/usr/share/acl2-4.3/books/cowles/acl2-agp.lisp is in acl2-books-source 4.3-3.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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 | #| This is the .lisp file for the Abelian Group book.
John Cowles, University of Wyoming, Summer 1993
Last modified 29 July 1994.
Modified A. Flatau 2-Nov-1994
Added a :verify-guards t hint to PRED for Acl2 1.8.
To use this book at the University of Wyoming:
1. Set the Connected Book Directory to a directory containing this book.
At one time, the argument to the following set-cbd named such a
directory.
(set-cbd "/meru0/cowles/acl2-libs/ver1.6/")
2. Execute the event:
(include-book
"acl2-agp")
========================================================
The following were used for certification of this book
at the University of Wyoming.
(set-cbd "/meru0/cowles/acl2-libs/ver1.6/")
(defpkg
"ACL2-ASG"
(set-difference-equal
(union-eq *acl2-exports*
*common-lisp-symbols-from-main-lisp-package*)
'(zero)))
(defpkg
"ACL2-AGP"
(set-difference-equal
(union-eq *acl2-exports*
*common-lisp-symbols-from-main-lisp-package*)
'(zero)))
(certify-book
"acl2-agp"
2
nil)
============================================
The following documentation is from the file
/meru0/cowles/acl2-libs/ver1.6/libs.doc
(deflabel
abelian-groups
:doc
":Doc-Section Libraries
Axiomatization of an associative and commutative binary operation
with an identity and an unary inverse operation.~/
Axiomatization by J. Cowles, University of Wyoming, Summer 1993.
Last modified 29 July 1994.
See :DOC ~/
Theory of Abelian Groups.
ACL2-AGP::op is an associative and commutative binary operation on the
set (of equivalence classes formed by the equivalence relation,
ACL2-AGP::equiv, on the set) GP = { x | (ACL2-AGP::pred x) not equal
nil }.
ACL2-AGP::id is a constant in the set GP which acts as an unit for
ACL2-AGP::op in GP.
ACL2-AGP::inv is an unary operation on the set (of equivalence classes
formed by the equivalence relation, ACL2-AGP::equiv, on the set) GP
which acts as an ACL2-AGP::op-inverse for ACL2-AGP:: id.
For example, let ACL2-AGP::pred = Booleanp,
ACL2-AGP::op = exclusive-or,
ACL2-AGP::id = nil, and
ACL2-AGP::inv = identity function.
Axioms of the theory of Abelian Groups.
Do :PE on the following events to see the details.
[Note. The actual names of these events are obtained by
adding the prefix ACL2-AGP:: to each name listed below.]
Equiv-is-an-equivalence
Equiv-1-implies-equiv-op
Equiv-2-implies-equiv-op
Closure-of-op-for-pred
Closure-of-id-for-pred
Closure-of-inv-for-pred
Commutativity-of-op
Associativity-of-op
Left-unicity-of-id-for-op
Right-inverse-of-inv-for-op
Theorems of the theory of Abelian Groups.
Do :PE on the following events to see the details.
[Note. The actual names of these events are obtained by
adding the prefix ACL2-AGP:: to each name listed below.]
Commutativity-2-of-op
Right-unicity-of-id-for-op
Left-inverse-of-inv-for-op
Right-cancellation-for-op
Left-cancellation-for-op
Uniqueness-of-id-as-op-idempotent
Uniqueness-of-op-inverses
Uniqueness-of-op-inverses-1
Involution-of-inv
Distributivity-of-inv-over-op
Id-is-its-own-invese
Inv-cancellation-on-right
Inv-cancellation-on-left~/
:cite libraries-location")
|#
(in-package "ACL2-AGP")
(include-book "acl2-asg" :load-compiled-file nil)
(encapsulate
((equiv ( x y ) t)
(pred ( x ) t)
(op ( x y ) t)
(id ( ) t)
(inv ( x ) t))
(local
(defun
equiv ( x y )
(equal x y)))
(local
(defun
pred ( x )
(declare (xargs :verify-guards t))
(or (equal x t)
(equal x nil))))
(local
(defun
op ( x y )
(declare (xargs :guard (and (pred x)
(pred y))))
(and (or x y)
(not (and x y)))))
(local
(defun
id ( )
nil))
(local
(defun
inv ( x )
(declare (xargs :guard (pred x)))
x))
(defthm
Equiv-is-an-equivalence
(and (acl2::booleanp (equiv x y))
(equiv x x)
(implies (equiv x y)
(equiv y x))
(implies (and (equiv x y)
(equiv y z))
(equiv x z)))
:rule-classes (:EQUIVALENCE
(:TYPE-PRESCRIPTION
:COROLLARY
(or (equal (equiv x y) t)
(equal (equiv x y) nil)))))
(defthm
Equiv-1-implies-equiv-op
(implies (equiv x1 x2)
(equiv (op x1 y)
(op x2 y)))
:rule-classes :CONGRUENCE)
(defthm
Equiv-2-implies-equiv-op
(implies (equiv y1 y2)
(equiv (op x y1)
(op x y2)))
:rule-classes :CONGRUENCE)
(defthm
Closure-of-op-for-pred
(implies (and (pred x)
(pred y))
(pred (op x y))))
(defthm
Closure-of-id-for-pred
(pred (id)))
(defthm
Closure-of-inv-for-pred
(implies (pred x)
(pred (inv x))))
(defthm
Commutativity-of-op
(implies (and (pred x)
(pred y))
(equiv (op x y)
(op y x))))
(defthm
Associativity-of-op
(implies (and (pred x)
(pred y)
(pred z))
(equiv (op (op x y) z)
(op x (op y z)))))
(defthm
Left-unicity-of-id-for-op
(implies (pred x)
(equiv (op (id) x)
x)))
(defthm
Right-inverse-of-inv-for-op
(implies (pred x)
(equiv (op x (inv x))
(id)))))
(acl2-asg::add-commutativity-2 equiv
pred
op
commutativity-of-op
commutativity-2-of-op)
(defthm
Right-unicity-of-id-for-op
(implies (pred x)
(equiv (op x (id))
x)))
(defthm
Left-inverse-of-inv-for-op
(implies (pred x)
(equiv (op (inv x) x)
(id))))
(local
(defthm
Right-cancellation-for-op-iff
(implies (and (pred x)
(pred y)
(pred z))
(iff (equiv (op x z)
(op y z))
(equiv x y)))
:rule-classes nil
:hints (("Subgoal 1"
:in-theory (disable Equiv-1-implies-equiv-op)
:use (:instance
Equiv-1-implies-equiv-op
(x1 (op x z))
(x2 (op y z))
(y (inv z)))))))
(defthm
Right-cancellation-for-op
(implies (and (pred x)
(pred y)
(pred z))
(equal (equiv (op x z)
(op y z))
(equiv x y)))
:rule-classes nil
:hints (("Goal"
:use Right-cancellation-for-op-iff)))
(local
(defthm
Left-cancellation-for-op-iff
(implies (and (pred x)
(pred y)
(pred z))
(iff (equiv (op x y)
(op x z))
(equiv z y)))
:rule-classes nil
:hints (("Goal"
:use ((:instance
Right-cancellation-for-op
(x z)
(z x)))))))
(defthm
Left-cancellation-for-op
(implies (and (pred x)
(pred y)
(pred z))
(equal (equiv (op x y)
(op x z))
(equiv y z)))
;rule-classes nil
:hints (("Goal"
:use Left-cancellation-for-op-iff)))
(defthm
Uniqueness-of-id-as-op-idempotent
(implies (and (pred x)
(equiv (op x x)
x))
(equiv x (id)))
:rule-classes nil
:hints (("Goal"
:use (:instance
Right-cancellation-for-op
(y (id))
(z x)))))
(defthm
Uniqueness-of-op-inverses
(implies (and (pred x)
(pred y)
(equiv (op x y)
(id)))
(equiv y (inv x)))
:rule-classes nil
:hints (("Goal"
:use (:instance
Right-cancellation-for-op
(x y)
(y (inv x))
(z x)))))
(defthm
Involution-of-inv
(implies (pred x)
(equiv (inv (inv x))
x))
:hints (("Goal"
:use (:instance
Uniqueness-of-op-inverses
(x (inv x))
(y x)))))
(defthm
Uniqueness-of-op-inverses-1
(implies (and (pred x)
(pred y)
(equiv (op x (inv y))
(id)))
(equiv x y))
:rule-classes nil
:hints (("Goal"
:use (:instance
Uniqueness-of-op-inverses
(y x)
(x (inv y))))))
(defthm
Distributivity-of-inv-over-op
(implies (and (pred x)
(pred y))
(equiv (inv (op x y))
(op (inv x)
(inv y))))
:hints (("Goal"
:use (:instance
Uniqueness-of-op-inverses
(x (op x y))
(y (op (inv x)(inv y)))))))
(defthm
id-is-its-own-invese
(equiv (inv (id))
(id))
:hints (("Goal"
:use (:instance
Uniqueness-of-op-inverses
(x (id))
(y (id))))))
(local
(defthm
obvious-inv-cancellation
(implies (and (pred x)
(pred y))
(equiv (op (op x (inv x)) y) y))
:rule-classes nil))
(defthm
inv-cancellation-on-right
(implies (and (pred x)
(pred y))
(equiv (op x (op y (inv x)))
y))
:hints (("Goal"
:use obvious-inv-cancellation
:in-theory (acl2::disable
right-inverse-of-inv-for-op))))
(defthm
inv-cancellation-on-left
(implies (and (pred x)
(pred y))
(equiv (op x (op (inv x) y))
y)))
|