1% Examples taken from the manual: 2 3% 1 Introduction 4oo := torder({x,y},lex)$ 5 6 7cgb {a*x,x+y}; 8 9 10{x + y,a*x,a*y} 11 12gsys {a*x,x+y}; 13 14 15{{a <> 0,{x + y,a*x,a*y}}, 16 17 {a = 0,{x + y}}} 18 19torder oo; 20 21 22{{x,y},lex} 23 24 25% 4 CGB: Comprehensive Groebner Basis 26oo := torder({x,y},lex)$ 27 28 29cgb {a*x+y,x+b*y}; 30 31 32{x + b*y,a*x + y,(a*b - 1)*y} 33 34torder oo; 35 36 37{{x,y},lex} 38 39 40% 5 GSYS: Groebner System 41oo := torder({x,y},lex)$ 42 43 44gsys {a*x+y,x+b*y}; 45 46 47{{a*b - 1 <> 0 and a <> 0, 48 49 {a*x + y,x + b*y,(a*b - 1)*y}}, 50 51 {a <> 0 and a*b - 1 = 0, 52 53 {a*x + y,x + b*y}}, 54 55 {a = 0,{x + b*y,y}}} 56 57torder oo; 58 59 60{{x,y},lex} 61 62 63% 6 GSYS2CGB: Groebner System to CGB 64oo := torder({x,y},lex)$ 65 66 67gsys {a*x+y,x+b*y}; 68 69 70{{a*b - 1 <> 0 and a <> 0, 71 72 {a*x + y,x + b*y,(a*b - 1)*y}}, 73 74 {a <> 0 and a*b - 1 = 0, 75 76 {a*x + y,x + b*y}}, 77 78 {a = 0,{x + b*y,y}}} 79 80gsys2cgb ws; 81 82 83{x + b*y,a*x + y,y,(a*b - 1)*y} 84 85torder oo; 86 87 88{{x,y},lex} 89 90 91% 7 Switch CGBREAL: Computing over the Real Numbers 92oo := torder({x,y},lex)$ 93 94 95off cgbreal; 96 97 98gsys {a*x+y,x-a*y}; 99 100 101 2 102{{a + 1 <> 0 and a <> 0, 103 104 2 105 {a*x + y,x - a*y,(a + 1)*y}}, 106 107 2 108 {a <> 0 and a + 1 = 0, 109 110 {a*x + y,x - a*y}}, 111 112 {a = 0,{x - a*y,y}}} 113 114on cgbreal; 115 116 117gsys({a*x+y,x-a*y}); 118 119 120{{a <> 0, 121 122 2 123 {a*x + y,x - a*y,(a + 1)*y}}, 124 125 {a = 0,{x - a*y,y}}} 126 127torder oo; 128 129 130{{x,y},lex} 131 132 133% Miscellaneous examples: 134 135% Dolzmann's Example 136oo := torder({x,y,z},lex); 137 138 139oo := {{},lex} 140 141cgb({a*x+b*y,c*x+d*y,(a*d-b*c)*z}); 142 143 144{c*x + d*y, 145 146 a*x + b*y, 147 148 (a*d - b*c)*y, 149 150 (a*d - b*c)*z} 151 152gsys({a*x+b*y,c*x+d*y,(a*d-b*c)*z}); 153 154 155{{a*d - b*c <> 0 and a <> 0 and c <> 0, 156 157 {a*x + b*y, 158 159 c*x + d*y, 160 161 (a*d - b*c)*y, 162 163 (a*d - b*c)*z}}, 164 165 {a <> 0 and c <> 0 and a*d - b*c = 0, 166 167 {a*x + b*y,c*x + d*y}}, 168 169 {a <> 0 and d <> 0 and c = 0, 170 171 {a*x + b*y,d*y,(a*d - b*c)*z}}, 172 173 {a <> 0 and c = 0 and d = 0, 174 175 {a*x + b*y}}, 176 177 {b <> 0 and c <> 0 and a = 0, 178 179 {c*x + d*y,b*y,(a*d - b*c)*z}}, 180 181 {b <> 0 and d <> 0 and a = 0 and c = 0, 182 183 {b*y,d*y}}, 184 185 {b <> 0 and a = 0 and c = 0 and d = 0, 186 187 {b*y}}, 188 189 {c <> 0 and a = 0 and b = 0, 190 191 {c*x + d*y}}, 192 193 {d <> 0 and a = 0 and b = 0 and c = 0, 194 195 {d*y}}, 196 197 {a = 0 and b = 0 and c = 0 and d = 0,{}}} 198 199gsys2cgb ws; 200 201 202{c*x + d*y, 203 204 a*x + b*y, 205 206 b*y, 207 208 d*y, 209 210 (a*d - b*c)*y, 211 212 (a*d - b*c)*z} 213 214torder oo; 215 216 217{{x,y,z},lex} 218 219 220% Forsman's Example (hybrid control system). 221oo := torder({x1,x2,y2,y1,y0},lex); 222 223 224oo := {{},lex} 225 226gsys({(u1*u2-u1)*x1+u2*x2+y2,(u2-1)*x1+u2*x2+y1,-x2+y0}); 227 228 229{{u1*u2 - u1 <> 0, 230 231 {(u1*u2 - u1)*x1 + u2*x2 + y2, 232 233 (u2 - 1)*x1 + u2*x2 + y1, 234 235 x2 - y0, 236 237 y2 - u1*y1 - (u1*u2 - u2)*y0}}, 238 239 {u2 - 1 <> 0 and u2 <> 0 and u1*u2 - u1 = 0, 240 241 {(u2 - 1)*x1 + u2*x2 + y1,u2*x2 + y2,x2 - y0,y2 + u2*y0}}, 242 243 {u1 = 0 and u2 = 0, 244 245 {(u2 - 1)*x1 + u2*x2 + y1,x2 - y0,y2}}, 246 247 {u2 - 1 = 0, 248 249 {u2*x2 + y2,u2*x2 + y1,x2 - y0,y2 - y1,y1 + u2*y0}}} 250 251torder oo; 252 253 254{{x1,x2,y2,y1,y0},lex} 255 256 257% Weispfenning's Example 258oo := torder({x,y},lex); 259 260 261oo := {{},lex} 262 263gsys({v*x*y + x,u*y^2 + x^2}); 264 265 266{{u*v <> 0 and v <> 0, 267 268 2 2 269 {x + u*y , 270 271 v*x*y + x, 272 273 3 2 274 (u*v)*y + u*y }}, 275 276 {u <> 0 and v = 0, 277 278 2 2 2 279 {x + u*y ,x,u*y }}, 280 281 {v <> 0 and u*v = 0, 282 283 2 2 284 {x + u*y ,v*x*y + x}}, 285 286 2 2 287 {u = 0 and v = 0,{x + u*y ,x}}} 288 289torder oo; 290 291 292{{x,y},lex} 293 294 295% The folllowing three examples are taken from 296% Weispfenning, Comprehensive Groebner Bases, 297% J. Symbolic Computation (1992) 14, 1-29 298 299% Weispfenning's Example 7.1 300oo := torder({x},lex); 301 302 303oo := {{},lex} 304 305gsys({a0*x**2 + a1*x + a2,b0*x**2 + b1*x + b2}); 306 307 308 3 2 2 2 2 2 2 309{{a0 *b2 - a0 *a1*b1*b2 - 2*a0 *a2*b0*b2 + a0 *a2*b1 + a0*a1 *b0*b2 310 311 2 2 312 - a0*a1*a2*b0*b1 + a0*a2 *b0 <> 0 and a0*b1 - a1*b0 <> 0 and a0 <> 0 313 314 and b0 <> 0, 315 316 3 2 2 2 2 2 2 317 {a0 *b2 - a0 *a1*b1*b2 - 2*a0 *a2*b0*b2 + a0 *a2*b1 + a0*a1 *b0*b2 318 319 2 2 320 - a0*a1*a2*b0*b1 + a0*a2 *b0 }}, 321 322 3 2 2 323 {a0*b1 - a1*b0 <> 0 and a0 <> 0 and b0 <> 0 and a0 *b2 - a0 *a1*b1*b2 324 325 2 2 2 2 2 2 326 - 2*a0 *a2*b0*b2 + a0 *a2*b1 + a0*a1 *b0*b2 - a0*a1*a2*b0*b1 + a0*a2 *b0 327 328 = 0, 329 330 2 331 {a0*x + a1*x + a2, 332 333 2 334 b0*x + b1*x + b2, 335 336 (a0*b1 - a1*b0)*x + (a0*b2 - a2*b0)}}, 337 338 2 2 339 {a0*b2 - a1*b1*b2 + a2*b1 <> 0 and a0 <> 0 and b1 <> 0 and b0 = 0, 340 341 2 2 342 {a0*b2 - a1*b1*b2 + a2*b1 }}, 343 344 {a0*b2 - a2*b0 <> 0 and a0 <> 0 and b0 <> 0 and a0*b1 - a1*b0 = 0, 345 346 {a0*b2 - a2*b0}}, 347 348 {a0 <> 0 and b0 <> 0 and a0*b1 - a1*b0 = 0 and a0*b2 - a2*b0 = 0, 349 350 2 2 351 {a0*x + a1*x + a2,b0*x + b1*x + b2}}, 352 353 2 2 354 {a0 <> 0 and b1 <> 0 and a0*b2 - a1*b1*b2 + a2*b1 = 0 and b0 = 0, 355 356 2 357 {a0*x + a1*x + a2,b1*x + b2}}, 358 359 {a0 <> 0 and b2 <> 0 and b0 = 0 and b1 = 0,{b2}}, 360 361 {a0 <> 0 and b0 = 0 and b1 = 0 and b2 = 0, 362 363 2 364 {a0*x + a1*x + a2}}, 365 366 2 2 367 {a1 *b2 - a1*a2*b1 + a2 *b0 <> 0 and a1 <> 0 and b0 <> 0 and a0 = 0, 368 369 2 2 370 {a1 *b2 - a1*a2*b1 + a2 *b0}}, 371 372 {a1*b2 - a2*b1 <> 0 and a1 <> 0 and b1 <> 0 and a0 = 0 and b0 = 0, 373 374 {a1*b2 - a2*b1}}, 375 376 2 2 377 {a1 <> 0 and b0 <> 0 and a0 = 0 and a1 *b2 - a1*a2*b1 + a2 *b0 = 0, 378 379 2 380 {b0*x + b1*x + b2,a1*x + a2}}, 381 382 {a1 <> 0 and b1 <> 0 and a0 = 0 and a1*b2 - a2*b1 = 0 and b0 = 0, 383 384 {a1*x + a2,b1*x + b2}}, 385 386 {a1 <> 0 and b2 <> 0 and a0 = 0 and b0 = 0 and b1 = 0, 387 388 {b2}}, 389 390 {a1 <> 0 and a0 = 0 and b0 = 0 and b1 = 0 and b2 = 0, 391 392 {a1*x + a2}}, 393 394 {a2 <> 0 and a0 = 0 and a1 = 0,{a2}}, 395 396 {b0 <> 0 and a0 = 0 and a1 = 0 and a2 = 0, 397 398 2 399 {b0*x + b1*x + b2}}, 400 401 {b1 <> 0 and a0 = 0 and a1 = 0 and a2 = 0 and b0 = 0, 402 403 {b1*x + b2}}, 404 405 {b2 <> 0 and a0 = 0 and a1 = 0 and a2 = 0 and b0 = 0 and b1 = 0, 406 407 {b2}}, 408 409 {a0 = 0 and a1 = 0 and a2 = 0 and b0 = 0 and b1 = 0 and b2 = 0, 410 411 {}}} 412 413torder oo; 414 415 416{{x},lex} 417 418 419% Weispfenning's Example 7.2 420oo := torder({x,y},lex); 421 422 423oo := {{},lex} 424 425gsys({v*x*y + u*x**2 + x,u*y**2 + x**2}); 426 427 428 4 3 429{{u *v + u*v <> 0 and u <> 0 and v <> 0, 430 431 2 432 {u*x + v*x*y + x, 433 434 2 2 435 x + u*y , 436 437 2 2 438 v*x*y + x - u *y , 439 440 3 5 2 3 3 5 2 2 2 441 u *x + (u *v + u *v )*y - (u - u *v )*y , 442 443 7 2 4 4 4 4 3 3 4 2 2 444 (u *v + u *v )*y + (2*u *v )*y + (u *v )*y }}, 445 446 4 3 447 {u <> 0 and v <> 0 and u *v + u*v = 0, 448 449 2 450 {u*x + v*x*y + x, 451 452 2 2 453 x + u*y , 454 455 2 2 456 v*x*y + x - u *y , 457 458 3 5 2 3 3 5 2 2 2 459 u *x + (u *v + u *v )*y - (u - u *v )*y , 460 461 4 3 3 4 2 2 462 (2*u *v )*y + (u *v )*y }}, 463 464 {u <> 0 and v = 0, 465 466 2 467 {u*x + v*x*y + x, 468 469 2 2 470 x + u*y , 471 472 2 2 473 x - u *y , 474 475 5 4 2 2 476 u *y + u *y }}, 477 478 {v <> 0 and u = 0, 479 480 2 2 481 {x + u*y ,v*x*y + x}}, 482 483 2 2 484 {u = 0 and v = 0,{x + u*y ,x}}} 485 486torder oo; 487 488 489{{x,y},lex} 490 491 492% Weispfenning's Example 7.3 493oo := torder({x1,x2,x3,x4},lex); 494 495 496oo := {{},lex} 497 498gsys {x4 - (a4-a2),x1 + x2 + x3 + x4 + (a1 + a3 + a4), 499 x1*x3 + x1*x4 + x2*x3 + x3*x4 - (a1*a4 + a1*a3 + a3*a4),x1*x3*x4 - a1*a3*a4}; 500 501 502 2 2 3 2 2 503{{a1*a2 - a1*a2*a3 - 3*a1*a2*a4 + 2*a1*a4 - a2 + a2 *a3 + 4*a2 *a4 504 505 2 2 3 506 - 3*a2*a3*a4 - 5*a2*a4 + 2*a3*a4 + 2*a4 <> 0 and a2 - a4 = 0, 507 508 2 2 3 2 2 509 {a1*a2 - a1*a2*a3 - 3*a1*a2*a4 + 2*a1*a4 - a2 + a2 *a3 + 4*a2 *a4 510 511 2 2 3 512 - 3*a2*a3*a4 - 5*a2*a4 + 2*a3*a4 + 2*a4 }}, 513 514 {a2 - a4 <> 0, 515 516 {x1*x3*x4 - a1*a3*a4, 517 518 x1*x3 + x1*x4 + x2*x3 + x3*x4 - (a1*a3 + a1*a4 + a3*a4), 519 520 x1 + x2 + x3 + x4 + (a1 + a3 + a4), 521 522 2 523 (a2 - a4)*x2 - x3 - (a1 - a2 + a3 + 2*a4)*x3 524 525 2 2 526 + (a1*a2 - a1*a3 - 2*a1*a4 - a2 + a2*a3 + 3*a2*a4 - 2*a3*a4 - 2*a4 ), 527 528 3 2 529 x3 + (a1 + a3 + a4)*x3 + (a1*a3 + a1*a4 + a3*a4)*x3 - a1*a3*a4, 530 531 x4 + (a2 - a4)}}, 532 533 2 2 3 2 2 534 {a1*a2 - a1*a2*a3 - 3*a1*a2*a4 + 2*a1*a4 - a2 + a2 *a3 + 4*a2 *a4 535 536 2 2 3 537 - 3*a2*a3*a4 - 5*a2*a4 + 2*a3*a4 + 2*a4 = 0 and a2 - a4 = 0, 538 539 {x1*x3*x4 - a1*a3*a4, 540 541 x1*x3 + x1*x4 + x2*x3 + x3*x4 - (a1*a3 + a1*a4 + a3*a4), 542 543 x1 + x2 + x3 + x4 + (a1 + a3 + a4), 544 545 2 546 - x3 - (a1 - a2 + a3 + 2*a4)*x3 547 548 2 2 549 + (a1*a2 - a1*a3 - 2*a1*a4 - a2 + a2*a3 + 3*a2*a4 - 2*a3*a4 - 2*a4 ), 550 551 x4 + (a2 - a4)}}} 552 553torder oo; 554 555 556{{x1,x2,x3,x4},lex} 557 558 559% Pesch's example (Circle through three points) 560oo := torder({y,x},revgradlex); 561 562 563oo := {{},lex} 564 565gsys({2*b2*y + 2*a2*x - b2**2 + a2**2,2*b1*y + 2*a1*x - b1**2 + a1**2}); 566 567 568 2 2 569{{a1 *a2 - a1*a2 <> 0 and a1 <> 0 and a2 <> 0 and b1 = 0 and b2 = 0, 570 571 2 2 2 2 572 {a1 *a2 - a1*a2 + a1*b2 - a2*b1 }}, 573 574 2 2 2 2 575 {a1 *b2 - a2 *b1 - b1 *b2 + b1*b2 <> 0 and b1 <> 0 and b2 <> 0 576 577 and a1*b2 - a2*b1 = 0, 578 579 2 2 2 2 580 {a1 *b2 - a2 *b1 - b1 *b2 + b1*b2 }}, 581 582 {a1*b2 - a2*b1 <> 0 and b1 <> 0 and b2 <> 0, 583 584 2 2 585 {(2*b1)*y + (2*a1)*x + (a1 - b1 ), 586 587 2 2 588 (2*b2)*y + (2*a2)*x + (a2 - b2 ), 589 590 2 2 2 2 591 (2*a1*b2 - 2*a2*b1)*x + (a1 *b2 - a2 *b1 - b1 *b2 + b1*b2 )}}, 592 593 2 2 594 {a1 <> 0 and a2 <> 0 and a1 *a2 - a1*a2 = 0 and b1 = 0 and b2 = 0, 595 596 2 2 597 {(2*a1)*x + (a1 - b1 ), 598 599 2 2 600 (2*a2)*x + (a2 - b2 )}}, 601 602 {a1 <> 0 and b2 <> 0 and b1 = 0, 603 604 2 2 605 {(2*b2)*y + (2*a2)*x + (a2 - b2 ), 606 607 2 2 608 (2*a1)*x + (a1 - b1 )}}, 609 610 {a1 <> 0 and a2 = 0 and b1 = 0 and b2 = 0, 611 612 2 2 613 {(2*a1)*x + (a1 - b1 )}}, 614 615 {a2 <> 0 and b1 <> 0 and b2 = 0, 616 617 2 2 618 {(2*b1)*y + (2*a1)*x + (a1 - b1 ), 619 620 2 2 621 (2*a2)*x + (a2 - b2 )}}, 622 623 {a2 <> 0 and a1 = 0 and b1 = 0 and b2 = 0, 624 625 2 2 626 {(2*a2)*x + (a2 - b2 )}}, 627 628 2 2 2 2 629 {b1 <> 0 and b2 <> 0 and a1 *b2 - a2 *b1 - b1 *b2 + b1*b2 = 0 630 631 and a1*b2 - a2*b1 = 0, 632 633 2 2 634 {(2*b1)*y + (2*a1)*x + (a1 - b1 ), 635 636 2 2 637 (2*b2)*y + (2*a2)*x + (a2 - b2 )}}, 638 639 {b1 <> 0 and a2 = 0 and b2 = 0, 640 641 2 2 642 {(2*b1)*y + (2*a1)*x + (a1 - b1 )}}, 643 644 {b2 <> 0 and a1 = 0 and b1 = 0, 645 646 2 2 647 {(2*b2)*y + (2*a2)*x + (a2 - b2 )}}, 648 649 {a1 = 0 and a2 = 0 and b1 = 0 and b2 = 0,{}}} 650 651torder oo; 652 653 654{{y,x},revgradlex} 655 656 657% Effelterre's example (Aspect graphs) 658f1 := -4-4*v**2-4*u**2+40*v*v1+24*v-120*v1+8*u-40*v2-68*v1**2-100*v2**2+40*u*v2+ 65924*v1*v2-4*v1**2*u-4*v2**2*v**2+24*v2**2*v-24*v1*u*v2+8*v*v1*u*v2$ 660 661 662f2 := 8*v*v1*u*v2-4*v1**2*u**2+4*v1**2-4*v2**2*v**2+4*v2**2-16*v**2-16*u**2+16$ 663 664 665f3 := 16*v-48*u+16*v*v1**2-48*u*v2**2-12*v1**2*u+4*v2**2*v-36*v*v1*v2+ 66612*v1*u*v2+12*v*v2**2*u- 66780*u*v1+80*v2*v-20*v1*u*v2**2+20*v2*v*v1**2-20*v1**3*u+20*v2**3*v-12*v1**2*v*u+ 66812*v2*v**2*v1-12*v1*u**2*v2$ 669 670 671f4 := -160u*v2-1596v2**2+3200*v2-1596-4*u**2+160*u$ 672 673 674 675% Special case I2, v1=0 676oo := torder({v,u},lex); 677 678 679oo := {{},lex} 680 681gsys(sub(v1=0,{f1,f2,f4})); 682 683 684 16 15 14 685{{1000557799569*v2 - 8971728968760*v2 + 32332553961916*v2 686 687 13 12 11 688 - 56816638983720*v2 + 43557281160966*v2 - 2232864400680*v2 689 690 10 9 8 691 - 2141149653636*v2 - 24710286679320*v2 + 22557177598385*v2 692 693 7 6 5 694 - 4364899181280*v2 + 1970637124608*v2 - 2915313822720*v2 695 696 4 3 2 697 + 1467702460416*v2 - 1340557025280*v2 + 685570158592*v2 - 140796887040*v2 698 699 7 6 5 4 700 + 63647907840 <> 0 and 55175*v2 - 184787*v2 + 144885*v2 + 44895*v2 701 702 3 2 703 - 13020*v2 - 35580*v2 + 320*v2 - 6368 <> 0, 704 705 16 15 14 706 {1000557799569*v2 - 8971728968760*v2 + 32332553961916*v2 707 708 13 12 11 709 - 56816638983720*v2 + 43557281160966*v2 - 2232864400680*v2 710 711 10 9 8 712 - 2141149653636*v2 - 24710286679320*v2 + 22557177598385*v2 713 714 7 6 5 715 - 4364899181280*v2 + 1970637124608*v2 - 2915313822720*v2 716 717 4 3 2 718 + 1467702460416*v2 - 1340557025280*v2 + 685570158592*v2 - 140796887040*v2 719 720 + 63647907840}}, 721 722 8 7 6 5 4 723 {3389663*v2 - 14658420*v2 + 20230730*v2 - 6945060*v2 - 3218385*v2 724 725 3 2 7 726 - 497040*v2 + 1656704*v2 - 257280*v2 + 255872 <> 0 and 55175*v2 727 728 6 5 4 3 2 729 - 184787*v2 + 144885*v2 + 44895*v2 - 13020*v2 - 35580*v2 + 320*v2 - 6368 730 731 = 0, 732 733 8 7 6 5 4 734 {3389663*v2 - 14658420*v2 + 20230730*v2 - 6945060*v2 - 3218385*v2 735 736 3 2 737 - 497040*v2 + 1656704*v2 - 257280*v2 + 255872}}, 738 739 7 6 5 4 3 2 740 {55175*v2 - 184787*v2 + 144885*v2 + 44895*v2 - 13020*v2 - 35580*v2 741 742 16 15 743 + 320*v2 - 6368 <> 0 and 1000557799569*v2 - 8971728968760*v2 744 745 14 13 12 746 + 32332553961916*v2 - 56816638983720*v2 + 43557281160966*v2 747 748 11 10 9 749 - 2232864400680*v2 - 2141149653636*v2 - 24710286679320*v2 750 751 8 7 6 752 + 22557177598385*v2 - 4364899181280*v2 + 1970637124608*v2 753 754 5 4 3 755 - 2915313822720*v2 + 1467702460416*v2 - 1340557025280*v2 756 757 2 758 + 685570158592*v2 - 140796887040*v2 + 63647907840 = 0, 759 760 2 2 2 2 2 761 {(v2 + 1)*v - (6*v2 + 6)*v + u - (10*v2 + 2)*u + (25*v2 + 10*v2 + 1), 762 763 2 2 2 2 764 (v2 + 4)*v + 4*u - (v2 + 4), 765 766 4 2 3 2 767 (6*v2 + 30*v2 + 24)*v - (110*v2 - 122*v2 - 40*v2 - 8)*u 768 769 4 3 2 770 - (1223*v2 - 2390*v2 + 1303*v2 + 40*v2 + 8), 771 772 2 2 773 u + (40*v2 - 40)*u + (399*v2 - 800*v2 + 399), 774 775 7 6 5 4 3 2 776 (220700*v2 - 739148*v2 + 579540*v2 + 179580*v2 - 52080*v2 - 142320*v2 777 778 8 7 6 779 + 1280*v2 - 25472)*u + (3389663*v2 - 14658420*v2 + 20230730*v2 780 781 5 4 3 2 782 - 6945060*v2 - 3218385*v2 - 497040*v2 + 1656704*v2 - 257280*v2 783 784 + 255872)}}, 785 786 8 7 6 5 4 787 {3389663*v2 - 14658420*v2 + 20230730*v2 - 6945060*v2 - 3218385*v2 788 789 3 2 7 790 - 497040*v2 + 1656704*v2 - 257280*v2 + 255872 = 0 and 55175*v2 791 792 6 5 4 3 2 793 - 184787*v2 + 144885*v2 + 44895*v2 - 13020*v2 - 35580*v2 + 320*v2 - 6368 794 795 = 0, 796 797 2 2 2 2 2 798 {(v2 + 1)*v - (6*v2 + 6)*v + u - (10*v2 + 2)*u + (25*v2 + 10*v2 + 1), 799 800 2 2 2 2 801 (v2 + 4)*v + 4*u - (v2 + 4), 802 803 4 2 3 2 804 (6*v2 + 30*v2 + 24)*v - (110*v2 - 122*v2 - 40*v2 - 8)*u 805 806 4 3 2 807 - (1223*v2 - 2390*v2 + 1303*v2 + 40*v2 + 8), 808 809 2 2 810 u + (40*v2 - 40)*u + (399*v2 - 800*v2 + 399)}}} 811 812torder oo; 813 814 815{{v,u},lex} 816 817clear f1,f2,f3,f4; 818 819 820 821% Sit's Example 2.2 822oo := torder({z2,z2},revgradlex); 823 824 825oo := {{},lex} 826 827gsys({d*z2 + c*z1 - v,b*z2 + a*z1 - u}); 828 829 830{{a*d*z1 - b*c*z1 + b*v - d*u <> 0 and b <> 0 and d <> 0, 831 832 {a*d*z1 - b*c*z1 + b*v - d*u}}, 833 834 {a*z1 - u <> 0 and b = 0,{a*z1 - u}}, 835 836 {b <> 0 and c*z1 - v <> 0 and d = 0, 837 838 {c*z1 - v}}, 839 840 {b <> 0 and d <> 0 and a*d*z1 - b*c*z1 + b*v - d*u = 0, 841 842 {b*z2 + (a*z1 - u),d*z2 + (c*z1 - v)}}, 843 844 {b <> 0 and c*z1 - v = 0 and d = 0, 845 846 {b*z2 + (a*z1 - u)}}, 847 848 {c*z1 - v <> 0 and a*z1 - u = 0 and b = 0 and d = 0, 849 850 {c*z1 - v}}, 851 852 {d <> 0 and a*z1 - u = 0 and b = 0, 853 854 {d*z2 + (c*z1 - v)}}, 855 856 {a*z1 - u = 0 and b = 0 and c*z1 - v = 0 and d = 0, 857 858 {}}} 859 860torder oo; 861 862 863{{z2,z2},revgradlex} 864 865 866% Sit's Example 2.3 867oo := torder({z2,z2},revgradlex); 868 869 870oo := {{},lex} 871 872gsys({x**3*z2 + (x**2+1)*z1,x**2*z2 + x*z1 - 1}); 873 874 875{{x + z1 <> 0 and x <> 0,{x + z1}}, 876 877 {x <> 0 and x + z1 = 0, 878 879 3 2 2 880 {x *z2 + (x *z1 + z1),x *z2 + (x*z1 - 1)}}, 881 882 2 883 {z1 <> 0 and x = 0,{x *z1 + z1}}, 884 885 {x = 0 and z1 = 0,{x*z1 - 1}}} 886 887torder oo; 888 889 890{{z2,z2},revgradlex} 891 892 893% Sit's Example 3.3 894oo := torder({z3,z2,z2},revgradlex); 895 896 897oo := {{},lex} 898 899gsys({z3 + b*z2 + a*z1 - 1,a*z3 + z2 + b*z1 - 1,b*z3 + a*z2 + z1 - 1}); 900 901 902 3 2 3 2 2 903{{a *z1 - a - 3*a*b*z1 + a*b + a + b *z1 - b + b + z1 - 1 <> 0 and a - b <> 0 904 905 and a <> 0 and b <> 0, 906 907 3 2 3 2 908 {a *z1 - a - 3*a*b*z1 + a*b + a + b *z1 - b + b + z1 - 1}}, 909 910 3 2 911 {a *z1 - a + a + z1 - 1 <> 0 and a <> 0 and b = 0, 912 913 3 2 914 {a *z1 - a - 2*a*b*z1 + a*b + a + z1 - 1}}, 915 916 2 917 {a *z1 - a - b*z1 + 1 <> 0 and a <> 0 and b <> 0 and a*b*z1 - b - z1 + 1 = 0 918 919 2 920 and a*b - 1 = 0 and a - b = 0, 921 922 2 923 {a *z1 - a - b*z1 + 1}}, 924 925 2 926 {a*b*z1 - b - z1 + 1 <> 0 and a <> 0 and b <> 0 and a - b = 0, 927 928 { - (a*b*z1 - b - z1 + 1)}}, 929 930 2 931 {a*b - 1 <> 0 and a <> 0 and b <> 0 and a*b*z1 - b - z1 + 1 = 0 and a - b = 0, 932 933 {z3 + b*z2 + (a*z1 - 1), 934 935 b*z3 + a*z2 + (z1 - 1), 936 937 a*z3 + z2 + (b*z1 - 1), 938 939 2 940 (a*b - 1)*z2 + (a *z1 - a - b*z1 + 1)}}, 941 942 2 943 {a - b <> 0 and a <> 0 and b <> 0 944 945 3 2 3 2 946 and a *z1 - a - 3*a*b*z1 + a*b + a + b *z1 - b + b + z1 - 1 = 0, 947 948 {z3 + b*z2 + (a*z1 - 1), 949 950 b*z3 + a*z2 + (z1 - 1), 951 952 a*z3 + z2 + (b*z1 - 1), 953 954 2 955 (a - b )*z2 - (a*b*z1 - b - z1 + 1)}}, 956 957 2 958 {a <> 0 and b <> 0 and a *z1 - a - b*z1 + 1 = 0 and a*b*z1 - b - z1 + 1 = 0 959 960 2 961 and a*b - 1 = 0 and a - b = 0, 962 963 {z3 + b*z2 + (a*z1 - 1), 964 965 b*z3 + a*z2 + (z1 - 1), 966 967 a*z3 + z2 + (b*z1 - 1)}}, 968 969 3 2 970 {a <> 0 and a *z1 - a + a + z1 - 1 = 0 and b = 0, 971 972 {z3 + b*z2 + (a*z1 - 1), 973 974 a*z3 + z2 + (b*z1 - 1), 975 976 a*z2 + (z1 - 1)}}, 977 978 3 2 979 {b *z1 - b + b + z1 - 1 <> 0 and b <> 0 and a = 0, 980 981 3 2 982 {2*a*b*z1 - a - b *z1 + b - b - z1 + 1}}, 983 984 3 2 985 {b <> 0 and a = 0 and b *z1 - b + b + z1 - 1 = 0, 986 987 {z3 + b*z2 + (a*z1 - 1), 988 989 b*z3 + a*z2 + (z1 - 1), 990 991 z2 + (b*z1 - 1)}}, 992 993 {z1 - 1 <> 0 and a = 0 and b = 0,{z1 - 1}}, 994 995 {a = 0 and b = 0 and z1 - 1 = 0, 996 997 {z3 + b*z2 + (a*z1 - 1),z2 + (b*z1 - 1)}}} 998 999torder oo; 1000 1001 1002{{z3,z2,z2},revgradlex} 1003 1004 1005% Sit's Example 8.3 1006oo := torder({z4,z3,z2,z2},revgradlex); 1007 1008 1009oo := {{},lex} 1010 1011gsys({z4 + c*z3 + b*z2 + a*z1 - w2,2*z4 + z2 - w1,a*z4 - z3 - w4,d*z4 + z3 + 10122*z1 - w3,z4 + z1 - w5}); 1013 1014 1015 3 2 2 2 2 2 2 2 2 1016{{a *c*z1 + a *b*c*w1 + a *c *w3 - 2*a *c *z1 + a *c*d*z1 - a *c*w2 + a *z1 1017 1018 2 1019 + a*b*c*d*w1 - 2*a*b*c*w3 - 2*a*b*c*w4 + 4*a*b*c*z1 + a*b*w1 - a*c *d*w4 1020 1021 - a*c*d*w2 + 2*a*c*w3 + a*c*w4 - 4*a*c*z1 + a*d*z1 - a*w2 + b*d*w1 - 2*b*w3 1022 1023 - 2*b*w4 + 4*b*z1 - c*d*w4 - d*w2 + w3 + w4 - 2*z1 <> 0 and a*b + b*d <> 0 1024 1025 and a*c + 1 <> 0 and a <> 0 and d <> 0, 1026 1027 3 2 2 2 2 2 2 2 2 1028 {a *c*z1 + a *b*c*w1 + a *c *w3 - 2*a *c *z1 + a *c*d*z1 - a *c*w2 + a *z1 1029 1030 2 1031 + a*b*c*d*w1 - 2*a*b*c*w3 - 2*a*b*c*w4 + 4*a*b*c*z1 + a*b*w1 - a*c *d*w4 1032 1033 - a*c*d*w2 + 2*a*c*w3 + a*c*w4 - 4*a*c*z1 + a*d*z1 - a*w2 + b*d*w1 - 2*b*w3 1034 1035 - 2*b*w4 + 4*b*z1 - c*d*w4 - d*w2 + w3 + w4 - 2*z1}}, 1036 1037 2 2 2 2 2 2 1038 {a *c *w5 - a *c *z1 + a *c*z1 + a*b*c*w1 - 2*a*b*c*w5 + 2*a*b*c*z1 - a*c *w4 1039 1040 - a*c*w2 + 2*a*c*w5 - 2*a*c*z1 + a*z1 + b*w1 - 2*b*w5 + 2*b*z1 - c*w4 - w2 1041 1042 + w5 - z1 <> 0 and a*c - 2*b + 1 <> 0 and a*c + 1 <> 0 and a <> 0 and d <> 0 1043 1044 2 1045 and a *z1 + a*c*w3 - 2*a*c*z1 + a*d*z1 - a*w2 - c*d*w4 - d*w2 + w3 + w4 - 2*z1 1046 1047 = 0 and a*b + b*d = 0, 1048 1049 2 2 2 2 2 2 1050 {a *c *w5 - a *c *z1 + a *c*z1 + a*b*c*w1 - 2*a*b*c*w5 + 2*a*b*c*z1 - a*c *w4 1051 1052 - a*c*w2 + 2*a*c*w5 - 2*a*c*z1 + a*z1 + b*w1 - 2*b*w5 + 2*b*z1 - c*w4 - w2 1053 1054 + w5 - z1}}, 1055 1056 2 2 1057 {a *c*d*z1 - a *z1 + a*b*c*d*w1 - 2*a*b*c*w3 + 4*a*b*c*z1 - a*b*w1 - a*c*d*w2 1058 1059 + a*w2 + 2*b*w4 + c*d*w4 - w4 <> 0 and a*b <> 0 and a*c*d - a <> 0 and a <> 0 1060 1061 and d <> 0 and a*c + 1 = 0, 1062 1063 2 2 1064 {a *c*d*z1 - a *z1 + a*b*c*d*w1 - 2*a*b*c*w3 + 4*a*b*c*z1 - a*b*w1 - a*c*d*w2 1065 1066 + a*w2 + 2*b*w4 + c*d*w4 - w4}}, 1067 1068 2 2 1069 {a *c*w5 - a *c*z1 + a*c*d*w5 - a*c*d*z1 - a*c*w3 - a*c*w4 + 2*a*c*z1 + a*w5 1070 1071 - a*z1 + d*w5 - d*z1 - w3 - w4 + 2*z1 <> 0 and a*b + b*d <> 0 1072 1073 3 2 2 2 1074 and a*c + 1 <> 0 and a <> 0 and d <> 0 and a *c*z1 + a *b*c*w1 + a *c *w3 1075 1076 2 2 2 2 2 1077 - 2*a *c *z1 + a *c*d*z1 - a *c*w2 + a *z1 + a*b*c*d*w1 - 2*a*b*c*w3 1078 1079 2 1080 - 2*a*b*c*w4 + 4*a*b*c*z1 + a*b*w1 - a*c *d*w4 - a*c*d*w2 + 2*a*c*w3 + a*c*w4 1081 1082 - 4*a*c*z1 + a*d*z1 - a*w2 + b*d*w1 - 2*b*w3 - 2*b*w4 + 4*b*z1 - c*d*w4 1083 1084 - d*w2 + w3 + w4 - 2*z1 = 0, 1085 1086 2 2 1087 {a *c*w5 - a *c*z1 + a*c*d*w5 - a*c*d*z1 - a*c*w3 - a*c*w4 + 2*a*c*z1 + a*w5 1088 1089 - a*z1 + d*w5 - d*z1 - w3 - w4 + 2*z1}}, 1090 1091 2 1092 {a *z1 + a*b*w1 - 2*a*b*w5 + 2*a*b*z1 - a*w2 + w4 <> 0 and a*b*c <> 0 1093 1094 and a*b <> 0 and a <> 0 and d <> 0 and a*c*d - a = 0 and a*c + 1 = 0 1095 1096 and a*w3 - 2*a*z1 - d*w4 = 0, 1097 1098 2 1099 {a *z1 + a*b*w1 - 2*a*b*w5 + 2*a*b*z1 - a*w2 + w4}}, 1100 1101 2 1102 {a *z1 + a*b*w1 + a*c*w3 - 2*a*c*z1 - a*w2 - 2*b*w3 - 2*b*w4 + 4*b*z1 + w3 + w4 1103 1104 - 2*z1 <> 0 and a*b <> 0 and a <> 0 and d = 0, 1105 1106 2 1107 {a *z1 + a*b*w1 + a*c*w3 - 2*a*c*z1 - a*w2 - 2*b*w3 - 2*b*w4 + 4*b*z1 + w3 1108 1109 + w4 - 2*z1}}, 1110 1111 2 1112 {a *z1 + a*b*w1 - a*w2 - 2*b*w4 + w4 <> 0 and a*b <> 0 and a <> 0 and d <> 0 1113 1114 and a*b*c = 0 and a*c*d - a = 0 and a*c + 1 = 0 and a*w3 - 2*a*z1 - d*w4 = 0, 1115 1116 2 1117 {a *z1 + a*b*w1 - a*w2 - 2*b*w4 + w4}}, 1118 1119 2 1120 {a *z1 + a*c*w3 - 2*a*c*z1 + a*d*z1 - a*w2 - c*d*w4 - d*w2 + w3 + w4 - 2*z1 1121 1122 <> 0 and a*c + 1 <> 0 and a <> 0 and d <> 0 and a*b + b*d = 0, 1123 1124 2 1125 {a *z1 + a*c*w3 - 2*a*c*z1 + a*d*z1 - a*w2 - c*d*w4 - d*w2 + w3 + w4 - 2*z1}}, 1126 1127 2 1128 {a *z1 + a*c*w3 - 2*a*c*z1 - a*w2 + w3 + w4 - 2*z1 <> 0 and a <> 0 and a*b = 0 1129 1130 and d = 0, 1131 1132 2 1133 {a *z1 + a*c*w3 - 2*a*c*z1 - a*w2 + w3 + w4 - 2*z1}}, 1134 1135 2 1136 {a *z1 - a*w2 + w4 <> 0 and a <> 0 and d <> 0 and a*b = 0 and a*c + 1 = 0, 1137 1138 2 1139 {a *z1 - a*w2 + w4}}, 1140 1141 {a*b*c <> 0 and a*b <> 0 and a <> 0 and d <> 0 1142 1143 2 1144 and a *z1 + a*b*w1 - 2*a*b*w5 + 2*a*b*z1 - a*w2 + w4 = 0 and a*c*d - a = 0 1145 1146 and a*c + 1 = 0 and a*w3 - 2*a*z1 - d*w4 = 0, 1147 1148 {z4 + c*z3 + b*z2 + (a*z1 - w2), 1149 1150 a*z4 - z3 - w4, 1151 1152 d*z4 + z3 - (w3 - 2*z1), 1153 1154 2*z4 + z2 - w1, 1155 1156 z4 - (w5 - z1), 1157 1158 2 1159 (2*a*b*c)*z3 + (a *z1 + a*b*w1 - a*w2 - 2*b*w4 + w4), 1160 1161 2 1162 (a*b)*z2 + (a *z1 - a*w2 + w4)}}, 1163 1164 3 2 1165 {a*b + b*d <> 0 and a*c + 1 <> 0 and a <> 0 and d <> 0 and a *c*z1 + a *b*c*w1 1166 1167 2 2 2 2 2 2 2 1168 + a *c *w3 - 2*a *c *z1 + a *c*d*z1 - a *c*w2 + a *z1 + a*b*c*d*w1 1169 1170 2 1171 - 2*a*b*c*w3 - 2*a*b*c*w4 + 4*a*b*c*z1 + a*b*w1 - a*c *d*w4 - a*c*d*w2 1172 1173 + 2*a*c*w3 + a*c*w4 - 4*a*c*z1 + a*d*z1 - a*w2 + b*d*w1 - 2*b*w3 - 2*b*w4 1174 1175 2 2 1176 + 4*b*z1 - c*d*w4 - d*w2 + w3 + w4 - 2*z1 = 0 and a *c*w5 - a *c*z1 1177 1178 + a*c*d*w5 - a*c*d*z1 - a*c*w3 - a*c*w4 + 2*a*c*z1 + a*w5 - a*z1 + d*w5 1179 1180 - d*z1 - w3 - w4 + 2*z1 = 0, 1181 1182 {z4 + c*z3 + b*z2 + (a*z1 - w2), 1183 1184 a*z4 - z3 - w4, 1185 1186 d*z4 + z3 - (w3 - 2*z1), 1187 1188 2*z4 + z2 - w1, 1189 1190 z4 - (w5 - z1), 1191 1192 2 1193 (a*c + 1)*z3 + (a*b)*z2 + (a *z1 - a*w2 + w4), 1194 1195 2 1196 (a*b + b*d)*z2 + (a *z1 + a*c*w3 - 2*a*c*z1 + a*d*z1 - a*w2 - c*d*w4 - d*w2 1197 1198 + w3 + w4 - 2*z1)}}, 1199 1200 {a*b <> 0 and a*c*d*w5 - a*c*d*z1 - a*c*w3 + 2*a*c*z1 - a*w5 + a*z1 + w4 <> 0 1201 1202 2 2 1203 and a*c*d - a <> 0 and a <> 0 and d <> 0 and a *c*d*z1 - a *z1 + a*b*c*d*w1 1204 1205 - 2*a*b*c*w3 + 4*a*b*c*z1 - a*b*w1 - a*c*d*w2 + a*w2 + 2*b*w4 + c*d*w4 - w4 1206 1207 = 0 and a*c + 1 = 0, 1208 1209 {a*c*d*w5 - a*c*d*z1 - a*c*w3 + 2*a*c*z1 - a*w5 + a*z1 + w4}}, 1210 1211 2 2 1212 {a*b <> 0 and a*c*d - a <> 0 and a <> 0 and d <> 0 and a *c*d*z1 - a *z1 1213 1214 + a*b*c*d*w1 - 2*a*b*c*w3 + 4*a*b*c*z1 - a*b*w1 - a*c*d*w2 + a*w2 + 2*b*w4 1215 1216 + c*d*w4 - w4 = 0 1217 1218 and a*c*d*w5 - a*c*d*z1 - a*c*w3 + 2*a*c*z1 - a*w5 + a*z1 + w4 = 0 1219 1220 and a*c + 1 = 0, 1221 1222 {z4 + c*z3 + b*z2 + (a*z1 - w2), 1223 1224 a*z4 - z3 - w4, 1225 1226 d*z4 + z3 - (w3 - 2*z1), 1227 1228 2*z4 + z2 - w1, 1229 1230 z4 - (w5 - z1), 1231 1232 (a*c*d - a)*z3 + (a*w3 - 2*a*z1 - d*w4), 1233 1234 2 1235 (a*b)*z2 + (a *z1 - a*w2 + w4)}}, 1236 1237 {a*b <> 0 and a*w3 - 2*a*z1 - d*w4 <> 0 and a <> 0 and d <> 0 and a*c*d - a = 0 1238 1239 and a*c + 1 = 0, 1240 1241 {a*w3 - 2*a*z1 - d*w4}}, 1242 1243 2 1244 {a*b <> 0 and a*w5 - a*z1 - w3 - w4 + 2*z1 <> 0 and a <> 0 and a *z1 + a*b*w1 1245 1246 + a*c*w3 - 2*a*c*z1 - a*w2 - 2*b*w3 - 2*b*w4 + 4*b*z1 + w3 + w4 - 2*z1 = 0 1247 1248 and d = 0, 1249 1250 {a*w5 - a*z1 - w3 - w4 + 2*z1}}, 1251 1252 2 1253 {a*b <> 0 and a <> 0 and d <> 0 and a *z1 + a*b*w1 - a*w2 - 2*b*w4 + w4 = 0 1254 1255 and a*b*c = 0 and a*c*d - a = 0 and a*c + 1 = 0 and a*w3 - 2*a*z1 - d*w4 = 0, 1256 1257 {z4 + c*z3 + b*z2 + (a*z1 - w2), 1258 1259 a*z4 - z3 - w4, 1260 1261 d*z4 + z3 - (w3 - 2*z1), 1262 1263 2*z4 + z2 - w1, 1264 1265 z4 - (w5 - z1), 1266 1267 (a*c)*z3 + (a*w5 - a*z1 - w4), 1268 1269 2 1270 (a*b)*z2 + (a *z1 - a*w2 + w4)}}, 1271 1272 2 1273 {a*b <> 0 and a <> 0 and a *z1 + a*b*w1 + a*c*w3 - 2*a*c*z1 - a*w2 - 2*b*w3 1274 1275 - 2*b*w4 + 4*b*z1 + w3 + w4 - 2*z1 = 0 and a*w5 - a*z1 - w3 - w4 + 2*z1 = 0 1276 1277 and d = 0, 1278 1279 {z4 + c*z3 + b*z2 + (a*z1 - w2), 1280 1281 a*z4 - z3 - w4, 1282 1283 2*z4 + z2 - w1, 1284 1285 z4 - (w5 - z1), 1286 1287 z3 - (w3 - 2*z1), 1288 1289 2 1290 (a*b)*z2 + (a *z1 + a*c*w3 - 2*a*c*z1 - a*w2 + w3 + w4 - 2*z1)}}, 1291 1292 {a*c*d*z1 - a*z1 + b*c*d*w1 - 2*b*c*d*w5 + 2*b*c*d*z1 - b*w1 + 2*b*w5 - 2*b*z1 1293 1294 2 2 2 2 2 2 1295 - c *d *w5 + c *d *z1 + c *d*w3 - 2*c *d*z1 - c*d*w2 + 2*c*d*w5 - 2*c*d*z1 1296 1297 - c*w3 + 2*c*z1 + w2 - w5 + z1 <> 0 and a <> 0 and 2*b + c*d - 1 <> 0 1298 1299 2 1300 and c*d - 1 <> 0 and d <> 0 and a *z1 - a*w2 + w4 = 0 and a*b = 0 1301 1302 and a*c + 1 = 0, 1303 1304 {a*c*d*z1 - a*z1 + b*c*d*w1 - 2*b*c*d*w5 + 2*b*c*d*z1 - b*w1 + 2*b*w5 - 2*b*z1 1305 1306 2 2 2 2 2 2 1307 - c *d *w5 + c *d *z1 + c *d*w3 - 2*c *d*z1 - c*d*w2 + 2*c*d*w5 - 2*c*d*z1 1308 1309 - c*w3 + 2*c*z1 + w2 - w5 + z1}}, 1310 1311 {a*c*w1 + 2*a*z1 - 2*c*w4 + w1 - 2*w2 <> 0 and a*c + 1 <> 0 and a <> 0 1312 1313 and d <> 0 and 1314 1315 2 1316 a *z1 + a*c*w3 - 2*a*c*z1 + a*d*z1 - a*w2 - c*d*w4 - d*w2 + w3 + w4 - 2*z1 = 0 1317 1318 and a*b + b*d = 0 and a*c - 2*b + 1 = 0, 1319 1320 { - (a*c*w1 + 2*a*z1 - 2*c*w4 + w1 - 2*w2)}}, 1321 1322 2 2 1323 {a*c - 2*b + 1 <> 0 and a*c + 1 <> 0 and a <> 0 and d <> 0 and a *c *w5 1324 1325 2 2 2 2 1326 - a *c *z1 + a *c*z1 + a*b*c*w1 - 2*a*b*c*w5 + 2*a*b*c*z1 - a*c *w4 - a*c*w2 1327 1328 + 2*a*c*w5 - 2*a*c*z1 + a*z1 + b*w1 - 2*b*w5 + 2*b*z1 - c*w4 - w2 + w5 - z1 1329 1330 = 0 and 1331 1332 2 1333 a *z1 + a*c*w3 - 2*a*c*z1 + a*d*z1 - a*w2 - c*d*w4 - d*w2 + w3 + w4 - 2*z1 = 0 1334 1335 and a*b + b*d = 0, 1336 1337 {z4 + c*z3 + b*z2 + (a*z1 - w2), 1338 1339 a*z4 - z3 - w4, 1340 1341 d*z4 + z3 - (w3 - 2*z1), 1342 1343 2*z4 + z2 - w1, 1344 1345 z4 - (w5 - z1), 1346 1347 2 1348 (a*c + 1)*z3 + (a*b)*z2 + (a *z1 - a*w2 + w4), 1349 1350 (a*c - 2*b + 1)*z2 - (a*c*w1 + 2*a*z1 - 2*c*w4 + w1 - 2*w2)}}, 1351 1352 {a*c + 1 <> 0 and a <> 0 and b <> 0 and d <> 0 and 1353 1354 2 1355 a *z1 + a*c*w3 - 2*a*c*z1 + a*d*z1 - a*w2 - c*d*w4 - d*w2 + w3 + w4 - 2*z1 = 0 1356 1357 and a*b + b*d = 0 and a*c*w1 + 2*a*z1 - 2*c*w4 + w1 - 2*w2 = 0 1358 1359 and a*c - 2*b + 1 = 0, 1360 1361 {z4 + c*z3 + b*z2 + (a*z1 - w2), 1362 1363 a*z4 - z3 - w4, 1364 1365 d*z4 + z3 - (w3 - 2*z1), 1366 1367 2*z4 + z2 - w1, 1368 1369 z4 - (w5 - z1), 1370 1371 2 1372 (a*c + 1)*z3 + (a*b)*z2 + (a *z1 - a*w2 + w4), 1373 1374 b*z2 + (a*c*w5 - a*c*z1 + a*z1 - c*w4 - w2 + w5 - z1)}}, 1375 1376 {a*c + 1 <> 0 and a <> 0 and d <> 0 and 1377 1378 2 1379 a *z1 + a*c*w3 - 2*a*c*z1 + a*d*z1 - a*w2 - c*d*w4 - d*w2 + w3 + w4 - 2*z1 = 0 1380 1381 and a*c*w1 + 2*a*z1 - 2*c*w4 + w1 - 2*w2 = 0 and a*c + 1 = 0 and b = 0, 1382 1383 {a*c*w5 - a*c*z1 + a*z1 - c*w4 - w2 + w5 - z1}}, 1384 1385 {a*d*z1 + b*d*w1 - 2*b*d*w5 + 2*b*d*z1 - d*w2 + w3 - 2*z1 <> 0 and a <> 0 1386 1387 2 1388 and b*c*d <> 0 and b*d <> 0 and d <> 0 and a *z1 - a*w2 + w4 = 0 and a*b = 0 1389 1390 and a*c + 1 = 0 and c*d - 1 = 0, 1391 1392 {a*d*z1 + b*d*w1 - 2*b*d*w5 + 2*b*d*z1 - d*w2 + w3 - 2*z1}}, 1393 1394 {a*d*z1 + b*d*w1 - 2*b*w3 + 4*b*z1 - d*w2 + w3 - 2*z1 <> 0 and a <> 0 1395 1396 2 1397 and b*d <> 0 and d <> 0 and a *z1 - a*w2 + w4 = 0 and a*b = 0 and a*c + 1 = 0 1398 1399 and b*c*d = 0 and c*d - 1 = 0, 1400 1401 {a*d*z1 + b*d*w1 - 2*b*w3 + 4*b*z1 - d*w2 + w3 - 2*z1}}, 1402 1403 2 1404 {a*d*z1 - d*w2 + w3 - 2*z1 <> 0 and a <> 0 and d <> 0 and a *z1 - a*w2 + w4 = 0 1405 1406 and a*b = 0 and a*c + 1 = 0 and b*d = 0 and c*d - 1 = 0, 1407 1408 {a*d*z1 - d*w2 + w3 - 2*z1}}, 1409 1410 {2*a*z1 - c*d*w1 + 2*c*w3 - 4*c*z1 + w1 - 2*w2 <> 0 and a <> 0 and c*d - 1 <> 0 1411 1412 2 1413 and d <> 0 and a *z1 - a*w2 + w4 = 0 and a*b = 0 and a*c + 1 = 0 1414 1415 and 2*b + c*d - 1 = 0, 1416 1417 {2*a*z1 - c*d*w1 + 2*c*w3 - 4*c*z1 + w1 - 2*w2}}, 1418 1419 {a*z1 + b*w1 - 2*b*w5 + 2*b*z1 + c*w3 - 2*c*z1 - w2 + w5 - z1 <> 0 and a <> 0 1420 1421 2 1422 and 2*b - 1 <> 0 and a *z1 + a*c*w3 - 2*a*c*z1 - a*w2 + w3 + w4 - 2*z1 = 0 1423 1424 and a*b = 0 and d = 0, 1425 1426 {a*z1 + b*w1 - 2*b*w5 + 2*b*z1 + c*w3 - 2*c*z1 - w2 + w5 - z1}}, 1427 1428 {a*z1 - c*d*w5 + c*d*z1 + c*w3 - 2*c*z1 - w2 + w5 - z1 <> 0 and a <> 0 1429 1430 2 1431 and c*d - 1 <> 0 and d <> 0 and a *z1 - a*w2 + w4 = 0 and a*b = 0 1432 1433 and a*c + 1 = 0 and 2*a*z1 - c*d*w1 + 2*c*w3 - 4*c*z1 + w1 - 2*w2 = 0 1434 1435 and 2*b + c*d - 1 = 0, 1436 1437 {a*z1 - c*d*w5 + c*d*z1 + c*w3 - 2*c*z1 - w2 + w5 - z1}}, 1438 1439 2 1440 {a <> 0 and b*c*d <> 0 and b*d <> 0 and d <> 0 and a *z1 - a*w2 + w4 = 0 1441 1442 and a*b = 0 and a*c + 1 = 0 1443 1444 and a*d*z1 + b*d*w1 - 2*b*d*w5 + 2*b*d*z1 - d*w2 + w3 - 2*z1 = 0 1445 1446 and c*d - 1 = 0, 1447 1448 {z4 + c*z3 + b*z2 + (a*z1 - w2), 1449 1450 a*z4 - z3 - w4, 1451 1452 d*z4 + z3 - (w3 - 2*z1), 1453 1454 2*z4 + z2 - w1, 1455 1456 z4 - (w5 - z1), 1457 1458 (2*b*c*d)*z3 + (a*d*z1 + b*d*w1 - 2*b*w3 + 4*b*z1 - d*w2 + w3 - 2*z1), 1459 1460 (b*d)*z2 + (a*d*z1 - d*w2 + w3 - 2*z1)}}, 1461 1462 2 1463 {a <> 0 and b*d <> 0 and d <> 0 and a *z1 - a*w2 + w4 = 0 and a*b = 0 1464 1465 and a*c + 1 = 0 and a*d*z1 + b*d*w1 - 2*b*w3 + 4*b*z1 - d*w2 + w3 - 2*z1 = 0 1466 1467 and b*c*d = 0 and c*d - 1 = 0, 1468 1469 {z4 + c*z3 + b*z2 + (a*z1 - w2), 1470 1471 a*z4 - z3 - w4, 1472 1473 d*z4 + z3 - (w3 - 2*z1), 1474 1475 2*z4 + z2 - w1, 1476 1477 z4 - (w5 - z1), 1478 1479 (c*d)*z3 + (d*w5 - d*z1 - w3 + 2*z1), 1480 1481 (b*d)*z2 + (a*d*z1 - d*w2 + w3 - 2*z1)}}, 1482 1483 {a <> 0 and 2*b + c*d - 1 <> 0 and c*d - 1 <> 0 and d <> 0 1484 1485 2 1486 and a *z1 - a*w2 + w4 = 0 and a*b = 0 and a*c*d*z1 - a*z1 + b*c*d*w1 1487 1488 2 2 2 2 1489 - 2*b*c*d*w5 + 2*b*c*d*z1 - b*w1 + 2*b*w5 - 2*b*z1 - c *d *w5 + c *d *z1 1490 1491 2 2 1492 + c *d*w3 - 2*c *d*z1 - c*d*w2 + 2*c*d*w5 - 2*c*d*z1 - c*w3 + 2*c*z1 + w2 1493 1494 - w5 + z1 = 0 and a*c + 1 = 0, 1495 1496 {z4 + c*z3 + b*z2 + (a*z1 - w2), 1497 1498 a*z4 - z3 - w4, 1499 1500 d*z4 + z3 - (w3 - 2*z1), 1501 1502 2*z4 + z2 - w1, 1503 1504 z4 - (w5 - z1), 1505 1506 (c*d - 1)*z3 + (b*d)*z2 + (a*d*z1 - d*w2 + w3 - 2*z1), 1507 1508 (2*b + c*d - 1)*z2 + (2*a*z1 - c*d*w1 + 2*c*w3 - 4*c*z1 + w1 - 2*w2)}}, 1509 1510 {a <> 0 and 2*b - 1 <> 0 1511 1512 2 1513 and a *z1 + a*c*w3 - 2*a*c*z1 - a*w2 + w3 + w4 - 2*z1 = 0 and a*b = 0 1514 1515 and a*z1 + b*w1 - 2*b*w5 + 2*b*z1 + c*w3 - 2*c*z1 - w2 + w5 - z1 = 0 1516 1517 and d = 0, 1518 1519 {z4 + c*z3 + b*z2 + (a*z1 - w2), 1520 1521 a*z4 - z3 - w4, 1522 1523 2*z4 + z2 - w1, 1524 1525 z4 - (w5 - z1), 1526 1527 z3 - (w3 - 2*z1), 1528 1529 (2*b - 1)*z2 + (2*a*z1 + 2*c*w3 - 4*c*z1 + w1 - 2*w2)}}, 1530 1531 2 1532 {a <> 0 and c*d - 1 <> 0 and d <> 0 and a *z1 - a*w2 + w4 = 0 and a*b = 0 1533 1534 and a*c + 1 = 0 and 2*a*z1 - c*d*w1 + 2*c*w3 - 4*c*z1 + w1 - 2*w2 = 0 1535 1536 and a*z1 - c*d*w5 + c*d*z1 + c*w3 - 2*c*z1 - w2 + w5 - z1 = 0 1537 1538 and 2*b + c*d - 1 = 0, 1539 1540 {z4 + c*z3 + b*z2 + (a*z1 - w2), 1541 1542 a*z4 - z3 - w4, 1543 1544 d*z4 + z3 - (w3 - 2*z1), 1545 1546 2*z4 + z2 - w1, 1547 1548 z4 - (w5 - z1), 1549 1550 (c*d - 1)*z3 + (b*d)*z2 + (a*d*z1 - d*w2 + w3 - 2*z1)}}, 1551 1552 2 1553 {a <> 0 and d <> 0 and a *z1 - a*w2 + w4 = 0 and a*b = 0 and a*c + 1 = 0 1554 1555 and a*d*z1 - d*w2 + w3 - 2*z1 = 0 and b*d = 0 and c*d - 1 = 0, 1556 1557 {z4 + c*z3 + b*z2 + (a*z1 - w2), 1558 1559 a*z4 - z3 - w4, 1560 1561 d*z4 + z3 - (w3 - 2*z1), 1562 1563 2*z4 + z2 - w1, 1564 1565 z4 - (w5 - z1), 1566 1567 (2*c)*z3 + (2*b - 1)*z2 + (2*a*z1 + w1 - 2*w2), 1568 1569 z2 - (w1 - 2*w5 + 2*z1)}}, 1570 1571 2 1572 {a <> 0 and a *z1 + a*c*w3 - 2*a*c*z1 - a*w2 + w3 + w4 - 2*z1 = 0 and a = 0 1573 1574 and 2*b - 1 = 0 and d = 0, 1575 1576 {2*a*z1 + 2*c*w3 - 4*c*z1 + w1 - 2*w2}}, 1577 1578 {b*d*w1 - 2*b*w3 - 2*b*w4 + 4*b*z1 - c*d*w4 - d*w2 + w3 + w4 - 2*z1 <> 0 1579 1580 and b*d <> 0 and d <> 0 and a = 0, 1581 1582 {a*d*z1 + b*d*w1 - 2*b*w3 - 2*b*w4 + 4*b*z1 - c*d*w4 - d*w2 + w3 + w4 - 2*z1}} 1583 1584 , 1585 1586 {b*d <> 0 and d*w5 - d*z1 - w3 - w4 + 2*z1 <> 0 and d <> 0 and a = 0 1587 1588 and b*d*w1 - 2*b*w3 - 2*b*w4 + 4*b*z1 - c*d*w4 - d*w2 + w3 + w4 - 2*z1 = 0, 1589 1590 {d*w5 - d*z1 - w3 - w4 + 2*z1}}, 1591 1592 {b*d <> 0 and d <> 0 and a = 0 1593 1594 and b*d*w1 - 2*b*w3 - 2*b*w4 + 4*b*z1 - c*d*w4 - d*w2 + w3 + w4 - 2*z1 = 0 1595 1596 and d*w5 - d*z1 - w3 - w4 + 2*z1 = 0, 1597 1598 {z4 + c*z3 + b*z2 + (a*z1 - w2), 1599 1600 d*z4 + z3 - (w3 - 2*z1), 1601 1602 2*z4 + z2 - w1, 1603 1604 z4 - (w5 - z1), 1605 1606 - z3 - w4, 1607 1608 (b*d)*z2 + (a*d*z1 - c*d*w4 - d*w2 + w3 + w4 - 2*z1)}}, 1609 1610 {b*w1 - 2*b*w5 + 2*b*z1 - c*w4 - w2 + w5 - z1 <> 0 and 2*b - 1 <> 0 and d <> 0 1611 1612 and a = 0 and b*d = 0 and c*d*w4 + d*w2 - w3 - w4 + 2*z1 = 0, 1613 1614 {a*z1 + b*w1 - 2*b*w5 + 2*b*z1 - c*w4 - w2 + w5 - z1}}, 1615 1616 {b*w1 - 2*b*w5 + 2*b*z1 - c*w4 - w2 + w5 - z1 <> 0 and 2*b - 1 <> 0 and a = 0 1617 1618 and d = 0 and w3 + w4 - 2*z1 = 0, 1619 1620 {a*z1 + b*w1 - 2*b*w5 + 2*b*z1 - c*w4 - w2 + w5 - z1}}, 1621 1622 {2*b - 1 <> 0 and d <> 0 and a = 0 and b*d = 0 1623 1624 and b*w1 - 2*b*w5 + 2*b*z1 - c*w4 - w2 + w5 - z1 = 0 1625 1626 and c*d*w4 + d*w2 - w3 - w4 + 2*z1 = 0, 1627 1628 {z4 + c*z3 + b*z2 + (a*z1 - w2), 1629 1630 d*z4 + z3 - (w3 - 2*z1), 1631 1632 2*z4 + z2 - w1, 1633 1634 z4 - (w5 - z1), 1635 1636 - z3 - w4, 1637 1638 (2*b - 1)*z2 + (2*a*z1 - 2*c*w4 + w1 - 2*w2)}}, 1639 1640 {2*b - 1 <> 0 and a = 0 and b*w1 - 2*b*w5 + 2*b*z1 - c*w4 - w2 + w5 - z1 = 0 1641 1642 and d = 0 and w3 + w4 - 2*z1 = 0, 1643 1644 {z4 + c*z3 + b*z2 + (a*z1 - w2), 1645 1646 2*z4 + z2 - w1, 1647 1648 z4 - (w5 - z1), 1649 1650 - z3 - w4, 1651 1652 z3 - (w3 - 2*z1), 1653 1654 (2*b - 1)*z2 + (2*a*z1 - 2*c*w4 + w1 - 2*w2)}}, 1655 1656 {c*d*w4 + d*w2 - w3 - w4 + 2*z1 <> 0 and d <> 0 and a = 0 and b*d = 0, 1657 1658 {a*d*z1 - c*d*w4 - d*w2 + w3 + w4 - 2*z1}}, 1659 1660 {2*c*w4 - w1 + 2*w2 <> 0 and a = 0 and 2*b - 1 = 0 and d = 0 1661 1662 and w3 + w4 - 2*z1 = 0, 1663 1664 {2*a*z1 - 2*c*w4 + w1 - 2*w2}}, 1665 1666 {d <> 0 and a = 0 and 2*b - 1 = 0 and c*d*w4 + d*w2 - w3 - w4 + 2*z1 = 0 1667 1668 and d = 0, 1669 1670 {2*a*z1 - 2*c*w4 + w1 - 2*w2}}, 1671 1672 {w3 + w4 - 2*z1 <> 0 and a = 0 and d = 0, 1673 1674 {w3 + w4 - 2*z1}}, 1675 1676 {a = 0 and 2*b - 1 = 0 and 2*c*w4 - w1 + 2*w2 = 0 and d = 0 1677 1678 and w3 + w4 - 2*z1 = 0, 1679 1680 {z4 + c*z3 + b*z2 + (a*z1 - w2), 1681 1682 2*z4 + z2 - w1, 1683 1684 z4 - (w5 - z1), 1685 1686 - z3 - w4, 1687 1688 z3 - (w3 - 2*z1), 1689 1690 b*z2 + (a*z1 - c*w4 - w2 + w5 - z1)}}} 1691 1692torder oo; 1693 1694 1695{{z4,z3,z2,z2},revgradlex} 1696 1697 1698% Two dimensional transportation problem 1699oo := torder({x33,x32,x31,x23,x22,x21,x13,x12,x11},lex); 1700 1701 1702oo := {{},lex} 1703 1704gsys({x11+x12+x13-a1,x11+x21+x31-b1,x12+x22+x32-b2,x13+x23+x33-b3, 1705x21+x22+x23-a2,x31+x32+x33-a3}); 1706 1707 1708{{a1 + a2 + a3 - b1 - b2 - b3 <> 0, 1709 1710 {a1 + a2 + a3 - b1 - b2 - b3}}, 1711 1712 {a1 + a2 + a3 - b1 - b2 - b3 = 0, 1713 1714 {x33 + x32 + x31 - a3, 1715 1716 x33 + x23 + x13 - b3, 1717 1718 x32 + x22 + x12 - b2, 1719 1720 x31 + x21 + x11 - b1, 1721 1722 x23 + x22 + x21 - a2, 1723 1724 x13 + x12 + x11 - a1}}} 1725 1726torder oo; 1727 1728 1729{{x33,x32,x31,x23,x22,x21,x13,x12,x11},lex} 1730 1731 1732% Thomas Weis's Example 1 1733oo := torder({x,y,z},lex); 1734 1735 1736oo := {{},lex} 1737 1738gsys({z*y*x-b*y*x-b*z*x+b**2*x-b*z*y+b**2*y+b**2*z-(n3+b**3), 1739z*y*x-a*y*x-a*z*x+a**2*x-a*z*y+a**2*y+a**2*z-(n3+a**3), 1740z*y*x-n1}); 1741 1742 1743 3 3 1744{{a *b - a*b + a*n1 - a*n3 - b*n1 + b*n3 <> 0 and a - b <> 0 1745 1746 2 2 1747 and a *b - a*b = 0, 1748 1749 3 3 1750 { - (a *b - a*b + a*n1 - a*n3 - b*n1 + b*n3)}}, 1751 1752 2 2 1753 {a *b - a*b <> 0 and a - b <> 0, 1754 1755 2 2 2 3 1756 {x*y*z - a*x*y - a*x*z + a *x - a*y*z + a *y + a *z - (a + n3), 1757 1758 2 2 2 3 1759 x*y*z - b*x*y - b*x*z + b *x - b*y*z + b *y + b *z - (b + n3), 1760 1761 x*y*z - n1, 1762 1763 2 2 2 2 1764 (a - b)*x*y + (a - b)*x*z - (a - b )*x + (a - b)*y*z - (a - b )*y 1765 1766 2 2 3 3 1767 - (a - b )*z + (a - b ), 1768 1769 2 2 2 2 2 2 1770 (a *b - a*b )*x + (a *b - a*b )*y + (a *b - a*b )*z 1771 1772 3 3 1773 - (a *b - a*b + a*n1 - a*n3 - b*n1 + b*n3), 1774 1775 2 2 2 2 2 1776 (a *b - a*b )*y + (a *b - a*b )*y*z 1777 1778 3 3 2 2 2 1779 - (a *b - a*b + a*n1 - a*n3 - b*n1 + b*n3)*y + (a *b - a*b )*z 1780 1781 3 3 1782 - (a *b - a*b + a*n1 - a*n3 - b*n1 + b*n3)*z 1783 1784 3 2 2 3 2 2 2 2 1785 + (a *b - a *b + a *n1 - a *n3 - b *n1 + b *n3), 1786 1787 2 2 3 3 3 2 1788 (a *b - a*b )*z - (a *b - a*b + a*n1 - a*n3 - b*n1 + b*n3)*z 1789 1790 3 2 2 3 2 2 2 2 2 2 1791 + (a *b - a *b + a *n1 - a *n3 - b *n1 + b *n3)*z - (a *b*n1 - a*b *n1)}}, 1792 1793 3 3 2 2 1794 {a - b <> 0 and a *b - a*b + a*n1 - a*n3 - b*n1 + b*n3 = 0 and a *b - a*b = 0 1795 1796 , 1797 1798 2 2 2 3 1799 {x*y*z - a*x*y - a*x*z + a *x - a*y*z + a *y + a *z - (a + n3), 1800 1801 2 2 2 3 1802 x*y*z - b*x*y - b*x*z + b *x - b*y*z + b *y + b *z - (b + n3), 1803 1804 x*y*z - n1, 1805 1806 2 2 2 2 1807 (a - b)*x*y + (a - b)*x*z - (a - b )*x + (a - b)*y*z - (a - b )*y 1808 1809 2 2 3 3 1810 - (a - b )*z + (a - b ), 1811 1812 2 2 2 2 2 2 2 2 2 1813 (a - b)*x*z - (a - b )*x*z + (a - b)*y*z - (a - b )*y*z - (a - b )*z 1814 1815 3 2 2 3 3 3 1816 + (a + a *b - a*b - b )*z - (a *b - a*b - a*n3 + b*n3), 1817 1818 2 2 2 2 2 2 2 2 3 2 2 3 1819 (a - b)*y *z - (a - b )*y *z - (a - b )*y*z + (a + a *b - a*b - b )*y*z 1820 1821 2 2 3 3 3 1822 - (a *b - a*b - a*n3 + b*n3)*y - (a *b - a*b - a*n3 + b*n3)*z 1823 1824 3 2 2 3 2 2 1825 + (a *b - a *b - a *n3 + b *n3)}}, 1826 1827 {a <> 0 and a - b = 0, 1828 1829 2 2 2 3 1830 {x*y*z - a*x*y - a*x*z + a *x - a*y*z + a *y + a *z - (a + n3), 1831 1832 2 2 2 3 1833 x*y*z - b*x*y - b*x*z + b *x - b*y*z + b *y + b *z - (b + n3), 1834 1835 x*y*z - n1, 1836 1837 2 2 2 3 1838 a*x*y + a*x*z - a *x + a*y*z - a *y - a *z + (a - n1 + n3), 1839 1840 2 2 2 2 2 2 3 1841 a*x*z - a *x*z + a*y*z - a *y*z - a *z + (a - n1 + n3)*z + a*n1, 1842 1843 2 2 2 2 2 2 3 1844 a*y *z - a *y *z - a *y*z + (a - n1 + n3)*y*z + (a*n1)*y + (a*n1)*z 1845 1846 2 1847 - a *n1}}, 1848 1849 {n1 - n3 <> 0 and a = 0 and b = 0, 1850 1851 3 1852 {a - n1 + n3}}, 1853 1854 {a = 0 and b = 0 and n1 - n3 = 0, 1855 1856 2 2 2 3 1857 {x*y*z - a*x*y - a*x*z + a *x - a*y*z + a *y + a *z - (a + n3), 1858 1859 2 2 2 3 1860 x*y*z - b*x*y - b*x*z + b *x - b*y*z + b *y + b *z - (b + n3), 1861 1862 x*y*z - n1}}} 1863 1864torder oo; 1865 1866 1867{{x,y,z},lex} 1868 1869 1870% Thomas Weis's Example 2 1871oo := torder({z,y,x,w},lex); 1872 1873 1874oo := {{},lex} 1875 1876gsys({w*x*y*z-x*y*z-w*y*z+y*z-w*x*z+x*z+w*z-z-w*x*y+x*y+w*y- 1877y+w*x-x-w-(b-1), 1878w*x*y*z-2*x*y*z-2*w*y*z+4*y*z-2*w*x*z+4*x*z+4*w*z-8*z-2*w*x*y+4x*y+ 18794*w*y-8*y+4*w*x-8*x-8*w-(c-16), 1880w*x*y*z-a,z+y+x+w-v}); 1881 1882 1883{{true, 1884 1885 {z*y*x*w - z*y*x - z*y*w + z*y - z*x*w + z*x + z*w - z - y*x*w + y*x + y*w - y 1886 1887 + x*w - x - w - (b - 1), 1888 1889 z*y*x*w - 2*z*y*x - 2*z*y*w + 4*z*y - 2*z*x*w + 4*z*x + 4*z*w - 8*z - 2*y*x*w 1890 1891 + 4*y*x + 4*y*w - 8*y + 4*x*w - 8*x - 8*w - (c - 16), 1892 1893 z*y*x*w - a, 1894 1895 z + y + x + w - v, 1896 1897 2 2 2 2 2 1898 y *x + y *w - 3*y + y*x + 2*y*x*w - (v + 3)*y*x + y*w - (v + 3)*y*w 1899 1900 2 2 2 2 1901 + (3*v)*y + x *w - 3*x + x*w - (v + 3)*x*w + (3*v)*x - 3*w + (3*v)*w 1902 1903 + (b - c - 7*v + 15), 1904 1905 2 2 2 1906 2*y + 2*y*x + 2*y*w - (2*v)*y + 2*x + 2*x*w - (2*v)*x + 2*w - (2*v)*w 1907 1908 + (a - 2*b + c + 6*v - 14), 1909 1910 3 2 2 2 1911 2*x + 2*x *w - (2*v)*x + 2*x*w - (2*v)*x*w + (a - 2*b + c + 6*v - 14)*x 1912 1913 3 2 1914 + 2*w - (2*v)*w + (a - 2*b + c + 6*v - 14)*w - (3*a - 4*b + c + 4*v - 12), 1915 1916 4 3 2 1917 2*w - (2*v)*w + (a - 2*b + c + 6*v - 14)*w - (3*a - 4*b + c + 4*v - 12)*w 1918 1919 + 2*a}}} 1920 1921torder oo; 1922 1923 1924{{z,y,x,w},lex} 1925 1926 1927end; 1928 1929Tested on x86_64-pc-windows CSL 1930Time (counter 1): 62 ms plus GC time: 16 ms 1931 1932End of Lisp run after 0.06+0.07 seconds 1933real 0.30 1934user 0.00 1935sys 0.07 1936