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-(************************************************************************)
-(* 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 *)
-(************************************************************************)
-
-Require Import BinInt.
-
-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.
- Local 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 := Z.to_N (to_Z n).
-
- Definition eq n m := [n] = [m].
- Definition lt n m := [n] < [m].
- Definition le n m := [n] <= [m].
-
- Parameter compare : t -> t -> comparison.
- Parameter eqb : t -> t -> bool.
- Parameter ltb : t -> t -> bool.
- Parameter leb : t -> t -> bool.
- Parameter max : t -> t -> t.
- Parameter min : t -> t -> t.
- Parameter zero : t.
- Parameter one : t.
- Parameter two : t.
- Parameter succ : t -> t.
- Parameter pred : t -> t.
- Parameter add : t -> t -> t.
- Parameter sub : t -> t -> t.
- Parameter mul : t -> t -> t.
- Parameter square : t -> t.
- Parameter pow_pos : t -> positive -> t.
- Parameter pow_N : t -> N -> t.
- Parameter pow : t -> t -> t.
- Parameter sqrt : t -> t.
- Parameter log2 : t -> t.
- Parameter div_eucl : t -> t -> t * t.
- Parameter div : t -> t -> t.
- Parameter modulo : t -> t -> t.
- Parameter gcd : t -> t -> t.
- Parameter even : t -> bool.
- Parameter odd : t -> bool.
- Parameter testbit : t -> t -> bool.
- Parameter shiftr : t -> t -> t.
- Parameter shiftl : t -> t -> t.
- Parameter land : t -> t -> t.
- Parameter lor : t -> t -> t.
- Parameter ldiff : t -> t -> t.
- Parameter lxor : t -> t -> t.
- Parameter div2 : t -> t.
-
- Parameter spec_compare: forall x y, compare x y = ([x] ?= [y]).
- Parameter spec_eqb : forall x y, eqb x y = ([x] =? [y]).
- Parameter spec_ltb : forall x y, ltb x y = ([x] <? [y]).
- Parameter spec_leb : forall x y, leb x y = ([x] <=? [y]).
- Parameter spec_max : forall x y, [max x y] = Z.max [x] [y].
- Parameter spec_min : forall x y, [min x y] = Z.min [x] [y].
- Parameter spec_0: [zero] = 0.
- Parameter spec_1: [one] = 1.
- Parameter spec_2: [two] = 2.
- Parameter spec_succ: forall n, [succ n] = [n] + 1.
- Parameter spec_add: forall x y, [add x y] = [x] + [y].
- Parameter spec_pred: forall x, [pred x] = Z.max 0 ([x] - 1).
- Parameter spec_sub: forall x y, [sub x y] = Z.max 0 ([x] - [y]).
- Parameter spec_mul: forall x y, [mul x y] = [x] * [y].
- Parameter spec_square: forall x, [square x] = [x] * [x].
- Parameter spec_pow_pos: forall x n, [pow_pos x n] = [x] ^ Zpos n.
- Parameter spec_pow_N: forall x n, [pow_N x n] = [x] ^ Z.of_N n.
- Parameter spec_pow: forall x n, [pow x n] = [x] ^ [n].
- Parameter spec_sqrt: forall x, [sqrt x] = Z.sqrt [x].
- Parameter spec_log2: forall x, [log2 x] = Z.log2 [x].
- Parameter spec_div_eucl: forall x y,
- let (q,r) := div_eucl x y in ([q], [r]) = Z.div_eucl [x] [y].
- Parameter spec_div: forall x y, [div x y] = [x] / [y].
- Parameter spec_modulo: forall x y, [modulo x y] = [x] mod [y].
- Parameter spec_gcd: forall a b, [gcd a b] = Z.gcd [a] [b].
- Parameter spec_even: forall x, even x = Z.even [x].
- Parameter spec_odd: forall x, odd x = Z.odd [x].
- Parameter spec_testbit: forall x p, testbit x p = Z.testbit [x] [p].
- Parameter spec_shiftr: forall x p, [shiftr x p] = Z.shiftr [x] [p].
- Parameter spec_shiftl: forall x p, [shiftl x p] = Z.shiftl [x] [p].
- Parameter spec_land: forall x y, [land x y] = Z.land [x] [y].
- Parameter spec_lor: forall x y, [lor x y] = Z.lor [x] [y].
- Parameter spec_ldiff: forall x y, [ldiff x y] = Z.ldiff [x] [y].
- Parameter spec_lxor: forall x y, [lxor x y] = Z.lxor [x] [y].
- Parameter spec_div2: forall x, [div2 x] = Z.div2 [x].
-
-End NType.
-
-Module Type NType_Notation (Import N:NType).
- Notation "[ x ]" := (to_Z x).
- Infix "==" := eq (at level 70).
- Notation "0" := zero.
- Notation "1" := one.
- Notation "2" := two.
- Infix "+" := add.
- Infix "-" := sub.
- Infix "*" := mul.
- Infix "^" := pow.
- Infix "<=" := le.
- Infix "<" := lt.
-End NType_Notation.
-
-Module Type NType' := NType <+ NType_Notation.