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
|
(in-package "ACL2")
(local ; ACL2 primitive
(defun natp (x)
(declare (xargs :guard t))
(and (integerp x)
(<= 0 x))))
(local (include-book "bvecp-proofs"))
;; New stuff:
(defund bvecp (x k)
(declare (xargs :guard (integerp k)))
(and (integerp x)
(<= 0 x)
(< x (expt 2 k))))
(defthm bvecp-with-n-not-a-positive-integer
(implies (or (not (integerp k))
(<= k 0))
(equal (bvecp x k)
(equal 0 x))))
(defthm bvecp-0
(bvecp 0 k))
;drop?
;just a special case of bvecp-with-n-not-a-positive-integer
(defthm bvecp-0-thm
(equal (bvecp x 0)
(equal x 0)))
(defthm bvecp-ones
(implies (case-split (<= 0 k))
(bvecp (1- (expt 2 k)) k)))
;k1 is a free var
(defthm bvecp-longer
(implies (and (bvecp x k1)
(<= k1 k2)
(case-split (integerp k2))
)
(bvecp x k2))
:rule-classes ((:rewrite :match-free :all)))
;expensive and so disabled
;no free var
(defthmd bvecp-one-longer
(implies (and (integerp k)
(bvecp x (- k 1)))
(bvecp x k))
:rule-classes ((:rewrite :backchain-limit-lst (nil 2))))
(defthm bvecp-of-non-integer
(implies (not (integerp x))
(not (bvecp x k))))
;gen (replace n+1 with an arbitrary integer > n)?
(defthm bvecp-expt-2-n
(implies (and (case-split (integerp n))
(case-split (<= 0 n))
)
(bvecp (expt 2 n) (+ 1 n))))
;Can help in back-chaining (sometimes ACL2 will refuse to split an IF during backchaining). Imagine that ACL2
;backchains to (bvecp (if test x y) k) and we know (bvecp x k) and (bvecp y k). ACL2 may fail to relieve the
;hyp because it refuses to split into cases (due to some heuristics that limit the cost of backchaining). But
;if this rule fires, both (bvecp x k) and (bvecp y k) can rewrite to T, and the hyp can be relieved. (At
;least, I think that's how this works.)
(defthm bvecp-if
(equal (bvecp (if test x y) k)
(if test (bvecp x k) (bvecp y k))))
; The following are analogous to mk-bvarr etc. in rtlarr.lisp.
;better name?
(defund mk-bvec (r k)
(declare (xargs :guard (integerp k)))
(if (bvecp r k) r 0))
(defthm mk-bvec-is-bvecp
(bvecp (mk-bvec r k) k))
(defthm mk-bvec-identity
(implies (bvecp r k)
(equal (mk-bvec r k) r)))
;BOZO make a version to shift by a constant!
(defthm bvecp-shift
(implies (and (integerp x) ;note!
(<= 0 m)
(case-split (integerp m))
(case-split (integerp n))
)
(equal (bvecp (* x (expt 2 m)) n)
(bvecp x (- n m)))))
(defthm bvecp-shift-alt
(implies (and (integerp x) ;note!
(<= 0 m)
(case-split (integerp m))
(case-split (integerp n))
)
(equal (bvecp (* (expt 2 m) x) n)
(bvecp x (- n m)))))
;gen this!
;BOZO will this unify (* 2 x) with 0??
(defthm bvecp-shift-by-2
(implies (and (syntaxp (not (quotep x))) ;prevents loops...
(integerp x)
(<= 0 m) ;gen?
(case-split (integerp m))
(case-split (integerp n))
)
(equal (bvecp (* 2 x) n)
(bvecp x (- n 1)))))
;gen?
;in general, rewrite (bvecp k n) where k is a constant to a fact about n
(defthm bvecp-1
(implies (and (<= 1 n)
(integerp n))
(bvecp 1 n)))
;n is a free variable
;Disabled since may cause expensive backchaining.
(defthmd natp-bvecp
(implies (bvecp x n)
(natp x))
:rule-classes ((:rewrite :match-free :once)))
(defthmd bvecp-forward
(implies (bvecp x k)
(and (integerp x)
(<= 0 x)
(< x (expt 2 k)))) ;tigher-bound?
:rule-classes :forward-chaining)
(defthm bvecp-product
(implies (and (bvecp x m)
(bvecp y n)
)
(bvecp (* x y) (+ m n)))
:rule-classes ())
(defthmd bvecp-1-rewrite
(equal (bvecp x 1)
(or (equal x 0) (equal x 1))))
;make another for not-equal-0 implies equal-1?
(defthm bvecp-1-0
(implies (and (bvecp x 1)
(not (equal x 1)))
(equal x 0))
:rule-classes :forward-chaining)
(defthm bvecp+1
(implies (and (natp n)
(bvecp x n))
(bvecp x (+ 1 n))))
;same as bvecp-longer.decide which param names to use. j and k??
(defthmd bvecp-monotone
(implies (and (bvecp x n)
(<= n m)
(case-split (integerp m))
)
(bvecp x m)))
;This bounds the amount of carry out that we can have from the sum.
(defthm bvecp-sum-of-bvecps
(implies (and (bvecp x (1- k))
(bvecp y (1- k))
(case-split (integerp k)))
(bvecp (+ x y) k)))
;add rule that (not (natp x)) implies (not (bvecp x k)) ??
;exported in lib/
(defthmd bvecp-0-1
(implies (and (bvecp x 1)
(not (equal x 0)))
(equal x 1))
:rule-classes :forward-chaining)
|
