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diff --git a/theories/Numbers/Cyclic/Int31/Cyclic31.v b/theories/Numbers/Cyclic/Int31/Cyclic31.v
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+(************************************************************************)
+(*  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/(2^31)Z *)
+
+(**
+Author: Arnaud Spiwack
+*)
+
+Require Export Int31.
+Require Import CyclicAxioms.
+
+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 Int31Cyclic <: CyclicType.
+  Definition w := int31.
+  Definition w_op := int31_op.
+  Definition w_spec := int31_spec.
+End Int31Cyclic.