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
|
(in-package "ACL2")
(local ; ACL2 primitive
(defun natp (x)
(declare (xargs :guard t))
(and (integerp x)
(<= 0 x))))
(defund bvecp (x k)
(declare (xargs :guard (integerp k)))
(and (integerp x)
(<= 0 x)
(< x (expt 2 k))))
(defund fl (x)
(declare (xargs :guard (real/rationalp x)))
(floor x 1))
(defun bits (x i j)
(declare (xargs :guard (and (natp x)
(natp i)
(natp j))
:verify-guards nil))
(mbe :logic (if (or (not (integerp i))
(not (integerp j)))
0
(fl (/ (mod x (expt 2 (1+ i))) (expt 2 j))))
:exec (if (< i j)
0
(logand (ash x (- j)) (1- (ash 1 (1+ (- i j))))))))
(defund bitn (x n)
(declare (xargs :guard (and (natp x)
(natp n))
:verify-guards nil))
(mbe :logic (bits x n n)
:exec (if (evenp (ash x (- n))) 0 1)))
(local (include-book "lnot-proofs"))
#| old definition:
(defun LNOT (x n)
(1- (- (expt 2 n) x)))
|#
;used to be called COMP1
(defund lnot (x n)
(declare (xargs :guard (and (natp x)
(integerp n)
(< 0 n))
:verify-guards nil))
(if (natp n)
(+ -1 (expt 2 n) (- (bits x (1- n) 0)))
0))
;note that this isn't a rewrite rule b/c we believe it will never need to be
;BOZO make it one anyway?
(defthm lnot-nonnegative-integer-type
(and (integerp (lnot x n))
(<= 0 (lnot x n)))
:rule-classes ((:type-prescription :typed-term (lnot x n))))
;lnot-nonnegative-integer-type is strictly better, and we don't need both
(in-theory (disable (:type-prescription lnot)))
(defthm lnot-natp
(natp (lnot x n)))
(defthm lnot-upper-bound
(< (lnot x n) (expt 2 n))
:rule-classes (:rewrite :linear)
)
;why is bvecp enabled here?
(defthm lnot-bvecp-simple
(bvecp (lnot x n) n))
(defthm lnot-bvecp
(implies (and (<= n k)
(case-split (integerp k)))
(bvecp (lnot x n) k)))
;perhaps conclude with bits of x and drop the bvecp hyp?
(defthm lnot-lnot
(implies (and (case-split (natp n))
(case-split (bvecp x n))
)
(equal (lnot (lnot x n) n)
x)))
;reorient this rule?
(defthmd lnot-times-2
(implies (and (case-split (natp x))
(case-split (natp n))
)
(equal (+ 1 (* 2 (lnot x n)))
(lnot (* 2 x) (1+ n)))))
(defthm lnot-with-n-0
(equal (lnot x 0)
0))
;gen?
;make a by-2 version?
;change param name?
;make a better rewrite rule
;RHS isn't simplified!
(defthmd lnot-fl
(implies (and (<= k n)
;(bvecp x n)
(<= 0 k)
(integerp n)
(integerp k)
)
(equal (fl (* (/ (expt 2 k)) (lnot x n)))
(lnot (fl (/ x (expt 2 k))) (- n k)))))
;gen
;add case-splits
(defthm mod-lnot
(implies (and (<= n m)
; (integerp x)
(integerp n)
(<= 0 n) ;gen
(integerp m)
)
(equal (mod (lnot x m) (expt 2 n))
(lnot (mod x (expt 2 n)) n))))
(defthm mod-lnot-by-2
(implies (and (< 0 n)
(integerp x) ;gen?
(integerp n)
)
(equal (mod (lnot x n) 2)
(lnot (mod x 2) 1))))
;disable?
(defthm lnot-ignores-bits
(equal (lnot (bits x (1- n) 0) n)
(lnot x n)))
(defthmd lnot-ignores-bits-2
(implies (and (integerp i)
(<= (1- n) i))
(equal (lnot (bits x i 0) n)
(lnot x n))))
;gen?
;formal m should be n
(defthm bits-lnot
(implies (and (< i m)
(case-split (natp j))
(case-split (integerp m))
(case-split (integerp i))
)
(equal (bits (lnot x m) i j)
(lnot (bits x i j) (1+ (- i j))))))
;gen?
(defthm bitn-lnot
(implies (and (case-split (natp m))
(case-split (natp n))
)
(equal (bitn (lnot x m) n)
(if (< n m)
(lnot (bitn x n) 1)
0))))
;do we still need this, given bitn-lnot?
(defthm bitn-lnot-not-equal
(implies (and (< k n)
(integerp n)
(<= 0 n)
(integerp k)
(<= 0 k)
)
(not (= (bitn (lnot x n) k)
(bitn x k))))
:rule-classes ())
;could generalize these a lot (when lnot equals a constant, take the lnot of both sides)
;drop bvecp hyp by wrapping bits around conclusion?
(defthm lnot-bvecp-equal-0
(implies (case-split (bvecp x 1))
(equal (equal (lnot x 1) 0)
(not (equal x 0)))))
(defthm lnot-bvecp-equal-1
(implies (case-split (bvecp x 1))
(equal (equal (lnot x 1) 1)
(equal x 0))))
;consider enabling?
(defthmd lnot-ignores-mod-special
(equal (lnot (mod x (expt 2 m)) m)
(lnot x m)))
;consider enabling?
(defthmd lnot-ignores-mod
(implies (and (<= m n)
(case-split (integerp n)))
(equal (lnot (mod x (expt 2 n)) m)
(lnot x m))))
;rename to shift-by-2?
;consider enabling?
(defthmd lnot-shift
(implies (and (syntaxp (not (quotep x))) ;prevents loops
(case-split (integerp x))
(case-split (< 0 n))
(case-split (integerp n))
)
(equal (lnot (* 2 x) n)
(+ 1 (* 2 (lnot x (1- n)))))))
;disable?
;BOZO rename the other lnot-fl. this one should be called lnot-fl.
(defthm lnot-fl-eric
(equal (lnot (fl x) n)
(lnot x n)))
(defthm lnot-with-n-not-an-integer
(implies (not (integerp n))
(equal (lnot x n)
0)))
(defthm lnot-with-n-not-positive
(implies (<= n 0)
(equal (lnot x n)
0)))
|
