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
|
(in-package "ACL2")
; block-listp.lisp Mihir Mehta
; Here we define the size of a block to be 8 characters, and define functions
; for making blocks from text and retrieving text from blocks, with proofs of
; their correctness and their one-way inverse relationship.
(local (include-book "../file-system-lemmas"))
;; I don't think blocks are 8 characters long in any system; I simply set this
;; in order to actually get fragmentation without having to make unreasonably
;; large examples.
(defconst *blocksize* 8)
;; This fragments a character-list into blocks that are *blocksize*-character
;; long. If the character-list is not exactly aligned to a block boundary, we
;; fill the space with null characters.
;; It will be used in wrchs.
(defund make-blocks (text)
(declare (xargs :guard (character-listp text)
:measure (len text)))
(if (atom text)
nil
(cons (make-character-list (take *blocksize* text))
(make-blocks (nthcdr *blocksize* text)))))
(defthm
make-blocks-correctness-5
(iff (consp (make-blocks text))
(consp text))
:rule-classes
(:rewrite
(:rewrite
:corollary (iff (equal (len (make-blocks text)) 0)
(atom text))
:hints (("goal''" :expand (len (make-blocks text))))))
:hints (("goal" :in-theory (enable make-blocks))))
;; Characterisation of a disk, which is a list of blocks as described before.
(defun block-listp (block-list)
(declare (xargs :guard t))
(if (atom block-list)
(eq block-list nil)
(and (character-listp (car block-list))
(equal (len (car block-list)) *blocksize*)
(block-listp (cdr block-list)))))
;; Proving that we get a proper block-list out of make-blocks.
(defthm make-blocks-correctness-2
(implies (character-listp text)
(block-listp (make-blocks text)))
:hints (("Goal" :in-theory (enable make-blocks))))
(defthm block-listp-correctness-1
(implies (block-listp block-list)
(true-listp block-list))
:rule-classes (:forward-chaining))
(defthm block-listp-correctness-2
(implies (true-listp block-list1)
(equal (block-listp (binary-append block-list1 block-list2))
(and (block-listp block-list1)
(block-listp block-list2)))))
;; This function spells out how many characters can be in a file given the
;; number of blocks associated with it. It is kept disabled in order to avoid
;; huge arithmetic-heavy subgoals where they're not wanted.
(defund feasible-file-length-p (index-list-length file-length)
(declare (xargs :guard (and (natp file-length) (natp index-list-length))))
(and (> file-length
(* *blocksize* (- index-list-length 1)))
(<= file-length
(* *blocksize* index-list-length))))
;; This is the counterpart of make-blocks that collapses blocks into a
;; character-list of the appropriate length.
;; It will be used in stat and, by extension, in rdchs.
(defun
unmake-blocks (blocks n)
(declare
(xargs
:guard (and (block-listp blocks)
(natp n)
(feasible-file-length-p (len blocks) n))
:guard-hints
(("goal" :in-theory (enable feasible-file-length-p)))))
(if (atom blocks)
nil
(if (atom (cdr blocks))
(take n (car blocks))
(binary-append (car blocks)
(unmake-blocks (cdr blocks)
(- n *blocksize*))))))
;; Proving that we get a proper character-list out provided we don't ask for
;; more characters than we have.
(defthm unmake-blocks-correctness-1
(implies (and (block-listp blocks)
(natp n)
(feasible-file-length-p (len blocks) n))
(character-listp (unmake-blocks blocks n)))
:hints (("Goal" :in-theory (enable feasible-file-length-p)) ))
(defthm
unmake-blocks-correctness-2
(implies (and (block-listp blocks)
(natp n)
(feasible-file-length-p (len blocks) n))
(equal (len (unmake-blocks blocks n))
n))
:rule-classes
((:rewrite :corollary (implies (and (block-listp blocks)
(natp n)
(feasible-file-length-p (len blocks) n))
(iff (consp (unmake-blocks blocks n))
(not (zp n))))))
:hints (("goal" :in-theory (enable feasible-file-length-p))
("subgoal *1/5'''" :expand (len (cdr blocks)))))
(defthm unmake-make-blocks-lemma-1
(implies (natp n)
(iff (consp (nthcdr n l)) (> (len l) n)))
:hints (("Goal" :induct (nthcdr n l))))
(encapsulate ()
(local (include-book "std/lists/repeat" :dir :system))
;; Proving that make and unmake are, in a sense, inverse functions of each
;; other.
(defthm
unmake-make-blocks
(implies (and (character-listp text))
(equal (unmake-blocks (make-blocks text)
(len text))
text))
:hints
(("goal" :in-theory (enable make-blocks))
("subgoal *1/3.3"
:in-theory (disable take-of-make-character-list
take-of-too-many)
:use ((:instance take-of-make-character-list
(i (len text))
(l (first-n-ac 8 text nil)))
(:instance take-of-too-many (x text)
(n *blocksize*)))))))
;; This is a constant that might be needed later.
;; This is to be returned when a block is not found. It's full of null
;; characters and is *blocksize* long.
(defconst *nullblock* (make-character-list (take *blocksize* nil)))
(defthm
make-blocks-correctness-1
(implies (character-listp text)
(and (< (- (* *blocksize* (len (make-blocks text)))
*blocksize*)
(len text))
(not (< (* *blocksize* (len (make-blocks text)))
(len text)))))
:hints (("goal" :in-theory (enable make-blocks)
:induct t)))
(defthm
make-blocks-correctness-3
(implies (and (character-listp cl))
(feasible-file-length-p (len (make-blocks cl))
(len cl)))
:hints
(("goal"
:in-theory (e/d (feasible-file-length-p)
(make-blocks-correctness-1))
:use (:instance make-blocks-correctness-1 (text cl)))))
(encapsulate ()
(local (include-book "std/lists/nthcdr" :dir :system))
(local (defun nthcdr-*blocksize*-induction-2 (text1 text2)
(cond ((or (atom text1) (atom text2))
(list text1 text2))
(t (nthcdr-*blocksize*-induction-2 (nthcdr *blocksize* text1)
(nthcdr *blocksize* text2))))))
(defthm
make-blocks-correctness-4
(implies (equal (len text1) (len text2))
(equal (len (make-blocks text1))
(len (make-blocks text2))))
:hints (("goal" :in-theory (enable make-blocks)
:induct (nthcdr-*blocksize*-induction-2 text1 text2)))))
(defthm
len-of-make-unmake
(implies (and (block-listp blocks)
(natp n)
(feasible-file-length-p (len blocks) n))
(equal (len (make-blocks (unmake-blocks blocks n)))
(len blocks)))
:hints
(("goal" :in-theory (e/d (make-blocks feasible-file-length-p)
(consp-of-append)))
("subgoal *1/8"
:expand (append (car blocks)
(unmake-blocks (cdr blocks) (+ -8 n))))
("subgoal *1/5.2" :cases ((atom (cdr blocks)))
:expand (len (cdr blocks)))
("subgoal *1/5.1'"
:expand (append (car blocks)
(unmake-blocks (cdr blocks) (+ -8 n))))
("subgoal *1/2"
:expand (make-blocks (first-n-ac n (car blocks) nil)))))
|
