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
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
|
; Copyright (C) 2012 Centaur Technology
;
; Contact:
; Centaur Technology Formal Verification Group
; 7600-C N. Capital of Texas Highway, Suite 300, Austin, TX 78731, USA.
; http://www.centtech.com/
;
; License: (An MIT/X11-style license)
;
; Permission is hereby granted, free of charge, to any person obtaining a
; copy of this software and associated documentation files (the "Software"),
; to deal in the Software without restriction, including without limitation
; the rights to use, copy, modify, merge, publish, distribute, sublicense,
; and/or sell copies of the Software, and to permit persons to whom the
; Software is furnished to do so, subject to the following conditions:
;
; The above copyright notice and this permission notice shall be included in
; all copies or substantial portions of the Software.
;
; THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
; IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
; FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
; AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
; LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
; FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
; DEALINGS IN THE SOFTWARE.
;
; Original author: Sol Swords <sswords@centtech.com>
(in-package "ACL2")
; cert_param: (non-acl2r)
(include-book "system/sublis-var" :dir :system)
(include-book "tools/flag" :dir :system)
(include-book "tools/mv-nth" :dir :system)
(include-book "tools/defevaluator-fast" :dir :system)
;; Prove CONS-TERM and CONS-TERM1-MV2 are simply applying the fn to the args.
(defevaluator-fast cterm-ev cterm-ev-lst
((acl2-numberp x)
(binary-* x y)
(binary-+ x y)
(unary-- x)
(unary-/ x)
(< x y)
(car x)
(cdr x)
(char-code x)
(characterp x)
(code-char x)
(complex x y)
(complex-rationalp x)
(coerce x y)
(cons x y)
(consp x)
(denominator x)
(equal x y)
(imagpart x)
(integerp x)
(intern-in-package-of-symbol x y)
(numerator x)
(rationalp x)
(realpart x)
(stringp x)
(symbol-name x)
(symbol-package-name x)
(symbolp x)
(if x y z)
(not x))
:namedp t)
(defthm cterm-ev-cons-term-correct
(equal (cterm-ev (cons-term fn args) a)
(cterm-ev (cons fn args) a))
:hints ((and stable-under-simplificationp
'(:expand ((quote-listp (cdr args))
(quote-listp (cddr args)))))))
(defthm cterm-ev-cons-term1-mv2-correct
(implies (and (quote-listp args)
(equal (cterm-ev form a)
(cterm-ev (cons fn args) a)))
(mv-let (flg result)
(cons-term1-mv2 fn args form)
(declare (ignore flg))
(equal (cterm-ev result a)
(cterm-ev (cons fn args) a))))
:hints (("goal" :do-not-induct t)
(and stable-under-simplificationp
'(:expand ((quote-listp (cdr args))
(quote-listp (cddr args)))))))
(defthm cons-term1-mv2-when-unchanged
(mv-let (changedp result)
(cons-term1-mv2 fn args form)
(implies (not changedp)
(equal result form))))
(defthm pseudo-termp-cons-term
(implies (and (symbolp fn)
(not (eq fn 'quote))
(pseudo-term-listp args))
(pseudo-termp (cons-term fn args))))
(local ;; because this is gross
(defthm pseudo-termp-cons-term-lambda
(implies (and (consp fn)
(consp (cdr fn))
(true-listp (cddr fn))
(equal (+ 2 (len (cddr fn))) 3) ;; gross
(eq (car fn) 'lambda)
(symbol-listp (cadr fn))
(pseudo-termp (caddr fn))
(equal (len args) (len (cadr fn)))
(pseudo-term-listp args))
(pseudo-termp (cons-term fn args)))))
(defthm pseudo-termp-cons-term1-mv2
(implies (and (symbolp fn)
(not (eq fn 'quote))
(pseudo-term-listp args)
(pseudo-termp x))
(pseudo-termp (mv-nth 1 (cons-term1-mv2 fn args x)))))
(in-theory (disable cons-term cons-term1-mv2))
; Goal now is to prove that sublis-var1 evaluates to the same thing as this
; much simpler function. (Unfortunately we can't use substitute-into-term from
; unify-subst because it behaves differently on unbound variables.)
(mutual-recursion
(defun term-subst (x alist)
(declare (xargs :guard (and (symbol-alistp alist)
(pseudo-termp x))))
(cond ((variablep x)
(let ((a (assoc-eq x alist)))
(if a (cdr a) x)))
((fquotep x) x)
(t (cons (ffn-symb x)
(termlist-subst (fargs x) alist)))))
(defun termlist-subst (x alist)
(declare (xargs :guard (and (symbol-alistp alist)
(pseudo-term-listp x))))
(if (endp x)
x
(cons (term-subst (car x) alist)
(termlist-subst (cdr x) alist)))))
(make-flag term-subst-flg term-subst
:flag-mapping ((term-subst term)
(termlist-subst list)))
;; (defthm-term-subst-flg
;; (defthm term-subst-when-unchanged
;; (implies (not (mv-nth 0 (sublis-var1 alist x)))
;; (equal (cterm-ev (term-subst x alist) a)
;; (cterm-ev x a)))
;; :hints ((and stable-under-simplificationp
;; '(:in-theory (enable cterm-ev-of-fncall-args))))
;; :flag term)
;; (defthm sublis-var1lst-is-termlist-subst
;; (implies (not (mv-nth 0 (sublis-var1-lst alist x)))
;; (equal (cterm-ev-lst (termlist-subst x alist) a)
;; (cterm-ev-lst x a)))
;; :flag list))
(defthm-term-subst-flg
(defthm sublis-var1-when-unchanged
(implies (not (mv-nth 0 (sublis-var1 alist x)))
(equal (mv-nth 1 (sublis-var1 alist x))
x))
:flag term)
(defthm sublis-var1-lst-when-unchanged
(implies (not (mv-nth 0 (sublis-var1-lst alist x)))
(equal (mv-nth 1 (sublis-var1-lst alist x))
x))
:flag list))
(defthm len-of-sublis-var1-lst
(equal (len (mv-nth 1 (sublis-var1-lst alist x)))
(len x)))
(defthm-term-subst-flg
(defthm pseudo-termp-of-sublis-var1
(implies (and (pseudo-termp x)
(pseudo-term-listp (strip-cdrs alist)))
(pseudo-termp (mv-nth 1 (sublis-var1 alist x))))
:hints ('(:expand ((pseudo-termp x)
(sublis-var1 alist x))))
:flag term)
(defthm pseudo-term-listp-of-sublis-var1-lst
(implies (and (pseudo-term-listp x)
(pseudo-term-listp (strip-cdrs alist)))
(pseudo-term-listp (mv-nth 1 (sublis-var1-lst alist x))))
:flag list))
(defthm pseudo-termp-of-sublis-var
(implies (and (pseudo-termp x)
(pseudo-term-listp (strip-cdrs alist)))
(pseudo-termp (sublis-var alist x)))
:hints(("Goal" :in-theory (enable sublis-var))))
(defthm-term-subst-flg
(defthm sublis-var1-is-term-subst
(equal (cterm-ev (mv-nth 1 (sublis-var1 alist x)) a)
(cterm-ev (term-subst x alist) a))
:hints ((and stable-under-simplificationp
'(:in-theory (enable cterm-ev-of-fncall-args))))
:flag term)
(defthm sublis-var1-lst-is-termlist-subst
(equal (cterm-ev-lst (mv-nth 1 (sublis-var1-lst alist x)) a)
(cterm-ev-lst (termlist-subst x alist) a))
:flag list))
(defthm len-of-termlist-subst
(equal (len (termlist-subst x alist))
(len x))
:hints (("goal" :induct (len x)
:expand ((termlist-subst x alist)))))
(defthm pseudo-termp-of-lookup
(implies (and (pseudo-term-listp (strip-cdrs alist))
(assoc x alist))
(pseudo-termp (cdr (assoc x alist)))))
(defthm-term-subst-flg
(defthm pseudo-termp-of-term-subst
(implies (and (pseudo-termp x)
(pseudo-term-listp (strip-cdrs alist)))
(pseudo-termp (term-subst x alist)))
:hints ((and stable-under-simplificationp
'(:expand ((:free (x y) (pseudo-termp (cons x y)))))))
:flag term)
(defthm pseudo-term-listp-of-termlist-subst
(implies (and (pseudo-term-listp x)
(pseudo-term-listp (strip-cdrs alist)))
(pseudo-term-listp (termlist-subst x alist)))
:flag list))
(defun cterm-ev-alist (x a)
(if (atom x)
nil
(cons (cons (caar x) (cterm-ev (cdar x) a))
(cterm-ev-alist (cdr x) a))))
(defthm lookup-in-cterm-ev-alist
(implies k
(equal (assoc k (cterm-ev-alist x a))
(and (assoc k x)
(cons k (cterm-ev (cdr (assoc k x)) a))))))
(defthm assoc-of-append-when-alistp-x
(implies (alistp x)
(equal (assoc k (append x y))
(or (assoc k x)
(assoc k y)))))
(defthm alistp-of-cterm-ev-alist
(alistp (cterm-ev-alist x a)))
(defthm-term-subst-flg
(defthm eval-of-term-subst
(implies (and (pseudo-termp x)
(not (assoc nil alist)))
(equal (cterm-ev (term-subst x alist) a)
(cterm-ev x (append (cterm-ev-alist alist a) a))))
:hints ((and stable-under-simplificationp
'(:in-theory (enable cterm-ev-of-fncall-args))))
:flag term)
(defthm eval-of-termlist-subst
(implies (and (pseudo-term-listp x)
(not (assoc nil alist)))
(equal (cterm-ev-lst (termlist-subst x alist) a)
(cterm-ev-lst x (append (cterm-ev-alist alist a) a))))
:flag list))
(defthm eval-of-sublis-var
(implies (and (pseudo-termp x)
(not (assoc nil alist)))
(equal (cterm-ev (sublis-var alist x) a)
(cterm-ev x (append (cterm-ev-alist alist a) a)))))
(in-theory (disable sublis-var))
(defthm subcor-var1-in-terms-of-cdr-assoc
(equal (subcor-var1 vars terms var)
(let ((look (assoc var (pairlis$ vars terms))))
(if look (cdr look) var))))
(local (defthm cons-term1-equals-cons-term1-mv2
(equal (cons-term1 fn args)
(mv-nth 1 (cons-term1-mv2 fn args (cons fn args))))
:hints(("Goal" :in-theory (enable cons-term1-mv2)))))
(local (defthm cons-term1-mv2-when-not-symbolp
(implies (not (symbolp fn))
(equal (mv-nth 1 (cons-term1-mv2 fn args x))
x))
:hints(("Goal" :in-theory (enable cons-term1-mv2)))))
(defthm-term-subst-flg
(defthm subcor-var-is-sublis-var1
(implies (pseudo-termp x)
(equal (subcor-var vars terms x)
(mv-nth 1 (sublis-var1 (pairlis$ vars terms) x))))
:hints ((and stable-under-simplificationp
'(:in-theory (e/d (cons-term) (cons-term1)))))
:flag term)
(defthm subcor-var-lst-is-sublis-var1-lst
(implies (pseudo-term-listp x)
(equal (subcor-var-lst vars terms x)
(mv-nth 1 (sublis-var1-lst (pairlis$ vars terms) x))))
:flag list))
|
