More BigNum cleanup:

 * View of Int31 as a Z/nZ moved to file Z31Z.v (TO FINISH: specs are still admitted!)
 * Modular specification of Z/nZ moved to ZnZ and renamed CyclicType
 * More isolation between Cyclic/Abstract and Cyclic/DoubleCyclic
 * A few comments



git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@10936 85f007b7-540e-0410-9357-904b9bb8a0f7

5 files changed, 153 insertions, 17 deletions

diff --git a/theories/Numbers/Cyclic/Abstract/ZnZ.v b/theories/Numbers/Cyclic/Abstract/ZnZ.v
index 877e49c11e..9ec7648e61 100644
--- a/theories/Numbers/Cyclic/Abstract/ZnZ.v
+++ b/theories/Numbers/Cyclic/Abstract/ZnZ.v
@@ -16,11 +16,13 @@ Set Implicit Arguments.
 
 Require Import ZArith.
 Require Import Znumtheory.
+Require Import BigNumPrelude.
 Require Import Basic_type.
-Require Import GenBase.
 
 Open Local Scope Z_scope.
 
+(** First, a description via operator records and spec records. *)
+
 Section ZnZ_Op.
 
  Variable znz : Set.
@@ -90,7 +92,7 @@ Section ZnZ_Op.
 
 End ZnZ_Op.
 
-Section Spec.
+Section ZnZ_Spec.
  Variable w : Set.
  Variable w_op : znz_op w.
 
@@ -164,9 +166,6 @@ Section Spec.
 
  Notation "[|| x ||]" :=
    (zn2z_to_Z wB w_to_Z x)  (at level 0, x at level 99).
- 
- Notation "[! n | x !]" := (gen_to_Z w_digits w_to_Z n x) 
-       (at level 0, x at level 99).
 
  Record znz_spec : Set := mk_znz_spec {
 
@@ -271,8 +270,11 @@ Section Spec.
        [|w_sqrt x|] ^ 2 <= [|x|] < ([|w_sqrt x|] + 1) ^ 2
   }.
 
-End Spec.
+End ZnZ_Spec.
+
+
 
+(** Injecting [Z] numbers into a cyclic structure *)
 
 Section znz_of_pos.
  
@@ -315,3 +317,12 @@ Section znz_of_pos.
  intros p1 (H1, _); contradict H1; apply Zlt_not_le; red; simpl; auto.
  Qed.
 End znz_of_pos.
+
+
+(** A modular specification grouping the earlier records. *)
+
+Module Type CyclicType.
+ Parameter w : Set.
+ Parameter w_op : znz_op w.
+ Parameter w_spec : znz_spec w_op.
+End CyclicType.
diff --git a/theories/Numbers/Cyclic/DoubleCyclic/Basic_type.v b/theories/Numbers/Cyclic/DoubleCyclic/Basic_type.v
index 60fa09584f..95f3c88e61 100644
--- a/theories/Numbers/Cyclic/DoubleCyclic/Basic_type.v
+++ b/theories/Numbers/Cyclic/DoubleCyclic/Basic_type.v
@@ -15,6 +15,8 @@ Set Implicit Arguments.
 Require Import ZArith.
 Open Local Scope Z_scope.
 
+Definition base digits := Zpower 2 (Zpos digits).
+
 Section Carry.
 
  Variable A : Set.
@@ -29,6 +31,12 @@ Section Zn2Z.
 
  Variable znz : Set.
 
+ (** From a type [znz] representing a cyclic structure Z/nZ, 
+     we produce a representation of Z/2nZ by pairs of elements of [znz]
+     (plus a special case for zero). High half of the new number comes 
+     first. 
+ *)
+
  Inductive zn2z : Set :=
   | W0 : zn2z
   | WW : znz -> znz -> zn2z.
@@ -39,8 +47,6 @@ Section Zn2Z.
   | WW xh xl => w_to_Z xh * wB + w_to_Z xl
   end.
 
- Definition base digits := Zpower 2 (Zpos digits).
-
  Definition interp_carry sign B (interp:znz -> Z) c :=
   match c with
   | C0 x => interp x
@@ -51,11 +57,11 @@ End Zn2Z.
 
 Implicit Arguments W0 [znz].
 
-Fixpoint word_tr (w:Set) (n:nat) {struct n} : Set :=
- match n with
- | O => w
- | S n => word_tr (zn2z w) n
- end.
+(** From a cyclic representation [w], we iterate the [zn2z] construct 
+    [n] times, gaining the type of binary trees of depth at most [n], 
+    whose leafs are either W0 (if depth < n) or elements of w 
+    (if depth = n). 
+*)
 
 Fixpoint word (w:Set) (n:nat) {struct n} : Set :=
  match n with
diff --git a/theories/Numbers/Cyclic/DoubleCyclic/GenBase.v b/theories/Numbers/Cyclic/DoubleCyclic/GenBase.v
index eb517c0607..8d07c6dee9 100644
--- a/theories/Numbers/Cyclic/DoubleCyclic/GenBase.v
+++ b/theories/Numbers/Cyclic/DoubleCyclic/GenBase.v
@@ -15,7 +15,6 @@ Set Implicit Arguments.
 Require Import ZArith.
 Require Import BigNumPrelude.
 Require Import Basic_type.
-Require Import JMeq.
 
 Open Local Scope Z_scope.
 
diff --git a/theories/Numbers/Cyclic/Int31/Int31.v b/theories/Numbers/Cyclic/Int31/Int31.v
index 032326c81f..5cd504a34a 100644
--- a/theories/Numbers/Cyclic/Int31/Int31.v
+++ b/theories/Numbers/Cyclic/Int31/Int31.v
@@ -16,11 +16,17 @@ Require Export Basic_type.
 
 Unset Boxed Definitions.
 
-Inductive digits : Type := |D0 |D1.
+Inductive digits : Type := D0 | D1.
 
 Inductive int31 : Type :=
-| I31 : digits ->digits ->digits ->digits ->digits ->digits ->digits ->digits ->digits ->digits ->digits ->digits ->digits ->digits ->digits ->digits ->digits ->digits ->digits ->digits ->digits ->digits ->digits ->digits ->digits ->digits ->digits ->digits ->digits ->digits ->digits -> int31
-.
+ | I31 : digits -> digits -> digits -> digits ->
+         digits -> digits -> digits -> digits ->
+         digits -> digits -> digits -> digits ->
+         digits -> digits -> digits -> digits ->
+         digits -> digits -> digits -> digits ->
+         digits -> digits -> digits -> digits ->
+         digits -> digits -> digits -> digits ->
+         digits -> digits -> digits -> int31.
 
 (* spiwack: Registration of the type of integers, so that the matchs in
    the functions below perform dynamic decompilation (otherwise some segfault
diff --git a/theories/Numbers/Cyclic/Int31/Z31Z.v b/theories/Numbers/Cyclic/Int31/Z31Z.v
new file mode 100644
index 0000000000..1a0046f8f6
--- /dev/null
+++ b/theories/Numbers/Cyclic/Int31/Z31Z.v
@@ -0,0 +1,114 @@
+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+
+(*i $Id$ i*)
+
+(** * Int31 numbers defines indeed a cyclic structure : Z/31Z *)
+
+(**
+Author: Arnaud Spiwack
+*)
+
+Require Export Int31.
+Require Import ZnZ.
+
+Open Scope int31_scope.
+
+Definition int31_op : znz_op int31.
+ split.
+
+ (* Conversion functions with Z *)
+ exact (31%positive). (* number of digits *)
+ exact (31). (* number of digits *)
+ exact (phi). (* conversion to Z *)
+ exact (positive_to_int31). (* positive -> N*int31 :  p => N,i where p = N*2^31+phi i *)
+ exact head031.  (* number of head 0 *)
+ exact tail031.  (* number of tail 0 *)
+
+ (* Basic constructors *)
+ exact 0. (* 0 *)
+ exact 1. (* 1 *)
+ exact Tn. (* 2^31 - 1 *)
+ (* A function which given two int31 i and j, returns a double word
+    which is worth i*2^31+j *)
+ exact (fun i j => match (match i ?= 0 with | Eq =>  j ?= 0 | not0 => not0 end) with | Eq => W0 | _ => WW i j end).
+ (* two special cases where i and j are respectively taken equal to 0 *)
+ exact (fun i => match i ?= 0 with | Eq => W0 | _ => WW i 0 end).
+ exact (fun j => match j ?= 0 with | Eq => W0 | _ => WW 0 j end).
+
+ (* Comparison *)
+ exact compare31.
+ exact (fun i => match i ?= 0 with | Eq => true | _ => false end).
+
+ (* Basic arithmetic operations *)
+ (* opposite functions *)
+ exact (fun i => 0 -c i).
+ exact (fun i => 0 - i).
+ exact (fun i => 0-i-1). (* the carry is always -1*)
+ (* successor and addition functions *)
+ exact (fun i => i +c 1).
+ exact add31c.
+ exact add31carryc.
+ exact (fun i => i + 1).
+ exact add31.
+ exact (fun i j => i + j + 1).
+ (* predecessor and subtraction functions *)
+ exact (fun i => i -c 1).
+ exact sub31c.
+ exact sub31carryc.
+ exact (fun i => i - 1).
+ exact sub31.
+ exact (fun i j => i - j - 1).
+ (* multiplication functions *)
+ exact mul31c.
+ exact mul31.
+ exact (fun x => x *c x).
+
+ (* special (euclidian) division operations *)
+ exact div3121.
+ exact div31. (* this is supposed to be the special case of division a/b where a > b *)
+ exact div31.
+ (* euclidian division remainder *)
+ (* again special case for a > b *)
+ exact (fun i j => let (_,r) := i/j in r).
+ exact (fun i j => let (_,r) := i/j in r).
+ (* gcd functions *)
+ exact gcd31. (*gcd_gt*)
+ exact gcd31. (*gcd*)
+ 
+ (* shift operations *)
+ exact addmuldiv31. (*add_mul_div  *)
+(*modulo 2^p  *)
+ exact (fun p i => 
+     match compare31 p 32 with
+     | Lt => addmuldiv31 p 0 (addmuldiv31 (31-p) i 0)
+     | _ => i
+     end).
+
+ (* is i even ? *)
+ exact (fun i => let (_,r) := i/2 in 
+                          match r ?= 0 with
+                          | Eq => true
+                          | _ => false
+                          end).
+
+ (* square root operations *)
+ exact sqrt312. (* sqrt2 *)
+ exact sqrt31. (* sqr *)
+Defined.
+
+
+Definition int31_spec : znz_spec int31_op.
+Admitted.
+
+
+Module Int31_words <: CyclicType.
+  Definition w := int31.
+  Definition w_op := int31_op.
+  Definition w_spec := int31_spec.
+End Int31_words.