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
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
|
/**************************************************************************/
/* */
/* The Why platform for program certification */
/* Copyright (C) 2002-2008 */
/* Romain BARDOU */
/* Jean-Franois COUCHOT */
/* Mehdi DOGGUY */
/* Jean-Christophe FILLITRE */
/* Thierry HUBERT */
/* Claude MARCH */
/* Yannick MOY */
/* Christine PAULIN */
/* Yann RGIS-GIANAS */
/* Nicolas ROUSSET */
/* Xavier URBAIN */
/* */
/* This software is free software; you can redistribute it and/or */
/* modify it under the terms of the GNU General Public */
/* License version 2, as published by the Free Software Foundation. */
/* */
/* This software is distributed in the hope that it will be useful, */
/* but WITHOUT ANY WARRANTY; without even the implied warranty of */
/* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. */
/* */
/* See the GNU General Public License version 2 for more details */
/* (enclosed in the file GPL). */
/* */
/**************************************************************************/
// TODO: does Caduceus permit verification where N and t are
// parameters?
// TODO: assert that the initial queue state models Qnew() -- not
// clear that Caduceus supports this
// TODO: create separate interface functions that implement the spec,
// that call internal functions to do the work, put the algebraic
// queue into a separate file
// TODO: head and dequeue require q_size>0 but also handle the case
// where this is not true -- can and should the spec say something
// about this case?
// TODO: is the set of axioms sound, complete, and minimal? would be
// nice to check this.
typedef unsigned char uint8_t;
typedef uint8_t bool;
typedef uint8_t error_t;
typedef int t;
#define N 10
// Simplify has no model for mod and verifying this queue requires
// only these special cases
/*@ axiom int_mod_1: \forall int x; \forall int y;
@ ((0 <= x < y) => ((x%y) == x))
@*/
/*@ axiom int_mod_2: \forall int x; \forall int y;
@ ((y > 0) && (y <= x < (y+y))) => ((x%y) == (x-y))
@*/
/*@ type Q */
/*@ logic Q Qnew() */
/*@ logic Q Qerror() */
/*@ logic t Ierror() */
/*@ logic int QmaxSize(Q q) */
/*@ logic int Qsize(Q q) */
/*@ logic Q Qenqueue(Q q, t x) */
/*@ logic int Qempty(Q q) */
/*@ logic t Qelement(Q q, int x) */
/*@ logic t Qhead(Q q) */
/*@ logic t Qdequeue_t(Q q) */
/*@ logic Q Qdequeue_Q(Q q) */
/*@ axiom Q01 : \forall Q q; QmaxSize(q) == N */
/*@ axiom Q02 : \forall Q q; \forall t x; (N <= Qsize(q)) => (Qenqueue(q,x) == Qerror()) */
/*@ axiom Q02a : \forall Q q; \forall t x; (Qsize(q) < N) => (Qenqueue(q,x) != Qerror()) */
/*@ axiom Q02b : \forall Q q; (Qsize(q) == 0) => (Qdequeue_t(q) == Ierror()) */
/*@ axiom Q02c : \forall Q q; (0 < Qsize(q)) => !(Qdequeue_t(q) == Ierror()) */
/*@ axiom Q02d : \forall Q q; (Qsize(q) == 0) => (Qdequeue_Q(q) == Qerror()) */
/*@ axiom Q02e : \forall Q q; (0 < Qsize(q)) => (Qdequeue_Q(q) != Qerror()) */
/*@ axiom Q09 : \forall Q q; \forall int x; (x >= Qsize(q)) => (Qelement(q,x) == Ierror()) */
/*@ axiom Q03 : \forall Q q; Qsize(Qnew()) == 0 */
/*@ axiom Q04 : \forall Q q; \forall t x; (Qsize(q) < N) => Qsize(Qenqueue(q,x)) == 1+Qsize(q) */
/*@ axiom Q04a : \forall Q q; ((0 < Qsize(q))) => Qsize(Qdequeue_Q(q)) == Qsize(q) - 1 */
/*@ axiom Q05 : \forall Q q; (Qsize(q) == 0) => (Qempty(q) == 1) */
/*@ axiom Q06 : \forall Q q; (0 < Qsize(q)) => (Qempty(q) == 0) */
/*@ axiom Q08 : \forall t y; \forall Q q; \forall int x; (0 <= x < Qsize(q)) =>
@ ( Qelement(Qenqueue(q,y),x) == Qelement(q,x))
@*/
/*@ axiom Q08a : \forall Q q; \forall int x; ((0 <= x < Qsize(q) - 1) ) =>
@ (Qelement(Qdequeue_Q(q),x) == Qelement(q,x + 1))
@*/
/*@ axiom Q08b : \forall Q q; \forall int x; ((x == Qsize(q)) && (Qsize(q) < N)) =>
@ (\forall t y; Qelement(Qenqueue(q,y),x) == y)
@*/
/*@ axiom Q10 : \forall Q q; Qhead(q) == Qelement(q,0) */
/*@ axiom Q11 : \forall Q q; Qdequeue_t(q) == Qelement(q,0) */
/*@ axiom Q12 : \forall Q q; (Qempty(q) == 1) =>
@ (\forall t x; Qdequeue_Q(Qenqueue(q,x)) == Qnew())
@*/
/*@ axiom Q13 : \forall Q q; (Qempty(q) == 0) =>
@ (\forall t x; Qdequeue_Q(Qenqueue(q,x)) ==
@ Qenqueue(Qdequeue_Q(q),x))
@*/
static t queue[N];
static unsigned char q_head = 0;
static unsigned char q_tail = 0;
static unsigned char q_size = 0;
/*@ predicate models(Q q) {
@ (q != Qerror()) &&
@ (q_size == Qsize(q)) &&
@ (\forall int i; (0 <= i < q_size) => (Qelement(q,i) == queue[(i+q_head)%N]))
@ }
@*/
/*@ predicate valid_slot (uint8_t i) {
@ (i >= 0) && (i < N) &&
@ \valid_index(queue,i)
@ }
@*/
/*@ predicate occupied_slot (uint8_t i) {
@ ((q_tail > q_head) && (q_head <= i < q_tail)) ||
@ ((q_tail < q_head) && ((q_head <= i) || (i < q_tail))) ||
@ ((q_head == q_tail) && (q_size == N))
@ }
@*/
/* should say 2**(8*sizeof(uint8_t)) instead of 256 */
/*@ invariant q0 : 0 <= N < 256 */
/*@ invariant q1 : 0 <= q_size <= N */
/*@ invariant q2 : valid_slot(q_head) */
/*@ invariant q3 : valid_slot(q_tail) */
/*@ invariant size_1 : (q_tail > q_head) => (q_size == (q_tail - q_head)) */
/*@ invariant size_2 : (q_tail < q_head) => (q_size == (N + q_tail - q_head)) */
/*@ invariant size_3 : (q_tail == q_head) <=> ((q_size == 0) || (q_size == N)) */
/*@ assigns \nothing
@ ensures (\result == N) &&
@ (\forall Q q; \old(models(q)) => (\result == QmaxSize(q)))
@*/
uint8_t maxSize (void)
{
return N;
}
/*@ assigns \nothing
@ ensures (\result == q_size) &&
@ (\forall Q q; \old(models(q)) => (\result == Qsize(q)))
@*/
uint8_t size (void)
{
return q_size;
}
/*@ assigns \nothing
@ ensures ((q_size == 0) => (\result == 1)) &&
@ ((q_size != 0) => (\result == 0)) &&
@ (\forall Q q; \old(models(q)) => (\result == Qempty(q)))
@*/
bool empty (void)
{
return (q_size == 0);
}
/*@ requires q_size > 0
@ assigns \nothing
@ ensures (\forall Q q; \old(models(q)) => ((Qhead(q) == \result) &&
@ (Qhead(q) != Ierror())))
@*/
t head (void)
{
/*@ assert occupied_slot(q_head) */
return queue[q_head];
}
/*@ requires 0 <= e < q_size
@ assigns \nothing
@ ensures (\forall Q q; \old(models(q)) => ((Qelement(q,e) == \result)))
@*/
t element (uint8_t e)
{
uint8_t tmp = e + q_head;
tmp = tmp % N;
/*@ assert valid_slot(tmp) */
/*@ assert occupied_slot(tmp) */
return queue[tmp];
}
/*@ assigns queue[\old(q_tail)], q_tail, q_size
@ ensures (((\old(q_size) <N) =>
@ (
@ (\result == 1) &&
@ (q_size == (\old(q_size)+1)) &&
@ (occupied_slot(\old(q_tail))) &&
@ (\forall Q q; \old(models(q)) =>
@ (
@ (Qelement(Qenqueue(q,e),Qsize(q)) ==queue[(Qsize(q)+q_head)%N]) &&
@ (models(Qenqueue(q,e)))
@ )
@ )
@ )
@ ) &&
@ ((\old(q_size) == N) =>
@ ((\result == 0) && (q_size == \old(q_size)) && (\forall Q q; \old(models(q)) => (Qenqueue(q,e) == Qerror())))
@ )
@ )
@*/
error_t enqueue (t e)
{
if (q_size < N) {
/*@ assert !occupied_slot(q_tail) */
queue[q_tail] = e;
q_tail = q_tail + 1;
q_tail = q_tail % N;
q_size = q_size + 1;
return 1;
} else {
return 0;
}
}
/*@ requires q_size > 0
@ assigns q_head, q_size
@ ensures (\result == queue[\old(q_head)]) &&
@ (q_size == (\old(q_size)-1)) &&
@ (!occupied_slot(\old(q_head))) &&
@ (\forall Q q; \old(models(q)) => models(Qdequeue_Q(q))) &&
@ (\forall Q q; \old(models(q)) => ((Qdequeue_t(q) == \result) &&
@ (Qdequeue_t(q) != Ierror())))
@*/
t dequeue (void)
{
int tmp;
/*@ assert occupied_slot(q_head) */
tmp = queue[q_head];
if (!(q_size == 0)) {
q_head = q_head + 1;
q_head = q_head % N;
q_size = q_size - 1;
}
return tmp;
}
|
