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
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
|
(* -*- coding: utf-8 -*- *)
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* $Id: Ascii.v 14641 2011-11-06 11:59:10Z herbelin $ *)
(** Contributed by Laurent Théry (INRIA);
Adapted to Coq V8 by the Coq Development Team *)
Require Import Bool BinPos BinNat Nnat.
Declare ML Module "ascii_syntax_plugin".
(** * Definition of ascii characters *)
(** Definition of ascii character as a 8 bits constructor *)
Inductive ascii : Set := Ascii (_ _ _ _ _ _ _ _ : bool).
Delimit Scope char_scope with char.
Bind Scope char_scope with ascii.
Definition zero := Ascii false false false false false false false false.
Definition one := Ascii true false false false false false false false.
Definition shift (c : bool) (a : ascii) :=
match a with
| Ascii a1 a2 a3 a4 a5 a6 a7 a8 => Ascii c a1 a2 a3 a4 a5 a6 a7
end.
(** Definition of a decidable function that is effective *)
Definition ascii_dec : forall a b : ascii, {a = b} + {a <> b}.
decide equality; apply bool_dec.
Defined.
(** * Conversion between natural numbers modulo 256 and ascii characters *)
(** Auxillary function that turns a positive into an ascii by
looking at the last 8 bits, ie z mod 2^8 *)
Definition ascii_of_pos : positive -> ascii :=
let loop := fix loop n p :=
match n with
| O => zero
| S n' =>
match p with
| xH => one
| xI p' => shift true (loop n' p')
| xO p' => shift false (loop n' p')
end
end
in loop 8.
(** Conversion from [N] to [ascii] *)
Definition ascii_of_N (n : N) :=
match n with
| N0 => zero
| Npos p => ascii_of_pos p
end.
(** Same for [nat] *)
Definition ascii_of_nat (a : nat) := ascii_of_N (N_of_nat a).
(** The opposite functions *)
Local Open Scope list_scope.
Fixpoint N_of_digits (l:list bool) : N :=
match l with
| nil => 0
| b :: l' => (if b then 1 else 0) + 2*(N_of_digits l')
end%N.
Definition N_of_ascii (a : ascii) : N :=
let (a0,a1,a2,a3,a4,a5,a6,a7) := a in
N_of_digits (a0::a1::a2::a3::a4::a5::a6::a7::nil).
Definition nat_of_ascii (a : ascii) : nat := nat_of_N (N_of_ascii a).
(** Proofs that we have indeed opposite function (below 256) *)
Theorem ascii_N_embedding :
forall a : ascii, ascii_of_N (N_of_ascii a) = a.
Proof.
destruct a as [[|][|][|][|][|][|][|][|]]; vm_compute; reflexivity.
Qed.
Theorem N_ascii_embedding :
forall n:N, (n < 256)%N -> N_of_ascii (ascii_of_N n) = n.
Proof.
destruct n.
reflexivity.
do 8 (destruct p; [ | | intros; vm_compute; reflexivity ]);
intro H; vm_compute in H; destruct p; discriminate.
Qed.
Theorem ascii_nat_embedding :
forall a : ascii, ascii_of_nat (nat_of_ascii a) = a.
Proof.
destruct a as [[|][|][|][|][|][|][|][|]]; compute; reflexivity.
Qed.
Theorem nat_ascii_embedding :
forall n : nat, n < 256 -> nat_of_ascii (ascii_of_nat n) = n.
Proof.
intros. unfold nat_of_ascii, ascii_of_nat.
rewrite N_ascii_embedding.
apply nat_of_N_of_nat.
unfold Nlt.
change 256%N with (N_of_nat 256).
rewrite <- N_of_nat_compare.
rewrite <- Compare_dec.nat_compare_lt. auto.
Qed.
(** * Concrete syntax *)
(**
Ascii characters can be represented in scope char_scope as follows:
- ["c"] represents itself if c is a character of code < 128,
- [""""] is an exception: it represents the ascii character 34
(double quote),
- ["nnn"] represents the ascii character of decimal code nnn.
For instance, both ["065"] and ["A"] denote the character `uppercase
A', and both ["034"] and [""""] denote the character `double quote'.
Notice that the ascii characters of code >= 128 do not denote
stand-alone utf8 characters so that only the notation "nnn" is
available for them (unless your terminal is able to represent them,
which is typically not the case in coqide).
*)
Open Local Scope char_scope.
Example Space := " ".
Example DoubleQuote := """".
Example Beep := "007".
|
