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
|
; Copyright (C) 2017, Regents of the University of Texas
; Written by Mihir Mehta
; License: A 3-clause BSD license. See the LICENSE file distributed with ACL2.
(in-package "ACL2")
; find-n-free-blocks.lisp Mihir Mehta
(local (include-book "file-system-lemmas"))
(include-book "bounded-nat-listp")
(defthm mv-nth-replacement
(equal (mv-nth n (cons a b))
(if (zp n) a (mv-nth (- n 1) b))))
(in-theory (disable mv-nth))
(defun count-free-blocks (alv)
(declare (xargs :guard (and (boolean-listp alv))))
(if (atom alv)
0
(if (car alv)
(count-free-blocks (cdr alv))
(+ (count-free-blocks (cdr alv)) 1))))
(defthm count-free-blocks-correctness-1
(equal (count-free-blocks (binary-append x y))
(+ (count-free-blocks x)
(count-free-blocks y))))
(defthm count-free-blocks-correctness-2
(equal (count-free-blocks (revappend x y))
(+ (count-free-blocks x)
(count-free-blocks y))))
(defthm count-free-blocks-correctness-3
(implies (and (nth key l) (not val))
(equal (count-free-blocks (update-nth key val l))
(+ 1 (count-free-blocks l)))))
(defund find-n-free-blocks-helper (alv n start)
(declare (xargs :guard (and (boolean-listp alv)
(natp n)
(natp start))))
(if (or (atom alv) (zp n))
nil
(if (car alv)
;; this block is taken
(find-n-free-blocks-helper (cdr alv) n (+ start 1))
;; this block isn't taken
(cons start (find-n-free-blocks-helper (cdr alv) (- n 1) (+ start 1))))))
(defthmd
find-n-free-blocks-helper-correctness-1
(implies (and (boolean-listp alv) (natp n))
(<= (len (find-n-free-blocks-helper alv n start))
n))
:rule-classes (:rewrite :linear)
:hints (("goal" :in-theory (enable find-n-free-blocks-helper))))
(defthmd
find-n-free-blocks-helper-correctness-2
(implies (and (boolean-listp alv)
(natp n))
(equal (len (find-n-free-blocks-helper alv n start))
(min (count-free-blocks alv) n)))
:hints (("goal" :in-theory (enable find-n-free-blocks-helper))))
(defthmd
find-n-free-blocks-helper-correctness-3
(implies (and (boolean-listp alv)
(natp n)
(natp start))
(nat-listp (find-n-free-blocks-helper alv n start)))
:rule-classes (:rewrite :type-prescription)
:hints (("goal" :in-theory (enable find-n-free-blocks-helper))))
(defthmd
find-n-free-blocks-helper-correctness-4
(implies (and (natp n)
(natp start)
(member-equal m
(find-n-free-blocks-helper alv n start)))
(<= start m))
:hints (("goal" :in-theory (enable find-n-free-blocks-helper))))
(defthmd
find-n-free-blocks-helper-correctness-5
(implies (and (boolean-listp alv)
(natp n)
(natp start)
(member-equal m
(find-n-free-blocks-helper alv n start)))
(not (nth (- m start) alv)))
:hints
(("goal" :in-theory (enable find-n-free-blocks-helper
find-n-free-blocks-helper-correctness-3))
("subgoal *1/6.1'"
:use (:instance find-n-free-blocks-helper-correctness-4
(alv (cdr alv))
(start (+ 1 start))))))
(defthmd
find-n-free-blocks-helper-correctness-6
(implies (and (natp n) (natp start))
(no-duplicatesp-equal (find-n-free-blocks-helper alv n start)))
:hints (("goal" :in-theory (enable find-n-free-blocks-helper))
("subgoal *1/9''"
:in-theory (disable find-n-free-blocks-helper-correctness-4)
:use (:instance find-n-free-blocks-helper-correctness-4
(m start)
(alv (cdr alv))
(n (+ -1 n))
(start (+ 1 start))))))
(defthmd
find-n-free-blocks-helper-correctness-7
(implies (and (boolean-listp alv)
(natp n)
(natp start))
(bounded-nat-listp (find-n-free-blocks-helper alv n start)
(+ start (len alv))))
:hints (("goal" :in-theory (enable find-n-free-blocks-helper))))
(defund find-n-free-blocks (alv n)
(declare (xargs :guard (and (boolean-listp alv)
(natp n))))
(find-n-free-blocks-helper alv n 0))
;; Here are some examples showing how this works.
;; ACL2 !>(find-n-free-blocks (list t nil t) 1)
;; (1)
;; ACL2 !>(find-n-free-blocks (list t nil t) 2)
;; (1)
;; ACL2 !>(find-n-free-blocks (list t nil nil) 2)
;; (1 2)
(defthm
find-n-free-blocks-correctness-1
(implies (and (boolean-listp alv) (natp n))
(<= (len (find-n-free-blocks alv n)) n))
:rule-classes (:rewrite :linear)
:hints
(("goal" :in-theory (enable find-n-free-blocks
find-n-free-blocks-helper-correctness-1))))
(defthm
find-n-free-blocks-correctness-2
(implies (and (boolean-listp alv)
(natp n))
(equal (len (find-n-free-blocks alv n))
(min (count-free-blocks alv) n)))
:hints
(("goal" :in-theory (enable find-n-free-blocks
find-n-free-blocks-helper-correctness-2))))
(defthm
find-n-free-blocks-correctness-3
(implies (and (boolean-listp alv) (natp n))
(nat-listp (find-n-free-blocks alv n)))
:rule-classes (:rewrite :type-prescription)
:hints
(("goal" :in-theory (enable find-n-free-blocks
find-n-free-blocks-helper-correctness-3))))
(defthm
find-n-free-blocks-correctness-5
(implies (and (member-equal m (find-n-free-blocks alv n))
(boolean-listp alv)
(natp n))
(not (nth m alv)))
:rule-classes (:forward-chaining)
:hints (("goal" :in-theory (enable find-n-free-blocks)
:use (:instance find-n-free-blocks-helper-correctness-5
(start 0)))))
(defthm
find-n-free-blocks-correctness-6
(implies (and (natp n))
(no-duplicatesp-equal (find-n-free-blocks alv n)))
:hints
(("goal" :in-theory (enable find-n-free-blocks
find-n-free-blocks-helper-correctness-6))))
(defthm
find-n-free-blocks-correctness-7
(implies (and (boolean-listp alv)
(natp n)
(equal m (len alv)))
(bounded-nat-listp (find-n-free-blocks alv n)
m))
:hints (("goal" :in-theory (enable find-n-free-blocks)
:use (:instance find-n-free-blocks-helper-correctness-7
(start 0)))))
(defun count-free-blocks-alt (alv n)
(declare (xargs :guard (and (natp n) (boolean-listp alv))))
(if (zp n)
0
(+ (if (nth (- n 1) alv) 0 1)
(count-free-blocks-alt alv (- n 1)))))
(defthm
count-free-blocks-alt-correctness-1
(implies (and (boolean-listp alv)
(boolean-listp ac)
(natp n)
(<= n (len alv)))
(equal (count-free-blocks-alt (revappend ac alv)
(+ n (len ac)))
(count-free-blocks (first-n-ac n alv ac))))
:hints (("goal" :induct (first-n-ac n alv ac))))
(defthm
count-free-blocks-alt-correctness-2
(implies (and (boolean-listp alv)
(equal n (len alv)))
(equal (count-free-blocks-alt alv n)
(count-free-blocks alv)))
:hints (("goal" :in-theory (disable count-free-blocks-alt-correctness-1)
:use (:instance count-free-blocks-alt-correctness-1
(ac nil)))))
(defun count-before-n (l b)
(declare (xargs :guard (and (natp b) (nat-listp l))))
(if (atom l) 0 (+ (if (< (car l) b) 1 0) (count-before-n (cdr l) b))))
(defthm count-before-n-correctness-1
(<= (count-before-n l b) (len l))
:rule-classes (:linear))
(defthm count-before-n-correctness-2
(implies (nat-listp l)
(iff (equal (count-before-n l b) (len l))
(bounded-nat-listp l b))))
(defthm count-before-n-correctness-3
(implies (nat-listp l)
(equal (count-before-n l 0) 0)))
|
