1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
|
(************************************************************************)
(* 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.
|
