1 files changed, 217 insertions, 269 deletions

diff --git a/theories/Numbers/Cyclic/Abstract/CyclicAxioms.v b/theories/Numbers/Cyclic/Abstract/CyclicAxioms.v
index fa097802..59656eed 100644
--- a/theories/Numbers/Cyclic/Abstract/CyclicAxioms.v
+++ b/theories/Numbers/Cyclic/Abstract/CyclicAxioms.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -8,8 +8,6 @@
 (*            Benjamin Gregoire, Laurent Thery, INRIA, 2007             *)
 (************************************************************************)
 
-(* $Id: CyclicAxioms.v 14641 2011-11-06 11:59:10Z herbelin $ *)
-
 (** * Signature and specification of a bounded integer structure *)
 
 (** This file specifies how to represent [Z/nZ] when [n=2^d],
@@ -26,352 +24,300 @@ Local Open Scope Z_scope.
 
 (** First, a description via an operator record and a spec record. *)
 
-Section Z_nZ_Op.
-
- Variable znz : Type.
+Module ZnZ.
 
- Record znz_op := mk_znz_op {
+ Class Ops (t:Type) := MkOps {
 
     (* Conversion functions with Z *)
-    znz_digits : positive;
-    znz_zdigits: znz;
-    znz_to_Z   : znz -> Z;
-    znz_of_pos : positive -> N * znz; (* Euclidean division by [2^digits] *)
-    znz_head0  : znz -> znz; (* number of digits 0 in front of the number *)
-    znz_tail0  : znz -> znz; (* number of digits 0 at the bottom of the number *)
+    digits : positive;
+    zdigits: t;
+    to_Z   : t -> Z;
+    of_pos : positive -> N * t; (* Euclidean division by [2^digits] *)
+    head0  : t -> t; (* number of digits 0 in front of the number *)
+    tail0  : t -> t; (* number of digits 0 at the bottom of the number *)
 
     (* Basic numbers *)
-    znz_0   : znz;
-    znz_1   : znz;
-    znz_Bm1 : znz;  (* [2^digits-1], which is equivalent to [-1] *)
+    zero : t;
+    one  : t;
+    minus_one : t;  (* [2^digits-1], which is equivalent to [-1] *)
 
     (* Comparison *)
-    znz_compare     : znz -> znz -> comparison;
-    znz_eq0         : znz -> bool;
+    compare     : t -> t -> comparison;
+    eq0         : t -> bool;
 
     (* Basic arithmetic operations *)
-    znz_opp_c       : znz -> carry znz;
-    znz_opp         : znz -> znz;
-    znz_opp_carry   : znz -> znz; (* the carry is known to be -1 *)
-
-    znz_succ_c      : znz -> carry znz;
-    znz_add_c       : znz -> znz -> carry znz;
-    znz_add_carry_c : znz -> znz -> carry znz;
-    znz_succ        : znz -> znz;
-    znz_add         : znz -> znz -> znz;
-    znz_add_carry   : znz -> znz -> znz;
-
-    znz_pred_c      : znz -> carry znz;
-    znz_sub_c       : znz -> znz -> carry znz;
-    znz_sub_carry_c : znz -> znz -> carry znz;
-    znz_pred        : znz -> znz;
-    znz_sub         : znz -> znz -> znz;
-    znz_sub_carry   : znz -> znz -> znz;
-
-    znz_mul_c       : znz -> znz -> zn2z znz;
-    znz_mul         : znz -> znz -> znz;
-    znz_square_c    : znz -> zn2z znz;
+    opp_c       : t -> carry t;
+    opp         : t -> t;
+    opp_carry   : t -> t; (* the carry is known to be -1 *)
+
+    succ_c      : t -> carry t;
+    add_c       : t -> t -> carry t;
+    add_carry_c : t -> t -> carry t;
+    succ        : t -> t;
+    add         : t -> t -> t;
+    add_carry   : t -> t -> t;
+
+    pred_c      : t -> carry t;
+    sub_c       : t -> t -> carry t;
+    sub_carry_c : t -> t -> carry t;
+    pred        : t -> t;
+    sub         : t -> t -> t;
+    sub_carry   : t -> t -> t;
+
+    mul_c       : t -> t -> zn2z t;
+    mul         : t -> t -> t;
+    square_c    : t -> zn2z t;
 
     (* Special divisions operations *)
-    znz_div21       : znz -> znz -> znz -> znz*znz;
-    znz_div_gt      : znz -> znz -> znz * znz; (* specialized version of [znz_div] *)
-    znz_div         : znz -> znz -> znz * znz;
+    div21       : t -> t -> t -> t*t;
+    div_gt      : t -> t -> t * t; (* specialized version of [div] *)
+    div         : t -> t -> t * t;
 
-    znz_mod_gt      : znz -> znz -> znz; (* specialized version of [znz_mod] *)
-    znz_mod         : znz -> znz -> znz;
+    modulo_gt   : t -> t -> t; (* specialized version of [mod] *)
+    modulo      : t -> t -> t;
 
-    znz_gcd_gt      : znz -> znz -> znz; (* specialized version of [znz_gcd] *)
-    znz_gcd         : znz -> znz -> znz;
-    (* [znz_add_mul_div p i j] is a combination of the [(digits-p)]
+    gcd_gt      : t -> t -> t; (* specialized version of [gcd] *)
+    gcd         : t -> t -> t;
+    (* [add_mul_div p i j] is a combination of the [(digits-p)]
        low bits of [i] above the [p] high bits of [j]:
-       [znz_add_mul_div p i j = i*2^p+j/2^(digits-p)] *)
-    znz_add_mul_div : znz -> znz -> znz -> znz;
-    (* [znz_pos_mod p i] is [i mod 2^p] *)
-    znz_pos_mod     : znz -> znz -> znz;
+       [add_mul_div p i j = i*2^p+j/2^(digits-p)] *)
+    add_mul_div : t -> t -> t -> t;
+    (* [pos_mod p i] is [i mod 2^p] *)
+    pos_mod     : t -> t -> t;
 
-    znz_is_even     : znz -> bool;
+    is_even     : t -> bool;
     (* square root *)
-    znz_sqrt2       : znz -> znz -> znz * carry znz;
-    znz_sqrt        : znz -> znz  }.
-
-End Z_nZ_Op.
-
-Section Z_nZ_Spec.
- Variable w : Type.
- Variable w_op : znz_op w.
-
- Let w_digits      := w_op.(znz_digits).
- Let w_zdigits     := w_op.(znz_zdigits).
- Let w_to_Z        := w_op.(znz_to_Z).
- Let w_of_pos      := w_op.(znz_of_pos).
- Let w_head0       := w_op.(znz_head0).
- Let w_tail0       := w_op.(znz_tail0).
-
- Let w0            := w_op.(znz_0).
- Let w1            := w_op.(znz_1).
- Let wBm1          := w_op.(znz_Bm1).
-
- Let w_compare     := w_op.(znz_compare).
- Let w_eq0         := w_op.(znz_eq0).
-
- Let w_opp_c       := w_op.(znz_opp_c).
- Let w_opp         := w_op.(znz_opp).
- Let w_opp_carry   := w_op.(znz_opp_carry).
-
- Let w_succ_c      := w_op.(znz_succ_c).
- Let w_add_c       := w_op.(znz_add_c).
- Let w_add_carry_c := w_op.(znz_add_carry_c).
- Let w_succ        := w_op.(znz_succ).
- Let w_add         := w_op.(znz_add).
- Let w_add_carry   := w_op.(znz_add_carry).
+    sqrt2       : t -> t -> t * carry t;
+    sqrt        : t -> t  }.
 
- Let w_pred_c      := w_op.(znz_pred_c).
- Let w_sub_c       := w_op.(znz_sub_c).
- Let w_sub_carry_c := w_op.(znz_sub_carry_c).
- Let w_pred        := w_op.(znz_pred).
- Let w_sub         := w_op.(znz_sub).
- Let w_sub_carry   := w_op.(znz_sub_carry).
+ Section Specs.
+ Context {t : Type}{ops : Ops t}.
 
- Let w_mul_c       := w_op.(znz_mul_c).
- Let w_mul         := w_op.(znz_mul).
- Let w_square_c    := w_op.(znz_square_c).
+ Notation "[| x |]" := (to_Z x)  (at level 0, x at level 99).
 
- Let w_div21       := w_op.(znz_div21).
- Let w_div_gt      := w_op.(znz_div_gt).
- Let w_div         := w_op.(znz_div).
-
- Let w_mod_gt      := w_op.(znz_mod_gt).
- Let w_mod         := w_op.(znz_mod).
-
- Let w_gcd_gt      := w_op.(znz_gcd_gt).
- Let w_gcd         := w_op.(znz_gcd).
-
- Let w_add_mul_div := w_op.(znz_add_mul_div).
-
- Let w_pos_mod     := w_op.(znz_pos_mod).
-
- Let w_is_even     := w_op.(znz_is_even).
- Let w_sqrt2       := w_op.(znz_sqrt2).
- Let w_sqrt        := w_op.(znz_sqrt).
-
- Notation "[| x |]" := (w_to_Z x)  (at level 0, x at level 99).
-
- Let wB := base w_digits.
+ Let wB := base digits.
 
  Notation "[+| c |]" :=
-   (interp_carry 1 wB w_to_Z c)  (at level 0, x at level 99).
+   (interp_carry 1 wB to_Z c)  (at level 0, x at level 99).
 
  Notation "[-| c |]" :=
-   (interp_carry (-1) wB w_to_Z c)  (at level 0, x at level 99).
+   (interp_carry (-1) wB to_Z c)  (at level 0, x at level 99).
 
  Notation "[|| x ||]" :=
-   (zn2z_to_Z wB w_to_Z x)  (at level 0, x at level 99).
+   (zn2z_to_Z wB to_Z x)  (at level 0, x at level 99).
 
- Record znz_spec := mk_znz_spec {
+ Class Specs := MkSpecs {
 
     (* Conversion functions with Z *)
     spec_to_Z   : forall x, 0 <= [| x |] < wB;
     spec_of_pos : forall p,
-           Zpos p = (Z_of_N (fst (w_of_pos p)))*wB + [|(snd (w_of_pos p))|];
-    spec_zdigits : [| w_zdigits |] = Zpos w_digits;
-    spec_more_than_1_digit: 1 < Zpos w_digits;
+           Zpos p = (Z_of_N (fst (of_pos p)))*wB + [|(snd (of_pos p))|];
+    spec_zdigits : [| zdigits |] = Zpos digits;
+    spec_more_than_1_digit: 1 < Zpos digits;
 
     (* Basic numbers *)
-    spec_0   : [|w0|] = 0;
-    spec_1   : [|w1|] = 1;
-    spec_Bm1 : [|wBm1|] = wB - 1;
+    spec_0   : [|zero|] = 0;
+    spec_1   : [|one|] = 1;
+    spec_m1 : [|minus_one|] = wB - 1;
 
     (* Comparison *)
-    spec_compare :
-     forall x y,
-       match w_compare x y with
-       | Eq => [|x|] = [|y|]
-       | Lt => [|x|] < [|y|]
-       | Gt => [|x|] > [|y|]
-       end;
-    spec_eq0 : forall x, w_eq0 x = true -> [|x|] = 0;
+    spec_compare : forall x y, compare x y = ([|x|] ?= [|y|]);
+    (* NB: the spec of [eq0] is deliberately partial,
+       see DoubleCyclic where [eq0 x = true <-> x = W0] *)
+    spec_eq0 : forall x, eq0 x = true -> [|x|] = 0;
     (* Basic arithmetic operations *)
-    spec_opp_c : forall x, [-|w_opp_c x|] = -[|x|];
-    spec_opp : forall x, [|w_opp x|] = (-[|x|]) mod wB;
-    spec_opp_carry : forall x, [|w_opp_carry x|] = wB - [|x|] - 1;
-
-    spec_succ_c : forall x, [+|w_succ_c x|] = [|x|] + 1;
-    spec_add_c  : forall x y, [+|w_add_c x y|] = [|x|] + [|y|];
-    spec_add_carry_c : forall x y, [+|w_add_carry_c x y|] = [|x|] + [|y|] + 1;
-    spec_succ : forall x, [|w_succ x|] = ([|x|] + 1) mod wB;
-    spec_add : forall x y, [|w_add x y|] = ([|x|] + [|y|]) mod wB;
+    spec_opp_c : forall x, [-|opp_c x|] = -[|x|];
+    spec_opp : forall x, [|opp x|] = (-[|x|]) mod wB;
+    spec_opp_carry : forall x, [|opp_carry x|] = wB - [|x|] - 1;
+
+    spec_succ_c : forall x, [+|succ_c x|] = [|x|] + 1;
+    spec_add_c  : forall x y, [+|add_c x y|] = [|x|] + [|y|];
+    spec_add_carry_c : forall x y, [+|add_carry_c x y|] = [|x|] + [|y|] + 1;
+    spec_succ : forall x, [|succ x|] = ([|x|] + 1) mod wB;
+    spec_add : forall x y, [|add x y|] = ([|x|] + [|y|]) mod wB;
     spec_add_carry :
-	 forall x y, [|w_add_carry x y|] = ([|x|] + [|y|] + 1) mod wB;
+	 forall x y, [|add_carry x y|] = ([|x|] + [|y|] + 1) mod wB;
 
-    spec_pred_c : forall x, [-|w_pred_c x|] = [|x|] - 1;
-    spec_sub_c : forall x y, [-|w_sub_c x y|] = [|x|] - [|y|];
-    spec_sub_carry_c : forall x y, [-|w_sub_carry_c x y|] = [|x|] - [|y|] - 1;
-    spec_pred : forall x, [|w_pred x|] = ([|x|] - 1) mod wB;
-    spec_sub : forall x y, [|w_sub x y|] = ([|x|] - [|y|]) mod wB;
+    spec_pred_c : forall x, [-|pred_c x|] = [|x|] - 1;
+    spec_sub_c : forall x y, [-|sub_c x y|] = [|x|] - [|y|];
+    spec_sub_carry_c : forall x y, [-|sub_carry_c x y|] = [|x|] - [|y|] - 1;
+    spec_pred : forall x, [|pred x|] = ([|x|] - 1) mod wB;
+    spec_sub : forall x y, [|sub x y|] = ([|x|] - [|y|]) mod wB;
     spec_sub_carry :
-	 forall x y, [|w_sub_carry x y|] = ([|x|] - [|y|] - 1) mod wB;
+	 forall x y, [|sub_carry x y|] = ([|x|] - [|y|] - 1) mod wB;
 
-    spec_mul_c : forall x y, [|| w_mul_c x y ||] = [|x|] * [|y|];
-    spec_mul : forall x y, [|w_mul x y|] = ([|x|] * [|y|]) mod wB;
-    spec_square_c : forall x, [|| w_square_c x||] = [|x|] * [|x|];
+    spec_mul_c : forall x y, [|| mul_c x y ||] = [|x|] * [|y|];
+    spec_mul : forall x y, [|mul x y|] = ([|x|] * [|y|]) mod wB;
+    spec_square_c : forall x, [|| square_c x||] = [|x|] * [|x|];
 
     (* Special divisions operations *)
     spec_div21 : forall a1 a2 b,
       wB/2 <= [|b|] ->
       [|a1|] < [|b|] ->
-      let (q,r) := w_div21 a1 a2 b in
+      let (q,r) := div21 a1 a2 b in
       [|a1|] *wB+ [|a2|] = [|q|] * [|b|] + [|r|] /\
       0 <= [|r|] < [|b|];
     spec_div_gt : forall a b, [|a|] > [|b|] -> 0 < [|b|] ->
-      let (q,r) := w_div_gt a b in
+      let (q,r) := div_gt a b in
       [|a|] = [|q|] * [|b|] + [|r|] /\
       0 <= [|r|] < [|b|];
     spec_div : forall a b, 0 < [|b|] ->
-      let (q,r) := w_div a b in
+      let (q,r) := div a b in
       [|a|] = [|q|] * [|b|] + [|r|] /\
       0 <= [|r|] < [|b|];
 
-    spec_mod_gt : forall a b, [|a|] > [|b|] -> 0 < [|b|] ->
-      [|w_mod_gt a b|] = [|a|] mod [|b|];
-    spec_mod :  forall a b, 0 < [|b|] ->
-      [|w_mod a b|] = [|a|] mod [|b|];
+    spec_modulo_gt : forall a b, [|a|] > [|b|] -> 0 < [|b|] ->
+      [|modulo_gt a b|] = [|a|] mod [|b|];
+    spec_modulo :  forall a b, 0 < [|b|] ->
+      [|modulo a b|] = [|a|] mod [|b|];
 
     spec_gcd_gt : forall a b, [|a|] > [|b|] ->
-      Zis_gcd [|a|] [|b|] [|w_gcd_gt a b|];
-    spec_gcd : forall a b, Zis_gcd [|a|] [|b|] [|w_gcd a b|];
+      Zis_gcd [|a|] [|b|] [|gcd_gt a b|];
+    spec_gcd : forall a b, Zis_gcd [|a|] [|b|] [|gcd a b|];
 
 
     (* shift operations *)
-    spec_head00:  forall x, [|x|] = 0 -> [|w_head0 x|] = Zpos w_digits;
+    spec_head00:  forall x, [|x|] = 0 -> [|head0 x|] = Zpos digits;
     spec_head0  : forall x,  0 < [|x|] ->
-	 wB/ 2 <= 2 ^ ([|w_head0 x|]) * [|x|] < wB;
-    spec_tail00:  forall x, [|x|] = 0 -> [|w_tail0 x|] = Zpos w_digits;
+	 wB/ 2 <= 2 ^ ([|head0 x|]) * [|x|] < wB;
+    spec_tail00:  forall x, [|x|] = 0 -> [|tail0 x|] = Zpos digits;
     spec_tail0  : forall x, 0 < [|x|] ->
-         exists y, 0 <= y /\ [|x|] = (2 * y + 1) * (2 ^ [|w_tail0 x|]) ;
+         exists y, 0 <= y /\ [|x|] = (2 * y + 1) * (2 ^ [|tail0 x|]) ;
     spec_add_mul_div : forall x y p,
-       [|p|] <= Zpos w_digits ->
-       [| w_add_mul_div p x y |] =
+       [|p|] <= Zpos digits ->
+       [| add_mul_div p x y |] =
          ([|x|] * (2 ^ [|p|]) +
-          [|y|] / (2 ^ ((Zpos w_digits) - [|p|]))) mod wB;
+          [|y|] / (2 ^ ((Zpos digits) - [|p|]))) mod wB;
     spec_pos_mod : forall w p,
-       [|w_pos_mod p w|] = [|w|] mod (2 ^ [|p|]);
+       [|pos_mod p w|] = [|w|] mod (2 ^ [|p|]);
     (* sqrt *)
     spec_is_even : forall x,
-      if w_is_even x then [|x|] mod 2 = 0 else [|x|] mod 2 = 1;
+      if is_even x then [|x|] mod 2 = 0 else [|x|] mod 2 = 1;
     spec_sqrt2 : forall x y,
        wB/ 4 <= [|x|] ->
-       let (s,r) := w_sqrt2 x y in
+       let (s,r) := sqrt2 x y in
           [||WW x y||] = [|s|] ^ 2 + [+|r|] /\
           [+|r|] <= 2 * [|s|];
     spec_sqrt : forall x,
-       [|w_sqrt x|] ^ 2 <= [|x|] < ([|w_sqrt x|] + 1) ^ 2
+       [|sqrt x|] ^ 2 <= [|x|] < ([|sqrt x|] + 1) ^ 2
   }.
 
-End Z_nZ_Spec.
+ End Specs.
+
+ Arguments Specs {t} ops.
+
+ (** Generic construction of double words *)
 
-(** Generic construction of double words *)
+ Section WW.
 
-Section WW.
+ Context {t : Type}{ops : Ops t}{specs : Specs ops}.
 
- Variable w : Type.
- Variable w_op : znz_op w.
- Variable op_spec : znz_spec w_op.
+ Let wB := base digits.
 
- Let wB := base w_op.(znz_digits).
- Let w_to_Z := w_op.(znz_to_Z).
- Let w_eq0 := w_op.(znz_eq0).
- Let w_0 := w_op.(znz_0).
+ Definition WO' (eq0:t->bool) zero h :=
+  if eq0 h then W0 else WW h zero.
 
- Definition znz_W0 h :=
-  if w_eq0 h then W0 else WW h w_0.
+ Definition WO := Eval lazy beta delta [WO'] in
+   let eq0 := ZnZ.eq0 in
+   let zero := ZnZ.zero in
+   WO' eq0 zero.
 
- Definition znz_0W l :=
-  if w_eq0 l then W0 else WW w_0 l.
+ Definition OW' (eq0:t->bool) zero l :=
+  if eq0 l then W0 else WW zero l.
 
- Definition znz_WW h l :=
-  if w_eq0 h then znz_0W l else WW h l.
+ Definition OW := Eval lazy beta delta [OW'] in
+   let eq0 := ZnZ.eq0 in
+   let zero := ZnZ.zero in
+   OW' eq0 zero.
 
- Lemma spec_W0 : forall h,
-   zn2z_to_Z wB w_to_Z (znz_W0 h) = (w_to_Z h)*wB.
+ Definition WW'  (eq0:t->bool) zero h l :=
+  if eq0 h then OW' eq0 zero l else WW h l.
+
+ Definition WW := Eval lazy beta delta [WW' OW'] in
+   let eq0 := ZnZ.eq0 in
+   let zero := ZnZ.zero in
+   WW' eq0 zero.
+
+ Lemma spec_WO : forall h,
+   zn2z_to_Z wB to_Z (WO h) = (to_Z h)*wB.
  Proof.
- unfold zn2z_to_Z, znz_W0, w_to_Z; simpl; intros.
- case_eq (w_eq0 h); intros.
- rewrite (op_spec.(spec_eq0) _ H); auto.
- unfold w_0; rewrite op_spec.(spec_0); auto with zarith.
+ unfold zn2z_to_Z, WO; simpl; intros.
+ case_eq (eq0 h); intros.
+ rewrite (spec_eq0 _ H); auto.
+ rewrite spec_0; auto with zarith.
  Qed.
 
- Lemma spec_0W : forall l,
-   zn2z_to_Z wB w_to_Z (znz_0W l) = w_to_Z l.
+ Lemma spec_OW : forall l,
+   zn2z_to_Z wB to_Z (OW l) = to_Z l.
  Proof.
- unfold zn2z_to_Z, znz_0W, w_to_Z; simpl; intros.
- case_eq (w_eq0 l); intros.
- rewrite (op_spec.(spec_eq0) _ H); auto.
- unfold w_0; rewrite op_spec.(spec_0); auto with zarith.
+ unfold zn2z_to_Z, OW; simpl; intros.
+ case_eq (eq0 l); intros.
+ rewrite (spec_eq0 _ H); auto.
+ rewrite spec_0; auto with zarith.
  Qed.
 
  Lemma spec_WW : forall h l,
-  zn2z_to_Z wB w_to_Z (znz_WW h l) = (w_to_Z h)*wB + w_to_Z l.
+  zn2z_to_Z wB to_Z (WW h l) = (to_Z h)*wB + to_Z l.
  Proof.
- unfold znz_WW, w_to_Z; simpl; intros.
- case_eq (w_eq0 h); intros.
- rewrite (op_spec.(spec_eq0) _ H); auto.
- rewrite spec_0W; auto.
+ unfold WW; simpl; intros.
+ case_eq (eq0 h); intros.
+ rewrite (spec_eq0 _ H); auto.
+ fold (OW l).
+ rewrite spec_OW; auto.
  simpl; auto.
  Qed.
 
-End WW.
-
-(** Injecting [Z] numbers into a cyclic structure *)
+ End WW.
 
-Section znz_of_pos.
+ (** Injecting [Z] numbers into a cyclic structure *)
 
- Variable w : Type.
- Variable w_op : znz_op w.
- Variable op_spec : znz_spec w_op.
+ Section Of_Z.
 
- Notation "[| x |]" := (znz_to_Z w_op x)  (at level 0, x at level 99).
+ Context {t : Type}{ops : Ops t}{specs : Specs ops}.
 
- Definition znz_of_Z (w:Type) (op:znz_op w) z :=
- match z with
- | Zpos p => snd (op.(znz_of_pos) p)
- | _ => op.(znz_0)
- end.
+ Notation "[| x |]" := (to_Z x)  (at level 0, x at level 99).
 
- Theorem znz_of_pos_correct:
-   forall p, Zpos p < base (znz_digits w_op) -> [|(snd (znz_of_pos w_op p))|] = Zpos p.
+ Theorem of_pos_correct:
+   forall p, Zpos p < base digits -> [|(snd (of_pos p))|] = Zpos p.
+ Proof.
  intros p Hp.
- generalize (spec_of_pos op_spec p).
- case (znz_of_pos w_op p); intros n w1; simpl.
+ generalize (spec_of_pos p).
+ case (of_pos p); intros n w1; simpl.
  case n; simpl Npos; auto with zarith.
  intros p1 Hp1; contradict Hp; apply Zle_not_lt.
- rewrite Hp1; auto with zarith.
- match goal with |- _ <= ?X + ?Y =>
-  apply Zle_trans with X; auto with zarith
- end.
- match goal with |- ?X <= _ =>
-  pattern X at 1; rewrite <- (Zmult_1_l);
-  apply Zmult_le_compat_r; auto with zarith
- end.
+ replace (base digits) with (1 * base digits + 0) by ring.
+ rewrite Hp1.
+ apply Zplus_le_compat.
+ apply Zmult_le_compat; auto with zarith.
  case p1; simpl; intros; red; simpl; intros; discriminate.
  unfold base; auto with zarith.
- case (spec_to_Z op_spec w1); auto with zarith.
+ case (spec_to_Z w1); auto with zarith.
  Qed.
 
- Theorem znz_of_Z_correct:
-   forall p, 0 <= p < base (znz_digits w_op) -> [|znz_of_Z w_op p|] = p.
+ Definition of_Z z :=
+ match z with
+ | Zpos p => snd (of_pos p)
+ | _ => zero
+ end.
+
+ Theorem of_Z_correct:
+   forall p, 0 <= p < base digits -> [|of_Z p|] = p.
+ Proof.
  intros p; case p; simpl; try rewrite spec_0; auto.
- intros; rewrite znz_of_pos_correct; auto with zarith.
+ intros; rewrite of_pos_correct; auto with zarith.
  intros p1 (H1, _); contradict H1; apply Zlt_not_le; red; simpl; auto.
  Qed.
-End znz_of_pos.
 
+ End Of_Z.
+
+End ZnZ.
 
 (** A modular specification grouping the earlier records. *)
 
 Module Type CyclicType.
- Parameter w : Type.
- Parameter w_op : znz_op w.
- Parameter w_spec : znz_spec w_op.
+ Parameter t : Type.
+ Declare Instance ops : ZnZ.Ops t.
+ Declare Instance specs : ZnZ.Specs ops.
 End CyclicType.
 
 
@@ -379,38 +325,29 @@ End CyclicType.
 
 Module CyclicRing (Import Cyclic : CyclicType).
 
-Definition t := w.
-
-Local Notation "[| x |]" := (w_op.(znz_to_Z) x) (at level 0, x at level 99).
+Local Notation "[| x |]" := (ZnZ.to_Z x) (at level 0, x at level 99).
 
 Definition eq (n m : t) := [| n |] = [| m |].
-Definition zero : t := w_op.(znz_0).
-Definition one := w_op.(znz_1).
-Definition add := w_op.(znz_add).
-Definition sub := w_op.(znz_sub).
-Definition mul := w_op.(znz_mul).
-Definition opp := w_op.(znz_opp).
 
 Local Infix "=="  := eq (at level 70).
-Local Notation "0" := zero.
-Local Notation "1" := one.
-Local Infix "+" := add.
-Local Infix "-" := sub.
-Local Infix "*" := mul.
-Local Notation "!!" := (base (znz_digits w_op)).
-
-Hint Rewrite
- w_spec.(spec_0) w_spec.(spec_1)
- w_spec.(spec_add) w_spec.(spec_mul) w_spec.(spec_opp) w_spec.(spec_sub)
+Local Notation "0" := ZnZ.zero.
+Local Notation "1" := ZnZ.one.
+Local Infix "+" := ZnZ.add.
+Local Infix "-" := ZnZ.sub.
+Local Notation "- x" := (ZnZ.opp x).
+Local Infix "*" := ZnZ.mul.
+Local Notation wB := (base ZnZ.digits).
+
+Hint Rewrite ZnZ.spec_0 ZnZ.spec_1 ZnZ.spec_add ZnZ.spec_mul
+ ZnZ.spec_opp ZnZ.spec_sub
  : cyclic.
 
-Ltac zify :=
- unfold eq, zero, one, add, sub, mul, opp in *; autorewrite with cyclic.
+Ltac zify := unfold eq in *; autorewrite with cyclic.
 
 Lemma add_0_l : forall x, 0 + x == x.
 Proof.
 intros. zify. rewrite Zplus_0_l.
-apply Zmod_small. apply w_spec.(spec_to_Z).
+apply Zmod_small. apply ZnZ.spec_to_Z.
 Qed.
 
 Lemma add_comm : forall x y, x + y == y + x.
@@ -426,7 +363,7 @@ Qed.
 Lemma mul_1_l : forall x, 1 * x == x.
 Proof.
 intros. zify. rewrite Zmult_1_l.
-apply Zmod_small. apply w_spec.(spec_to_Z).
+apply Zmod_small. apply ZnZ.spec_to_Z.
 Qed.
 
 Lemma mul_comm : forall x y, x * y == y * x.
@@ -444,22 +381,22 @@ Proof.
 intros. zify. now rewrite <- Zplus_mod, Zmult_mod_idemp_l, Zmult_plus_distr_l.
 Qed.
 
-Lemma add_opp_r : forall x y, x + opp y == x-y.
+Lemma add_opp_r : forall x y, x + - y == x-y.
 Proof.
 intros. zify. rewrite <- Zminus_mod_idemp_r. unfold Zminus.
-destruct (Z_eq_dec ([|y|] mod !!) 0) as [EQ|NEQ].
+destruct (Z_eq_dec ([|y|] mod wB) 0) as [EQ|NEQ].
 rewrite Z_mod_zero_opp_full, EQ, 2 Zplus_0_r; auto.
 rewrite Z_mod_nz_opp_full by auto.
 rewrite <- Zplus_mod_idemp_r, <- Zminus_mod_idemp_l.
 rewrite Z_mod_same_full. simpl. now rewrite Zplus_mod_idemp_r.
 Qed.
 
-Lemma add_opp_diag_r : forall x, x + opp x == 0.
+Lemma add_opp_diag_r : forall x, x + - x == 0.
 Proof.
 intros. red. rewrite add_opp_r. zify. now rewrite Zminus_diag, Zmod_0_l.
 Qed.
 
-Lemma CyclicRing : ring_theory 0 1 add mul sub opp eq.
+Lemma CyclicRing : ring_theory 0 1 ZnZ.add ZnZ.mul ZnZ.sub ZnZ.opp eq.
 Proof.
 constructor.
 exact add_0_l. exact add_comm. exact add_assoc.
@@ -470,15 +407,26 @@ exact add_opp_diag_r.
 Qed.
 
 Definition eqb x y :=
- match w_op.(znz_compare) x y with Eq => true | _ => false end.
+ match ZnZ.compare x y with Eq => true | _ => false end.
+
+Lemma eqb_eq : forall x y, eqb x y = true <-> x == y.
+Proof.
+ intros. unfold eqb, eq.
+ rewrite ZnZ.spec_compare.
+ case Zcompare_spec; intuition; try discriminate.
+Qed.
 
+(* POUR HUGO:
 Lemma eqb_eq : forall x y, eqb x y = true <-> x == y.
 Proof.
- intros. unfold eqb, eq. generalize (w_spec.(spec_compare) x y).
- destruct (w_op.(znz_compare) x y); intuition; try discriminate.
+ intros. unfold eqb, eq. generalize (ZnZ.spec_compare x y).
+ case (ZnZ.compare x y); intuition; try discriminate.
+ (* BUG ?! using destruct instead of case won't work:
+   it gives 3 subcases, but ZnZ.compare x y is still there in them! *)
 Qed.
+*)
 
 Lemma eqb_correct : forall x y, eqb x y = true -> x==y.
 Proof. now apply eqb_eq. Qed.
 
-End CyclicRing.
\ No newline at end of file
+End CyclicRing.