small; auto with zarith.
-    split; auto with zarith.
-    apply Z.lt_le_trans with (Zpos w_digits); auto with zarith.
-    unfold base; apply Zpower2_le_lin; auto with zarith.
-    exists wB; unfold base; rewrite <- Zpower_exp; auto with zarith.
-    rewrite spec_ww_digits;
-      apply f_equal with (f := Z.pow 2); rewrite Pos2Z.inj_xO; auto with zarith.
-   simpl ww_to_Z; autorewrite with w_rewrite.
-   rewrite spec_pos_mod; rewrite HH0.
-   pattern [|xh|] at 2;
-     rewrite Z_div_mod_eq with (b := 2 ^ ([[p]] - Zpos w_digits));
-     auto with zarith.
-   rewrite (fun x => (Z.mul_comm (2 ^ x))); rewrite Z.mul_add_distr_r.
-   unfold base; rewrite <- Z.mul_assoc; rewrite <- Zpower_exp;
-    auto with zarith.
-   rewrite F0; auto with zarith.
-   rewrite <- Z.add_assoc; rewrite Zplus_mod; auto with zarith.
-   rewrite Z_mod_mult; auto with zarith.
-   autorewrite with rm10.
-   rewrite Zmod_mod; auto with zarith.
-   symmetry; apply Zmod_small; auto with zarith.
-   case (spec_to_Z xh); intros U1 U2.
-   case (spec_to_Z xl); intros U3 U4.
-   split; auto with zarith.
-   apply Z.add_nonneg_nonneg; auto with zarith.
-   apply Z.mul_nonneg_nonneg; auto with zarith.
-   match goal with |- 0 <= ?X mod ?Y =>
-    case (Z_mod_lt X Y); auto with zarith
-   end.
-   match goal with |- ?X mod ?Y * ?U + ?Z < ?T =>
-    apply Z.le_lt_trans with ((Y - 1) * U + Z );
-     [case (Z_mod_lt X Y); auto with zarith | idtac]
-   end.
-   match goal with |- ?X * ?U + ?Y < ?Z =>
-    apply Z.le_lt_trans with (X * U + (U - 1))
-   end.
-   apply Z.add_le_mono_l; auto with zarith.
-   case (spec_to_Z xl); unfold base; auto with zarith.
-   rewrite Z.mul_sub_distr_r; rewrite <- Zpower_exp; auto with zarith.
-   rewrite F0; auto with zarith.
-  rewrite Zmod_small; auto with zarith.
-  case (spec_to_w_Z (WW xh xl)); intros U1 U2.
-  split; auto with zarith.
-  apply Z.lt_le_trans with (1:= U2).
-  unfold base; rewrite spec_ww_digits.
-  apply Zpower_le_monotone; auto with zarith.
-  split; auto with zarith.
-  rewrite Pos2Z.inj_xO; auto with zarith.
- Qed.
-
-End POS_MOD.
-
-Section DoubleDiv32.
-
- Variable w             : Type.
- Variable w_0           : w.
- Variable w_Bm1         : w.
- Variable w_Bm2         : w.
- Variable w_WW          : w -> w -> zn2z w.
- Variable w_compare     : w -> w -> comparison.
- Variable w_add_c       : w -> w -> carry w.
- Variable w_add_carry_c : w -> w -> carry w.
- Variable w_add         : w -> w -> w.
- Variable w_add_carry   : w -> w -> w.
- Variable w_pred        : w -> w.
- Variable w_sub         : w -> w -> w.
- Variable w_mul_c       : w -> w -> zn2z w.
- Variable w_div21       : w -> w -> w -> w*w.
- Variable ww_sub_c      : zn2z w -> zn2z w -> carry (zn2z w).
-
- Definition w_div32_body a1 a2 a3 b1 b2 :=
-  match w_compare a1 b1 with
-  | Lt =>
-    let (q,r) := w_div21 a1 a2 b1 in
-    match ww_sub_c (w_WW r a3) (w_mul_c q b2) with
-    | C0 r1 => (q,r1)
-    | C1 r1 =>
-      let q := w_pred q in
-      ww_add_c_cont w_WW w_add_c w_add_carry_c
-      (fun r2=>(w_pred q, ww_add w_add_c w_add w_add_carry r2 (WW b1 b2)))
-      (fun r2 => (q,r2))
-      r1 (WW b1 b2)
-    end
-  | Eq =>
-    ww_add_c_cont w_WW w_add_c w_add_carry_c
-    (fun r => (w_Bm2, ww_add w_add_c w_add w_add_carry r (WW b1 b2)))
-    (fun r => (w_Bm1,r))
-    (WW (w_sub a2 b2) a3) (WW b1 b2)
-  | Gt => (w_0, W0) (* cas absurde *)
-  end.
-
- Definition w_div32 a1 a2 a3 b1 b2 :=
-  Eval lazy beta iota delta [ww_add_c_cont ww_add w_div32_body] in
-     w_div32_body a1 a2 a3 b1 b2.
-
- (* Proof *)
-
-  Variable w_digits      : positive.
-  Variable w_to_Z : w -> Z.
-
-  Notation wB  := (base w_digits).
-  Notation wwB := (base (ww_digits w_digits)).
-  Notation "[| x |]" := (w_to_Z x)  (at level 0, x at level 99).
-  Notation "[+| c |]" :=
-   (interp_carry 1 wB w_to_Z c) (at level 0, c at level 99).
-  Notation "[-| c |]" :=
-   (interp_carry (-1) wB w_to_Z c) (at level 0, c at level 99).
-
-  Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99).
-  Notation "[-[ c ]]" :=
-   (interp_carry (-1) wwB (ww_to_Z w_digits w_to_Z) c)
-   (at level 0, c at level 99).
-
-
-  Variable spec_w_0   : [|w_0|] = 0.
-  Variable spec_w_Bm1   : [|w_Bm1|] = wB - 1.
-  Variable spec_w_Bm2   : [|w_Bm2|] = wB - 2.
-
-  Variable spec_to_Z  : forall x, 0 <= [|x|] < wB.
-
-  Variable spec_w_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|].
-  Variable spec_compare :
-     forall x y, w_compare x y = Z.compare [|x|] [|y|].
-  Variable spec_w_add_c  : forall x y, [+|w_add_c x y|] = [|x|] + [|y|].
-  Variable spec_w_add_carry_c :
-         forall x y, [+|w_add_carry_c x y|] = [|x|] + [|y|] + 1.
-
-  Variable spec_w_add : forall x y, [|w_add x y|] = ([|x|] + [|y|]) mod wB.
-  Variable spec_w_add_carry :
-	 forall x y, [|w_add_carry x y|] = ([|x|] + [|y|] + 1) mod wB.
-
-  Variable spec_pred : forall x, [|w_pred x|] = ([|x|] - 1) mod wB.
-  Variable spec_sub : forall x y, [|w_sub x y|] = ([|x|] - [|y|]) mod wB.
-
-  Variable spec_mul_c : forall x y, [[ w_mul_c x y ]] = [|x|] * [|y|].
-  Variable spec_div21 : forall a1 a2 b,
-      wB/2 <= [|b|] ->
-      [|a1|] < [|b|] ->
-      let (q,r) := w_div21 a1 a2 b in
-      [|a1|] *wB+ [|a2|] = [|q|] * [|b|] + [|r|] /\
-      0 <= [|r|] < [|b|].
-
-  Variable spec_ww_sub_c : forall x y, [-[ww_sub_c x y]] = [[x]] - [[y]].
-
-  Ltac Spec_w_to_Z x :=
-   let H:= fresh "HH" in
-   assert (H:= spec_to_Z x).
-  Ltac Spec_ww_to_Z x :=
-   let H:= fresh "HH" in
-   assert (H:= spec_ww_to_Z w_digits w_to_Z spec_to_Z x).
-
-  Theorem wB_div2: forall x, wB/2  <= x -> wB <= 2 * x.
-   intros x H; rewrite <- wB_div_2; apply Z.mul_le_mono_nonneg_l; auto with zarith.
-  Qed.
-
-  Lemma Zmult_lt_0_reg_r_2 : forall n m : Z, 0 <= n -> 0 < m * n -> 0 < m.
-  Proof.
-   intros n m H1 H2;apply Z.mul_pos_cancel_r with n;trivial.
-   Z.le_elim H1; trivial.
-   subst;rewrite Z.mul_0_r in H2;discriminate H2.
-  Qed.
-
-  Theorem spec_w_div32 : forall a1 a2 a3 b1 b2,
-     wB/2 <= [|b1|] ->
-     [[WW a1 a2]] < [[WW b1 b2]] ->
-     let (q,r) := w_div32 a1 a2 a3 b1 b2 in
-     [|a1|] * wwB + [|a2|] * wB  + [|a3|] =
-        [|q|] *  ([|b1|] * wB + [|b2|])  + [[r]] /\
-     0 <= [[r]] < [|b1|] * wB + [|b2|].
-  Proof.
-   intros a1 a2 a3 b1 b2 Hle Hlt.
-   assert (U:= lt_0_wB w_digits); assert (U1:= lt_0_wwB w_digits).
-   Spec_w_to_Z a1;Spec_w_to_Z a2;Spec_w_to_Z a3;Spec_w_to_Z b1;Spec_w_to_Z b2.
-   rewrite wwB_wBwB; rewrite Z.pow_2_r; rewrite Z.mul_assoc;rewrite <- Z.mul_add_distr_r.
-   change (w_div32 a1 a2 a3 b1 b2) with (w_div32_body a1 a2 a3 b1 b2).
-   unfold w_div32_body.
-   rewrite spec_compare. case Z.compare_spec; intro Hcmp.
-   simpl in Hlt.
-   rewrite Hcmp in Hlt;assert ([|a2|] < [|b2|]). omega.
-   assert ([[WW (w_sub a2 b2) a3]] = ([|a2|]-[|b2|])*wB + [|a3|] + wwB).
-    simpl;rewrite spec_sub.
-    assert ([|a2|] - [|b2|] = wB*(-1) + ([|a2|] - [|b2|] + wB)). ring.
-    assert (0 <= [|a2|] - [|b2|] + wB < wB). omega.
-    rewrite <-(Zmod_unique ([|a2|]-[|b2|]) wB (-1) ([|a2|]-[|b2|]+wB) H1 H0).
-    rewrite wwB_wBwB;ring.
-   assert (U2 := wB_pos w_digits).
-   eapply spec_ww_add_c_cont with (P :=
-     fun (x y:zn2z w) (res:w*zn2z w) =>
-     let (q, r) := res in
-     ([|a1|] * wB + [|a2|]) * wB + [|a3|] =
-     [|q|] * ([|b1|] * wB + [|b2|]) + [[r]] /\
-     0 <= [[r]] < [|b1|] * wB + [|b2|]);eauto.
-   rewrite H0;intros r.
-   repeat
-    (rewrite spec_ww_add;eauto || rewrite spec_w_Bm1 || rewrite spec_w_Bm2);
-   simpl ww_to_Z;try rewrite Z.mul_1_l;intros H1.
-   assert (0<= ([[r]] + ([|b1|] * wB + [|b2|])) - wwB < [|b1|] * wB + [|b2|]).
-   Spec_ww_to_Z r;split;zarith.
-   rewrite H1.
-   assert (H12:= wB_div2 Hle). assert (wwB <= 2 * [|b1|] * wB).
-   rewrite wwB_wBwB; rewrite Z.pow_2_r; zarith.
-   assert (-wwB < ([|a2|] - [|b2|]) * wB + [|a3|] < 0).
-   split. apply Z.lt_le_trans with (([|a2|] - [|b2|]) * wB);zarith.
-   rewrite wwB_wBwB;replace (-(wB^2)) with (-wB*wB);[zarith | ring].
-   apply Z.mul_lt_mono_pos_r;zarith.
-   apply Z.le_lt_trans with (([|a2|] - [|b2|]) * wB + (wB -1));zarith.
-   replace ( ([|a2|] - [|b2|]) * wB + (wB - 1)) with
-     (([|a2|] - [|b2|] + 1) * wB + - 1);[zarith | ring].
-   assert (([|a2|] - [|b2|] + 1) * wB <= 0);zarith.
-   replace 0 with (0*wB);zarith.
-   replace (([|a2|] - [|b2|]) * wB + [|a3|] + wwB + ([|b1|] * wB + [|b2|]) +
-              ([|b1|] * wB + [|b2|]) - wwB) with
-         (([|a2|] - [|b2|]) * wB + [|a3|] + 2*[|b1|] * wB + 2*[|b2|]);
-   [zarith | ring].
-   rewrite <- (Zmod_unique ([[r]] + ([|b1|] * wB + [|b2|])) wwB
-             1 ([[r]] + ([|b1|] * wB + [|b2|]) - wwB));zarith;try (ring;fail).
-   split. rewrite H1;rewrite Hcmp;ring. trivial.
-   Spec_ww_to_Z (WW b1 b2). simpl in HH4;zarith.
-   rewrite H0;intros r;repeat
-    (rewrite spec_w_Bm1 || rewrite spec_w_Bm2);
-   simpl ww_to_Z;try rewrite Z.mul_1_l;intros H1.
-   assert ([[r]]=([|a2|]-[|b2|])*wB+[|a3|]+([|b1|]*wB+[|b2|])). zarith.
-   split. rewrite H2;rewrite Hcmp;ring.
-   split. Spec_ww_to_Z r;zarith.
-   rewrite H2.
-   assert (([|a2|] - [|b2|]) * wB + [|a3|] < 0);zarith.
-   apply Z.le_lt_trans with (([|a2|] - [|b2|]) * wB + (wB -1));zarith.
-   replace ( ([|a2|] - [|b2|]) * wB + (wB - 1)) with
-     (([|a2|] - [|b2|] + 1) * wB + - 1);[zarith|ring].
-   assert (([|a2|] - [|b2|] + 1) * wB <= 0);zarith.
-   replace 0 with (0*wB);zarith.
-   (* Cas Lt *)
-   assert (Hdiv21 := spec_div21 a2 Hle Hcmp);
-    destruct (w_div21 a1 a2 b1) as (q, r);destruct Hdiv21.
-   rewrite H.
-   assert (Hq := spec_to_Z q).
-   generalize
-    (spec_ww_sub_c (w_WW r a3) (w_mul_c q b2));
-    destruct (ww_sub_c (w_WW r a3) (w_mul_c q b2))
-    as [r1|r1];repeat (rewrite spec_w_WW || rewrite spec_mul_c);
-    unfold interp_carry;intros H1.
-    rewrite H1.
-   split. ring. split.
-   rewrite <- H1;destruct (spec_ww_to_Z w_digits w_to_Z spec_to_Z r1);trivial.
-   apply Z.le_lt_trans with ([|r|] * wB + [|a3|]).
-   assert ( 0 <= [|q|] * [|b2|]);zarith.
-   apply beta_lex_inv;zarith.
-   assert ([[r1]] = [|r|] * wB + [|a3|] - [|q|] * [|b2|] + wwB).
-   rewrite <- H1;ring.
-   Spec_ww_to_Z r1; assert (0 <= [|r|]*wB). zarith.
-   assert (0 < [|q|] * [|b2|]). zarith.
-   assert (0 < [|q|]).
-    apply Zmult_lt_0_reg_r_2 with [|b2|];zarith.
-   eapply spec_ww_add_c_cont with (P :=
-     fun (x y:zn2z w) (res:w*zn2z w) =>
-     let (q0, r0) := res in
-      ([|q|] * [|b1|] + [|r|]) * wB + [|a3|] =
-      [|q0|] * ([|b1|] * wB + [|b2|]) + [[r0]] /\
-      0 <= [[r0]] < [|b1|] * wB + [|b2|]);eauto.
-   intros r2;repeat (rewrite spec_pred || rewrite spec_ww_add;eauto);
-   simpl ww_to_Z;intros H7.
-   assert (0 < [|q|] - 1).
-   assert (H6 : 1 <= [|q|]) by zarith.
-   Z.le_elim H6;zarith.
-   rewrite <- H6 in H2;rewrite H2 in H7.
-   assert (0 < [|b1|]*wB). apply Z.mul_pos_pos;zarith.
-   Spec_ww_to_Z r2. zarith.
-   rewrite (Zmod_small ([|q|] -1));zarith.
-   rewrite (Zmod_small ([|q|] -1 -1));zarith.
-   assert ([[r2]] + ([|b1|] * wB + [|b2|]) =
-       wwB * 1 +
-       ([|r|] * wB + [|a3|] - [|q|] * [|b2|] + 2 * ([|b1|] * wB + [|b2|]))).
-   rewrite H7;rewrite H2;ring.
-   assert
-    ([|r|]*wB + [|a3|] - [|q|]*[|b2|] + 2 * ([|b1|]*wB + [|b2|])
-        < [|b1|]*wB + [|b2|]).
-   Spec_ww_to_Z r2;omega.
-   Spec_ww_to_Z (WW b1 b2). simpl in HH5.
-   assert
-    (0 <= [|r|]*wB + [|a3|] - [|q|]*[|b2|] + 2 * ([|b1|]*wB + [|b2|])
-       < wwB). split;try omega.
-   replace (2*([|b1|]*wB+[|b2|])) with ((2*[|b1|])*wB+2*[|b2|]). 2:ring.
-   assert (H12:= wB_div2 Hle). assert (wwB <= 2 * [|b1|] * wB).
-   rewrite wwB_wBwB; rewrite Z.pow_2_r; zarith. omega.
-   rewrite <- (Zmod_unique
-            ([[r2]] + ([|b1|] * wB + [|b2|]))
-            wwB
-            1
-            ([|r|] * wB + [|a3|] - [|q|] * [|b2|] + 2*([|b1|] * wB + [|b2|]))
-            H10 H8).
-   split. ring. zarith.
-   intros r2;repeat (rewrite spec_pred);simpl ww_to_Z;intros H7.
-   rewrite (Zmod_small ([|q|] -1));zarith.
-   split.
-   replace [[r2]] with  ([[r1]] + ([|b1|] * wB + [|b2|]) -wwB).
-   rewrite H2; ring. rewrite <- H7; ring.
-   Spec_ww_to_Z r2;Spec_ww_to_Z r1. omega.
-   simpl in Hlt.
-   assert ([|a1|] * wB + [|a2|] <= [|b1|] * wB + [|b2|]). zarith.
-   assert (H1 := beta_lex _ _ _ _ _ H HH0 HH3). rewrite spec_w_0;simpl;zarith.
-  Qed.
-
-
-End DoubleDiv32.
-
-Section DoubleDiv21.
- Variable w : Type.
- Variable w_0 : w.
-
- Variable w_0W : w -> zn2z w.
- Variable w_div32 : w -> w -> w -> w -> w -> w * zn2z w.
-
- Variable ww_1 : zn2z w.
- Variable ww_compare : zn2z w -> zn2z w -> comparison.
- Variable ww_sub : zn2z w -> zn2z w -> zn2z w.
-
-
- Definition ww_div21 a1 a2 b :=
-  match a1 with
-  | W0 =>
-    match ww_compare a2 b with
-    | Gt => (ww_1, ww_sub a2 b)
-    | Eq => (ww_1, W0)
-    | Lt => (W0, a2)
-    end
-  | WW a1h a1l =>
-    match a2 with
-    | W0 =>
-      match b with
-      | W0 => (W0,W0) (* cas absurde *)
-      | WW b1 b2 =>
-        let (q1, r)  := w_div32 a1h a1l w_0 b1 b2 in
-        match r with
-        | W0 => (WW q1 w_0, W0)
-        | WW r1 r2 =>
-	  let (q2, s)  := w_div32 r1 r2 w_0 b1 b2 in
-          (WW q1 q2, s)
-        end
-      end
-    | WW a2h a2l =>
-      match b with
-      | W0 => (W0,W0) (* cas absurde *)
-      | WW b1 b2 =>
-        let (q1, r)  := w_div32 a1h a1l a2h b1 b2 in
-        match r with
-        | W0 => (WW q1 w_0, w_0W a2l)
-        | WW r1 r2 =>
-	  let (q2, s)  := w_div32 r1 r2 a2l b1 b2 in
-          (WW q1 q2, s)
-        end
-      end
-    end
-  end.
-
- (* Proof *)
-
-  Variable w_digits : positive.
-  Variable w_to_Z : w -> Z.
-  Notation wB  := (base w_digits).
-  Notation wwB := (base (ww_digits w_digits)).
-  Notation "[| x |]" := (w_to_Z x)  (at level 0, x at level 99).
-  Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99).
-  Notation "[-[ c ]]" :=
-   (interp_carry (-1) wwB (ww_to_Z w_digits w_to_Z) c)
-   (at level 0, c at level 99).
-
-  Variable spec_w_0   : [|w_0|] = 0.
-  Variable spec_to_Z  : forall x, 0 <= [|x|] < wB.
-  Variable spec_w_0W : forall l, [[w_0W l]] = [|l|].
-  Variable spec_w_div32 : forall a1 a2 a3 b1 b2,
-     wB/2 <= [|b1|] ->
-     [[WW a1 a2]] < [[WW b1 b2]] ->
-     let (q,r) := w_div32 a1 a2 a3 b1 b2 in
-     [|a1|] * wwB + [|a2|] * wB  + [|a3|] =
-        [|q|] *  ([|b1|] * wB + [|b2|])  + [[r]] /\
-     0 <= [[r]] < [|b1|] * wB + [|b2|].
-  Variable spec_ww_1 : [[ww_1]] = 1.
-  Variable spec_ww_compare :  forall x y,
-    ww_compare x y = Z.compare [[x]] [[y]].
-  Variable spec_ww_sub : forall x y, [[ww_sub x y]] = ([[x]] - [[y]]) mod wwB.
-
-  Theorem wwB_div: wwB  = 2 * (wwB / 2).
-  Proof.
-   rewrite wwB_div_2; rewrite  Z.mul_assoc; rewrite  wB_div_2; auto.
-   rewrite <- Z.pow_2_r; apply wwB_wBwB.
-  Qed.
-
-  Ltac Spec_w_to_Z x :=
-   let H:= fresh "HH" in
-   assert (H:= spec_to_Z x).
-  Ltac Spec_ww_to_Z x :=
-   let H:= fresh "HH" in
-   assert (H:= spec_ww_to_Z w_digits w_to_Z spec_to_Z x).
-
-  Theorem   spec_ww_div21 : forall a1 a2 b,
-     wwB/2 <= [[b]] ->
-     [[a1]] < [[b]] ->
-     let (q,r) := ww_div21 a1 a2 b in
-     [[a1]] *wwB+[[a2]] = [[q]] *  [[b]] + [[r]] /\ 0 <= [[r]] < [[b]].
-  Proof.
-   assert (U:= lt_0_wB w_digits).
-   assert (U1:= lt_0_wwB w_digits).
-   intros a1 a2 b H Hlt; unfold ww_div21.
-   Spec_ww_to_Z b; assert (Eq: 0 < [[b]]).    Spec_ww_to_Z a1;omega.
-   generalize Hlt H ;clear Hlt H;case a1.
-   intros H1 H2;simpl in H1;Spec_ww_to_Z a2.
-   rewrite spec_ww_compare. case Z.compare_spec;
-   simpl;try rewrite spec_ww_1;autorewrite with rm10; intros;zarith.
-   rewrite spec_ww_sub;simpl. rewrite Zmod_small;zarith.
-   split. ring.
-   assert (wwB <= 2*[[b]]);zarith.
-   rewrite wwB_div;zarith.
-   intros a1h a1l.  Spec_w_to_Z a1h;Spec_w_to_Z a1l. Spec_ww_to_Z a2.
-   destruct a2 as [ |a3 a4];
-    (destruct b as [ |b1 b2];[unfold Z.le in Eq;discriminate Eq|idtac]);
-   try (Spec_w_to_Z a3; Spec_w_to_Z a4); Spec_w_to_Z b1; Spec_w_to_Z b2;
-   intros Hlt H;  match goal with |-context [w_div32 ?X ?Y ?Z ?T ?U] =>
-     generalize (@spec_w_div32 X Y Z T U); case (w_div32 X Y Z T U);
-     intros q1 r H0
-   end; (assert (Eq1: wB / 2 <= [|b1|]);[
-    apply (@beta_lex (wB / 2) 0 [|b1|] [|b2|] wB); auto with zarith;
-    autorewrite with rm10;repeat rewrite (Z.mul_comm wB);
-    rewrite <- wwB_div_2; trivial
-   | generalize (H0 Eq1 Hlt);clear H0;destruct r as [ |r1 r2];simpl;
-     try rewrite spec_w_0; try rewrite spec_w_0W;repeat rewrite Z.add_0_r;
-     intros (H1,H2) ]).
-   split;[rewrite wwB_wBwB; rewrite Z.pow_2_r | trivial].
-   rewrite Z.mul_assoc;rewrite Z.mul_add_distr_r;rewrite <- Z.mul_assoc;
-   rewrite <- Z.pow_2_r; rewrite <- wwB_wBwB;rewrite H1;ring.
-   destruct H2 as (H2,H3);match goal with |-context [w_div32 ?X ?Y ?Z ?T ?U] =>
-     generalize (@spec_w_div32 X Y Z T U); case (w_div32 X Y Z T U);
-     intros q r H0;generalize (H0 Eq1 H3);clear H0;intros (H4,H5) end.
-   split;[rewrite wwB_wBwB | trivial].
-   rewrite Z.pow_2_r.
-   rewrite Z.mul_assoc;rewrite Z.mul_add_distr_r;rewrite <- Z.mul_assoc;
-   rewrite <- Z.pow_2_r.
-   rewrite <- wwB_wBwB;rewrite H1.
-   rewrite spec_w_0 in H4;rewrite Z.add_0_r in H4.
-   repeat rewrite Z.mul_add_distr_r. rewrite <- (Z.mul_assoc [|r1|]).
-   rewrite <- Z.pow_2_r; rewrite <- wwB_wBwB;rewrite H4;simpl;ring.
-   split;[rewrite wwB_wBwB | split;zarith].
-   replace (([|a1h|] * wB + [|a1l|]) * wB^2 + ([|a3|] * wB + [|a4|]))
-   with (([|a1h|] * wwB + [|a1l|] * wB + [|a3|])*wB+ [|a4|]).
-   rewrite H1;ring. rewrite wwB_wBwB;ring.
-   change [|a4|] with (0*wB+[|a4|]);apply beta_lex_inv;zarith.
-   assert (1 <= wB/2);zarith.
-    assert (H_:= wB_pos w_digits);apply Zdiv_le_lower_bound;zarith.
-   destruct H2 as (H2,H3);match goal with |-context [w_div32 ?X ?Y ?Z ?T ?U] =>
-     generalize (@spec_w_div32 X Y Z T U); case (w_div32 X Y Z T U);
-     intros q r H0;generalize (H0 Eq1 H3);clear H0;intros (H4,H5) end.
-   split;trivial.
-   replace (([|a1h|] * wB + [|a1l|]) * wwB + ([|a3|] * wB + [|a4|])) with
-   (([|a1h|] * wwB + [|a1l|] * wB + [|a3|])*wB + [|a4|]);
-   [rewrite H1 | rewrite wwB_wBwB;ring].
-   replace (([|q1|]*([|b1|]*wB+[|b2|])+([|r1|]*wB+[|r2|]))*wB+[|a4|]) with
-   (([|q1|]*([|b1|]*wB+[|b2|]))*wB+([|r1|]*wwB+[|r2|]*wB+[|a4|]));
-   [rewrite H4;simpl|rewrite wwB_wBwB];ring.
-  Qed.
-
-End DoubleDiv21.
-
-Section DoubleDivGt.
- Variable w : Type.
- Variable w_digits : positive.
- Variable w_0 : w.
-
- Variable w_WW : w -> w -> zn2z w.
- Variable w_0W : w -> zn2z w.
- Variable w_compare : w -> w -> comparison.
- Variable w_eq0 : w -> bool.
- Variable w_opp_c : w -> carry w.
- Variable w_opp w_opp_carry : w -> w.
- Variable w_sub_c : w -> w -> carry w.
- Variable w_sub w_sub_carry : w -> w -> w.
-
- Variable w_div_gt : w -> w -> w*w.
- Variable w_mod_gt : w -> w -> w.
- Variable w_gcd_gt : w -> w -> w.
- Variable w_add_mul_div : w -> w -> w -> w.
- Variable w_head0 : w -> w.
- Variable w_div21 : w -> w -> w -> w * w.
- Variable w_div32 : w -> w -> w -> w -> w -> w * zn2z w.
-
-
- Variable _ww_zdigits : zn2z w.
- Variable ww_1 : zn2z w.
- Variable ww_add_mul_div : zn2z w -> zn2z w -> zn2z w -> zn2z w.
-
- Variable w_zdigits : w.
-
- Definition ww_div_gt_aux ah al bh bl :=
-  Eval lazy beta iota delta [ww_sub ww_opp] in
-    let p := w_head0 bh in
-    match w_compare p w_0 with
-    | Gt =>
-      let b1 := w_add_mul_div p bh bl in
-      let b2 := w_add_mul_div p bl w_0 in
-      let a1 := w_add_mul_div p w_0 ah in
-      let a2 := w_add_mul_div p ah al in
-      let a3 := w_add_mul_div p al w_0 in
-      let (q,r) := w_div32 a1 a2 a3 b1 b2 in
-      (WW w_0 q, ww_add_mul_div
-        (ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c
-            w_opp w_sub w_sub_carry _ww_zdigits (w_0W p)) W0 r)
-    | _ => (ww_1, ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c
-            w_opp w_sub w_sub_carry (WW ah al) (WW bh bl))
-    end.
-
- Definition ww_div_gt a b :=
-  Eval lazy beta iota delta [ww_div_gt_aux double_divn1
-  double_divn1_p double_divn1_p_aux double_divn1_0 double_divn1_0_aux
-  double_split double_0 double_WW] in
-  match a, b with
-  | W0, _ => (W0,W0)
-  | _, W0 => (W0,W0)
-  | WW ah al, WW bh bl =>
-    if w_eq0 ah then
-     let (q,r) := w_div_gt al bl in
-     (WW w_0 q, w_0W r)
-    else
-     match w_compare w_0 bh with
-     | Eq =>
-       let(q,r):=
-        double_divn1 w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21
-                  w_compare w_sub 1 a bl in
-       (q, w_0W r)
-     | Lt => ww_div_gt_aux ah al bh bl
-     | Gt => (W0,W0) (* cas absurde *)
-     end
-  end.
-
- Definition ww_mod_gt_aux ah al bh bl :=
-  Eval lazy beta iota delta [ww_sub ww_opp] in
-    let p := w_head0 bh in
-    match w_compare p w_0 with
-    | Gt =>
-      let b1 := w_add_mul_div p bh bl in
-      let b2 := w_add_mul_div p bl w_0 in
-      let a1 := w_add_mul_div p w_0 ah in
-      let a2 := w_add_mul_div p ah al in
-      let a3 := w_add_mul_div p al w_0 in
-      let (q,r) := w_div32 a1 a2 a3 b1 b2 in
-      ww_add_mul_div (ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c
-         w_opp w_sub w_sub_carry _ww_zdigits (w_0W p)) W0 r
-    | _ =>
-      ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c
-         w_opp w_sub w_sub_carry (WW ah al) (WW bh bl)
-    end.
-
- Definition ww_mod_gt a b :=
-  Eval lazy beta iota delta [ww_mod_gt_aux double_modn1
-  double_modn1_p double_modn1_p_aux double_modn1_0 double_modn1_0_aux
-  double_split double_0 double_WW snd] in
-  match a, b with
-  | W0, _ => W0
-  | _, W0 => W0
-  | WW ah al, WW bh bl =>
-    if w_eq0 ah then w_0W (w_mod_gt al bl)
-    else
-     match w_compare w_0 bh with
-     | Eq =>
-       w_0W (double_modn1 w_zdigits w_0 w_head0 w_add_mul_div w_div21
-             w_compare w_sub 1 a bl)
-     | Lt => ww_mod_gt_aux ah al bh bl
-     | Gt => W0 (* cas absurde *)
-     end
-  end.
-
-  Definition ww_gcd_gt_body (cont: w->w->w->w->zn2z w) (ah al bh bl: w) :=
-   Eval lazy beta iota delta [ww_mod_gt_aux double_modn1
-   double_modn1_p double_modn1_p_aux double_modn1_0 double_modn1_0_aux
-   double_split double_0 double_WW snd] in
-    match w_compare w_0 bh with
-    | Eq =>
-      match w_compare w_0 bl with
-      | Eq => WW ah al (* normalement n'arrive pas si forme normale *)
-      | Lt =>
-	let m := double_modn1 w_zdigits w_0 w_head0 w_add_mul_div w_div21
-                   w_compare w_sub 1 (WW ah al) bl in
-        WW w_0 (w_gcd_gt bl m)
-      | Gt => W0 (* absurde *)
-      end
-    | Lt =>
-      let m := ww_mod_gt_aux ah al bh bl in
-      match m with
-      | W0 => WW bh bl
-      | WW mh ml =>
-        match w_compare w_0 mh with
-        | Eq =>
-          match w_compare w_0 ml with
-          | Eq => WW bh bl
-          | _  =>
-            let r := double_modn1 w_zdigits w_0 w_head0 w_add_mul_div w_div21
-                      w_compare w_sub 1 (WW bh bl) ml in
-            WW w_0 (w_gcd_gt ml r)
-          end
-        | Lt =>
-          let r := ww_mod_gt_aux bh bl mh ml in
-          match r with
-          | W0 => m
-          | WW rh rl => cont mh ml rh rl
-          end
-        | Gt => W0 (* absurde *)
-        end
-      end
-    | Gt => W0 (* absurde *)
-    end.
-
-  Fixpoint ww_gcd_gt_aux
-     (p:positive) (cont: w -> w -> w -> w -> zn2z w) (ah al bh bl : w)
-        {struct p} : zn2z w :=
-    ww_gcd_gt_body
-       (fun mh ml rh rl => match p with
-        | xH => cont mh ml rh rl
-        | xO p => ww_gcd_gt_aux p (ww_gcd_gt_aux p cont) mh ml rh rl
-        | xI p => ww_gcd_gt_aux p (ww_gcd_gt_aux p cont) mh ml rh rl
-        end) ah al bh bl.
-
-
-  (* Proof *)
-
-  Variable w_to_Z : w -> Z.
-  Notation wB  := (base w_digits).
-  Notation wwB := (base (ww_digits w_digits)).
-  Notation "[| x |]" := (w_to_Z x)  (at level 0, x at level 99).
-  Notation "[-| c |]" :=
-   (interp_carry (-1) wB w_to_Z c) (at level 0, c at level 99).
-
-  Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99).
-
-  Variable spec_w_0   : [|w_0|] = 0.
-  Variable spec_to_Z  : forall x, 0 <= [|x|] < wB.
-  Variable spec_to_w_Z  : forall x, 0 <= [[x]] < wwB.
-
-  Variable spec_w_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|].
-  Variable spec_w_0W : forall l, [[w_0W l]] = [|l|].
-  Variable spec_compare :
-     forall x y, w_compare x y = Z.compare [|x|] [|y|].
-  Variable spec_eq0 : forall x, w_eq0 x = true -> [|x|] = 0.
-
-  Variable spec_opp_c : forall x, [-|w_opp_c x|] = -[|x|].
-  Variable spec_opp : forall x, [|w_opp x|] = (-[|x|]) mod wB.
-  Variable spec_opp_carry : forall x, [|w_opp_carry x|] = wB - [|x|] - 1.
-
-  Variable spec_sub_c : forall x y, [-|w_sub_c x y|] = [|x|] - [|y|].
-  Variable spec_sub : forall x y, [|w_sub x y|] = ([|x|] - [|y|]) mod wB.
-  Variable spec_sub_carry :
-	 forall x y, [|w_sub_carry x y|] = ([|x|] - [|y|] - 1) mod wB.
-
-  Variable spec_div_gt : forall a b, [|a|] > [|b|] -> 0 < [|b|] ->
-      let (q,r) := w_div_gt a b in
-      [|a|] = [|q|] * [|b|] + [|r|] /\
-      0 <= [|r|] < [|b|].
-  Variable spec_mod_gt : forall a b, [|a|] > [|b|] -> 0 < [|b|] ->
-      [|w_mod_gt a b|] = [|a|] mod [|b|].
-  Variable spec_gcd_gt : forall a b, [|a|] > [|b|] ->
-      Zis_gcd [|a|] [|b|] [|w_gcd_gt a b|].
-
-  Variable spec_add_mul_div : forall x y p,
-       [|p|] <= Zpos w_digits ->
-       [| w_add_mul_div p x y |] =
-         ([|x|] * (2 ^ ([|p|])) +
-          [|y|] / (2 ^ ((Zpos w_digits) - [|p|]))) mod wB.
-  Variable spec_head0  : forall x,  0 < [|x|] ->
-	 wB/ 2 <= 2 ^ [|w_head0 x|] * [|x|] < wB.
-
-  Variable spec_div21 : forall a1 a2 b,
-      wB/2 <= [|b|] ->
-      [|a1|] < [|b|] ->
-      let (q,r) := w_div21 a1 a2 b in
-      [|a1|] *wB+ [|a2|] = [|q|] * [|b|] + [|r|] /\
-      0 <= [|r|] < [|b|].
-
-  Variable spec_w_div32 : forall a1 a2 a3 b1 b2,
-     wB/2 <= [|b1|] ->
-     [[WW a1 a2]] < [[WW b1 b2]] ->
-     let (q,r) := w_div32 a1 a2 a3 b1 b2 in
-     [|a1|] * wwB + [|a2|] * wB  + [|a3|] =
-        [|q|] *  ([|b1|] * wB + [|b2|])  + [[r]] /\
-     0 <= [[r]] < [|b1|] * wB + [|b2|].
-
-  Variable spec_w_zdigits: [|w_zdigits|] = Zpos w_digits.
-
-  Variable spec_ww_digits_ : [[_ww_zdigits]] = Zpos (xO w_digits).
-  Variable spec_ww_1 : [[ww_1]] = 1.
-  Variable spec_ww_add_mul_div : forall x y p,
-       [[p]] <= Zpos (xO w_digits) ->
-       [[ ww_add_mul_div p x y ]] =
-         ([[x]] * (2^[[p]]) +
-          [[y]] / (2^(Zpos (xO w_digits) - [[p]]))) mod wwB.
-
-  Ltac Spec_w_to_Z x :=
-   let H:= fresh "HH" in
-   assert (H:= spec_to_Z x).
-
-  Ltac Spec_ww_to_Z x :=
-   let H:= fresh "HH" in
-   assert (H:= spec_ww_to_Z w_digits w_to_Z spec_to_Z x).
-
-  Lemma to_Z_div_minus_p : forall x p,
-   0 < [|p|] < Zpos w_digits  ->
-   0 <= [|x|] / 2 ^ (Zpos w_digits - [|p|]) < 2 ^ [|p|].
-  Proof.
-   intros x p H;Spec_w_to_Z x.
-   split. apply Zdiv_le_lower_bound;zarith.
-   apply Zdiv_lt_upper_bound;zarith.
-   rewrite <- Zpower_exp;zarith.
-   ring_simplify ([|p|] + (Zpos w_digits - [|p|])); unfold base in HH;zarith.
-  Qed.
-  Hint Resolve to_Z_div_minus_p : zarith.
-
-  Lemma spec_ww_div_gt_aux : forall ah al bh bl,
-      [[WW ah al]] > [[WW bh bl]] ->
-      0 < [|bh|] ->
-      let (q,r) := ww_div_gt_aux ah al bh bl in
-      [[WW ah al]] = [[q]] * [[WW bh bl]] + [[r]] /\
-      0 <= [[r]] < [[WW bh bl]].
-  Proof.
-   intros ah al bh bl Hgt Hpos;unfold ww_div_gt_aux.
-   change
-   (let (q, r) := let p := w_head0 bh in
-    match w_compare p w_0 with
-    | Gt =>
-      let b1 := w_add_mul_div p bh bl in
-      let b2 := w_add_mul_div p bl w_0 in
-      let a1 := w_add_mul_div p w_0 ah in
-      let a2 := w_add_mul_div p ah al in
-      let a3 := w_add_mul_div p al w_0 in
-      let (q,r) := w_div32 a1 a2 a3 b1 b2 in
-      (WW w_0 q, ww_add_mul_div
-        (ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c
-            w_opp w_sub w_sub_carry _ww_zdigits (w_0W p)) W0 r)
-    | _ => (ww_1, ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c
-            w_opp w_sub w_sub_carry (WW ah al) (WW bh bl))
-    end in [[WW ah al]]=[[q]]*[[WW bh bl]]+[[r]] /\ 0 <=[[r]]< [[WW bh bl]]).
-   assert (Hh := spec_head0 Hpos).
-   lazy zeta.
-   rewrite spec_compare; case Z.compare_spec;
-    rewrite spec_w_0; intros HH.
-   generalize Hh; rewrite HH; simpl Z.pow;
-        rewrite Z.mul_1_l; intros (HH1, HH2); clear HH.
-   assert (wwB <= 2*[[WW bh bl]]).
-    apply Z.le_trans with (2*[|bh|]*wB).
-    rewrite wwB_wBwB; rewrite Z.pow_2_r; apply Z.mul_le_mono_nonneg_r; zarith.
-    rewrite <- wB_div_2; apply Z.mul_le_mono_nonneg_l; zarith.
-    simpl ww_to_Z;rewrite Z.mul_add_distr_l;rewrite Z.mul_assoc.
-    Spec_w_to_Z bl;zarith.
-   Spec_ww_to_Z (WW ah al).
-   rewrite spec_ww_sub;eauto.
-   simpl;rewrite spec_ww_1;rewrite Z.mul_1_l;simpl.
-   simpl ww_to_Z in Hgt, H, HH;rewrite Zmod_small;split;zarith.
-   case (spec_to_Z (w_head0 bh)); auto with zarith.
-   assert ([|w_head0 bh|] < Zpos w_digits).
-    destruct (Z_lt_ge_dec [|w_head0 bh|]  (Zpos w_digits));trivial.
-    exfalso.
-    assert (2 ^ [|w_head0 bh|] * [|bh|] >= wB);auto with zarith.
-    apply Z.le_ge; replace wB with (wB * 1);try ring.
-    Spec_w_to_Z bh;apply Z.mul_le_mono_nonneg;zarith.
-    unfold base;apply Zpower_le_monotone;zarith.
-   assert (HHHH : 0 < [|w_head0 bh|] < Zpos w_digits); auto with zarith.
-   assert (Hb:= Z.lt_le_incl _ _ H).
-   generalize (spec_add_mul_div w_0 ah Hb)
-    (spec_add_mul_div ah al Hb)
-    (spec_add_mul_div al w_0 Hb)
-    (spec_add_mul_div bh bl Hb)
-    (spec_add_mul_div bl w_0  Hb);
-   rewrite spec_w_0; repeat rewrite Z.mul_0_l;repeat rewrite Z.add_0_l;
-   rewrite Zdiv_0_l;repeat rewrite Z.add_0_r.
-   Spec_w_to_Z ah;Spec_w_to_Z bh.
-   unfold base;repeat rewrite Zmod_shift_r;zarith.
-   assert (H3:=to_Z_div_minus_p ah HHHH);assert(H4:=to_Z_div_minus_p al HHHH);
-   assert (H5:=to_Z_div_minus_p bl HHHH).
-   rewrite Z.mul_comm in Hh.
-   assert (2^[|w_head0 bh|] < wB). unfold base;apply Zpower_lt_monotone;zarith.
-   unfold base in H0;rewrite Zmod_small;zarith.
-   fold wB; rewrite (Zmod_small ([|bh|] * 2 ^ [|w_head0 bh|]));zarith.
-   intros U1 U2 U3 V1 V2.
-   generalize (@spec_w_div32 (w_add_mul_div (w_head0 bh) w_0 ah)
-               (w_add_mul_div (w_head0 bh) ah al)
-               (w_add_mul_div (w_head0 bh) al w_0)
-               (w_add_mul_div (w_head0 bh) bh bl)
-               (w_add_mul_div (w_head0 bh) bl w_0)).
-   destruct (w_div32 (w_add_mul_div (w_head0 bh) w_0 ah)
-              (w_add_mul_div (w_head0 bh) ah al)
-              (w_add_mul_div (w_head0 bh) al w_0)
-              (w_add_mul_div (w_head0 bh) bh bl)
-              (w_add_mul_div (w_head0 bh) bl w_0)) as (q,r).
-   rewrite V1;rewrite V2. rewrite Z.mul_add_distr_r.
-   rewrite <- (Z.add_assoc ([|bh|] * 2 ^ [|w_head0 bh|] * wB)).
-   unfold base;rewrite <- shift_unshift_mod;zarith. fold wB.
-   replace ([|bh|] * 2 ^ [|w_head0 bh|] * wB + [|bl|] * 2 ^ [|w_head0 bh|]) with
-    ([[WW bh bl]] * 2^[|w_head0 bh|]). 2:simpl;ring.
-   fold wwB. rewrite wwB_wBwB. rewrite Z.pow_2_r. rewrite U1;rewrite U2;rewrite U3.
-   rewrite Z.mul_assoc. rewrite Z.mul_add_distr_r.
-   rewrite (Z.add_assoc ([|ah|] / 2^(Zpos(w_digits) - [|w_head0 bh|])*wB * wB)).
-   rewrite <- Z.mul_add_distr_r.  rewrite <- Z.add_assoc.
-   unfold base;repeat rewrite <- shift_unshift_mod;zarith. fold wB.
-   replace ([|ah|] * 2 ^ [|w_head0 bh|] * wB + [|al|] * 2 ^ [|w_head0 bh|]) with
-    ([[WW ah al]] * 2^[|w_head0 bh|]). 2:simpl;ring.
-   intros Hd;destruct Hd;zarith.
-   simpl. apply beta_lex_inv;zarith. rewrite U1;rewrite V1.
-   assert ([|ah|] / 2 ^ (Zpos (w_digits) - [|w_head0 bh|]) < wB/2);zarith.
-   apply Zdiv_lt_upper_bound;zarith.
-   unfold base.
-   replace (2^Zpos (w_digits)) with (2^(Zpos (w_digits) - 1)*2).
-   rewrite Z_div_mult;zarith.  rewrite <- Zpower_exp;zarith.
-   apply Z.lt_le_trans with wB;zarith.
-   unfold base;apply Zpower_le_monotone;zarith.
-   pattern 2 at 2;replace 2 with (2^1);trivial.
-   rewrite <- Zpower_exp;zarith. ring_simplify (Zpos (w_digits) - 1 + 1);trivial.
-   change [[WW w_0 q]] with ([|w_0|]*wB+[|q|]);rewrite spec_w_0;rewrite
-   Z.mul_0_l;rewrite Z.add_0_l.
-   replace [[ww_add_mul_div (ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c w_opp w_sub w_sub_carry
-       _ww_zdigits (w_0W (w_head0 bh))) W0 r]] with ([[r]]/2^[|w_head0 bh|]).
-   assert (0 < 2^[|w_head0 bh|]). apply Z.pow_pos_nonneg;zarith.
-   split.
-   rewrite <- (Z_div_mult [[WW ah al]] (2^[|w_head0 bh|]));zarith.
-   rewrite H1;rewrite Z.mul_assoc;apply Z_div_plus_l;trivial.
-   split;[apply Zdiv_le_lower_bound| apply Zdiv_lt_upper_bound];zarith.
-   rewrite spec_ww_add_mul_div.
-   rewrite spec_ww_sub; auto with zarith.
-   rewrite spec_ww_digits_.
-   change (Zpos (xO (w_digits))) with (2*Zpos (w_digits));zarith.
-   simpl ww_to_Z;rewrite Z.mul_0_l;rewrite Z.add_0_l.
-   rewrite spec_w_0W.
-   rewrite (fun x y => Zmod_small (x-y)); auto with zarith.
-   ring_simplify (2 * Zpos w_digits - (2 * Zpos w_digits - [|w_head0 bh|])).
-   rewrite Zmod_small;zarith.
-   split;[apply Zdiv_le_lower_bound| apply Zdiv_lt_upper_bound];zarith.
-   Spec_ww_to_Z r.
-   apply Z.lt_le_trans with wwB;zarith.
-   rewrite <- (Z.mul_1_r wwB);apply Z.mul_le_mono_nonneg;zarith.
-   split; auto with zarith.
-   apply Z.le_lt_trans with (2 * Zpos w_digits); auto with zarith.
-   unfold base, ww_digits; rewrite (Pos2Z.inj_xO w_digits).
-   apply Zpower2_lt_lin; auto with zarith.
-   rewrite spec_ww_sub; auto with zarith.
-   rewrite spec_ww_digits_; rewrite spec_w_0W.
-   rewrite Zmod_small;zarith.
-   rewrite Pos2Z.inj_xO; split; auto with zarith.
-   apply Z.le_lt_trans with (2 * Zpos w_digits); auto with zarith.
-   unfold base, ww_digits; rewrite (Pos2Z.inj_xO w_digits).
-   apply Zpower2_lt_lin; auto with zarith.
-   Qed.
-
-  Lemma spec_ww_div_gt : forall a b, [[a]] > [[b]] -> 0 < [[b]] ->
-      let (q,r) := ww_div_gt a b in
-      [[a]] = [[q]] * [[b]] + [[r]] /\
-      0 <= [[r]] < [[b]].
-  Proof.
-   intros a b Hgt Hpos;unfold ww_div_gt.
-   change (let (q,r) :=   match a, b with
-  | W0, _ => (W0,W0)
-  | _, W0 => (W0,W0)
-  | WW ah al, WW bh bl =>
-    if w_eq0 ah then
-     let (q,r) := w_div_gt al bl in
-     (WW w_0 q, w_0W r)
-    else
-     match w_compare w_0 bh with
-     | Eq =>
-       let(q,r):=
-        double_divn1 w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21
-                  w_compare w_sub 1 a bl in
-       (q, w_0W r)
-     | Lt => ww_div_gt_aux ah al bh bl
-     | Gt => (W0,W0) (* cas absurde *)
-     end
-  end in [[a]] = [[q]] * [[b]] + [[r]] /\ 0 <= [[r]] < [[b]]).
-   destruct a as [ |ah al]. simpl in Hgt;omega.
-   destruct b as [ |bh bl]. simpl in Hpos;omega.
-   Spec_w_to_Z ah; Spec_w_to_Z al; Spec_w_to_Z bh; Spec_w_to_Z bl.
-   assert (H:=@spec_eq0 ah);destruct (w_eq0 ah).
-   simpl ww_to_Z;rewrite H;trivial. simpl in Hgt;rewrite H in Hgt;trivial.
-   assert ([|bh|] <= 0).
-   apply beta_lex with (d:=[|al|])(b:=[|bl|]) (beta := wB);zarith.
-   assert ([|bh|] = 0);zarith. rewrite H1 in Hgt;rewrite H1;simpl in Hgt.
-   simpl. simpl in Hpos;rewrite H1 in Hpos;simpl in Hpos.
-   assert (H2:=spec_div_gt Hgt Hpos);destruct (w_div_gt al bl).
-   repeat rewrite spec_w_0W;simpl;rewrite spec_w_0;simpl;trivial.
-   clear H.
-   rewrite spec_compare; case Z.compare_spec; intros Hcmp.
-   rewrite spec_w_0 in Hcmp. change [[WW bh bl]] with ([|bh|]*wB+[|bl|]).
-   rewrite <- Hcmp;rewrite Z.mul_0_l;rewrite Z.add_0_l.
-   simpl in Hpos;rewrite <- Hcmp in Hpos;simpl in Hpos.
-   assert (H2:= @spec_double_divn1 w w_digits w_zdigits w_0 w_WW w_head0 w_add_mul_div
-    w_div21 w_compare w_sub w_to_Z spec_to_Z spec_w_zdigits spec_w_0 spec_w_WW spec_head0
-    spec_add_mul_div spec_div21 spec_compare spec_sub 1 (WW ah al) bl Hpos).
-   destruct (double_divn1 w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21
-              w_compare w_sub 1
-             (WW ah al) bl).
-   rewrite spec_w_0W;unfold ww_to_Z;trivial.
-   apply spec_ww_div_gt_aux;trivial. rewrite spec_w_0 in Hcmp;trivial.
-   rewrite spec_w_0 in Hcmp;exfalso;omega.
-  Qed.
-
-  Lemma spec_ww_mod_gt_aux_eq : forall ah al bh bl,
-   ww_mod_gt_aux ah al bh bl = snd (ww_div_gt_aux ah al bh bl).
-  Proof.
-   intros ah al bh bl. unfold ww_mod_gt_aux, ww_div_gt_aux.
-   case w_compare; auto.
-   case w_div32; auto.
-  Qed.
-
-  Lemma spec_ww_mod_gt_aux : forall ah al bh bl,
-   [[WW ah al]] > [[WW bh bl]] ->
-    0 < [|bh|] ->
-    [[ww_mod_gt_aux ah al bh bl]] = [[WW ah al]] mod [[WW bh bl]].
-  Proof.
-   intros. rewrite spec_ww_mod_gt_aux_eq;trivial.
-   assert (H3 := spec_ww_div_gt_aux ah al bl H H0).
-   destruct (ww_div_gt_aux ah al bh bl) as (q,r);simpl. simpl in H,H3.
-   destruct H3;apply Zmod_unique with [[q]];zarith.
-   rewrite H1;ring.
-  Qed.
-
-  Lemma spec_w_mod_gt_eq : forall a b, [|a|] > [|b|] -> 0 <[|b|] ->
-    [|w_mod_gt a b|] = [|snd (w_div_gt a b)|].
-  Proof.
-   intros a b Hgt Hpos.
-   rewrite spec_mod_gt;trivial.
-   assert (H:=spec_div_gt Hgt Hpos).
-   destruct (w_div_gt a b) as (q,r);simpl.
-   rewrite Z.mul_comm in H;destruct H.
-   symmetry;apply Zmod_unique with [|q|];trivial.
-  Qed.
-
-  Lemma spec_ww_mod_gt_eq : forall a b, [[a]] > [[b]] -> 0 < [[b]] ->
-      [[ww_mod_gt a b]] = [[snd (ww_div_gt a b)]].
-  Proof.
-   intros a b Hgt Hpos.
-   change (ww_mod_gt a b) with
-   (match a, b with
-  | W0, _ => W0
-  | _, W0 => W0
-  | WW ah al, WW bh bl =>
-    if w_eq0 ah then w_0W (w_mod_gt al bl)
-    else
-     match w_compare w_0 bh with
-     | Eq =>
-       w_0W (double_modn1 w_zdigits w_0 w_head0 w_add_mul_div w_div21
-             w_compare w_sub 1 a bl)
-     | Lt => ww_mod_gt_aux ah al bh bl
-     | Gt => W0 (* cas absurde *)
-     end end).
-   change (ww_div_gt a b) with
-   (match a, b with
-  | W0, _ => (W0,W0)
-  | _, W0 => (W0,W0)
-  | WW ah al, WW bh bl =>
-    if w_eq0 ah then
-     let (q,r) := w_div_gt al bl in
-     (WW w_0 q, w_0W r)
-    else
-     match w_compare w_0 bh with
-     | Eq =>
-       let(q,r):=
-        double_divn1 w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21
-                  w_compare w_sub 1 a bl in
-       (q, w_0W r)
-     | Lt => ww_div_gt_aux ah al bh bl
-     | Gt => (W0,W0) (* cas absurde *)
-     end
-  end).
-   destruct a as [ |ah al];trivial.
-   destruct b as [ |bh bl];trivial.
-   Spec_w_to_Z ah; Spec_w_to_Z al; Spec_w_to_Z bh; Spec_w_to_Z bl.
-   assert (H:=@spec_eq0 ah);destruct (w_eq0 ah).
-   simpl in Hgt;rewrite H in Hgt;trivial.
-   assert ([|bh|] <= 0).
-   apply beta_lex with (d:=[|al|])(b:=[|bl|]) (beta := wB);zarith.
-   assert ([|bh|] = 0);zarith. rewrite H1 in Hgt;simpl in Hgt.
-   simpl in Hpos;rewrite H1 in Hpos;simpl in Hpos.
-   rewrite spec_w_0W;rewrite spec_w_mod_gt_eq;trivial.
-   destruct (w_div_gt al bl);simpl;rewrite spec_w_0W;trivial.
-   clear H.
-   rewrite spec_compare; case Z.compare_spec; intros H2.
-   rewrite (@spec_double_modn1_aux w w_zdigits w_0 w_WW w_head0 w_add_mul_div
-            w_div21 w_compare w_sub w_to_Z spec_w_0 spec_compare 1 (WW ah al) bl).
-   destruct (double_divn1 w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21 w_compare w_sub 1
-             (WW ah al) bl);simpl;trivial.
-   rewrite spec_ww_mod_gt_aux_eq;trivial;symmetry;trivial.
-   trivial.
-  Qed.
-
-  Lemma spec_ww_mod_gt : forall a b, [[a]] > [[b]] -> 0 < [[b]] ->
-      [[ww_mod_gt a b]] = [[a]] mod [[b]].
-  Proof.
-   intros a b Hgt Hpos.
-   assert (H:= spec_ww_div_gt a b Hgt Hpos).
-   rewrite (spec_ww_mod_gt_eq a b Hgt Hpos).
-   destruct (ww_div_gt a b)as(q,r);destruct H.
-   apply Zmod_unique with[[q]];simpl;trivial.
-   rewrite Z.mul_comm;trivial.
-  Qed.
-
-  Lemma Zis_gcd_mod : forall a b d,
-   0 < b -> Zis_gcd b (a mod b) d -> Zis_gcd a b d.
-  Proof.
-   intros a b d H H1; apply Zis_gcd_for_euclid with (a/b).
-   pattern a at 1;rewrite (Z_div_mod_eq a b).
-   ring_simplify (b * (a / b) + a mod b - a / b * b);trivial. zarith.
-  Qed.
-
-  Lemma spec_ww_gcd_gt_aux_body :
-   forall ah al bh bl n cont,
-    [[WW bh bl]] <= 2^n ->
-    [[WW ah al]] > [[WW bh bl]] ->
-    (forall xh xl yh yl,
-     [[WW xh xl]] > [[WW yh yl]] -> [[WW yh yl]] <= 2^(n-1) ->
-     Zis_gcd [[WW xh xl]] [[WW yh yl]] [[cont xh xl yh yl]]) ->
-    Zis_gcd [[WW ah al]] [[WW bh bl]] [[ww_gcd_gt_body cont ah al bh bl]].
-  Proof.
-   intros ah al bh bl n cont Hlog Hgt Hcont.
-   change (ww_gcd_gt_body cont ah al bh bl) with (match w_compare w_0 bh with
-    | Eq =>
-      match w_compare w_0 bl with
-      | Eq => WW ah al (* normalement n'arrive pas si forme normale *)
-      | Lt =>
-	let m := double_modn1 w_zdigits w_0 w_head0 w_add_mul_div w_div21
-                   w_compare w_sub 1 (WW ah al) bl in
-        WW w_0 (w_gcd_gt bl m)
-      | Gt => W0 (* absurde *)
-      end
-    | Lt =>
-      let m := ww_mod_gt_aux ah al bh bl in
-      match m with
-      | W0 => WW bh bl
-      | WW mh ml =>
-        match w_compare w_0 mh with
-        | Eq =>
-          match w_compare w_0 ml with
-          | Eq => WW bh bl
-          | _  =>
-            let r := double_modn1 w_zdigits w_0 w_head0 w_add_mul_div w_div21
-                      w_compare w_sub 1 (WW bh bl) ml in
-            WW w_0 (w_gcd_gt ml r)
-          end
-        | Lt =>
-          let r := ww_mod_gt_aux bh bl mh ml in
-          match r with
-          | W0 => m
-          | WW rh rl => cont mh ml rh rl
-          end
-        | Gt => W0 (* absurde *)
-        end
-      end
-    | Gt => W0 (* absurde *)
-    end).
-   rewrite spec_compare, spec_w_0.
-   case Z.compare_spec; intros Hbh.
-   simpl ww_to_Z in *. rewrite <- Hbh.
-   rewrite Z.mul_0_l;rewrite Z.add_0_l.
-   rewrite spec_compare, spec_w_0.
-   case Z.compare_spec; intros Hbl.
-   rewrite <- Hbl;apply Zis_gcd_0.
-   simpl;rewrite spec_w_0;rewrite Z.mul_0_l;rewrite Z.add_0_l.
-   apply Zis_gcd_mod;zarith.
-   change ([|ah|] * wB + [|al|]) with (double_to_Z w_digits w_to_Z 1 (WW ah al)).
-   rewrite <- (@spec_double_modn1 w w_digits w_zdigits w_0 w_WW w_head0 w_add_mul_div
-    w_div21 w_compare w_sub w_to_Z spec_to_Z spec_w_zdigits spec_w_0 spec_w_WW spec_head0 spec_add_mul_div
-    spec_div21 spec_compare spec_sub 1 (WW ah al) bl Hbl).
-   apply spec_gcd_gt.
-   rewrite  (@spec_double_modn1 w w_digits w_zdigits w_0 w_WW); trivial.
-   apply Z.lt_gt;match goal with | |- ?x mod ?y < ?y =>
-    destruct (Z_mod_lt x y);zarith end.
-   Spec_w_to_Z bl;exfalso;omega.
-   assert (H:= spec_ww_mod_gt_aux _ _ _ Hgt Hbh).
-   assert (H2 : 0 < [[WW bh bl]]).
-   simpl;Spec_w_to_Z bl. apply Z.lt_le_trans with ([|bh|]*wB);zarith.
-   apply Z.mul_pos_pos;zarith.
-   apply Zis_gcd_mod;trivial. rewrite <- H.
-   simpl in *;destruct (ww_mod_gt_aux ah al bh bl) as [ |mh ml].
-   simpl;apply Zis_gcd_0;zarith.
-   rewrite spec_compare, spec_w_0; case Z.compare_spec; intros Hmh.
-   simpl;rewrite <- Hmh;simpl.
-   rewrite spec_compare, spec_w_0; case Z.compare_spec; intros Hml.
-   rewrite <- Hml;simpl;apply Zis_gcd_0.
-   simpl; rewrite spec_w_0; simpl.
-   apply Zis_gcd_mod;zarith.
-   change ([|bh|] * wB + [|bl|]) with (double_to_Z w_digits w_to_Z 1 (WW bh bl)).
-   rewrite <- (@spec_double_modn1 w w_digits w_zdigits w_0 w_WW w_head0 w_add_mul_div
-   w_div21 w_compare w_sub w_to_Z spec_to_Z spec_w_zdigits spec_w_0 spec_w_WW spec_head0 spec_add_mul_div
-   spec_div21 spec_compare spec_sub 1 (WW bh bl) ml Hml).
-   apply spec_gcd_gt.
-   rewrite  (@spec_double_modn1 w w_digits w_zdigits w_0 w_WW); trivial.
-   apply Z.lt_gt;match goal with | |- ?x mod ?y < ?y =>
-   destruct (Z_mod_lt x y);zarith end.
-   Spec_w_to_Z ml;exfalso;omega.
-   assert ([[WW bh bl]] > [[WW mh ml]]).
-   rewrite H;simpl; apply Z.lt_gt;match goal with | |- ?x mod ?y < ?y =>
-   destruct (Z_mod_lt x y);zarith end.
-   assert (H1:= spec_ww_mod_gt_aux _ _ _ H0 Hmh).
-   assert (H3 : 0 < [[WW mh ml]]).
-   simpl;Spec_w_to_Z ml. apply Z.lt_le_trans with ([|mh|]*wB);zarith.
-   apply Z.mul_pos_pos;zarith.
-   apply Zis_gcd_mod;zarith. simpl in *;rewrite <- H1.
-   destruct (ww_mod_gt_aux bh bl mh ml) as [ |rh rl]. simpl; apply Zis_gcd_0.
-   simpl;apply Hcont. simpl in H1;rewrite H1.
-   apply Z.lt_gt;match goal with | |- ?x mod ?y < ?y =>
-   destruct (Z_mod_lt x y);zarith end.
-   apply Z.le_trans with (2^n/2).
-   apply Zdiv_le_lower_bound;zarith.
-   apply Z.le_trans with ([|bh|] * wB + [|bl|]);zarith.
-   assert (H3' := Z_div_mod_eq [[WW bh bl]] [[WW mh ml]] (Z.lt_gt _ _ H3)).
-   assert (H4 : 0 <= [[WW bh bl]]/[[WW mh ml]]).
-   apply Z.ge_le;apply Z_div_ge0;zarith. simpl in *;rewrite H1.
-   pattern ([|bh|] * wB + [|bl|]) at 2;rewrite H3'.
-   Z.le_elim H4.
-   assert (H6' : [[WW bh bl]] mod [[WW mh ml]] =
-                  [[WW bh bl]] - [[WW mh ml]] * ([[WW bh bl]]/[[WW mh ml]])).
-   simpl;pattern ([|bh|] * wB + [|bl|]) at 2;rewrite H3';ring. simpl in H6'.
-   assert ([[WW mh ml]] <= [[WW mh ml]] * ([[WW bh bl]]/[[WW mh ml]])).
-   simpl;pattern ([|mh|]*wB+[|ml|]) at 1;rewrite <- Z.mul_1_r;zarith.
-   simpl in *;assert (H8 := Z_mod_lt [[WW bh bl]] [[WW mh ml]]);simpl in H8;
-    zarith.
-   assert (H8 := Z_mod_lt [[WW bh bl]] [[WW mh ml]]);simpl in *;zarith.
-   rewrite <- H4 in H3';rewrite Z.mul_0_r in H3';simpl in H3';zarith.
-   pattern n at 1;replace n with (n-1+1);try ring.
-   rewrite Zpower_exp;zarith. change (2^1) with 2.
-   rewrite Z_div_mult;zarith.
-   assert (2^1 <= 2^n). change (2^1) with 2;zarith.
-   assert (H7 := @Zpower_le_monotone_inv 2 1 n);zarith.
-   Spec_w_to_Z mh;exfalso;zarith.
-   Spec_w_to_Z bh;exfalso;zarith.
-  Qed.
-
-  Lemma spec_ww_gcd_gt_aux :
-   forall p cont n,
-   (forall xh xl yh yl,
-     [[WW xh xl]] > [[WW yh yl]] ->
-     [[WW yh yl]] <= 2^n ->
-      Zis_gcd [[WW xh xl]] [[WW yh yl]] [[cont xh xl yh yl]]) ->
-   forall ah al bh bl , [[WW ah al]] > [[WW bh bl]] ->
-     [[WW bh bl]] <= 2^(Zpos p + n) ->
-     Zis_gcd [[WW ah al]] [[WW bh bl]]
-        [[ww_gcd_gt_aux p cont ah al bh bl]].
-  Proof.
-   induction p;intros cont n Hcont ah al bh bl Hgt Hs;simpl ww_gcd_gt_aux.
-   assert (0 < Zpos p). unfold Z.lt;reflexivity.
-   apply spec_ww_gcd_gt_aux_body with (n := Zpos (xI p) + n);
-   trivial;rewrite Pos2Z.inj_xI.
-   intros. apply IHp with (n := Zpos p + n);zarith.
-   intros. apply IHp with (n := n );zarith.
-   apply Z.le_trans with (2 ^ (2* Zpos p + 1+ n -1));zarith.
-   apply Z.pow_le_mono_r;zarith.
-   assert (0 < Zpos p). unfold Z.lt;reflexivity.
-   apply spec_ww_gcd_gt_aux_body with (n := Zpos (xO p) + n );trivial.
-   rewrite (Pos2Z.inj_xO p).
-   intros. apply IHp with (n := Zpos p + n - 1);zarith.
-   intros. apply IHp with (n := n -1 );zarith.
-   intros;apply Hcont;zarith.
-   apply Z.le_trans with (2^(n-1));zarith.
-   apply Z.pow_le_mono_r;zarith.
-   apply Z.le_trans with (2 ^ (Zpos p + n -1));zarith.
-   apply Z.pow_le_mono_r;zarith.
-   apply Z.le_trans with (2 ^ (2*Zpos p + n -1));zarith.
-   apply Z.pow_le_mono_r;zarith.
-   apply spec_ww_gcd_gt_aux_body with (n := n+1);trivial.
-   rewrite Z.add_comm;trivial.
-   ring_simplify (n + 1 - 1);trivial.
-  Qed.
-
-End DoubleDivGt.
-
-Section DoubleDiv.
-
- Variable w : Type.
- Variable w_digits : positive.
- Variable ww_1 : zn2z w.
- Variable ww_compare : zn2z w -> zn2z w -> comparison.
-
- Variable ww_div_gt : zn2z w -> zn2z w -> zn2z w * zn2z w.
- Variable ww_mod_gt : zn2z w -> zn2z w -> zn2z w.
-
- Definition ww_div a b :=
-  match ww_compare a b with
-  | Gt => ww_div_gt a b
-  | Eq => (ww_1, W0)
-  | Lt => (W0, a)
-  end.
-
- Definition ww_mod a b :=
-  match ww_compare a b with
-  | Gt => ww_mod_gt a b
-  | Eq => W0
-  | Lt => a
-  end.
-
-  Variable w_to_Z : w -> Z.
-  Notation wB  := (base w_digits).
-  Notation wwB := (base (ww_digits w_digits)).
-  Notation "[| x |]" := (w_to_Z x)  (at level 0, x at level 99).
-  Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99).
-  Variable spec_to_Z  : forall x, 0 <= [|x|] < wB.
-  Variable spec_ww_1 : [[ww_1]] = 1.
-  Variable spec_ww_compare :  forall x y,
-    ww_compare x y = Z.compare [[x]] [[y]].
-  Variable spec_ww_div_gt : forall a b, [[a]] > [[b]] -> 0 < [[b]] ->
-      let (q,r) := ww_div_gt a b in
-      [[a]] = [[q]] * [[b]] + [[r]] /\
-      0 <= [[r]] < [[b]].
-  Variable spec_ww_mod_gt : forall a b, [[a]] > [[b]] -> 0 < [[b]] ->
-      [[ww_mod_gt a b]] = [[a]] mod [[b]].
-
-  Ltac Spec_w_to_Z x :=
-   let H:= fresh "HH" in
-   assert (H:= spec_to_Z x).
-
-  Ltac Spec_ww_to_Z x :=
-   let H:= fresh "HH" in
-   assert (H:= spec_ww_to_Z w_digits w_to_Z spec_to_Z x).
-
-  Lemma spec_ww_div : forall a b, 0 < [[b]] ->
-      let (q,r) := ww_div a b in
-      [[a]] = [[q]] * [[b]] + [[r]] /\
-      0 <= [[r]] < [[b]].
-  Proof.
-   intros a b Hpos;unfold ww_div.
-   rewrite spec_ww_compare; case Z.compare_spec; intros.
-   simpl;rewrite spec_ww_1;split;zarith.
-   simpl;split;[ring|Spec_ww_to_Z a;zarith].
-   apply spec_ww_div_gt;auto with zarith.
-  Qed.
-
-  Lemma  spec_ww_mod :  forall a b, 0 < [[b]] ->
-      [[ww_mod a b]] = [[a]] mod [[b]].
-  Proof.
-   intros a b Hpos;unfold ww_mod.
-   rewrite spec_ww_compare; case Z.compare_spec; intros.
-   simpl;apply Zmod_unique with 1;try rewrite H;zarith.
-   Spec_ww_to_Z a;symmetry;apply Zmod_small;zarith.
-   apply spec_ww_mod_gt;auto with zarith.
-  Qed.
-
-
- Variable w_0 : w.
- Variable w_1 : w.
- Variable w_compare : w -> w -> comparison.
- Variable w_eq0 : w -> bool.
- Variable w_gcd_gt : w -> w -> w.
- Variable _ww_digits : positive.
- Variable spec_ww_digits_ : _ww_digits = xO w_digits.
- Variable ww_gcd_gt_fix :
-   positive -> (w -> w -> w -> w -> zn2z w) ->
-             w -> w -> w -> w -> zn2z w.
-
-  Variable spec_w_0   : [|w_0|] = 0.
-  Variable spec_w_1   : [|w_1|] = 1.
-  Variable spec_compare :
-    forall x y, w_compare x y = Z.compare [|x|] [|y|].
-  Variable spec_eq0 : forall x, w_eq0 x = true -> [|x|] = 0.
-  Variable spec_gcd_gt : forall a b, [|a|] > [|b|] ->
-      Zis_gcd [|a|] [|b|] [|w_gcd_gt a b|].
-  Variable spec_gcd_gt_fix :
-   forall p cont n,
-   (forall xh xl yh yl,
-     [[WW xh xl]] > [[WW yh yl]] ->
-     [[WW yh yl]] <= 2^n ->
-      Zis_gcd [[WW xh xl]] [[WW yh yl]] [[cont xh xl yh yl]]) ->
-   forall ah al bh bl , [[WW ah al]] > [[WW bh bl]] ->
-     [[WW bh bl]] <= 2^(Zpos p + n) ->
-     Zis_gcd [[WW ah al]] [[WW bh bl]]
-        [[ww_gcd_gt_fix p cont ah al bh bl]].
-
- Definition gcd_cont (xh xl yh yl:w) :=
-  match w_compare w_1 yl with
-  | Eq => ww_1
-  | _ => WW xh xl
-  end.
-
-  Lemma spec_gcd_cont : forall xh xl yh yl,
-     [[WW xh xl]] > [[WW yh yl]] ->
-     [[WW yh yl]] <= 1 ->
-      Zis_gcd [[WW xh xl]] [[WW yh yl]] [[gcd_cont xh xl yh yl]].
-  Proof.
-   intros xh xl yh yl Hgt' Hle. simpl in Hle.
-   assert ([|yh|] = 0).
-    change 1 with (0*wB+1) in Hle.
-    assert (0 <= 1 < wB). split;zarith. apply wB_pos.
-    assert (H1:= beta_lex _ _ _ _ _ Hle (spec_to_Z yl) H).
-    Spec_w_to_Z yh;zarith.
-   unfold gcd_cont; rewrite spec_compare, spec_w_1.
-   case Z.compare_spec; intros Hcmpy.
-   simpl;rewrite H;simpl;
-   rewrite spec_ww_1;rewrite <- Hcmpy;apply Zis_gcd_mod;zarith.
-   rewrite <- (Zmod_unique ([|xh|]*wB+[|xl|]) 1 ([|xh|]*wB+[|xl|]) 0);zarith.
-   rewrite H in Hle; exfalso;zarith.
-   assert (H0 : [|yl|] = 0) by (Spec_w_to_Z yl;zarith).
-   simpl. rewrite H0, H;simpl;apply Zis_gcd_0;trivial.
-  Qed.
-
-
-  Variable cont : w -> w -> w -> w -> zn2z w.
-  Variable spec_cont : forall xh xl yh yl,
-     [[WW xh xl]] > [[WW yh yl]] ->
-     [[WW yh yl]] <= 1 ->
-      Zis_gcd [[WW xh xl]] [[WW yh yl]] [[cont xh xl yh yl]].
-
-  Definition ww_gcd_gt a b :=
-   match a, b with
-   | W0, _ => b
-   | _, W0 => a
-   | WW ah al, WW bh bl =>
-     if w_eq0 ah then (WW w_0 (w_gcd_gt al bl))
-     else ww_gcd_gt_fix _ww_digits cont ah al bh bl
-   end.
-
-  Definition ww_gcd a b :=
-   Eval lazy beta delta [ww_gcd_gt] in
-   match ww_compare a b with
-   | Gt => ww_gcd_gt a b
-   | Eq => a
-   | Lt => ww_gcd_gt b a
-   end.
-
-  Lemma spec_ww_gcd_gt : forall a b, [[a]] > [[b]] ->
-      Zis_gcd [[a]] [[b]] [[ww_gcd_gt a b]].
-  Proof.
-   intros a b Hgt;unfold ww_gcd_gt.
-   destruct a as [ |ah al]. simpl;apply Zis_gcd_sym;apply Zis_gcd_0.
-   destruct b as [ |bh bl]. simpl;apply Zis_gcd_0.
-   simpl in Hgt. generalize (@spec_eq0 ah);destruct (w_eq0 ah);intros.
-   simpl;rewrite H in Hgt;trivial;rewrite H;trivial;rewrite spec_w_0;simpl.
-   assert ([|bh|] <= 0).
-   apply beta_lex with (d:=[|al|])(b:=[|bl|]) (beta := wB);zarith.
-   Spec_w_to_Z bh;assert ([|bh|] = 0);zarith. rewrite H1 in Hgt;simpl in Hgt.
-   rewrite H1;simpl;auto. clear H.
-   apply spec_gcd_gt_fix with (n:= 0);trivial.
-   rewrite Z.add_0_r;rewrite spec_ww_digits_.
-   change (2 ^ Zpos (xO w_digits)) with wwB. Spec_ww_to_Z (WW bh bl);zarith.
-  Qed.
-
-  Lemma spec_ww_gcd : forall a b, Zis_gcd [[a]] [[b]] [[ww_gcd a b]].
-  Proof.
-   intros a b.
-   change (ww_gcd a b) with
-    (match ww_compare a b with
-     | Gt => ww_gcd_gt a b
-     | Eq => a
-     | Lt => ww_gcd_gt b a
-     end).
-   rewrite spec_ww_compare; case Z.compare_spec; intros Hcmp.
-   Spec_ww_to_Z b;rewrite Hcmp.
-   apply Zis_gcd_for_euclid with 1;zarith.
-   ring_simplify ([[b]] - 1 * [[b]]). apply Zis_gcd_0;zarith.
-   apply Zis_gcd_sym;apply spec_ww_gcd_gt;zarith.
-   apply spec_ww_gcd_gt;zarith.
-  Qed.
-
-End DoubleDiv.
-
diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleDivn1.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleDivn1.v
deleted file mode 100644
index 195527dd5..000000000
--- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleDivn1.v
+++ /dev/null
@@ -1,519 +0,0 @@
-(************************************************************************)
-(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016     *)
-(*   \VV/  **************************************************************)
-(*    //   *      This file is distributed under the terms of the       *)
-(*         *       GNU Lesser General Public License Version 2.1        *)
-(************************************************************************)
-(*            Benjamin Gregoire, Laurent Thery, INRIA, 2007             *)
-(************************************************************************)
-
-Set Implicit Arguments.
-
-Require Import ZArith Ndigits.
-Require Import BigNumPrelude.
-Require Import DoubleType.
-Require Import DoubleBase.
-
-Local Open Scope Z_scope.
-
-Local Infix "<<" := Pos.shiftl_nat (at level 30).
-
-Section GENDIVN1.
-
- Variable w             : Type.
- Variable w_digits      : positive.
- Variable w_zdigits     : w.
- Variable w_0           : w.
- Variable w_WW          : w -> w -> zn2z w.
- Variable w_head0       : w -> w.
- Variable w_add_mul_div : w -> w -> w -> w.
- Variable w_div21       : w -> w -> w -> w * w.
- Variable w_compare     : w -> w -> comparison.
- Variable w_sub         : w -> w -> w.
-
-
-
- (* ** For proofs ** *)
- Variable w_to_Z        : w -> Z.
-
- Notation wB := (base w_digits).
-
- Notation "[| x |]" := (w_to_Z x)  (at level 0, x at level 99).
- Notation "[! n | x !]" := (double_to_Z w_digits w_to_Z n x)
-                                      (at level 0, x at level 99).
- Notation "[[ x ]]" := (zn2z_to_Z wB w_to_Z x)  (at level 0, x at level 99).
-
- Variable spec_to_Z   : forall x, 0 <= [| x |] < wB.
- Variable spec_w_zdigits: [|w_zdigits|] = Zpos w_digits.
- Variable spec_0   : [|w_0|] = 0.
- Variable spec_WW  : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|].
- Variable spec_head0  : forall x,  0 < [|x|] ->
-	 wB/ 2 <= 2 ^ [|w_head0 x|] * [|x|] < wB.
- Variable spec_add_mul_div : forall x y p,
-        [|p|] <= Zpos w_digits ->
-       [| w_add_mul_div p x y |] =
-         ([|x|] * (2 ^ [|p|]) +
-          [|y|] / (2 ^ ((Zpos w_digits) - [|p|]))) mod wB.
- Variable spec_div21 : forall a1 a2 b,
-     wB/2 <= [|b|] ->
-     [|a1|] < [|b|] ->
-     let (q,r) := w_div21 a1 a2 b in
-     [|a1|] *wB+ [|a2|] = [|q|] * [|b|] + [|r|] /\
-     0 <= [|r|] < [|b|].
- Variable spec_compare :
-   forall x y, w_compare x y = Z.compare [|x|] [|y|].
- Variable spec_sub: forall x y,
-   [|w_sub x y|] = ([|x|] - [|y|]) mod wB.
-
-
-
- Section DIVAUX.
-  Variable b2p : w.
-  Variable b2p_le : wB/2 <= [|b2p|].
-
-  Definition double_divn1_0_aux n (divn1: w -> word w n -> word w n * w) r h :=
-   let (hh,hl) := double_split w_0 n h in
-   let (qh,rh) := divn1 r hh in
-   let (ql,rl) := divn1 rh hl in
-   (double_WW w_WW n qh ql, rl).
-
-  Fixpoint double_divn1_0 (n:nat) : w -> word w n -> word w n * w :=
-   match n return w -> word w n -> word w n * w with
-   | O => fun r x => w_div21 r x b2p
-   | S n => double_divn1_0_aux n (double_divn1_0 n)
-   end.
-
-  Lemma spec_split : forall (n : nat) (x : zn2z (word w n)),
-       let (h, l) := double_split w_0 n x in
-       [!S n | x!] = [!n | h!] * double_wB w_digits n + [!n | l!].
-  Proof (spec_double_split w_0 w_digits w_to_Z spec_0).
-
-  Lemma spec_double_divn1_0 : forall n r a,
-    [|r|] < [|b2p|] ->
-    let (q,r') := double_divn1_0 n r a in
-    [|r|] * double_wB w_digits n + [!n|a!] = [!n|q!] * [|b2p|] + [|r'|] /\
-    0 <= [|r'|] < [|b2p|].
-  Proof.
-   induction n;intros.
-   exact (spec_div21 a b2p_le H).
-   simpl (double_divn1_0 (S n) r a); unfold double_divn1_0_aux.
-   assert (H1 := spec_split n a);destruct (double_split w_0 n a) as (hh,hl).
-   rewrite H1.
-   assert (H2 := IHn r hh H);destruct (double_divn1_0 n r hh) as (qh,rh).
-   destruct H2.
-   assert ([|rh|] < [|b2p|]). omega.
-   assert (H4 := IHn rh hl H3);destruct (double_divn1_0 n rh hl) as (ql,rl).
-   destruct H4;split;trivial.
-   rewrite spec_double_WW;trivial.
-   rewrite <- double_wB_wwB.
-   rewrite Z.mul_assoc;rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r.
-   rewrite H0;rewrite Z.mul_add_distr_r;rewrite <- Z.add_assoc.
-   rewrite H4;ring.
-  Qed.
-
-  Definition double_modn1_0_aux n (modn1:w -> word w n -> w) r h :=
-   let (hh,hl) := double_split w_0 n h in modn1 (modn1 r hh) hl.
-
-  Fixpoint double_modn1_0 (n:nat) : w -> word w n -> w :=
-   match n return w -> word w n -> w with
-   | O => fun r x => snd (w_div21 r x b2p)
-   | S n => double_modn1_0_aux n (double_modn1_0 n)
-   end.
-
-  Lemma spec_double_modn1_0 : forall n r x,
-     double_modn1_0 n r x = snd (double_divn1_0 n r x).
-  Proof.
-   induction n;simpl;intros;trivial.
-   unfold double_modn1_0_aux, double_divn1_0_aux.
-   destruct (double_split w_0 n x) as (hh,hl).
-   rewrite (IHn r hh).
-   destruct (double_divn1_0 n r hh) as (qh,rh);simpl.
-   rewrite IHn. destruct (double_divn1_0 n rh hl);trivial.
-  Qed.
-
-  Variable p : w.
-  Variable p_bounded : [|p|] <= Zpos w_digits.
-
-  Lemma spec_add_mul_divp : forall x y,
-    [| w_add_mul_div p x y |] =
-       ([|x|] * (2 ^ [|p|]) +
-          [|y|] / (2 ^ ((Zpos w_digits) - [|p|]))) mod wB.
-  Proof.
-   intros;apply spec_add_mul_div;auto.
-  Qed.
-
-  Definition double_divn1_p_aux n
-   (divn1 : w -> word w n -> word w n -> word w n * w) r h l :=
-   let (hh,hl) := double_split w_0 n h in
-   let (lh,ll) := double_split w_0 n l in
-   let (qh,rh) := divn1 r hh hl in
-   let (ql,rl) := divn1 rh hl lh in
-   (double_WW w_WW n qh ql, rl).
-
-  Fixpoint double_divn1_p (n:nat) : w -> word w n -> word w n -> word w n * w :=
-   match n return w -> word w n -> word w n ->  word w n * w with
-   | O => fun r h l => w_div21 r (w_add_mul_div p h l) b2p
-   | S n => double_divn1_p_aux n (double_divn1_p n)
-   end.
-
-  Lemma p_lt_double_digits : forall n, [|p|] <= Zpos (w_digits << n).
-  Proof.
-   induction n;simpl. trivial.
-   case (spec_to_Z p); rewrite Pos2Z.inj_xO;auto with zarith.
-  Qed.
-
-  Lemma spec_double_divn1_p : forall n r h l,
-    [|r|] < [|b2p|] ->
-    let (q,r') := double_divn1_p n r h l in
-    [|r|] * double_wB w_digits n +
-      ([!n|h!]*2^[|p|] +
-        [!n|l!] / (2^(Zpos(w_digits << n) - [|p|])))
-        mod double_wB w_digits n = [!n|q!] * [|b2p|] + [|r'|] /\
-    0 <= [|r'|] < [|b2p|].
-  Proof.
-   case (spec_to_Z p); intros HH0 HH1.
-   induction n;intros.
-   simpl (double_divn1_p 0 r h l).
-   unfold double_to_Z, double_wB, "<<".
-   rewrite <- spec_add_mul_divp.
-   exact (spec_div21 (w_add_mul_div p h l) b2p_le H).
-   simpl (double_divn1_p (S n) r h l).
-   unfold double_divn1_p_aux.
-   assert (H1 := spec_split n h);destruct (double_split w_0 n h) as (hh,hl).
-   rewrite H1. rewrite <- double_wB_wwB.
-   assert (H2 := spec_split n l);destruct (double_split w_0 n l) as (lh,ll).
-   rewrite H2.
-   replace ([|r|] * (double_wB w_digits n * double_wB w_digits n) +
-    (([!n|hh!] * double_wB w_digits n + [!n|hl!]) * 2 ^ [|p|] +
-    ([!n|lh!] * double_wB w_digits n + [!n|ll!]) /
-     2^(Zpos (w_digits << (S n)) - [|p|])) mod
-      (double_wB w_digits n * double_wB w_digits n)) with
-    (([|r|] * double_wB w_digits n + ([!n|hh!] * 2^[|p|] +
-      [!n|hl!] / 2^(Zpos (w_digits << n) - [|p|])) mod
-                      double_wB w_digits n) * double_wB w_digits n +
-     ([!n|hl!] * 2^[|p|] +
-      [!n|lh!] / 2^(Zpos (w_digits << n) - [|p|])) mod
-              double_wB w_digits n).
-   generalize (IHn r hh hl H);destruct (double_divn1_p n r hh hl) as (qh,rh);
-   intros (H3,H4);rewrite H3.
-   assert ([|rh|] < [|b2p|]). omega.
-   replace (([!n|qh!] * [|b2p|] + [|rh|]) * double_wB w_digits n +
-     ([!n|hl!] * 2 ^ [|p|] +
-      [!n|lh!] / 2 ^ (Zpos (w_digits << n) - [|p|])) mod
-      double_wB w_digits n)  with
-    ([!n|qh!] * [|b2p|] *double_wB w_digits n + ([|rh|]*double_wB w_digits n +
-      ([!n|hl!] * 2 ^ [|p|] +
-       [!n|lh!] / 2 ^ (Zpos (w_digits << n) - [|p|])) mod
-      double_wB w_digits n)). 2:ring.
-   generalize (IHn rh hl lh H0);destruct (double_divn1_p n rh hl lh) as (ql,rl);
-   intros (H5,H6);rewrite H5.
-   split;[rewrite spec_double_WW;trivial;ring|trivial].
-   assert (Uhh := spec_double_to_Z w_digits w_to_Z spec_to_Z n hh);
-    unfold double_wB,base in Uhh.
-   assert (Uhl := spec_double_to_Z w_digits w_to_Z spec_to_Z n hl);
-    unfold double_wB,base in Uhl.
-   assert (Ulh := spec_double_to_Z w_digits w_to_Z spec_to_Z n lh);
-    unfold double_wB,base in Ulh.
-   assert (Ull := spec_double_to_Z w_digits w_to_Z spec_to_Z n ll);
-    unfold double_wB,base in Ull.
-   unfold double_wB,base.
-   assert (UU:=p_lt_double_digits n).
-   rewrite Zdiv_shift_r;auto with zarith.
-   2:change (Zpos (w_digits << (S n)))
-     with (2*Zpos (w_digits << n));auto with zarith.
-   replace (2 ^ (Zpos (w_digits << (S n)) - [|p|])) with
-    (2^(Zpos (w_digits << n) - [|p|])*2^Zpos (w_digits << n)).
-   rewrite Zdiv_mult_cancel_r;auto with zarith.
-   rewrite Z.mul_add_distr_r with (p:=  2^[|p|]).
-   pattern  ([!n|hl!] * 2^[|p|]) at 2;
-   rewrite (shift_unshift_mod (Zpos(w_digits << n))([|p|])([!n|hl!]));
-    auto with zarith.
-   rewrite Z.add_assoc.
-   replace
-    ([!n|hh!] * 2^Zpos (w_digits << n)* 2^[|p|] +
-      ([!n|hl!] / 2^(Zpos (w_digits << n)-[|p|])*
-       2^Zpos(w_digits << n)))
-   with
-    (([!n|hh!] *2^[|p|] + double_to_Z w_digits w_to_Z n hl /
-       2^(Zpos (w_digits << n)-[|p|]))
-      * 2^Zpos(w_digits << n));try (ring;fail).
-   rewrite <- Z.add_assoc.
-   rewrite <- (Zmod_shift_r ([|p|]));auto with zarith.
-   replace
-     (2 ^ Zpos (w_digits << n) * 2 ^ Zpos (w_digits << n)) with
-     (2 ^ (Zpos (w_digits << n) + Zpos (w_digits << n))).
-   rewrite (Zmod_shift_r (Zpos (w_digits << n)));auto with zarith.
-   replace (2 ^ (Zpos (w_digits << n) + Zpos (w_digits << n)))
-   with (2^Zpos(w_digits << n) *2^Zpos(w_digits << n)).
-   rewrite (Z.mul_comm (([!n|hh!] * 2 ^ [|p|] +
-      [!n|hl!] / 2 ^ (Zpos (w_digits << n) - [|p|])))).
-   rewrite  Zmult_mod_distr_l;auto with zarith.
-   ring.
-   rewrite Zpower_exp;auto with zarith.
-   assert (0 < Zpos (w_digits << n)). unfold Z.lt;reflexivity.
-   auto with zarith.
-   apply Z_mod_lt;auto with zarith.
-   rewrite Zpower_exp;auto with zarith.
-   split;auto with zarith.
-   apply Zdiv_lt_upper_bound;auto with zarith.
-   rewrite <- Zpower_exp;auto with zarith.
-   replace ([|p|] + (Zpos (w_digits << n) - [|p|])) with
-     (Zpos(w_digits << n));auto with zarith.
-   rewrite <- Zpower_exp;auto with zarith.
-   replace (Zpos (w_digits << (S n)) - [|p|]) with
-       (Zpos (w_digits << n) - [|p|] +
-         Zpos (w_digits << n));trivial.
-   change (Zpos (w_digits << (S n))) with
-    (2*Zpos (w_digits << n)). ring.
-  Qed.
-
-  Definition double_modn1_p_aux n (modn1 : w -> word w n -> word w n -> w) r h l:=
-   let (hh,hl) := double_split w_0 n h in
-   let (lh,ll) := double_split w_0 n l in
-   modn1 (modn1 r hh hl) hl lh.
-
-  Fixpoint double_modn1_p (n:nat) : w -> word w n -> word w n -> w :=
-   match n return w -> word w n -> word w n -> w with
-   | O => fun r h l => snd (w_div21 r (w_add_mul_div p h l) b2p)
-   | S n => double_modn1_p_aux n (double_modn1_p n)
-   end.
-
-  Lemma spec_double_modn1_p : forall n r h l ,
-    double_modn1_p n r h l = snd (double_divn1_p n r h l).
-  Proof.
-   induction n;simpl;intros;trivial.
-   unfold double_modn1_p_aux, double_divn1_p_aux.
-   destruct(double_split w_0 n h)as(hh,hl);destruct(double_split w_0 n l) as (lh,ll).
-   rewrite (IHn r hh hl);destruct (double_divn1_p n r hh hl) as (qh,rh).
-   rewrite IHn;simpl;destruct (double_divn1_p n rh hl lh);trivial.
-  Qed.
-
- End DIVAUX.
-
- Fixpoint high (n:nat) : word w n -> w :=
-  match n return word w n -> w with
-  | O => fun a => a
-  | S n =>
-    fun (a:zn2z (word w n)) =>
-     match a with
-     | W0 => w_0
-     | WW h l => high n h
-     end
-  end.
-
- Lemma spec_double_digits:forall n, Zpos w_digits <= Zpos (w_digits << n).
- Proof.
-  induction n;simpl;auto with zarith.
-  change (Zpos (xO (w_digits << n))) with
-    (2*Zpos (w_digits << n)).
-  assert (0 < Zpos w_digits) by reflexivity.
-  auto with zarith.
- Qed.
-
- Lemma spec_high : forall n (x:word w n),
-   [|high n x|] = [!n|x!] / 2^(Zpos (w_digits << n) - Zpos w_digits).
- Proof.
-  induction n;intros.
-  unfold high,double_to_Z. rewrite Pshiftl_nat_0.
-  replace (Zpos w_digits - Zpos w_digits) with 0;try ring.
-  simpl. rewrite <- (Zdiv_unique [|x|] 1 [|x|] 0);auto with zarith.
-  assert (U2 := spec_double_digits n).
-  assert (U3 : 0 < Zpos w_digits). exact (eq_refl Lt).
-  destruct x;unfold high;fold high.
-  unfold double_to_Z,zn2z_to_Z;rewrite spec_0.
-  rewrite Zdiv_0_l;trivial.
-  assert (U0 := spec_double_to_Z w_digits w_to_Z spec_to_Z n w0);
-   assert (U1 := spec_double_to_Z w_digits w_to_Z spec_to_Z n w1).
-  simpl [!S n|WW w0 w1!].
-  unfold double_wB,base;rewrite Zdiv_shift_r;auto with zarith.
-  replace (2 ^ (Zpos (w_digits << (S n)) - Zpos w_digits)) with
-   (2^(Zpos (w_digits << n) - Zpos w_digits) *
-        2^Zpos (w_digits << n)).
-  rewrite Zdiv_mult_cancel_r;auto with zarith.
-  rewrite <- Zpower_exp;auto with zarith.
-  replace (Zpos (w_digits << n) - Zpos w_digits +
-    Zpos (w_digits << n)) with
-   (Zpos (w_digits << (S n)) - Zpos w_digits);trivial.
-  change (Zpos (w_digits << (S n))) with
-     (2*Zpos (w_digits << n));ring.
-  change (Zpos (w_digits << (S n))) with
-  (2*Zpos (w_digits << n)); auto with zarith.
- Qed.
-
- Definition double_divn1 (n:nat) (a:word w n) (b:w) :=
-  let p := w_head0 b in
-  match w_compare p w_0 with
-  | Gt =>
-    let b2p := w_add_mul_div p b w_0 in
-    let ha := high n a in
-    let k := w_sub w_zdigits p in
-    let lsr_n := w_add_mul_div k w_0 in
-    let r0 := w_add_mul_div p w_0 ha in
-    let (q,r) := double_divn1_p b2p p n r0 a (double_0 w_0 n) in
-    (q, lsr_n r)
-  | _ => double_divn1_0 b n w_0 a
-  end.
-
- Lemma spec_double_divn1 : forall n a b,
-    0 < [|b|] ->
-    let (q,r) := double_divn1 n a b in
-    [!n|a!] = [!n|q!] * [|b|] + [|r|] /\
-    0 <= [|r|] < [|b|].
-  Proof.
-   intros n a b H. unfold double_divn1.
-   case (spec_head0 H); intros H0 H1.
-   case (spec_to_Z (w_head0 b)); intros HH1 HH2.
-   rewrite spec_compare; case Z.compare_spec;
-     rewrite spec_0; intros H2; auto with zarith.
-   assert (Hv1: wB/2 <= [|b|]).
-     generalize H0; rewrite H2; rewrite Z.pow_0_r;
-       rewrite Z.mul_1_l; auto.
-   assert (Hv2: [|w_0|] < [|b|]).
-     rewrite spec_0; auto.
-   generalize (spec_double_divn1_0 Hv1 n a Hv2).
-   rewrite spec_0;rewrite Z.mul_0_l; rewrite Z.add_0_l; auto.
-   contradict H2; auto with zarith.
-   assert (HHHH : 0 < [|w_head0 b|]); auto with zarith.
-   assert ([|w_head0 b|] < Zpos w_digits).
-    case (Z.le_gt_cases (Zpos w_digits) [|w_head0 b|]); auto; intros HH.
-    assert (2 ^ [|w_head0 b|] < wB).
-     apply Z.le_lt_trans with (2 ^ [|w_head0 b|] * [|b|]);auto with zarith.
-     replace (2 ^ [|w_head0 b|]) with (2^[|w_head0 b|] * 1);try (ring;fail).
-     apply Z.mul_le_mono_nonneg;auto with zarith.
-     assert (wB <= 2^[|w_head0 b|]).
-     unfold base;apply Zpower_le_monotone;auto with zarith. omega.
-   assert ([|w_add_mul_div (w_head0 b) b w_0|] =
-               2 ^ [|w_head0 b|] * [|b|]).
-    rewrite (spec_add_mul_div b w_0); auto with zarith.
-    rewrite spec_0;rewrite Zdiv_0_l; try omega.
-    rewrite Z.add_0_r; rewrite Z.mul_comm.
-    rewrite Zmod_small; auto with zarith.
-   assert (H5 := spec_to_Z (high n a)).
-   assert
-    ([|w_add_mul_div (w_head0 b) w_0 (high n a)|]
-          <[|w_add_mul_div (w_head0 b) b w_0|]).
-    rewrite H4.
-    rewrite spec_add_mul_div;auto with zarith.
-    rewrite spec_0;rewrite Z.mul_0_l;rewrite Z.add_0_l.
-    assert (([|high n a|]/2^(Zpos w_digits - [|w_head0 b|])) < wB).
-     apply Zdiv_lt_upper_bound;auto with zarith.
-     apply Z.lt_le_trans with wB;auto with zarith.
-     pattern wB at 1;replace wB with (wB*1);try ring.
-     apply Z.mul_le_mono_nonneg;auto with zarith.
-     assert (H6 := Z.pow_pos_nonneg 2 (Zpos w_digits - [|w_head0 b|]));
-       auto with zarith.
-    rewrite Zmod_small;auto with zarith.
-    apply Zdiv_lt_upper_bound;auto with zarith.
-    apply Z.lt_le_trans with wB;auto with zarith.
-    apply Z.le_trans with (2 ^ [|w_head0 b|] * [|b|] * 2).
-    rewrite <- wB_div_2; try omega.
-    apply Z.mul_le_mono_nonneg;auto with zarith.
-    pattern 2 at 1;rewrite <- Z.pow_1_r.
-    apply Zpower_le_monotone;split;auto with zarith.
-   rewrite <- H4 in H0.
-   assert (Hb3: [|w_head0 b|] <= Zpos w_digits); auto with zarith.
-   assert (H7:= spec_double_divn1_p H0 Hb3 n a (double_0 w_0 n) H6).
-   destruct (double_divn1_p (w_add_mul_div (w_head0 b) b w_0) (w_head0 b) n
-             (w_add_mul_div (w_head0 b) w_0 (high n a)) a
-             (double_0 w_0 n)) as (q,r).
-   assert (U:= spec_double_digits n).
-   rewrite spec_double_0 in H7;trivial;rewrite Zdiv_0_l in H7.
-   rewrite Z.add_0_r in H7.
-   rewrite spec_add_mul_div in H7;auto with zarith.
-   rewrite spec_0 in H7;rewrite Z.mul_0_l in H7;rewrite Z.add_0_l in H7.
-   assert (([|high n a|] / 2 ^ (Zpos w_digits - [|w_head0 b|])) mod wB
-     = [!n|a!] / 2^(Zpos (w_digits << n) - [|w_head0 b|])).
-   rewrite Zmod_small;auto with zarith.
-   rewrite spec_high. rewrite Zdiv_Zdiv;auto with zarith.
-   rewrite <- Zpower_exp;auto with zarith.
-   replace (Zpos (w_digits << n) - Zpos w_digits +
-                       (Zpos w_digits - [|w_head0 b|]))
-    with (Zpos (w_digits << n) - [|w_head0 b|]);trivial;ring.
-   assert (H8 := Z.pow_pos_nonneg 2  (Zpos w_digits - [|w_head0 b|]));auto with zarith.
-   split;auto with zarith.
-   apply Z.le_lt_trans with ([|high n a|]);auto with zarith.
-   apply Zdiv_le_upper_bound;auto with zarith.
-   pattern ([|high n a|]) at 1;rewrite <- Z.mul_1_r.
-   apply Z.mul_le_mono_nonneg;auto with zarith.
-   rewrite H8 in H7;unfold double_wB,base in H7.
-   rewrite <- shift_unshift_mod in H7;auto with zarith.
-   rewrite H4 in H7.
-   assert ([|w_add_mul_div (w_sub w_zdigits (w_head0 b)) w_0 r|]
-               = [|r|]/2^[|w_head0 b|]).
-   rewrite spec_add_mul_div.
-   rewrite spec_0;rewrite Z.mul_0_l;rewrite Z.add_0_l.
-   replace (Zpos w_digits - [|w_sub w_zdigits (w_head0 b)|])
-      with ([|w_head0 b|]).
-   rewrite Zmod_small;auto with zarith.
-   assert (H9 := spec_to_Z r).
-   split;auto with zarith.
-   apply Z.le_lt_trans with ([|r|]);auto with zarith.
-   apply Zdiv_le_upper_bound;auto with zarith.
-   pattern ([|r|]) at 1;rewrite <- Z.mul_1_r.
-   apply Z.mul_le_mono_nonneg;auto with zarith.
-   assert (H10 := Z.pow_pos_nonneg 2  ([|w_head0 b|]));auto with zarith.
-   rewrite spec_sub.
-   rewrite Zmod_small; auto with zarith.
-   split; auto with zarith.
-   case (spec_to_Z w_zdigits); auto with zarith.
-   rewrite spec_sub.
-   rewrite Zmod_small; auto with zarith.
-   split; auto with zarith.
-   case (spec_to_Z w_zdigits); auto with zarith.
-  case H7; intros H71 H72.
-   split.
-   rewrite <- (Z_div_mult [!n|a!] (2^[|w_head0 b|]));auto with zarith.
-   rewrite H71;rewrite H9.
-   replace ([!n|q!] * (2 ^ [|w_head0 b|] * [|b|]))
-      with ([!n|q!] *[|b|] * 2^[|w_head0 b|]);
-     try (ring;fail).
-   rewrite Z_div_plus_l;auto with zarith.
-   assert (H10 := spec_to_Z
-       (w_add_mul_div (w_sub w_zdigits (w_head0 b)) w_0 r));split;
-    auto with zarith.
-   rewrite H9.
-   apply Zdiv_lt_upper_bound;auto with zarith.
-   rewrite Z.mul_comm;auto with zarith.
-   exact (spec_double_to_Z w_digits w_to_Z spec_to_Z n a).
- Qed.
-
-
- Definition double_modn1 (n:nat) (a:word w n) (b:w) :=
-  let p := w_head0 b in
-  match w_compare p w_0 with
-  | Gt =>
-    let b2p := w_add_mul_div p b w_0 in
-    let ha := high n a in
-    let k := w_sub w_zdigits p in
-    let lsr_n := w_add_mul_div k w_0 in
-    let r0 := w_add_mul_div p w_0 ha in
-    let r := double_modn1_p b2p p n r0 a (double_0 w_0 n) in
-    lsr_n r
-  | _ => double_modn1_0 b n w_0 a
-  end.
-
- Lemma spec_double_modn1_aux : forall n a b,
-    double_modn1 n a b = snd (double_divn1 n a b).
- Proof.
-  intros n a b;unfold double_divn1,double_modn1.
-  rewrite spec_compare; case Z.compare_spec;
-     rewrite spec_0; intros H2; auto with zarith.
-  apply spec_double_modn1_0.
-  apply spec_double_modn1_0.
-  rewrite spec_double_modn1_p.
-  destruct (double_divn1_p (w_add_mul_div (w_head0 b) b w_0) (w_head0 b) n
-            (w_add_mul_div (w_head0 b) w_0 (high n a)) a (double_0 w_0 n));simpl;trivial.
- Qed.
-
- Lemma spec_double_modn1 : forall n a b, 0 < [|b|] ->
-  [|double_modn1 n a b|] = [!n|a!] mod [|b|].
- Proof.
-  intros n a b H;assert (H1 := spec_double_divn1 n a H).
-  assert (H2 := spec_double_modn1_aux n a b).
-  rewrite H2;destruct (double_divn1 n a b) as (q,r).
-  simpl;apply Zmod_unique with (double_to_Z w_digits w_to_Z n q);auto with zarith.
-  destruct H1 as (h1,h2);rewrite h1;ring.
- Qed.
-
-End GENDIVN1.
diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleLift.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleLift.v
deleted file mode 100644
index f65b47c8c..000000000
--- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleLift.v
+++ /dev/null
@@ -1,475 +0,0 @@
-(************************************************************************)
-(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016     *)
-(*   \VV/  **************************************************************)
-(*    //   *      This file is distributed under the terms of the       *)
-(*         *       GNU Lesser General Public License Version 2.1        *)
-(************************************************************************)
-(*            Benjamin Gregoire, Laurent Thery, INRIA, 2007             *)
-(************************************************************************)
-
-Set Implicit Arguments.
-
-Require Import ZArith.
-Require Import BigNumPrelude.
-Require Import DoubleType.
-Require Import DoubleBase.
-
-Local Open Scope Z_scope.
-
-Section DoubleLift.
- Variable w : Type.
- Variable w_0 : w.
- Variable w_WW : w -> w -> zn2z w.
- Variable w_W0 : w ->  zn2z w.
- Variable w_0W : w ->  zn2z w.
- Variable w_compare : w -> w -> comparison.
- Variable ww_compare : zn2z w -> zn2z w -> comparison.
- Variable w_head0 : w -> w.
- Variable w_tail0 : w -> w.
- Variable w_add: w -> w -> zn2z w.
- Variable w_add_mul_div : w -> w -> w -> w.
- Variable ww_sub: zn2z w -> zn2z w -> zn2z w.
- Variable w_digits : positive.
- Variable ww_Digits : positive.
- Variable w_zdigits : w.
- Variable ww_zdigits : zn2z w.
- Variable low: zn2z w -> w.
-
- Definition ww_head0 x :=
-  match x with
-  | W0 => ww_zdigits
-  | WW xh xl =>
-    match w_compare w_0 xh with
-    | Eq => w_add w_zdigits (w_head0 xl)
-    | _ => w_0W (w_head0 xh)
-    end
-  end.
-
-
- Definition ww_tail0 x :=
-  match x with
-  | W0 => ww_zdigits
-  | WW xh xl =>
-    match w_compare w_0 xl with
-    | Eq => w_add w_zdigits (w_tail0 xh)
-    | _ => w_0W (w_tail0 xl)
-    end
-  end.
-
-
-  (* 0 < p < ww_digits *)
- Definition ww_add_mul_div p x y :=
-  let zdigits := w_0W w_zdigits in
-  match x, y with
-  | W0, W0 => W0
-  | W0, WW yh yl =>
-    match ww_compare p zdigits with
-    | Eq => w_0W yh
-    | Lt => w_0W (w_add_mul_div (low p) w_0 yh)
-    | Gt =>
-      let n := low (ww_sub p zdigits) in
-      w_WW (w_add_mul_div n w_0 yh) (w_add_mul_div n yh yl)
-    end
-  | WW xh xl, W0 =>
-    match ww_compare p zdigits with
-    | Eq => w_W0 xl
-    | Lt => w_WW (w_add_mul_div (low p) xh xl) (w_add_mul_div (low p) xl w_0)
-    | Gt =>
-      let n := low (ww_sub p zdigits) in
-      w_W0 (w_add_mul_div n xl w_0)
-    end
-  | WW xh xl, WW yh yl =>
-    match ww_compare p zdigits with
-    | Eq => w_WW xl yh
-    | Lt => w_WW (w_add_mul_div (low p) xh xl) (w_add_mul_div (low p) xl yh)
-    | Gt =>
-      let n := low (ww_sub p zdigits) in
-      w_WW (w_add_mul_div n xl yh) (w_add_mul_div n yh yl)
-    end
-  end.
-
- Section DoubleProof.
-  Variable w_to_Z : w -> Z.
-
-  Notation wB  := (base w_digits).
-  Notation wwB := (base (ww_digits w_digits)).
-  Notation "[| x |]" := (w_to_Z x)  (at level 0, x at level 99).
-  Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99).
-
-  Variable spec_w_0   : [|w_0|] = 0.
-  Variable spec_to_Z  : forall x, 0 <= [|x|] < wB.
-  Variable spec_to_w_Z  : forall x, 0 <= [[x]] < wwB.
-  Variable spec_w_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|].
-  Variable spec_w_W0 : forall h, [[w_W0 h]] = [|h|] * wB.
-  Variable spec_w_0W : forall l, [[w_0W l]] = [|l|].
-  Variable spec_compare : forall x y,
-   w_compare x y = Z.compare [|x|] [|y|].
-  Variable spec_ww_compare : forall x y,
-   ww_compare x y = Z.compare [[x]] [[y]].
-  Variable spec_ww_digits : ww_Digits = xO w_digits.
-  Variable spec_w_head00  : forall x, [|x|] = 0 -> [|w_head0 x|] = Zpos w_digits.
-  Variable spec_w_head0  : forall x,  0 < [|x|] ->
-	 wB/ 2 <= 2 ^ ([|w_head0 x|]) * [|x|] < wB.
-  Variable spec_w_tail00  : forall x, [|x|] = 0 -> [|w_tail0 x|] = Zpos w_digits.
-  Variable spec_w_tail0 : forall x, 0 < [|x|] ->
-         exists y, 0 <= y /\ [|x|] = (2* y + 1) * (2 ^ [|w_tail0 x|]).
-  Variable spec_w_add_mul_div : forall x y p,
-       [|p|] <= Zpos w_digits ->
-       [| w_add_mul_div p x y |] =
-         ([|x|] * (2 ^ [|p|]) +
-          [|y|] / (2 ^ ((Zpos w_digits) - [|p|]))) mod wB.
- Variable spec_w_add: forall x y,
-   [[w_add x y]] = [|x|] + [|y|].
- Variable spec_ww_sub: forall x y,
-   [[ww_sub x y]] = ([[x]] - [[y]]) mod wwB.
-
- Variable spec_zdigits : [| w_zdigits |] = Zpos w_digits.
- Variable spec_low: forall x, [| low x|] = [[x]] mod wB.
-
- Variable spec_ww_zdigits : [[ww_zdigits]] = Zpos ww_Digits.
-
-  Hint Resolve div_le_0 div_lt w_to_Z_wwB: lift.
-  Ltac zarith := auto with zarith lift.
-
-  Lemma spec_ww_head00  : forall x, [[x]] = 0 -> [[ww_head0 x]] = Zpos ww_Digits.
-  Proof.
-  intros x; case x; unfold ww_head0.
-    intros HH; rewrite spec_ww_zdigits; auto.
-  intros xh xl; simpl; intros Hx.
-  case (spec_to_Z xh); intros Hx1 Hx2.
-  case (spec_to_Z xl); intros Hy1 Hy2.
-  assert (F1: [|xh|] = 0).
-  { Z.le_elim Hy1; auto.
-    - absurd (0 < [|xh|] * wB + [|xl|]); auto with zarith.
-      apply Z.lt_le_trans with (1 := Hy1); auto with zarith.
-      pattern [|xl|] at 1; rewrite <- (Z.add_0_l [|xl|]).
-      apply Z.add_le_mono_r; auto with zarith.
-    - Z.le_elim Hx1; auto.
-      absurd (0 < [|xh|] * wB + [|xl|]); auto with zarith.
-      rewrite <- Hy1; rewrite Z.add_0_r; auto with zarith.
-      apply Z.mul_pos_pos; auto with zarith. }
-  rewrite spec_compare. case Z.compare_spec.
-    intros H; simpl.
-    rewrite spec_w_add; rewrite spec_w_head00.
-    rewrite spec_zdigits; rewrite spec_ww_digits.
-    rewrite Pos2Z.inj_xO; auto with zarith.
-  rewrite F1 in Hx; auto with zarith.
-  rewrite spec_w_0; auto with zarith.
-  rewrite spec_w_0; auto with zarith.
-  Qed.
-
-  Lemma spec_ww_head0  : forall x,  0 < [[x]] ->
-	 wwB/ 2 <= 2 ^ [[ww_head0 x]] * [[x]] < wwB.
-  Proof.
-   clear spec_ww_zdigits.
-   rewrite wwB_div_2;rewrite Z.mul_comm;rewrite wwB_wBwB.
-   assert (U:= lt_0_wB w_digits); destruct x as [ |xh xl];simpl ww_to_Z;intros H.
-   unfold Z.lt in H;discriminate H.
-   rewrite spec_compare, spec_w_0. case Z.compare_spec; intros H0.
-   rewrite <- H0 in *. simpl Z.add. simpl in H.
-   case (spec_to_Z w_zdigits);
-     case (spec_to_Z (w_head0 xl)); intros HH1 HH2 HH3 HH4.
-   rewrite spec_w_add.
-   rewrite spec_zdigits; rewrite Zpower_exp; auto with zarith.
-   case (spec_w_head0 H); intros H1 H2.
-   rewrite Z.pow_2_r; fold wB; rewrite <- Z.mul_assoc; split.
-     apply Z.mul_le_mono_nonneg_l; auto with zarith.
-   apply Z.mul_lt_mono_pos_l; auto with zarith.
-   assert (H1 := spec_w_head0 H0).
-   rewrite spec_w_0W.
-   split.
-   rewrite Z.mul_add_distr_l;rewrite Z.mul_assoc.
-   apply Z.le_trans with (2 ^ [|w_head0 xh|] * [|xh|] * wB).
-   rewrite Z.mul_comm; zarith.
-   assert (0 <= 2 ^ [|w_head0 xh|] * [|xl|]);zarith.
-   assert (H2:=spec_to_Z xl);apply Z.mul_nonneg_nonneg;zarith.
-   case (spec_to_Z (w_head0 xh)); intros H2 _.
-   generalize ([|w_head0 xh|]) H1 H2;clear H1 H2;
-     intros p H1 H2.
-   assert (Eq1 : 2^p < wB).
-     rewrite <- (Z.mul_1_r (2^p));apply Z.le_lt_trans with (2^p*[|xh|]);zarith.
-   assert (Eq2: p < Zpos w_digits).
-    destruct (Z.le_gt_cases (Zpos w_digits) p);trivial;contradict  Eq1.
-   apply Z.le_ngt;unfold base;apply Zpower_le_monotone;zarith.
-   assert (Zpos w_digits = p + (Zpos w_digits - p)). ring.
-   rewrite Z.pow_2_r.
-   unfold base at 2;rewrite H3;rewrite Zpower_exp;zarith.
-   rewrite <- Z.mul_assoc; apply Z.mul_lt_mono_pos_l; zarith.
-   rewrite <- (Z.add_0_r (2^(Zpos w_digits - p)*wB));apply beta_lex_inv;zarith.
-   apply Z.mul_lt_mono_pos_r with (2 ^ p); zarith.
-   rewrite <- Zpower_exp;zarith.
-   rewrite Z.mul_comm;ring_simplify (Zpos w_digits - p + p);fold wB;zarith.
-   assert (H1 := spec_to_Z xh);zarith.
-  Qed.
-
-  Lemma spec_ww_tail00  : forall x, [[x]] = 0 -> [[ww_tail0 x]] = Zpos ww_Digits.
-  Proof.
-  intros x; case x; unfold ww_tail0.
-    intros HH; rewrite spec_ww_zdigits; auto.
-  intros xh xl; simpl; intros Hx.
-  case (spec_to_Z xh); intros Hx1 Hx2.
-  case (spec_to_Z xl); intros Hy1 Hy2.
-  assert (F1: [|xh|] = 0).
-  { Z.le_elim Hy1; auto.
-    - absurd (0 < [|xh|] * wB + [|xl|]); auto with zarith.
-      apply Z.lt_le_trans with (1 := Hy1); auto with zarith.
-      pattern [|xl|] at 1; rewrite <- (Z.add_0_l [|xl|]).
-      apply Z.add_le_mono_r; auto with zarith.
-    - Z.le_elim Hx1; auto.
-      absurd (0 < [|xh|] * wB + [|xl|]); auto with zarith.
-      rewrite <- Hy1; rewrite Z.add_0_r; auto with zarith.
-      apply Z.mul_pos_pos; auto with zarith. }
-  assert (F2: [|xl|] = 0).
-    rewrite F1 in Hx; auto with zarith.
-  rewrite spec_compare; case Z.compare_spec.
-    intros H; simpl.
-    rewrite spec_w_add; rewrite spec_w_tail00; auto.
-    rewrite spec_zdigits; rewrite spec_ww_digits.
-    rewrite Pos2Z.inj_xO; auto with zarith.
-  rewrite spec_w_0; auto with zarith.
-  rewrite spec_w_0; auto with zarith.
-  Qed.
-
-  Lemma spec_ww_tail0  : forall x,  0 < [[x]] ->
-	 exists y, 0 <= y /\ [[x]] = (2 * y + 1) * 2 ^ [[ww_tail0 x]].
-  Proof.
-   clear spec_ww_zdigits.
-   destruct x as [ |xh xl];simpl ww_to_Z;intros H.
-   unfold Z.lt in H;discriminate H.
-   rewrite spec_compare, spec_w_0. case Z.compare_spec; intros H0.
-   rewrite <- H0; rewrite Z.add_0_r.
-   case (spec_to_Z (w_tail0 xh)); intros HH1 HH2.
-   generalize H; rewrite <- H0; rewrite Z.add_0_r; clear H; intros H.
-   case (@spec_w_tail0 xh).
-     apply Z.mul_lt_mono_pos_r with wB; auto with zarith.
-     unfold base; auto with zarith.
-   intros z (Hz1, Hz2); exists z; split; auto.
-   rewrite spec_w_add; rewrite (fun x => Z.add_comm [|x|]).
-   rewrite spec_zdigits; rewrite Zpower_exp; auto with zarith.
-   rewrite Z.mul_assoc; rewrite <- Hz2; auto.
-
-   case (spec_to_Z (w_tail0 xh)); intros HH1 HH2.
-   case (spec_w_tail0 H0); intros z (Hz1, Hz2).
-   assert (Hp: [|w_tail0 xl|] < Zpos w_digits).
-     case (Z.le_gt_cases (Zpos w_digits) [|w_tail0 xl|]); auto; intros H1.
-     absurd (2 ^  (Zpos w_digits) <= 2 ^ [|w_tail0 xl|]).
-       apply Z.lt_nge.
-       case (spec_to_Z xl); intros HH3 HH4.
-       apply Z.le_lt_trans with (2 := HH4).
-       apply Z.le_trans with (1 * 2 ^ [|w_tail0 xl|]); auto with zarith.
-       rewrite Hz2.
-       apply Z.mul_le_mono_nonneg_r; auto with zarith.
-     apply Zpower_le_monotone; auto with zarith.
-   exists ([|xh|] * (2 ^ ((Zpos w_digits - [|w_tail0 xl|]) - 1)) + z); split.
-     apply Z.add_nonneg_nonneg; auto.
-     apply Z.mul_nonneg_nonneg; auto with zarith.
-     case (spec_to_Z xh); auto.
-   rewrite spec_w_0W.
-   rewrite (Z.mul_add_distr_l 2); rewrite <- Z.add_assoc.
-   rewrite Z.mul_add_distr_r; rewrite <- Hz2.
-   apply f_equal2 with (f := Z.add); auto.
-   rewrite (Z.mul_comm 2).
-   repeat rewrite <- Z.mul_assoc.
-   apply f_equal2 with (f := Z.mul); auto.
-   case (spec_to_Z (w_tail0 xl)); intros HH3 HH4.
-   pattern 2 at 2; rewrite <- Z.pow_1_r.
-   lazy beta; repeat rewrite <- Zpower_exp; auto with zarith.
-   unfold base; apply f_equal with (f := Z.pow 2); auto with zarith.
-
-   contradict H0; case (spec_to_Z xl); auto with zarith.
-  Qed.
-
-  Hint Rewrite Zdiv_0_l Z.mul_0_l Z.add_0_l Z.mul_0_r Z.add_0_r
-    spec_w_W0 spec_w_0W spec_w_WW spec_w_0
-    (wB_div w_digits w_to_Z spec_to_Z)
-    (wB_div_plus w_digits w_to_Z spec_to_Z) : w_rewrite.
-  Ltac w_rewrite := autorewrite with w_rewrite;trivial.
-
-  Lemma spec_ww_add_mul_div_aux : forall xh xl yh yl p,
-   let zdigits := w_0W w_zdigits in
-    [[p]] <= Zpos (xO w_digits) ->
-      [[match ww_compare p zdigits with
-        | Eq => w_WW xl yh
-        | Lt => w_WW (w_add_mul_div (low p) xh xl)
-                     (w_add_mul_div (low p) xl yh)
-        | Gt =>
-              let n := low (ww_sub p zdigits) in
-            w_WW (w_add_mul_div n xl yh) (w_add_mul_div n yh yl)
-        end]] =
-      ([[WW xh xl]] * (2^[[p]]) +
-       [[WW yh yl]] / (2^(Zpos (xO w_digits) - [[p]]))) mod wwB.
-  Proof.
-   clear spec_ww_zdigits.
-   intros xh xl yh yl p zdigits;assert (HwwB := wwB_pos w_digits).
-   case (spec_to_w_Z p); intros Hv1 Hv2.
-   replace (Zpos (xO w_digits)) with (Zpos w_digits + Zpos w_digits).
-   2 : rewrite Pos2Z.inj_xO;ring.
-   replace (Zpos w_digits + Zpos w_digits - [[p]]) with
-     (Zpos w_digits + (Zpos w_digits - [[p]])). 2:ring.
-   intros Hp; assert (Hxh := spec_to_Z xh);assert (Hxl:=spec_to_Z xl);
-   assert (Hx := spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW xh xl));
-   simpl in Hx;assert (Hyh := spec_to_Z yh);assert (Hyl:=spec_to_Z yl);
-   assert (Hy:=spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW yh yl));simpl in Hy.
-   rewrite spec_ww_compare; case Z.compare_spec; intros H1.
-   rewrite H1; unfold zdigits; rewrite spec_w_0W.
-   rewrite spec_zdigits; rewrite Z.sub_diag; rewrite Z.add_0_r.
-   simpl ww_to_Z; w_rewrite;zarith.
-   fold wB.
-   rewrite Z.mul_add_distr_r;rewrite <- Z.mul_assoc;rewrite <- Z.add_assoc.
-   rewrite <- Z.pow_2_r.
-   rewrite <- wwB_wBwB;apply Zmod_unique with [|xh|].
-   exact (spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW xl yh)). ring.
-   simpl ww_to_Z; w_rewrite;zarith.
-   assert (HH0: [|low p|] = [[p]]).
-     rewrite spec_low.
-     apply Zmod_small.
-     case (spec_to_w_Z p); intros HH1 HH2; split; auto.
-     generalize H1; unfold zdigits; rewrite spec_w_0W;
-      rewrite spec_zdigits; intros tmp.
-     apply Z.lt_le_trans with (1 := tmp).
-     unfold base.
-     apply Zpower2_le_lin; auto with zarith.
-   2: generalize H1; unfold zdigits; rewrite spec_w_0W;
-      rewrite spec_zdigits; auto with zarith.
-   generalize H1; unfold zdigits; rewrite spec_w_0W;
-      rewrite spec_zdigits; auto; clear H1; intros H1.
-   assert (HH: [|low p|] <= Zpos w_digits).
-     rewrite HH0; auto with zarith.
-   repeat rewrite spec_w_add_mul_div with (1 := HH).
-   rewrite HH0.
-   rewrite Z.mul_add_distr_r.
-   pattern ([|xl|] * 2 ^ [[p]]) at 2;
-   rewrite shift_unshift_mod with (n:= Zpos w_digits);fold wB;zarith.
-   replace ([|xh|] * wB * 2^[[p]]) with ([|xh|] * 2^[[p]] * wB). 2:ring.
-   rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. rewrite <- Z.add_assoc.
-   unfold base at 5;rewrite <- Zmod_shift_r;zarith.
-   unfold base;rewrite Zmod_shift_r with (b:= Zpos (ww_digits w_digits));
-   fold wB;fold wwB;zarith.
-   rewrite wwB_wBwB;rewrite Z.pow_2_r; rewrite  Zmult_mod_distr_r;zarith.
-   unfold ww_digits;rewrite Pos2Z.inj_xO;zarith. apply Z_mod_lt;zarith.
-   split;zarith. apply Zdiv_lt_upper_bound;zarith.
-   rewrite <- Zpower_exp;zarith.
-   ring_simplify ([[p]] + (Zpos w_digits - [[p]]));fold wB;zarith.
-   assert (Hv: [[p]] > Zpos w_digits).
-     generalize H1; clear H1.
-     unfold zdigits; rewrite spec_w_0W; rewrite spec_zdigits; auto with zarith.
-   clear H1.
-   assert (HH0: [|low (ww_sub p zdigits)|] = [[p]] - Zpos w_digits).
-     rewrite spec_low.
-     rewrite spec_ww_sub.
-     unfold zdigits; rewrite spec_w_0W; rewrite spec_zdigits.
-     rewrite <- Zmod_div_mod; auto with zarith.
-     rewrite Zmod_small; auto with zarith.
-     split; auto with zarith.
-     apply Z.le_lt_trans with (Zpos w_digits); auto with zarith.
-     unfold base; apply Zpower2_lt_lin; auto with zarith.
-     exists wB; unfold base.
-     unfold ww_digits; rewrite (Pos2Z.inj_xO w_digits).
-     rewrite <- Zpower_exp; auto with zarith.
-     apply f_equal with (f := fun x => 2 ^ x); auto with zarith.
-   assert (HH: [|low (ww_sub p zdigits)|] <= Zpos w_digits).
-     rewrite HH0; auto with zarith.
-   replace (Zpos w_digits + (Zpos w_digits - [[p]])) with
-       (Zpos w_digits - ([[p]] - Zpos w_digits)); zarith.
-   lazy zeta; simpl ww_to_Z; w_rewrite;zarith.
-   repeat rewrite spec_w_add_mul_div;zarith.
-   rewrite HH0.
-   pattern wB at 5;replace wB with
-    (2^(([[p]] - Zpos w_digits)
-         + (Zpos w_digits - ([[p]] - Zpos w_digits)))).
-   rewrite Zpower_exp;zarith. rewrite Z.mul_assoc.
-   rewrite Z_div_plus_l;zarith.
-   rewrite shift_unshift_mod with (a:= [|yh|]) (p:= [[p]] - Zpos w_digits)
-    (n := Zpos w_digits);zarith. fold wB.
-   set (u := [[p]] - Zpos w_digits).
-   replace [[p]] with (u + Zpos w_digits);zarith.
-   rewrite Zpower_exp;zarith. rewrite Z.mul_assoc. fold wB.
-   repeat rewrite Z.add_assoc. rewrite <- Z.mul_add_distr_r.
-   repeat rewrite <- Z.add_assoc.
-   unfold base;rewrite Zmod_shift_r with (b:= Zpos (ww_digits w_digits));
-   fold wB;fold wwB;zarith.
-   unfold base;rewrite Zmod_shift_r with (a:= Zpos w_digits)
-   (b:= Zpos w_digits);fold wB;fold wwB;zarith.
-   rewrite wwB_wBwB; rewrite Z.pow_2_r; rewrite  Zmult_mod_distr_r;zarith.
-   rewrite Z.mul_add_distr_r.
-   replace ([|xh|] * wB * 2 ^ u) with
-     ([|xh|]*2^u*wB). 2:ring.
-   repeat rewrite <- Z.add_assoc.
-   rewrite (Z.add_comm ([|xh|] * 2 ^ u * wB)).
-   rewrite Z_mod_plus;zarith. rewrite Z_mod_mult;zarith.
-   unfold base;rewrite <- Zmod_shift_r;zarith. fold base;apply Z_mod_lt;zarith.
-   unfold u; split;zarith.
-   split;zarith. unfold u; apply Zdiv_lt_upper_bound;zarith.
-   rewrite <- Zpower_exp;zarith.
-   fold u.
-   ring_simplify (u + (Zpos w_digits - u)); fold
-   wB;zarith. unfold ww_digits;rewrite Pos2Z.inj_xO;zarith.
-   unfold base;rewrite <- Zmod_shift_r;zarith. fold base;apply Z_mod_lt;zarith.
-   unfold u; split;zarith.
-   unfold u; split;zarith.
-   apply Zdiv_lt_upper_bound;zarith.
-   rewrite <- Zpower_exp;zarith.
-   fold u.
-   ring_simplify (u + (Zpos w_digits - u)); fold wB; auto with zarith.
-   unfold u;zarith.
-   unfold u;zarith.
-   set (u := [[p]] - Zpos w_digits).
-   ring_simplify (u + (Zpos w_digits - u)); fold wB; auto with zarith.
-  Qed.
-
-  Lemma spec_ww_add_mul_div : forall x y p,
-       [[p]] <= Zpos (xO w_digits) ->
-       [[ ww_add_mul_div p x y ]] =
-         ([[x]] * (2^[[p]]) +
-          [[y]] / (2^(Zpos (xO w_digits) - [[p]]))) mod wwB.
-  Proof.
-   clear spec_ww_zdigits.
-   intros x y p H.
-   destruct x as [ |xh xl];
-   [assert (H1 := @spec_ww_add_mul_div_aux w_0 w_0)
-   |assert (H1 := @spec_ww_add_mul_div_aux xh xl)];
-   (destruct y as [ |yh yl];
-    [generalize (H1 w_0 w_0 p H) | generalize (H1 yh yl p H)];
-    clear H1;w_rewrite);simpl ww_add_mul_div.
-   replace [[WW w_0 w_0]] with 0;[w_rewrite|simpl;w_rewrite;trivial].
-   intros Heq;rewrite <- Heq;clear Heq; auto.
-   rewrite spec_ww_compare. case Z.compare_spec; intros H1; w_rewrite.
-   rewrite (spec_w_add_mul_div w_0 w_0);w_rewrite;zarith.
-   generalize H1; w_rewrite; rewrite spec_zdigits; clear H1; intros H1.
-   assert (HH0: [|low p|] = [[p]]).
-     rewrite spec_low.
-     apply Zmod_small.
-     case (spec_to_w_Z p); intros HH1 HH2; split; auto.
-     apply Z.lt_le_trans with (1 := H1).
-     unfold base; apply Zpower2_le_lin; auto with zarith.
-   rewrite HH0; auto with zarith.
-   replace [[WW w_0 w_0]] with 0;[w_rewrite|simpl;w_rewrite;trivial].
-   intros Heq;rewrite <- Heq;clear Heq.
-   generalize (spec_ww_compare p (w_0W w_zdigits));
-     case ww_compare; intros H1; w_rewrite.
-   rewrite (spec_w_add_mul_div w_0 w_0);w_rewrite;zarith.
-   rewrite Pos2Z.inj_xO in H;zarith.
-   assert (HH: [|low (ww_sub p (w_0W w_zdigits)) |] = [[p]] - Zpos w_digits).
-     symmetry in H1; change ([[p]] > [[w_0W w_zdigits]]) in H1.
-     revert H1.
-     rewrite spec_low.
-     rewrite spec_ww_sub; w_rewrite; intros H1.
-     rewrite <- Zmod_div_mod; auto with zarith.
-     rewrite Zmod_small; auto with zarith.
-     split; auto with zarith.
-     apply Z.le_lt_trans with (Zpos w_digits); auto with zarith.
-     unfold base; apply Zpower2_lt_lin; auto with zarith.
-     unfold base; auto with zarith.
-     unfold base; auto with zarith.
-     exists wB; unfold base.
-     unfold ww_digits; rewrite (Pos2Z.inj_xO w_digits).
-     rewrite <- Zpower_exp; auto with zarith.
-     apply f_equal with (f := fun x => 2 ^ x); auto with zarith.
-  case (spec_to_Z xh); auto with zarith.
-  Qed.
-
- End DoubleProof.
-
-End DoubleLift.
-
diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleMul.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleMul.v
deleted file mode 100644
index b99013900..000000000
--- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleMul.v
+++ /dev/null
@@ -1,621 +0,0 @@
-(************************************************************************)
-(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016     *)
-(*   \VV/  **************************************************************)
-(*    //   *      This file is distributed under the terms of the       *)
-(*         *       GNU Lesser General Public License Version 2.1        *)
-(************************************************************************)
-(*            Benjamin Gregoire, Laurent Thery, INRIA, 2007             *)
-(************************************************************************)
-
-Set Implicit Arguments.
-
-Require Import ZArith.
-Require Import BigNumPrelude.
-Require Import DoubleType.
-Require Import DoubleBase.
-
-Local Open Scope Z_scope.
-
-Section DoubleMul.
- Variable w : Type.
- Variable w_0 : w.
- Variable w_1 : w.
- Variable w_WW : w -> w -> zn2z w.
- Variable w_W0 : w -> zn2z w.
- Variable w_0W : w -> zn2z w.
- Variable w_compare : w -> w -> comparison.
- Variable w_succ : w -> w.
- Variable w_add_c : w -> w -> carry w.
- Variable w_add : w -> w -> w.
- Variable w_sub: w -> w -> w.
- Variable w_mul_c : w -> w -> zn2z w.
- Variable w_mul : w -> w -> w.
- Variable w_square_c : w -> zn2z w.
- Variable ww_add_c : zn2z w -> zn2z w -> carry (zn2z w).
- Variable ww_add   : zn2z w -> zn2z w -> zn2z w.
- Variable ww_add_carry : zn2z w -> zn2z w -> zn2z w.
- Variable ww_sub_c : zn2z w -> zn2z w -> carry (zn2z w).
- Variable ww_sub   : zn2z w -> zn2z w -> zn2z w.
-
- (* ** Multiplication ** *)
-
- (*   (xh*B+xl) (yh*B + yl)
-   xh*yh         = hh  = |hhh|hhl|B2
-   xh*yl +xl*yh  = cc  =     |cch|ccl|B
-   xl*yl         = ll  =         |llh|lll
-  *)
-
- Definition double_mul_c (cross:w->w->w->w->zn2z w -> zn2z w -> w*zn2z w) x y :=
-  match x, y with
-  | W0, _ => W0
-  | _, W0 => W0
-  | WW xh xl, WW yh yl =>
-    let hh := w_mul_c xh yh in
-    let ll := w_mul_c xl yl in
-    let (wc,cc) := cross xh xl yh yl hh ll in
-    match cc with
-    | W0 => WW (ww_add hh (w_W0 wc)) ll
-    | WW cch ccl =>
-      match ww_add_c (w_W0 ccl) ll with
-      | C0 l => WW (ww_add hh (w_WW wc cch)) l
-      | C1 l => WW (ww_add_carry hh (w_WW wc cch)) l
-      end
-    end
-  end.
-
- Definition ww_mul_c :=
-  double_mul_c
-    (fun xh xl yh yl hh ll=>
-      match ww_add_c (w_mul_c xh yl) (w_mul_c xl yh) with
-      | C0 cc => (w_0, cc)
-      | C1 cc => (w_1, cc)
-      end).
-
- Definition w_2 := w_add w_1 w_1.
-
- Definition kara_prod xh xl yh yl hh ll :=
-    match ww_add_c hh ll with
-      C0 m =>
-         match w_compare xl xh with
-           Eq => (w_0, m)
-         | Lt =>
-           match w_compare yl yh with
-             Eq => (w_0, m)
-           | Lt => (w_0, ww_sub m (w_mul_c (w_sub xh xl) (w_sub yh yl)))
-           | Gt => match ww_add_c m (w_mul_c (w_sub xh xl) (w_sub yl yh)) with
-                      C1 m1 => (w_1, m1) | C0 m1 => (w_0, m1)
-                   end
-           end
-         | Gt =>
-           match w_compare yl yh with
-             Eq => (w_0, m)
-           | Lt => match ww_add_c m (w_mul_c (w_sub xl xh)  (w_sub yh yl)) with
-                      C1 m1 => (w_1, m1) | C0 m1 => (w_0, m1)
-                   end
-           | Gt => (w_0, ww_sub m (w_mul_c (w_sub xl xh)  (w_sub yl yh)))
-           end
-         end
-    | C1 m =>
-         match w_compare xl xh with
-           Eq => (w_1, m)
-         | Lt =>
-           match w_compare yl yh with
-             Eq => (w_1, m)
-           | Lt => match ww_sub_c m (w_mul_c (w_sub xh xl) (w_sub yh yl)) with
-                    C0 m1 => (w_1, m1) | C1 m1 => (w_0, m1)
-                   end
-           | Gt => match ww_add_c m (w_mul_c (w_sub xh xl) (w_sub yl yh)) with
-                      C1 m1 => (w_2, m1) | C0 m1 => (w_1, m1)
-                   end
-           end
-         | Gt =>
-           match w_compare yl yh with
-             Eq => (w_1, m)
-           | Lt => match ww_add_c m (w_mul_c (w_sub xl xh) (w_sub yh yl)) with
-                      C1 m1 => (w_2, m1) | C0 m1 => (w_1, m1)
-                   end
-           | Gt => match ww_sub_c m (w_mul_c (w_sub xl xh) (w_sub yl yh)) with
-                     C1 m1 => (w_0, m1) | C0 m1 => (w_1, m1)
-                   end
-           end
-         end
-    end.
-
- Definition ww_karatsuba_c :=  double_mul_c kara_prod.
-
- Definition ww_mul x y :=
-  match x, y with
-  | W0, _ => W0
-  | _, W0 => W0
-  | WW xh xl, WW yh yl =>
-    let ccl := w_add (w_mul xh yl) (w_mul xl yh) in
-    ww_add (w_W0 ccl) (w_mul_c xl yl)
-  end.
-
- Definition ww_square_c x  :=
-  match x with
-  | W0 => W0
-  | WW xh xl =>
-    let hh := w_square_c xh in
-    let ll := w_square_c xl in
-    let xhxl := w_mul_c xh xl in
-    let (wc,cc) :=
-      match ww_add_c xhxl xhxl with
-      | C0 cc => (w_0, cc)
-      | C1 cc => (w_1, cc)
-      end in
-    match cc with
-    | W0 => WW (ww_add hh (w_W0 wc)) ll
-    | WW cch ccl =>
-      match ww_add_c (w_W0 ccl) ll with
-      | C0 l => WW (ww_add hh (w_WW wc cch)) l
-      | C1 l => WW (ww_add_carry hh (w_WW wc cch)) l
-      end
-    end
-  end.
-
- Section DoubleMulAddn1.
-  Variable w_mul_add : w -> w -> w -> w * w.
-
-  Fixpoint double_mul_add_n1 (n:nat) : word w n -> w -> w -> w * word w n :=
-   match n return word w n -> w -> w -> w * word w n with
-   | O => w_mul_add
-   | S n1 =>
-     let mul_add := double_mul_add_n1 n1 in
-     fun x y r =>
-     match x with
-     | W0 => (w_0,extend w_0W n1 r)
-     | WW xh xl =>
-       let (rl,l) := mul_add xl y r in
-       let (rh,h) := mul_add xh y rl in
-       (rh, double_WW w_WW n1 h l)
-     end
-   end.
-
- End DoubleMulAddn1.
-
- Section DoubleMulAddmn1.
-  Variable wn: Type.
-  Variable extend_n : w -> wn.
-  Variable wn_0W : wn -> zn2z wn.
-  Variable wn_WW : wn -> wn -> zn2z wn.
-  Variable w_mul_add_n1 : wn -> w -> w -> w*wn.
-  Fixpoint double_mul_add_mn1 (m:nat) :
-	word wn m -> w -> w -> w*word wn m :=
-   match m return word wn m -> w -> w -> w*word wn m with
-   | O => w_mul_add_n1
-   | S m1 =>
-     let mul_add := double_mul_add_mn1 m1 in
-     fun x y r =>
-     match x with
-     | W0 => (w_0,extend wn_0W m1 (extend_n r))
-     | WW xh xl =>
-       let (rl,l) := mul_add xl y r in
-       let (rh,h) := mul_add xh y rl in
-       (rh, double_WW wn_WW m1 h l)
-     end
-   end.
-
- End DoubleMulAddmn1.
-
- Definition w_mul_add x y r :=
-  match w_mul_c x y with
-  | W0 => (w_0, r)
-  | WW h l =>
-    match w_add_c l r with
-    | C0 lr => (h,lr)
-    | C1 lr => (w_succ h, lr)
-    end
-  end.
-
-
- (*Section DoubleProof. *)
-  Variable w_digits : positive.
-  Variable w_to_Z : w -> Z.
-
-  Notation wB  := (base w_digits).
-  Notation wwB := (base (ww_digits w_digits)).
-  Notation "[| x |]" := (w_to_Z x)  (at level 0, x at level 99).
-  Notation "[+| c |]" :=
-   (interp_carry 1 wB w_to_Z c) (at level 0, c at level 99).
-  Notation "[-| c |]" :=
-   (interp_carry (-1) wB w_to_Z c) (at level 0, c at level 99).
-
-  Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99).
-  Notation "[+[ c ]]" :=
-   (interp_carry 1 wwB (ww_to_Z w_digits w_to_Z) c)
-   (at level 0, c at level 99).
-  Notation "[-[ c ]]" :=
-   (interp_carry (-1) wwB (ww_to_Z w_digits w_to_Z) c)
-   (at level 0, c at level 99).
-
-  Notation "[|| x ||]" :=
-    (zn2z_to_Z wwB (ww_to_Z w_digits w_to_Z) x)  (at level 0, x at level 99).
-
-  Notation "[! n | x !]" := (double_to_Z w_digits w_to_Z n x)
-    (at level 0, x at level 99).
-
-  Variable spec_more_than_1_digit: 1 < Zpos w_digits.
-  Variable spec_w_0   : [|w_0|] = 0.
-  Variable spec_w_1   : [|w_1|] = 1.
-
-  Variable spec_to_Z  : forall x, 0 <= [|x|] < wB.
-
-  Variable spec_w_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|].
-  Variable spec_w_W0 : forall h, [[w_W0 h]] = [|h|] * wB.
-  Variable spec_w_0W : forall l, [[w_0W l]] = [|l|].
-  Variable spec_w_compare :
-     forall x y, w_compare x y = Z.compare [|x|] [|y|].
-  Variable spec_w_succ : forall x, [|w_succ x|] = ([|x|] + 1) mod wB.
-  Variable spec_w_add_c  : forall x y, [+|w_add_c x y|] = [|x|] + [|y|].
-  Variable spec_w_add : forall x y, [|w_add x y|] = ([|x|] + [|y|]) mod wB.
-  Variable spec_w_sub : forall x y, [|w_sub x y|] = ([|x|] - [|y|]) mod wB.
-
-  Variable spec_w_mul_c : forall x y, [[ w_mul_c x y ]] = [|x|] * [|y|].
-  Variable spec_w_mul : forall x y, [|w_mul x y|] = ([|x|] * [|y|]) mod wB.
-  Variable spec_w_square_c : forall x, [[ w_square_c x]] = [|x|] * [|x|].
-
-  Variable spec_ww_add_c  : forall x y, [+[ww_add_c x y]] = [[x]] + [[y]].
-  Variable spec_ww_add : forall x y, [[ww_add x y]] = ([[x]] + [[y]]) mod wwB.
-  Variable spec_ww_add_carry :
-	 forall x y, [[ww_add_carry x y]] = ([[x]] + [[y]] + 1) mod wwB.
-  Variable spec_ww_sub : forall x y, [[ww_sub x y]] = ([[x]] - [[y]]) mod wwB.
-  Variable spec_ww_sub_c : forall x y, [-[ww_sub_c x y]] = [[x]] - [[y]].
-
-
-  Lemma spec_ww_to_Z : forall x, 0 <= [[x]] < wwB.
-  Proof. intros x;apply spec_ww_to_Z;auto. Qed.
-
-  Lemma spec_ww_to_Z_wBwB : forall x, 0 <= [[x]] < wB^2.
-  Proof. rewrite <- wwB_wBwB;apply spec_ww_to_Z. Qed.
-
-  Hint Resolve spec_ww_to_Z spec_ww_to_Z_wBwB : mult.
-  Ltac zarith := auto with zarith mult.
-
-  Lemma wBwB_lex: forall a b c d,
-      a * wB^2 + [[b]] <= c * wB^2 + [[d]] ->
-      a <= c.
-  Proof.
-   intros a b c d H; apply beta_lex with [[b]] [[d]] (wB^2);zarith.
-  Qed.
-
-  Lemma wBwB_lex_inv: forall a b c d,
-      a < c ->
-      a * wB^2 + [[b]] < c * wB^2 + [[d]].
-  Proof.
-   intros a b c d H; apply beta_lex_inv; zarith.
-  Qed.
-
-  Lemma sum_mul_carry : forall xh xl yh yl wc cc,
-   [|wc|]*wB^2 + [[cc]] = [|xh|] * [|yl|] + [|xl|] * [|yh|] ->
-   0 <= [|wc|] <= 1.
-  Proof.
-   intros.
-   apply (sum_mul_carry [|xh|] [|xl|] [|yh|] [|yl|] [|wc|][[cc]] wB);zarith.
-   apply wB_pos.
-  Qed.
-
-  Theorem mult_add_ineq: forall xH yH crossH,
-               0 <= [|xH|] * [|yH|] + [|crossH|] < wwB.
-  Proof.
-   intros;rewrite wwB_wBwB;apply mult_add_ineq;zarith.
-  Qed.
-
-  Hint Resolve mult_add_ineq : mult.
-
-  Lemma spec_mul_aux : forall xh xl yh yl wc (cc:zn2z w) hh ll,
-   [[hh]] = [|xh|] * [|yh|] ->
-   [[ll]] = [|xl|] * [|yl|] ->
-   [|wc|]*wB^2 + [[cc]] = [|xh|] * [|yl|] + [|xl|] * [|yh|] ->
-    [||match cc with
-      | W0 => WW (ww_add hh (w_W0 wc)) ll
-      | WW cch ccl =>
-          match ww_add_c (w_W0 ccl) ll with
-          | C0 l => WW (ww_add hh (w_WW wc cch)) l
-          | C1 l => WW (ww_add_carry hh (w_WW wc cch)) l
-          end
-      end||] = ([|xh|] * wB + [|xl|]) * ([|yh|] * wB + [|yl|]).
-  Proof.
-   intros;assert (U1 := wB_pos w_digits).
-   replace (([|xh|] * wB + [|xl|]) * ([|yh|] * wB + [|yl|])) with
-   ([|xh|]*[|yh|]*wB^2 + ([|xh|]*[|yl|] + [|xl|]*[|yh|])*wB + [|xl|]*[|yl|]).
-   2:ring. rewrite <- H1;rewrite <- H;rewrite <- H0.
-   assert (H2 := sum_mul_carry _ _ _ _ _ _ H1).
-   destruct cc as [ | cch ccl]; simpl zn2z_to_Z; simpl ww_to_Z.
-   rewrite spec_ww_add;rewrite spec_w_W0;rewrite Zmod_small;
-    rewrite wwB_wBwB. ring.
-   rewrite <- (Z.add_0_r ([|wc|]*wB));rewrite H;apply mult_add_ineq3;zarith.
-   simpl ww_to_Z in H1. assert (U:=spec_to_Z cch).
-   assert ([|wc|]*wB + [|cch|] <= 2*wB - 3).
-    destruct (Z_le_gt_dec ([|wc|]*wB + [|cch|]) (2*wB - 3)) as [Hle|Hgt];trivial.
-    assert ([|xh|] * [|yl|] + [|xl|] * [|yh|] <= (2*wB - 4)*wB + 2).
-     ring_simplify ((2*wB - 4)*wB + 2).
-     assert (H4 := Zmult_lt_b _ _ _ (spec_to_Z xh) (spec_to_Z yl)).
-     assert (H5 := Zmult_lt_b _ _ _ (spec_to_Z xl) (spec_to_Z yh)).
-     omega.
-   generalize H3;clear H3;rewrite <- H1.
-   rewrite Z.add_assoc; rewrite Z.pow_2_r; rewrite Z.mul_assoc;
-     rewrite <- Z.mul_add_distr_r.
-    assert (((2 * wB - 4) + 2)*wB <= ([|wc|] * wB + [|cch|])*wB).
-     apply Z.mul_le_mono_nonneg;zarith.
-    rewrite Z.mul_add_distr_r in H3.
-    intros. assert (U2 := spec_to_Z ccl);omega.
-   generalize (spec_ww_add_c (w_W0 ccl) ll);destruct (ww_add_c (w_W0 ccl) ll)
-   as [l|l];unfold interp_carry;rewrite spec_w_W0;try rewrite Z.mul_1_l;
-   simpl zn2z_to_Z;
-   try rewrite spec_ww_add;try rewrite spec_ww_add_carry;rewrite spec_w_WW;
-   rewrite Zmod_small;rewrite wwB_wBwB;intros.
-   rewrite H4;ring. rewrite H;apply mult_add_ineq2;zarith.
-   rewrite Z.add_assoc;rewrite Z.mul_add_distr_r.
-   rewrite Z.mul_1_l;rewrite <- Z.add_assoc;rewrite H4;ring.
-   repeat rewrite <- Z.add_assoc;rewrite H;apply mult_add_ineq2;zarith.
-  Qed.
-
-  Lemma spec_double_mul_c : forall cross:w->w->w->w->zn2z w -> zn2z w -> w*zn2z w,
-     (forall xh xl yh yl hh ll,
-        [[hh]] = [|xh|]*[|yh|] ->
-        [[ll]] = [|xl|]*[|yl|] ->
-        let (wc,cc) :=  cross xh xl yh yl hh ll in
-        [|wc|]*wwB + [[cc]] = [|xh|]*[|yl|] + [|xl|]*[|yh|]) ->
-     forall x y, [||double_mul_c cross x y||] = [[x]] * [[y]].
-  Proof.
-   intros cross Hcross x y;destruct x as [ |xh xl];simpl;trivial.
-   destruct y as [ |yh yl];simpl. rewrite Z.mul_0_r;trivial.
-   assert (H1:= spec_w_mul_c xh yh);assert (H2:= spec_w_mul_c xl yl).
-   generalize (Hcross _ _ _ _ _ _ H1 H2).
-   destruct (cross xh xl yh yl (w_mul_c xh yh) (w_mul_c xl yl)) as (wc,cc).
-   intros;apply spec_mul_aux;trivial.
-   rewrite <- wwB_wBwB;trivial.
-  Qed.
-
-  Lemma spec_ww_mul_c : forall x y, [||ww_mul_c x y||] = [[x]] * [[y]].
-  Proof.
-   intros x y;unfold ww_mul_c;apply spec_double_mul_c.
-   intros xh xl yh yl hh ll H1 H2.
-   generalize (spec_ww_add_c (w_mul_c xh yl) (w_mul_c xl yh));
-   destruct (ww_add_c (w_mul_c xh yl) (w_mul_c xl yh)) as [c|c];
-   unfold interp_carry;repeat rewrite spec_w_mul_c;intros H;
-   (rewrite spec_w_0 || rewrite spec_w_1);rewrite H;ring.
-  Qed.
-
-  Lemma spec_w_2: [|w_2|] = 2.
-  unfold w_2; rewrite spec_w_add; rewrite spec_w_1; simpl.
-  apply Zmod_small; split; auto with zarith.
-  rewrite <- (Z.pow_1_r 2); unfold base; apply Zpower_lt_monotone; auto with zarith.
-  Qed.
-
-  Lemma kara_prod_aux : forall xh xl yh yl,
-   xh*yh + xl*yl - (xh-xl)*(yh-yl) = xh*yl + xl*yh.
-  Proof. intros;ring. Qed.
-
-  Lemma spec_kara_prod : forall xh xl yh yl hh ll,
-   [[hh]] = [|xh|]*[|yh|] ->
-   [[ll]] = [|xl|]*[|yl|] ->
-   let (wc,cc) := kara_prod xh xl yh yl hh ll in
-   [|wc|]*wwB + [[cc]] = [|xh|]*[|yl|] + [|xl|]*[|yh|].
-  Proof.
-   intros xh xl yh yl hh ll H H0; rewrite <- kara_prod_aux;
-     rewrite <- H; rewrite <- H0; unfold kara_prod.
-   assert (Hxh := (spec_to_Z xh)); assert (Hxl := (spec_to_Z xl));
-     assert (Hyh := (spec_to_Z yh)); assert (Hyl := (spec_to_Z yl)).
-   generalize (spec_ww_add_c hh ll); case (ww_add_c hh ll);
-     intros z Hz; rewrite <- Hz; unfold interp_carry; assert (Hz1 := (spec_ww_to_Z z)).
-   rewrite spec_w_compare; case Z.compare_spec; intros Hxlh;
-    try rewrite Hxlh; try rewrite spec_w_0; try (ring; fail).
-   rewrite spec_w_compare; case Z.compare_spec; intros Hylh.
-   rewrite Hylh; rewrite spec_w_0; try (ring; fail).
-   rewrite spec_w_0; try (ring; fail).
-   repeat (rewrite spec_ww_sub || rewrite spec_w_sub || rewrite spec_w_mul_c).
-   repeat rewrite Zmod_small; auto with zarith; try (ring; fail).
-   split; auto with zarith.
-   simpl in Hz; rewrite Hz; rewrite H; rewrite H0.
-   rewrite kara_prod_aux; apply Z.add_nonneg_nonneg; apply Z.mul_nonneg_nonneg; auto with zarith.
-   apply Z.le_lt_trans with ([[z]]-0); auto with zarith.
-   unfold Z.sub; apply Z.add_le_mono_l; apply Z.le_0_sub; simpl; rewrite Z.opp_involutive.
-   apply Z.mul_nonneg_nonneg; auto with zarith.
-   match goal with |- context[ww_add_c ?x ?y] =>
-     generalize (spec_ww_add_c x y); case (ww_add_c x y); try rewrite spec_w_0;
-     intros z1 Hz2
-   end.
-   simpl in Hz2; rewrite Hz2; repeat (rewrite spec_w_sub || rewrite spec_w_mul_c).
-   repeat rewrite Zmod_small; auto with zarith; try (ring; fail).
-   rewrite spec_w_1; unfold interp_carry in Hz2; rewrite Hz2;
-     repeat (rewrite spec_w_sub || rewrite spec_w_mul_c).
-   repeat rewrite Zmod_small; auto with zarith; try (ring; fail).
-   rewrite spec_w_compare; case Z.compare_spec; intros Hylh.
-   rewrite Hylh; rewrite spec_w_0; try (ring; fail).
-   match goal with |- context[ww_add_c ?x ?y] =>
-     generalize (spec_ww_add_c x y); case (ww_add_c x y); try rewrite spec_w_0;
-     intros z1 Hz2
-   end.
-   simpl in Hz2; rewrite Hz2; repeat (rewrite spec_w_sub || rewrite spec_w_mul_c).
-   repeat rewrite Zmod_small; auto with zarith; try (ring; fail).
-   rewrite spec_w_1; unfold interp_carry in Hz2; rewrite Hz2;
-     repeat (rewrite spec_w_sub || rewrite spec_w_mul_c).
-   repeat rewrite Zmod_small; auto with zarith; try (ring; fail).
-   rewrite spec_w_0; try (ring; fail).
-   repeat (rewrite spec_ww_sub || rewrite spec_w_sub || rewrite spec_w_mul_c).
-   repeat rewrite Zmod_small; auto with zarith; try (ring; fail).
-   split.
-   match goal with |- context[(?x - ?y) * (?z - ?t)] =>
-     replace ((x - y) * (z - t)) with ((y - x) * (t - z)); [idtac | ring]
-   end.
-   simpl in Hz; rewrite Hz; rewrite H; rewrite H0.
-   rewrite kara_prod_aux; apply Z.add_nonneg_nonneg; apply Z.mul_nonneg_nonneg; auto with zarith.
-   apply Z.le_lt_trans with ([[z]]-0); auto with zarith.
-   unfold Z.sub; apply Z.add_le_mono_l; apply Z.le_0_sub; simpl; rewrite Z.opp_involutive.
-   apply Z.mul_nonneg_nonneg; auto with zarith.
-   (** there is a carry in hh + ll **)
-   rewrite Z.mul_1_l.
-   rewrite spec_w_compare; case Z.compare_spec; intros Hxlh;
-    try rewrite Hxlh; try rewrite spec_w_1; try (ring; fail).
-   rewrite spec_w_compare; case Z.compare_spec; intros Hylh;
-     try rewrite Hylh; try rewrite spec_w_1; try (ring; fail).
-   match goal with |- context[ww_sub_c ?x ?y] =>
-     generalize (spec_ww_sub_c x y); case (ww_sub_c x y); try rewrite spec_w_1;
-     intros z1 Hz2
-   end.
-   simpl in Hz2; rewrite Hz2; repeat (rewrite spec_w_sub || rewrite spec_w_mul_c).
-   repeat rewrite Zmod_small; auto with zarith; try (ring; fail).
-   rewrite spec_w_0; rewrite Z.mul_0_l; rewrite Z.add_0_l.
-   generalize Hz2; clear Hz2; unfold interp_carry.
-   repeat (rewrite spec_w_sub || rewrite spec_w_mul_c).
-   repeat rewrite Zmod_small; auto with zarith; try (ring; fail).
-   match goal with |- context[ww_add_c ?x ?y] =>
-     generalize (spec_ww_add_c x y); case (ww_add_c x y); try rewrite spec_w_1;
-     intros z1 Hz2
-   end.
-   simpl in Hz2; rewrite Hz2; repeat (rewrite spec_w_sub || rewrite spec_w_mul_c).
-   repeat rewrite Zmod_small; auto with zarith; try (ring; fail).
-   rewrite spec_w_2; unfold interp_carry in Hz2.
-   transitivity (wwB + (1 * wwB + [[z1]])).
-   ring.
-   rewrite Hz2; repeat (rewrite spec_w_sub || rewrite spec_w_mul_c).
-   repeat rewrite Zmod_small; auto with zarith; try (ring; fail).
-   rewrite spec_w_compare; case Z.compare_spec; intros Hylh;
-     try rewrite Hylh; try rewrite spec_w_1; try (ring; fail).
-   match goal with |- context[ww_add_c ?x ?y] =>
-     generalize (spec_ww_add_c x y); case (ww_add_c x y); try rewrite spec_w_1;
-     intros z1 Hz2
-   end.
-   simpl in Hz2; rewrite Hz2; repeat (rewrite spec_w_sub || rewrite spec_w_mul_c).
-   repeat rewrite Zmod_small; auto with zarith; try (ring; fail).
-   rewrite spec_w_2; unfold interp_carry in Hz2.
-   transitivity (wwB + (1 * wwB + [[z1]])).
-   ring.
-   rewrite Hz2; repeat (rewrite spec_w_sub || rewrite spec_w_mul_c).
-   repeat rewrite Zmod_small; auto with zarith; try (ring; fail).
-   match goal with |- context[ww_sub_c ?x ?y] =>
-     generalize (spec_ww_sub_c x y); case (ww_sub_c x y); try rewrite spec_w_1;
-     intros z1 Hz2
-   end.
-   simpl in Hz2; rewrite Hz2; repeat (rewrite spec_w_sub || rewrite spec_w_mul_c).
-   repeat rewrite Zmod_small; auto with zarith; try (ring; fail).
-   rewrite spec_w_0; rewrite Z.mul_0_l; rewrite Z.add_0_l.
-   match goal with |- context[(?x - ?y) * (?z - ?t)] =>
-     replace ((x - y) * (z - t)) with ((y - x) * (t - z)); [idtac | ring]
-   end.
-   generalize Hz2; clear Hz2; unfold interp_carry.
-   repeat (rewrite spec_w_sub || rewrite spec_w_mul_c).
-   repeat rewrite Zmod_small; auto with zarith; try (ring; fail).
-  Qed.
-
-  Lemma sub_carry : forall xh xl yh yl z,
-    0 <= z ->
-    [|xh|]*[|yl|] + [|xl|]*[|yh|] = wwB + z ->
-    z < wwB.
-  Proof.
-   intros xh xl yh yl z Hle Heq.
-   destruct (Z_le_gt_dec wwB z);auto with zarith.
-   generalize (Zmult_lt_b _ _ _ (spec_to_Z xh) (spec_to_Z yl)).
-   generalize (Zmult_lt_b _ _ _ (spec_to_Z xl) (spec_to_Z yh)).
-   rewrite <- wwB_wBwB;intros H1 H2.
-   assert (H3 := wB_pos w_digits).
-   assert (2*wB <= wwB).
-    rewrite wwB_wBwB; rewrite Z.pow_2_r; apply Z.mul_le_mono_nonneg;zarith.
-   omega.
-  Qed.
-
-  Ltac Spec_ww_to_Z x :=
-   let H:= fresh "H" in
-   assert (H:= spec_ww_to_Z x).
-
-  Ltac Zmult_lt_b x y :=
-   let H := fresh "H" in
-   assert (H := Zmult_lt_b _ _ _ (spec_to_Z x) (spec_to_Z y)).
-
-  Lemma spec_ww_karatsuba_c : forall x y, [||ww_karatsuba_c x y||]=[[x]]*[[y]].
-  Proof.
-   intros x y; unfold ww_karatsuba_c;apply spec_double_mul_c.
-   intros; apply spec_kara_prod; auto.
-  Qed.
-
-  Lemma spec_ww_mul : forall x y, [[ww_mul x y]] = [[x]]*[[y]] mod wwB.
-  Proof.
-   assert (U:= lt_0_wB w_digits).
-   assert (U1:= lt_0_wwB w_digits).
-   intros x y; case x; auto; intros xh xl.
-   case y; auto.
-   simpl; rewrite Z.mul_0_r; rewrite Zmod_small; auto with zarith.
-   intros yh yl;simpl.
-   repeat (rewrite spec_ww_add || rewrite spec_w_W0 || rewrite spec_w_mul_c
-    || rewrite spec_w_add || rewrite spec_w_mul).
-   rewrite <- Zplus_mod; auto with zarith.
-   repeat (rewrite Z.mul_add_distr_r || rewrite Z.mul_add_distr_l).
-   rewrite <- Zmult_mod_distr_r; auto with zarith.
-   rewrite <- Z.pow_2_r; rewrite <- wwB_wBwB; auto with zarith.
-   rewrite Zplus_mod; auto with zarith.
-   rewrite Zmod_mod; auto with zarith.
-   rewrite <- Zplus_mod; auto with zarith.
-   match goal with |- ?X mod _ = _ =>
-     rewrite <- Z_mod_plus with (a := X) (b := [|xh|] * [|yh|])
-   end; auto with zarith.
-   f_equal; auto; rewrite wwB_wBwB; ring.
-  Qed.
-
-  Lemma spec_ww_square_c : forall x, [||ww_square_c x||] = [[x]]*[[x]].
-  Proof.
-   destruct x as [ |xh xl];simpl;trivial.
-   case_eq match ww_add_c (w_mul_c xh xl) (w_mul_c xh xl) with
-          | C0 cc => (w_0, cc)
-          | C1 cc => (w_1, cc)
-          end;intros wc cc Heq.
-   apply (spec_mul_aux xh xl xh xl wc cc);trivial.
-   generalize Heq (spec_ww_add_c (w_mul_c xh xl) (w_mul_c xh xl));clear Heq.
-   rewrite spec_w_mul_c;destruct (ww_add_c (w_mul_c xh xl) (w_mul_c xh xl));
-   unfold interp_carry;try rewrite Z.mul_1_l;intros Heq Heq';inversion Heq;
-   rewrite (Z.mul_comm [|xl|]);subst.
-   rewrite spec_w_0;rewrite Z.mul_0_l;rewrite Z.add_0_l;trivial.
-   rewrite spec_w_1;rewrite Z.mul_1_l;rewrite <- wwB_wBwB;trivial.
-  Qed.
-
-  Section DoubleMulAddn1Proof.
-
-   Variable w_mul_add : w -> w -> w -> w * w.
-   Variable spec_w_mul_add : forall x y r,
-    let (h,l):= w_mul_add x y r in
-    [|h|]*wB+[|l|] = [|x|]*[|y|] + [|r|].
-
-   Lemma spec_double_mul_add_n1 : forall n x y r,
-     let (h,l) := double_mul_add_n1 w_mul_add n x y r in
-     [|h|]*double_wB w_digits n + [!n|l!] = [!n|x!]*[|y|]+[|r|].
-   Proof.
-    induction n;intros x y r;trivial.
-    exact (spec_w_mul_add x y r).
-    unfold double_mul_add_n1;destruct x as[ |xh xl];
-     fold(double_mul_add_n1 w_mul_add).
-    rewrite spec_w_0;rewrite spec_extend;simpl;trivial.
-    assert(H:=IHn xl y r);destruct (double_mul_add_n1 w_mul_add n xl y r)as(rl,l).
-   assert(U:=IHn xh y rl);destruct(double_mul_add_n1 w_mul_add n xh y rl)as(rh,h).
-    rewrite <- double_wB_wwB. rewrite spec_double_WW;simpl;trivial.
-    rewrite Z.mul_add_distr_r;rewrite <- Z.add_assoc;rewrite <- H.
-    rewrite Z.mul_assoc;rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r.
-    rewrite U;ring.
-   Qed.
-
-  End DoubleMulAddn1Proof.
-
-  Lemma spec_w_mul_add : forall x y r,
-    let (h,l):= w_mul_add x y r in
-    [|h|]*wB+[|l|] = [|x|]*[|y|] + [|r|].
-  Proof.
-   intros x y r;unfold w_mul_add;assert (H:=spec_w_mul_c x y);
-   destruct (w_mul_c x y) as [ |h l];simpl;rewrite <- H.
-   rewrite spec_w_0;trivial.
-   assert (U:=spec_w_add_c l r);destruct (w_add_c l r) as [lr|lr];unfold
-   interp_carry in U;try rewrite Z.mul_1_l in H;simpl.
-   rewrite U;ring. rewrite spec_w_succ. rewrite Zmod_small.
-   rewrite <- Z.add_assoc;rewrite <- U;ring.
-   simpl in H;assert (H1:= Zmult_lt_b _ _ _ (spec_to_Z x) (spec_to_Z y)).
-   rewrite <- H in H1.
-   assert (H2:=spec_to_Z h);split;zarith.
-   case H1;clear H1;intro H1;clear H1.
-   replace (wB ^ 2 - 2 * wB) with ((wB - 2)*wB). 2:ring.
-   intros H0;assert (U1:= wB_pos w_digits).
-   assert (H1 := beta_lex _ _ _ _ _ H0 (spec_to_Z l));zarith.
-  Qed.
-
-(* End DoubleProof. *)
-
-End DoubleMul.
diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleSqrt.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleSqrt.v
deleted file mode 100644
index d07ce3018..000000000
--- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleSqrt.v
+++ /dev/null
@@ -1,1369 +0,0 @@
-(************************************************************************)
-(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016     *)
-(*   \VV/  **************************************************************)
-(*    //   *      This file is distributed under the terms of the       *)
-(*         *       GNU Lesser General Public License Version 2.1        *)
-(************************************************************************)
-(*            Benjamin Gregoire, Laurent Thery, INRIA, 2007             *)
-(************************************************************************)
-
-Set Implicit Arguments.
-
-Require Import ZArith.
-Require Import BigNumPrelude.
-Require Import DoubleType.
-Require Import DoubleBase.
-
-Local Open Scope Z_scope.
-
-Section DoubleSqrt.
- Variable w              : Type.
- Variable w_is_even      : w -> bool.
- Variable w_compare      : w -> w -> comparison.
- Variable w_0            : w.
- Variable w_1            : w.
- Variable w_Bm1          : w.
- Variable w_WW           : w -> w -> zn2z w.
- Variable w_W0           : w -> zn2z w.
- Variable w_0W           : w -> zn2z w.
- Variable w_sub          : w -> w -> w.
- Variable w_sub_c        : w -> w -> carry w.
- Variable w_square_c     : w -> zn2z w.
- Variable w_div21        : w -> w -> w -> w * w.
- Variable w_add_mul_div  : w -> w -> w -> w.
- Variable w_digits       : positive.
- Variable w_zdigits      : w.
- Variable ww_zdigits     : zn2z w.
- Variable w_add_c        : w -> w -> carry w.
- Variable w_sqrt2        : w -> w -> w * carry w.
- Variable w_pred         : w -> w.
- Variable ww_pred_c      : zn2z w -> carry (zn2z w).
- Variable ww_pred        : zn2z w -> zn2z w.
- Variable ww_add_c       : zn2z w -> zn2z w -> carry (zn2z w).
- Variable ww_add         : zn2z w -> zn2z w -> zn2z w.
- Variable ww_sub_c       : zn2z w -> zn2z w -> carry (zn2z w).
- Variable ww_add_mul_div : zn2z w -> zn2z w -> zn2z w -> zn2z w.
- Variable ww_head0       : zn2z w -> zn2z w.
- Variable ww_compare     : zn2z w -> zn2z w -> comparison.
- Variable low            : zn2z w -> w.
-
- Let wwBm1 := ww_Bm1 w_Bm1.
-
- Definition ww_is_even x :=
-   match x with
-   | W0 => true
-   | WW xh xl => w_is_even xl
-   end.
-
- Let w_div21c x y z :=
-   match w_compare x z with
-   | Eq =>
-      match w_compare y z with
-        Eq => (C1 w_1, w_0)
-      | Gt => (C1 w_1, w_sub y z)
-      | Lt => (C1 w_0, y)
-      end
-   | Gt =>
-      let x1 := w_sub x z in
-      let (q, r) := w_div21 x1 y z in
-        (C1 q, r)
-   | Lt =>
-      let (q, r) := w_div21 x y z in
-        (C0 q, r)
-   end.
-
- Let w_div2s x y s :=
-  match x with
-   C1 x1 =>
-     let x2 := w_sub x1 s in
-     let (q, r) := w_div21c x2 y s in
-     match q with
-       C0 q1 =>
-         if w_is_even q1 then
-          (C0 (w_add_mul_div (w_pred w_zdigits) w_1 q1), C0 r)
-         else
-          (C0 (w_add_mul_div (w_pred w_zdigits) w_1 q1), w_add_c r s)
-     | C1 q1 =>
-         if w_is_even q1 then
-          (C1 (w_add_mul_div (w_pred w_zdigits) w_0 q1), C0 r)
-         else
-          (C1 (w_add_mul_div (w_pred w_zdigits) w_0 q1), w_add_c r s)
-     end
-  | C0 x1 =>
-     let (q, r) := w_div21c x1 y s in
-     match q with
-       C0 q1 =>
-         if w_is_even q1 then
-          (C0 (w_add_mul_div (w_pred w_zdigits) w_0 q1), C0 r)
-         else
-          (C0 (w_add_mul_div (w_pred w_zdigits) w_0 q1), w_add_c r s)
-     | C1 q1 =>
-         if w_is_even q1 then
-          (C0 (w_add_mul_div (w_pred w_zdigits) w_1 q1), C0 r)
-         else
-          (C0 (w_add_mul_div (w_pred w_zdigits) w_1 q1), w_add_c r s)
-     end
-  end.
-
- Definition split x :=
- match x with
-  | W0 => (w_0,w_0)
-  | WW h l => (h,l)
-  end.
-
- Definition ww_sqrt2 x y :=
-   let (x1, x2) := split x in
-   let (y1, y2) := split y in
-   let ( q, r)  := w_sqrt2 x1 x2 in
-   let (q1, r1) := w_div2s r y1 q in
-   match q1 with
-     C0 q1 =>
-      let q2 := w_square_c q1 in
-      let a  := WW q q1 in
-        match r1 with
-          C1 r2 =>
-           match ww_sub_c (WW r2 y2) q2 with
-             C0 r3 => (a, C1 r3)
-           | C1 r3 => (a, C0 r3)
-           end
-        | C0 r2 =>
-           match ww_sub_c (WW r2 y2) q2 with
-             C0 r3 => (a, C0 r3)
-           | C1 r3 =>
-              let a2 := ww_add_mul_div (w_0W w_1) a W0 in
-              match ww_pred_c a2 with
-                C0 a3 =>
-                  (ww_pred a, ww_add_c a3 r3)
-              | C1 a3 =>
-                  (ww_pred a, C0 (ww_add a3 r3))
-              end
-            end
-         end
-   | C1 q1 =>
-         let a1 := WW q w_Bm1 in
-         let a2 := ww_add_mul_div (w_0W w_1) a1 wwBm1 in
-            (a1, ww_add_c a2 y)
-   end.
-
- Definition ww_is_zero x :=
-  match ww_compare W0 x with
-   Eq => true
-  | _ => false
-  end.
-
- Definition ww_head1 x :=
-   let p := ww_head0 x in
-   if (ww_is_even p) then p else ww_pred p.
-
- Definition ww_sqrt x :=
-   if (ww_is_zero x) then W0
-   else
-    let p := ww_head1 x in
-    match ww_compare p W0 with
-    | Gt =>
-        match ww_add_mul_div p x W0 with
-         W0 => W0
-       | WW x1 x2 =>
-          let (r, _) := w_sqrt2 x1 x2 in
-            WW w_0 (w_add_mul_div
-                     (w_sub w_zdigits
-                     (low (ww_add_mul_div (ww_pred ww_zdigits)
-                              W0 p))) w_0 r)
-        end
-     | _ =>
-        match x with
-          W0 => W0
-        | WW x1 x2 => WW w_0 (fst (w_sqrt2 x1 x2))
-        end
-    end.
-
-
-  Variable w_to_Z : w -> Z.
-
-  Notation wB  := (base w_digits).
-  Notation wwB := (base (ww_digits w_digits)).
-  Notation "[| x |]" := (w_to_Z x)  (at level 0, x at level 99).
-  Notation "[+| c |]" :=
-   (interp_carry 1 wB w_to_Z c) (at level 0, c at level 99).
-  Notation "[-| c |]" :=
-   (interp_carry (-1) wB w_to_Z c) (at level 0, c at level 99).
-
-  Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99).
-  Notation "[+[ c ]]" :=
-   (interp_carry 1 wwB (ww_to_Z w_digits w_to_Z) c)
-   (at level 0, c at level 99).
-  Notation "[-[ c ]]" :=
-   (interp_carry (-1) wwB (ww_to_Z w_digits w_to_Z) c)
-   (at level 0, c at level 99).
-
-  Notation "[|| x ||]" :=
-    (zn2z_to_Z wwB (ww_to_Z w_digits w_to_Z) x)  (at level 0, x at level 99).
-
-  Notation "[! n | x !]" := (double_to_Z w_digits w_to_Z n x)
-    (at level 0, x at level 99).
-
-  Variable spec_w_0   : [|w_0|] = 0.
-  Variable spec_w_1   : [|w_1|] = 1.
-  Variable spec_w_Bm1 : [|w_Bm1|] = wB - 1.
-  Variable spec_w_zdigits : [|w_zdigits|] = Zpos w_digits.
-  Variable spec_more_than_1_digit: 1 < Zpos w_digits.
-
-  Variable spec_ww_zdigits : [[ww_zdigits]] = Zpos (xO w_digits).
-  Variable spec_to_Z  : forall x, 0 <= [|x|] < wB.
-  Variable spec_to_w_Z  : forall x, 0 <= [[x]] < wwB.
-
-  Variable spec_w_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|].
-  Variable spec_w_W0 : forall h, [[w_W0 h]] = [|h|] * wB.
-  Variable spec_w_0W : forall l, [[w_0W l]] = [|l|].
-  Variable spec_w_is_even : forall x,
-      if w_is_even x then [|x|] mod 2 = 0 else [|x|] mod 2 = 1.
-  Variable spec_w_compare : forall x y,
-       w_compare x y = Z.compare [|x|] [|y|].
- Variable spec_w_sub : forall x y, [|w_sub x y|] = ([|x|] - [|y|]) mod wB.
- Variable spec_w_square_c : forall x, [[ w_square_c x]] = [|x|] * [|x|].
- Variable spec_w_div21 : forall a1 a2 b,
-     wB/2 <= [|b|] ->
-     [|a1|] < [|b|] ->
-     let (q,r) := w_div21 a1 a2 b in
-     [|a1|] *wB+ [|a2|] = [|q|] * [|b|] + [|r|] /\
-     0 <= [|r|] < [|b|].
- Variable spec_w_add_mul_div : forall x y p,
-        [|p|] <= Zpos w_digits ->
-       [| w_add_mul_div p x y |] =
-         ([|x|] * (2 ^ [|p|]) +
-          [|y|] / (Z.pow 2 ((Zpos w_digits) - [|p|]))) mod wB.
- Variable spec_ww_add_mul_div : forall x y p,
-       [[p]] <= Zpos (xO w_digits) ->
-       [[ ww_add_mul_div p x y ]] =
-         ([[x]] * (2^ [[p]]) +
-          [[y]] / (2^ (Zpos (xO w_digits) - [[p]]))) mod wwB.
- Variable spec_w_add_c  : forall x y, [+|w_add_c x y|] = [|x|] + [|y|].
- Variable spec_ww_add : forall x y, [[ww_add x y]] = ([[x]] + [[y]]) mod wwB.
- Variable spec_w_sqrt2 : forall x y,
-       wB/ 4 <= [|x|] ->
-       let (s,r) := w_sqrt2 x y in
-          [[WW x y]] = [|s|] ^ 2 + [+|r|] /\
-          [+|r|] <= 2 * [|s|].
- Variable spec_ww_sub_c : forall x y, [-[ww_sub_c x y]] = [[x]] - [[y]].
- Variable spec_ww_pred_c : forall x, [-[ww_pred_c x]] = [[x]] - 1.
- Variable spec_pred : forall x, [|w_pred x|] = ([|x|] - 1) mod wB.
- Variable spec_ww_pred : forall x, [[ww_pred x]] = ([[x]] - 1) mod wwB.
- Variable spec_ww_add_c  : forall x y, [+[ww_add_c x y]] = [[x]] + [[y]].
- Variable spec_ww_compare : forall x y,
-    ww_compare x y = Z.compare [[x]] [[y]].
- Variable spec_ww_head0  : forall x,  0 < [[x]] ->
-	 wwB/ 2 <= 2 ^ [[ww_head0 x]] * [[x]] < wwB.
- Variable spec_low: forall x, [|low x|] = [[x]] mod wB.
-
- Let spec_ww_Bm1 : [[wwBm1]] = wwB - 1.
- Proof. refine (spec_ww_Bm1 w_Bm1 w_digits w_to_Z _);auto. Qed.
-
- Hint Rewrite spec_w_0 spec_w_1 spec_w_WW spec_w_sub
-   spec_w_add_mul_div spec_ww_Bm1 spec_w_add_c : w_rewrite.
-
- Lemma spec_ww_is_even : forall x,
-      if ww_is_even x then [[x]] mod 2 = 0 else [[x]] mod 2 = 1.
-clear spec_more_than_1_digit.
-intros x; case x; simpl ww_is_even.
- reflexivity.
- simpl.
- intros w1 w2; simpl.
- unfold base.
- rewrite Zplus_mod; auto with zarith.
- rewrite (fun x y => (Zdivide_mod (x * y))); auto with zarith.
- rewrite Z.add_0_l; rewrite Zmod_mod; auto with zarith.
- apply spec_w_is_even; auto with zarith.
- apply Z.divide_mul_r; apply Zpower_divide; auto with zarith.
- Qed.
-
-
- Theorem  spec_w_div21c : forall a1 a2 b,
-     wB/2 <= [|b|] ->
-     let (q,r) := w_div21c a1 a2 b in
-     [|a1|] * wB + [|a2|] = [+|q|] *  [|b|] + [|r|] /\ 0 <= [|r|] < [|b|].
- intros a1 a2 b Hb; unfold w_div21c.
- assert (H: 0 < [|b|]); auto with zarith.
- assert (U := wB_pos w_digits).
- apply Z.lt_le_trans with (2 := Hb); auto with zarith.
- apply Z.lt_le_trans with 1; auto with zarith.
- apply Zdiv_le_lower_bound; auto with zarith.
- rewrite !spec_w_compare. repeat case Z.compare_spec.
- intros H1 H2; split.
- unfold interp_carry; autorewrite with w_rewrite rm10; auto with zarith.
- rewrite H1; rewrite H2; ring.
- autorewrite with w_rewrite; auto with zarith.
- intros H1 H2; split.
- unfold interp_carry; autorewrite with w_rewrite rm10; auto with zarith.
- rewrite H2; ring.
- destruct (spec_to_Z a2);auto with zarith.
- intros H1 H2; split.
- unfold interp_carry; autorewrite with w_rewrite rm10; auto with zarith.
- rewrite H2; rewrite Zmod_small; auto with zarith.
- ring.
- destruct (spec_to_Z a2);auto with zarith.
- rewrite spec_w_sub; auto with zarith.
- destruct (spec_to_Z a2) as [H3 H4];auto with zarith.
- rewrite Zmod_small; auto with zarith.
- split; auto with zarith.
- assert ([|a2|] < 2 * [|b|]); auto with zarith.
- apply Z.lt_le_trans with (2 * (wB / 2)); auto with zarith.
- rewrite wB_div_2; auto.
- intros H1.
- match goal with |- context[w_div21 ?y ?z ?t] =>
-   generalize (@spec_w_div21 y z t Hb H1);
-   case (w_div21 y z t); simpl; autorewrite with w_rewrite;
-   auto
- end.
- intros H1.
- assert (H2: [|w_sub a1 b|] < [|b|]).
- rewrite spec_w_sub; auto with zarith.
- rewrite Zmod_small; auto with zarith.
- assert ([|a1|] < 2 * [|b|]); auto with zarith.
- apply Z.lt_le_trans with (2 * (wB / 2)); auto with zarith.
- rewrite wB_div_2; auto.
- destruct (spec_to_Z a1);auto with zarith.
- destruct (spec_to_Z a1);auto with zarith.
- match goal with |- context[w_div21 ?y ?z ?t] =>
-   generalize (@spec_w_div21 y z t Hb H2);
-   case (w_div21 y z t); autorewrite with w_rewrite;
-   auto
- end.
- intros w0 w1; replace [+|C1 w0|] with (wB + [|w0|]).
- rewrite Zmod_small; auto with zarith.
- intros (H3, H4); split; auto.
- rewrite Z.mul_add_distr_r.
- rewrite <- Z.add_assoc; rewrite <- H3; ring.
- split; auto with zarith.
- assert ([|a1|] < 2 * [|b|]); auto with zarith.
- apply Z.lt_le_trans with (2 * (wB / 2)); auto with zarith.
- rewrite wB_div_2; auto.
- destruct (spec_to_Z a1);auto with zarith.
- destruct (spec_to_Z a1);auto with zarith.
- simpl; case wB; auto.
- Qed.
-
- Theorem C0_id: forall p, [+|C0 p|] = [|p|].
- intros p; simpl; auto.
- Qed.
-
- Theorem add_mult_div_2: forall w,
-    [|w_add_mul_div (w_pred w_zdigits) w_0 w|] = [|w|] / 2.
- intros w1.
- assert (Hp: [|w_pred w_zdigits|] = Zpos w_digits - 1).
-   rewrite spec_pred; rewrite spec_w_zdigits.
-   rewrite Zmod_small; auto with zarith.
-   split; auto with zarith.
-   apply Z.lt_le_trans with (Zpos w_digits); auto with zarith.
-   unfold base; apply Zpower2_le_lin; auto with zarith.
- rewrite spec_w_add_mul_div; auto with zarith.
- autorewrite with w_rewrite rm10.
- match goal with |- context[?X - ?Y] =>
-  replace (X - Y) with 1
- end.
- rewrite Z.pow_1_r; rewrite Zmod_small; auto with zarith.
- destruct (spec_to_Z w1) as [H1 H2];auto with zarith.
- split; auto with zarith.
- apply Zdiv_lt_upper_bound; auto with zarith.
- rewrite Hp; ring.
- Qed.
-
- Theorem add_mult_div_2_plus_1: forall w,
-    [|w_add_mul_div (w_pred w_zdigits) w_1 w|] =
-      [|w|] / 2 + 2 ^ Zpos (w_digits - 1).
- intros w1.
- assert (Hp: [|w_pred w_zdigits|] = Zpos w_digits - 1).
-   rewrite spec_pred; rewrite spec_w_zdigits.
-   rewrite Zmod_small; auto with zarith.
-   split; auto with zarith.
-   apply Z.lt_le_trans with (Zpos w_digits); auto with zarith.
-   unfold base; apply Zpower2_le_lin; auto with zarith.
- autorewrite with w_rewrite rm10; auto with zarith.
- match goal with |- context[?X - ?Y] =>
-  replace (X - Y) with 1
- end; rewrite Hp; try ring.
- rewrite Pos2Z.inj_sub_max; auto with zarith.
- rewrite Z.max_r; auto with zarith.
- rewrite Z.pow_1_r; rewrite Zmod_small; auto with zarith.
- destruct (spec_to_Z w1) as [H1 H2];auto with zarith.
- split; auto with zarith.
- unfold base.
- match goal with |- _ < _ ^ ?X =>
- assert (tmp: forall p, 1 + (p - 1) = p); auto with zarith;
-  rewrite <- (tmp X); clear tmp
- end.
- rewrite Zpower_exp; try rewrite Z.pow_1_r; auto with zarith.
- assert (tmp: forall p, 1 + (p -1) - 1 = p - 1); auto with zarith;
-  rewrite tmp; clear tmp; auto with zarith.
- match goal with |- ?X + ?Y < _ =>
- assert (Y < X); auto with zarith
- end.
- apply Zdiv_lt_upper_bound; auto with zarith.
- pattern 2 at 2; rewrite <- Z.pow_1_r; rewrite <- Zpower_exp;
-  auto with zarith.
- assert (tmp: forall p, (p - 1) + 1 = p); auto with zarith;
-  rewrite tmp; clear tmp; auto with zarith.
- Qed.
-
- Theorem add_mult_mult_2: forall w,
-    [|w_add_mul_div w_1 w w_0|] = 2 * [|w|] mod wB.
- intros w1.
- autorewrite with w_rewrite rm10; auto with zarith.
- rewrite Z.pow_1_r; auto with zarith.
- rewrite Z.mul_comm; auto.
- Qed.
-
- Theorem ww_add_mult_mult_2: forall w,
-    [[ww_add_mul_div (w_0W w_1) w W0]] = 2 * [[w]] mod wwB.
- intros w1.
- rewrite spec_ww_add_mul_div; auto with zarith.
- autorewrite with w_rewrite rm10.
- rewrite spec_w_0W; rewrite spec_w_1.
- rewrite Z.pow_1_r; auto with zarith.
- rewrite Z.mul_comm; auto.
- rewrite spec_w_0W; rewrite spec_w_1; auto with zarith.
- red; simpl; intros; discriminate.
- Qed.
-
- Theorem ww_add_mult_mult_2_plus_1: forall w,
-    [[ww_add_mul_div (w_0W w_1) w wwBm1]] =
-      (2 * [[w]] + 1) mod wwB.
- intros w1.
- rewrite spec_ww_add_mul_div; auto with zarith.
- rewrite spec_w_0W; rewrite spec_w_1; auto with zarith.
- rewrite Z.pow_1_r; auto with zarith.
- f_equal; auto.
- rewrite Z.mul_comm; f_equal; auto.
- autorewrite with w_rewrite rm10.
- unfold ww_digits, base.
- symmetry; apply Zdiv_unique with (r := 2 ^ (Zpos (ww_digits w_digits) - 1) -1);
-  auto with zarith.
- unfold ww_digits; split; auto with zarith.
- match goal with  |- 0 <= ?X - 1 =>
-   assert (0 < X); auto with zarith
- end.
- apply Z.pow_pos_nonneg; auto with zarith.
- match goal with  |- 0 <= ?X - 1 =>
-   assert (0 < X); auto with zarith; red; reflexivity
- end.
- unfold ww_digits; autorewrite with rm10.
- assert (tmp: forall p q r, p + (q - r) = p + q - r); auto with zarith;
-  rewrite tmp; clear tmp.
- assert (tmp: forall p, p + p = 2 * p); auto with zarith;
-  rewrite tmp; clear tmp.
- f_equal; auto.
- pattern 2 at 2; rewrite <- Z.pow_1_r; rewrite <- Zpower_exp;
-  auto with zarith.
- assert (tmp: forall p, 1 + (p - 1) = p); auto with zarith;
-  rewrite tmp; clear tmp; auto.
- match goal with  |- ?X - 1 >= 0 =>
-   assert (0 < X); auto with zarith; red; reflexivity
- end.
- rewrite spec_w_0W; rewrite spec_w_1; auto with zarith.
- red; simpl; intros; discriminate.
- Qed.
-
- Theorem Zplus_mod_one: forall a1 b1, 0 < b1 -> (a1 + b1) mod b1 = a1 mod b1.
- intros a1 b1 H; rewrite Zplus_mod; auto with zarith.
- rewrite Z_mod_same; try rewrite Z.add_0_r; auto with zarith.
- apply Zmod_mod; auto.
- Qed.
-
- Lemma C1_plus_wB: forall x, [+|C1 x|] = wB + [|x|].
- unfold interp_carry; auto with zarith.
- Qed.
-
- Theorem  spec_w_div2s : forall a1 a2 b,
-     wB/2 <= [|b|] -> [+|a1|] <= 2 * [|b|] ->
-     let (q,r) := w_div2s a1 a2 b in
-     [+|a1|] * wB + [|a2|] = [+|q|] *  (2 * [|b|]) + [+|r|] /\ 0 <= [+|r|] < 2 * [|b|].
- intros a1 a2 b H.
- assert (HH: 0 < [|b|]); auto with zarith.
- assert (U := wB_pos w_digits).
- apply Z.lt_le_trans with (2 := H); auto with zarith.
- apply Z.lt_le_trans with 1; auto with zarith.
- apply Zdiv_le_lower_bound; auto with zarith.
- unfold w_div2s; case a1; intros w0 H0.
- match goal with |- context[w_div21c ?y ?z ?t] =>
-   generalize (@spec_w_div21c y z t H);
-   case (w_div21c y z t); autorewrite with w_rewrite;
-   auto
- end.
- intros c w1; case c.
- simpl interp_carry; intros w2 (Hw1, Hw2).
- match goal with |- context[w_is_even ?y] =>
-   generalize (spec_w_is_even y);
-   case (w_is_even y)
- end.
- repeat rewrite C0_id.
- rewrite add_mult_div_2.
- intros H1; split; auto with zarith.
- rewrite Hw1.
- pattern [|w2|] at 1; rewrite (Z_div_mod_eq [|w2|] 2);
-  auto with zarith.
- rewrite H1; ring.
- repeat rewrite C0_id.
- rewrite add_mult_div_2.
- rewrite spec_w_add_c; auto with zarith.
- intros H1; split; auto with zarith.
- rewrite Hw1.
- pattern [|w2|] at 1; rewrite (Z_div_mod_eq [|w2|] 2);
-  auto with zarith.
- rewrite H1; ring.
- intros w2; rewrite C1_plus_wB.
- intros (Hw1, Hw2).
- match goal with |- context[w_is_even ?y] =>
-   generalize (spec_w_is_even y);
-   case (w_is_even y)
- end.
- repeat rewrite C0_id.
- intros H1; split; auto with zarith.
- rewrite Hw1.
- pattern [|w2|] at 1; rewrite (Z_div_mod_eq [|w2|] 2);
-  auto with zarith.
- rewrite H1.
- repeat rewrite C0_id.
- rewrite add_mult_div_2_plus_1; unfold base.
- match goal with |- context[_ ^ ?X] =>
- assert (tmp: forall p, 1 + (p - 1) = p); auto with zarith;
-  rewrite <- (tmp X); clear tmp; rewrite Zpower_exp;
-  try rewrite Z.pow_1_r; auto with zarith
- end.
- rewrite Pos2Z.inj_sub_max; auto with zarith.
- rewrite Z.max_r; auto with zarith.
- ring.
- repeat rewrite C0_id.
- rewrite spec_w_add_c; auto with zarith.
- intros H1; split; auto with zarith.
- rewrite add_mult_div_2_plus_1.
- rewrite Hw1.
- pattern [|w2|] at 1; rewrite (Z_div_mod_eq [|w2|] 2);
-  auto with zarith.
- rewrite H1.
- unfold base.
- match goal with |- context[_ ^ ?X] =>
- assert (tmp: forall p, 1 + (p - 1) = p); auto with zarith;
-  rewrite <- (tmp X); clear tmp; rewrite Zpower_exp;
-  try rewrite Z.pow_1_r; auto with zarith
- end.
- rewrite Pos2Z.inj_sub_max; auto with zarith.
- rewrite Z.max_r; auto with zarith.
- ring.
- repeat rewrite C1_plus_wB in H0.
- rewrite C1_plus_wB.
- match goal with |- context[w_div21c ?y ?z ?t] =>
-   generalize (@spec_w_div21c y z t H);
-   case (w_div21c y z t); autorewrite with w_rewrite;
-   auto
- end.
- intros c w1; case c.
- intros w2 (Hw1, Hw2); rewrite C0_id in Hw1.
- rewrite <- Zplus_mod_one in Hw1; auto with zarith.
- rewrite Zmod_small in Hw1; auto with zarith.
- match goal with |- context[w_is_even ?y] =>
-   generalize (spec_w_is_even y);
-   case (w_is_even y)
- end.
- repeat rewrite C0_id.
- intros H1; split; auto with zarith.
- rewrite add_mult_div_2_plus_1.
- replace (wB + [|w0|]) with ([|b|] + ([|w0|] - [|b|] + wB));
-  auto with zarith.
- rewrite Z.mul_add_distr_r; rewrite <- Z.add_assoc.
- rewrite Hw1.
- pattern [|w2|] at 1; rewrite (Z_div_mod_eq [|w2|] 2);
-  auto with zarith.
- rewrite H1; unfold base.
- match goal with |- context[_ ^ ?X] =>
- assert (tmp: forall p, 1 + (p - 1) = p); auto with zarith;
-  rewrite <- (tmp X); clear tmp; rewrite Zpower_exp;
-  try rewrite Z.pow_1_r; auto with zarith
- end.
- rewrite Pos2Z.inj_sub_max; auto with zarith.
- rewrite Z.max_r; auto with zarith.
- ring.
- repeat rewrite C0_id.
- rewrite add_mult_div_2_plus_1.
- rewrite spec_w_add_c; auto with zarith.
- intros H1; split; auto with zarith.
- replace (wB + [|w0|]) with ([|b|] + ([|w0|] - [|b|] + wB));
-  auto with zarith.
- rewrite Z.mul_add_distr_r; rewrite <- Z.add_assoc.
- rewrite Hw1.
- pattern [|w2|] at 1; rewrite (Z_div_mod_eq [|w2|] 2);
-  auto with zarith.
- rewrite H1; unfold base.
- match goal with |- context[_ ^ ?X] =>
- assert (tmp: forall p, 1 + (p - 1) = p); auto with zarith;
-  rewrite <- (tmp X); clear tmp; rewrite Zpower_exp;
-  try rewrite Z.pow_1_r; auto with zarith
- end.
- rewrite Pos2Z.inj_sub_max; auto with zarith.
- rewrite Z.max_r; auto with zarith.
- ring.
- split; auto with zarith.
- destruct (spec_to_Z b);auto with zarith.
- destruct (spec_to_Z w0);auto with zarith.
- destruct (spec_to_Z b);auto with zarith.
- destruct (spec_to_Z b);auto with zarith.
- intros w2; rewrite C1_plus_wB.
- rewrite <- Zplus_mod_one; auto with zarith.
- rewrite Zmod_small; auto with zarith.
- intros (Hw1, Hw2).
- match goal with |- context[w_is_even ?y] =>
-   generalize (spec_w_is_even y);
-   case (w_is_even y)
- end.
- repeat (rewrite C0_id || rewrite C1_plus_wB).
- intros H1; split; auto with zarith.
- rewrite add_mult_div_2.
- replace (wB + [|w0|]) with ([|b|] + ([|w0|] - [|b|] + wB));
-  auto with zarith.
- rewrite Z.mul_add_distr_r; rewrite <- Z.add_assoc.
- rewrite Hw1.
- pattern [|w2|] at 1; rewrite (Z_div_mod_eq [|w2|] 2);
-  auto with zarith.
- rewrite H1; ring.
- repeat (rewrite C0_id || rewrite C1_plus_wB).
- rewrite spec_w_add_c; auto with zarith.
- intros H1; split; auto with zarith.
- rewrite add_mult_div_2.
- replace (wB + [|w0|]) with ([|b|] + ([|w0|] - [|b|] + wB));
-  auto with zarith.
- rewrite Z.mul_add_distr_r; rewrite <- Z.add_assoc.
- rewrite Hw1.
- pattern [|w2|] at 1; rewrite (Z_div_mod_eq [|w2|] 2);
-  auto with zarith.
- rewrite H1; ring.
- split; auto with zarith.
- destruct (spec_to_Z b);auto with zarith.
- destruct (spec_to_Z w0);auto with zarith.
- destruct (spec_to_Z b);auto with zarith.
- destruct (spec_to_Z b);auto with zarith.
- Qed.
-
- Theorem wB_div_4:  4 * (wB / 4) = wB.
- Proof.
-  unfold base.
-  assert (2 ^ Zpos w_digits =
-              4 * (2 ^ (Zpos w_digits - 2))).
-  change 4 with (2 ^ 2).
-  rewrite <- Zpower_exp; auto with zarith.
-  f_equal; auto with zarith.
-  rewrite H.
-  rewrite (fun x => (Z.mul_comm 4 (2 ^x))).
-  rewrite Z_div_mult; auto with zarith.
- Qed.
-
- Theorem Zsquare_mult: forall p, p ^ 2 = p * p.
- intros p; change 2 with (1 + 1); rewrite Zpower_exp;
-  try rewrite Z.pow_1_r; auto with zarith.
- Qed.
-
- Theorem Zsquare_pos: forall p, 0 <= p ^ 2.
- intros p; case (Z.le_gt_cases 0 p); intros H1.
- rewrite Zsquare_mult; apply Z.mul_nonneg_nonneg; auto with zarith.
- rewrite Zsquare_mult; replace (p * p) with ((- p) * (- p)); try ring.
- apply Z.mul_nonneg_nonneg; auto with zarith.
- Qed.
-
- Lemma spec_split: forall x,
-  [|fst (split x)|] * wB + [|snd (split x)|] = [[x]].
- intros x; case x; simpl; autorewrite with w_rewrite;
-  auto with zarith.
- Qed.
-
- Theorem mult_wwB: forall x y, [|x|] * [|y|] < wwB.
- Proof.
-  intros x y; rewrite wwB_wBwB; rewrite Z.pow_2_r.
-  generalize (spec_to_Z x); intros U.
-  generalize (spec_to_Z y); intros U1.
-  apply Z.le_lt_trans with ((wB -1 ) * (wB - 1)); auto with zarith.
-  apply Z.mul_le_mono_nonneg; auto with zarith.
-  rewrite ?Z.mul_sub_distr_l, ?Z.mul_sub_distr_r; auto with zarith.
- Qed.
- Hint Resolve mult_wwB.
-
- Lemma spec_ww_sqrt2 : forall x y,
-       wwB/ 4 <= [[x]] ->
-       let (s,r) := ww_sqrt2 x y in
-          [||WW x y||] = [[s]] ^ 2 + [+[r]] /\
-          [+[r]] <= 2 * [[s]].
- intros x y H; unfold ww_sqrt2.
- repeat match goal with |- context[split ?x] =>
-   generalize (spec_split x); case (split x)
- end; simpl @fst; simpl @snd.
- intros w0 w1 Hw0 w2 w3 Hw1.
- assert (U: wB/4 <= [|w2|]).
- case (Z.le_gt_cases (wB / 4) [|w2|]); auto; intros H1.
- contradict H; apply Z.lt_nge.
- rewrite wwB_wBwB; rewrite Z.pow_2_r.
- pattern wB at 1; rewrite <- wB_div_4; rewrite <- Z.mul_assoc;
-   rewrite Z.mul_comm.
- rewrite Z_div_mult; auto with zarith.
- rewrite <- Hw1.
- match goal with |- _  < ?X =>
-  pattern X; rewrite <- Z.add_0_r; apply beta_lex_inv;
-  auto with zarith
- end.
- destruct (spec_to_Z w3);auto with zarith.
- generalize (@spec_w_sqrt2 w2 w3 U); case (w_sqrt2 w2 w3).
- intros w4 c (H1, H2).
- assert (U1: wB/2 <= [|w4|]).
- case (Z.le_gt_cases (wB/2) [|w4|]); auto with zarith.
- intros U1.
- assert (U2 : [|w4|] <= wB/2 -1); auto with zarith.
- assert (U3 : [|w4|] ^ 2 <= wB/4 * wB - wB + 1); auto with zarith.
- match goal with |- ?X ^ 2 <= ?Y =>
-  rewrite Zsquare_mult;
-  replace Y with ((wB/2 - 1) * (wB/2 -1))
- end.
- apply Z.mul_le_mono_nonneg; auto with zarith.
- destruct (spec_to_Z w4);auto with zarith.
- destruct (spec_to_Z w4);auto with zarith.
- pattern wB at 4 5; rewrite <- wB_div_2.
- rewrite Z.mul_assoc.
- replace ((wB / 4) * 2) with (wB / 2).
- ring.
- pattern wB at 1; rewrite <- wB_div_4.
- change 4 with (2 * 2).
- rewrite <- Z.mul_assoc; rewrite (Z.mul_comm 2).
- rewrite Z_div_mult; try ring; auto with zarith.
- assert (U4 : [+|c|]  <= wB -2); auto with zarith.
- apply Z.le_trans with (1 := H2).
- match goal with |- ?X <= ?Y =>
-  replace Y with (2 * (wB/ 2 - 1)); auto with zarith
- end.
- pattern wB at 2; rewrite <- wB_div_2; auto with zarith.
- match type of H1 with ?X = _ =>
-  assert (U5: X < wB / 4 * wB)
- end.
- rewrite H1; auto with zarith.
- contradict U; apply Z.lt_nge.
- apply Z.mul_lt_mono_pos_r with wB; auto with zarith.
- destruct (spec_to_Z w4);auto with zarith.
- apply Z.le_lt_trans with (2 := U5).
- unfold ww_to_Z, zn2z_to_Z.
- destruct (spec_to_Z w3);auto with zarith.
- generalize (@spec_w_div2s c w0 w4 U1 H2).
- case (w_div2s c w0 w4).
- intros c0; case c0; intros w5;
-   repeat (rewrite C0_id || rewrite C1_plus_wB).
- intros c1; case c1; intros w6;
-   repeat (rewrite C0_id || rewrite C1_plus_wB).
- intros (H3, H4).
- match goal with |- context [ww_sub_c ?y ?z] =>
-  generalize (spec_ww_sub_c y z); case (ww_sub_c y z)
- end.
- intros z; change [-[C0 z]] with ([[z]]).
- change [+[C0 z]] with ([[z]]).
- intros H5; rewrite spec_w_square_c in H5;
-  auto.
- split.
- unfold zn2z_to_Z; rewrite <- Hw1.
- unfold ww_to_Z, zn2z_to_Z in H1. rewrite H1.
- rewrite <- Hw0.
- match goal with |- (?X ^2 + ?Y) * wwB + (?Z * wB + ?T) = ?U =>
-  transitivity ((X * wB) ^ 2 + (Y * wB + Z) * wB + T)
- end.
- repeat rewrite Zsquare_mult.
- rewrite wwB_wBwB; ring.
- rewrite H3.
- rewrite H5.
- unfold ww_to_Z, zn2z_to_Z.
- repeat rewrite Zsquare_mult; ring.
- rewrite H5.
- unfold ww_to_Z, zn2z_to_Z.
- match goal with |- ?X - ?Y * ?Y <= _ =>
-  assert (V := Zsquare_pos Y);
-  rewrite Zsquare_mult in V;
-  apply Z.le_trans with X; auto with zarith;
-  clear V
- end.
- match goal with |- ?X * wB + ?Y <= 2 * (?Z * wB + ?T) =>
-  apply Z.le_trans with ((2 * Z - 1) * wB + wB); auto with zarith
- end.
- destruct (spec_to_Z w1);auto with zarith.
- match goal with |- ?X <= _ =>
-  replace X with (2 * [|w4|] * wB); auto with zarith
- end.
- rewrite Z.mul_add_distr_l; rewrite Z.mul_assoc.
- destruct (spec_to_Z w5); auto with zarith.
- ring.
- intros z; replace [-[C1 z]] with (- wwB + [[z]]).
- 2: simpl; case wwB; auto with zarith.
- intros H5; rewrite spec_w_square_c in H5;
-  auto.
- match goal with |- context [ww_pred_c ?y] =>
-  generalize (spec_ww_pred_c y); case (ww_pred_c y)
- end.
- intros z1; change [-[C0 z1]] with ([[z1]]).
- rewrite ww_add_mult_mult_2.
- rewrite spec_ww_add_c.
- rewrite spec_ww_pred.
- rewrite <- Zmod_unique with (q := 1) (r := -wwB + 2 * [[WW w4 w5]]);
-  auto with zarith.
- intros Hz1; rewrite Zmod_small; auto with zarith.
- match type of H5 with -?X + ?Y = ?Z =>
-  assert (V: Y = Z + X);
-  try (rewrite <- H5; ring)
- end.
- split.
- unfold zn2z_to_Z; rewrite <- Hw1.
- unfold ww_to_Z, zn2z_to_Z in H1; rewrite H1.
- rewrite <- Hw0.
- match goal with |- (?X ^2 + ?Y) * wwB + (?Z * wB + ?T) = ?U =>
-  transitivity ((X * wB) ^ 2 + (Y * wB + Z) * wB + T)
- end.
- repeat rewrite Zsquare_mult.
- rewrite wwB_wBwB; ring.
- rewrite H3.
- rewrite V.
- rewrite Hz1.
- unfold ww_to_Z; simpl zn2z_to_Z.
- repeat rewrite Zsquare_mult; ring.
- rewrite Hz1.
- destruct (spec_ww_to_Z w_digits w_to_Z spec_to_Z z);auto with zarith.
- assert (V1 := spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW w4 w5)).
- assert (0 <  [[WW w4 w5]]); auto with zarith.
- apply Z.lt_le_trans with (wB/ 2 * wB + 0); auto with zarith.
- autorewrite with rm10; apply Z.mul_pos_pos; auto with zarith.
- apply Z.mul_lt_mono_pos_r with 2; auto with zarith.
- autorewrite with rm10.
- rewrite Z.mul_comm; rewrite wB_div_2; auto with zarith.
- case (spec_to_Z w5);auto with zarith.
- case (spec_to_Z w5);auto with zarith.
- simpl.
- assert (V2 := spec_to_Z w5);auto with zarith.
- assert (V1 := spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW w4 w5)); auto with zarith.
- split; auto with zarith.
- assert (wwB <= 2 * [[WW w4 w5]]); auto with zarith.
- apply Z.le_trans with (2 * ([|w4|] * wB)).
- rewrite wwB_wBwB; rewrite Z.pow_2_r.
- rewrite Z.mul_assoc; apply Z.mul_le_mono_nonneg_r; auto with zarith.
- assert (V2 := spec_to_Z w5);auto with zarith.
- rewrite <- wB_div_2; auto with zarith.
- simpl ww_to_Z; assert (V2 := spec_to_Z w5);auto with zarith.
- assert (V1 := spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW w4 w5)); auto with zarith.
- intros z1; change [-[C1 z1]] with (-wwB + [[z1]]).
- match goal with |- context[([+[C0 ?z]])] =>
-   change [+[C0 z]] with ([[z]])
- end.
- rewrite spec_ww_add; auto with zarith.
- rewrite spec_ww_pred; auto with zarith.
- rewrite ww_add_mult_mult_2.
- rename V1 into VV1.
- assert (VV2: 0 <  [[WW w4 w5]]); auto with zarith.
- apply Z.lt_le_trans with (wB/ 2 * wB + 0); auto with zarith.
- autorewrite with rm10; apply Z.mul_pos_pos; auto with zarith.
- apply Z.mul_lt_mono_pos_r with 2; auto with zarith.
- autorewrite with rm10.
- rewrite Z.mul_comm; rewrite wB_div_2; auto with zarith.
- assert (VV3 := spec_to_Z w5);auto with zarith.
- assert (VV3 := spec_to_Z w5);auto with zarith.
- simpl.
- assert (VV3 := spec_to_Z w5);auto with zarith.
- assert (VV3: wwB <= 2 * [[WW w4 w5]]); auto with zarith.
- apply Z.le_trans with (2 * ([|w4|] * wB)).
- rewrite wwB_wBwB; rewrite Z.pow_2_r.
- rewrite Z.mul_assoc; apply Z.mul_le_mono_nonneg_r; auto with zarith.
- case (spec_to_Z w5);auto with zarith.
- rewrite <- wB_div_2; auto with zarith.
- simpl ww_to_Z; assert (V4 := spec_to_Z w5);auto with zarith.
- rewrite <- Zmod_unique with (q := 1) (r := -wwB + 2 * [[WW w4 w5]]);
-  auto with zarith.
- intros Hz1; rewrite Zmod_small; auto with zarith.
- match type of H5 with -?X + ?Y = ?Z =>
-  assert (V: Y = Z + X);
-  try (rewrite <- H5; ring)
- end.
- match type of Hz1 with -?X + ?Y = -?X + ?Z - 1 =>
-  assert (V1: Y = Z - 1);
-  [replace (Z - 1) with (X + (-X + Z -1));
-    [rewrite <- Hz1 | idtac]; ring
-    | idtac]
- end.
- rewrite <- Zmod_unique with (q := 1) (r := -wwB + [[z1]] + [[z]]);
-  auto with zarith.
- unfold zn2z_to_Z; rewrite <- Hw1.
- unfold ww_to_Z, zn2z_to_Z in H1; rewrite H1.
- rewrite <- Hw0.
- split.
- match goal with |- (?X ^2 + ?Y) * wwB + (?Z * wB + ?T) = ?U =>
-  transitivity ((X * wB) ^ 2 + (Y * wB + Z) * wB + T)
- end.
- repeat rewrite Zsquare_mult.
- rewrite wwB_wBwB; ring.
- rewrite H3.
- rewrite V.
- rewrite Hz1.
- unfold ww_to_Z; simpl zn2z_to_Z.
- repeat rewrite Zsquare_mult; ring.
- assert (V2 := spec_ww_to_Z w_digits w_to_Z spec_to_Z z);auto with zarith.
- assert (V2 := spec_ww_to_Z w_digits w_to_Z spec_to_Z z);auto with zarith.
- assert (V3 := spec_ww_to_Z w_digits w_to_Z spec_to_Z z1);auto with zarith.
- split; auto with zarith.
- rewrite (Z.add_comm (-wwB)); rewrite <- Z.add_assoc.
- rewrite H5.
- match goal with |- 0 <= ?X + (?Y - ?Z)  =>
-  apply Z.le_trans with (X - Z); auto with zarith
- end.
- 2: generalize (spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW w6 w1)); unfold ww_to_Z; auto with zarith.
- rewrite V1.
- match goal with |- 0 <= ?X - 1 - ?Y =>
-   assert (Y < X); auto with zarith
- end.
- apply Z.lt_le_trans with wwB; auto with zarith.
- intros (H3, H4).
- match goal with |- context [ww_sub_c ?y ?z] =>
-  generalize (spec_ww_sub_c y z); case (ww_sub_c y z)
- end.
- intros z; change [-[C0 z]] with ([[z]]).
- match goal with |- context[([+[C1 ?z]])] =>
-   replace [+[C1 z]] with (wwB + [[z]])
- end.
- 2: simpl; case wwB; auto.
- intros H5; rewrite spec_w_square_c in H5;
-  auto.
- split.
- change ([||WW x y||]) with ([[x]] * wwB + [[y]]).
- rewrite <- Hw1.
- unfold ww_to_Z, zn2z_to_Z in H1; rewrite H1.
- rewrite <- Hw0.
- match goal with |- (?X ^2 + ?Y) * wwB + (?Z * wB + ?T) = ?U =>
-  transitivity ((X * wB) ^ 2 + (Y * wB + Z) * wB + T)
- end.
- repeat rewrite Zsquare_mult.
- rewrite wwB_wBwB; ring.
- rewrite H3.
- rewrite H5.
- unfold ww_to_Z; simpl zn2z_to_Z.
- rewrite wwB_wBwB.
- repeat rewrite Zsquare_mult; ring.
- simpl ww_to_Z.
- rewrite H5.
- simpl ww_to_Z.
- rewrite wwB_wBwB; rewrite Z.pow_2_r.
- match goal with |- ?X * ?Y + (?Z * ?Y + ?T - ?U) <= _ =>
-  apply Z.le_trans with (X * Y + (Z * Y + T - 0));
-  auto with zarith
- end.
- assert (V := Zsquare_pos [|w5|]);
- rewrite Zsquare_mult in V; auto with zarith.
- autorewrite with rm10.
- match goal with |- _ <= 2 * (?U * ?V + ?W) =>
-  apply Z.le_trans with (2 * U * V + 0);
-  auto with zarith
- end.
- match goal with |- ?X * ?Y + (?Z * ?Y + ?T) <= _ =>
-  replace (X * Y + (Z * Y + T)) with ((X + Z) * Y + T);
-  try ring
- end.
- apply Z.lt_le_incl; apply beta_lex_inv; auto with zarith.
- destruct (spec_to_Z w1);auto with zarith.
- destruct (spec_to_Z w5);auto with zarith.
- rewrite Z.mul_add_distr_l; auto with zarith.
- rewrite Z.mul_assoc; auto with zarith.
- intros z; replace [-[C1 z]] with (- wwB + [[z]]).
- 2: simpl; case wwB; auto with zarith.
- intros H5; rewrite spec_w_square_c in H5;
-  auto.
- match goal with |- context[([+[C0 ?z]])] =>
-   change [+[C0 z]] with ([[z]])
- end.
- match type of H5 with -?X + ?Y = ?Z =>
-  assert (V: Y = Z + X);
-  try (rewrite <- H5; ring)
- end.
- change ([||WW x y||]) with ([[x]] * wwB + [[y]]).
- simpl ww_to_Z.
- rewrite <- Hw1.
- simpl ww_to_Z in H1; rewrite H1.
- rewrite <- Hw0.
- split.
- match goal with |- (?X ^2 + ?Y) * wwB + (?Z * wB + ?T) = ?U =>
-  transitivity ((X * wB) ^ 2 + (Y * wB + Z) * wB + T)
- end.
- repeat rewrite Zsquare_mult.
- rewrite wwB_wBwB; ring.
- rewrite H3.
- rewrite V.
- simpl ww_to_Z.
- rewrite wwB_wBwB.
- repeat rewrite Zsquare_mult; ring.
- rewrite V.
- simpl ww_to_Z.
- rewrite wwB_wBwB; rewrite Z.pow_2_r.
- match goal with |- (?Z * ?Y + ?T - ?U) + ?X * ?Y <= _ =>
-  apply Z.le_trans with ((Z * Y + T - 0) + X * Y);
-  auto with zarith
- end.
- assert (V1 := Zsquare_pos [|w5|]);
- rewrite Zsquare_mult in V1; auto with zarith.
- autorewrite with rm10.
- match goal with |- _ <= 2 * (?U * ?V + ?W) =>
-  apply Z.le_trans with (2 * U * V + 0);
-  auto with zarith
- end.
- match goal with |- (?Z * ?Y + ?T) + ?X * ?Y  <= _ =>
-  replace ((Z * Y + T) + X * Y) with ((X + Z) * Y + T);
-  try ring
- end.
- apply Z.lt_le_incl; apply beta_lex_inv; auto with zarith.
- destruct (spec_to_Z w1);auto with zarith.
- destruct (spec_to_Z w5);auto with zarith.
- rewrite Z.mul_add_distr_l; auto with zarith.
- rewrite Z.mul_assoc; auto with zarith.
- Z.le_elim H2.
- intros c1 (H3, H4).
- match type of H3 with ?X = ?Y => absurd (X < Y) end.
- apply Z.le_ngt; rewrite <- H3; auto with zarith.
- rewrite Z.mul_add_distr_r.
- apply Z.lt_le_trans with ((2 * [|w4|]) * wB + 0);
-  auto with zarith.
- apply beta_lex_inv; auto with zarith.
- destruct (spec_to_Z w0);auto with zarith.
- assert (V1 := spec_to_Z w5);auto with zarith.
- rewrite (Z.mul_comm wB); auto with zarith.
- assert (0 <= [|w5|] * (2 * [|w4|])); auto with zarith.
- intros c1 (H3, H4); rewrite H2 in H3.
- match type of H3 with ?X + ?Y = (?Z + ?T) * ?U + ?V =>
-  assert (VV: (Y = (T * U) + V));
-  [replace Y with ((X + Y) - X);
-    [rewrite H3; ring | ring] | idtac]
- end.
- assert (V1 := spec_to_Z w0);auto with zarith.
- assert (V2 := spec_to_Z w5);auto with zarith.
- case V2; intros V3 _.
- Z.le_elim V3; auto with zarith.
- match type of VV with ?X = ?Y => absurd (X < Y) end.
- apply Z.le_ngt; rewrite <- VV; auto with zarith.
- apply Z.lt_le_trans with wB; auto with zarith.
- match goal with |- _ <= ?X + _ =>
-  apply Z.le_trans with X; auto with zarith
- end.
- match goal with |- _ <= _ * ?X =>
-  apply Z.le_trans with (1 * X); auto with zarith
- end.
- autorewrite with rm10.
- rewrite <- wB_div_2; apply Z.mul_le_mono_nonneg_l; auto with zarith.
- rewrite <- V3 in VV; generalize VV; autorewrite with rm10;
- clear VV; intros VV.
- rewrite spec_ww_add_c; auto with zarith.
- rewrite ww_add_mult_mult_2_plus_1.
- match goal with |- context[?X mod wwB] =>
-   rewrite <- Zmod_unique with (q := 1) (r := -wwB + X)
- end; auto with zarith.
- simpl ww_to_Z.
- rewrite spec_w_Bm1; auto with zarith.
- split.
- change ([||WW x y||]) with ([[x]] * wwB + [[y]]).
- rewrite <- Hw1.
- simpl ww_to_Z in H1; rewrite H1.
- rewrite <- Hw0.
- match goal with |- (?X ^2 + ?Y) * wwB + (?Z * wB + ?T) = ?U =>
-  transitivity ((X * wB) ^ 2 + (Y * wB + Z) * wB + T)
- end.
- repeat rewrite Zsquare_mult.
- rewrite wwB_wBwB; ring.
- rewrite H2.
- rewrite wwB_wBwB.
- repeat rewrite Zsquare_mult; ring.
- assert (V4 := spec_ww_to_Z w_digits w_to_Z spec_to_Z y);auto with zarith.
- assert (V4 := spec_ww_to_Z w_digits w_to_Z spec_to_Z y);auto with zarith.
- simpl ww_to_Z; unfold ww_to_Z.
- rewrite spec_w_Bm1; auto with zarith.
- split.
- rewrite wwB_wBwB; rewrite Z.pow_2_r.
- match goal with |- _ <= -?X + (2 * (?Z * ?T + ?U) + ?V) =>
-  assert (X <= 2 * Z * T); auto with zarith
- end.
- apply Z.mul_le_mono_nonneg_r; auto with zarith.
- rewrite <- wB_div_2; apply Z.mul_le_mono_nonneg_l; auto with zarith.
- rewrite Z.mul_add_distr_l; auto with zarith.
- rewrite Z.mul_assoc; auto with zarith.
- match goal with |- _ + ?X  < _ =>
-  replace X with ((2 * (([|w4|]) + 1) * wB) - 1); try ring
- end.
- assert (2 * ([|w4|] + 1) * wB <= 2 * wwB); auto with zarith.
- rewrite <- Z.mul_assoc; apply Z.mul_le_mono_nonneg_l; auto with zarith.
- rewrite wwB_wBwB; rewrite Z.pow_2_r.
- apply Z.mul_le_mono_nonneg_r; auto with zarith.
- case (spec_to_Z w4);auto with zarith.
-Qed.
-
- Lemma spec_ww_is_zero: forall x,
-   if ww_is_zero x then [[x]] = 0 else 0 < [[x]].
-  intro x; unfold ww_is_zero.
-  rewrite spec_ww_compare. case Z.compare_spec;
-   auto with zarith.
-  simpl ww_to_Z.
-  assert (V4 := spec_ww_to_Z w_digits w_to_Z spec_to_Z x);auto with zarith.
-  Qed.
-
-  Lemma wwB_4_2: 2 * (wwB / 4) = wwB/ 2.
-  pattern wwB at 1; rewrite wwB_wBwB; rewrite Z.pow_2_r.
-  rewrite <- wB_div_2.
-  match goal with |- context[(2 * ?X) * (2 * ?Z)] =>
-    replace ((2 * X) * (2 * Z)) with ((X * Z) * 4); try ring
-  end.
-  rewrite Z_div_mult; auto with zarith.
-  rewrite Z.mul_assoc; rewrite wB_div_2.
-  rewrite wwB_div_2; ring.
-  Qed.
-
-
-  Lemma spec_ww_head1
-       : forall x : zn2z w,
-         (ww_is_even (ww_head1 x) = true) /\
-         (0 < [[x]] -> wwB / 4 <= 2 ^ [[ww_head1 x]] * [[x]] < wwB).
-  assert (U := wB_pos w_digits).
-  intros x; unfold ww_head1.
-  generalize (spec_ww_is_even (ww_head0 x)); case_eq (ww_is_even (ww_head0 x)).
-    intros HH H1; rewrite HH; split; auto.
-  intros H2.
-  generalize (spec_ww_head0 x H2); case (ww_head0 x);  autorewrite with rm10.
-  intros (H3, H4); split; auto with zarith.
-  apply Z.le_trans with (2 := H3).
-  apply Zdiv_le_compat_l; auto with zarith.
-  intros xh xl (H3, H4); split; auto with zarith.
-  apply Z.le_trans with (2 := H3).
-  apply Zdiv_le_compat_l; auto with zarith.
-  intros H1.
-  case (spec_to_w_Z (ww_head0 x)); intros Hv1 Hv2.
-  assert (Hp0: 0 < [[ww_head0 x]]).
-    generalize (spec_ww_is_even (ww_head0 x)); rewrite H1.
-    generalize Hv1; case [[ww_head0 x]].
-    rewrite Zmod_small; auto with zarith.
-    intros; assert (0 < Zpos p); auto with zarith.
-    red; simpl; auto.
-    intros p H2; case H2; auto.
-  assert (Hp: [[ww_pred (ww_head0 x)]] = [[ww_head0 x]] - 1).
-    rewrite spec_ww_pred.
-    rewrite Zmod_small; auto with zarith.
-  intros H2; split.
-    generalize (spec_ww_is_even (ww_pred (ww_head0 x)));
-      case ww_is_even; auto.
-    rewrite Hp.
-    rewrite Zminus_mod; auto with zarith.
-    rewrite H2; repeat rewrite Zmod_small; auto with zarith.
-  intros H3; rewrite Hp.
-  case (spec_ww_head0 x); auto; intros Hv3 Hv4.
-  assert (Hu: forall u, 0 < u -> 2 *  2 ^ (u - 1) = 2 ^u).
-    intros u Hu.
-    pattern 2 at 1; rewrite <- Z.pow_1_r.
-    rewrite <- Zpower_exp; auto with zarith.
-    ring_simplify (1 + (u - 1)); auto with zarith.
-  split; auto with zarith.
-  apply Z.mul_le_mono_pos_r with 2; auto with zarith.
-  repeat rewrite (fun x => Z.mul_comm x 2).
-  rewrite wwB_4_2.
-  rewrite Z.mul_assoc; rewrite Hu; auto with zarith.
-  apply Z.le_lt_trans with (2 * 2 ^ ([[ww_head0 x]] - 1) * [[x]]); auto with zarith;
-    rewrite Hu; auto with zarith.
-  apply Z.mul_le_mono_nonneg_r; auto with zarith.
-  apply Zpower_le_monotone; auto with zarith.
-  Qed.
-
-  Theorem wwB_4_wB_4: wwB / 4 = wB / 4 * wB.
-  Proof.
-  symmetry; apply Zdiv_unique with 0; auto with zarith.
-  rewrite Z.mul_assoc; rewrite wB_div_4; auto with zarith.
-  rewrite wwB_wBwB; ring.
-  Qed.
-
-  Lemma spec_ww_sqrt : forall x,
-       [[ww_sqrt x]] ^ 2 <= [[x]] < ([[ww_sqrt x]] + 1) ^ 2.
-  assert (U := wB_pos w_digits).
-  intro x; unfold ww_sqrt.
-  generalize (spec_ww_is_zero x); case (ww_is_zero x).
-  simpl ww_to_Z; simpl Z.pow; unfold Z.pow_pos; simpl;
-    auto with zarith.
-  intros H1.
-  rewrite spec_ww_compare. case Z.compare_spec;
-    simpl ww_to_Z; autorewrite with rm10.
-  generalize H1; case x.
-  intros HH; contradict HH; simpl ww_to_Z; auto with zarith.
-  intros w0 w1; simpl ww_to_Z; autorewrite with w_rewrite rm10.
-  intros H2; case (spec_ww_head1 (WW w0 w1)); intros H3 H4 H5.
-  generalize (H4 H2); clear H4; rewrite H5; clear H5; autorewrite with rm10.
-  intros (H4, H5).
-  assert (V: wB/4 <= [|w0|]).
-  apply beta_lex with 0 [|w1|] wB; auto with zarith; autorewrite with rm10.
-  rewrite <- wwB_4_wB_4; auto.
-  generalize (@spec_w_sqrt2 w0 w1 V);auto with zarith.
-  case (w_sqrt2 w0 w1); intros w2 c.
-  simpl ww_to_Z; simpl @fst.
-  case c; unfold interp_carry; autorewrite with rm10.
-  intros w3 (H6, H7); rewrite H6.
-  assert (V1 := spec_to_Z w3);auto with zarith.
-  split; auto with zarith.
-  apply Z.le_lt_trans with ([|w2|] ^2  + 2 * [|w2|]); auto with zarith.
-  match goal with |- ?X < ?Z =>
-    replace Z with (X + 1); auto with zarith
-  end.
-  repeat rewrite Zsquare_mult; ring.
-  intros w3 (H6, H7); rewrite H6.
-  assert (V1 := spec_to_Z w3);auto with zarith.
-  split; auto with zarith.
-  apply Z.le_lt_trans with ([|w2|] ^2  + 2 * [|w2|]); auto with zarith.
-  match goal with |- ?X < ?Z =>
-    replace Z with (X + 1); auto with zarith
-  end.
-  repeat rewrite Zsquare_mult; ring.
-  intros HH; case (spec_to_w_Z (ww_head1 x)); auto with zarith.
-  intros Hv1.
-  case (spec_ww_head1 x); intros Hp1 Hp2.
-  generalize (Hp2 H1); clear Hp2; intros Hp2.
-  assert (Hv2: [[ww_head1 x]] <= Zpos (xO w_digits)).
-    case (Z.le_gt_cases (Zpos (xO w_digits)) [[ww_head1 x]]); auto with zarith; intros HH1.
-    case Hp2; intros _ HH2; contradict HH2.
-    apply Z.le_ngt; unfold base.
-    apply Z.le_trans with (2 ^ [[ww_head1 x]]).
-      apply Zpower_le_monotone; auto with zarith.
-    pattern (2 ^ [[ww_head1 x]]) at 1;
-      rewrite <- (Z.mul_1_r (2 ^ [[ww_head1 x]])).
-    apply Z.mul_le_mono_nonneg_l; auto with zarith.
-  generalize (spec_ww_add_mul_div x W0 (ww_head1 x) Hv2);
-     case ww_add_mul_div.
-  simpl ww_to_Z; autorewrite with w_rewrite rm10.
-  rewrite Zmod_small; auto with zarith.
-  intros H2. symmetry in H2. rewrite Z.mul_eq_0 in H2. destruct H2 as [H2|H2].
-  rewrite H2; unfold Z.pow, Z.pow_pos; simpl; auto with zarith.
-  match type of H2 with ?X = ?Y =>
-   absurd (Y < X); try (rewrite H2; auto with zarith; fail)
-  end.
-  apply Z.pow_pos_nonneg; auto with zarith.
-  split; auto with zarith.
-  case Hp2; intros _ tmp; apply Z.le_lt_trans with (2 := tmp);
-   clear tmp.
-  rewrite Z.mul_comm; apply Z.mul_le_mono_nonneg_r; auto with zarith.
-  assert (Hv0: [[ww_head1 x]] = 2 * ([[ww_head1 x]]/2)).
-    pattern [[ww_head1 x]] at 1; rewrite (Z_div_mod_eq [[ww_head1 x]] 2);
-      auto with zarith.
-    generalize (spec_ww_is_even (ww_head1 x)); rewrite Hp1;
-      intros tmp; rewrite tmp; rewrite Z.add_0_r; auto.
-  intros w0 w1; autorewrite with w_rewrite rm10.
-  rewrite Zmod_small; auto with zarith.
-  2: rewrite Z.mul_comm; auto with zarith.
-  intros H2.
-  assert (V: wB/4 <= [|w0|]).
-  apply beta_lex with 0 [|w1|] wB; auto with zarith; autorewrite with rm10.
-  simpl ww_to_Z in H2; rewrite H2.
-  rewrite <- wwB_4_wB_4; auto with zarith.
-  rewrite Z.mul_comm; auto with zarith.
-  assert (V1 := spec_to_Z w1);auto with zarith.
-  generalize (@spec_w_sqrt2 w0 w1 V);auto with zarith.
-  case (w_sqrt2 w0 w1); intros w2 c.
-  case (spec_to_Z w2); intros HH1 HH2.
-  simpl ww_to_Z; simpl @fst.
-  assert (Hv3: [[ww_pred ww_zdigits]]
-                 = Zpos (xO w_digits) - 1).
-    rewrite spec_ww_pred; rewrite spec_ww_zdigits.
-    rewrite Zmod_small; auto with zarith.
-    split; auto with zarith.
-    apply Z.lt_le_trans with (Zpos (xO w_digits)); auto with zarith.
-    unfold base; apply Zpower2_le_lin; auto with zarith.
-  assert (Hv4: [[ww_head1 x]]/2 < wB).
-    apply Z.le_lt_trans with (Zpos w_digits).
-    apply Z.mul_le_mono_pos_r with 2; auto with zarith.
-    repeat rewrite (fun x => Z.mul_comm x 2).
-    rewrite <- Hv0;  rewrite <- Pos2Z.inj_xO; auto.
-    unfold base; apply Zpower2_lt_lin; auto with zarith.
-  assert (Hv5: [[(ww_add_mul_div (ww_pred ww_zdigits) W0 (ww_head1 x))]]
-                 = [[ww_head1 x]]/2).
-    rewrite spec_ww_add_mul_div.
-    simpl ww_to_Z; autorewrite with rm10.
-    rewrite Hv3.
-    ring_simplify (Zpos (xO w_digits) - (Zpos (xO w_digits) - 1)).
-    rewrite Z.pow_1_r.
-    rewrite Zmod_small; auto with zarith.
-    split; auto with zarith.
-    apply Z.lt_le_trans with (1 := Hv4); auto with zarith.
-    unfold base; apply Zpower_le_monotone; auto with zarith.
-    split; unfold ww_digits; try rewrite Pos2Z.inj_xO; auto with zarith.
-    rewrite Hv3; auto with zarith.
-  assert (Hv6: [|low(ww_add_mul_div (ww_pred ww_zdigits) W0 (ww_head1 x))|]
-                 = [[ww_head1 x]]/2).
-    rewrite spec_low.
-    rewrite Hv5; rewrite Zmod_small; auto with zarith.
-  rewrite spec_w_add_mul_div; auto with zarith.
-  rewrite spec_w_sub; auto with zarith.
-  rewrite spec_w_0.
-  simpl ww_to_Z; autorewrite with rm10.
-  rewrite Hv6; rewrite spec_w_zdigits.
-  rewrite (fun x y => Zmod_small (x - y)).
-  ring_simplify (Zpos w_digits - (Zpos w_digits - [[ww_head1 x]] / 2)).
-  rewrite Zmod_small.
-  simpl ww_to_Z in H2; rewrite H2; auto with zarith.
-  intros (H4, H5); split.
-  apply Z.mul_le_mono_pos_r with (2 ^ [[ww_head1 x]]); auto with zarith.
-  rewrite H4.
-  apply Z.le_trans with ([|w2|] ^ 2); auto with zarith.
-  rewrite Z.mul_comm.
-  pattern [[ww_head1 x]] at 1;
-    rewrite Hv0; auto with zarith.
-  rewrite (Z.mul_comm 2); rewrite Z.pow_mul_r;
-    auto with zarith.
-  assert (tmp: forall p q, p ^ 2 * q ^ 2 = (p * q) ^2);
-   try (intros; repeat rewrite Zsquare_mult; ring);
-   rewrite tmp; clear tmp.
-  apply Zpower_le_monotone3; auto with zarith.
-  split; auto with zarith.
-  pattern [|w2|] at 2;
-     rewrite (Z_div_mod_eq [|w2|] (2 ^ ([[ww_head1 x]] / 2)));
-     auto with zarith.
-  match goal with |- ?X <= ?X + ?Y =>
-    assert (0 <= Y); auto with zarith
-  end.
-  case (Z_mod_lt [|w2|] (2 ^ ([[ww_head1 x]] / 2))); auto with zarith.
-  case c; unfold interp_carry; autorewrite with rm10;
-    intros w3; assert (V3 := spec_to_Z w3);auto with zarith.
-  apply Z.mul_lt_mono_pos_r with (2 ^ [[ww_head1 x]]); auto with zarith.
-  rewrite H4.
-  apply Z.le_lt_trans with ([|w2|] ^ 2 + 2 * [|w2|]); auto with zarith.
-  apply Z.lt_le_trans with (([|w2|] + 1) ^ 2); auto with zarith.
-  match goal with |- ?X < ?Y =>
-    replace Y with (X + 1); auto with zarith
-  end.
-  repeat rewrite (Zsquare_mult); ring.
-  rewrite Z.mul_comm.
-  pattern [[ww_head1 x]] at 1; rewrite Hv0.
-  rewrite (Z.mul_comm 2); rewrite Z.pow_mul_r;
-   auto with zarith.
-  assert (tmp: forall p q, p ^ 2 * q ^ 2 = (p * q) ^2);
-   try (intros; repeat rewrite Zsquare_mult; ring);
-   rewrite tmp; clear tmp.
-  apply Zpower_le_monotone3; auto with zarith.
-  split; auto with zarith.
-  pattern [|w2|] at 1; rewrite (Z_div_mod_eq [|w2|] (2 ^ ([[ww_head1 x]]/2)));
-    auto with zarith.
-  rewrite <- Z.add_assoc; rewrite Z.mul_add_distr_l.
-  autorewrite with rm10; apply Z.add_le_mono_l; auto with zarith.
-  case (Z_mod_lt [|w2|] (2 ^ ([[ww_head1 x]]/2))); auto with zarith.
-  split; auto with zarith.
-  apply Z.le_lt_trans with ([|w2|]); auto with zarith.
-  apply Zdiv_le_upper_bound; auto with zarith.
-  pattern [|w2|] at 1; replace [|w2|] with ([|w2|] * 2 ^0);
-    auto with zarith.
-  apply Z.mul_le_mono_nonneg_l; auto with zarith.
-  apply Zpower_le_monotone; auto with zarith.
-  rewrite Z.pow_0_r; autorewrite with rm10; auto.
-  split; auto with zarith.
-  rewrite Hv0 in Hv2; rewrite (Pos2Z.inj_xO w_digits) in Hv2; auto with zarith.
-  apply Z.le_lt_trans with (Zpos w_digits); auto with zarith.
-  unfold base; apply Zpower2_lt_lin; auto with zarith.
-  rewrite spec_w_sub; auto with zarith.
-  rewrite Hv6; rewrite spec_w_zdigits; auto with zarith.
-  assert (Hv7: 0 < [[ww_head1 x]]/2); auto with zarith.
-  rewrite Zmod_small; auto with zarith.
-  split; auto with zarith.
-  assert ([[ww_head1 x]]/2 <= Zpos w_digits); auto with zarith.
-  apply Z.mul_le_mono_pos_r with 2; auto with zarith.
-  repeat rewrite (fun x => Z.mul_comm x 2).
-  rewrite <- Hv0; rewrite <- Pos2Z.inj_xO; auto with zarith.
-  apply Z.le_lt_trans with (Zpos w_digits); auto with zarith.
-  unfold base; apply Zpower2_lt_lin; auto with zarith.
-  Qed.
-
-End DoubleSqrt.
diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleSub.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleSub.v
deleted file mode 100644
index a2df26002..000000000
--- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleSub.v
+++ /dev/null
@@ -1,356 +0,0 @@
-
-(************************************************************************)
-(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
-(*   \VV/  **************************************************************)
-(*    //   *      This file is distributed under the terms of the       *)
-(*         *       GNU Lesser General Public License Version 2.1        *)
-(************************************************************************)
-(*            Benjamin Gregoire, Laurent Thery, INRIA, 2007             *)
-(************************************************************************)
-
-Set Implicit Arguments.
-
-Require Import ZArith.
-Require Import BigNumPrelude.
-Require Import DoubleType.
-Require Import DoubleBase.
-
-Local Open Scope Z_scope.
-
-Section DoubleSub.
- Variable w : Type.
- Variable w_0 : w.
- Variable w_Bm1 : w.
- Variable w_WW : w -> w -> zn2z w.
- Variable ww_Bm1 : zn2z w.
- Variable w_opp_c : w -> carry w.
- Variable w_opp_carry : w -> w.
- Variable w_pred_c : w -> carry w.
- Variable w_sub_c : w -> w -> carry w.
- Variable w_sub_carry_c : w -> w -> carry w.
- Variable w_opp : w -> w.
- Variable w_pred : w -> w.
- Variable w_sub : w -> w -> w.
- Variable w_sub_carry : w -> w -> w.
-
- (* ** Opposites ** *)
- Definition ww_opp_c x :=
-  match x with
-  | W0 => C0 W0
-  | WW xh xl =>
-    match w_opp_c xl with
-    | C0 _ =>
-      match w_opp_c xh with
-      | C0 h => C0 W0
-      | C1 h => C1 (WW h w_0)
-      end
-    | C1 l => C1 (WW (w_opp_carry xh) l)
-    end
-  end.
-
- Definition ww_opp x :=
-  match x with
-  | W0 => W0
-  | WW xh xl =>
-    match w_opp_c xl with
-    | C0 _ => WW (w_opp xh) w_0
-    | C1 l => WW (w_opp_carry xh) l
-    end
-  end.
-
- Definition ww_opp_carry x :=
-  match x with
-  | W0 => ww_Bm1
-  | WW xh xl => w_WW (w_opp_carry xh) (w_opp_carry xl)
-  end.
-
- Definition ww_pred_c x :=
-  match x with
-  | W0 => C1 ww_Bm1
-  | WW xh xl =>
-    match w_pred_c xl with
-    | C0 l => C0 (w_WW xh l)
-    | C1 _ =>
-      match w_pred_c xh with
-      | C0 h => C0 (WW h w_Bm1)
-      | C1 _ => C1 ww_Bm1
-      end
-    end
-  end.
-
- Definition ww_pred x :=
-  match x with
-  | W0 => ww_Bm1
-  | WW xh xl =>
-    match w_pred_c xl with
-    | C0 l => w_WW xh l
-    | C1 l => WW (w_pred xh) w_Bm1
-    end
-  end.
-
- Definition ww_sub_c x y :=
-  match y, x with
-  | W0, _ => C0 x
-  | WW yh yl, W0 => ww_opp_c (WW yh yl)
-  | WW yh yl, WW xh xl =>
-    match w_sub_c xl yl with
-    | C0 l =>
-      match w_sub_c xh yh with
-      | C0 h => C0 (w_WW h l)
-      | C1 h => C1 (WW h l)
-      end
-    | C1 l =>
-      match w_sub_carry_c xh yh with
-      | C0 h => C0 (WW h l)
-      | C1 h => C1 (WW h l)
-      end
-    end
-  end.
-
- Definition ww_sub x y :=
-  match y, x with
-  | W0, _ => x
-  | WW yh yl, W0 => ww_opp (WW yh yl)
-  | WW yh yl, WW xh xl =>
-    match w_sub_c xl yl with
-    | C0 l => w_WW (w_sub xh yh) l
-    | C1 l => WW (w_sub_carry xh yh) l
-    end
-  end.
-
- Definition ww_sub_carry_c x y :=
-  match y, x with
-  | W0, W0 => C1 ww_Bm1
-  | W0, WW xh xl => ww_pred_c (WW xh xl)
-  | WW yh yl, W0 => C1 (ww_opp_carry (WW yh yl))
-  | WW yh yl, WW xh xl =>
-    match w_sub_carry_c xl yl with
-    | C0 l =>
-      match w_sub_c xh yh with
-      | C0 h => C0 (w_WW h l)
-      | C1 h => C1 (WW h l)
-      end
-    | C1 l =>
-      match w_sub_carry_c xh yh with
-      | C0 h => C0 (w_WW h l)
-      | C1 h => C1 (w_WW h l)
-      end
-    end
-  end.
-
- Definition ww_sub_carry x y :=
-  match y, x with
-  | W0, W0 => ww_Bm1
-  | W0, WW xh xl => ww_pred (WW xh xl)
-  | WW yh yl, W0 => ww_opp_carry (WW yh yl)
-  | WW yh yl, WW xh xl =>
-    match w_sub_carry_c xl yl with
-    | C0 l => w_WW (w_sub xh yh) l
-    | C1 l => w_WW (w_sub_carry xh yh) l
-    end
-  end.
-
- (*Section DoubleProof.*)
-  Variable w_digits : positive.
-  Variable w_to_Z : w -> Z.
-
-
-  Notation wB  := (base w_digits).
-  Notation wwB := (base (ww_digits w_digits)).
-  Notation "[| x |]" := (w_to_Z x)  (at level 0, x at level 99).
-  Notation "[+| c |]" :=
-   (interp_carry 1 wB w_to_Z c) (at level 0, c at level 99).
-  Notation "[-| c |]" :=
-   (interp_carry (-1) wB w_to_Z c) (at level 0, c at level 99).
-
-  Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99).
-  Notation "[+[ c ]]" :=
-   (interp_carry 1 wwB (ww_to_Z w_digits w_to_Z) c)
-   (at level 0, c at level 99).
-  Notation "[-[ c ]]" :=
-   (interp_carry (-1) wwB (ww_to_Z w_digits w_to_Z) c)
-   (at level 0, c at level 99).
-
-  Variable spec_w_0   : [|w_0|] = 0.
-  Variable spec_w_Bm1   : [|w_Bm1|] = wB - 1.
-  Variable spec_ww_Bm1   : [[ww_Bm1]] = wwB - 1.
-  Variable spec_to_Z  : forall x, 0 <= [|x|] < wB.
-  Variable spec_w_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|].
-
-  Variable spec_opp_c : forall x, [-|w_opp_c x|] = -[|x|].
-  Variable spec_opp : forall x, [|w_opp x|] = (-[|x|]) mod wB.
-  Variable spec_opp_carry : forall x, [|w_opp_carry x|] = wB - [|x|] - 1.
-
-  Variable spec_pred_c : forall x, [-|w_pred_c x|] = [|x|] - 1.
-  Variable spec_sub_c : forall x y, [-|w_sub_c x y|] = [|x|] - [|y|].
-  Variable spec_sub_carry_c :
-   forall x y, [-|w_sub_carry_c x y|] = [|x|] - [|y|] - 1.
-
-  Variable spec_pred : forall x, [|w_pred x|] = ([|x|] - 1) mod wB.
-  Variable spec_sub : forall x y, [|w_sub x y|] = ([|x|] - [|y|]) mod wB.
-  Variable spec_sub_carry :
-	 forall x y, [|w_sub_carry x y|] = ([|x|] - [|y|] - 1) mod wB.
-
-
-  Lemma spec_ww_opp_c : forall x, [-[ww_opp_c x]] = -[[x]].
-  Proof.
-   destruct x as [ |xh xl];simpl. reflexivity.
-   rewrite Z.opp_add_distr;generalize (spec_opp_c xl);destruct (w_opp_c xl)
-   as [l|l];intros H;unfold interp_carry in H;rewrite <- H;
-   rewrite <- Z.mul_opp_l.
-   assert ([|l|] = 0).
-    assert (H1:= spec_to_Z l);assert (H2 := spec_to_Z xl);omega.
-   rewrite H0;generalize (spec_opp_c xh);destruct (w_opp_c xh)
-   as [h|h];intros H1;unfold interp_carry in *;rewrite <- H1.
-   assert ([|h|] = 0).
-    assert (H3:= spec_to_Z h);assert (H2 := spec_to_Z xh);omega.
-   rewrite H2;reflexivity.
-   simpl ww_to_Z;rewrite wwB_wBwB;rewrite spec_w_0;ring.
-   unfold interp_carry;simpl ww_to_Z;rewrite wwB_wBwB;rewrite spec_opp_carry;
-   ring.
-  Qed.
-
-  Lemma spec_ww_opp : forall x, [[ww_opp x]] = (-[[x]]) mod wwB.
-  Proof.
-   destruct x as [ |xh xl];simpl. reflexivity.
-   rewrite Z.opp_add_distr, <- Z.mul_opp_l.
-   generalize (spec_opp_c xl);destruct (w_opp_c xl)
-   as [l|l];intros H;unfold interp_carry in H;rewrite <- H;simpl ww_to_Z.
-   rewrite spec_w_0;rewrite Z.add_0_r;rewrite wwB_wBwB.
-   assert ([|l|] = 0).
-    assert (H1:= spec_to_Z l);assert (H2 := spec_to_Z xl);omega.
-   rewrite H0;rewrite Z.add_0_r; rewrite Z.pow_2_r;
-    rewrite Zmult_mod_distr_r;try apply lt_0_wB.
-   rewrite spec_opp;trivial.
-   apply Zmod_unique with (q:= -1).
-   exact (spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW (w_opp_carry xh) l)).
-   rewrite spec_opp_carry;rewrite wwB_wBwB;ring.
-  Qed.
-
-  Lemma spec_ww_opp_carry : forall x, [[ww_opp_carry x]] = wwB - [[x]] - 1.
-  Proof.
-   destruct x as [ |xh xl];simpl. rewrite spec_ww_Bm1;ring.
-   rewrite spec_w_WW;simpl;repeat rewrite spec_opp_carry;rewrite wwB_wBwB;ring.
-  Qed.
-
-  Lemma spec_ww_pred_c : forall x, [-[ww_pred_c x]] = [[x]] - 1.
-  Proof.
-   destruct x as [ |xh xl];unfold ww_pred_c.
-   unfold interp_carry;rewrite spec_ww_Bm1;simpl ww_to_Z;ring.
-   simpl ww_to_Z;replace (([|xh|]*wB+[|xl|])-1) with ([|xh|]*wB+([|xl|]-1)).
-   2:ring. generalize (spec_pred_c xl);destruct (w_pred_c xl) as [l|l];
-   intros H;unfold interp_carry in H;rewrite <- H. simpl;apply spec_w_WW.
-   rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r.
-   assert ([|l|] = wB - 1).
-     assert (H1:= spec_to_Z l);assert (H2 := spec_to_Z xl);omega.
-   rewrite H0;change ([|xh|] + -1) with ([|xh|] - 1).
-   generalize (spec_pred_c xh);destruct (w_pred_c xh) as [h|h];
-   intros H1;unfold interp_carry in H1;rewrite <- H1.
-   simpl;rewrite spec_w_Bm1;ring.
-   assert ([|h|] = wB - 1).
-     assert (H3:= spec_to_Z h);assert (H2 := spec_to_Z xh);omega.
-   rewrite H2;unfold interp_carry;rewrite spec_ww_Bm1;rewrite wwB_wBwB;ring.
-  Qed.
-
-  Lemma spec_ww_sub_c : forall x y, [-[ww_sub_c x y]] = [[x]] - [[y]].
-  Proof.
-   destruct y as [ |yh yl];simpl. ring.
-   destruct x as [ |xh xl];simpl. exact (spec_ww_opp_c (WW yh yl)).
-   replace ([|xh|] * wB + [|xl|] - ([|yh|] * wB + [|yl|]))
-   with (([|xh|]-[|yh|])*wB + ([|xl|]-[|yl|])). 2:ring.
-   generalize (spec_sub_c xl yl);destruct (w_sub_c xl yl) as [l|l];intros H;
-   unfold interp_carry in H;rewrite <- H.
-   generalize (spec_sub_c xh yh);destruct (w_sub_c xh yh) as [h|h];intros H1;
-   unfold interp_carry in H1;rewrite <- H1;unfold interp_carry;
-   try rewrite spec_w_WW;simpl ww_to_Z;try rewrite wwB_wBwB;ring.
-   rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r.
-   change ([|xh|] - [|yh|] + -1) with ([|xh|] - [|yh|] - 1).
-   generalize (spec_sub_carry_c xh yh);destruct (w_sub_carry_c xh yh) as [h|h];
-   intros H1;unfold interp_carry in *;rewrite <- H1;simpl ww_to_Z;
-   try rewrite wwB_wBwB;ring.
-  Qed.
-
-  Lemma spec_ww_sub_carry_c :
-     forall x y, [-[ww_sub_carry_c x y]] = [[x]] - [[y]] - 1.
-  Proof.
-   destruct y as [ |yh yl];simpl.
-   unfold Z.sub;simpl;rewrite Z.add_0_r;exact (spec_ww_pred_c x).
-   destruct x as [ |xh xl].
-   unfold interp_carry;rewrite spec_w_WW;simpl ww_to_Z;rewrite wwB_wBwB;
-   repeat rewrite spec_opp_carry;ring.
-   simpl ww_to_Z.
-   replace ([|xh|] * wB + [|xl|] - ([|yh|] * wB + [|yl|]) - 1)
-   with (([|xh|]-[|yh|])*wB + ([|xl|]-[|yl|]-1)). 2:ring.
-   generalize (spec_sub_carry_c xl yl);destruct (w_sub_carry_c xl yl)
-   as [l|l];intros H;unfold interp_carry in H;rewrite <- H.
-   generalize (spec_sub_c xh yh);destruct (w_sub_c xh yh) as [h|h];intros H1;
-   unfold interp_carry in H1;rewrite <- H1;unfold interp_carry;
-   try rewrite spec_w_WW;simpl ww_to_Z;try rewrite wwB_wBwB;ring.
-   rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r.
-   change ([|xh|] - [|yh|] + -1) with ([|xh|] - [|yh|] - 1).
-   generalize (spec_sub_carry_c xh yh);destruct (w_sub_carry_c xh yh) as [h|h];
-   intros H1;unfold interp_carry in *;rewrite <- H1;try rewrite spec_w_WW;
-   simpl ww_to_Z; try rewrite wwB_wBwB;ring.
-  Qed.
-
-  Lemma spec_ww_pred : forall x, [[ww_pred x]] = ([[x]] - 1) mod wwB.
-  Proof.
-   destruct x as [ |xh xl];simpl.
-   apply Zmod_unique with (-1). apply spec_ww_to_Z;trivial.
-   rewrite spec_ww_Bm1;ring.
-   replace ([|xh|]*wB + [|xl|] - 1) with ([|xh|]*wB + ([|xl|] - 1)). 2:ring.
-   generalize (spec_pred_c xl);destruct (w_pred_c xl) as [l|l];intro H;
-   unfold interp_carry in H;rewrite <- H;simpl ww_to_Z.
-   rewrite Zmod_small. apply spec_w_WW.
-   exact (spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW xh l)).
-   rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r.
-   change ([|xh|] + -1) with ([|xh|] - 1).
-   assert ([|l|] = wB - 1).
-    assert (H1:= spec_to_Z l);assert (H2:= spec_to_Z xl);omega.
-   rewrite (mod_wwB w_digits w_to_Z);trivial.
-   rewrite spec_pred;rewrite spec_w_Bm1;rewrite <- H0;trivial.
-  Qed.
-
-  Lemma spec_ww_sub : forall x y, [[ww_sub x y]] = ([[x]] - [[y]]) mod wwB.
-  Proof.
-   destruct y as [ |yh yl];simpl.
-   ring_simplify ([[x]] - 0);rewrite Zmod_small;trivial. apply spec_ww_to_Z;trivial.
-   destruct x as [ |xh xl];simpl. exact (spec_ww_opp (WW yh yl)).
-   replace ([|xh|] * wB + [|xl|] - ([|yh|] * wB + [|yl|]))
-   with (([|xh|] - [|yh|]) * wB + ([|xl|] - [|yl|])). 2:ring.
-   generalize (spec_sub_c xl yl);destruct (w_sub_c xl yl)as[l|l];intros H;
-   unfold interp_carry in H;rewrite <- H.
-   rewrite spec_w_WW;rewrite (mod_wwB w_digits w_to_Z spec_to_Z).
-   rewrite spec_sub;trivial.
-   simpl ww_to_Z;rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r.
-   rewrite (mod_wwB w_digits w_to_Z spec_to_Z);rewrite spec_sub_carry;trivial.
-  Qed.
-
-  Lemma spec_ww_sub_carry :
-	 forall x y, [[ww_sub_carry x y]] = ([[x]] - [[y]] - 1) mod wwB.
-  Proof.
-   destruct y as [ |yh yl];simpl.
-   ring_simplify ([[x]] - 0);exact (spec_ww_pred x).
-   destruct x as [ |xh xl];simpl.
-   apply Zmod_unique with (-1).
-   apply spec_ww_to_Z;trivial.
-   fold (ww_opp_carry (WW yh yl)).
-   rewrite (spec_ww_opp_carry (WW yh yl));simpl ww_to_Z;ring.
-   replace ([|xh|] * wB + [|xl|] - ([|yh|] * wB + [|yl|]) - 1)
-   with (([|xh|] - [|yh|]) * wB + ([|xl|] - [|yl|] - 1)). 2:ring.
-   generalize (spec_sub_carry_c xl yl);destruct (w_sub_carry_c xl yl)as[l|l];
-   intros H;unfold interp_carry in H;rewrite <- H;rewrite spec_w_WW.
-   rewrite (mod_wwB w_digits w_to_Z spec_to_Z);rewrite spec_sub;trivial.
-   rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r.
-   rewrite (mod_wwB w_digits w_to_Z spec_to_Z);rewrite spec_sub_carry;trivial.
-  Qed.
-
-(* End DoubleProof. *)
-
-End DoubleSub.
-
-
-
-
-
diff --git a/theories/Numbers/Cyclic/Int31/Cyclic31.v b/theories/Numbers/Cyclic/Int31/Cyclic31.v
index 0e58b8155..ba55003f7 100644
--- a/theories/Numbers/Cyclic/Int31/Cyclic31.v
+++ b/theories/Numbers/Cyclic/Int31/Cyclic31.v
@@ -18,13 +18,16 @@ Require Export Int31.
 Require Import Znumtheory.
 Require Import Zgcd_alt.
 Require Import Zpow_facts.
-Require Import BigNumPrelude.
 Require Import CyclicAxioms.
 Require Import ROmega.
 
+Declare ML Module "int31_syntax_plugin".
+
 Local Open Scope nat_scope.
 Local Open Scope int31_scope.
 
+Local Hint Resolve Z.lt_gt Z.div_pos : zarith.
+
 Section Basics.
 
  (** * Basic results about [iszero], [shiftl], [shiftr] *)
@@ -455,12 +458,19 @@ Section Basics.
  rewrite Z.succ_double_spec; auto with zarith.
  Qed.
 
- Lemma phi_bounded  : forall x, (0 <= phi x < 2 ^ (Z.of_nat size))%Z.
+ Lemma phi_nonneg : forall x, (0 <= phi x)%Z.
  Proof.
  intros.
  rewrite <- phibis_aux_equiv.
- split.
  apply phibis_aux_pos.
+ Qed.
+
+ Hint Resolve phi_nonneg : zarith.
+
+ Lemma phi_bounded  : forall x, (0 <= phi x < 2 ^ (Z.of_nat size))%Z.
+ Proof.
+ intros. split; [auto with zarith|].
+ rewrite <- phibis_aux_equiv.
  change x with (nshiftr x (size-size)).
  apply phibis_aux_bounded; auto.
  Qed.
@@ -1624,6 +1634,37 @@ Section Int31_Specs.
  rewrite Z.mul_comm, Z_div_mult; auto with zarith.
  Qed.
 
+ Lemma shift_unshift_mod_2 : forall n p a, 0 <= p <= n ->
+   ((a * 2 ^ (n - p)) mod (2^n) / 2 ^ (n - p)) mod (2^n) =
+   a mod 2 ^ p.
+ Proof.
+ intros.
+ rewrite Zmod_small.
+ rewrite Zmod_eq by (auto with zarith).
+ unfold Z.sub at 1.
+ rewrite Z_div_plus_full_l
+  by (cut (0 < 2^(n-p)); auto with zarith).
+ assert (2^n = 2^(n-p)*2^p).
+  rewrite <- Zpower_exp by (auto with zarith).
+  replace (n-p+p) with n; auto with zarith.
+ rewrite H0.
+ rewrite <- Zdiv_Zdiv, Z_div_mult by (auto with zarith).
+ rewrite (Z.mul_comm (2^(n-p))), Z.mul_assoc.
+ rewrite <- Z.mul_opp_l.
+ rewrite Z_div_mult by (auto with zarith).
+ symmetry; apply Zmod_eq; auto with zarith.
+
+ remember (a * 2 ^ (n - p)) as b.
+ destruct (Z_mod_lt b (2^n)); auto with zarith.
+ split.
+ apply Z_div_pos; auto with zarith.
+ apply Zdiv_lt_upper_bound; auto with zarith.
+ apply Z.lt_le_trans with (2^n); auto with zarith.
+ rewrite <- (Z.mul_1_r (2^n)) at 1.
+ apply Z.mul_le_mono_nonneg; auto with zarith.
+ cut (0 < 2 ^ (n-p)); auto with zarith.
+ Qed.
+
  Lemma spec_pos_mod : forall w p,
        [|ZnZ.pos_mod p w|] = [|w|] mod (2 ^ [|p|]).
  Proof.
@@ -1654,7 +1695,7 @@ Section Int31_Specs.
  rewrite spec_add_mul_div by (rewrite H4; auto with zarith).
  change [|0|] with 0%Z; rewrite Zdiv_0_l, Z.add_0_r.
  rewrite H4.
- apply shift_unshift_mod_2; auto with zarith.
+ apply shift_unshift_mod_2; simpl; auto with zarith.
  Qed.
 
 
@@ -1973,32 +2014,24 @@ Section Int31_Specs.
  assert (Hp2: 0 < [|2|]) by exact (eq_refl Lt).
  intros Hi Hj Hij H31 Hrec; rewrite sqrt31_step_def.
  rewrite spec_compare, div31_phi; auto.
-   case Z.compare_spec; auto; intros Hc;
+ case Z.compare_spec; auto; intros Hc;
    try (split; auto; apply sqrt_test_true; auto with zarith; fail).
- apply Hrec; repeat rewrite div31_phi; auto with zarith.
- replace [|(j + fst (i / j)%int31)|] with ([|j|] + [|i|] / [|j|]).
- split.
+ assert (E : [|(j + fst (i / j)%int31)|] = [|j|] + [|i|] / [|j|]).
+ { rewrite spec_add, div31_phi; auto using Z.mod_small with zarith. }
+ apply Hrec; rewrite !div31_phi, E; auto using sqrt_main with zarith.
+ split; try apply sqrt_test_false; auto with zarith.
  apply Z.le_succ_l in Hj. change (1 <= [|j|]) in Hj.
  Z.le_elim Hj.
- replace ([|j|] + [|i|]/[|j|]) with
-        (1 * 2 + (([|j|] - 2) + [|i|] / [|j|])); try ring.
- rewrite Z_div_plus_full_l; auto with zarith.
- assert (0 <= [|i|]/ [|j|]) by (apply Z_div_pos; auto with zarith).
- assert (0 <= ([|j|] - 2 + [|i|] / [|j|]) / [|2|]) ; auto with zarith.
- rewrite <- Hj, Zdiv_1_r.
- replace (1 + [|i|])%Z with (1 * 2 + ([|i|] - 1))%Z; try ring.
- rewrite Z_div_plus_full_l; auto with zarith.
- assert (0 <= ([|i|] - 1) /2)%Z by (apply Z_div_pos; auto with zarith).
- change ([|2|]) with 2%Z; auto with zarith.
- apply sqrt_test_false; auto with zarith.
- rewrite spec_add, div31_phi; auto.
- symmetry; apply Zmod_small.
- split; auto with zarith.
- replace [|j + fst (i / j)%int31|] with ([|j|] + [|i|] / [|j|]).
- apply sqrt_main; auto with zarith.
- rewrite spec_add, div31_phi; auto.
- symmetry; apply Zmod_small.
- split; auto with zarith.
+ - replace ([|j|] + [|i|]/[|j|]) with
+   (1 * 2 + (([|j|] - 2) + [|i|] / [|j|])) by ring.
+   rewrite Z_div_plus_full_l; auto with zarith.
+   assert (0 <= [|i|]/ [|j|]) by auto with zarith.
+   assert (0 <= ([|j|] - 2 + [|i|] / [|j|]) / [|2|]); auto with zarith.
+ - rewrite <- Hj, Zdiv_1_r.
+   replace (1 + [|i|]) with (1 * 2 + ([|i|] - 1)) by ring.
+   rewrite Z_div_plus_full_l; auto with zarith.
+   assert (0 <= ([|i|] - 1) /2) by auto with zarith.
+   change ([|2|]) with 2; auto with zarith.
  Qed.
 
  Lemma iter31_sqrt_correct n rec i j: 0 < [|i|] -> 0 < [|j|] ->
@@ -2078,11 +2111,12 @@ Section Int31_Specs.
  case (phi_bounded j); intros Hbj _.
  case (phi_bounded il); intros Hbil _.
  case (phi_bounded ih); intros Hbih Hbih1.
- assert (([|ih|] < [|j|] + 1)%Z); auto with zarith.
+ assert ([|ih|] < [|j|] + 1); auto with zarith.
  apply Z.square_lt_simpl_nonneg; auto with zarith.
- repeat rewrite <-Z.pow_2_r; apply Z.le_lt_trans with (2 := H1).
- apply Z.le_trans with ([|ih|] * base)%Z; unfold phi2, base;
-  try rewrite Z.pow_2_r; auto with zarith.
+ rewrite <- ?Z.pow_2_r; apply Z.le_lt_trans with (2 := H1).
+ apply Z.le_trans with ([|ih|] * wB).
+ - rewrite ? Z.pow_2_r; auto with zarith.
+ - unfold phi2. change base with wB; auto with zarith.
  Qed.
 
  Lemma div312_phi ih il j: (2^30 <= [|j|] -> [|ih|] < [|j|] ->
@@ -2104,90 +2138,89 @@ Section Int31_Specs.
  Proof.
  assert (Hp2: (0 < [|2|])%Z) by exact (eq_refl Lt).
  intros Hih Hj Hij Hrec; rewrite sqrt312_step_def.
- assert (H1: ([|ih|] <= [|j|])%Z) by (apply sqrt312_lower_bound with il; auto).
+ assert (H1: ([|ih|] <= [|j|])) by (apply sqrt312_lower_bound with il; auto).
  case (phi_bounded ih); intros Hih1 _.
  case (phi_bounded il); intros Hil1 _.
  case (phi_bounded j); intros _ Hj1.
  assert (Hp3: (0 < phi2 ih il)).
- unfold phi2; apply Z.lt_le_trans with ([|ih|] * base)%Z; auto with zarith.
- apply Z.mul_pos_pos; auto with zarith.
- apply Z.lt_le_trans with (2:= Hih); auto with zarith.
+ { unfold phi2; apply Z.lt_le_trans with ([|ih|] * base); auto with zarith.
+   apply Z.mul_pos_pos; auto with zarith.
+   apply Z.lt_le_trans with (2:= Hih); auto with zarith. }
  rewrite spec_compare. case Z.compare_spec; intros Hc1.
- split; auto.
- apply sqrt_test_true; auto.
- unfold phi2, base; auto with zarith.
- unfold phi2; rewrite Hc1.
- assert (0 <= [|il|]/[|j|]) by (apply Z_div_pos; auto with zarith).
- rewrite Z.mul_comm, Z_div_plus_full_l; unfold base; auto with zarith.
- simpl wB in Hj1. unfold Z.pow_pos in Hj1. simpl in Hj1. auto with zarith.
- case (Z.le_gt_cases (2 ^ 30) [|j|]); intros Hjj.
- rewrite spec_compare; case Z.compare_spec;
-  rewrite div312_phi; auto; intros Hc;
-  try (split; auto; apply sqrt_test_true; auto with zarith; fail).
- apply Hrec.
- assert (Hf1: 0 <= phi2 ih il/ [|j|]) by (apply Z_div_pos; auto with zarith).
- apply Z.le_succ_l in Hj. change (1 <= [|j|]) in Hj.
- Z.le_elim Hj.
- 2: contradict Hc; apply Z.le_ngt; rewrite <- Hj, Zdiv_1_r; auto with zarith.
- assert (Hf3: 0 < ([|j|] + phi2 ih il / [|j|]) / 2).
- replace ([|j|] + phi2 ih il/ [|j|])%Z with
-        (1 * 2 + (([|j|] - 2) + phi2 ih il / [|j|])); try ring.
- rewrite Z_div_plus_full_l; auto with zarith.
- assert (0 <= ([|j|] - 2 + phi2 ih il / [|j|]) / 2) ; auto with zarith.
- assert (Hf4: ([|j|] + phi2 ih il / [|j|]) / 2 < [|j|]).
- apply sqrt_test_false; auto with zarith.
- generalize (spec_add_c j (fst (div3121 ih il j))).
- unfold interp_carry;  case add31c; intros r;
-  rewrite div312_phi; auto with zarith.
- rewrite div31_phi; change [|2|] with 2%Z; auto with zarith.
- intros HH; rewrite HH; clear HH; auto with zarith.
- rewrite spec_add, div31_phi; change [|2|] with 2%Z; auto.
- rewrite Z.mul_1_l; intros HH.
- rewrite Z.add_comm, <- Z_div_plus_full_l; auto with zarith.
- change (phi v30 * 2) with (2 ^ Z.of_nat size).
- rewrite HH, Zmod_small; auto with zarith.
- replace (phi
-    match j +c fst (div3121 ih il j) with
-    | C0 m1 => fst (m1 / 2)%int31
-    | C1 m1 => fst (m1 / 2)%int31 + v30
-    end) with ((([|j|] + (phi2 ih il)/([|j|]))/2)).
- apply sqrt_main; auto with zarith.
- generalize (spec_add_c j (fst (div3121 ih il j))).
- unfold interp_carry;  case add31c; intros r;
-  rewrite div312_phi; auto with zarith.
- rewrite div31_phi; auto with zarith.
- intros HH; rewrite HH; auto with zarith.
- intros HH; rewrite <- HH.
- change (1 * 2 ^ Z.of_nat size) with (phi (v30) * 2).
- rewrite Z_div_plus_full_l; auto with zarith.
- rewrite Z.add_comm.
- rewrite spec_add, Zmod_small.
- rewrite div31_phi; auto.
- split; auto with zarith.
- case (phi_bounded (fst (r/2)%int31));
-   case (phi_bounded v30); auto with zarith.
- rewrite div31_phi; change (phi 2) with 2%Z; auto.
- change (2 ^Z.of_nat size) with (base/2 + phi v30).
- assert (phi r / 2 < base/2); auto with zarith.
- apply Z.mul_lt_mono_pos_r with 2; auto with zarith.
- change (base/2 * 2) with base.
- apply Z.le_lt_trans with (phi r).
- rewrite Z.mul_comm; apply Z_mult_div_ge; auto with zarith.
- case (phi_bounded r); auto with zarith.
- contradict Hij; apply Z.le_ngt.
- assert ((1 + [|j|]) <= 2 ^ 30); auto with zarith.
- apply Z.le_trans with ((2 ^ 30) * (2 ^ 30)); auto with zarith.
- assert (0 <= 1 + [|j|]); auto with zarith.
- apply Z.mul_le_mono_nonneg; auto with zarith.
- change ((2 ^ 30) * (2 ^ 30)) with ((2 ^ 29) * base).
- apply Z.le_trans with ([|ih|] * base); auto with zarith.
- unfold phi2, base; auto with zarith.
- split; auto.
- apply sqrt_test_true; auto.
- unfold phi2, base; auto with zarith.
- apply Z.le_ge; apply Z.le_trans with (([|j|] * base)/[|j|]).
- rewrite Z.mul_comm, Z_div_mult; auto with zarith.
- apply Z.ge_le; apply Z_div_ge; auto with zarith.
+ - split; auto.
+   apply sqrt_test_true; auto.
+   + unfold phi2, base; auto with zarith.
+   + unfold phi2; rewrite Hc1.
+     assert (0 <= [|il|]/[|j|]) by (apply Z_div_pos; auto with zarith).
+     rewrite Z.mul_comm, Z_div_plus_full_l; auto with zarith.
+     change base with wB. auto with zarith.
+ - case (Z.le_gt_cases (2 ^ 30) [|j|]); intros Hjj.
+   + rewrite spec_compare; case Z.compare_spec;
+     rewrite div312_phi; auto; intros Hc;
+     try (split; auto; apply sqrt_test_true; auto with zarith; fail).
+     apply Hrec.
+     * assert (Hf1: 0 <= phi2 ih il/ [|j|]) by auto with zarith.
+       apply Z.le_succ_l in Hj. change (1 <= [|j|]) in Hj.
+       Z.le_elim Hj;
+         [ | contradict Hc; apply Z.le_ngt;
+             rewrite <- Hj, Zdiv_1_r; auto with zarith ].
+       assert (Hf3: 0 < ([|j|] + phi2 ih il / [|j|]) / 2).
+       { replace ([|j|] + phi2 ih il/ [|j|]) with
+         (1 * 2 + (([|j|] - 2) + phi2 ih il / [|j|])); try ring.
+         rewrite Z_div_plus_full_l; auto with zarith.
+         assert (0 <= ([|j|] - 2 + phi2 ih il / [|j|]) / 2) ;
+         auto with zarith. }
+       assert (Hf4: ([|j|] + phi2 ih il / [|j|]) / 2 < [|j|]).
+       { apply sqrt_test_false; auto with zarith. }
+       generalize (spec_add_c j (fst (div3121 ih il j))).
+       unfold interp_carry;  case add31c; intros r;
+       rewrite div312_phi; auto with zarith.
+       { rewrite div31_phi; change [|2|] with 2; auto with zarith.
+         intros HH; rewrite HH; clear HH; auto with zarith. }
+       { rewrite spec_add, div31_phi; change [|2|] with 2; auto.
+         rewrite Z.mul_1_l; intros HH.
+         rewrite Z.add_comm, <- Z_div_plus_full_l; auto with zarith.
+         change (phi v30 * 2) with (2 ^ Z.of_nat size).
+         rewrite HH, Zmod_small; auto with zarith. }
+     * replace (phi _) with (([|j|] + (phi2 ih il)/([|j|]))/2);
+       [ apply sqrt_main; auto with zarith | ].
+       generalize (spec_add_c j (fst (div3121 ih il j))).
+       unfold interp_carry;  case add31c; intros r;
+       rewrite div312_phi; auto with zarith.
+       { rewrite div31_phi; auto with zarith.
+         intros HH; rewrite HH; auto with zarith. }
+       { intros HH; rewrite <- HH.
+         change (1 * 2 ^ Z.of_nat size) with (phi (v30) * 2).
+         rewrite Z_div_plus_full_l; auto with zarith.
+         rewrite Z.add_comm.
+         rewrite spec_add, Zmod_small.
+         - rewrite div31_phi; auto.
+         - split; auto with zarith.
+           + case (phi_bounded (fst (r/2)%int31));
+             case (phi_bounded v30); auto with zarith.
+           + rewrite div31_phi; change (phi 2) with 2; auto.
+             change (2 ^Z.of_nat size) with (base/2 + phi v30).
+             assert (phi r / 2 < base/2); auto with zarith.
+             apply Z.mul_lt_mono_pos_r with 2; auto with zarith.
+             change (base/2 * 2) with base.
+             apply Z.le_lt_trans with (phi r).
+             * rewrite Z.mul_comm; apply Z_mult_div_ge; auto with zarith.
+             * case (phi_bounded r); auto with zarith. }
+   + contradict Hij; apply Z.le_ngt.
+     assert ((1 + [|j|]) <= 2 ^ 30); auto with zarith.
+     apply Z.le_trans with ((2 ^ 30) * (2 ^ 30)); auto with zarith.
+     * assert (0 <= 1 + [|j|]); auto with zarith.
+       apply Z.mul_le_mono_nonneg; auto with zarith.
+     * change ((2 ^ 30) * (2 ^ 30)) with ((2 ^ 29) * base).
+       apply Z.le_trans with ([|ih|] * base);
+         change wB with base in *; auto with zarith.
+       unfold phi2, base; auto with zarith.
+ - split; auto.
+   apply sqrt_test_true; auto.
+   + unfold phi2, base; auto with zarith.
+   + apply Z.le_ge; apply Z.le_trans with (([|j|] * base)/[|j|]).
+     * rewrite Z.mul_comm, Z_div_mult; auto with zarith.
+     * apply Z.ge_le; apply Z_div_ge; auto with zarith.
  Qed.
 
  Lemma iter312_sqrt_correct n rec ih il j:
@@ -2209,7 +2242,7 @@ Section Int31_Specs.
  intros j3 Hj3 Hpj3.
  apply HHrec; auto.
  rewrite Nat2Z.inj_succ, Z.pow_succ_r.
- apply Z.le_trans with (2 ^Z.of_nat n + [|j2|])%Z; auto with zarith.
+ apply Z.le_trans with (2 ^Z.of_nat n + [|j2|]); auto with zarith.
  apply Nat2Z.is_nonneg.
  Qed.
 
diff --git a/theories/Numbers/Cyclic/ZModulo/ZModulo.v b/theories/Numbers/Cyclic/ZModulo/ZModulo.v
index 04fc5a8df..a3d7edbf4 100644
--- a/theories/Numbers/Cyclic/ZModulo/ZModulo.v
+++ b/theories/Numbers/Cyclic/ZModulo/ZModulo.v
@@ -18,7 +18,7 @@ Set Implicit Arguments.
 Require Import Bool.
 Require Import ZArith.
 Require Import Znumtheory.
-Require Import BigNumPrelude.
+Require Import Zpow_facts.
 Require Import DoubleType.
 Require Import CyclicAxioms.
 
@@ -48,13 +48,14 @@ Section ZModulo.
 
  Lemma spec_more_than_1_digit: 1 < Zpos digits.
  Proof.
-  generalize digits_ne_1; destruct digits; auto.
+  generalize digits_ne_1; destruct digits; red; auto.
   destruct 1; auto.
  Qed.
  Let digits_gt_1 := spec_more_than_1_digit.
 
  Lemma wB_pos : wB > 0.
  Proof.
+  apply Z.lt_gt.
   unfold wB, base; auto with zarith.
  Qed.
  Hint Resolve wB_pos.
@@ -558,7 +559,7 @@ Section ZModulo.
  apply Zmod_small.
  generalize (Z_mod_lt [|w|] (2 ^ [|p|])); intros.
  split.
- destruct H; auto with zarith.
+ destruct H; auto using Z.lt_gt with zarith.
  apply Z.le_lt_trans with [|w|]; auto with zarith.
  apply Zmod_le; auto with zarith.
  Qed.