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
|
; Copyright (C) 2024, Matt Kaufmann
; Written by Matt Kaufmann
; License: A 3-clause BSD license. See the LICENSE file distributed with ACL2.
; See README.txt for an overview of this example.
(in-package "ACL2")
(defstobj mem{ht}$c ; concrete stobj representing memory with a hash table
(mem{ht}$c-ht :type (hash-table eql)))
(defun list-alist-subsetp (index lst alist)
; At the top level, index is 0 and this predicate defines when a list, lst
; (representing an array) corresponds to an alist, alist (representing a hash
; table). It leaves open the possibility that the hash table has "junk" not
; corresponding to pairs <i,v> where v is the value at position i in lst
; (again, assuming we are at the top level).
(declare (xargs :measure (len lst)))
(cond ((atom lst) t)
(t (and (equal (car lst)
(cdr (hons-assoc-equal index alist)))
(list-alist-subsetp (1+ index) (cdr lst) alist)))))
(defun-nx mem{ht}$corr (mem{ht}$c x)
(declare (xargs :stobjs mem{ht}$c
:verify-guards nil))
(list-alist-subsetp 0 x (nth 0 mem{ht}$c)))
(include-book "logic")
(defun lookup{ht}$c (k mem{ht}$c)
; This is a little wrapper for mem{ht}$c-ht-get.
; We use cw to print evidence that mem{ht}$c-ht-get is being called.
(declare (xargs :stobjs mem{ht}$c
:guard (eqlablep k)))
(prog2$ (cw "@@ Calling: ~x0~|" `(mem{ht}$c-ht-get ,k mem{ht}$c))
(mem{ht}$c-ht-get k mem{ht}$c)))
(defun update{ht}$c (k v mem{ht}$c)
; This is a little wrapper for mem{ht}$c-ht-put.
; We use cw to print evidence that mem{ht}$c-ht-put is being called.
(declare (xargs :stobjs mem{ht}$c
:guard (eqlablep k)))
(prog2$ (cw "@@ Calling: ~x0~|" `(mem{ht}$c-ht-put ,k ,v mem{ht}$c))
(mem{ht}$c-ht-put k v mem{ht}$c)))
(progn
; The events below include the events that are automatically generated by the
; defabsstobj event below, which are all in upper case. Their proofs are
; supported by the lower-case events, which are all local.
(DEFTHM CREATE-MEM{HT}{CORRESPONDENCE}
(MEM{HT}$CORR (CREATE-MEM{HT}$C)
(CREATE-MEM$A))
:RULE-CLASSES NIL)
(DEFTHM CREATE-MEM{HT}{PRESERVED}
(MEM$AP (CREATE-MEM$A))
:RULE-CLASSES NIL)
(local (defthm list-alist-subsetp-works-1
(implies (and (natp index)
(natp k)
(< k (len lst))
(list-alist-subsetp index lst alist))
(equal (nth k lst)
(cdr (hons-assoc-equal (+ index k) alist))))
:rule-classes nil))
(local (defthm list-alist-subsetp-works
(implies (and (natp k)
(< k (len lst))
(list-alist-subsetp 0 lst alist))
(equal (equal (cdr (hons-assoc-equal k alist))
(nth k lst))
t))
:hints (("Goal" :use ((:instance list-alist-subsetp-works-1
(index 0)))))))
(DEFTHM LOOKUP{HT}{CORRESPONDENCE}
(IMPLIES (AND (MEM{HT}$CORR MEM{HT}$C MEM{HT})
(MEM-INDEXP K)
(MEM$AP MEM{HT}))
(EQUAL (LOOKUP{HT}$C K MEM{HT}$C)
(LOOKUP$A K MEM{HT})))
:hints (("Goal" :in-theory (disable hons-assoc-equal)))
:RULE-CLASSES NIL)
(DEFTHM LOOKUP{HT}{GUARD-THM}
(IMPLIES (AND (MEM{HT}$CORR MEM{HT}$C MEM{HT})
(MEM-INDEXP K)
(MEM$AP MEM{HT}))
(EQLABLEP K))
:RULE-CLASSES NIL)
(local (defun list-alist-subsetp-preserved-1-induction (index lst alist k)
(cond ((zp k) (list index lst alist k)) ; avoid irrelevant formals
((atom lst) t)
(t (list-alist-subsetp-preserved-1-induction
(1+ index) (cdr lst) alist (1- k))))))
(local (defthm list-alist-subsetp-preserved-1-1
(implies (and (integerp index)
(natp n)
(< n index)
(list-alist-subsetp index lst alist))
(list-alist-subsetp index
lst
(cons (cons n v) alist)))))
(local (defthm list-alist-subsetp-preserved-1
(implies (and (natp index)
(list-alist-subsetp index lst alist)
(integerp k)
(<= 0 k)
(< k (len lst)))
(list-alist-subsetp index
(update-nth k v lst)
(cons (cons (+ index k) v) alist)))
:hints (("Goal" :induct (list-alist-subsetp-preserved-1-induction
index lst alist k)))
:rule-classes nil))
(local (defthm list-alist-subsetp-preserved
(implies (and (list-alist-subsetp 0 lst alist)
(integerp k)
(<= 0 k)
(< k (len lst)))
(list-alist-subsetp 0
(update-nth k v lst)
(cons (cons k v) alist)))
:hints (("Goal" :use ((:instance list-alist-subsetp-preserved-1
(index 0)))))))
(DEFTHM UPDATE{HT}{CORRESPONDENCE}
(IMPLIES (AND (MEM{HT}$CORR MEM{HT}$C MEM{HT})
(MEM-INDEXP K)
(MEM$AP MEM{HT}))
(MEM{HT}$CORR (UPDATE{HT}$C K V MEM{HT}$C)
(UPDATE$A K V MEM{HT})))
:RULE-CLASSES NIL)
(DEFTHM UPDATE{HT}{GUARD-THM}
(IMPLIES (AND (MEM{HT}$CORR MEM{HT}$C MEM{HT})
(MEM-INDEXP K)
(MEM$AP MEM{HT}))
(EQLABLEP K))
:RULE-CLASSES NIL)
(DEFTHM UPDATE{HT}{PRESERVED}
(IMPLIES (AND (MEM-INDEXP K) (MEM$AP MEM{HT}))
(MEM$AP (UPDATE$A K V MEM{HT})))
:RULE-CLASSES NIL)
)
(defabsstobj mem{ht} ; abstract stobj, based on a hash table
:foundation mem{ht}$c
:recognizer (mem{ht}p :logic mem$ap :exec mem{ht}$cp)
:creator (create-mem{ht} :logic create-mem$a :exec create-mem{ht}$c)
:corr-fn mem{ht}$corr
:exports ((lookup{ht} :logic lookup$a :exec lookup{ht}$c)
(update{ht} :logic update$a :exec update{ht}$c)))
; This little test at certification time is skipped by include-book:
(value-triple (update{ht} 4 'four mem{ht})
:stobjs-out '(mem{ht}))
; This little test at certification time is skipped by include-book:
(assert-event (equal (lookup{ht} 4 mem{ht})
'four))
|
