Annotation of OpenXM/src/kan96xx/Doc/restall_s.sm1, Revision 1.1
1.1 ! maekawa 1: %% restall_sn.sm1, New restall_s.sm1 is developed here.
! 2: %% /BFmessage 0 def controlled from cohom.sm1
! 3: BFmessage {
! 4: (**********************************************) message
! 5: [$ New restall_s (restall_s.sm1) $
! 6: $ 1999, 1/29: only Schreyer 1 works. For more than one unknowns. $
! 7: $ 1999, 5/21: Schreyer 2 is working for more than one unknowns.$
! 8: $ Schryer 2 still contains a bug when we truncate from the below.See gbhg3/Int/bug.sm1 and Example 5.6$
! 9: $ of Oaku-Takayama, math.AG/980506 $
! 10: $ 1999, 9/9: make restall1_s to vector clean program $
! 11: ] {message} map
! 12: (**********************************************) message
! 13: } { } ifelse
! 14: %%changed the following symbols.
! 15: %% complement ==> oaku.complement
! 16: %% syz ==> o.syz
! 17:
! 18: %%%%%%%%%%%%%%%%%%%%%%% restall.sm1 (Version 19980415) %%%%%%%%%%%%%%%%%%%%%%%
! 19: (restall_s.sm1...compute all the cohomology groups of the restriction) message-quiet
! 20: ( of a D-module to tt = (t_1,...,t_d) = (0,...,0).) message-quiet
! 21: (Schreyer Version: 19990521 by N.Takayama & T.Oaku) message-quiet
! 22: (usage: [(P1)...] [(t1)...] k0 k1 deg restall_s -> cohomologies of restriction)
! 23: message-quiet
! 24: ( [(P1)...] [(t1)...] k0 k1 deg intall_s --> cohomologies of integration)
! 25: message-quiet
! 26: % History: Nov.10, 1997, Apr.15,1998
! 27: %%%%%%%%%%%%%%%%%%%%%%%%%%%% Global variables %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! 28: %/BFvarlist %% Set all the variables (except s and the parameters) here.
! 29: /BFs (s) def
! 30: /BFth (s) def
! 31: /BFu (u) def
! 32:
! 33: /BFunknowns 1 def
! 34: [(x) (y)] /BFvarlist set
! 35: [ ] /BFparlist set
! 36:
! 37: /BFff
! 38: [ $x^3-y^2$ , $2*x*Dx + 3*y*Dy + 6$ , $2*y*Dx + 3*x^2*Dy$ ]
! 39: def
! 40:
! 41: %% 1 /Schreyer set Controlled from cohom.sm1
! 42: 0 /Cheat.restall_sn set
! 43: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! 44: %% The cohomology groups of the restriction
! 45: %% [(P1)...] [(t1)...] k0 k1 degmax restall
! 46: %% --> [0-th cohomology -1-th cohomology, ...,-degmax-th cohomology]
! 47:
! 48: /restall_s {
! 49: /arg5 set %% degmax
! 50: /arg4 set %% k1
! 51: /arg3 set %% k0
! 52: /arg2 set %% [(t1) ... (td)]
! 53: /arg1 set %% BFequations
! 54: [
! 55: /ff /bftt /k0 /k1 /degmax /syzlist /mveclist /cohomlist
! 56: /ideg /gbase /o.syz /m1vec /m2vec /r1 /r2
! 57: /i /syzi /j /syzij /maxtmp /max0 /ee /psi1index /zerolist
! 58: /psi1 /psi1ker /psi2image
! 59: /gbase1 /m1i /emonoi /nmono /bfDttmonoi /eei /dtp /k /psi1kervec
! 60: /pn /pn0 /psi1i /psi1keri /m2i /nker /nim /cohm /psiall /psisyz /cohom0
! 61: /syz0
! 62: ] pushVariables
! 63: [
! 64: /ff arg1 def /bftt arg2 def /k0 arg3 def /k1 arg4 def
! 65: /degmax arg5 def
! 66: bftt length /d set
! 67:
! 68: (Computing a free resolution ... ) message
! 69:
! 70: Schreyer 1 eq {ff bftt degmax resolution_SV /syzlist set}
! 71: { Cheat.restall_sn
! 72: {
! 73: Schreyer 0 eq {ff bftt degmax resolution_nsV /syzlist set}{ } ifelse
! 74: }
! 75: {
! 76: Schreyer 2 eq {ff bftt degmax resolution_Sh dehomogenize /syzlist set}
! 77: { (This version of restall_s works only for Schyreyer 1 and 2) message
! 78: error
! 79: } ifelse
! 80: } ifelse
! 81: } ifelse
! 82:
! 83: syzlist /BFresolution set
! 84: (A free resolution obtained.) message
! 85:
! 86: BFvarlist /ttxx set
! 87: BFparlist /aa set
! 88: [BFs] ttxx join aa join /allvarlist set
! 89: ttxx length /dn set
! 90: ttxx {xtoDx} map /Dttxx set
! 91:
! 92: /BFs_weight
! 93: [ [ BFs 1 ]
! 94: [ 0 1 dn 1 sub
! 95: { /i set Dttxx i get 1 }
! 96: for
! 97: 0 1 dn 1 sub
! 98: { /i set ttxx i get 1 }
! 99: for ]
! 100: ] def
! 101:
! 102: [ allvarlist listtostring ring_of_differential_operators
! 103: BFs_weight weight_vector 0 ] define_ring
! 104:
! 105: %% Reformatting the free resolution:
! 106: %% [[f1,f2,..],[syz1,...]] --> [[[f1],[f2],...],[syz,...]] (strings)
! 107: %% (to be modified for the case with more than one unknowns.)
! 108:
! 109: Schreyer 0 gt {
! 110: /syzlist1 [
! 111: syzlist 0 get /syz0 set
! 112: %% start N.T.
! 113: [BFunknowns syz0 { toString . } map]
! 114: toVectors2 { {toString} map } map
! 115: %% end N.T.
! 116: 1 1 degmax {/i set
! 117: syzlist 1 get i 1 sub get {toStrings} map
! 118: } for
! 119: ] def
! 120: syzlist1 /syzlist set
! 121: }{
! 122: /syzlist1 [
! 123: syzlist 0 get /syz0 set
! 124: [ 0 1 syz0 length 1 sub {/i set
! 125: [ syz0 i get (string) dc ]
! 126: } for ]
! 127: 1 1 degmax {/i set
! 128: syzlist i get {toStrings} map
! 129: } for
! 130: ] def
! 131: syzlist1 /syzlist set
! 132: } ifelse
! 133:
! 134: [ ] /cohomlist set
! 135:
! 136: %% Start the loop:
! 137: 0 1 degmax {/ideg set
! 138:
! 139: %(new loop: ) messagen ideg ::
! 140:
! 141: ideg 0 eq {
! 142: 1 /r0 set
! 143: %% N.T.
! 144: BFunknowns /r1 set
! 145: [ 1 1 BFunknowns { pop [ (0) ]} for ] /gbase set
! 146: [ 0 ] /m0vec set
! 147: [ 1 1 BFunknowns { pop 0} for ] /m1vec set
! 148: %% end N.T.
! 149: }{
! 150: syzlist << ideg 1 sub >> get /gbase set
! 151: r0 /r1 set
! 152: } ifelse
! 153: syzlist ideg get /o.syz set
! 154:
! 155: %% o.syz gbase
! 156: %% D^{r2} --> D^{r1} --> D^{r0}
! 157: %% with weight vectors: m2vec m1vec m0vec
! 158: %% which will induce a complex
! 159: %% psi2 psi1
! 160: %% D_{Y->X}^{r2} --> D_{Y->X}^{r1} --> D_{Y->X}^{r0}
! 161:
! 162: gbase length /r1 set
! 163: o.syz length /r2 set
! 164: /m2vec null def %% initialize.
! 165: BFmessage {
! 166: (m2vec = ) messagen m2vec message
! 167: (o.syz = ) messagen o.syz pmat
! 168: (m1vec = ) messagen m1vec message
! 169: (gbase = ) messagen gbase pmat
! 170: (m0vec = ) messagen m0vec message
! 171: } { } ifelse
! 172:
! 173:
! 174: %% (Computing the weight vector m2vec from m1vec and syz) message
! 175: /m2vec [
! 176: 0 1 r2 1 sub {/i set
! 177: o.syz i get /syzi set
! 178: 0 /nonzero set
! 179: 0 1 r1 1 sub {/j set
! 180: syzi j get expand /syzij set
! 181: syzij (0). eq { }{
! 182: syzij bftt fwh_order m1vec j get add /maxtmp set
! 183: nonzero 0 eq { maxtmp /max0 set }{
! 184: maxtmp max0 gt { maxtmp /max0 set }{ } ifelse
! 185: } ifelse
! 186: 1 /nonzero set
! 187: } ifelse
! 188: } for
! 189: max0 } for ] def
! 190:
! 191: %% ee = [u1,...,ud] corresponds to [Dt1,...,Dtd] (for graduation)
! 192: BFu /estr set
! 193: /ee
! 194: [ 1 1 d {/i set estr i toString 2 cat_n} for ]
! 195: def
! 196: [@@@.esymbol ] ee join /eee set
! 197:
! 198: %%(Setting up a ring that represents D_{Y->X}^{r1}) message
! 199: eee length /neee set
! 200: /eeemvec [ 1 1 neee {pop 1} for ] def
! 201: eee [ ] [BFs] BFvarlist join eeemvec setupDringVshift
! 202: bftt {xtoDx expand} map /bfDtt set
! 203: [ ] /psi1 set
! 204: [ ] /psi1index set
! 205: [ ] /zerolist set
! 206:
! 207: %%(converting gbase to a list of polynomials) message
! 208: /gbase1
! 209: [ 0 1 r1 1 sub {/i set
! 210: gbase i get {expand [[BFs expand (1).]] replace} map vector_to_poly
! 211: } for ] def
! 212:
! 213: gbase1 /gbase set
! 214:
! 215: %%(ideg =) messagen ideg ::
! 216: %%(Computing psi1) message
! 217: %% psi1
! 218: %% Computes D_{Y->X}^{r1} --> D_{Y->X}^{r0} induced by gbase
! 219: %% with weight k0 - m1vec <= k <= k1 - m1vec
! 220: 0 1 r1 1 sub {/i set
! 221: m1vec i get /m1i set
! 222: ee {expand} map k0 m1i sub k1 m1i sub monomials /emonoi set
! 223: bfDtt k0 m1i sub k1 m1i sub monomials /bfDttmonoi set
! 224: emonoi length /nmono set
! 225: 0 1 nmono 1 sub {/j set
! 226: @@@.esymbol expand i npower /eei set
! 227: emonoi j get eei mul /eei set
! 228: gbase i get /dtp set
! 229: bfDttmonoi j get dtp mul /dtp set
! 230: 0 1 d 1 sub {/k set
! 231: dtp [[bftt k get expand (0).]] replace /dtp set
! 232: dtp [[bfDtt k get ee k get expand]] replace /dtp set
! 233: } for
! 234: dtp [[(h). (1).]] replace /dtp set
! 235: dtp << ee {expand} map >> m0vec k0 Vtruncate_below /dtp set
! 236: dtp (0). eq {
! 237: zerolist [eei] join /zerolist set
! 238: }{
! 239: psi1index [eei] join /psi1index set
! 240: psi1 [dtp] join /psi1 set
! 241: } ifelse
! 242: } for
! 243: } for
! 244:
! 245: %%(ideg =) messagen ideg ::
! 246: %%(psi1 obtained.) message
! 247: %%(Computing psi1ker) message
! 248:
! 249: %% Computing psi1ker := Ker psi1 :
! 250: psi1 length 0 eq {
! 251: [ ] /psi1ker set
! 252: }{
! 253: psi1 {[[(h). (1).]] replace homogenize} map /psi1 set
! 254: [psi1 [(needSyz)]] groebner 2 get /psi1kervec set
! 255: psi1kervec length /pn set
! 256: psi1index length /pn0 set
! 257: [ ] /psi1ker set
! 258: 0 1 pn 1 sub {/i set
! 259: psi1kervec i get /psi1i set
! 260: (0). /psi1keri set
! 261: 0 1 pn0 1 sub {/j set
! 262: psi1index j get psi1i j get mul psi1keri add /psi1keri set
! 263: } for
! 264: psi1ker [ psi1keri [[(h). (1).]] replace ] join /psi1ker set
! 265: } for
! 266: } ifelse
! 267: zerolist psi1ker join /psi1ker set
! 268: % Is it all right to use reducedBase here?
! 269: % psi1ker length 0 eq { }{
! 270: % psi1ker reducedBase /psi1ker set
! 271: % } ifelse
! 272: %%(ideg =) messagen ideg ::
! 273: %%(psi1ker obtained.) message
! 274: %%(Computing psi2image ...) message
! 275:
! 276: %% psi2
! 277: %% Computes the image of D_{Y->X}^{r2} --> D_{Y->X}^{r1} induced by syz
! 278: %% with weight k0 - m2vec <= k <= k1 - m2vec
! 279: /psi2image [
! 280: 0 1 r2 1 sub {/i set
! 281: o.syz i get {expand [[BFs expand (1).]] replace} map /syzi set
! 282: syzi vector_to_poly /syzi set
! 283: m2vec i get /m2i set
! 284: bfDtt k0 m2i sub k1 m2i sub monomials /bfDttmonoi set
! 285: bfDttmonoi length /nmono set
! 286: 0 1 nmono 1 sub {/j set
! 287: bfDttmonoi j get syzi mul /syzij set
! 288: 0 1 d 1 sub {/k set
! 289: syzij [[bftt k get expand (0).]] replace /syzij set
! 290: syzij [[bfDtt k get ee k get expand]] replace /syzij set
! 291: } for
! 292: syzij [[(h). (1).]] replace /syzij set
! 293: syzij << ee {expand} map >> m1vec k0 Vtruncate_below /syzij set
! 294: syzij (0). eq { }{syzij} ifelse
! 295: } for
! 296: } for
! 297: ] def
! 298:
! 299: %(psi2image obtained.) message
! 300: %(ideg = ) messagen ideg ::
! 301: %(psi1ker = ) message psi1ker ::
! 302: %(psi2image =) message psi2image ::
! 303:
! 304: %% Computes the quotient module psi1ker/psi2image
! 305: psi1ker length /nker set
! 306: nker 0 eq {
! 307: [0 [ ]] /cohom set
! 308: }{
! 309: psi2image length /nim set
! 310: psi1ker psi2image join /psiall set
! 311: psiall {homogenize} map /psiall set
! 312: [psiall [(needSyz)]] groebner 2 get /psisyz set
! 313: psisyz {nker proj vector_to_poly [[(h). (1).]] replace} map /cohom set
! 314: cohom {remove0} map /cohom set
! 315: cohom length 0 eq {
! 316: [nker [ ]] /cohom set
! 317: }{
! 318: cohom {homogenize} map /cohom set
! 319: [cohom] groebner 0 get reducedBase /cohom set
! 320: cohom {[[(h). (1).]] replace} map /cohom set
! 321: [nker cohom] trimModule /cohom set
! 322: } ifelse
! 323: } ifelse
! 324: cohomlist [cohom] join /cohomlist set
! 325: 0 ideg sub print (-th cohomology: ) messagen
! 326: cohom ::
! 327: r1 /r0 set
! 328: r2 /r1 set
! 329: m1vec /m0vec set
! 330: m2vec /m1vec set
! 331: } for
! 332:
! 333: cohomlist /arg1 set
! 334: ] pop
! 335: popVariables
! 336: arg1
! 337: } def
! 338:
! 339: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! 340: %% The cohomology groups of the restriction without truncation from below
! 341: %% [(P1)...] [(t1)...] k1 degmax restall
! 342: %% --> [0-th cohomology -1-th cohomology, ...,-degmax-th cohomology]
! 343:
! 344: /restall1_s {
! 345: /arg5 set %% degmax
! 346: /arg4 set %% k1
! 347: /arg2 set %% [(t1) ... (td)]
! 348: /arg1 set %% BFequations
! 349: [
! 350: /ff /bftt /k1 /degmax /syzlist /mveclist /cohomlist
! 351: /ideg /gbase /o.syz /m1vec /m2vec /r1 /r2
! 352: /i /syzi /j /syzij /maxtmp /max0 /ee /psi1index /zerolist
! 353: /psi1 /psi1ker /psi2image
! 354: /gbase1 /m1i /emonoi /nmono /bfDttmonoi /eei /dtp /k /psi1kervec
! 355: /pn /pn0 /psi1i /psi1keri /m2i /nker /nim /cohm /psiall /psisyz /cohom0
! 356: /syz0
! 357: ] pushVariables
! 358: [
! 359: /ff arg1 def /bftt arg2 def /k1 arg4 def
! 360: /degmax arg5 def
! 361: bftt length /d set
! 362:
! 363: (Computing a free resolution ... ) message
! 364:
! 365: Schreyer 1 eq {ff bftt degmax resolution_SV /syzlist set}
! 366: { Cheat.restall_sn
! 367: {
! 368: Schreyer 0 eq {ff bftt degmax resolution_nsV /syzlist set}{ } ifelse
! 369: }
! 370: {
! 371: Schreyer 2 eq {ff bftt degmax resolution_Sh dehomogenize /syzlist set}
! 372: { (This version of restall_s works only for Schyreyer 1 and 2) message
! 373: error
! 374: } ifelse
! 375: } ifelse
! 376: } ifelse
! 377:
! 378: syzlist /BFresolution set
! 379: (A free resolution obtained.) message
! 380:
! 381:
! 382: BFvarlist /ttxx set
! 383: BFparlist /aa set
! 384: [BFs] ttxx join aa join /allvarlist set
! 385: ttxx length /dn set
! 386: ttxx {xtoDx} map /Dttxx set
! 387:
! 388: /BFs_weight
! 389: [ [ BFs 1 ]
! 390: [ 0 1 dn 1 sub
! 391: { /i set Dttxx i get 1 }
! 392: for
! 393: 0 1 dn 1 sub
! 394: { /i set ttxx i get 1 }
! 395: for ]
! 396: ] def
! 397:
! 398: [ allvarlist listtostring ring_of_differential_operators
! 399: BFs_weight weight_vector 0 ] define_ring
! 400:
! 401: %% Reformatting the free resolution:
! 402: %% [[f1,f2,..],[syz1,...]] --> [[[f1],[f2],...],[syz,...]] (strings)
! 403: %% (to be modified for the case with more than one unknowns.)
! 404:
! 405: Schreyer 0 gt {
! 406: /syzlist1 [
! 407: syzlist 0 get /syz0 set
! 408: %% start N.T.
! 409: [BFunknowns syz0 { toString . } map]
! 410: toVectors2 { {toString} map } map
! 411: %% end N.T.
! 412: 1 1 degmax {/i set
! 413: syzlist 1 get i 1 sub get {toStrings} map
! 414: } for
! 415: ] def
! 416: syzlist1 /syzlist set
! 417: }{
! 418: /syzlist1 [
! 419: syzlist 0 get /syz0 set
! 420: [ 0 1 syz0 length 1 sub {/i set
! 421: [ syz0 i get (string) dc ]
! 422: } for ]
! 423: 1 1 degmax {/i set
! 424: syzlist i get {toStrings} map
! 425: } for
! 426: ] def
! 427: syzlist1 /syzlist set
! 428: } ifelse
! 429:
! 430: [ ] /cohomlist set
! 431:
! 432: %% Start the loop:
! 433: 0 1 degmax {/ideg set
! 434:
! 435: %(new loop: ) messagen ideg ::
! 436:
! 437: ideg 0 eq {
! 438: 1 /r0 set
! 439: %% N.T.
! 440: BFunknowns /r1 set
! 441: [ 1 1 BFunknowns { pop [ (0) ]} for ] /gbase set
! 442: [ 0 ] /m0vec set
! 443: [ 1 1 BFunknowns { pop 0} for ] /m1vec set
! 444: %% end N.T.
! 445: }{
! 446: syzlist << ideg 1 sub >> get /gbase set
! 447: r0 /r1 set
! 448: } ifelse
! 449: syzlist ideg get /o.syz set
! 450:
! 451: %% o.syz gbase
! 452: %% D^{r2} --> D^{r1} --> D^{r0}
! 453: %% with weight vectors: m2vec m1vec m0vec
! 454: %% which will induce a complex
! 455: %% psi2 psi1
! 456: %% D_{Y->X}^{r2} --> D_{Y->X}^{r1} --> D_{Y->X}^{r0}
! 457:
! 458: gbase length /r1 set
! 459: o.syz length /r2 set
! 460:
! 461: BFmessage {
! 462: (m2vec = ) messagen m2vec message
! 463: (o.syz = ) messagen o.syz pmat
! 464: (m1vec = ) messagen m1vec message
! 465: (gbase = ) messagen gbase pmat
! 466: (m0vec = ) messagen m0vec message
! 467: } { } ifelse
! 468:
! 469: %% (Computing the weight vector m2vec from m1vec and syz) message
! 470: /m2vec [
! 471: 0 1 r2 1 sub {/i set
! 472: o.syz i get /syzi set
! 473: 0 /nonzero set
! 474: 0 1 r1 1 sub {/j set
! 475: syzi j get expand /syzij set
! 476: syzij (0). eq { }{
! 477: syzij bftt fwh_order m1vec j get add /maxtmp set
! 478: nonzero 0 eq { maxtmp /max0 set }{
! 479: maxtmp max0 gt { maxtmp /max0 set }{ } ifelse
! 480: } ifelse
! 481: 1 /nonzero set
! 482: } ifelse
! 483: } for
! 484: max0 } for ] def
! 485:
! 486: %% ee = [u1,...,ud] corresponds to [Dt1,...,Dtd] (for graduation)
! 487: BFu /estr set
! 488: /ee
! 489: [ 1 1 d {/i set estr i toString 2 cat_n} for ]
! 490: def
! 491: [@@@.esymbol ] ee join /eee set
! 492:
! 493: %%(Setting up a ring that represents D_{Y->X}^{r1}) message
! 494: eee length /neee set
! 495: /eeemvec [ 1 1 neee {pop 1} for ] def
! 496: eee [ ] [BFs] BFvarlist join eeemvec setupDringVshift
! 497: bftt {xtoDx expand} map /bfDtt set
! 498: [ ] /psi1 set
! 499: [ ] /psi1index set
! 500: [ ] /zerolist set
! 501:
! 502: %%(converting gbase to a list of polynomials) message
! 503: /gbase1
! 504: [ 0 1 r1 1 sub {/i set
! 505: gbase i get {expand [[BFs expand (1).]] replace} map vector_to_poly
! 506: } for ] def
! 507:
! 508: gbase1 /gbase set
! 509:
! 510: %%(ideg =) messagen ideg ::
! 511: %%(Computing psi1) message
! 512: %% psi1
! 513: %% Computes D_{Y->X}^{r1} --> D_{Y->X}^{r0} induced by gbase
! 514: %% with weight = k <= k1 - m1vec
! 515: 0 1 r1 1 sub {/i set
! 516: m1vec i get /m1i set
! 517: ee {expand} map 0 k1 m1i sub monomials /emonoi set
! 518: bfDtt 0 k1 m1i sub monomials /bfDttmonoi set
! 519: emonoi length /nmono set
! 520: 0 1 nmono 1 sub {/j set
! 521: @@@.esymbol expand i npower /eei set
! 522: emonoi j get eei mul /eei set
! 523: gbase i get /dtp set
! 524: bfDttmonoi j get dtp mul /dtp set
! 525: 0 1 d 1 sub {/k set
! 526: dtp [[bftt k get expand (0).]] replace /dtp set
! 527: dtp [[bfDtt k get ee k get expand]] replace /dtp set
! 528: } for
! 529: dtp [[(h). (1).]] replace /dtp set
! 530: dtp (0). eq {
! 531: zerolist [eei] join /zerolist set
! 532: }{
! 533: psi1index [eei] join /psi1index set
! 534: psi1 [dtp] join /psi1 set
! 535: } ifelse
! 536: } for
! 537: } for
! 538:
! 539: %%(ideg =) messagen ideg ::
! 540: %%(psi1 obtained.) message
! 541: %%(Computing psi1ker) message
! 542:
! 543: %% Computing psi1ker := Ker psi1 :
! 544: psi1 length 0 eq {
! 545: [ ] /psi1ker set
! 546: }{
! 547: psi1 {[[(h). (1).]] replace homogenize} map /psi1 set
! 548: [psi1 [(needSyz)]] groebner 2 get /psi1kervec set
! 549: psi1kervec length /pn set
! 550: psi1index length /pn0 set
! 551: [ ] /psi1ker set
! 552: 0 1 pn 1 sub {/i set
! 553: psi1kervec i get /psi1i set
! 554: (0). /psi1keri set
! 555: 0 1 pn0 1 sub {/j set
! 556: psi1index j get psi1i j get mul psi1keri add /psi1keri set
! 557: } for
! 558: psi1ker [ psi1keri [[(h). (1).]] replace ] join /psi1ker set
! 559: } for
! 560: } ifelse
! 561: zerolist psi1ker join /psi1ker set
! 562: % Is it all right to use reducedBase here?
! 563: % psi1ker length 0 eq { }{
! 564: % psi1ker reducedBase /psi1ker set
! 565: % } ifelse
! 566: %%(ideg =) messagen ideg ::
! 567: %%(psi1ker obtained.) message
! 568: %%(Computing psi2image ...) message
! 569:
! 570: %% psi2
! 571: %% Computes the image of D_{Y->X}^{r2} --> D_{Y->X}^{r1} induced by syz
! 572: %% with weight m2vec <= k <= k1 - m2vec
! 573: /psi2image [
! 574: 0 1 r2 1 sub {/i set
! 575: o.syz i get {expand [[BFs expand (1).]] replace} map /syzi set
! 576: syzi vector_to_poly /syzi set
! 577: m2vec i get /m2i set
! 578: bfDtt 0 k1 m2i sub monomials /bfDttmonoi set
! 579: bfDttmonoi length /nmono set
! 580: 0 1 nmono 1 sub {/j set
! 581: bfDttmonoi j get syzi mul /syzij set
! 582: 0 1 d 1 sub {/k set
! 583: syzij [[bftt k get expand (0).]] replace /syzij set
! 584: syzij [[bfDtt k get ee k get expand]] replace /syzij set
! 585: } for
! 586: syzij [[(h). (1).]] replace /syzij set
! 587: syzij (0). eq { }{syzij} ifelse
! 588: } for
! 589: } for
! 590: ] def
! 591:
! 592: %(psi2image obtained.) message
! 593: %(ideg = ) messagen ideg ::
! 594: %(psi1ker = ) message psi1ker ::
! 595: %(psi2image =) message psi2image ::
! 596:
! 597: %% Computes the quotient module psi1ker/psi2image
! 598: psi1ker length /nker set
! 599: nker 0 eq {
! 600: [0 [ ]] /cohom set
! 601: }{
! 602: psi2image length /nim set
! 603: psi1ker psi2image join /psiall set
! 604: psiall {homogenize} map /psiall set
! 605: [psiall [(needSyz)]] groebner 2 get /psisyz set
! 606: psisyz {nker proj vector_to_poly [[(h). (1).]] replace} map /cohom set
! 607: cohom {remove0} map /cohom set
! 608: cohom length 0 eq {
! 609: [nker [ ]] /cohom set
! 610: }{
! 611: cohom {homogenize} map /cohom set
! 612: [cohom] groebner 0 get reducedBase /cohom set
! 613: cohom {[[(h). (1).]] replace} map /cohom set
! 614: [nker cohom] trimModule /cohom set
! 615: } ifelse
! 616: } ifelse
! 617: cohomlist [cohom] join /cohomlist set
! 618: 0 ideg sub print (-th cohomology: ) messagen
! 619: cohom ::
! 620: r1 /r0 set
! 621: r2 /r1 set
! 622: m1vec /m0vec set
! 623: m2vec /m1vec set
! 624: } for
! 625:
! 626: cohomlist /arg1 set
! 627: ] pop
! 628: popVariables
! 629: arg1
! 630: } def
! 631:
! 632: /intall_s {
! 633: /arg5 set %% degmax
! 634: /arg4 set %% k1
! 635: /arg3 set %% k0
! 636: /arg2 set %% [(t1) ... (td)]
! 637: /arg1 set %% BFequations
! 638: [ /ff /bftt /k0 /k1 /degmax /ffdx ] pushVariables
! 639: [
! 640: /ff arg1 def /bftt arg2 def /k0 arg3 def /k1 arg4 def
! 641: /degmax arg5 def
! 642: BFvarlist setupDring
! 643: ff {bftt fourier} map /ffdx set
! 644: ffdx bftt k0 k1 degmax restall_s /arg1 set
! 645: ] pop
! 646: popVariables
! 647: arg1
! 648: } def
! 649:
! 650: /intall1_s {
! 651: /arg5 set %% degmax
! 652: /arg4 set %% k1
! 653: /arg2 set %% [(t1) ... (td)]
! 654: /arg1 set %% BFequations
! 655: [ /ff /bftt /k1 /degmax /ffdx ] pushVariables
! 656: [
! 657: /ff arg1 def /bftt arg2 def /k0 arg3 def /k1 arg4 def
! 658: /degmax arg5 def
! 659: BFvarlist setupDring
! 660: ff {bftt fourier} map /ffdx set
! 661: ffdx bftt k1 degmax restall1_s /arg1 set
! 662: ] pop
! 663: popVariables
! 664: arg1
! 665: } def
! 666:
! 667: /resolution_Sh {
! 668: /arg3 set /arg2 set /arg1 set
! 669: [ /tt /ff /deg /ttxx /aa /allvarlist /d /n /m /Dtt /Dxx /xx
! 670: /i /V_weight /G
! 671: ] pushVariables
! 672: [
! 673: arg1 /ff set arg2 /tt set arg3 /deg set
! 674: BFvarlist /ttxx set
! 675: BFparlist /aa set
! 676: ttxx aa join /allvarlist set
! 677: tt length /d set
! 678: ttxx tt setminus /xx set
! 679: xx length /n set
! 680: aa length /m set
! 681: tt {xtoDx} map /Dtt set
! 682: xx {xtoDx} map /Dxx set
! 683:
! 684: /V_weight [
! 685: [ 0 1 d 1 sub {/i set Dtt i get 1} for
! 686: 0 1 d 1 sub {/i set tt i get -1} for ]
! 687: [ 0 1 n 1 sub {/i set Dxx i get 1} for
! 688: 0 1 n 1 sub {/i set xx i get 1} for ]
! 689: ] def
! 690:
! 691: ttxx aa join /allvarlist set
! 692:
! 693: %% start N.T.
! 694: ff 0 get isArray {
! 695: /BFunknowns ff 0 get length def
! 696: } { /BFunknowns 1 def } ifelse
! 697: ff 0 get isArray {
! 698: ff { {toString} map } map /ff set
! 699: }{ ff { toString } map /ff set } ifelse
! 700: BFmessage
! 701: { (Homogenized ff = ) messagen ff message
! 702: (BFvarlist = ) messagen BFvarlist message
! 703: (BFparlist = ) messagen BFparlist message
! 704: (BFs = ) messagen BFs message
! 705: } { } ifelse
! 706: %% end N.T.
! 707:
! 708: [ allvarlist listtostring s_ring_of_differential_operators
! 709: V_weight s_weight_vector 0 [(schreyer) 1]] define_ring
! 710: ff 0 get isArray {
! 711: deg ff { {tparse} map } map sResolution /G set
! 712: } {
! 713: deg ff {tparse} map sResolution /G set
! 714: } ifelse
! 715: G /arg1 set
! 716: ] pop
! 717: popVariables
! 718: arg1
! 719: } def
! 720:
! 721: /resolution_SV {
! 722: /arg3 set /arg2 set /arg1 set
! 723: [ /ff /tt /deg /ttxx /aa /allvarlist /xx /dn /Dttxx /BFs_weight /i /G
! 724: ] pushVariables
! 725: [
! 726: arg1 /ff set arg2 /tt set arg3 /deg set
! 727: BFvarlist /ttxx set
! 728: BFparlist /aa set
! 729: [BFs] ttxx join aa join /allvarlist set
! 730: ttxx tt setminus /xx set
! 731: ttxx length /dn set
! 732: ttxx {xtoDx} map /Dttxx set
! 733:
! 734: /BFs_weight
! 735: [ [ BFs 1 ]
! 736: [ 0 1 dn 1 sub
! 737: { /i set Dttxx i get 1 }
! 738: for
! 739: 0 1 dn 1 sub
! 740: { /i set ttxx i get 1 }
! 741: for ]
! 742: ] def
! 743:
! 744: %% start N.T.
! 745: ff 0 get isArray {
! 746: /BFunknowns ff 0 get length def
! 747: } { /BFunknowns 1 def } ifelse
! 748: [ allvarlist listtostring ring_of_differential_operators
! 749: BFs_weight s_weight_vector 0] define_ring
! 750: ff 0 get isArray {
! 751: ff { {toString .} map
! 752: [0 1 BFunknowns 1 sub { /i set @@@.esymbol . i npower } for]
! 753: mul toString
! 754: } map /ff set
! 755: }{ ff { toString } map /ff set } ifelse
! 756: ff {tt fwm_homogenize} map /ff set
! 757: BFmessage
! 758: { (Homogenized ff = ) messagen ff message
! 759: (BFvarlist = ) messagen BFvarlist message
! 760: (BFparlist = ) messagen BFparlist message
! 761: (BFs = ) messagen BFs message
! 762: } { } ifelse
! 763: %% end N.T.
! 764:
! 765: [ allvarlist listtostring s_ring_of_differential_operators
! 766: BFs_weight s_weight_vector 0 [(schreyer) 1]] define_ring
! 767:
! 768: deg ff {tparse [[(h).(1).]] replace } map sResolution /G set
! 769:
! 770: G /arg1 set
! 771: ] pop
! 772: popVariables
! 773: arg1
! 774: } def
! 775:
! 776: %% Computing a free resolution compatible with the V-filtration
! 777: %% w.r.t. tt
! 778: /resolution_nsV {
! 779: /arg3 set %% rdegmax
! 780: /arg2 set %% tt
! 781: /arg1 set %% ff
! 782: [
! 783: /ff /tt /rdegmax /ttxx /xx /aa /dn /d /Dttxx /i /syzlist /rdeg
! 784: /allvarlist /gbase /o.syz /gbase1 /syz2 /syz3 /nsyz /syz2i /syz2ij
! 785: ] pushVariables
! 786: [
! 787: arg1 /ff set
! 788: arg2 /tt set
! 789: arg3 /rdegmax set
! 790: BFvarlist /ttxx set
! 791: BFparlist /aa set
! 792: ttxx tt setminus /xx set
! 793: ttxx length /dn set
! 794: /allvarlist
! 795: [ BFs ] ttxx join aa join
! 796: def
! 797: ttxx {xtoDx} map /Dttxx set
! 798: /BFs_weight
! 799: [ [ BFs 1 ]
! 800: [ 0 1 dn 1 sub
! 801: { /i set Dttxx i get 1 }
! 802: for
! 803: 0 1 dn 1 sub
! 804: { /i set ttxx i get 1 }
! 805: for ]
! 806: ] def
! 807: [ allvarlist listtostring ring_of_differential_operators
! 808: BFs_weight weight_vector
! 809: 0] define_ring
! 810: BFs expand /bfs set
! 811: [ ] /syzlist set
! 812:
! 813: %% start the loop (the counter rdeg represents the degree of the resolution)
! 814: 0 1 rdegmax {/rdeg set
! 815: %% From
! 816: %% ff=syz
! 817: %% ... <--- D_X^{r0} <--- D_X^{#ff},
! 818: %% computes
! 819: %% gbase syz
! 820: %% ... <--- D_X^{r0} <--- D_X^{r1} <--- D_X^{#syz}.
! 821:
! 822: rdeg 0 eq {
! 823: 1 /r0 set
! 824: ff {tt fwm_homogenize expand} map /ff set
! 825: }{
! 826: r1 /r0 set
! 827: o.syz {vector_to_poly} map /ff set
! 828: } ifelse
! 829:
! 830: ff {[[(h). (1).]] replace homogenize} map /ff set
! 831: %% Is it OK to use reducedBase here?
! 832: [ff] groebner 0 get {[[(h). (1).]] replace} map /gbase set
! 833: gbase reducedBase {homogenize} map /gbase set
! 834: [gbase [(needSyz)]] groebner 2 get /o.syz set
! 835: gbase length /r1 set
! 836:
! 837: %% V-homogenize syz:
! 838: gbase {bfs coefficients 0 get 0 get} map /msvec set
! 839: o.syz length /nsyz set
! 840: o.syz /syz2 set
! 841: /syz3 [ 0 1 nsyz 1 sub {/i set
! 842: syz2 i get /syz2i set
! 843: [ 0 1 r1 1 sub {/j set
! 844: syz2i j get /syz2ij set
! 845: msvec j get /msj set
! 846: syz2ij << bfs msj npower >> mul
! 847: } for ]
! 848: } for ] def
! 849: syz3 /o.syz set
! 850:
! 851: %% Comment out % if you want the output to be string lists
! 852: gbase {[[(h). (1).]] replace} map /gbase set
! 853: rdeg 0 eq {
! 854: % gbase toStrings /gbase1 set
! 855: gbase /gbase1 set
! 856: }{
! 857: % gbase r0 n_toVectors {toStrings} map /gbase1 set
! 858: gbase r0 n_toVectors /gbase1 set
! 859: } ifelse
! 860: syzlist [gbase1] join /syzlist set
! 861: o.syz length 0 eq {
! 862: syzlist [o.syz] join /syzlist set
! 863: 1 break
! 864: }{ } ifelse
! 865: } for
! 866:
! 867: syzlist /arg1 set
! 868: ] pop
! 869: popVariables
! 870: arg1
! 871: } def
! 872: %%%%%%%%%%%%%%%%%%%%% Utilities %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! 873: %% [u1,...] [v1,...] setminus --> [u1,...] \setminus [v1,...]
! 874: /setminus {
! 875: /arg2 set /arg1 set
! 876: [ /Set1 /Set2 /n2 /i ] pushVariables
! 877: [
! 878: arg1 /Set1 set arg2 /Set2 set
! 879: Set2 length /n2 set
! 880: 0 1 n2 1 sub {/i set
! 881: Set1 Set2 i get complement.oaku /Set1 set
! 882: } for
! 883: Set1 /arg1 set
! 884: ] pop
! 885: popVariables
! 886: arg1
! 887: } def
! 888:
! 889: %% (list arg1) \setminus {(an element arg2)}
! 890: /complement.oaku {
! 891: /arg2 set /arg1 set
! 892: arg1 { arg2 notidentical } map
! 893: } def
! 894:
! 895: /notidentical {
! 896: /arg2 set
! 897: /arg1 set
! 898: arg1 arg2 eq
! 899: { } {arg1} ifelse
! 900: } def
! 901:
! 902: %% Convert a polynomial list to a list of vectors of length r
! 903: %% [(P1).,,,,] r n_toVectors
! 904: /n_toVectors {
! 905: /arg2 set /arg1 set
! 906: [ ] pushVariables
! 907: [
! 908: arg1 /Ps set
! 909: arg2 /r set
! 910: Ps length /n set
! 911: Ps toVectors /Vecs set
! 912: /Vecs1 [ 0 1 n 1 sub {/i set
! 913: Vecs i get /Veci set
! 914: Veci length /ri set
! 915: 1 1 r ri sub {pop Veci [(0).] join /Veci set} for
! 916: Veci
! 917: } for ] def
! 918: Vecs1 /arg1 set
! 919: ] pop
! 920: popVariables
! 921: arg1
! 922: } def
! 923:
! 924: /toStrings {
! 925: /arg1 set
! 926: arg1 {(string) dc} map /arg1 set
! 927: arg1
! 928: } def
! 929:
! 930: %% (x1) --> (Dx1)
! 931: /xtoDx {
! 932: /arg1 set
! 933: @@@.Dsymbol arg1 2 cat_n
! 934: } def
! 935:
! 936: %% [(x1) (x2) (x3)] ---> (x1,x2,x3)
! 937: /listtostring {
! 938: /arg1 set
! 939: [/n /j /ary /str] pushVariables
! 940: [
! 941: /ary arg1 def
! 942: /n ary length def
! 943: arg1 0 get /str set
! 944: n 1 gt
! 945: { str (,) 2 cat_n /str set }{ }
! 946: ifelse
! 947: 1 1 n 1 sub {
! 948: /j set
! 949: j n 1 sub eq
! 950: {str << ary j get >> 2 cat_n /str set}
! 951: {str << ary j get >> (,) 3 cat_n /str set}
! 952: ifelse
! 953: } for
! 954: /arg1 str def
! 955: ] pop
! 956: popVariables
! 957: arg1
! 958: } def
! 959:
! 960: %% FW-homogenization
! 961: %% Op (string) [(t1) (t2) ...] fw_homogenize ---> h(Op) (string)
! 962: /fwm_homogenize {
! 963: /arg2 set %% bft (string list)
! 964: /arg1 set %% an operator (string)
! 965: [ /bftt /bft /bfDt /bfht /bfhDt /Op /degs /m /mn /d /i ] pushVariables
! 966: [
! 967: /Op arg1 expand def
! 968: /bftt arg2 def
! 969: bftt length /d set
! 970:
! 971: 0 1 d 1 sub { /i set
! 972: bftt i get /bft set
! 973: bft xtoDx /bfDt set
! 974: BFs (^(-1)*) bft 3 cat_n /bfht set
! 975: BFs (*) bfDt 3 cat_n /bfhDt set
! 976: Op [[bft expand bfht expand][bfDt expand bfhDt expand]] replace
! 977: /Op set
! 978: } for
! 979: Op BFs expand coefficients 0 get
! 980: {(integer) data_conversion} map /degs set
! 981: degs << degs length 1 sub >> get /m set
! 982: 0 m sub /mn set
! 983: << BFs expand mn powerZ >> Op mul /Op set
! 984: Op (string) data_conversion /arg1 set
! 985: ] pop
! 986: popVariables
! 987: arg1
! 988: } def
! 989:
! 990: %% var (poly) m (integer) ---> var^m (poly)
! 991: /powerZ {
! 992: /arg2 set %% m
! 993: /arg1 set %% Var
! 994: [ /m /var /varstr /pow /nvar] pushVariables
! 995: [
! 996: arg1 /var set
! 997: arg2 /m set
! 998: var (string) data_conversion /varstr set
! 999: m -1 gt
! 1000: { var m npower /pow set}
! 1001: { varstr (^(-1)) 2 cat_n expand /nvar set
! 1002: nvar << 0 m sub >> npower /pow set
! 1003: }
! 1004: ifelse
! 1005: pow /arg1 set
! 1006: ] pop
! 1007: popVariables
! 1008: arg1
! 1009: } def
! 1010:
! 1011:
! 1012:
! 1013: [(restall_s)
! 1014: [(f v s_0 s_1 level restall_s c)
! 1015: (array f; array v; integer s_0, s_1, level; array c)
! 1016: (f is a set of generators. v is a set of variables.)
! 1017: (s_0 is the minimal integral root of the b-function.)
! 1018: (s_1 is the maximal integral root of the b-function.)
! 1019: (level is the truncation degree of the resolution.)
! 1020: (c is the cohomolgy.)
! 1021: (This command is vector clean for Schreyer 1 and 2.)
! 1022: ( )
! 1023: (Global variables: array BFvarlist : list of variables.)
! 1024: ( array BFparlist : list of parameters.)
! 1025: ( int BFmessage : verbose or not.)
! 1026: ( int Schreyer : Schreyer 1 with tower-sugar.sm1 -- V-homog.)
! 1027: ( Schreyer 2 with tower.sm1 --- H-homog.)
! 1028: ( Schreyer 0 is a step by step resolution.)
! 1029: (result global var:array BFresolution : resolution matrices.)
! 1030: ( )
! 1031: (cf. /tower.verbose 1 def /tower-sugar.verbose 1 def )
! 1032: ( /debug.sResolution 1 def)
! 1033: ( to see steps of resoluitons.)
! 1034: $ /Schreyer 2 def (tower.sm1) run to try Schreyer 2. Schreyer 2 is default.$
! 1035: (See bfm --- b-function, restriction.)
! 1036: (restall1_s is as same as restall_s, except it does not truncate from below.)
! 1037: (Example 1: /BFvarlist [(x1) (x2)] def /BFparlist [ ] def)
! 1038: ( [(x1) (Dx2-1)] [(x1)] -1 -1 1 restall_s )
! 1039: (Example 1a: /BFvarlist [(x1) (x2)] def /BFparlist [ ] def)
! 1040: ( [(x1) (Dx2-1)] [(x1)] bfm )
! 1041: (Example 2: /BFvarlist [(x) (y)] def /BFparlist [ ] def)
! 1042: $ [[(x Dx -1) (0)] [(y Dy -2) (0)] [(1) (1)] ] /ff set $
! 1043: $ ff [(x) (y)] 0 4 2 restall_s $
! 1044: ]] putUsages
! 1045:
! 1046: [(restall1_s)
! 1047: [(f v s_1 level restall1_s c)
! 1048: (array f; array v; integer s_0, s_1, level; array c)
! 1049: (f is a set of generators. v is a set of variables.)
! 1050: (s_1 is the maximal integral root of the b-function.)
! 1051: (level is the truncation degree of the resolution.)
! 1052: (c is the cohomolgy.)
! 1053: (This command is vector clean for Schreyer 1 and 2.)
! 1054: ( )
! 1055: (Global variables: array BFvarlist : list of variables.)
! 1056: ( array BFparlist : list of parameters.)
! 1057: ( int BFmessage : verbose or not.)
! 1058: ( int Schreyer : Schreyer 1 with tower-sugar.sm1 -- V-homog.)
! 1059: ( Schreyer 2 with tower.sm1 --- H-homog.)
! 1060: ( Schreyer 0 is a step by step resolution.)
! 1061: (result global var:array BFresolution : resolution matrices.)
! 1062: ( )
! 1063: (cf. /tower.verbose 1 def /tower-sugar.verbose 1 def )
! 1064: ( /debug.sResolution 1 def)
! 1065: ( to see steps of resoluitons.)
! 1066: $ /Schreyer 2 def (tower.sm1) run to try Schreyer 2. Schreyer 2 is default.$
! 1067: (See bfm --- b-function, restriction.)
! 1068: (restall1_s is as same as restall_s, except it does not truncate from below.)
! 1069: (Example 1: /BFvarlist [(x1) (x2)] def /BFparlist [ ] def)
! 1070: ( [(x1) (Dx2-1)] [(x1)] -1 1 restall1_s )
! 1071: (Example 1a: /BFvarlist [(x1) (x2)] def /BFparlist [ ] def)
! 1072: ( [(x1) (Dx2-1)] [(x1)] bfm )
! 1073: (Example 2: /BFvarlist [(x) (y)] def /BFparlist [ ] def)
! 1074: $ [[(x Dx -1) (0)] [(y Dy -2) (0)] [(1) (1)] ] /ff set $
! 1075: $ ff [(x) (y)] 4 2 restall1_s $
! 1076: ]] putUsages
! 1077:
! 1078:
! 1079: %%%%%%% code for test.
! 1080:
! 1081: /test2 {
! 1082: [(x) ring_of_differential_operators 0] define_ring
! 1083: [(-x*Dx^2-Dx). [(x) (Dx)] laplace0 (-2*x*Dx-2). [(x) (Dx)] laplace0]
! 1084: {toString} map /ff1 set
! 1085: [(-x^2*Dx^3+4*x^2*Dx-3*x*Dx^2+4*x*Dx+4*x-Dx+2). [(x) (Dx)] laplace0 (0).]
! 1086: {toString} map /ff2 set
! 1087: (ff1, ff2, BFresolution ) message
! 1088: /BFvarlist [(x)] def /BFparlist [ ] def
! 1089: [ff1 ff2] [(x)] 0 4 1 restall_s ::
! 1090: [ [ff1 ff2] [(x)] [[(x)] [ ]]] restriction
! 1091: } def
! 1092:
! 1093: /test3 {
! 1094: [(x,y) ring_of_differential_operators 0] define_ring
! 1095: [(x Dx -1). (0).]
! 1096: [(1). @@@.esymbol .] mul toString /ff1 set
! 1097: [(y Dy -2). (0).]
! 1098: [(1). @@@.esymbol .] mul toString /ff2 set
! 1099: [(1). (1).]
! 1100: [(1). @@@.esymbol .] mul toString /ff3 set
! 1101: (ff1, ff2, ff3, BFresolution ) message
! 1102: /BFvarlist [(x) (y)] def
! 1103: /BFparlist [ ] def
! 1104: [ff1 ff2 ff3] [(x) (y)] 0 2 2 restall_s
! 1105: } def
! 1106:
! 1107: /test {
! 1108: [(x,y) ring_of_differential_operators 0] define_ring
! 1109: [(x Dx -1) (0)] /ff1 set
! 1110: [(y Dy -2) (0)] /ff2 set
! 1111: [(1) (1)] /ff3 set
! 1112: (ff1, ff2, ff3, BFresolution ) message
! 1113: /BFvarlist [(x) (y)] def
! 1114: /BFparlist [ ] def
! 1115: [ff1 ff2 ff3] [(x) (y)] 0 4 2 restall_s
! 1116: } def
! 1117:
! 1118:
! 1119:
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