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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        *)
+(************************************************************************)
+(*            Benjamin Gregoire, Laurent Thery, INRIA, 2007             *)
+(************************************************************************)
+
+(*i $Id: NSig.v 11027 2008-06-01 13:28:59Z letouzey $ i*)
+
+Require Import ZArith Znumtheory.
+
+Open Scope Z_scope.
+
+(** * NSig *)
+
+(** Interface of a rich structure about natural numbers.
+    Specifications are written via translation to Z.
+*)
+
+Module Type NType.
+
+ Parameter t : Type.
+
+ Parameter to_Z : t -> Z.
+ Notation "[ x ]" := (to_Z x).
+ Parameter spec_pos: forall x, 0 <= [x].
+
+ Parameter of_N : N -> t.
+ Parameter spec_of_N: forall x, to_Z (of_N x) = Z_of_N x.
+ Definition to_N n := Zabs_N (to_Z n).
+
+ Definition eq n m := ([n] = [m]).
+
+ Parameter zero : t.
+ Parameter one : t.
+
+ Parameter spec_0: [zero] = 0.
+ Parameter spec_1: [one] = 1.
+
+ Parameter compare : t -> t -> comparison.
+
+ Parameter spec_compare: forall x y,
+   match compare x y with
+     | Eq => [x] = [y]
+     | Lt => [x] < [y]
+     | Gt => [x] > [y]
+   end.
+
+ Definition lt n m := compare n m = Lt.
+ Definition le n m := compare n m <> Gt.
+ Definition min n m := match compare n m with Gt => m | _ => n end.
+ Definition max n m := match compare n m with Lt => m | _ => n end.
+
+ Parameter eq_bool : t -> t -> bool.
+
+ Parameter spec_eq_bool: forall x y,
+    if eq_bool x y then [x] = [y] else [x] <> [y].
+ 
+ Parameter succ : t -> t.
+
+ Parameter spec_succ: forall n, [succ n] = [n] + 1.
+
+ Parameter add  : t -> t -> t.
+
+ Parameter spec_add: forall x y, [add x y] = [x] + [y].
+
+ Parameter pred : t -> t.
+
+ Parameter spec_pred: forall x, 0 < [x] -> [pred x] = [x] - 1.
+ Parameter spec_pred0: forall x, [x] = 0 -> [pred x] = 0.
+
+ Parameter sub : t -> t -> t.
+
+ Parameter spec_sub: forall x y, [y] <= [x] -> [sub x y] = [x] - [y].
+ Parameter spec_sub0: forall x y, [x] < [y]-> [sub x y] = 0.
+
+ Parameter mul : t -> t -> t.
+
+ Parameter spec_mul: forall x y, [mul x y] = [x] * [y].
+
+ Parameter square : t -> t.
+
+ Parameter spec_square: forall x, [square x] = [x] *  [x].
+
+ Parameter power_pos : t -> positive -> t.
+
+ Parameter spec_power_pos: forall x n, [power_pos x n] = [x] ^ Zpos n.
+
+ Parameter sqrt : t -> t.
+
+ Parameter spec_sqrt: forall x, [sqrt x] ^ 2 <= [x] < ([sqrt x] + 1) ^ 2.
+
+ Parameter div_eucl : t -> t -> t * t.
+
+ Parameter spec_div_eucl: forall x y,
+      0 < [y] ->
+      let (q,r) := div_eucl x y in ([q], [r]) = Zdiv_eucl [x] [y].
+ 
+ Parameter div : t -> t -> t.
+
+ Parameter spec_div: forall x y, 0 < [y] -> [div x y] = [x] / [y].
+
+ Parameter modulo : t -> t -> t.
+
+ Parameter spec_modulo:
+   forall x y, 0 < [y] -> [modulo x y] = [x] mod [y].
+
+ Parameter gcd : t -> t -> t.
+
+ Parameter spec_gcd: forall a b, [gcd a b] = Zgcd (to_Z a) (to_Z b).
+
+End NType.