/usr/share/acl2-7.2dfsg/books/meta/meta-times-equal.lisp is in acl2-books-source 7.2dfsg-3.
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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 | ; ACL2 books on arithmetic metafunctions
; Copyright (C) 1997 Computational Logic, Inc.
; License: A 3-clause BSD license. See the LICENSE file distributed with ACL2.
; Written by: Matt Kaufmann and John Cowles
; Computational Logic, Inc.
; 1717 West Sixth Street, Suite 290
; Austin, TX 78703-4776 U.S.A.
(in-package "ACL2")
(include-book "term-defuns")
(local (include-book "term-lemmas"
:load-compiled-file nil))
(local (include-book "arithmetic/equalities" :dir :system
:load-compiled-file nil))
(defevaluator ev-times-equal ev-times-equal-list
((binary-* x y)
(fix x)
(equal x y)
(acl2-numberp x)
(if x y z)))
(defun cancel_times-equal$1 (x)
(declare (xargs :guard (and (pseudo-termp x)
(consp x))))
; Not all of
; (and (consp (cadr x))
; (eq (car (cadr x)) 'binary-*)
; (consp (caddr x))
; (eq (car (caddr x)) 'binary-*))
; hold.
(mv-let
(elt term)
(cond ((and (consp (cadr x))
(eq (car (cadr x)) 'binary-*))
(mv (caddr x) (cadr x)))
((and (consp (caddr x))
(eq (car (caddr x)) 'binary-*))
(mv (cadr x) (caddr x)))
(t (mv nil nil)))
(cond
((and elt (fringe-occur 'binary-* elt term))
(list 'if
(list 'acl2-numberp elt)
(list 'if
(list 'equal elt *0*)
*t*
(list 'equal
*1*
(binary-op_tree
'binary-*
1
'fix
(del elt (binary-op_fringe 'binary-* term)))))
*nil*))
(t x))))
(defun formal-some-zerop (lst)
(declare (xargs :guard (and (consp lst)
(true-listp lst))))
(cond
((endp (cdr lst))
(list 'equal *0* (list 'fix (car lst))))
((memb (car lst) (cdr lst))
(formal-some-zerop (cdr lst)))
(t (list 'if
(list 'equal *0* (list 'fix (car lst)))
*t*
(formal-some-zerop (cdr lst))))))
(defun cancel_times-equal (x)
(declare (xargs :guard (pseudo-termp x)))
(if (and (consp x)
(eq (car x) 'equal))
(cond
((and (consp (cadr x))
(eq (car (cadr x)) 'binary-*)
(consp (caddr x))
(eq (car (caddr x)) 'binary-*))
(let* ((lt-side (binary-op_fringe 'binary-* (cadr x)))
(rt-side (binary-op_fringe 'binary-* (caddr x)))
(int (bagint lt-side rt-side)))
(if int
(list 'if
(formal-some-zerop int)
*t*
(list 'equal
(binary-op_tree 'binary-*
1
'fix
(bagdiff lt-side int))
(binary-op_tree 'binary-*
1
'fix
(bagdiff rt-side int))))
x)))
(t (cancel_times-equal$1 x)))
x))
(local
(defthm acl2-numberp-ev-times-equal
(acl2-numberp (ev-times-equal (binary-op_tree 'binary-* 1 'fix fringe) a))
:rule-classes :type-prescription))
(local (in-theory (disable binary-op_tree)))
(local
(defthm ev-times-equal-binary-op_tree-append
(equal (ev-times-equal (binary-op_tree 'binary-*
1 'fix
(append fringe1 fringe2))
a)
(* (ev-times-equal (binary-op_tree 'binary-*
1 'fix
fringe1)
a)
(ev-times-equal (binary-op_tree 'binary-*
1 'fix
fringe2)
a)))
:hints (("Goal" :induct (append fringe1 fringe2)))))
(local
(defthm ev-times-equal-binary-op_tree-fringe
(equal (ev-times-equal (binary-op_tree 'binary-*
1 'fix
(binary-op_fringe 'binary-* x))
a)
(fix (ev-times-equal x a)))))
(local
(defthm times-cancel-left
(equal (equal (* x y) (* x z))
(or (equal (fix x) 0)
(equal (fix y) (fix z))))))
(local
(defthm binary-op_tree-times-fringe-del-lemma
(implies (memb summand fringe)
(equal (* (ev-times-equal summand a)
(ev-times-equal (binary-op_tree 'binary-*
1 'fix
(del summand fringe))
a))
(ev-times-equal (binary-op_tree 'binary-*
1 'fix
fringe)
a)))
:rule-classes nil
:hints (("Goal" :expand ((binary-op_tree 'binary-*
1 'fix
(cdr fringe)))))))
(local
(encapsulate
()
(local
(defthm times-cancel-left-better
(implies (equal left (* x y))
(equal (equal left (* x z))
(or (equal (fix x) 0)
(equal (fix y) (fix z)))))))
(defthm binary-op_tree-times-fringe-del
(implies (and (memb summand fringe)
(acl2-numberp y)
(not (equal (fix (ev-times-equal summand a)) 0)))
(equal (equal y
(ev-times-equal (binary-op_tree 'binary-*
1 'fix
(del summand fringe))
a))
(equal (* (ev-times-equal summand a) y)
(ev-times-equal (binary-op_tree 'binary-*
1 'fix
fringe)
a))))
:hints (("Goal" :use binary-op_tree-times-fringe-del-lemma
:in-theory (disable commutativity-of-*))))))
(local
(defthm memb-of-fringe-non-zero
(implies (and (memb x (binary-op_fringe 'binary-* term))
(acl2-numberp (ev-times-equal term a))
(not (equal (ev-times-equal term a) 0)))
(and (not (equal (ev-times-equal x a) 0))
(acl2-numberp (ev-times-equal x a))))))
(local
(defthm cancel_times-equal$1-property
(implies (and (consp x)
(equal (car x) 'equal))
(equal (ev-times-equal (cancel_times-equal$1 x) a)
(ev-times-equal x a)))))
(local
(in-theory (disable cancel_times-equal$1)))
(local
(defthm ev-times-equal-binary-op_tree-is-zero
(implies (and (memb x fringe)
(equal (fix (ev-times-equal x a)) 0))
(equal (ev-times-equal (binary-op_tree 'binary-* 1 'fix fringe)
a)
0))))
(local
(defthm ev-times-equal-binary-op_tree-is-zero-from-del
(implies (equal (ev-times-equal (binary-op_tree 'binary-* 1 'fix (del x fringe))
a)
0)
(equal (ev-times-equal (binary-op_tree 'binary-* 1 'fix fringe)
a)
0))
:hints (("Goal" :expand ((binary-op_tree 'binary-* 1 'fix fringe)
(binary-op_tree 'binary-* 1 'fix (cdr fringe)))))))
(local
(encapsulate
()
(local
(defthm binary-op_tree-opener-extra-1
(implies (and (consp fringe)
(not (consp (cdr fringe))))
(equal (binary-op_tree 'binary-* 1 op fringe)
(list op (car fringe))))))
(local
(defthm cancel_equal-times-correct-lemma-1-lemma
(implies (memb x fringe)
(equal (* (ev-times-equal x a)
(ev-times-equal (binary-op_tree 'binary-*
1 'fix
(del x fringe))
a))
(fix (ev-times-equal (binary-op_tree 'binary-*
1 'fix
fringe)
a))))
:hints (("Goal" :use
((:instance
binary-op_tree-times-fringe-del
(summand x)
(y (ev-times-equal (binary-op_tree 'binary-*
1 'fix
(del x fringe))
a))
(fringe fringe)
(a a)))
:in-theory (disable binary-op_tree-times-fringe-del)))))
(defthm cancel_equal-times-correct-lemma-1
(implies (subbagp fringe2 fringe1)
(equal
(* (ev-times-equal (binary-op_tree
'binary-*
1
'fix
fringe2)
a)
(ev-times-equal (binary-op_tree
'binary-*
1
'fix
(bagdiff fringe1 fringe2))
a))
(ev-times-equal (binary-op_tree
'binary-*
1
'fix
fringe1)
a)))
:hints (("Goal" :expand ((binary-op_tree 'binary-*
1 'fix
fringe2)))))))
(local
(encapsulate
()
(local
(defthm times-cancel-right-left-better
(implies (equal left (* y x))
(equal (equal left (* x z))
(or (equal (fix x) 0)
(equal (fix y) (fix z)))))))
(defthm cancel_equal-times-correct-lemma-2
(implies (and (subbagp fringe2 fringe1)
(not (equal (ev-times-equal
(binary-op_tree
'binary-*
1
'fix
fringe2)
a)
0))
(acl2-numberp y))
(equal
(equal y
(ev-times-equal (binary-op_tree
'binary-*
1
'fix
(bagdiff fringe1 fringe2))
a))
(equal (* (ev-times-equal
(binary-op_tree
'binary-*
1
'fix
fringe2)
a)
y)
(ev-times-equal (binary-op_tree
'binary-*
1
'fix
fringe1)
a))))
:hints (("Goal" :use cancel_equal-times-correct-lemma-1
:in-theory (disable cancel_equal-times-correct-lemma-1))))))
(local
(defthm acl2-numberp-ev-times-equal-again
(acl2-numberp (ev-times-equal (cons 'binary-* x8)
a))
:rule-classes :type-prescription))
(local
(defthm ev-times-equal-binary-op_tree-is-zero-alternate
(implies (and (memb x fringe)
(equal (fix (ev-times-equal x a)) 0))
(ev-times-equal (formal-some-zerop fringe)
a))))
(local
(defthm ev-times-equal-formal-some-zerop-0
(implies (consp fringe)
(equal (ev-times-equal
(formal-some-zerop fringe)
a)
(equal
(fix (ev-times-equal (binary-op_tree 'binary-*
1 'fix
fringe)
a))
0)))
:hints (("Goal" :expand ((binary-op_tree 'binary-* 1 'fix fringe))
:restrict ((ev-times-equal-binary-op_tree-is-zero-alternate
((x (car fringe)))))))))
(local
(defthm consp-bagint
(iff (consp (bagint x y))
(bagint x y))))
(local
(defthm cancel_equal-times-correct-lemma
(let ((int (bagint (binary-op_fringe 'binary-* x1)
(binary-op_fringe 'binary-* x2))))
(implies
(and int
(not (ev-times-equal (formal-some-zerop int) a)))
(equal
(equal
(ev-times-equal (binary-op_tree
'binary-*
1
'fix
(bagdiff (binary-op_fringe 'binary-* x1)
int))
a)
(ev-times-equal (binary-op_tree
'binary-*
1
'fix
(bagdiff (binary-op_fringe 'binary-* x2)
int))
a))
(equal (fix (ev-times-equal x1 a))
(fix (ev-times-equal x2 a))))))
:hints (("Goal" :use
((:instance cancel_equal-times-correct-lemma-1
(fringe1 (binary-op_fringe 'binary-* x1))
(fringe2 (bagint (binary-op_fringe 'binary-* x1)
(binary-op_fringe 'binary-* x2))))
(:instance cancel_equal-times-correct-lemma-1
(fringe1 (binary-op_fringe 'binary-* x2))
(fringe2 (bagint (binary-op_fringe 'binary-* x1)
(binary-op_fringe 'binary-* x2)))))))))
(local
(defthm formal-some-zerop-bagint-yields-0
(implies
(and (ev-times-equal (formal-some-zerop fringe)
a)
(subbagp fringe (binary-op_fringe 'binary-*
term))
(acl2-numberp (ev-times-equal term a))
(consp fringe))
(equal (ev-times-equal term a)
0))))
(defthm cancel_times-equal-correct
(equal (ev-times-equal x a)
(ev-times-equal (cancel_times-equal x) a))
:rule-classes ((:meta :trigger-fns (equal)))
:hints (("Goal" :in-theory (disable ev-times-equal-constraint-6
ev-times-equal-formal-some-zerop-0))))
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