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Annotation of OpenXM/src/kan96xx/Kan/dr.sm1, Revision 1.34

1.34    ! takayama    1: % $OpenXM: OpenXM/src/kan96xx/Kan/dr.sm1,v 1.33 2004/08/31 05:30:20 takayama Exp $
1.1       maekawa     2: %% dr.sm1 (Define Ring) 1994/9/25, 26
                      3: %% This file is error clean.
                      4:
                      5: @@@.quiet {   }
1.5       takayama    6: { (macro package : dr.sm1,   9/26,1995 --- Version 12/10, 2000. ) message } ifelse
1.1       maekawa     7:
                      8: /ctrlC-hook {
                      9: %%% define your own routing in case of error.
                     10: } def
                     11: [(ctrlC-hook)
                     12: [(When ctrl-C is pressed, this function is executed.)
                     13:  (User can define one's own ctrlC-hook function.)
                     14: ]] putUsages
                     15:
                     16: %% n evenQ  bool
                     17: /evenQ {
                     18:   /arg1 set
                     19:   arg1 2 idiv  2 mul arg1 sub 0 eq
                     20:   { true }
                     21:   { false } ifelse
                     22: } def
                     23:
                     24: %% (x,y,z) polynomial_ring [x-list, d-list , paramList]
                     25: /ring_of_polynomials {
                     26:   /arg1 set
                     27:   [/vars /n /i /xList /dList /param] pushVariables
                     28:   %dup print (-----) message
                     29:   [
                     30:      (mmLarger) (matrix) switch_function
                     31:      (mpMult)   (poly) switch_function
                     32:      (red@)     (module1) switch_function
                     33:      (groebner) (standard) switch_function
                     34:      (isSameComponent) (x) switch_function
                     35:
                     36:      [arg1 to_records pop] /vars set
                     37:      vars length evenQ
                     38:      { }
                     39:      { vars [(PAD)] join /vars set }
                     40:      ifelse
                     41:      vars length 2 idiv /n set
                     42:      [ << n 1 sub >> -1 0
                     43:           { /i set
                     44:             vars i get
                     45:           } for
                     46:      ] /xList set
                     47:      [ << n 1 sub >> -1 0
                     48:           { /i set
                     49:             vars << i n add >> get
                     50:           } for
                     51:      ] /dList set
                     52:
1.28      takayama   53:      [@@@.Hsymbol] xList join [@@@.esymbol] join /xList set
1.1       maekawa    54:      [(h)] dList join [@@@.Esymbol] join /dList set
                     55:      [0 %% dummy characteristic
                     56:       << xList length >> << xList length >> << xList length >>
                     57:                                             << xList length >>
                     58:       << xList length 1 sub >> << xList length >> << xList length >>
                     59:                                                   << xList length >>
                     60:      ] /param set
                     61:
                     62:      [xList dList param] /arg1 set
                     63:    ] pop
                     64:    popVariables
                     65:    arg1
                     66: } def
                     67:
                     68: %% (x,y,z) polynomial_ring [x-list, d-list , paramList]
                     69: %% with no graduation and homogenization variables.
                     70: /ring_of_polynomials2 {
                     71:   /arg1 set
                     72:   [/vars /n /i /xList /dList /param] pushVariables
                     73:   %dup print (-----) message
                     74:   [
                     75:      (mmLarger) (matrix) switch_function
                     76:      (mpMult)   (poly) switch_function
                     77:      (red@)     (module1) switch_function
                     78:      (groebner) (standard) switch_function
                     79:      (isSameComponent) (x) switch_function
                     80:
                     81:      [arg1 to_records pop] /vars set
                     82:      vars length evenQ
                     83:      { }
                     84:      { vars [(PAD)] join /vars set }
                     85:      ifelse
                     86:      vars length 2 idiv /n set
                     87:      [ << n 1 sub >> -1 0
                     88:           { /i set
                     89:             vars i get
                     90:           } for
                     91:      ] /xList set
                     92:      [ << n 1 sub >> -1 0
                     93:           { /i set
                     94:             vars << i n add >> get
                     95:           } for
                     96:      ] /dList set
                     97:
                     98:      [0 %% dummy characteristic
                     99:       << xList length >> << xList length >> << xList length >>
                    100:                                             << xList length >>
                    101:       << xList length >> << xList length >> << xList length >>
                    102:                                             << xList length >>
                    103:      ] /param set
                    104:
                    105:      [xList dList param] /arg1 set
                    106:    ] pop
                    107:    popVariables
                    108:    arg1
                    109: } def
                    110:
                    111: %% (x,y,z) polynomial_ring [x-list, d-list , paramList]
                    112: %% with no homogenization variables.
                    113: /ring_of_polynomials3 {
                    114:   /arg1 set
                    115:   [/vars /n /i /xList /dList /param] pushVariables
                    116:   %dup print (-----) message
                    117:   [
                    118:      (mmLarger) (matrix) switch_function
                    119:      (mpMult)   (poly) switch_function
                    120:      (red@)     (module1) switch_function
                    121:      (groebner) (standard) switch_function
                    122:      (isSameComponent) (x) switch_function
                    123:
                    124:      [arg1 to_records pop] /vars set
                    125:      vars length evenQ
                    126:      { }
                    127:      { vars [(PAD)] join /vars set }
                    128:      ifelse
                    129:      vars length 2 idiv /n set
                    130:      [ << n 1 sub >> -1 0
                    131:           { /i set
                    132:             vars i get
                    133:           } for
                    134:      ] /xList set
                    135:      xList [@@@.esymbol] join /xList set
                    136:      [ << n 1 sub >> -1 0
                    137:           { /i set
                    138:             vars << i n add >> get
                    139:           } for
                    140:      ] /dList set
                    141:      dList [@@@.Esymbol] join /dList set
                    142:
                    143:      [0 %% dummy characteristic
                    144:       << xList length >> << xList length >> << xList length >>
                    145:                                             << xList length >>
                    146:       << xList length >> << xList length >> << xList length >>
                    147:                                             << xList length >>
                    148:      ] /param set
                    149:
                    150:      [xList dList param] /arg1 set
                    151:    ] pop
                    152:    popVariables
                    153:    arg1
                    154: } def
                    155:
                    156: /ring_of_differential_operators {
                    157:   /arg1 set
                    158:   [/vars /n /i /xList /dList /param] pushVariables
                    159:   [
                    160:      (mmLarger) (matrix) switch_function
                    161:      (mpMult)   (diff) switch_function
                    162:      (red@)     (module1) switch_function
                    163:      (groebner) (standard) switch_function
                    164:      (isSameComponent) (x) switch_function
                    165:
                    166:      [arg1 to_records pop] /vars set %[x y z]
                    167:      vars reverse /xList set         %[z y x]
                    168:      vars {@@@.Dsymbol 2 1 roll 2 cat_n} map
                    169:      reverse /dList set              %[Dz Dy Dx]
1.28      takayama  170:      [@@@.Hsymbol] xList join [@@@.esymbol] join /xList set
1.1       maekawa   171:      [(h)] dList join [@@@.Esymbol] join /dList set
                    172:      [0 1 1 1 << xList length >>
                    173:         1 1 1 << xList length 1 sub >> ] /param set
                    174:      [ xList dList param ] /arg1 set
                    175:   ] pop
                    176:   popVariables
                    177:   arg1
                    178: } def
                    179:
                    180: /ring_of_differential_operators3 {
                    181: %% with no homogenization variables.
                    182:   /arg1 set
                    183:   [/vars /n /i /xList /dList /param] pushVariables
                    184:   [
                    185:      (mmLarger) (matrix) switch_function
                    186:      (mpMult)   (diff) switch_function
                    187:      (red@)     (module1) switch_function
                    188:      (groebner) (standard) switch_function
                    189:      (isSameComponent) (x) switch_function
                    190:
                    191:      [arg1 to_records pop] /vars set %[x y z]
                    192:      vars reverse /xList set         %[z y x]
                    193:      vars {@@@.Dsymbol 2 1 roll 2 cat_n} map
                    194:      reverse /dList set              %[Dz Dy Dx]
                    195:      xList [@@@.esymbol] join /xList set
                    196:      dList [@@@.Esymbol] join /dList set
                    197:      [0 0 0 0 << xList length >>
                    198:         0 0 0 << xList length 1 sub >> ] /param set
                    199:      [ xList dList param ] /arg1 set
                    200:   ] pop
                    201:   popVariables
                    202:   arg1
                    203: } def
                    204:
                    205: /ring_of_q_difference_operators {
                    206:   /arg1 set
                    207:   [/vars /n /i /xList /dList /param] pushVariables
                    208:   [
                    209:      (mmLarger) (matrix) switch_function
                    210:      (mpMult)   (diff) switch_function
                    211:      (red@)     (module1) switch_function
                    212:      (groebner) (standard) switch_function
                    213:      (isSameComponent) (x) switch_function
                    214:
                    215:      [arg1 to_records pop] /vars set %[x y z]
                    216:      vars reverse /xList set         %[z y x]
                    217:      vars {@@@.Qsymbol 2 1 roll 2 cat_n} map
                    218:      reverse /dList set              %[Dz Dy Dx]
                    219:      [(q)] xList join [@@@.esymbol] join /xList set
                    220:      [(h)] dList join [@@@.Esymbol] join /dList set
                    221:      [0 1 << xList length >> << xList length >> << xList length >>
                    222:         1 << xList length 1 sub >> << xList length >> << xList length >> ]
                    223:      /param set
                    224:      [ xList dList param ] /arg1 set
                    225:   ] pop
                    226:   popVariables
                    227:   arg1
                    228: } def
                    229:
                    230: /ring_of_q_difference_operators3 {
                    231: %% with no homogenization and q variables.
                    232:   /arg1 set
                    233:   [/vars /n /i /xList /dList /param] pushVariables
                    234:   [
                    235:      (mmLarger) (matrix) switch_function
                    236:      (mpMult)   (diff) switch_function
                    237:      (red@)     (module1) switch_function
                    238:      (groebner) (standard) switch_function
                    239:      (isSameComponent) (x) switch_function
                    240:
                    241:      [arg1 to_records pop] /vars set %[x y z]
                    242:      vars reverse /xList set         %[z y x]
                    243:      vars {@@@.Qsymbol 2 1 roll 2 cat_n} map
                    244:      reverse /dList set              %[Dz Dy Dx]
                    245:      xList  [@@@.esymbol] join /xList set
                    246:      dList  [@@@.Esymbol] join /dList set
                    247:      [0 0 << xList length >> << xList length >> << xList length >>
                    248:         0 << xList length 1 sub >> << xList length >> << xList length >> ]
                    249:      /param set
                    250:      [ xList dList param ] /arg1 set
                    251:   ] pop
                    252:   popVariables
                    253:   arg1
                    254: } def
                    255:
                    256: /ring_of_difference_operators {
                    257:   /arg1 set
                    258:   [/vars /n /i /xList /dList /param] pushVariables
                    259:   [
1.8       takayama  260:      (This is an obsolete macro. Use ring_of_differential_difference_operators)
                    261:       error
1.1       maekawa   262:      (mmLarger) (matrix) switch_function
                    263:      (mpMult)   (difference) switch_function
                    264:      (red@)     (module1) switch_function
                    265:      (groebner) (standard) switch_function
                    266:      (isSameComponent) (x) switch_function
                    267:
                    268:      [arg1 to_records pop] /vars set %[x y z]
                    269:      vars reverse /xList set         %[z y x]
                    270:      vars {@@@.diffEsymbol 2 1 roll 2 cat_n} map
                    271:      reverse /dList set              %[Dz Dy Dx]
1.28      takayama  272:      [@@@.Hsymbol] xList join [@@@.esymbol] join /xList set
1.1       maekawa   273:      [(h)] dList join [@@@.Esymbol] join /dList set
                    274:      [0 1 1 << xList length >> << xList length >>
                    275:         1 1 << xList length 1 sub >> << xList length >> ] /param set
                    276:      [ xList dList param ] /arg1 set
                    277:   ] pop
                    278:   popVariables
                    279:   arg1
                    280: } def
                    281:
                    282:
1.8       takayama  283: /ring_of_differential_difference_operators {
                    284:   /arg1 set
                    285:   [/vars /n /i /xList /dList /param /dvar /evar /vars2 ] pushVariables
                    286:   [
                    287:      /vars arg1 def
                    288:      vars tag 6 eq not {
                    289:        ( List is expected as the argument for ring_of_differential_difference_operators ) error
                    290:      } { } ifelse
                    291:      vars 0 get /dvar set
                    292:      vars 1 get /evar set
                    293:      (mmLarger) (matrix) switch_function
                    294:      (mpMult)   (difference) switch_function
                    295:      (red@)     (module1) switch_function
                    296:      (groebner) (standard) switch_function
                    297:      (isSameComponent) (x) switch_function
                    298:
                    299:      [dvar to_records pop] /vars set %[x y z]
                    300:      vars reverse /xList set         %[z y x]
                    301:
                    302:      [evar to_records pop] /vars2 set %[s1 s2]
                    303:
                    304:      vars2 reverse  {@@@.Esymbol 2 1 roll 2 cat_n} map
                    305:      xList
                    306:      join /xList set   %[Es2 Es1 z y x]
                    307:
                    308:      vars2 reverse
                    309:      vars {@@@.Dsymbol 2 1 roll 2 cat_n} map
                    310:      reverse join /dList set              %[s2 s1 Dz Dy Dx]
1.28      takayama  311:      [@@@.Hsymbol] xList join [@@@.esymbol] join /xList set
1.8       takayama  312:      [(h)] dList join [@@@.Esymbol] join /dList set
                    313:      [0 1 1 << vars2 length 1 add >>  << xList length >>
                    314:         1 1 << vars2 length 1 add >> << xList length 1 sub >> ] /param set
                    315:      [ xList dList param ] /arg1 set
                    316:   ] pop
                    317:   popVariables
                    318:   arg1
                    319: } def
1.1       maekawa   320:
                    321: /reverse {
                    322:   /arg1 set
                    323:   arg1 length 1 lt
                    324:   { [ ] }
                    325:   {
                    326:     [
                    327:      <<  arg1 length 1 sub >> -1 0
                    328:      {
                    329:         arg1 2 1 roll get
                    330:       } for
                    331:      ]
                    332:    } ifelse
                    333: } def
                    334:
                    335: /memberQ {
                    336: %% a set0 memberQ bool
                    337:   /arg2 set  /arg1 set
                    338:   [/a /set0 /flag /i ] pushVariables
                    339:   [
                    340:      /a arg1 def  /set0 arg2 def
                    341:      /flag 0 def
                    342:      0 1 << set0 length 1 sub >>
                    343:      {
                    344:         /i set
                    345:         << set0 i get >> a eq
                    346:         {
                    347:            /flag 1 def
                    348:          }
                    349:         { }
                    350:         ifelse
                    351:      } for
                    352:   ] pop
                    353:   /arg1 flag def
                    354:   popVariables
                    355:   arg1
                    356: } def
                    357:
                    358: /transpose {
                    359:   /arg1 set
                    360:   [/mat /m /n /ans /i /j] pushVariables
                    361:   [
                    362:     /mat arg1 def
                    363:     /m mat length def
1.34    ! takayama  364:   {
        !           365:     m 0 eq { /ans [ ] def exit } { } ifelse
1.1       maekawa   366:     mat 0 get isArray
                    367:     {   }
                    368:     { (transpose: Argument must be an array of arrays.) error }
                    369:     ifelse
                    370:     /n mat 0 get length def
                    371:     /ans [ 1 1 n { pop [ 1 1 m { pop 0 } for ]} for ] def
                    372:     0 1 << m 1 sub >> {
                    373:        /i set
                    374:        0 1 << n 1 sub >> {
                    375:          /j set
                    376:          ans [ j i ]  <<  mat i get j get >> put
                    377:       } for
                    378:     } for
1.34    ! takayama  379:     exit
        !           380:    } loop
1.1       maekawa   381:    /arg1 ans def
                    382:   ] pop
                    383:   popVariables
                    384:   arg1
                    385: } def
                    386:
                    387:
                    388: /getPerm {
                    389: %% old new getPerm perm
                    390:   /arg2 set /arg1 set
                    391:   [/old /new /i /j /p] pushVariables
                    392:   [
                    393:     /old arg1 def
                    394:     /new arg2 def
                    395:     [
                    396:         /p old length def
                    397:         0 1 << p 1 sub >>
                    398:         {
                    399:            /i set
                    400:            0 1 << p 1 sub >>
                    401:            {
                    402:               /j set
                    403:               old i get
                    404:               new j get
                    405:               eq
                    406:               { j }
                    407:               {   } ifelse
                    408:             } for
                    409:          } for
                    410:      ] /arg1 set
                    411:    ] pop
                    412:    popVariables
                    413:    arg1
                    414: } def
                    415:
                    416: /permuteOrderMatrix {
                    417: %% order perm puermuteOrderMatrix newOrder
                    418:   /arg2 set /arg1 set
                    419:   [/order /perm /newOrder /k ] pushVariables
                    420:   [
                    421:     /order arg1 def
                    422:     /perm arg2 def
                    423:     order transpose /order set
                    424:     order 1 copy /newOrder set pop
                    425:
                    426:     0 1 << perm length 1 sub >>
                    427:     {
                    428:        /k set
                    429:        newOrder << perm k get >> << order k get >> put
                    430:     } for
                    431:     newOrder transpose /newOrder set
                    432:   ] pop
                    433:   /arg1 newOrder def
                    434:   popVariables
                    435:   arg1
                    436: } def
                    437:
                    438:
                    439:
                    440: /complement {
                    441: %% set0 universe complement compl
                    442:   /arg2 set /arg1 set
                    443:   [/set0 /universe /compl /i] pushVariables
                    444:    /set0 arg1 def  /universe arg2 def
                    445:   [
                    446:      0 1 << universe length 1 sub >>
                    447:      {
                    448:         /i set
                    449:         << universe i get >> set0 memberQ
                    450:         {   }
                    451:         { universe i get }
                    452:         ifelse
                    453:       } for
                    454:    ] /arg1 set
                    455:    popVariables
                    456:    arg1
                    457: } def
                    458:
                    459:
                    460: %%% from order.sm1
                    461:
                    462: %% size i evec [0 0 ... 0 1 0 ... 0]
                    463: /evec {
                    464:  /arg2 set /arg1 set
                    465:  [/size /iii] pushVariables
                    466:  /size arg1 def  /iii arg2 def
                    467:  [
                    468:    0 1 << size 1 sub >>
                    469:    {
                    470:       iii eq
                    471:       {  1 }
                    472:       {  0 }
                    473:       ifelse
                    474:    } for
                    475:   ] /arg1 set
                    476:   popVariables
                    477:   arg1
                    478: } def
                    479:
                    480: %% size i evec_neg [0 0 ... 0 -1 0 ... 0]
                    481: /evec_neg {
                    482:  /arg2 set /arg1 set
                    483:  [/size /iii] pushVariables
                    484:  /size arg1 def  /iii arg2 def
                    485:  [
                    486:    0 1 << size 1 sub >>
                    487:    {
                    488:       iii eq
                    489:       {  -1 }
                    490:       {  0 }
                    491:       ifelse
                    492:    } for
                    493:   ] /arg1 set
                    494:   popVariables
                    495:   arg1
                    496: } def
                    497:
                    498:
                    499: %% size i j e_ij  << matrix e(i,j) >>
                    500: /e_ij {
                    501:   /arg3 set /arg2 set /arg1 set
                    502:   [/size /k /i /j] pushVariables
                    503:   [
                    504:     /size arg1 def  /i arg2 def /j arg3 def
                    505:     [ 0 1 << size 1 sub >>
                    506:       {
                    507:          /k set
                    508:          k i eq
                    509:          { size j evec }
                    510:          {
                    511:             k j eq
                    512:             { size i evec }
                    513:             { size k evec }
                    514:             ifelse
                    515:           } ifelse
                    516:        } for
                    517:      ] /arg1 set
                    518:    ] pop
                    519:    popVariables
                    520:    arg1
                    521: } def
                    522:
                    523:
                    524: %% size i j d_ij  << matrix E_{ij} >>
                    525: /d_ij {
                    526:   /arg3 set /arg2 set /arg1 set
                    527:   [/size /k /i /j] pushVariables
                    528:   [
                    529:     /size arg1 def  /i arg2 def /j arg3 def
                    530:     [ 0 1 << size 1 sub >>
                    531:       {
                    532:          /k set
                    533:          k i eq
                    534:          { size j evec }
                    535:          {
                    536:             [ 0 1 << size 1 sub >> { pop 0} for ]
                    537:           } ifelse
                    538:        } for
                    539:      ] /arg1 set
                    540:    ] pop
                    541:    popVariables
                    542:    arg1
                    543: } def
                    544:
                    545: %% size matid << id matrix  >>
                    546: /matid {
                    547:   /arg1 set
                    548:   [/size /k ] pushVariables
                    549:   [
                    550:     /size arg1 def
                    551:     [ 0 1 << size 1 sub >>
                    552:       {
                    553:          /k set
                    554:          size k evec
                    555:        } for
                    556:      ] /arg1 set
                    557:    ] pop
                    558:    popVariables
                    559:    arg1
                    560: } def
                    561:
                    562:
                    563: %% m1 m2 oplus
                    564: /oplus {
                    565:   /arg2 set /arg1 set
                    566:   [/m1 /m2 /n /m  /k ] pushVariables
                    567:   [
                    568:     /m1 arg1 def  /m2 arg2 def
                    569:     m1 length /n set
                    570:     m2 length /m set
                    571:     [
                    572:       0 1 << n m add 1 sub >>
                    573:       {
                    574:         /k set
                    575:         k n lt
                    576:         {
                    577:             << m1 k get >> << m -1 evec >> join
                    578:         }
                    579:         {
                    580:             << n -1 evec >> << m2 << k n sub >> get >> join
                    581:         } ifelse
                    582:       } for
                    583:      ] /arg1 set
                    584:    ] pop
                    585:    popVariables
                    586:    arg1
                    587: } def
                    588:
                    589: %%%%%%%%%%%%%%%%%%%%%%%
                    590:
                    591: /eliminationOrderTemplate  { %% esize >= 1
                    592: %% if esize == 0, it returns reverse lexicographic order.
                    593: %%  m esize eliminationOrderTemplate mat
                    594:   /arg2 set /arg1 set
                    595:   [/m  /esize /m1 /m2 /k ] pushVariables
                    596:   [
                    597:     /m arg1 def  /esize arg2 def
                    598:     /m1 m esize sub 1 sub def
                    599:     /m2 esize 1 sub def
                    600:      [esize 0 gt
                    601:       {
                    602:        [1 1 esize
                    603:         { pop 1 } for
                    604:         esize 1 << m 1 sub >>
                    605:         { pop 0 } for
                    606:        ]  %% 1st vector
                    607:       }
                    608:       { } ifelse
                    609:
                    610:       m esize gt
                    611:       {
                    612:        [1 1  esize
                    613:         { pop 0 } for
                    614:         esize 1 << m 1 sub >>
                    615:         { pop 1 } for
                    616:        ]  %% 2nd vector
                    617:       }
                    618:       { } ifelse
                    619:
                    620:       m1 0 gt
                    621:       {
                    622:          m 1 sub -1 << m m1 sub >>
                    623:          {
                    624:               /k set
                    625:               m  k  evec_neg
                    626:          } for
                    627:       }
                    628:       { } ifelse
                    629:
                    630:       m2 0 gt
                    631:       {
                    632:          << esize 1 sub >> -1 1
                    633:          {
                    634:               /k set
                    635:               m  k  evec_neg
                    636:          } for
                    637:       }
                    638:       { } ifelse
                    639:
                    640:     ] /arg1 set
                    641:    ] pop
                    642:    popVariables
                    643:    arg1
                    644: } def
                    645:
                    646: /elimination_order {
                    647: %% [x-list d-list params]  (x,y,z) elimination_order
                    648: %%  vars                    evars
                    649: %% [x-list d-list params order]
                    650:   /arg2 set  /arg1 set
                    651:   [/vars /evars /univ /order /perm /univ0 /compl] pushVariables
                    652:   /vars arg1 def /evars [arg2 to_records pop] def
                    653:   [
                    654:     /univ vars 0 get reverse
                    655:           vars 1 get reverse join
                    656:     def
                    657:
                    658:     << univ length 2 sub >>
                    659:     << evars length >>
                    660:     eliminationOrderTemplate /order set
                    661:
                    662:     [[1]] order oplus [[1]] oplus /order set
                    663:
                    664:     /univ0 [univ reverse aload pop pop] reverse def %% [e,x,y,h] --> [x,y,h]
                    665:
                    666:     /compl
                    667:       [univ 0 get] evars join evars univ0 complement join
                    668:     def
                    669:     compl univ
                    670:     getPerm /perm set
                    671:     %%perm :: univ :: compl ::
                    672:
                    673:     order perm permuteOrderMatrix /order set
                    674:
                    675:
                    676:     vars [order] join /arg1 set
                    677:   ] pop
                    678:   popVariables
                    679:   arg1
                    680: } def
                    681:
                    682: /elimination_order2 {
                    683: %% [x-list d-list params]  (x,y,z) elimination_order
                    684: %%  vars                    evars
                    685: %% [x-list d-list params order]
                    686: %% with no graduation and homogenization variables.
                    687:   /arg2 set  /arg1 set
                    688:   [/vars /evars /univ /order /perm /compl] pushVariables
                    689:   /vars arg1 def /evars [arg2 to_records pop] def
                    690:   [
                    691:     /univ vars 0 get reverse
                    692:           vars 1 get reverse join
                    693:     def
                    694:
                    695:     << univ length  >>
                    696:     << evars length >>
                    697:     eliminationOrderTemplate /order set
                    698:     /compl
                    699:       evars << evars univ complement >> join
                    700:     def
                    701:     compl univ
                    702:     getPerm /perm set
                    703:     %%perm :: univ :: compl ::
                    704:
                    705:     order perm permuteOrderMatrix /order set
                    706:
                    707:     vars [order] join /arg1 set
                    708:   ] pop
                    709:   popVariables
                    710:   arg1
                    711: } def
                    712:
                    713:
                    714: /elimination_order3 {
                    715: %% [x-list d-list params]  (x,y,z) elimination_order
                    716: %%  vars                    evars
                    717: %% [x-list d-list params order]
                    718:   /arg2 set  /arg1 set
                    719:   [/vars /evars /univ /order /perm /univ0 /compl] pushVariables
                    720:   /vars arg1 def /evars [arg2 to_records pop] def
                    721:   [
                    722:     /univ vars 0 get reverse
                    723:           vars 1 get reverse join
                    724:     def
                    725:
                    726:     << univ length 1 sub >>
                    727:     << evars length >>
                    728:     eliminationOrderTemplate /order set
                    729:
                    730:     [[1]] order oplus  /order set
                    731:
                    732:     /univ0 [univ reverse aload pop pop] reverse def %% [e,x,y] --> [x,y]
                    733:
                    734:     /compl
                    735:       [univ 0 get] evars join evars univ0 complement join
                    736:     def
                    737:     compl univ
                    738:     getPerm /perm set
                    739:     %%perm :: univ :: compl ::
                    740:
                    741:     order perm permuteOrderMatrix /order set
                    742:
                    743:     vars [order] join /arg1 set
                    744:   ] pop
                    745:   popVariables
                    746:   arg1
                    747: } def
                    748:
                    749:
                    750: /define_ring {
                    751: %[  (x,y,z) ring_of_polynominals
                    752: %   (x,y) elimination_order
                    753: %   17
                    754: %] define_ring
                    755: % or
                    756: %[  (x,y,z) ring_of_polynominals
                    757: %   (x,y) elimination_order
                    758: %   17
                    759: %   [(keyword) value (keyword) value ...]
                    760: %] define_ring
                    761:    /arg1 set
                    762:    [/rp /param /foo] pushVariables
                    763:    [/rp arg1 def
                    764:
                    765:      rp 0 get length 3 eq {
                    766:        rp 0  [rp 0 get 0 get rp 0 get 1 get rp 0 get 2 get ]
                    767:              ( ) elimination_order put
                    768:      } { } ifelse
                    769:
                    770:     [
                    771:       rp 0 get 0 get             %% x-list
                    772:       rp 0 get 1 get             %% d-list
                    773:       rp 0 get 2 get /param set
                    774:       param 0 << rp 1 get >> put %% << rp 1 get >> is 17 in the example.
                    775:       param                      %% parameters.
                    776:       rp 0 get 3 get             %% order matrix.
                    777:       rp length 2 eq
                    778:       { [  ] }                   %% null optional argument.
                    779:       { rp 2 get }
                    780:       ifelse
                    781:     ]  /foo set
                    782:     foo aload pop set_up_ring@
                    783:    ] pop
                    784:    popVariables
                    785:    [(CurrentRingp)] system_variable
                    786: } def
                    787:
                    788:
                    789: [(define_qring)
                    790:   [( [varlist ring_of_q_difference_operators order characteristic] define_qring)
                    791:    (    Pointer to the ring. )
                    792:    (Example: [$x,y$ ring_of_q_difference_operators $Qx,Qy$ elimination_order)
                    793:    (          0] define_qring )
                    794:    (cf. define_ring, set_up_ring@ <coefficient ring>, ring_def, << ,, >>)
                    795:   ]
                    796: ] putUsages
                    797: /define_qring {
                    798: %[  (x,y,z) ring_of_q_difference_operators
                    799: %   (Qx,Qy) elimination_order
                    800: %   17
                    801: %] define_qring
                    802:    /arg1 set
                    803:    [/rp /param /foo /cring /ppp] pushVariables
                    804:    [/rp arg1 def
                    805:     /ppp rp 1 get def
                    806:     %% define coefficient ring.
                    807:     [(q) @@@.esymbol] [(h) @@@.Esymbol]
                    808:     [ppp 2 2 2 2 1 2 2 2]
                    809:     [[1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1]]
                    810:     [(mpMult) (poly)] set_up_ring@
                    811:     /cring  [(CurrentRingp)] system_variable def
                    812:
                    813:      rp 0 get length 3 eq {
                    814:        rp 0  [rp 0 get 0 get rp 0 get 1 get rp 0 get 2 get ]
                    815:              ( ) elimination_order put
                    816:      } { } ifelse
                    817:
                    818:     [
                    819:       rp 0 get 0 get             %% x-list
                    820:       rp 0 get 1 get             %% d-list
                    821:       rp 0 get 2 get /param set
                    822:       param 0 << rp 1 get >> put %% << rp 1 get >> is 17 in the example.
                    823:       param                      %% parameters.
                    824:       rp 0 get 3 get             %% order matrix.
                    825:       rp length 2 eq
                    826:       { [(mpMult) (diff) (coefficient ring) cring] }  %% optional argument.
                    827:       { [(mpMult) (diff) (coefficient ring) cring] rp 2 get join }
                    828:       ifelse
                    829:     ]  /foo set
                    830:     foo aload pop set_up_ring@
                    831:    ] pop
                    832:    popVariables
                    833:    [(CurrentRingp)] system_variable
                    834: } def
                    835:
                    836: [(ring_def)
                    837:  [(ring ring_def)
                    838:   (Set the current ring to the <<ring>>)
                    839:   (Example: [(x,y) ring_of_polynomials [[(x) 1]] weight_vector 0 ] define_ring)
                    840:   (          /R set)
                    841:   (          R ring_def)
                    842:   (In order to get the ring object R to which a given polynomial f belongs,)
                    843:   (one may use the command )
                    844:   (          f (ring) data_conversion /R set)
                    845:   (cf. define_ring, define_qring, system_variable, poly (ring) data_conversion)
                    846:   (cf. << ,, >>)
                    847:  ]
                    848: ] putUsages
                    849:
                    850: /ring_def {
                    851:   /arg1 set
                    852:   [(CurrentRingp) arg1] system_variable
                    853: } def
                    854:
                    855:
                    856:
                    857: /lexicographicOrderTemplate {
                    858: % size lexicographicOrderTemplate matrix
                    859:   /arg1 set
                    860:   [/k /size] pushVariables
                    861:   [
                    862:     /size arg1 def
                    863:     [ 0 1 << size 1 sub >>
                    864:       {
                    865:          /k set
                    866:          size k evec
                    867:        } for
                    868:     ] /arg1 set
                    869:   ] pop
                    870:   popVariables
                    871:   arg1
                    872: } def
                    873:
                    874: /lexicographic_order {
                    875: %% [x-list d-list params]  (x,y,z) lexicograhic_order
                    876: %%  vars                    evars
                    877: %% [x-list d-list params order]
                    878:   /arg2 set  /arg1 set
                    879:   [/vars /evars /univ /order /perm /univ0 /compl] pushVariables
                    880:   /vars arg1 def /evars [arg2 to_records pop] def
                    881:   [
                    882:     /univ vars 0 get reverse
                    883:           vars 1 get reverse join
                    884:     def
                    885:
                    886:     << univ length 2 sub >>
                    887:     lexicographicOrderTemplate /order set
                    888:
                    889:     [[1]] order oplus [[1]] oplus /order set
                    890:
                    891:     /univ0 [univ reverse aload pop pop] reverse def %% [e,x,y,h] --> [x,y,h]
                    892:
                    893:     /compl
                    894:       [univ 0 get] evars join evars univ0 complement join
                    895:     def
                    896:     compl univ
                    897:     getPerm /perm set
                    898:     %%perm :: univ :: compl ::
                    899:
                    900:     order perm permuteOrderMatrix /order set
                    901:
                    902:     vars [order] join /arg1 set
                    903:   ] pop
                    904:   popVariables
                    905:   arg1
                    906: } def
                    907:
                    908: /lexicographic_order2 {
                    909: %% [x-list d-list params]  (x,y,z) lexicograhic_order
                    910: %%  vars                    evars
                    911: %% [x-list d-list params order]
                    912: %% with no graduation and homogenization variables
                    913:   /arg2 set  /arg1 set
                    914:   [/vars /evars /univ /order /perm /compl] pushVariables
                    915:   /vars arg1 def /evars [arg2 to_records pop] def
                    916:   [
                    917:     /univ vars 0 get reverse
                    918:           vars 1 get reverse join
                    919:     def
                    920:
                    921:     << univ length  >>
                    922:     lexicographicOrderTemplate /order set
                    923:
                    924:     /compl
                    925:       evars << evars univ complement >> join
                    926:     def
                    927:     compl univ
                    928:     getPerm /perm set
                    929:
                    930:     order perm permuteOrderMatrix /order set
                    931:
                    932:     vars [order] join /arg1 set
                    933:   ] pop
                    934:   popVariables
                    935:   arg1
                    936: } def
                    937:
                    938: /lexicographic_order3 {
                    939: %% [x-list d-list params]  (x,y,z) lexicograhic_order
                    940: %%  vars                    evars
                    941: %% [x-list d-list params order]
                    942: %% with no homogenization variable.
                    943:   /arg2 set  /arg1 set
                    944:   [/vars /evars /univ /order /perm /univ0 /compl] pushVariables
                    945:   /vars arg1 def /evars [arg2 to_records pop] def
                    946:   [
                    947:     /univ vars 0 get reverse
                    948:           vars 1 get reverse join
                    949:     def
                    950:
                    951:     << univ length 1 sub >>
                    952:     lexicographicOrderTemplate /order set
                    953:
                    954:     [[1]] order oplus /order set
                    955:
                    956:     /univ0 [univ reverse aload pop pop] reverse def %% [e,x,y] --> [x,y]
                    957:
                    958:     /compl
                    959:       [univ 0 get] evars join evars univ0 complement join
                    960:     def
                    961:     compl univ
                    962:     getPerm /perm set
                    963:     %%perm :: univ :: compl ::
                    964:
                    965:     order perm permuteOrderMatrix /order set
                    966:
                    967:     vars [order] join /arg1 set
                    968:   ] pop
                    969:   popVariables
                    970:   arg1
                    971: } def
                    972:
                    973: %%%%%%   add_rings %%%%%%%%%%%%%% 10/5
                    974:
                    975: /graded_reverse_lexicographic_order {
                    976:   (  ) elimination_order
                    977: } def
                    978:
                    979:
                    980: /getX {
                    981: %% param [1|2|3|4] getX [var-lists]  ;  1->c,2->l,3->m,4->n
                    982:   /arg2 set /arg1 set
                    983:   [/k /param /func /low /top] pushVariables
                    984:   [
                    985:      /param arg1 def  /func arg2 def
                    986:      func 1 eq
                    987:      {
                    988:        /low 0 def
                    989:      }
                    990:      {
                    991:        /low << param 2 get >> << func 1 sub >> get def
                    992:      } ifelse
                    993:      /top << param 2 get >> << func 4 add >> get 1 sub def
                    994:      [
                    995:        low 1 top
                    996:        {
                    997:            /k set
                    998:           param 0 get k get
                    999:         } for
                   1000:      ] /arg1 set
                   1001:   ] pop
                   1002:   popVariables
                   1003:   arg1
                   1004: } def
                   1005:
                   1006: /getD {
                   1007: %% param [1|2|3|4] getD [var-lists]  ;  1->c,2->l,3->m,4->n
                   1008:   /arg2 set /arg1 set
                   1009:   [/k /param /func /low /top] pushVariables
                   1010:   [
                   1011:      /param arg1 def  /func arg2 def
                   1012:      func 1 eq
                   1013:      {
                   1014:        /low 0 def
                   1015:      }
                   1016:      {
                   1017:        /low << param 2 get >> << func 1 sub >> get def
                   1018:      } ifelse
                   1019:      /top << param 2 get >> << func 4 add >> get 1 sub def
                   1020:      [
                   1021:        low 1 top
                   1022:        {
                   1023:            /k set
                   1024:           param 1 get k get
                   1025:         } for
                   1026:      ] /arg1 set
                   1027:   ] pop
                   1028:   popVariables
                   1029:   arg1
                   1030: } def
                   1031:
                   1032: /getXV {
                   1033: %% param [1|2|3|4] getXV [var-lists]  ;  1->c,2->l,3->m,4->n
                   1034:   /arg2 set /arg1 set
                   1035:   [/k /param /func /low /top] pushVariables
                   1036:   [
                   1037:      /param arg1 def  /func arg2 def
                   1038:      /low << param 2 get >> << func 4 add >> get def
                   1039:      /top << param 2 get >>  func get 1 sub def
                   1040:      [
                   1041:        low 1 top
                   1042:        {
                   1043:            /k set
                   1044:           param 0 get k get
                   1045:         } for
                   1046:      ] /arg1 set
                   1047:   ] pop
                   1048:   popVariables
                   1049:   arg1
                   1050: } def
                   1051:
                   1052: /getDV {
                   1053: %% param [1|2|3|4] getDV [var-lists]  ;  1->c,2->l,3->m,4->n
                   1054:   /arg2 set /arg1 set
                   1055:   [/k /param /func /low /top] pushVariables
                   1056:   [
                   1057:      /param arg1 def  /func arg2 def
                   1058:      /low << param 2 get >> << func 4 add >> get def
                   1059:      /top << param 2 get >>  func get 1 sub def
                   1060:      [
                   1061:        low 1 top
                   1062:        {
                   1063:            /k set
                   1064:           param 1 get k get
                   1065:         } for
                   1066:      ] /arg1 set
                   1067:   ] pop
                   1068:   popVariables
                   1069:   arg1
                   1070: } def
                   1071:
                   1072: /reNaming {
                   1073:   %% It also changes oldx2 and oldd2, which are globals.
                   1074:   /arg1 set
                   1075:   [/i /j /new /count /ostr /k] pushVariables
                   1076:   [
                   1077:     /new arg1 def
                   1078:     /count 0 def
                   1079:     0 1 << new length 1 sub >> {
                   1080:        /i set
                   1081:       << i 1 add >> 1 << new length 1 sub >> {
                   1082:           /j set
                   1083:           << new i get >> << new j get >> eq
                   1084:           {
                   1085:              new j get /ostr set
                   1086:              (The two rings have the same name :) messagen
                   1087:              new i get messagen (.) message
                   1088:              (The name ) messagen
                   1089:              new i get messagen ( is changed into ) messagen
                   1090:              new j << new i get << 48 count add $string$ data_conversion >>
                   1091:                       2 cat_n >> put
                   1092:              new j get messagen (.) message
                   1093:              /oldx2 ostr << new j get >> reNaming2
                   1094:              /oldd2 ostr << new j get >> reNaming2
                   1095:              /count count 1 add def
                   1096:            }
                   1097:            { }
                   1098:            ifelse
                   1099:       } for
                   1100:     } for
                   1101:     /arg1 new def
                   1102:   ] pop
                   1103:   popVariables
                   1104:   arg1
                   1105: } def
                   1106:
                   1107: /reNaming2 {
                   1108:   %% array oldString newString reNaming2
                   1109:   %% /aa (x) (y) reNaming2
                   1110:   /arg3 set /arg2 set /arg1 set
                   1111:   [/array /oldString /newString /k] pushVariables
                   1112:   [
                   1113:     /array arg1 def /oldString arg2 def /newString arg3 def
                   1114:       0 1 << array load length 1 sub >>
                   1115:       {
                   1116:          /k set
                   1117:          << array load k get  >> oldString eq
                   1118:          {
                   1119:             array load k newString put
                   1120:           }
                   1121:           { } ifelse
                   1122:       } for
                   1123:    ] pop
                   1124:    popVariables
                   1125: } def
                   1126:
                   1127: /add_rings {
                   1128:   /arg2 set /arg1 set
                   1129:   [/param1 /param2
                   1130:    /newx /newd  /newv
                   1131:    /k /const /od1 /od2 /od
                   1132:    /oldx2 /oldd2  % these will be changed in reNaming.
                   1133:    /oldv
                   1134:   ] pushVariables
                   1135:   [
                   1136:      /param1 arg1 def /param2 arg2 def
                   1137:    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
                   1138:      /newx
                   1139:        [ ]
                   1140:        param2 1 getX join  param1 1 getX join
                   1141:        param2 1 getXV join param1 1 getXV join
                   1142:
                   1143:        param2 2 getX join  param1 2 getX join
                   1144:        param2 2 getXV join param1 2 getXV join
                   1145:
                   1146:        param2 3 getX join  param1 3 getX join
                   1147:        param2 3 getXV join param1 3 getXV join
                   1148:
                   1149:        param2 4 getX join  param1 4 getX join
                   1150:        param2 4 getXV join param1 4 getXV join
                   1151:      def
                   1152:      /newd
                   1153:        [ ]
                   1154:        param2 1 getD join  param1 1 getD join
                   1155:        param2 1 getDV join param1 1 getDV join
                   1156:
                   1157:        param2 2 getD join  param1 2 getD join
                   1158:        param2 2 getDV join param1 2 getDV join
                   1159:
                   1160:        param2 3 getD join  param1 3 getD join
                   1161:        param2 3 getDV join param1 3 getDV join
                   1162:
                   1163:        param2 4 getD join  param1 4 getD join
                   1164:        param2 4 getDV join param1 4 getDV join
                   1165:      def
                   1166:
                   1167:      /newv  newx newd join def
                   1168:      /oldx2 param2 0 get def  /oldd2 param2 1 get def
                   1169:      /oldx2 oldx2 {1 copy 2 1 roll pop} map def
                   1170:      /oldd2 oldd2 {1 copy 2 1 roll pop} map def
                   1171:      /newv newv reNaming def
                   1172:
                   1173:      /newx [
                   1174:        0 1 << newv length 2 idiv 1 sub >>
                   1175:        {
                   1176:           /k set
                   1177:           newv k get
                   1178:        } for
                   1179:      ] def
                   1180:      /newd [
                   1181:        0 1 << newv length 2 idiv 1 sub >>
                   1182:        {
                   1183:           /k set
                   1184:           newv << newv length 2 idiv k add >> get
                   1185:        } for
                   1186:      ] def
                   1187:      /const [
                   1188:         << param1 2 get 0 get >>
                   1189:         << param1 2 get 1 get  param2 2 get 1 get add >>
                   1190:         << param1 2 get 2 get  param2 2 get 2 get add >>
                   1191:         << param1 2 get 3 get  param2 2 get 3 get add >>
                   1192:         << param1 2 get 4 get  param2 2 get 4 get add >>
                   1193:         << param1 2 get 5 get  param2 2 get 5 get add >>
                   1194:         << param1 2 get 6 get  param2 2 get 6 get add >>
                   1195:         << param1 2 get 7 get  param2 2 get 7 get add >>
                   1196:         << param1 2 get 8 get  param2 2 get 8 get add >>
                   1197:     ] def
                   1198:
                   1199:     /od1 param1 3 get def /od2 param2 3 get def
                   1200:     od1 od2 oplus /od set
                   1201:
                   1202:     %%oldx2 :: oldd2 ::
                   1203:     << param1 0 get reverse >> << param1 1 get reverse >> join
                   1204:     << oldx2 reverse >> << oldd2 reverse >> join
                   1205:     join /oldv set
                   1206:
                   1207:
                   1208:     od << oldv << newx reverse newd reverse join >> getPerm >>
                   1209:     permuteOrderMatrix /od set
                   1210:
                   1211:      /arg1 [newx newd const od] def
                   1212:   ] pop
                   1213:   popVariables
                   1214:   arg1
                   1215: } def
                   1216:
                   1217:
                   1218: %%%% end of add_rings
                   1219:
                   1220:
                   1221:
                   1222: [(swap01) [
                   1223:    $[ .... ] swap01 [....]$
                   1224:    $Examples: [(x,y) ring_of_polynomials (x) elmination_order 0] swap01 $
                   1225:    $          define_ring$
                   1226: ]] putUsages
                   1227: %
                   1228: /swap01 {
                   1229:   /arg1 set
                   1230:   [/rg /ch ] pushVariables
                   1231:   [
                   1232:     arg1 0 get /rg set  % ring
                   1233:     arg1 1 get /ch set  % characteristics
                   1234:     [rg 0 get , rg 1 get , rg 2 get ,
                   1235:      << rg 3 get length >> 0 1 e_ij << rg 3 get >> mul ] /rg set
                   1236:     /arg1 [ rg ch ] def
                   1237:   ] pop
                   1238:   popVariables
                   1239:   arg1
                   1240: } def
                   1241:
                   1242: [(swap0k) [
                   1243:    $[ .... ] k swap0k [....]$
                   1244:    $Examples: [(x,y) ring_of_polynomials (x) elmination_order 0] 1 swap0k $
                   1245:    $          define_ring$
                   1246:    $swap01 == 1 swap0k$
                   1247: ]] putUsages
                   1248: %
                   1249: /swap0k {
                   1250:   /arg2 set
                   1251:   /arg1 set
                   1252:   [/rg /ch /kk] pushVariables
                   1253:   [
                   1254:     arg2 /kk set
                   1255:     arg1 0 get /rg set  % ring
                   1256:     arg1 1 get /ch set  % characteristics
                   1257:     [rg 0 get , rg 1 get , rg 2 get ,
                   1258:      << rg 3 get length >> 0 kk e_ij << rg 3 get >> mul ] /rg set
                   1259:     /arg1 [ rg ch ] def
                   1260:   ] pop
                   1261:   popVariables
                   1262:   arg1
                   1263: } def
                   1264:
                   1265: %%%%%%%%%%%%%   weight vector
                   1266: [(position)
                   1267:   [(set element position number)
                   1268:    (Example: [(cat) (dog) (hot chocolate)] (cat) position ===> 0.)
                   1269:   ]
                   1270: ] putUsages
                   1271: /position {
                   1272:   /arg2 set /arg1 set
                   1273:   [/univ /elem /num /flag] pushVariables
                   1274:   [
                   1275:      /univ arg1 def
                   1276:      /elem arg2 def
                   1277:      /num -1 def /flag -1 def
                   1278:      0 1 << univ length 1 sub >>
                   1279:      {
                   1280:         /num set
                   1281:         univ num get  elem  eq
                   1282:         { /flag 0 def exit }
                   1283:         {    }
                   1284:         ifelse
                   1285:      }  for
                   1286:      flag -1 eq
                   1287:      {/num -1 def}
                   1288:      {  }
                   1289:      ifelse
                   1290:   ] pop
                   1291:   /arg1 num def
                   1292:   popVariables
                   1293:   arg1
                   1294: } def
                   1295:
                   1296:
                   1297: [(evecw)
                   1298:   [(size position weight evecw  [0 0 ... 0 weight 0 ... 0] )
                   1299:    (Example: 3 0 113 evecw ===> [113  0  0])
                   1300:   ]
                   1301: ] putUsages
                   1302: /evecw {
                   1303:  /arg3 set /arg2 set /arg1 set
                   1304:  [/size /iii /www] pushVariables
                   1305:  /size arg1 def  /iii arg2 def /www arg3 def
                   1306:  [
                   1307:    0 1 << size 1 sub >>
                   1308:    {
                   1309:       iii eq
                   1310:       {  www }
                   1311:       {  0 }
                   1312:       ifelse
                   1313:    } for
                   1314:   ] /arg1 set
                   1315:   popVariables
                   1316:   arg1
                   1317: } def
                   1318:
                   1319: [(weight_vector)
                   1320:  [ ([x-list d-list params] [[(name) weight ...] [...] ...] weight_vector)
                   1321:    ([x-list d-list params order])
                   1322:    (Example:)
                   1323:    (   [(x,y,z) ring_of_polynomials [[(x) 100 (y) 10]] weight_vector 0] )
                   1324:    (   define_ring )
                   1325:   ]
                   1326: ] putUsages
                   1327: /weight_vector {
                   1328:   /arg2 set  /arg1 set
                   1329:   [/vars /univ /w-vectors /www /k /order1 /order2] pushVariables
                   1330:   /vars arg1 def /w-vectors arg2 def
                   1331:   [
                   1332:     /univ vars 0 get reverse
                   1333:           vars 1 get reverse join
                   1334:     def
1.33      takayama 1335:     w-vectors to_int32 /w-vectors set
1.1       maekawa  1336:     [
                   1337:     0 1 << w-vectors length 1 sub >>
                   1338:     {
                   1339:       /k set
                   1340:       univ w-vectors k get w_to_vec
                   1341:     } for
                   1342:     ] /order1 set
                   1343:     %% order1 ::
                   1344:
                   1345:     vars ( ) elimination_order 3 get /order2 set
                   1346:     vars [ << order1 order2 join >> ] join /arg1 set
                   1347:   ] pop
                   1348:   popVariables
                   1349:   arg1
                   1350: } def
                   1351:
                   1352: %% [@@@.esymbol (x) (y) (h)] [(x) 100 (y) 10] w_to_vec [0 100 10 0]
                   1353: %%  univ              www
                   1354: /w_to_vec {
                   1355:   /arg2 set  /arg1 set
                   1356:   [/univ /www /k /vname /vweight /ans] pushVariables
                   1357:   /univ arg1 def /www arg2 def
1.32      takayama 1358:   [
1.33      takayama 1359:     www to_int32 /www set
1.1       maekawa  1360:     /ans << univ length >> -1 0 evecw def
                   1361:     0  2  << www length 2 sub >>
                   1362:     {
                   1363:       %% ans ::
                   1364:       /k set
                   1365:       www k get /vname set
                   1366:       www << k 1 add >> get /vweight set
                   1367:       << univ length >>
                   1368:       << univ vname position >>
                   1369:       vweight evecw
                   1370:       ans add /ans set
                   1371:     } for
                   1372:     /arg1 ans def
                   1373:   ] pop
                   1374:   popVariables
                   1375:   arg1
                   1376: } def
                   1377:
                   1378: %%%%%%%%%% end of weight_vector macro
                   1379:
                   1380: %%%%%%%% eliminatev macro
                   1381: [(eliminatev)
                   1382:  [([g1 g2 g3 ...gm] [list of variables] eliminatev [r1 ... rp])
                   1383:   (Example: [(x y z - 1). (z-1). (y-1).] [(x) (y)] eliminatev [ z-1 ])
                   1384:  ]
                   1385: ] putUsages
                   1386: /eliminatev {
                   1387:  /arg2 set /arg1 set
                   1388:  [/gb /var /vars /ans /k] pushVariables
                   1389:  [
                   1390:    /gb arg1 def
                   1391:    /vars arg2 def
                   1392:    /ans gb def
                   1393:    0 1 << vars length 1 sub >> {
                   1394:      /k set
                   1395:      ans  << vars k get >> eliminatev.tmp
                   1396:      /ans set
                   1397:    } for
                   1398:    /arg1 ans def
                   1399:  ] pop
                   1400:  popVariables
                   1401:  arg1
                   1402: } def
                   1403: /eliminatev.tmp {
                   1404:   /arg2 set /arg1 set
                   1405:   [/gb /degs /ans /n /var /ff /rr /gg] pushVariables
                   1406:   [
                   1407:   /gb arg1 def
                   1408:   /var arg2 def
                   1409:   /degs gb {
                   1410:        /gg set
                   1411:        gg (0). eq
                   1412:        { 0 }
                   1413:        { gg (ring) data_conversion /rr set
                   1414:          gg  << var rr ,, >> degree
                   1415:        } ifelse
                   1416:     } map def
                   1417:   %%degs message
                   1418:   /ans [
                   1419:     0 1 << gb length 1 sub >> {
                   1420:       /n set
                   1421:       << degs n get  >>  0 eq
                   1422:       { gb n get /ff set
                   1423:         ff (0). eq
                   1424:         {  }
                   1425:         { ff } ifelse
                   1426:       }
                   1427:       {   } ifelse
                   1428:     } for
                   1429:   ] def
                   1430:   /arg1 ans def
                   1431:   ] pop
                   1432:   popVariables
                   1433:   arg1
                   1434: } def
                   1435:
                   1436: /eliminatev.tmp.org {
                   1437:   /arg2 set /arg1 set
                   1438:   [/gb /degs /ans /n /var /ff] pushVariables
                   1439:   [
                   1440:   /gb arg1 def
                   1441:   /var arg2 def
                   1442:   /degs gb {var . degree} map def
                   1443:   /ans [
                   1444:     0 1 << gb length 1 sub >> {
                   1445:       /n set
                   1446:       << degs n get  >>  0 eq
                   1447:       { gb n get /ff set
                   1448:         ff (0). eq
                   1449:         {  }
                   1450:         { ff } ifelse
                   1451:       }
                   1452:       {   } ifelse
                   1453:     } for
                   1454:   ] def
                   1455:   /arg1 ans def
                   1456:   ] pop
                   1457:   popVariables
                   1458:   arg1
                   1459: } def
                   1460: %%% end of eliminatev macro
                   1461:
                   1462: %%% macro for output
                   1463:
                   1464: [(isInteger)
                   1465:  [(obj isInteger bool) ]
                   1466: ] putUsages
                   1467: /isInteger {
                   1468:   (type?) data_conversion  << 0  (type?) data_conversion >> eq
                   1469: } def
                   1470:
                   1471: [(isArray)
                   1472:  [(obj isArray bool) ]
                   1473: ] putUsages
                   1474: /isArray {
                   1475:   (type?) data_conversion << [ ] (type?) data_conversion >>  eq
                   1476: } def
                   1477:
                   1478: [(isPolynomial)
                   1479:  [(obj isPolynomial bool) ]
                   1480: ] putUsages
                   1481: /isPolynomial {
                   1482:   (type?) data_conversion
                   1483:    << [(x) (var) 0] system_variable . (type?) data_conversion >> eq
                   1484: } def
                   1485:
                   1486: [(isString)
                   1487:  [(obj isString bool) ]
                   1488: ] putUsages
                   1489: /isString {
                   1490:   (type?) data_conversion
                   1491:    << (Hi) (type?) data_conversion >> eq
                   1492: } def
                   1493:
                   1494: [(isClass)
                   1495:  [(obj isClass bool) ]
                   1496: ] putUsages
                   1497: /isClass {
                   1498:   (type?) data_conversion  ClassP eq
                   1499: } def
                   1500:
                   1501: [(isUniversalNumber)
                   1502:  [(obj isUniversalNumber bool) ]
                   1503: ] putUsages
                   1504: /isUniversalNumber {
                   1505:   (type?) data_conversion  UniversalNumberP eq
                   1506: } def
                   1507:
                   1508: [(isDouble)
                   1509:  [(obj isDouble bool) ]
                   1510: ] putUsages
                   1511: /isDouble {
                   1512:   (type?) data_conversion  DoubleP eq
                   1513: } def
                   1514:
                   1515: [(isRational)
                   1516:  [(obj isRational bool) ]
                   1517: ] putUsages
                   1518: /isRational {
                   1519:   (type?) data_conversion  RationalFunctionP eq
1.7       takayama 1520: } def
                   1521:
                   1522: [(isRing)
                   1523:  [(obj isRing bool) ]
                   1524: ] putUsages
                   1525: /isRing {
                   1526:   (type?) data_conversion  RingP eq
1.1       maekawa  1527: } def
                   1528:
                   1529: /toString.tmp {
                   1530:   /arg1 set
                   1531:   [/obj /fname] pushVariables
                   1532:   /obj arg1 def
                   1533:   [
                   1534:     obj isArray
                   1535:     {
                   1536:        obj {toString.tmp} map
                   1537:     }
                   1538:     { } ifelse
                   1539:     obj isInteger
                   1540:     {
                   1541:        obj (dollar) data_conversion  %% not string. It returns the ascii code.
                   1542:     }
                   1543:     { } ifelse
                   1544:     obj isPolynomial
                   1545:     {
                   1546:        obj (string) data_conversion
                   1547:     }
                   1548:     { } ifelse
                   1549:     obj isString
                   1550:     { obj }
                   1551:     { } ifelse
                   1552:     obj isUniversalNumber
                   1553:     { obj (string) data_conversion } { } ifelse
                   1554:     obj isDouble
                   1555:     { obj (string) data_conversion } { } ifelse
                   1556:     obj isRational
                   1557:     { obj (string) data_conversion } { } ifelse
                   1558:     obj tag 0 eq
                   1559:     { (null) } { } ifelse
                   1560:
                   1561:     %%% New code that uses a file.
                   1562:     obj tag 2 eq obj tag 13 eq or obj tag 14 eq or obj tag 17 eq or
                   1563:     { [(getUniqueFileName) (/tmp/sm1_toString)] extension /fname set
                   1564:       [(outputObjectToFile) fname obj] extension pop
                   1565:       fname pushfile
                   1566:       [(/bin/rm -rf ) fname] cat system
                   1567:     } { } ifelse
                   1568:   ] /arg1 set
                   1569:   popVariables
                   1570:   arg1 aload pop
                   1571: } def
                   1572:
                   1573:
                   1574:
                   1575: %% [(xy) [(x+1) (2)]] toString.tmp2 ([ xy , [ x+1 , 2 ] ])
                   1576: /toString.tmp2 {
                   1577:   /arg1 set
                   1578:   [/obj /i /n /r] pushVariables
                   1579:   [
                   1580:     /obj arg1 def
                   1581:     obj isArray
                   1582:     {
1.2       takayama 1583:        [(LeftBracket)] system_variable %%( [ )
1.1       maekawa  1584:        obj {toString.tmp2} map /r set
                   1585:        /n r length 1 sub def
                   1586:        [0 1  n  {
                   1587:           /i set
                   1588:           i n eq {
                   1589:             r i get
                   1590:           }
                   1591:           { r i get ( , ) 2 cat_n }
                   1592:           ifelse
                   1593:         } for
                   1594:        ] aload length cat_n
1.2       takayama 1595:        [(RightBracket)] system_variable %%( ] )
1.1       maekawa  1596:        3 cat_n
                   1597:      }
                   1598:      {
                   1599:         obj
                   1600:      } ifelse
                   1601:    ] /arg1 set
                   1602:    popVariables
                   1603:    arg1 aload pop
                   1604: } def
                   1605:
                   1606:
                   1607: [(toString)
                   1608:  [(obj toString)
                   1609:   (Convert obj to a string.)
                   1610:   (Example: [ 1 (x+1). [ 2 (Hello)]] toString ==> $[ 1 , x+1 , [ 2 , Hello ]  ]$)
                   1611:  ]
                   1612: ] putUsages
                   1613: /toString {
                   1614:   /arg1 set
                   1615:   [/obj ] pushVariables
                   1616:   [
                   1617:     /obj arg1 def
                   1618:     obj isString
                   1619:     { obj }
                   1620:     { obj toString.tmp toString.tmp2 }
                   1621:     ifelse /arg1 set
                   1622:   ] pop
                   1623:   popVariables
                   1624:   arg1
                   1625: } def
                   1626:
                   1627: [(output)
                   1628:  [(obj output) (Output the object to the standard file sm1out.txt)]
                   1629: ] putUsages
                   1630: /output {
                   1631:   /arg1 set
                   1632:   [/obj /fd ] pushVariables
                   1633:   [
                   1634:     /obj arg1 def
                   1635:     (sm1out.txt) (a) file /fd set
                   1636:     (Writing to sm1out.txt  ...) messagen
                   1637:     [ fd << obj toString >> writestring ] pop
                   1638:     [ fd << 10 (string) data_conversion >> writestring ] pop
                   1639:     ( Done.) message
                   1640:     fd closefile
                   1641:   ] pop
                   1642:   popVariables
                   1643: } def
                   1644: %%%% end of macro for output.
                   1645: [(tag)
                   1646:  [(obj tag integer)
                   1647:   (tag returns datatype.)
                   1648:   (cf. data_conversion)
                   1649:   (Example: 2 tag IntegerP eq ---> 1)
                   1650:  ]
                   1651: ] putUsages
                   1652: /etag {(type??) data_conversion} def
                   1653: [(etag)
                   1654:  [(obj etag integer)
                   1655:   (etag returns extended object tag. cf. kclass.c)
                   1656:  ]
                   1657: ] putUsages
                   1658: /tag {(type?) data_conversion} def
                   1659: %% datatype constants
                   1660: /IntegerP 1  (type?) data_conversion def
                   1661: /LiteralP /arg1 (type?) data_conversion def   %Sstring
                   1662: /StringP (?) (type?) data_conversion def      %Sdollar
                   1663: /ExecutableArrayP  { 1 } (type?) data_conversion def
                   1664: /ArrayP [ 0 ] (type?) data_conversion def
                   1665: /PolyP  (1).  (type?) data_conversion def
                   1666: /FileP  13 def
                   1667: /RingP  14 def
                   1668: /UniversalNumberP 15 def
                   1669: /RationalFunctionP 16 def
                   1670: /ClassP 17 def
                   1671: /DoubleP 18 def
                   1672: /@.datatypeConstant.usage [
                   1673:  (IntegerP, LiteralP, StringP, ExecutableArrayP, ArrayP, PolyP, FileP, RingP,)
                   1674:  (UniversalNumberP, RationalFunctionP, ClassP, DoubleP)
                   1675:  (      return data type identifiers.)
                   1676:  (Example:  7 tag IntegerP eq  ---> 1)
                   1677: ] def
                   1678: [(IntegerP) @.datatypeConstant.usage ] putUsages
                   1679: [(LiteralP) @.datatypeConstant.usage ] putUsages
                   1680: [(StringP) @.datatypeConstant.usage ] putUsages
                   1681: [(ExecutableArrayP) @.datatypeConstant.usage ] putUsages
                   1682: [(ArrayP) @.datatypeConstant.usage ] putUsages
                   1683: [(PolyP) @.datatypeConstant.usage ] putUsages
                   1684: [(RingP) @.datatypeConstant.usage ] putUsages
                   1685: [(UniversalNumberP) @.datatypeConstant.usage ] putUsages
                   1686: [(RationalFunctionP) @.datatypeConstant.usage ] putUsages
                   1687: [(ClassP) @.datatypeConstant.usage ] putUsages
                   1688: [(DoubleP) @.datatypeConstant.usage ] putUsages
                   1689:
                   1690: [(,,)
                   1691:  [( string ring ,, polynomial)
                   1692:   (Parse the <<string>> as an element in the <<ring>> and returns)
                   1693:   (the polynomial.)
                   1694:   (cf. define_ring, define_qring, ring_def)
                   1695:   (Example: [(x,y) ring_of_polynomials [[(x) 1]] weight_vector 7]define_ring)
                   1696:   (         /myring set)
                   1697:   (         ((x+y)^4) myring ,, /f set)
                   1698: ]] putUsages
                   1699:
                   1700: /,, {
                   1701:   /arg2 set /arg1 set
                   1702:   [/rrr] pushVariables
                   1703:   [ arg1 tag StringP eq
                   1704:     arg2 tag RingP eq  and
                   1705:     { [(CurrentRingp)] system_variable /rrr set
                   1706:       [(CurrentRingp) arg2] system_variable
                   1707:       /arg1 arg1 expand def
                   1708:       [(CurrentRingp) rrr] system_variable
                   1709:     }
                   1710:     {(Argument Error for ,, ) error }
                   1711:     ifelse
                   1712:   ] pop
                   1713:   popVariables
                   1714:   arg1
                   1715: } def
                   1716:
                   1717: [(..)
                   1718:  [( string .. universalNumber)
                   1719:   (Parse the << string >> as a universalNumber.)
                   1720:   (Example:  (123431232123123).. /n set)
                   1721: ]] putUsages
                   1722: /.. { (universalNumber) data_conversion } def
                   1723:
                   1724: [(dc)
                   1725:  [(Abbreviation of data_conversion.)
                   1726: ]] putUsages
                   1727: /dc { data_conversion } def
                   1728:
                   1729:
                   1730: %%% start of shell sort macro.
                   1731: [(and) [(obj1 obj2 and bool)]] putUsages
                   1732: /and { add 1 copy 2 eq {pop 1} {pop 0} ifelse } def
                   1733:
                   1734: [(or) [(obj1 obj2 or bool)]] putUsages
                   1735: /or  { add 1 copy 2 eq {pop 1} { } ifelse} def
                   1736:
                   1737: [(ge) [(obj1 obj2 ge bool) (greater than or equal)]] putUsages
                   1738: %% 2 copy is equivalent to  dup 3 -1 roll dup 4 -2 roll 3 -1 roll 2 -1 roll
                   1739: /ge  { dup 3 -1 roll dup 4 -2 roll 3 -1 roll 2 -1 roll
                   1740:                eq {pop pop 1}
                   1741:                   { gt {1}
                   1742:                        {0}
                   1743:                        ifelse}
                   1744:                   ifelse} def
                   1745:
                   1746: [(le) [(obj1 obj2 le bool) (less than or equal)]] putUsages
                   1747: /le  { dup 3 -1 roll dup 4 -2 roll 3 -1 roll 2 -1 roll
                   1748:                eq {pop pop 1}
                   1749:                   { lt {1}
                   1750:                        {0}
                   1751:                        ifelse}
                   1752:                   ifelse} def
                   1753:
                   1754: [(break)
                   1755:  [(bool break)]
                   1756: ] putUsages
                   1757: /break { {exit} { } ifelse } def
                   1758:
                   1759: /not { 0 eq {1} {0} ifelse} def
                   1760: /append { /arg2 set [arg2] join } def
                   1761:
                   1762: [(power)
                   1763:  [(obj1 obj2 power obj3)
                   1764:   $obj3 is (obj1)^(obj2). cf. npower$
                   1765:   $Example:  (2). 8 power ::  ===>  256 $
                   1766:  ]
                   1767: ] putUsages
                   1768: %% From SSWork/yacc/incmac.sm1
                   1769: %% f k power f^k
                   1770: /power {
                   1771:   /arg2 set
                   1772:   /arg1 set
                   1773:   [/f /k /i /ans] pushVariables
                   1774:   [
1.26      takayama 1775:    /ans (1).. def
                   1776:    [(QuoteMode)] system_variable {
                   1777:      /f arg1 def   /k arg2 def
                   1778:      [(ooPower) f k] extension /ans set
                   1779:    } {
1.1       maekawa  1780:      /f arg1 def   /k arg2 ..int def
                   1781:      k 0 lt {
                   1782:        1 1 << 0 k sub >> {
                   1783:          /ans f ans {mul} sendmsg2 def
                   1784:        } for
                   1785:        /ans (1).. ans {div} sendmsg2 def
                   1786:      }
                   1787:      {
                   1788:        1 1 k {
                   1789:          /ans f ans {mul} sendmsg2 def
                   1790:        } for
                   1791:      } ifelse
1.26      takayama 1792:    } ifelse
                   1793:    /arg1 ans def
1.1       maekawa  1794:   ] pop
                   1795:   popVariables
                   1796:   arg1
                   1797: } def
                   1798: [(..int)
                   1799:  [ (universalNumber ..int int)]] putUsages
                   1800: /..int { %% universal number to int
                   1801:   (integer) data_conversion
                   1802: } def
                   1803: [(SmallRing) [(SmallRing is the ring of polynomials Q[t,x,T,h].)]] putUsages
                   1804: /SmallRing  [(CurrentRingp)] system_variable  def
                   1805:
                   1806: %%% From SSWork/yacc/lib/printSVector.modified.sm1
                   1807: %%% supporting code for printSVector.
                   1808: /greaterThanOrEqual {
                   1809:   /arg2 set /arg1 set
                   1810:   arg1 arg2 gt { 1 }
                   1811:   { arg1 arg2 eq {1} {0} ifelse} ifelse
                   1812: } def
                   1813:
                   1814: /lengthUniv {
                   1815:  length (universalNumber) dc
                   1816: } def
                   1817:
                   1818: /getUniv {
                   1819:  (integer) dc get
                   1820: } def  %% Do not forget to thow away /.
                   1821:
                   1822: %%[(@@@.printSVector)
                   1823: %% [( vector @@@.printSVector   outputs the <<vector>> in a pretty way.)
                   1824: %%  ( The elements of the vector must be strings.)
                   1825: %% ]
                   1826: %%] putUsages
                   1827:
                   1828: %%% compiled code by d0, 1996, 8/17.
                   1829: /@@@.printSVector {
                   1830:  /arg1 set
                   1831: [ %%start of local variables
                   1832: /keys /i /j /n /max /width /m /k /kk /tmp0 ] pushVariables [ %%local variables
                   1833: /keys arg1 def
                   1834: /n
                   1835: keys lengthUniv
                   1836:  def
                   1837: /max (0)..  def
                   1838: /i (0)..  def
                   1839: %%for init.
                   1840: %%for
                   1841: { i n  lt
                   1842:  {  } {exit} ifelse
                   1843: [ {%%increment
                   1844: /i  i (1).. add def
                   1845: } %%end of increment{A}
                   1846: {%%start of B part{B}
                   1847: keys i  getUniv lengthUniv
                   1848: max  gt
                   1849:  %% if-condition
                   1850:   { %%ifbody
                   1851: /max
                   1852: keys i   getUniv lengthUniv
                   1853:  def
                   1854:   }%%end if if body
                   1855:   { %%if- else part
                   1856:   } ifelse
                   1857: } %% end of B part. {B}
                   1858:  2 1 roll] {exec} map
                   1859: } loop %%end of for
                   1860: /max max (3)..  add
                   1861:  def
                   1862: /width (80)..  def
                   1863: /m (0)..  def
                   1864:
                   1865: %%while
                   1866: { m max  mul
                   1867: (80)..  lt
                   1868:  { } {exit} ifelse
                   1869:  /m m (1)..  add
                   1870:  def
                   1871: } loop
                   1872: /k (0)..  def
                   1873: /kk (0)..  def
                   1874: /i (0)..  def
                   1875: %%for init.
                   1876: %%for
                   1877: { i n  lt
                   1878:  {  } {exit} ifelse
                   1879: [ {%%increment
                   1880: /i  i (1).. add def
                   1881: } %%end of increment{A}
                   1882: {%%start of B part{B}
                   1883: keys i getUniv messagen
                   1884: /kk kk (1)..  add
                   1885:  def
                   1886: /k k
                   1887: keys i getUniv lengthUniv
                   1888:  add
                   1889:  def
                   1890: /tmp0 max
                   1891: keys i  getUniv lengthUniv
                   1892:  sub
                   1893:  def
                   1894: /j (0)..  def
                   1895: %%for init.
                   1896: %%for
                   1897: { j tmp0  lt
                   1898:  {  } {exit} ifelse
                   1899: [ {%%increment
                   1900: /j  j (1).. add def
                   1901: } %%end of increment{A}
                   1902: {%%start of B part{B}
                   1903: /k k (1)..  add
                   1904:  def
                   1905: kk m  lt
                   1906:  %% if-condition
                   1907:   { %%ifbody
                   1908: ( ) messagen
                   1909:   }%%end if if body
                   1910:   { %%if- else part
                   1911:   } ifelse
                   1912: } %% end of B part. {B}
                   1913:  2 1 roll] {exec} map
                   1914: } loop %%end of for
                   1915: kk m  greaterThanOrEqual
                   1916:  %% if-condition
                   1917:   { %%ifbody
                   1918: /kk (0)..  def
                   1919: /k (0)..  def
                   1920: newline
                   1921:   }%%end if if body
                   1922:   { %%if- else part
                   1923:   } ifelse
                   1924: } %% end of B part. {B}
                   1925:  2 1 roll] {exec} map
                   1926: } loop %%end of for
                   1927: newline
                   1928: /ExitPoint ]pop popVariables %%pop the local variables
                   1929: } def
                   1930: %%end of function
                   1931:
                   1932: /rest { % returns remainder of a given list
                   1933:   [ 2 1 roll  aload length -1 roll pop ]
                   1934: } def
                   1935: [(rest)
                   1936:  [(array rest the-rest-of-the-array)
                   1937:   (Ex. [1 2 [3 0]] rest ===> [2 [3 0]])
                   1938:  ]
                   1939: ] putUsages
                   1940:
                   1941: %% from SSkan/develop/minbase.sm1
                   1942: /reducedBase {
                   1943:   /arg1 set
                   1944:   [/base /minbase /n /i /j /myring /zero /f] pushVariables
                   1945:   [
                   1946:      /base arg1 def
                   1947:      base isArray { }
                   1948:      { (The argument of reducedBase must be an array of polynomials)
                   1949:         error
                   1950:      } ifelse
                   1951:      base 0 get isPolynomial { }
                   1952:      { (The element of the argument of reducedBase must be polynomials)
                   1953:         error
                   1954:      } ifelse
                   1955:      /myring  base 0 get (ring) dc def
                   1956:      /zero (0) myring ,, def
                   1957:      base length 1 sub /n set
                   1958:      /minbase [ 0 1 n { /i set base i get } for ] def
                   1959:      0 1 n {
                   1960:        /i set
                   1961:        minbase i get  /f set
                   1962:        f zero eq {
                   1963:        }
                   1964:        {
                   1965:            0 1 n {
                   1966:                /j set
                   1967:               << minbase j get zero eq >> << i j eq >> or {
                   1968:               }
                   1969:               {
                   1970:                  [(isReducible) << minbase j get >> f] gbext
                   1971:                  {
                   1972:                      minbase j zero put
                   1973:                   }
                   1974:                  {  } ifelse
                   1975:               } ifelse
                   1976:            } for
                   1977:        } ifelse
                   1978:      } for
                   1979:      minbase { minbase.iszero } map /arg1 set
                   1980:   ] pop
                   1981:   popVariables
                   1982:   arg1
                   1983: } def
                   1984:
                   1985: [(reducedBase)
                   1986:  [(base reducedBase reducedBase)
                   1987:   (<<reducedBase>> prunes redundant elements in the Grobner basis <<base>> and)
                   1988:   (returns <<reducedBase>>.)
                   1989:   (Ex. [(x^2+1). (x+1). (x^3).] reducedBase ---> [(x+1).])
                   1990:  ]
                   1991: ] putUsages
                   1992:
                   1993: %% package functions
                   1994: /minbase.iszero {
                   1995:   dup (0). eq {
                   1996:     pop
                   1997:   }
                   1998:   { } ifelse
                   1999: } def
                   2000:
                   2001: /== {
                   2002:   message
                   2003: } def
                   2004: [(==)
                   2005:  [(obj ==)
                   2006:   (Print obj)
                   2007:  ]
                   2008: ] putUsages
                   2009:
                   2010: /@@@.all_variables {
                   2011:   [/n /i] pushVariables
                   2012:   [
                   2013:      /n [(N)] system_variable def
                   2014:      [
                   2015:       0 1 n 1 sub {
                   2016:           /i set
                   2017:           [(x) (var) i] system_variable
                   2018:       } for
                   2019:       0 1 n 1 sub {
                   2020:           /i set
                   2021:           [(D) (var) i] system_variable
                   2022:       } for
                   2023:      ] /arg1 set
                   2024:   ] pop
                   2025:   popVariables
                   2026:   arg1
                   2027: } def
                   2028:
                   2029: /weightv {
                   2030:   @@@.all_variables
                   2031:   2 1 roll w_to_vec
                   2032: } def
                   2033:
                   2034: [(weightv)
                   2035:  [(array weightv weight_vector_for_init)
                   2036:   (cf. init)
                   2037:   (Example: /w [(x) 10 (h) 2] weightv def)
                   2038:   (         ((x-h)^10).  w init ::)
                   2039:  ]
                   2040: ] putUsages
                   2041:
                   2042: /output_order {
                   2043:   /arg1 set
                   2044:   [/vars /vlist /perm /total /ans] pushVariables
                   2045:   [
                   2046:     /vlist arg1 def
                   2047:     /vars @@@.all_variables def
                   2048:     vlist { vars 2 1 roll position } map  /perm set
                   2049:     perm ==
                   2050:     /total [ 0 1 [(N)] system_variable 2 mul 1 sub { } for ] def
                   2051:     perm perm total complement join /ans set
                   2052:     [(outputOrder) ans] system_variable
                   2053:   ] pop
                   2054:   popVariables
                   2055: } def
                   2056:
                   2057: [(output_order)
                   2058:  [$ [(v1) (v2) ...] output_order $
                   2059:   (Set the order of variables to print for the current ring.)
                   2060:   (cf. system_variable)
                   2061:   (Example:  [(y) (x)] output_order)
                   2062:   $           (x*y). ::   ===> y*x $
                   2063:  ]
                   2064: ] putUsages
                   2065:
                   2066: %% destraction.   SSkan/Kan/debug/des.sm1, 1998, 2/27 ,  3/1
                   2067: %% should be included in dr.sm1
                   2068:
                   2069: /factorial {
                   2070:   /arg2 set
                   2071:   /arg1 set
                   2072:   [ /f /n ] pushVariables
                   2073:   [
                   2074:     /f arg1 def
                   2075:     /n arg2 def
                   2076:     /ans (1).. def
                   2077:     n 0 lt { (f n factorial : n must be a non-negative integer)
                   2078:               error } { } ifelse
                   2079:     0 1 n 1 sub {
                   2080:        (universalNumber) dc /i set
                   2081:          ans  << f  i sub >> mul /ans set
                   2082:     } for
                   2083:     /arg1 ans def
                   2084:    ] pop
                   2085:    popVariables
                   2086:    arg1
                   2087: } def
                   2088:
                   2089: [(factorial)
                   2090:  [(f n factorial g)
                   2091:   $integer n,  g is f (f-1) ... (f-n+1)$
                   2092:  ]
                   2093: ] putUsages
                   2094:
                   2095:
                   2096: /destraction1 {
                   2097:   /arg4 set
                   2098:   /arg3 set
                   2099:   /arg2 set
                   2100:   /arg1 set
                   2101:   [/ww /f  /dx /ss /xx /coeff0 /expvec
                   2102:    /coeffvec /expvec2 /ans /one] pushVariables
                   2103:   [
                   2104:     /f arg1 def /xx arg2 def /dx arg3 def /ss arg4 def
                   2105:     /one (1). def %%
                   2106:     /ww [ xx toString -1 dx toString 1 ] weightv  def
                   2107:     f ww init f sub (0). eq {   }
                   2108:     { [(destraction1 : inhomogeneous with respect to )
                   2109:         xx  ( and )  dx ] cat error } ifelse
                   2110:     f [[xx one]] replace dx coefficients  /coeff0 set
                   2111:     /expvec coeff0 0 get { (integer) dc } map def
                   2112:     /coeffvec coeff0 1 get def
                   2113:     expvec { ss 2 -1 roll factorial } map /expvec2 set
                   2114:     expvec2 coeffvec mul  /ans set
                   2115:     /arg1 ans def
                   2116:    ] pop
                   2117:    popVariables
                   2118:    arg1
                   2119: } def
                   2120:
                   2121:
                   2122: /distraction {
                   2123:   /arg4 set
                   2124:   /arg3 set
                   2125:   /arg2 set
                   2126:   /arg1 set
                   2127:   [/f  /dx /ss /xx  /ans /n /i] pushVariables
                   2128:   [(CurrentRingp)] pushEnv
                   2129:   [
                   2130:     /f arg1 def /xx arg2 def /dx arg3 def /ss arg4 def
                   2131:     f (0). eq { /dist1.L goto } { f (ring) dc ring_def } ifelse
                   2132:     /n xx length  def
                   2133:     0 1 n 1 sub {
                   2134:        /i set
                   2135:        /f  f xx i get dx i get ss i get destraction1 /f set
                   2136:     } for
                   2137:     /dist1.L
                   2138:     /arg1 f def
                   2139:   ]pop
                   2140:   popEnv
                   2141:   popVariables
                   2142:   arg1
                   2143: } def
                   2144: [(distraction)
                   2145:  [(f [ list of x-variables ] [ list of D-variables ] [ list of s-variables ])
                   2146:   (   distraction  result )
                   2147:   $Example: (x Dx Dy + Dy). [(x). (y).] [(Dx). (Dy).] [(x). (y).] distraction$
                   2148:  ]
                   2149: ] putUsages
                   2150: /destraction { distraction } def
                   2151:
                   2152:
                   2153:
                   2154:
                   2155: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
                   2156: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
                   2157: %%%%%%%%%%%%%%%% sorting
                   2158: %/N 1000 def
                   2159: %/a.shell [N -1 0 { } for ]  def
                   2160: %a.shell 0 -1000 put
                   2161: %% You need gate keeper.
                   2162: [(shell)
                   2163:  [([gate-keeper f1 f2 ... fm] shell result)
                   2164:   (Sort the list. Gate-keeper should be the smallest element)]
                   2165: ] putUsages
                   2166: /shell {
                   2167:  /arg1 set
                   2168:  [/N /a.shell /h /i /v /j] pushVariables
                   2169:  [
                   2170:   /a.shell arg1 def
                   2171:   /N a.shell length  1 sub def
                   2172:
                   2173:   /h 1 def
                   2174:   {/h h 3 mul 1 add def
                   2175:    << h N ge >> break
                   2176:   } loop
                   2177:   {
                   2178:      /h << h 3 idiv >> def
                   2179:      << h 1 add >> 1 N {
                   2180:         /i set
                   2181:         /v a.shell i get def
                   2182:         /j i def
                   2183:         {
                   2184:            %% a.shell print newline
                   2185:            << a.shell << j h sub >> get >>  v  le  break
                   2186:            a.shell j << a.shell << j h sub >> get >> put
                   2187:            /j j h sub def
                   2188:            j  h le break
                   2189:          } loop
                   2190:          a.shell j v put
                   2191:      } for
                   2192:      h 1 lt break
                   2193:   } loop
                   2194:   /arg1 a.shell def
                   2195:  ] pop
                   2196:  popVariables
                   2197:  arg1
                   2198: } def
                   2199: %%%% end of shell sort macro
                   2200:
                   2201: /variableNames {
                   2202:   /arg1 set
                   2203:   [/in-variableNames /rrr /nnn /i /cp] pushVariables
                   2204:   [
                   2205:     /rrr arg1 def
                   2206:     [(CurrentRingp)] system_variable /cp set
                   2207:     [(CurrentRingp) rrr] system_variable
                   2208:     [(N)] system_variable /nnn set
                   2209:     [ 0 1 nnn 1 sub {
                   2210:          /i set [(x) (var) i] system_variable } for ]
                   2211:     [ 0 1 nnn 1 sub {
                   2212:          /i set [(D) (var) i] system_variable } for ]
                   2213:     join /arg1 set
                   2214:     [(CurrentRingp) cp] system_variable
                   2215:    ] pop
                   2216:    popVariables
                   2217:    arg1
                   2218: } def
                   2219:
                   2220:
                   2221: /makeRingMap {
                   2222:   /arg3 set /arg2 set /arg1 set
                   2223:   [/in-makeRingMap /corres /M /N /corresM /corresN
                   2224:     /vars /vars-org /i /p /ans /cp] pushVariables
                   2225:   [
                   2226:     /corres arg1 def /M arg2 def /N arg3 def
                   2227:     /corresM corres 0 get def
                   2228:     /corresN corres 1 get def
                   2229:     [(CurrentRingp)] system_variable /cp set
                   2230:     [(CurrentRingp) M] system_variable
                   2231:     M variableNames /vars set  vars 1 copy /vars-org set
                   2232:     0 1 corresM length 1 sub {
                   2233:       /i set
                   2234:       vars corresM i get position /p set
                   2235:       p -1 gt {
                   2236:         vars p  $($ corresN i get $)$ 3 cat_n put
                   2237:       }  {   } ifelse
                   2238:     } for
                   2239:     /arg1 [vars M  N vars-org] def
                   2240:     [(CurrentRingp) cp] system_variable
                   2241:   ] pop
                   2242:   popVariables
                   2243:   arg1
                   2244: } def
                   2245:
                   2246:
                   2247:
                   2248: /ringmap {
                   2249:   /arg2 set /arg1 set
                   2250:   [/in-ringmap /f /M2N /cp /f2] pushVariables
                   2251:   [
                   2252:     /f arg1 def /M2N arg2 def
                   2253:     [(CurrentRingp)] system_variable /cp set
                   2254:     f (0). eq { /f2 f def }
                   2255:     {
                   2256:        %f (ring) dc M2N 1 get eq
                   2257:        %{ }
                   2258:        %{ (The argument polynomial does not belong to the domain ring.) message
                   2259:        %  error
                   2260:        % } ifelse
                   2261:         [(CurrentRingp) M2N 1 get] system_variable
                   2262:         [(variableNames) M2N 0 get] system_variable
                   2263:         f toString /f2 set
                   2264:         [(variableNames) M2N 3 get] system_variable
                   2265:         f2 M2N 2 get ,, /f2 set
                   2266:     } ifelse
                   2267:     [(CurrentRingp) cp] system_variable
                   2268:     /arg1 f2 def
                   2269:   ] pop
                   2270:   popVariables
                   2271:   arg1
                   2272: } def
                   2273:
                   2274: [(makeRingMap)
                   2275:  [( rule  ring1 ring2 makeRingMap maptable   )
                   2276:   (makeRingMap is an auxiliary function for the macro ringmap. See ringmap)
                   2277:  ]
                   2278: ] putUsages
                   2279: [(ringmap)
                   2280:  [(f mapTable ringmap r)
                   2281:   (f is mapped to r where the map is defined by the mapTable, which is generated)
                   2282:   (by makeRingMap as follows:)
                   2283:   ( rule  ring1 ring2 makeRingMap maptable   )
                   2284:   $Example:$
                   2285:   $[(x,y) ring_of_differential_operators ( ) elimination_order 0] define_ring$
                   2286:   $/R1 set$
                   2287:   $[(t,y,z) ring_of_differential_operators ( ) elimination_order 0] define_ring$
                   2288:   $/R2 set$
                   2289:   $[[(x) (Dx)] [((t-1) Dt) (z)]] /r0 set$
                   2290:   $r0 R1 R2 makeRingMap /maptable set$
                   2291:   $(Dx-1) R1 ,, /ff set$
                   2292:   $ ff maptable ringmap :: $
                   2293:  ]
                   2294: ] putUsages
                   2295:
                   2296:
                   2297: /getVariableNames {
                   2298:   [/in-getVariableNames /ans /i /n] pushVariables
                   2299:   [
                   2300:      /n [(N)] system_variable def
                   2301:      [
                   2302:        n 1 sub -1 0 {
                   2303:          /i set
                   2304:          [(x) (var) i] system_variable
                   2305:        } for
                   2306:        n 1 sub -1 0{
                   2307:          /i set
                   2308:          [(D) (var) i] system_variable
                   2309:        } for
                   2310:       ] /arg1 set
                   2311:    ] pop
                   2312:   popVariables
                   2313:   arg1
                   2314: } def
                   2315: [(getVariableNames)
                   2316:  [(getVariableNames list-of-variables)
                   2317:   (Example: getVariableNames :: [e,x,y,E,H,Dx,Dy,h])
                   2318:  ]
                   2319: ] putUsages
                   2320:
                   2321: /tolower {
                   2322:   /arg1 set
                   2323:   [/in-tolower /s /sl] pushVariables
                   2324:   [
                   2325:     /s arg1 def
                   2326:     s (array) dc /s set
                   2327:     s { tolower.aux (string) dc } map /sl set
                   2328:     sl aload length cat_n /arg1 set
                   2329:   ] pop
                   2330:   popVariables
                   2331:   arg1
                   2332: } def
                   2333:
                   2334: /tolower.aux {
                   2335:   /arg1 set
1.9       takayama 2336:   arg1 64 gt  arg1 91 lt and
1.1       maekawa  2337:   { arg1 32 add }
                   2338:   { arg1 } ifelse
                   2339: } def
                   2340: [(tolower)
                   2341:  [(string tolower string2)
                   2342:   (Capital letters in string are converted to lower case letters.)
                   2343:   $Example:  (Hello World) tolower :: (hello world)$
                   2344:  ]
                   2345: ] putUsages
                   2346:
                   2347: /hilbert {
                   2348:   /arg2 set
                   2349:   /arg1 set
                   2350:   [/in-hilb /base /vlist /rrrorg /rrr /ff /strf] pushVariables
                   2351:   [
                   2352:      /base arg1 def
                   2353:      /vlist arg2 def
                   2354:      [(CurrentRingp)] system_variable /rrrorg set
                   2355:      /strf 0 def
                   2356:      vlist isString
                   2357:      {  /vlist [ vlist to_records pop ] def }
                   2358:      {  } ifelse
                   2359:      base isArray {  }
                   2360:      { (hilb : the first argument must be an array of polynomials.)
                   2361:         error
                   2362:      } ifelse
                   2363:      vlist isArray {  }
                   2364:      { (hilb : the second argument must be an array of polynomials.)
                   2365:         error
                   2366:      } ifelse
                   2367:
                   2368:      vlist 0 get isString{ /strf 1 def } { } ifelse
                   2369:      base 0 get isPolynomial {
                   2370:        base 0 get (ring) dc /rrr set
                   2371:      }
                   2372:      {
                   2373:        [  vlist { (,) } map  aload length cat_n ring_of_polynomials 0 ] define_ring
                   2374:        /rrr set
                   2375:        base { . } map /base set
                   2376:      } ifelse
                   2377:      vlist { dup isPolynomial {  } { rrr ,, } ifelse } map /vlist set
                   2378:
                   2379:      [(hilbert) base vlist] extension /ff set
                   2380:      [(CurrentRingp) rrrorg] system_variable
                   2381:      /arg1 ff def
                   2382:   ] pop
                   2383:   popVariables
                   2384:   arg1
                   2385: } def
                   2386:
                   2387: /hilbReduce {
                   2388:   /arg2 set
                   2389:   /arg1 set
                   2390:   [/hhh /f /d /vv /ans] pushVariables
                   2391:   [
                   2392:      /hhh arg1 def %% hilbert function
                   2393:      /vv arg2 def
                   2394:      /f hhh 1 get def
                   2395:      f (0). eq { /ans [0] def /hilbReduce.label goto } { } ifelse
                   2396:      f vv << f (ring) dc >> ,,  degree /vv set
                   2397:      hhh 0 get /d set
                   2398:      d   d  (integer) dc factorial /d set
                   2399:      d << vv (universalNumber) dc vv factorial >> idiv /d set
                   2400:      [(divByN) f d] gbext /ans set
                   2401:      ans 1 get (0). eq
                   2402:      {  }
                   2403:      { (hilbReduce : Invalid hilbert function ) error } ifelse
                   2404:      /hilbReduce.label
                   2405:      ans 0 get /arg1 set
                   2406:   ]  pop
                   2407:   popVariables
                   2408:   arg1
                   2409: } def
                   2410:
                   2411:
                   2412: [(hilbReduce)
                   2413:  [([f,g] v hilbReduce p)
                   2414:   (output of hilbert [f,g];  string v; poly p)
                   2415:   (p is  (g/(f!))*deg(g)!)
                   2416:   $ [(x) (y^3)] (x,y,z) hilbert (h) hilbReduce $
                   2417:  ]
                   2418: ] putUsages
                   2419: [(hilbert)
                   2420:  [(base vlist hilbert [m f])
                   2421:   (array of poly base; array of poly vlist; number m; poly f;)
                   2422:   (array of string base; array of string vlist; number m; poly f;)
                   2423:   (array of string base; string vlist; number m; poly f;)
                   2424:   ([m f] represents the hilbert function (a_d x^d + ...)/m! where f=a_d x^d + ...)
                   2425:   (The << base >> should be a reduced Grobner basis.)
                   2426:   (Or, when the << base >> is an array of string,)
                   2427:   (all entries should be monomials.)
                   2428:   (Example: [(x^2) (x y )] (x,y)  hilbert ::  [2, 2 h + 4] )
                   2429:   (Example: [(x^2) (y^2)] (x,y) hilbert (h) hilbReduce ::  4)
                   2430:   (Example: [(x^2) (y^2) (x y)] [(x) (y)] hilbert (h) hilbReduce ::  3)
                   2431:   (cf. hilb,    hilbReduce)
                   2432:  ]
                   2433: ] putUsages
                   2434:
                   2435: /hilb {
                   2436:   hilbert (h) hilbReduce
                   2437: } def
                   2438: [(hilb)
                   2439:  [(base vlist hilb f)
                   2440:   (array of poly base; array of poly vlist;  poly f;)
                   2441:   (array of string base; array of string vlist; poly f;)
                   2442:   (array of string base; string vlist; number m; poly f;)
                   2443:   (f is the hilbert function (a_d x^d + ...)/m!)
                   2444:   (The << base >> should be a reduced Grobner basis.)
                   2445:   (Or, when the << base >> is an array of string,)
                   2446:   (all entries should be monomials.)
                   2447:   (Example: [(x^2) (x y )] (x,y)  hilb ::   h + 2 )
                   2448:   (Example: [(x^2) (y^2)] (x,y) hilb  4)
                   2449:   (Example: [(x^2) (y^2) (x y)] [(x) (y)] hilb ::  3)
                   2450:   (cf. hilbert,    hilbReduce)
                   2451:  ]
                   2452: ] putUsages
                   2453:
                   2454: [(diff0)
                   2455:  [ (f v n diff0 fn)
                   2456:    (<poly> fn, v ; <integer>  n ; <poly> fn)
                   2457:    (fn = v^n f where v^n is the operator to take the n-th differential.)
                   2458:    (We can use diff0 only in the ring of differential operators.)
                   2459:    (Example: [(x) ring_of_differential_operators 0] define_ring )
                   2460:    (         (x^10-x). (Dx). 1 diff0 ::)
                   2461:  ]
                   2462: ] putUsages
                   2463: /diff0 {
                   2464:   /arg3 set /arg2 set /arg1 set
                   2465:   [/in-diff /f /v /n /fn /rrr] pushVariables
                   2466:   [
                   2467:     /f arg1 def /v arg2 def /n arg3 def
                   2468:     f (0). eq
                   2469:     { /fn (0). def }
                   2470:     {
                   2471:        f (ring) dc /rrr set
                   2472:        v toString (^) n toString 3 cat_n rrr ,,
                   2473:        f mul
                   2474:        [[v (0).] [(h) rrr ,, (1) rrr ,,]] replace /fn set
                   2475:      } ifelse
                   2476:      fn /arg1 set
                   2477:   ] pop
                   2478:   popVariables
                   2479:   arg1
                   2480: } def
                   2481:
                   2482: [(action)
                   2483:  [( f g action p )
                   2484:   (<poly> f,g,p)
                   2485:   (Act f on g. The result is p. The homogenization variable h is put to 1.)
                   2486:   (We can use diff0 only in the ring of differential operators.)
                   2487:   (Example: [(x) ring_of_differential_operators 0] define_ring )
                   2488:   (         (Dx^2). (x^2). action ::)
                   2489:  ]
                   2490: ] putUsages
                   2491: /action {
                   2492:   /arg2 set /arg1 set
                   2493:   [/in-action /f /g /h /rr /rr.org /rule] pushVariables
                   2494:   [
                   2495:     /f arg1 def /g arg2 def
                   2496:     /rr.org [(CurrentRingp)] system_variable def
                   2497:     f (0). eq
                   2498:     { /h (0). def }
                   2499:     {
                   2500:         f (ring) dc /rr set
                   2501:         [(CurrentRingp) rr] system_variable
                   2502:         f g mul /h set
                   2503:         /rule getVariableNames def
                   2504:         0 1 rule length 2 idiv { rule rest /rule set } for
                   2505:         rule { . [ 2 1 roll (0). ] } map /rule set
                   2506:         rule << rule length 1 sub >> [(h). (1).] put
                   2507:         %%ex. rule = [[(Dx1). (0).] [(Dx2). (0).] [(h). (1).]]
                   2508:         /h h rule replace def
                   2509:     } ifelse
                   2510:     [(CurrentRingp) rr.org ] system_variable
                   2511:     /arg1 h def
                   2512:   ] pop
                   2513:   popVariables
                   2514:   arg1
                   2515: } def
                   2516:
                   2517: [(ord_w)
                   2518:  [(ff [v1 w1 v2 w2 ... vm wm] ord_w d)
                   2519:   (poly ff; string v1; integer w1; ...)
1.11      takayama 2520:   (order of the initial of ff by the weight vector [w1 w2 ...])
1.1       maekawa  2521:   (Example: [(x,y) ring_of_polynomials 0] define_ring )
                   2522:   (          (x^2 y^3-x). [(x) 2 (y) 1] ord_w ::)
                   2523:  ]
                   2524: ] putUsages
                   2525: /ord_w {
                   2526:   /arg2 set /arg1 set
                   2527:   [/ord_w-in /fff /www /rrr /iii /ddd] pushVariables
                   2528:   [
                   2529:     /fff arg1 def
                   2530:     /www arg2 def
1.33      takayama 2531:     www to_int32 /www set
1.1       maekawa  2532:     fff (0). eq { /ddd -intInfinity def /ord_w.LLL goto} { } ifelse
                   2533:     fff (ring) dc /rrr set
                   2534:     fff init /fff set
                   2535:     /ddd 0 def
                   2536:     0 2 www length 1 sub {
                   2537:       /iii set
                   2538:       fff << www iii get rrr ,, >> degree
                   2539:       << www iii 1 add get >> mul
                   2540:       ddd add /ddd set
                   2541:     } for
1.12      takayama 2542:     /ord_w.LLL
                   2543:     /arg1 ddd def
                   2544:   ] pop
                   2545:   popVariables
                   2546:   arg1
                   2547: } def
                   2548:
                   2549: [(ord_w_all)
                   2550:  [(ff [v1 w1 v2 w2 ... vm wm] ord_w d)
                   2551:   (poly ff; string v1; integer w1; ...)
                   2552:   (order of ff by the weight vector [w1 w2 ...])
                   2553:   (Example: [(x,y,t) ring_of_polynomials 0] define_ring )
                   2554:   (          (x^2 y^3-x-t). [(t) 1 ] ord_w_all ::)
                   2555:  ]
                   2556: ] putUsages
                   2557: /ord_w_all {
                   2558:   /arg2 set /arg1 set
1.13      takayama 2559:   [/ord_w_all-in /fff /fff-in /www /rrr /iii /ddd /zzz /ddd-tmp] pushVariables
1.12      takayama 2560:   [
                   2561:     /fff arg1 def
                   2562:     /www arg2 def
1.33      takayama 2563:     www to_int32 /www set
1.13      takayama 2564:     fff (0). eq { /ddd -intInfinity def /ord_w_all.LLL goto} { } ifelse
                   2565:     /ddd -intInfinity def
1.12      takayama 2566:     fff (ring) dc /rrr set
                   2567:     /zzz (0) rrr ,, def
                   2568:     fff init /fff-in set
                   2569:     fff fff-in sub /fff set
                   2570:     {
1.13      takayama 2571:      /ddd-tmp 0 def
1.12      takayama 2572:      0 2 www length 1 sub {
                   2573:        /iii set
                   2574:        fff-in << www iii get rrr ,, >> degree
                   2575:        << www iii 1 add get >> mul
1.13      takayama 2576:        ddd-tmp add /ddd-tmp set
1.12      takayama 2577:      } for
1.13      takayama 2578:      ddd-tmp ddd gt { /ddd ddd-tmp def }  {  } ifelse
1.12      takayama 2579:      fff zzz eq { exit } {  } ifelse
                   2580:      fff init /fff-in set
                   2581:      fff fff-in sub /fff set
                   2582:     } loop
1.13      takayama 2583:     /ord_w_all.LLL
1.1       maekawa  2584:     /arg1 ddd def
                   2585:   ] pop
                   2586:   popVariables
                   2587:   arg1
                   2588: } def
                   2589:
                   2590: [(laplace0)
                   2591:  [
                   2592:  (f [v1 ... vn] laplace0 g)
                   2593:  (poly f ; string v1 ... vn ; poly g;)
                   2594:  (array of poly f ; string v1 ... vn ; array of poly g;)
                   2595:  ( g is the lapalce transform of f with respect to variables v1, ..., vn.)
                   2596:  $Example: (x Dx + y Dy + z Dz). [(x) (y) (Dx) (Dy)] laplace0$
                   2597:  $ x --> -Dx, Dx --> x,  y --> -Dy, Dy --> y. $
                   2598:  ]
                   2599: ] putUsages
                   2600: /laplace0 {
                   2601:   /arg2 set /arg1 set
                   2602:   [/in-laplace0 /ff /rule /vv /nn /ii /v0 /v1 /rr /ans1 /Dascii
                   2603:   ] pushVariables
                   2604:   [
                   2605:   /ff arg1 def /vv arg2 def
                   2606:   /Dascii @@@.Dsymbol (array) dc 0 get def %%D-clean
                   2607:   /rule [ ] def
                   2608:   ff isPolynomial {
                   2609:    ff (0). eq { /ans1 (0). def }
                   2610:    {
                   2611:      ff (ring) dc /rr set
                   2612:      /nn vv length def
                   2613:      0 1 nn 1 sub {
                   2614:       /ii set
                   2615:       vv ii get (type?) dc 1 eq
                   2616:       {  }  % skip, may be weight [(x) 2 ] is OK.
                   2617:       {
                   2618:          /v0 vv ii get (string) dc def
                   2619:          v0 (array) dc 0 get Dascii eq  %% If the first character is D?
                   2620:          {  rule  %% Dx-->x
                   2621:             [v0 rr ,,
                   2622:             v0 (array) dc rest { (string) dc} map aload length cat_n rr ,,]
                   2623:             append /rule set
                   2624:          }
                   2625:          { rule   %% x --> -Dx
                   2626:            [v0 rr ,,
                   2627:             (0).
                   2628:             [Dascii] v0 (array) dc join { (string) dc } map aload length
                   2629:             cat_n rr ,,  sub
                   2630:            ]
                   2631:            append /rule set
                   2632:          } ifelse
                   2633:       } ifelse
                   2634:      } for
                   2635:      % rule message
                   2636:      ff rule replace [[(h) rr ,, (1) rr ,,]] replace /ans1 set
                   2637:      } ifelse
                   2638:     }
                   2639:    {
                   2640:        ff isArray { /ans1 ff {vv laplace0 } map def }
                   2641:        {
                   2642:          (laplace0 : the first argument must be a polynomial.) error
                   2643:        }ifelse
                   2644:     } ifelse
                   2645:     /arg1 ans1 def
                   2646:   ] pop
                   2647:   popVariables
                   2648:   arg1
                   2649: } def
                   2650:
                   2651: [(ip1)
                   2652:  [( [v1 ... vn] [w1 ... wn] m ip1 [f1 ... fs])
                   2653:   (<poly> v1 ... vn ; <integer> w1 ... wn m)
                   2654:   (<poly> f1 ... fs )
                   2655:   (Example: [(x,y) ring_of_differential_operators 0] define_ring )
                   2656:   (         [(Dx). (Dy).] [2 1] 3 ip1  ::    [(2 Dx Dy). (Dy^3).])
                   2657:   (         Returns Dx^p Dy^q such that  2 p + 1 q = 3.)
                   2658:  ]
                   2659: ] putUsages
                   2660: /ip1 {
                   2661:   /arg3 set /arg2 set /arg1 set
                   2662:   [/in-ip1 /vv /ww /m /ans /k /tt /rr /rr.org /ff /tmp1] pushVariables
                   2663:   [
                   2664:      /vv arg1 def /ww arg2 def /m arg3 def
                   2665:      vv 0 get (ring) dc /rr set
                   2666:      /rr.org [(CurrentRingp)] system_variable def
                   2667:      [(CurrentRingp) rr] system_variable
                   2668:      [(x) (var) [(N)] system_variable 1 sub ] system_variable . /tt set
                   2669:      /ans [ ] def
                   2670:      m 0 lt
                   2671:      {  }
                   2672:      {
                   2673:        vv
                   2674:        ww { tt 2 1 roll power } map mul /tmp1 set
                   2675:  %%      (tmp1 = ) messagen tmp1 message
                   2676:        0 1 m {
                   2677:          /k set
                   2678:          k 0 eq {
                   2679:            /ff (1). def
                   2680:          }
                   2681:          { tmp1 k power /ff set } ifelse
                   2682:          ff [[(h). (1).]] replace /ff set
                   2683:  %%        ff message
                   2684:          {
                   2685:            ff init tt degree m eq {
                   2686:              /ans ans [ ff init [[tt (1).]] replace ] join def
                   2687:            } { } ifelse
                   2688:            ff ff init sub /ff set
                   2689:            ff (0). eq { exit } {  } ifelse
                   2690:          } loop
                   2691:         } for
                   2692:       } ifelse
                   2693:      [(CurrentRingp) rr.org] system_variable
                   2694:      /arg1 ans def
                   2695:   ] pop
                   2696:   popVariables
                   2697:   arg1
                   2698: } def
                   2699:
                   2700: [(findIntegralRoots)
                   2701:  [( f findIntegralRoots vlist)
                   2702:   (poly f; list of integers vlist;)
                   2703:   (string f; list of integers vlist;)
                   2704:   (f is a polynomials in one variable s. vlist the list of integral roots sorted.)
                   2705:   (Example: (s^4-1) findIntegralRoots )
                   2706:  ]
                   2707: ] putUsages
                   2708:
                   2709: /findIntegralRoots { findIntegralRoots.slow } def
                   2710:
                   2711: /findIntegralRoots.slow {  %% by a stupid algorithm
                   2712:   /arg1 set
                   2713:   [/in-findIntegralRoots
                   2714:    /ff /kk /roots /rrr /nn /k0 /d.find
                   2715:   ] pushVariables
                   2716:   [
                   2717:     /ff arg1 def
                   2718:     /roots [  ] def
                   2719:     /rrr [(CurrentRingp)] system_variable def
                   2720:     ff toString /ff set
                   2721:     [(s) ring_of_polynomials ( ) elimination_order 0] define_ring
                   2722:     ff . /ff set
                   2723:
                   2724:     %%ff message  %% Cancel the common numerical factor of the polynomial ff.
                   2725:     ff (s). coeff 1 get { (universalNumber) dc } map ngcd /d.find set
                   2726:     [(divByN) ff d.find] gbext 0 get /ff set
                   2727:     %% d.find message
                   2728:     %% ff message
                   2729:
                   2730:     ff [[(s). (0).]] replace /k0 set
                   2731:     k0 (universalNumber) dc /k0 set
                   2732:     k0 (0).. eq { roots (0).. append /roots set } { } ifelse
                   2733:
                   2734:     {
                   2735:       ff [[(s). (0).]] replace /nn set
                   2736:       nn (universalNumber) dc /nn set
                   2737:       nn (0).. eq
                   2738:       { (s^(-1)). ff mul /ff set }
                   2739:       { exit }
                   2740:       ifelse
                   2741:     } loop
                   2742:     ff [[(s). (0).]] replace /k0 set
                   2743:     k0 (universalNumber) dc /k0 set
                   2744:     k0 (-40000).. gt k0 (40000).. lt and not {
                   2745:      [(Roots of b-function cannot be obtained by a stupid method.) nl
                   2746:       (Use ox_asir for efficient factorizations, or restall and bfm manually.)
                   2747:        nl
                   2748:       (ox_asir server will be distributed from the asir ftp cite.) nl
                   2749:       (See lib/ttt.tex for details.) nl
                   2750:       ] cat
                   2751:       error
                   2752:     } {  } ifelse
                   2753:     nn (0).. lt { (0).. nn sub /nn set } {  } ifelse
                   2754:     /kk  (0).. nn sub  def
                   2755:     /roots [ kk (1).. sub ] roots join def
                   2756:     {
                   2757:        kk nn gt { exit } {  } ifelse
                   2758:        ff [[(s). kk (poly) dc]] replace
                   2759:        (0). eq
                   2760:        { /roots roots kk append def }
                   2761:        {  } ifelse
                   2762:        kk (1).. add /kk set
                   2763:     }  loop
                   2764:     [(CurrentRingp) rrr] system_variable
                   2765:     roots { (integer) dc } map /roots set %% ??  OK?
                   2766:     roots shell rest /roots set
                   2767:     /arg1 roots def
                   2768:   ] pop
                   2769:   popVariables
                   2770:   arg1
                   2771: } def
                   2772:
                   2773: /ngcd {
                   2774:   /arg1 set
                   2775:   [/in-ngcd /nlist /g.ngcd /ans] pushVariables
                   2776:   [
                   2777:      /nlist arg1 def
1.29      takayama 2778:      nlist to_univNum /nlist set
1.1       maekawa  2779:      nlist length 2 lt
                   2780:      { /ans nlist 0 get def /L.ngcd goto }
                   2781:      {
                   2782:         [(gcd) nlist 0 get nlist 1 get] mpzext /g.ngcd set
                   2783:         g.ngcd (1).. eq { /ans (1).. def /L.ngcd goto } { } ifelse
                   2784:         [g.ngcd] nlist rest rest join ngcd /ans set
                   2785:       } ifelse
                   2786:      /L.ngcd
                   2787:      ans /arg1 set
                   2788:   ] pop
                   2789:   popVariables
                   2790:   arg1
                   2791: } def
                   2792:
                   2793: [(ngcd)
                   2794:  [(nlist ngcd d )
                   2795:   (list of numbers nlist; number d;)
                   2796:   (d is the gcd of the numbers in nlist.)
                   2797:   (Example: [(12345).. (67890).. (98765)..] ngcd )
                   2798: ]] putUsages
                   2799:
                   2800: /dehomogenize {
                   2801:   /arg1 set
                   2802:   [/in-dehomogenize /f /rr /ans /cring] pushVariables
                   2803:   [
                   2804:      /f arg1 def
                   2805:      f isPolynomial {
                   2806:        f (0). eq
                   2807:        { f /ans set }
                   2808:        {
                   2809:           f (ring) dc /rr set
                   2810:           [(CurrentRingp)] system_variable /cring set
                   2811:           [(CurrentRingp) rr] system_variable
                   2812:           f [[[(D) (var) 0] system_variable . (1). ]] replace /ans set
                   2813:           [(CurrentRingp) cring] system_variable
                   2814:        } ifelse
                   2815:      }
                   2816:      {
                   2817:         f isArray {
                   2818:          f { dehomogenize } map /ans set
                   2819:         }
                   2820:         {(dehomogenize: argument should be a polynomial.) error }
                   2821:         ifelse
                   2822:      } ifelse
                   2823:      /arg1 ans def
                   2824:   ] pop
                   2825:   popVariables
                   2826:   arg1
                   2827: } def
                   2828:
                   2829: [(dehomogenize)
                   2830:  [(obj dehomogenize obj2)
                   2831:   (dehomogenize puts the homogenization variable to 1.)
                   2832:   (Example:  (x*h+h^2). dehomogenize ::    x+1 )
                   2833:  ]
                   2834: ] putUsages
                   2835:
                   2836:
                   2837: /from_records { { (,) } map aload length cat_n } def
                   2838: [(from_records)
                   2839:  [ ([s1 s2 s3 ... sn] from_records (s1,s2,...,sn,))
                   2840:    (Example : [(x) (y)] from_records ::    (x,y,))
                   2841:    (cf. to_records)
                   2842:  ]
                   2843: ] putUsages
                   2844: /popEnv {
                   2845:   { system_variable pop } map pop
                   2846: } def
                   2847:
                   2848: /pushEnv {
                   2849:    %% opt=[(CurrentRingp) (NN)] ==> [[(CurrentRingp) val] [(NN) val]]
                   2850:    { [ 2 1 roll dup [ 2 1 roll ] system_variable ] } map
                   2851: } def
                   2852: [(pushEnv)
                   2853:  [(keylist pushEnv envlist)
                   2854:   (array of string keylist, array of [string object] envlist;)
                   2855:   (Values <<envlist>> of the global system variables specified )
                   2856:   (by the <<keylist>> is push on the stack.)
                   2857:   (keylist is an array of keywords for system_variable.)
                   2858:   (cf. system_variable, popEnv)
                   2859:   (Example:  [(CurrentRingp) (KanGBmessage)] pushEnv)
                   2860:  ]
                   2861: ] putUsages
                   2862: [(popEnv)
                   2863:  [(envlist popEnv)
                   2864:   (cf. pushEnv)
                   2865:  ]
                   2866: ] putUsages
                   2867:
                   2868: /npower {
                   2869:   /arg2 set
                   2870:   /arg1 set
                   2871:   [/f /k /i /ans] pushVariables
                   2872:   [
                   2873:      /f arg1 def   /k arg2 ..int def
                   2874:      f tag PolyP eq {
                   2875:        /ans (1). def
                   2876:      } {
                   2877:        /ans (1).. def
                   2878:      } ifelse
                   2879:      k 0 lt {
                   2880:        1 1 << 0 k sub >> {
                   2881:          /ans f ans {mul} sendmsg2 def
                   2882:        } for
                   2883:        /ans (1).. ans {div} sendmsg2 def
                   2884:      }
                   2885:      {
                   2886:        1 1 k {
                   2887:          /ans f ans {mul} sendmsg2 def
                   2888:        } for
                   2889:      } ifelse
                   2890:      /arg1 ans def
                   2891:   ] pop
                   2892:   popVariables
                   2893:   arg1
                   2894: } def
                   2895: [(npower)
                   2896:  [(obj1 obj2 npower obj3)
                   2897:   (npower returns obj1^obj2 as obj3)
                   2898:   (The difference between power and npower occurs when we compute f^0)
                   2899:   (where f is a polynomial.)
                   2900:   $power returns number(universalNumber) 1, but npower returns 1$
                   2901:   (in the current ring.)
                   2902:  ]
                   2903: ] putUsages
                   2904:
                   2905: /gensym {
                   2906:   (dollar) dc 2 cat_n
                   2907: } def
                   2908: [(gensym)
                   2909:  [(x i gensym xi)
                   2910:   (string x; integer i; string xi)
                   2911:   (It generate a string x indexed with the number i.)
                   2912:   $Example:  (Dx) 12 gensym (Dx12)$
                   2913:  ]
                   2914: ] putUsages
                   2915:
                   2916: /cat {
                   2917:   { toString } map aload length cat_n
                   2918: } def
                   2919: [(cat)
                   2920:  [(a cat s)
                   2921:   (array a ; string s;)
                   2922:   (cat converts each entry of << a >> to a string and concatenates them.)
                   2923:   (Example: [ (x) 1 2] cat ==> (x12))
                   2924:  ]
                   2925: ] putUsages
                   2926:
                   2927:
                   2928: %%%%%%%%%%%%%%%%%%% pmat-level
                   2929: /pmat-level {
                   2930:   /arg2 set
                   2931:   /arg1 set
                   2932:   [/n /i /m /lev /flag] pushVariables
                   2933:   [
                   2934:     /m arg1 def
                   2935:     /lev arg2 def
                   2936:     m isArray {
                   2937:        /n m length def
                   2938:        n 0 eq { /flag 0 def }
                   2939:        { m 0 get isArray { /flag 1 def } { /flag 0 def} ifelse } ifelse
                   2940:     } {  /flag 0 def } ifelse
                   2941:
                   2942:     flag {
                   2943:       0 1 lev {
                   2944:         pop ( ) messagen
                   2945:       } for
                   2946:       ([ ) message
                   2947:       0 1 n 1 sub {
                   2948:         /i set
                   2949:         m i get lev 1 add pmat-level
                   2950:       } for
                   2951:       0 1 lev {
                   2952:         pop ( ) messagen
                   2953:       } for
                   2954:       (]) message
                   2955:     }
                   2956:     {
                   2957:        0 1 lev {
                   2958:          pop ( ) messagen
                   2959:        } for
                   2960:        ( ) messagen
                   2961:        m message
                   2962:     } ifelse
                   2963:   ] pop
                   2964:   popVariables
                   2965: } def
                   2966:
                   2967: /pmat {  0 pmat-level } def
                   2968:
                   2969: [(pmat)
                   2970:  [(f pmat)
                   2971:   (array f;)
                   2972:   (f is pretty printed.)
                   2973:  ]
                   2974: ] putUsages
                   2975:
                   2976:
                   2977: /adjoint1 {
                   2978:   /arg2 set
                   2979:   /arg1 set
                   2980:   [/in-adjoint1 /f /p /q /xx /dxx /ans /g /one] pushVariables
                   2981:   [
                   2982:      /f arg1 def
                   2983:      /xx arg2 def
                   2984:      f isPolynomial {  }
                   2985:      { (adjoint1: the first argument must be a polynomial.) message
                   2986:        pop popVariables
                   2987:        (adjoint1: the first argument must be a polynomial.)  error
                   2988:      } ifelse
                   2989:      /ans (0). def
                   2990:      f (0). eq {   }
                   2991:      {
                   2992:         /xx xx (string) dc def
                   2993:         /dxx [@@@.Dsymbol xx] cat def
                   2994:         /xx xx f (ring) dc ,, def
                   2995:         /dxx dxx f (ring) dc ,, def
                   2996:         /one (1) f (ring) dc ,, def
                   2997:
                   2998:         {
                   2999:           /g f init def
                   3000:           /f f g sub def
                   3001:           /p g xx degree def
                   3002:           /q g dxx degree def
                   3003:           g [[xx one] [dxx one]] replace /g set
                   3004:           g
                   3005:           << (0). dxx sub q npower    xx p npower mul >>
                   3006:           mul
                   3007:           ans add /ans set
                   3008:           f (0). eq { exit } { } ifelse
                   3009:         } loop
                   3010:         ans dehomogenize /ans set
                   3011:      } ifelse
                   3012:      /arg1 ans def
                   3013:   ] pop
                   3014:   popVariables
                   3015:   arg1
                   3016: } def
                   3017:
                   3018: /adjoint {
                   3019:   /arg2 set
                   3020:   /arg1 set
                   3021:   [/in-adjoint /f /xx /xx0] pushVariables
                   3022:   [
                   3023:      /f arg1 def /xx arg2 def
                   3024:      xx toString /xx set
                   3025:      [xx to_records pop] /xx set
                   3026:      xx { /xx0 set f xx0 adjoint1 /f set } map
                   3027:      /arg1 f def
                   3028:   ]pop
                   3029:   popVariables
                   3030:   arg1
                   3031: } def
                   3032:
                   3033: [(adjoint)
                   3034:  [(f xlist adjoint g)
                   3035:   (poly f; string xlist; poly g;)
                   3036:   (g is the adjoint operator of f.)
                   3037:   (The variables to take adjoint are specified by xlist.)
                   3038:   (Example: [(x,y) ring_of_differential_operators 0] define_ring)
                   3039:   (          (x^2 Dx - y x Dx Dy-2). (x,y) adjoint )
                   3040:   $          ((-Dx) x^2 - (-Dx) (-Dy) x y -2). dehomogenize sub :: ==> 0$
                   3041: ]] putUsages
                   3042:
                   3043: %%%%% diagonal for tensor products
                   3044: %% 1998, 12/4 (Sat)
                   3045: %% s_i = x_i, t_i = x_i - y_i,    Restrict to t_i = 0.
                   3046: %% x_i = x_i, y_i = s_i - t_i,
                   3047: %% Dx_i = Dt_i + Ds_i, Dy_i = -Dt_i.
                   3048: /diagonalx {
                   3049:   /arg2 set
                   3050:   /arg1 set
                   3051:   [/in-diagonalx /f] pushVariables
                   3052:   [
                   3053:     (Not implemented yet.) message
                   3054:   ] pop
                   3055:   popVariables
                   3056:   arg1
                   3057: } def
                   3058:
                   3059:
                   3060:
                   3061: %%%%%%%%%%%  distraction2 for b-function
                   3062: /distraction2 {
                   3063:   /arg4 set
                   3064:   /arg3 set
                   3065:   /arg2 set
                   3066:   /arg1 set
                   3067:   [/f  /dx /ss /xx  /ans /n /i /rr] pushVariables
                   3068:   [
                   3069:     /f arg1 def /xx arg2 def /dx arg3 def /ss arg4 def
                   3070:     f (0). eq {  }
                   3071:     {
                   3072:       /rr f (ring) dc def
                   3073:       xx {toString rr ,, } map /xx set
                   3074:       dx {toString rr ,, } map /dx set
                   3075:       ss {toString rr ,, } map /ss set
                   3076:       /n xx length  def
                   3077:       0 1 n 1 sub {
                   3078:          /i set
                   3079:          /f  f xx i get dx i get ss i get destraction2.1 /f set
                   3080:       } for
                   3081:     } ifelse
                   3082:     /arg1 f def
                   3083:   ]pop
                   3084:   popVariables
                   3085:   arg1
                   3086: } def
                   3087: [(distraction2)
                   3088:  [(f [ list of x-variables ] [ list of D-variables ] [ list of s-variables ])
                   3089:   (   distraction2  result )
                   3090:   $Example 1: [(x,y) ring_of_differential_operators 0] define_ring $
                   3091:   $  (x^2 Dx Dy + x Dy). [(x). (y).] [(Dx). (Dy).] [(x). (y).] distraction2$
                   3092:   $Example 2: (x^4 Dx^2 + x^2). [(x).] [(Dx). ] [(x).] distraction2$
                   3093:  ]
                   3094: ] putUsages
                   3095: /destraction2.1 {
                   3096:   /arg4 set
                   3097:   /arg3 set
                   3098:   /arg2 set
                   3099:   /arg1 set
                   3100:   [/ww /f  /dx /ss /xx /coeff0 /expvec
                   3101:    /coeffvec /expvec2 /ans /one /rr /dd] pushVariables
                   3102:   [
                   3103:     /f arg1 def /xx arg2 def /dx arg3 def /ss arg4 def
                   3104:     f (ring) dc /rr set
                   3105:     /one (1) rr ,, def %%
                   3106:     /ww [ xx toString -1 dx toString 1 ] weightv  def
                   3107:     f ww init f sub (0). eq {   }
                   3108:     { [(destraction2.1 : inhomogeneous with respect to )
                   3109:         xx  ( and )  dx  nl
                   3110:        (Your weight vector may not be generic.)
                   3111:       ] cat error } ifelse
                   3112:     /dd << f dx degree >> << f xx degree >> sub def
                   3113:     f [[xx one]] replace dx coefficients  /coeff0 set
                   3114:     /expvec coeff0 0 get { (integer) dc } map def
                   3115:     /coeffvec coeff0 1 get def
                   3116:     expvec { ss 2 -1 roll factorial } map /expvec2 set
                   3117:     expvec2 coeffvec mul  /ans set
                   3118:     %% x^p d^q, (p > q) case.  x^2( x^2 Dx^2 + x Dx + 1)
                   3119:     dd 0 lt {
                   3120:       %% (ss+1) (ss+2) ... (ss+d)
                   3121:       one 1 1 0 dd sub { (universalNumber) dc ss add mul} for
                   3122:       ans mul /ans set
                   3123:     }
                   3124:     {  } ifelse
                   3125:     /arg1 ans def
                   3126:    ] pop
                   3127:    popVariables
                   3128:    arg1
1.3       takayama 3129: } def
                   3130:
                   3131: /distraction2* {
                   3132:   /arg1 set
                   3133:   [/in-distraction2* /aa /f /vlist /xlist /dlist /slist ] pushVariables
                   3134:   [(CurrentRingp)] pushEnv
                   3135:   [
                   3136:      /aa arg1 def
                   3137:      /f aa 0 get def
                   3138:      /vlist aa 1 get def
                   3139:      /xlist aa 2 get def
                   3140:      /dlist aa 3 get def
                   3141:      /slist aa 4 get def
                   3142:      vlist isArray
                   3143:      {
                   3144:         vlist { toString } map   /vlist set
                   3145:      }
                   3146:      {
                   3147:         vlist toString to_records  /vlist set
                   3148:       } ifelse
                   3149:      xlist isArray
                   3150:      {
                   3151:         xlist { toString } map   /xlist set
                   3152:      }
                   3153:      {
                   3154:         xlist toString to_records  /xlist set
                   3155:       } ifelse
                   3156:      slist isArray
                   3157:      {
                   3158:         slist { toString } map   /slist set
                   3159:      }
                   3160:      {
                   3161:         slist toString to_records  /slist set
                   3162:       } ifelse
                   3163:      [vlist from_records ring_of_differential_operators 0] define_ring pop
                   3164:      f toString .
                   3165:      xlist { . } map
                   3166:      dlist { toString . } map
                   3167:      slist { toString . } map
                   3168:      distraction2 /arg1 set
                   3169:   ] pop
                   3170:   popEnv
                   3171:   popVariables
                   3172:   arg1
1.1       maekawa  3173: } def
                   3174:
                   3175: /message-quiet  {
                   3176:   @@@.quiet { pop } { message } ifelse
                   3177: } def
                   3178: [(message-quiet)
                   3179:  [(s message-quiet )
                   3180:  (string s;)
                   3181:  (It outputs the message s when @@@.quiet is not equal to 1.)
                   3182:  (@@@.quiet is set to 1 when you start sm1 with the option -q.)
                   3183: ]] putUsages
                   3184: /messagen-quiet  {
                   3185:   @@@.quiet { pop } { messagen } ifelse
                   3186: } def
                   3187: [(messagen-quiet)
                   3188:  [(s messagen-quiet )
                   3189:  (string s;)
                   3190:  (It outputs the message s without the newline when @@@.quiet is not equal to 1.)
                   3191:  (@@@.quiet is set to 1 when you start sm1 with the option -q.)
                   3192: ]] putUsages
                   3193:
                   3194: /getvNames0 {
                   3195:   /arg1 set
                   3196:   [/in-getvNames0 /nlist /nn /i] pushVariables
                   3197:   [
                   3198:     /nlist arg1 def
                   3199:     [(N)] system_variable /nn set
                   3200:     nlist { /i set
                   3201:        i nn lt {
                   3202:          [(x) (var) i] system_variable
                   3203:        } {
                   3204:          [(D) (var) i nn sub] system_variable
                   3205:        } ifelse
                   3206:     } map
                   3207:     /arg1 set
                   3208:   ] pop
                   3209:   popVariables
                   3210:   arg1
                   3211: } def
                   3212:
                   3213: /getvNames {
                   3214:   [/in-getvNames /nn] pushVariables
                   3215:   [
                   3216:     [(N)] system_variable /nn set
                   3217:     [0 1 nn 2 mul 1 sub {  } for] getvNames0 /arg1 set
                   3218:   ] pop
                   3219:   popVariables
                   3220:   arg1
                   3221: } def
                   3222: [(getvNames)
                   3223: [(getvNames vlist)
                   3224:  (list vlist)
                   3225:  (It returns of the list of the variables in the order x0, x1, ..., D0, ...)
                   3226:  (Use with [(variableNames) vlist] system_variable.)
                   3227:  (cf. nlist getvNames0 vlist is used internally. cf. getvNamesC)
                   3228: ]] putUsages
                   3229:
                   3230: /getvNamesC {
                   3231:   [/in-getvNamesC /nn /i] pushVariables
                   3232:   [
                   3233:     [(N)] system_variable /nn set
                   3234:     [nn 1 sub -1 0 {  } for nn 2 mul 1 sub -1 nn { } for ] getvNames0 /arg1 set
                   3235:   ] pop
                   3236:   popVariables
                   3237:   arg1
                   3238: } def
                   3239: [(getvNamesC)
                   3240: [(getvNamesC vlist)
                   3241:  (list vlist)
                   3242:  $It returns of the list of the variables in the order 0, 1, 2, ... $
                   3243:  $(cmo-order and output_order).$
                   3244:  (cf. getvNames)
                   3245: ]] putUsages
                   3246:
                   3247: /getvNamesCR {
                   3248:   /arg1 set
                   3249:   [/in-getvNamesCR /rrr] pushVariables
                   3250:   [(CurrentRingp)] pushEnv
                   3251:   [
                   3252:     /rrr arg1 def
                   3253:     rrr isPolynomial {
                   3254:       rrr (0). eq { (No name field for 0 polynomial.) error }
                   3255:       { rrr (ring) dc /rrr set } ifelse
                   3256:     } { } ifelse
                   3257:     [(CurrentRingp) rrr] system_variable
                   3258:     getvNamesC /arg1 set
                   3259:   ] pop
                   3260:   popEnv
                   3261:   popVariables
                   3262:   arg1
                   3263: } def
                   3264: [(getvNamesCR)
                   3265: [(obj getvNamesCR vlist)
                   3266:  (obj ring | poly ; list vlist)
                   3267:  $It returns of the list of the variables in the order 0, 1, 2, ... (cmo-order)$
                   3268:  (for <<obj>>.)
                   3269:  (Example: ( (x-2)^3 ). /ff set )
                   3270:  (         [(x) ring_of_differential_operators 0] define_ring ff getvNamesCR ::)
                   3271: ]] putUsages
                   3272:
                   3273:
                   3274: /reduction-noH {
                   3275:   /arg2 set
                   3276:   /arg1 set
                   3277:   [/in-reduction-noH /ff /gg] pushVariables
                   3278:   [(Homogenize)] pushEnv
                   3279:   [
                   3280:     /ff arg1 def
                   3281:     /gg arg2 def
                   3282:     [(Homogenize) 0] system_variable
                   3283:     ff gg reduction /arg1 set
                   3284:   ] pop
                   3285:   popEnv
                   3286:   popVariables
                   3287:   arg1
                   3288: } def
                   3289: [(reduction-noH)
                   3290: [(f g reduction-noH r)
                   3291:  (poly f; array g; array r;)
                   3292:  (Apply the normal form algorithm for f with the set g. All computations are)
                   3293:  (done with the rule Dx x = x Dx +1, i.e., no homogenization, but other)
                   3294:  (specifications are the same with reduction. cf. reduction)
                   3295:  (g should be dehomogenized.)
                   3296: ]] putUsages
                   3297:
                   3298: /-intInfinity -999999999  def
                   3299: /intInfinity   999999999  def
                   3300: [(intInfinity)
                   3301: [(intInfinity = 999999999)]
                   3302: ] putUsages
                   3303: [(-intInfinity)
                   3304: [(-intInfinity = -999999999)]
                   3305: ] putUsages
                   3306:
                   3307:
                   3308: /maxInArray {
                   3309:   /arg1 set
                   3310:   [/in-maxInArray /v /ans /i /n] pushVariables
                   3311:   [
                   3312:     /v arg1 def
                   3313:     /n v length def
                   3314:     /maxInArray.pos 0 def
                   3315:     n 0 eq {
                   3316:       /ans null def
                   3317:     } {
                   3318:       /ans v 0 get def
                   3319:       1 1 n 1 sub {
                   3320:         /i set
                   3321:         v i get ans gt {
                   3322:            /ans v i get def
                   3323:            /maxInArray.pos i def
                   3324:         } { } ifelse
                   3325:       } for
                   3326:     } ifelse
                   3327:     /arg1 ans def
                   3328:   ] pop
                   3329:   popVariables
                   3330:   arg1
                   3331: } def
                   3332: [(maxInArray)
                   3333: [( [v1 v2 ....] maxInArray m )
                   3334:  (m is the maximum in [v1 v2 ...].)
                   3335:  (The position of m is stored in the global variable maxInArray.pos.)
                   3336: ]] putUsages
                   3337:
                   3338: /cancelCoeff {
1.18      takayama 3339:   /arg1 set
                   3340:   [(reduceContent) arg1] gbext 0 get
                   3341: } def
                   3342: /cancelCoeff_org {
1.1       maekawa  3343:  /arg1 set
                   3344:  [/in-cancelCoeff /ff /gg /dd /dd2] pushVariables
                   3345:  [  /ff arg1 def
                   3346:     ff (0). eq {
                   3347:       /label.cancelCoeff2 goto
                   3348:     } {  } ifelse
                   3349:     /gg ff def
                   3350:     /dd [(lcoeff) ff init ] gbext (universalNumber) dc def
                   3351:     {
                   3352:        gg (0). eq { exit} {  } ifelse
                   3353:        [(lcoeff) gg init] gbext (universalNumber) dc  /dd2 set
                   3354:        [(gcd) dd dd2] mpzext /dd set
                   3355:        dd (1).. eq {
                   3356:          /label.cancelCoeff goto
                   3357:        } {  } ifelse
                   3358:        /gg gg gg init sub def
                   3359:      } loop
                   3360:      [(divByN)  ff dd] gbext 0 get /ff set
                   3361:     /label.cancelCoeff
                   3362:      [(lcoeff) ff init] gbext (universalNumber) dc (0).. lt
                   3363:      { ff (-1).. mul /ff set } {  } ifelse
                   3364:     /label.cancelCoeff2
                   3365:     /arg1 ff def
                   3366:  ] pop
                   3367:  popVariables
                   3368:  arg1
                   3369: } def
                   3370: [(cancelCoeff)
                   3371:  [(f cancelcoeff g)
                   3372:   (poly f,g;)
                   3373:   (Factor out the gcd of the coefficients.)
                   3374:   (Example: (6 x^2 - 10 x). cancelCoeff)
                   3375:   (See also gbext.)
                   3376: ]] putUsages
                   3377:
                   3378:
                   3379: /flatten {
                   3380:   /arg1 set
                   3381:   [/in-flatten /mylist] pushVariables
                   3382:   [
                   3383:      /mylist arg1 def
                   3384:      mylist isArray {
                   3385:        mylist { dup isArray { aload pop } { } ifelse } map /mylist set
                   3386:      }{ } ifelse
                   3387:      /arg1 mylist def
                   3388:   ] pop
                   3389:   popVariables
                   3390:   arg1
                   3391: } def
                   3392: [(flatten)
                   3393:  [(list flatten list2)
                   3394:   (Flatten the list.)
                   3395:   (Example 1: [ [1 2 3] 4 [2]] flatten    ===> [1 2 3 4 2])
                   3396: ]] putUsages
                   3397:
                   3398: %% Take first N elements.
                   3399: /carN {
                   3400:   /arg2 set
                   3401:   /arg1 set
                   3402:   [/in-res-getN /pp /nn /ans] pushVariables
                   3403:   [
                   3404:      /nn arg2 def
                   3405:      /pp arg1 def
                   3406:      pp isArray {
                   3407:        pp length nn lt {
                   3408:          /ans pp def
                   3409:        } {
                   3410:          [pp aload length nn sub /nn set 1 1 nn { pop pop } for ] /ans set
                   3411:        } ifelse
                   3412:      } {
                   3413:        /ans pp def
                   3414:      } ifelse
                   3415:      /arg1 ans def
                   3416:   ] pop
                   3417:   popVariables
                   3418:   arg1
                   3419: } def
                   3420: [(carN)
                   3421: [([f1 ... fm]  n carN  [f1 ... fn])
                   3422:  (carN extracts the first n elements from the list.)
                   3423: ]] putUsages
                   3424:
                   3425: /getRing {
                   3426:   /arg1 set
                   3427:   [/in-getRing /aa /n /i /ans] pushVariables
                   3428:   [
                   3429:     /aa arg1 def
                   3430:     /ans null def
                   3431:     aa isPolynomial {
                   3432:       aa (0). eq {
                   3433:       } {
                   3434:          /ans aa (ring) dc def
                   3435:       } ifelse
                   3436:     } {
                   3437:      aa isArray {
                   3438:        /n aa length 1 sub def
                   3439:        0 1 n { /i set aa i get getRing /ans set
                   3440:                ans tag 0 eq {  } { /getRing.LLL goto } ifelse
                   3441:        } for
                   3442:      }{ } ifelse
                   3443:     } ifelse
                   3444:     /getRing.LLL
                   3445:     /arg1 ans def
                   3446:   ] pop
                   3447:   popVariables
                   3448:   arg1
                   3449: } def
                   3450: [(getRing)
                   3451: [(obj getRing rr)
                   3452:  (ring rr;)
                   3453:  (getRing obtains the ring structure from obj.)
                   3454:  (If obj is a polynomial, it returns the ring structure associated to)
                   3455:  (the polynomial.)
                   3456:  (If obj is an array, it recursively looks for the ring structure.)
                   3457: ]] putUsages
                   3458: /toVectors {
                   3459:   /arg1 set
                   3460:   [/in-toVectors /gg /n /ans] pushVariables
                   3461:   [
                   3462:     /gg arg1 def
                   3463:     gg isArray {
                   3464:       gg length 0 eq {
                   3465:         /ans [ ] def
                   3466:         /toVectors.LLL goto
                   3467:       } {
                   3468:         gg 0 get isInteger {
                   3469:           gg @@@.toVectors2 /ans set
                   3470:         } {
                   3471:           gg @@@.toVectors /ans set
                   3472:         } ifelse
                   3473:         /toVectors.LLL goto
                   3474:       } ifelse
                   3475:     } {
                   3476:       %% It is not array.
                   3477:       gg (array) dc /ans set
                   3478:     } ifelse
                   3479:     /toVectors.LLL
                   3480:     /arg1 ans def
                   3481:    ] pop
                   3482:    popVariables
                   3483:    arg1
                   3484: } def
                   3485: /@@@.toVectors2 {
                   3486:   /arg1 set
                   3487:   [/in-@@@.toVectors2 /gg /ans /n /tmp /notarray] pushVariables
                   3488:   [
                   3489:     /gg arg1 def
                   3490:     /ans gg 1 get @@@.toVectors def
                   3491:     /n   gg 0 get def
                   3492:     gg 1 get isArray not {
                   3493:        /ans [ans] def
                   3494:        /notarray 1 def
                   3495:     }{ /notarray 0 def} ifelse
                   3496:     ans {
                   3497:       /tmp set
                   3498:       tmp length n lt {
                   3499:         tmp
                   3500:         [1 1 n tmp length sub { pop (0). } for ]
                   3501:         join /tmp set
                   3502:       } {  } ifelse
                   3503:       tmp
                   3504:     } map
                   3505:     /ans set
                   3506:     notarray { ans 0 get /ans set } { } ifelse
                   3507:     /arg1 ans def
                   3508:   ] pop
                   3509:   popVariables
                   3510:   arg1
                   3511: } def
                   3512:
                   3513: /@@@.toVectors {
                   3514:   /arg1 set
                   3515:   [/in-@@@.toVectors /gg ] pushVariables
                   3516:   [
                   3517:     /gg arg1 def
                   3518:     gg isArray {
                   3519:       gg { $array$ data_conversion } map
                   3520:     } {
                   3521:       gg (array) data_conversion
                   3522:     }ifelse
                   3523:     /arg1 set
                   3524:    ] pop
                   3525:    popVariables
                   3526:    arg1
                   3527: } def
                   3528:
                   3529: /toVectors2 { toVectors } def
                   3530:
                   3531: /fromVectors { { fromVectors.aux } map } def
                   3532: /fromVectors.aux {
                   3533:   /arg1 set
                   3534:   [/in-fromVector.aux /vv /mm /ans /i /ee] pushVariables
                   3535:   [(CurrentRingp)] pushEnv
                   3536:   [
                   3537:     /vv arg1 def
                   3538:     /mm vv length def
                   3539:     /ans (0). def
                   3540:     /ee (0). def
                   3541:     0 1 mm 1 sub {
                   3542:       /i set
                   3543:       vv i get (0). eq {
                   3544:       } {
                   3545:         [(CurrentRingp) vv i get (ring) dc] system_variable
                   3546:         [(x) (var) [(N)] system_variable 1 sub] system_variable . /ee set
                   3547:         /fromVector.LLL  goto
                   3548:       } ifelse
                   3549:     } for
                   3550:     /fromVector.LLL
                   3551:     %% vv message
                   3552:     0 1 mm 1 sub {
                   3553:       /i set
                   3554:       vv i get (0). eq {
                   3555:       } {
                   3556:         /ans ans
                   3557:             << vv i get >> << ee i npower >> mul
                   3558:          add def
                   3559:       } ifelse
                   3560:       %% [i ans] message
                   3561:     } for
                   3562:     /arg1 ans def
                   3563:   ] pop
                   3564:   popEnv
                   3565:   popVariables
                   3566:   arg1
                   3567: } def
                   3568: [(fromVectors)
                   3569: [
                   3570: ([v1 v2 ...] fromVectors [s1 s2 ...])
                   3571: (array of poly : v1, v2, ... ; poly : s1, s2 ....)
                   3572: (cf. toVectors. <<e_>> varaible is assumed to be the last )
                   3573: (    variable in x.  @@@.esymbol)
                   3574: $Example: [(x,y) ring_of_differential_operators 0] define_ring$
                   3575: $ [(x).  (y).] /ff set  $
                   3576: $ [ff ff] fromVectors :: $
                   3577: ]] putUsages
                   3578:
                   3579: /getOrderMatrix {
                   3580:   /arg1 set
                   3581:   [/in-getOrderMatrix /obj /rr /ans /ans2 /i] pushVariables
                   3582:   [(CurrentRingp)] pushEnv
                   3583:   [
                   3584:     /obj arg1 def
                   3585:     obj isArray {
                   3586:       obj { getOrderMatrix } map /ans set
                   3587:       ans length 0 {
                   3588:          /ans null def
                   3589:       } {
                   3590:          /ans2 null def
                   3591:          0 1 ans length 1 sub {
                   3592:            /i set
                   3593:            ans i get tag 0 eq
                   3594:            {   }
                   3595:            { /ans2 ans i get def } ifelse
                   3596:          } for
                   3597:          /ans ans2 def
                   3598:       } ifelse
                   3599:       /getOrderMatrix.LLL goto
                   3600:     } {  } ifelse
                   3601:     obj tag 14 eq {
                   3602:       [(CurrentRingp) obj] system_variable
                   3603:       [(orderMatrix)] system_variable /ans set
                   3604:       /getOrderMatrix.LLL goto
                   3605:     } {  } ifelse
                   3606:     obj isPolynomial {
                   3607:       obj (0). eq
                   3608:       { /ans null def
                   3609:       } { obj getRing /rr set
                   3610:         [(CurrentRingp) rr] system_variable
                   3611:         [(orderMatrix)] system_variable /ans set
                   3612:       } ifelse
                   3613:       /getOrderMatrix.LLL goto
                   3614:     } { (getOrderMatrix: wrong argument.)  error } ifelse
                   3615:     /getOrderMatrix.LLL
                   3616:     /arg1 ans def
                   3617:   ] pop
                   3618:   popEnv
                   3619:   popVariables
                   3620:   arg1
                   3621: } def
                   3622:
                   3623:
                   3624: [(getOrderMatrix)
                   3625: [(obj getOrderMatrix m)
                   3626:  (array  m)
                   3627:  (getOrderMatrix obtains the order matrix from obj.)
                   3628:  (If obj is a polynomial, it returns the order matrix associated to)
                   3629:  (the polynomial.)
                   3630:  (If obj is an array, it returns an order matrix of an element.)
                   3631: ]] putUsages
                   3632:
                   3633: /nl {
                   3634:    10 $string$ data_conversion
                   3635: } def
                   3636: [(nl)
                   3637: [(nl is the newline character.)
                   3638:  $Example: [(You can break line) nl (here.)] cat message$
1.4       takayama 3639: ]] putUsages
                   3640:
                   3641: /to_int {
                   3642:   /arg1 set
                   3643:   [/to-int /ob /ans] pushVariables
                   3644:   [
                   3645:     /ob arg1 def
                   3646:     /ans ob def
                   3647:     ob isArray {
                   3648:       ob {to_int} map /ans set
                   3649:       /LLL.to_int goto
                   3650:     } {  } ifelse
                   3651:     ob isInteger {
                   3652:       ob (universalNumber) dc /ans set
                   3653:       /LLL.to_int goto
                   3654:     } {  } ifelse
                   3655:     /LLL.to_int
                   3656:     /arg1 ans def
                   3657:   ] pop
                   3658:   popVariables
                   3659:   arg1
                   3660: } def
                   3661: [(to_int)
                   3662: [(obj to_int obj2)
                   3663:  (All integers in obj are changed to universalNumber.)
                   3664:  (Example: /ff [1 2 [(hello) (0).]] def  ff { tag } map ::)
                   3665:  (         ff to_int { tag } map :: )
1.5       takayama 3666: ]] putUsages
                   3667:
1.33      takayama 3668: /to_int32 {
                   3669:   /arg1 set
                   3670:   [/to-int32 /ob /ans] pushVariables
                   3671:   [
                   3672:     /ob arg1 def
                   3673:     /ans ob def
                   3674:     ob isArray {
                   3675:       ob {to_int32} map /ans set
                   3676:       /LLL.to_int32 goto
                   3677:     } {  } ifelse
                   3678:     ob isUniversalNumber {
                   3679:       ob (integer) dc /ans set
                   3680:       /LLL.to_int32 goto
                   3681:     } {  } ifelse
                   3682:     /LLL.to_int32
                   3683:     /arg1 ans def
                   3684:   ] pop
                   3685:   popVariables
                   3686:   arg1
                   3687: } def
                   3688: [(to_int32)
                   3689: [(obj to_int32 obj2)
                   3690:  $All universalNumber in obj are changed to integer (int32).$
                   3691:  (Example: /ff [1 (2).. [(hello) (0).]] def  ff { tag } map ::)
                   3692:  (         ff to_int32 { tag } map :: )
                   3693:  (cf. to_int, to_univNum )
                   3694: ]] putUsages
                   3695:
1.5       takayama 3696: /define_ring_variables {
1.6       takayama 3697:   [/in-define_ring_variables /drv._v /drv._p /drv._v0] pushVariables
                   3698: %% You cannot use these names for names for polynomials.
1.5       takayama 3699:   [
1.6       takayama 3700:      /drv._v getVariableNames def
                   3701:      /drv._v0 drv._v def
                   3702:      drv._v { dup /drv._p set (/) 2 1 roll ( $) drv._p ($. def ) } map cat
                   3703:      /drv._v set
                   3704: %     drv._v message
                   3705:      [(parse) drv._v] extension
1.5       takayama 3706:   ] pop
                   3707:   popVariables
                   3708: } def
                   3709: [(define_ring_variables)
                   3710: [(It binds  a variable <<a>> in the current ring to the sm1 variable <<a>>.)
                   3711:  (For example, if x is a variable in the current ring, it defines the sm1)
                   3712:  (variable x by /x (x) def)
                   3713: ]] putUsages
                   3714:
                   3715: /boundp {
                   3716:   /arg1 set
                   3717:   [/a /ans] pushVariables
                   3718:   [
                   3719:     /a arg1 def
                   3720:     [(parse) [(/) a ( load tag 0 eq { /ans 0 def } )
                   3721:                     (               { /ans 1 def } ifelse )] cat ] extension
                   3722:     /arg1 ans def
                   3723:   ] pop
                   3724:   popVariables
                   3725:   arg1
                   3726: } def
                   3727: [(boundp)
                   3728:  [( a boundp b)
                   3729:   (string a, b is 0 or 1.)
                   3730:   (If the variable named << a >> is bounded to a value,)
                   3731:   (it returns 1 else it returns 0.)
                   3732:   $Example: (hoge) boundp ::$
1.1       maekawa  3733: ]] putUsages
1.10      takayama 3734: [(isSubstr)
                   3735:  [
                   3736:   (s1 s2 isSubstr pos)
                   3737:   (If s1 is a substring of s2, isSubstr returns the position in s2 from which)
                   3738:   (s1 is contained in s2.)
                   3739:   (If s1 is not a substring of s2, then isSubstr returns -1.)
                   3740:  ]
                   3741: ] putUsages
                   3742: /isSubstr {
                   3743:   /arg2 set /arg1 set
                   3744:   [/in-isSubstr /s1 /s2 /i1 /i2 /n1 /n2
                   3745:    /ans /flg
                   3746:   ]  pushVariables
                   3747:   [
                   3748:     /s1 arg1 def
                   3749:     /s2 arg2 def
                   3750:     s1 (array) dc /s1 set
                   3751:     s2 (array) dc /s2 set
                   3752:     /n1 s1 length def
                   3753:     /n2 s2 length def
                   3754:     /ans -1 def
                   3755:     0 1 n2 n1 sub {
                   3756:       /i2 set
                   3757:       /flg 1 def
                   3758:       0 1 n1 1 sub {
                   3759:         /i1 set
                   3760:         s1 i1 get s2 i2 i1 add get eq {
                   3761:         } {
                   3762:           /flg 0 def exit
                   3763:         } ifelse
                   3764:       } for
                   3765:       flg {
                   3766:         /ans i2 def
                   3767:         /isSubstr.L2 goto
                   3768:       } { /ans -1 def } ifelse
                   3769:     } for
                   3770:     /isSubstr.L2
                   3771:     /arg1 ans def
                   3772:   ] pop
                   3773:   popVariables
                   3774:   arg1
1.14      takayama 3775: } def
                   3776:
                   3777: [(execve)
                   3778:  [
                   3779:    (command execve)
                   3780:    ([arg0 arg1 arg2 ...] execve )
                   3781:    (It executes the command by the system call execve.)
                   3782:    (cf. system, forkExec)
                   3783:  ]
                   3784: ] putUsages
                   3785:
                   3786: /execve {
                   3787:   /execve.arg set
                   3788:   [(forkExec) execve.arg [ ] 1] extension
1.15      takayama 3789: } def
                   3790:
                   3791: [(beginEcart)
                   3792:  [
                   3793:    (beginEcart)
                   3794:    (Set the environments for the ecart division algorithm.)
                   3795:  ]
                   3796: ] putUsages
                   3797:
1.23      takayama 3798: /ecart.debug_reduction1 0 def
1.15      takayama 3799: /beginEcart {
                   3800:   (red@) (ecart) switch_function
                   3801:   [(Ecart) 1] system_variable
1.16      takayama 3802:   [(CheckHomogenization) 0] system_variable
                   3803:   [(ReduceLowerTerms) 0] system_variable
                   3804:   [(AutoReduce) 0] system_variable
1.17      takayama 3805:   [(EcartAutomaticHomogenization) 0] system_variable
1.23      takayama 3806:   ecart.debug_reduction1 {
                   3807:     (red@) (debug) switch_function
                   3808:   } {  } ifelse
1.15      takayama 3809: } def
                   3810:
                   3811: [(endEcart)
                   3812:  [
                   3813:    (endEcart)
                   3814:    (End of using the ecart division algorithm.)
                   3815:  ]
                   3816: ] putUsages
                   3817:
                   3818: /endEcart {
                   3819:   (red@) (standard) switch_function
                   3820:   [(Ecart) 0] system_variable
1.21      takayama 3821:   [(degreeShift) (reset)] homogenize pop
1.10      takayama 3822: } def
1.19      takayama 3823:
                   3824: /ord_ws_all {
                   3825:   /arg2 set /arg1 set
                   3826:   [(ord_ws_all) arg1 arg2] gbext
                   3827: } def
                   3828: [(ord_ws_all)
                   3829:  [
                   3830:    (fv wv ord_ws_all degree)
                   3831:    (  ord_ws_all returns the ord with respect to the weight vector wv.)
                   3832:    $Example: [(x,y) ring_of_differential_operators 0] define_ring  $
                   3833:    $        (Dx^2+x*Dx*Dy+2). [(Dx) 1 (Dy) 1] weightv ord_ws_all ::  $
                   3834:    (  )
                   3835:    (fv [wv shiftv] ord_ws_all degree)
                   3836:    (  ord_ws_all returns the ord with respect to the weight vector wv and)
                   3837:    (  the shift vector shiftv.)
                   3838:    $Example: [(x,y) ring_of_differential_operators 0] define_ring  $
                   3839:    $        [(Dx^2+x*Dx*Dy+2). (Dx).] [[(Dx) 1 (Dy) 1] weightv [0 2]] ord_ws_all ::$
                   3840:    (  )
                   3841:    (cf: init, gbext.   Obsolete: ord_w, ord_w_all)
                   3842:  ]
                   3843: ] putUsages
1.22      takayama 3844:
                   3845: [(newVector)
                   3846:  [( n newVector vec)
                   3847: ]] putUsages
                   3848: /newVector {
                   3849:   /arg1 set
                   3850:   [/in-newVector /n] pushVariables
                   3851:   [
                   3852:     /n arg1 def
                   3853:     [(newVector) n] extension /arg1 set
                   3854:   ] pop
                   3855:   popVariables
                   3856:   arg1
                   3857: } def
                   3858:
                   3859: [(newMatrix)
                   3860:  [( [m n] newMatrix mat)
                   3861: ]] putUsages
                   3862: /newMatrix {
                   3863:   /arg1 set
                   3864:   [/in-newMatrix /n] pushVariables
                   3865:   [
                   3866:     /n arg1 def
                   3867:     [(newMatrix) n 0 get n 1 get] extension /arg1 set
                   3868:   ] pop
                   3869:   popVariables
                   3870:   arg1
1.25      takayama 3871: } def
                   3872:
                   3873: /addStdoutStderr {
                   3874:   [(>) (stringOut://@@@stdout) (2>) (stringOut://@@@stderr)] join
1.27      takayama 3875: } def
                   3876:
                   3877: [(,,,)
                   3878: [(reparse a polynomial or polynomials)]
                   3879: ] putUsages
                   3880: /,,, {
                   3881:   /arg1 set
                   3882:   [/in-reparse /ff] pushVariables
                   3883:   [
                   3884:     /ff arg1 def
                   3885:     ff tag 6 eq {
                   3886:       ff { ,,, } map /arg1 set
                   3887:     } {
                   3888:       ff toString . /arg1 set
                   3889:     } ifelse
1.29      takayama 3890:   ] pop
                   3891:   popVariables
                   3892:   arg1
                   3893: } def
                   3894:
                   3895: /to_univNum {
                   3896:   /arg1 set
                   3897:   [/rr  ] pushVariables
                   3898:   [
                   3899:     /rr arg1 def
                   3900:     rr isArray {
                   3901:       rr { to_univNum } map /rr set
                   3902:     } {
                   3903:     } ifelse
                   3904:     rr isInteger {
                   3905:       rr (universalNumber) dc /rr set
                   3906:     } {
                   3907:     } ifelse
                   3908:     /arg1 rr def
                   3909:   ] pop
                   3910:   popVariables
                   3911:   arg1
                   3912: } def
                   3913: [(to_univNum)
                   3914: [(obj to_univNum obj2)
                   3915:  (Example. [ 2 (3).. ] to_univNum)
1.33      takayama 3916:  $cf. to_int32. (to_int)$
1.29      takayama 3917: ]] putUsages
                   3918:
                   3919: [(lcm)
                   3920:  [ ([a b c ...] lcm r)
                   3921:    (cf. polylcm, mpzext)
                   3922:  ]
                   3923: ] putUsages
                   3924: /lcm {
                   3925:   /arg1 set
                   3926:   [/aa /bb /rr /pp /i] pushVariables
                   3927:   [
                   3928:     /aa arg1 def
                   3929:     /rr (1).. def
                   3930:     /pp 0 def % isPolynomial array?
                   3931:     0 1 aa length 1 sub {
                   3932:       /i set
                   3933:       aa i get  isPolynomial {
                   3934:         /pp 1 def
                   3935:         exit
                   3936:       } {  } ifelse
                   3937:     } for
                   3938:
                   3939:     0 1 aa length 1 sub {
                   3940:       /i set
                   3941:       pp {
                   3942:         [rr aa i get] polylcm /rr set
                   3943:       } {
                   3944:         [(lcm) rr aa i get ] mpzext /rr set
                   3945:       } ifelse
                   3946:     } for
                   3947:
                   3948:     /arg1 rr def
                   3949:   ] pop
                   3950:   popVariables
                   3951:   arg1
                   3952: } def
                   3953: [(gcd)
                   3954:  [ ([a b c ...] gcd r)
                   3955:    (cf. polygcd, mpzext)
                   3956:  ]
                   3957: ] putUsages
                   3958: /gcd {
                   3959:   /arg1 set
                   3960:   [/aa /bb /rr /pp /i] pushVariables
                   3961:   [
                   3962:     /aa arg1 def
                   3963:     /rr (1).. def
                   3964:     /pp 0 def % isPolynomial array?
                   3965:     0 1 aa length 1 sub {
                   3966:       /i set
                   3967:       aa i get  isPolynomial {
                   3968:         /pp 1 def
                   3969:         /rr aa i get def
                   3970:         exit
                   3971:       } {  } ifelse
                   3972:     } for
                   3973:
                   3974:     pp {
                   3975:      0 1 aa length 1 sub {
                   3976:        /i set
                   3977:        [rr aa i get] polygcd /rr set
                   3978:      } for
                   3979:     } {
                   3980:       aa ngcd /rr set
                   3981:     } ifelse
                   3982:
                   3983:     /arg1 rr def
                   3984:   ] pop
                   3985:   popVariables
                   3986:   arg1
                   3987: } def
                   3988:
                   3989: [(denominator)
                   3990:  [ ([a b c ...] denominator r)
                   3991:    ( a denominator r )
                   3992:    (cf. dc, numerator)
1.30      takayama 3993:    (Output is Z or a polynomial.)
1.29      takayama 3994:  ]
                   3995: ] putUsages
                   3996: % test data.
                   3997: %  [(1).. (2).. div (1).. (3).. div ] denominator
                   3998: %  [(2).. (3).. (4).. ] denominator
                   3999: /denominator {
                   4000:   /arg1 set
                   4001:   [/pp /dd /ii /rr] pushVariables
                   4002:   [
                   4003:     /pp arg1 def
1.30      takayama 4004:     pp to_univNum /pp set
1.29      takayama 4005:     {
                   4006:       pp isArray {
                   4007:         pp { denominator } map /dd set
                   4008:         /rr dd lcm def % rr = lcm(dd[0], dd[1], ... )
                   4009:         rr /dd set
                   4010:         exit
                   4011:       } {  } ifelse
                   4012:
                   4013:       pp (denominator) dc /dd set
                   4014:       exit
                   4015:
                   4016:     } loop
                   4017:     /arg1 dd def
                   4018:   ] pop
                   4019:   popVariables
                   4020:   arg1
                   4021: } def
                   4022:
                   4023: [(numerator)
                   4024:  [ ([a b c ...] numerator r)
                   4025:    ( a numerator r )
                   4026:    (cf. dc, denominator)
1.30      takayama 4027:    (Output is a list of Z or polynomials.)
1.29      takayama 4028:  ]
                   4029: ] putUsages
                   4030: % test data.
                   4031: /numerator {
                   4032:   /arg1 set
                   4033:   [/pp /dd /ii /rr] pushVariables
                   4034:   [
                   4035:     /pp arg1 def
1.30      takayama 4036:     pp to_univNum /pp set
1.29      takayama 4037:     {
                   4038:       pp isArray {
                   4039:         pp denominator /dd set
                   4040:         pp dd mul /rr set
1.30      takayama 4041:         rr cancel /rr set
1.29      takayama 4042:         exit
                   4043:       } {  } ifelse
                   4044:
                   4045:       pp (numerator) dc /rr set
                   4046:       exit
                   4047:
                   4048:     } loop
                   4049:     /arg1 rr def
                   4050:   ] pop
                   4051:   popVariables
                   4052:   arg1
                   4053: } def
                   4054:
1.30      takayama 4055: /cancel.Q {
1.29      takayama 4056:   /arg1 set
                   4057:   [/aa /rr /nn /dd /gg]  pushVariables
                   4058:   [
                   4059:     /aa arg1 def
                   4060:     {
                   4061:       aa isRational {
                   4062:          [(cancel) aa] mpzext /rr set
                   4063:          rr (denominator) dc (1).. eq {
                   4064:             /rr rr (numerator) dc def
                   4065:             exit
                   4066:          } { } ifelse
                   4067:          rr (denominator) dc (-1).. eq {
                   4068:             /rr rr (numerator) dc (-1).. mul def
                   4069:          } { } ifelse
                   4070:          exit
                   4071:       } { } ifelse
                   4072:
                   4073:       /rr aa def
                   4074:       exit
                   4075:     } loop
                   4076:     /arg1 rr def
                   4077:   ] pop
                   4078:   popVariables
                   4079:   arg1
                   4080: } def
                   4081:
1.30      takayama 4082: /cancel.one {
1.29      takayama 4083:   /arg1 set
                   4084:   [/aa /rr /nn /dd /gg]  pushVariables
                   4085:   [
                   4086:     /aa arg1 def
                   4087:     {
                   4088:       aa isRational {
                   4089:         aa (numerator) dc /nn set
                   4090:         aa (denominator) dc /dd set
                   4091:         nn isUniversalNumber dd isUniversalNumber and {
1.30      takayama 4092:           /rr aa cancel.Q def
1.29      takayama 4093:           exit
1.30      takayama 4094:         } { (cancel: not implemented) error } ifelse
1.29      takayama 4095:       } { } ifelse
                   4096:
                   4097:       /rr aa def
                   4098:       exit
                   4099:     } loop
                   4100:     /arg1 rr def
                   4101:   ] pop
                   4102:   popVariables
                   4103:   arg1
                   4104: } def
                   4105:
1.30      takayama 4106: [(cancel)
                   4107:  [ (obj cancel r)
1.29      takayama 4108:    (Cancel numerators and denominators)
                   4109:    (The implementation has not yet been completed. It works only for Q.)
                   4110: ]] putUsages
1.30      takayama 4111: /cancel {
1.29      takayama 4112:   /arg1 set
                   4113:   [/aa /rr] pushVariables
                   4114:   [
                   4115:     /aa arg1 def
                   4116:     aa isArray {
1.30      takayama 4117:       aa {cancel} map /rr set
1.29      takayama 4118:     } {
1.30      takayama 4119:       aa cancel.one /rr set
1.29      takayama 4120:     } ifelse
                   4121:     /arg1 rr def
1.27      takayama 4122:   ] pop
                   4123:   popVariables
                   4124:   arg1
1.22      takayama 4125: } def
1.30      takayama 4126:
                   4127: /nnormalize_vec {
                   4128:   /arg1 set
                   4129:   [/pp /rr /dd ] pushVariables
                   4130:   [
                   4131:     /pp arg1 def
                   4132:     pp denominator /dd set
                   4133:     dd (0).. lt { (nnormalize_vec: internal error) error } { } ifelse
                   4134:     pp numerator dd mul cancel /pp set
                   4135:     /@@@.nnormalize_vec_c dd def
                   4136:     pp gcd /dd set
                   4137:     dd (0).. lt { (nnormalize_vec: internal error) error } { } ifelse
                   4138:     pp (1).. dd div mul cancel /rr set
                   4139:     @@@.nnormalize_vec_c dd div cancel /@@@.nnormalize_vec_c set
                   4140:     /arg1 rr def
                   4141:   ] pop
                   4142:   popVariables
                   4143:   arg1
                   4144: } def
                   4145: [(nnormalize_vec)
                   4146: [(pp nnormalize_vec npp)
                   4147:  (It normalizes a given vector of Q into a vector of Z with relatively prime)
                   4148:  (entries by multiplying a postive number.)
                   4149: ]] putUsages
1.31      takayama 4150:
                   4151: /getNode {
                   4152:   /arg2 set
                   4153:   /arg1 set
                   4154:   [/in-getNode /ob /key /rr /rr /ii] pushVariables
                   4155:   [
                   4156:     /ob arg1 def
                   4157:     /key arg2 def
                   4158:     /rr null def
                   4159:     {
                   4160:       ob isClass {
                   4161:         ob (array) dc /ob set
                   4162:       } { exit } ifelse
                   4163:       ob 0 get key eq {
                   4164:         /rr ob def
                   4165:         exit
                   4166:       } {  } ifelse
                   4167:       ob 2 get /ob set
                   4168:       0 1 ob length 1 sub {
                   4169:          /ii set
                   4170:          ob ii get key getNode /rr set
                   4171:          rr tag 0 eq { } { exit } ifelse
                   4172:       } for
                   4173:       exit
                   4174:     } loop
                   4175:     /arg1 rr def
                   4176:   ] pop
                   4177:   popVariables
                   4178:   arg1
                   4179: } def
                   4180: [(getNode)
                   4181: [(ob key getNode)
                   4182:  (ob is a class object.)
                   4183:  (The operator getNode returns the node with the key in ob.)
                   4184:  (The node is an array of the format [key attr-list node-list])
                   4185:  (Example:)
                   4186:  (  /dog [(dog) [[(legs) 4] ] [ ]] [(class) (tree)] dc def)
                   4187:  (  /man [(man) [[(legs) 2] ] [ ]] [(class) (tree)] dc def)
                   4188:  (  /ma [(mammal) [ ] [man dog]] [(class) (tree)] dc def)
                   4189:  (  ma (dog) getNode )
                   4190: ]] putUsages
                   4191:
1.22      takayama 4192:
1.24      takayama 4193: /usages {
                   4194:   /arg1 set
                   4195:   [/name /flag /n /k /slist /m /i /sss] pushVariables
                   4196:   [
                   4197:     /name arg1 def
                   4198:     /flag true def
                   4199:
                   4200:  %BUG: should use regular expression in a future.
                   4201:  name tag 1 eq {
                   4202:    @.usages { 0 get } map shell { (, ) nl } map /sss set
                   4203:  } {
                   4204:
                   4205:     /sss [ ] def
                   4206:    @.usages length /n set
                   4207:    0 1 << n 1 sub >>
                   4208:    {
                   4209:       /k set
                   4210:       name << @.usages k get 0 get >> eq
                   4211:       {
                   4212:         /slist @.usages k get 1 get def
                   4213:         /m slist length def
                   4214:         0 1 << m 1 sub >> {
                   4215:           /i set
                   4216:           sss slist i get append nl append /sss set
                   4217:         } for
                   4218:         /flag false def
                   4219:       }
                   4220:       { }
                   4221:       ifelse
                   4222:    } for
                   4223:
                   4224:    %BUG:  cannot get usages of primitives.
                   4225:    flag
                   4226:    {name Usage  /sss [(Usage of ) name ( could not obtained.) nl ] def}
                   4227:    { }
                   4228:    ifelse
                   4229:  } ifelse
                   4230:    /arg1 sss cat def
                   4231:    ] pop
                   4232:    popVariables
                   4233:    arg1
                   4234: } def
                   4235: [(usages)
                   4236:  [(key usages usages-as-a-string)
                   4237:   (num usages list-of-key-words)
                   4238: ]] putUsages
1.1       maekawa  4239:
                   4240: ;
                   4241:
                   4242:
                   4243:
                   4244:
                   4245:
                   4246:

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