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Annotation of OpenXM_contrib2/asir2000/lib/bfct, Revision 1.14

1.2       noro        1: /*
                      2:  * Copyright (c) 1994-2000 FUJITSU LABORATORIES LIMITED
                      3:  * All rights reserved.
                      4:  *
                      5:  * FUJITSU LABORATORIES LIMITED ("FLL") hereby grants you a limited,
                      6:  * non-exclusive and royalty-free license to use, copy, modify and
                      7:  * redistribute, solely for non-commercial and non-profit purposes, the
                      8:  * computer program, "Risa/Asir" ("SOFTWARE"), subject to the terms and
                      9:  * conditions of this Agreement. For the avoidance of doubt, you acquire
                     10:  * only a limited right to use the SOFTWARE hereunder, and FLL or any
                     11:  * third party developer retains all rights, including but not limited to
                     12:  * copyrights, in and to the SOFTWARE.
                     13:  *
                     14:  * (1) FLL does not grant you a license in any way for commercial
                     15:  * purposes. You may use the SOFTWARE only for non-commercial and
                     16:  * non-profit purposes only, such as academic, research and internal
                     17:  * business use.
                     18:  * (2) The SOFTWARE is protected by the Copyright Law of Japan and
                     19:  * international copyright treaties. If you make copies of the SOFTWARE,
                     20:  * with or without modification, as permitted hereunder, you shall affix
                     21:  * to all such copies of the SOFTWARE the above copyright notice.
                     22:  * (3) An explicit reference to this SOFTWARE and its copyright owner
                     23:  * shall be made on your publication or presentation in any form of the
                     24:  * results obtained by use of the SOFTWARE.
                     25:  * (4) In the event that you modify the SOFTWARE, you shall notify FLL by
1.3       noro       26:  * e-mail at risa-admin@sec.flab.fujitsu.co.jp of the detailed specification
1.2       noro       27:  * for such modification or the source code of the modified part of the
                     28:  * SOFTWARE.
                     29:  *
                     30:  * THE SOFTWARE IS PROVIDED AS IS WITHOUT ANY WARRANTY OF ANY KIND. FLL
                     31:  * MAKES ABSOLUTELY NO WARRANTIES, EXPRESSED, IMPLIED OR STATUTORY, AND
                     32:  * EXPRESSLY DISCLAIMS ANY IMPLIED WARRANTY OF MERCHANTABILITY, FITNESS
                     33:  * FOR A PARTICULAR PURPOSE OR NONINFRINGEMENT OF THIRD PARTIES'
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                     38:  * DIRECT, INDIRECT, SPECIAL, INCIDENTAL, PUNITIVE OR CONSEQUENTIAL
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                     40:  * ARISING OUT OF OR RELATING TO THE SOFTWARE OR THIS AGREEMENT, DAMAGES
                     41:  * FOR LOSS OF GOODWILL, WORK STOPPAGE, OR LOSS OF DATA, OR FOR ANY
                     42:  * DAMAGES, EVEN IF FLL SHALL HAVE BEEN INFORMED OF THE POSSIBILITY OF
                     43:  * SUCH DAMAGES, OR FOR ANY CLAIM BY ANY OTHER PARTY. EVEN IF A PART
                     44:  * OF THE SOFTWARE HAS BEEN DEVELOPED BY A THIRD PARTY, THE THIRD PARTY
                     45:  * DEVELOPER SHALL HAVE NO LIABILITY IN CONNECTION WITH THE USE,
                     46:  * PERFORMANCE OR NON-PERFORMANCE OF THE SOFTWARE.
                     47:  *
1.14    ! noro       48:  * $OpenXM: OpenXM_contrib2/asir2000/lib/bfct,v 1.13 2000/12/27 07:17:39 noro Exp $
1.10      noro       49:  */
1.1       noro       50: /* requires 'primdec' */
                     51:
1.6       noro       52: /* annihilating ideal of F^s */
1.1       noro       53:
                     54: def ann(F)
                     55: {
                     56:        V = vars(F);
                     57:        N = length(V);
1.8       noro       58:        D = newvect(N);
                     59:
                     60:        for ( I = 0; I < N; I++ )
                     61:                D[I] = [deg(F,V[I]),V[I]];
                     62:        qsort(D,compare_first);
                     63:        for ( V = [], I = N-1; I >= 0; I-- )
                     64:                V = cons(D[I][1],V);
                     65:
1.1       noro       66:        for ( I = N-1, DV = []; I >= 0; I-- )
                     67:                DV = cons(strtov("d"+rtostr(V[I])),DV);
1.8       noro       68:
                     69:        W = append([y1,y2,t],V);
1.1       noro       70:        DW = append([dy1,dy2,dt],DV);
1.8       noro       71:
                     72:        B = [1-y1*y2,t-y1*F];
1.1       noro       73:        for ( I = 0; I < N; I++ ) {
                     74:                B = cons(DV[I]+y1*diff(F,V[I])*dt,B);
                     75:        }
1.10      noro       76:
                     77:        /* homogenized (heuristics) */
1.1       noro       78:        dp_nelim(2);
1.10      noro       79:        G0 = dp_weyl_gr_main(B,append(W,DW),1,0,6);
1.1       noro       80:        G1 = [];
                     81:        for ( T = G0; T != []; T = cdr(T) ) {
                     82:                E = car(T); VL = vars(E);
                     83:                if ( !member(y1,VL) && !member(y2,VL) )
                     84:                        G1 = cons(E,G1);
                     85:        }
1.12      noro       86:        G2 = map(psi,G1,t,dt);
                     87:        G3 = map(subst,G2,t,-1-s);
                     88:        return G3;
1.1       noro       89: }
                     90:
1.10      noro       91: /*
                     92:  * compute J_f|s=r, where r = the minimal integral root of global b_f(s)
                     93:  * ann0(F) returns [MinRoot,Ideal]
                     94:  */
                     95:
                     96: def ann0(F)
                     97: {
                     98:        V = vars(F);
                     99:        N = length(V);
                    100:        D = newvect(N);
                    101:
                    102:        for ( I = 0; I < N; I++ )
                    103:                D[I] = [deg(F,V[I]),V[I]];
                    104:        qsort(D,compare_first);
                    105:        for ( V = [], I = 0; I < N; I++ )
                    106:                V = cons(D[I][1],V);
                    107:
                    108:        for ( I = N-1, DV = []; I >= 0; I-- )
                    109:                DV = cons(strtov("d"+rtostr(V[I])),DV);
                    110:
                    111:        /* XXX : heuristics */
                    112:        W = append([y1,y2,t],reverse(V));
                    113:        DW = append([dy1,dy2,dt],reverse(DV));
                    114:        WDW = append(W,DW);
                    115:
                    116:        B = [1-y1*y2,t-y1*F];
                    117:        for ( I = 0; I < N; I++ ) {
                    118:                B = cons(DV[I]+y1*diff(F,V[I])*dt,B);
                    119:        }
                    120:
                    121:        /* homogenized (heuristics) */
                    122:        dp_nelim(2);
                    123:        G0 = dp_weyl_gr_main(B,WDW,1,0,6);
                    124:        G1 = [];
                    125:        for ( T = G0; T != []; T = cdr(T) ) {
                    126:                E = car(T); VL = vars(E);
                    127:                if ( !member(y1,VL) && !member(y2,VL) )
                    128:                        G1 = cons(E,G1);
                    129:        }
1.12      noro      130:        G2 = map(psi,G1,t,dt);
                    131:        G3 = map(subst,G2,t,-1-s);
1.10      noro      132:
1.12      noro      133:        /* G3 = J_f(s) */
1.10      noro      134:
                    135:        V1 = cons(s,V); DV1 = cons(ds,DV); V1DV1 = append(V1,DV1);
1.12      noro      136:        G4 = dp_weyl_gr_main(cons(F,G3),V1DV1,0,1,0);
                    137:        Bf = weyl_minipoly(G4,V1DV1,0,s);
1.10      noro      138:
                    139:        FList = cdr(fctr(Bf));
                    140:        for ( T = FList, Min = 0; T != []; T = cdr(T) ) {
                    141:                LF = car(car(T));
                    142:                Root = -coef(LF,0)/coef(LF,1);
                    143:                if ( dn(Root) == 1 && Root < Min )
                    144:                        Min = Root;
                    145:        }
1.12      noro      146:        return [Min,map(subst,G3,s,Min)];
1.10      noro      147: }
                    148:
1.7       noro      149: def indicial1(F,V)
1.6       noro      150: {
                    151:        W = append([y1,t],V);
                    152:        N = length(V);
                    153:        B = [t-y1*F];
                    154:        for ( I = N-1, DV = []; I >= 0; I-- )
                    155:                DV = cons(strtov("d"+rtostr(V[I])),DV);
                    156:        DW = append([dy1,dt],DV);
                    157:        for ( I = 0; I < N; I++ ) {
                    158:                B = cons(DV[I]+y1*diff(F,V[I])*dt,B);
                    159:        }
                    160:        dp_nelim(1);
1.10      noro      161:
                    162:        /* homogenized (heuristics) */
1.7       noro      163:        G0 = dp_weyl_gr_main(B,append(W,DW),1,0,6);
1.6       noro      164:        G1 = map(subst,G0,y1,1);
                    165:        G2 = map(psi,G1,t,dt);
                    166:        G3 = map(subst,G2,t,-s-1);
                    167:        return G3;
                    168: }
                    169:
                    170: def psi(F,T,DT)
                    171: {
                    172:        D = dp_ptod(F,[T,DT]);
                    173:        Wmax = weight(D);
                    174:        D1 = dp_rest(D);
                    175:        for ( ; D1; D1 = dp_rest(D1) )
                    176:                if ( weight(D1) > Wmax )
                    177:                        Wmax = weight(D1);
                    178:        for ( D1 = D, Dmax = 0; D1; D1 = dp_rest(D1) )
                    179:                if ( weight(D1) == Wmax )
                    180:                        Dmax += dp_hm(D1);
                    181:        if ( Wmax >= 0 )
                    182:                Dmax = dp_weyl_mul(<<Wmax,0>>,Dmax);
                    183:        else
                    184:                Dmax = dp_weyl_mul(<<0,-Wmax>>,Dmax);
                    185:        Rmax = dp_dtop(Dmax,[T,DT]);
                    186:        R = b_subst(subst(Rmax,DT,1),T);
                    187:        return R;
                    188: }
                    189:
                    190: def weight(D)
                    191: {
                    192:        V = dp_etov(D);
                    193:        return V[1]-V[0];
                    194: }
                    195:
                    196: def compare_first(A,B)
                    197: {
                    198:        A0 = car(A);
                    199:        B0 = car(B);
                    200:        if ( A0 > B0 )
                    201:                return 1;
                    202:        else if ( A0 < B0 )
                    203:                return -1;
                    204:        else
                    205:                return 0;
                    206: }
                    207:
1.13      noro      208: /* generic b-function w.r.t. weight vector W */
                    209:
                    210: def generic_bfct(F,V,DV,W)
                    211: {
                    212:        N = length(V);
                    213:        N2 = N*2;
                    214:
1.14    ! noro      215:        /* create a term order M in D<x,d> (DRL) */
1.13      noro      216:        M = newmat(N2,N2);
                    217:        for ( J = 0; J < N2; J++ )
                    218:                M[0][J] = 1;
                    219:        for ( I = 1; I < N2; I++ )
                    220:                M[I][N2-I] = -1;
                    221:
                    222:        VDV = append(V,DV);
                    223:
                    224:        /* create a non-term order MW in D<x,d> */
                    225:        MW = newmat(N2+1,N2);
                    226:        for ( J = 0; J < N; J++ )
                    227:                MW[0][J] = -W[J];
                    228:        for ( ; J < N2; J++ )
                    229:                MW[0][J] = W[J-N];
                    230:        for ( I = 1; I <= N2; I++ )
                    231:                for ( J = 0; J < N2; J++ )
                    232:                        MW[I][J] = M[I-1][J];
                    233:
                    234:        /* create a homogenized term order MWH in D<x,d,h> */
                    235:        MWH = newmat(N2+2,N2+1);
                    236:        for ( J = 0; J <= N2; J++ )
                    237:                MWH[0][J] = 1;
                    238:        for ( I = 1; I <= N2+1; I++ )
                    239:                for ( J = 0; J < N2; J++ )
                    240:                        MWH[I][J] = MW[I-1][J];
                    241:
                    242:        /* homogenize F */
                    243:        VDVH = append(VDV,[h]);
                    244:        FH = map(dp_dtop,map(dp_homo,map(dp_ptod,F,VDV)),VDVH);
                    245:
                    246:        /* compute a groebner basis of FH w.r.t. MWH */
1.14    ! noro      247:        GH = dp_weyl_gr_main(FH,VDVH,0,1,MWH);
1.13      noro      248:
                    249:        /* dehomigenize GH */
                    250:        G = map(subst,GH,h,1);
                    251:
                    252:        /* G is a groebner basis w.r.t. a non term order MW */
                    253:        /* take the initial part w.r.t. (-W,W) */
                    254:        GIN = map(initial_part,G,VDV,MW,W);
                    255:
                    256:        /* GIN is a groebner basis w.r.t. a term order M */
                    257:        /* As -W+W=0, gr_(-W,W)(D<x,d>) = D<x,d> */
                    258:
                    259:        /* find b(W1*x1*d1+...+WN*xN*dN) in Id(GIN) */
                    260:        for ( I = 0, T = 0; I < N; I++ )
                    261:                T += W[I]*V[I]*DV[I];
1.14    ! noro      262:        B = weyl_minipoly(GIN,VDV,0,T); /* M represents DRL order */
1.13      noro      263:        return B;
                    264: }
                    265:
                    266: def initial_part(F,V,MW,W)
                    267: {
                    268:        N2 = length(V);
                    269:        N = N2/2;
                    270:        dp_ord(MW);
                    271:        DF = dp_ptod(F,V);
                    272:        R = dp_hm(DF);
                    273:        DF = dp_rest(DF);
                    274:
                    275:        E = dp_etov(R);
                    276:        for ( I = 0, TW = 0; I < N; I++ )
                    277:                TW += W[I]*(-E[I]+E[N+I]);
                    278:        RW = TW;
                    279:
                    280:        for ( ; DF; DF = dp_rest(DF) ) {
                    281:                E = dp_etov(DF);
                    282:                for ( I = 0, TW = 0; I < N; I++ )
                    283:                        TW += W[I]*(-E[I]+E[N+I]);
                    284:                if ( TW == RW )
                    285:                        R += dp_hm(DF);
                    286:                else if ( TW < RW )
                    287:                        break;
                    288:                else
                    289:                        error("initial_part : cannot happen");
                    290:        }
                    291:        return dp_dtop(R,V);
                    292:
                    293: }
                    294:
1.1       noro      295: /* b-function of F ? */
                    296:
                    297: def bfct(F)
                    298: {
                    299:        V = vars(F);
                    300:        N = length(V);
1.6       noro      301:        D = newvect(N);
1.7       noro      302:
1.6       noro      303:        for ( I = 0; I < N; I++ )
                    304:                D[I] = [deg(F,V[I]),V[I]];
                    305:        qsort(D,compare_first);
                    306:        for ( V = [], I = 0; I < N; I++ )
                    307:                V = cons(D[I][1],V);
1.1       noro      308:        for ( I = N-1, DV = []; I >= 0; I-- )
                    309:                DV = cons(strtov("d"+rtostr(V[I])),DV);
1.6       noro      310:        V1 = cons(s,V); DV1 = cons(ds,DV);
1.7       noro      311:
                    312:        G0 = indicial1(F,reverse(V));
                    313:        G1 = dp_weyl_gr_main(G0,append(V1,DV1),0,1,0);
                    314:        Minipoly = weyl_minipoly(G1,append(V1,DV1),0,s);
1.6       noro      315:        return Minipoly;
                    316: }
                    317:
1.14    ! noro      318: /* b-function computation via generic_bfct() (experimental) */
        !           319:
        !           320: def bfct_via_gbfct(F)
        !           321: {
        !           322:        V = vars(F);
        !           323:        N = length(V);
        !           324:        D = newvect(N);
        !           325:
        !           326:        for ( I = 0; I < N; I++ )
        !           327:                D[I] = [deg(F,V[I]),V[I]];
        !           328:        qsort(D,compare_first);
        !           329:        for ( V = [], I = 0; I < N; I++ )
        !           330:                V = cons(D[I][1],V);
        !           331:        V = reverse(V);
        !           332:        for ( I = N-1, DV = []; I >= 0; I-- )
        !           333:                DV = cons(strtov("d"+rtostr(V[I])),DV);
        !           334:
        !           335:        B = [t-F];
        !           336:        for ( I = 0; I < N; I++ ) {
        !           337:                B = cons(DV[I]+diff(F,V[I])*dt,B);
        !           338:        }
        !           339:        V1 = cons(t,V); DV1 = cons(dt,DV);
        !           340:        W = newvect(N+1);
        !           341:        W[0] = 1;
        !           342:        R = generic_bfct(B,V1,DV1,W);
        !           343:
        !           344:        return subst(R,s,-s-1);
        !           345: }
        !           346:
1.6       noro      347: def weyl_minipolym(G,V,O,M,V0)
                    348: {
                    349:        N = length(V);
                    350:        Len = length(G);
                    351:        dp_ord(O);
                    352:        setmod(M);
                    353:        PS = newvect(Len);
                    354:        PS0 = newvect(Len);
                    355:
                    356:        for ( I = 0, T = G; T != []; T = cdr(T), I++ )
                    357:                PS0[I] = dp_ptod(car(T),V);
                    358:        for ( I = 0, T = G; T != []; T = cdr(T), I++ )
                    359:                PS[I] = dp_mod(dp_ptod(car(T),V),M,[]);
                    360:
                    361:        for ( I = Len - 1, GI = []; I >= 0; I-- )
                    362:                GI = cons(I,GI);
                    363:
                    364:        U = dp_mod(dp_ptod(V0,V),M,[]);
                    365:
                    366:        T = dp_mod(<<0>>,M,[]);
                    367:        TT = dp_mod(dp_ptod(1,V),M,[]);
                    368:        G = H = [[TT,T]];
                    369:
                    370:        for ( I = 1; ; I++ ) {
1.14    ! noro      371:                if ( dp_gr_print() )
        !           372:                        print(".",2);
1.6       noro      373:                T = dp_mod(<<I>>,M,[]);
                    374:
                    375:                TT = dp_weyl_nf_mod(GI,dp_weyl_mul_mod(TT,U,M),PS,1,M);
                    376:                H = cons([TT,T],H);
                    377:                L = dp_lnf_mod([TT,T],G,M);
1.14    ! noro      378:                if ( !L[0] ) {
        !           379:                        if ( dp_gr_print() )
        !           380:                                print("");
1.13      noro      381:                        return dp_dtop(L[1],[t]); /* XXX */
1.14    ! noro      382:                } else
1.6       noro      383:                        G = insert(G,L);
                    384:        }
                    385: }
                    386:
1.13      noro      387: def weyl_minipoly(G0,V0,O0,P)
1.6       noro      388: {
1.11      noro      389:        HM = hmlist(G0,V0,O0);
1.13      noro      390:
                    391:        N = length(V0);
                    392:        Len = length(G0);
                    393:        dp_ord(O0);
                    394:        PS = newvect(Len);
                    395:        for ( I = 0, T = G0, HL = []; T != []; T = cdr(T), I++ )
                    396:                PS[I] = dp_ptod(car(T),V0);
                    397:        for ( I = Len - 1, GI = []; I >= 0; I-- )
                    398:                GI = cons(I,GI);
                    399:        DP = dp_ptod(P,V0);
                    400:
1.6       noro      401:        for ( I = 0; ; I++ ) {
                    402:                Prime = lprime(I);
1.11      noro      403:                if ( !valid_modulus(HM,Prime) )
                    404:                        continue;
1.13      noro      405:                MP = weyl_minipolym(G0,V0,O0,Prime,P);
                    406:                D = deg(MP,var(MP));
                    407:
                    408:                NFP = weyl_nf(GI,DP,1,PS);
                    409:                NF = [[dp_ptod(1,V0),1]];
                    410:                LCM = 1;
                    411:
                    412:                for ( J = 1; J <= D; J++ ) {
1.14    ! noro      413:                        if ( dp_gr_print() )
        !           414:                                print(".",2);
1.13      noro      415:                        NFPrev = car(NF);
                    416:                        NFJ = weyl_nf(GI,
                    417:                                dp_weyl_mul(NFP[0],NFPrev[0]),NFP[1]*NFPrev[1],PS);
                    418:                        NFJ = remove_cont(NFJ);
                    419:                        NF = cons(NFJ,NF);
                    420:                        LCM = ilcm(LCM,NFJ[1]);
                    421:                }
1.14    ! noro      422:                if ( dp_gr_print() )
        !           423:                        print("");
1.13      noro      424:                U = NF[0][0]*idiv(LCM,NF[0][1]);
                    425:                Coef = [];
                    426:                for ( J = D-1; J >= 0; J-- ) {
                    427:                        Coef = cons(strtov("u"+rtostr(J)),Coef);
                    428:                        U += car(Coef)*NF[D-J][0]*idiv(LCM,NF[D-J][1]);
                    429:                }
1.6       noro      430:
1.13      noro      431:                for ( UU = U, Eq = []; UU; UU = dp_rest(UU) )
                    432:                        Eq = cons(dp_hc(UU),Eq);
                    433:                M = etom([Eq,Coef]);
                    434:                B = henleq(M,Prime);
                    435:                if ( dp_gr_print() )
                    436:                        print("");
1.6       noro      437:                if ( B ) {
1.13      noro      438:                        R = 0;
                    439:                        for ( I = 0; I < D; I++ )
                    440:                                R += B[0][I]*s^I;
                    441:                        R += B[1]*s^D;
1.6       noro      442:                        return R;
                    443:                }
                    444:        }
                    445: }
                    446:
                    447: def weyl_nf(B,G,M,PS)
                    448: {
                    449:        for ( D = 0; G; ) {
                    450:                for ( U = 0, L = B; L != []; L = cdr(L) ) {
                    451:                        if ( dp_redble(G,R=PS[car(L)]) > 0 ) {
                    452:                                GCD = igcd(dp_hc(G),dp_hc(R));
                    453:                                CG = idiv(dp_hc(R),GCD); CR = idiv(dp_hc(G),GCD);
                    454:                                U = CG*G-dp_weyl_mul(CR*dp_subd(G,R),R);
                    455:                                if ( !U )
                    456:                                        return [D,M];
                    457:                                D *= CG; M *= CG;
                    458:                                break;
                    459:                        }
                    460:                }
                    461:                if ( U )
                    462:                        G = U;
                    463:                else {
                    464:                        D += dp_hm(G); G = dp_rest(G);
                    465:                }
                    466:        }
                    467:        return [D,M];
                    468: }
                    469:
                    470: def weyl_nf_mod(B,G,PS,Mod)
                    471: {
                    472:        for ( D = 0; G; ) {
                    473:                for ( U = 0, L = B; L != []; L = cdr(L) ) {
                    474:                        if ( dp_redble(G,R=PS[car(L)]) > 0 ) {
                    475:                                CR = dp_hc(G)/dp_hc(R);
                    476:                                U = G-dp_weyl_mul_mod(CR*dp_mod(dp_subd(G,R),Mod,[]),R,Mod);
                    477:                                if ( !U )
                    478:                                        return D;
1.1       noro      479:                                break;
1.6       noro      480:                        }
                    481:                }
                    482:                if ( U )
                    483:                        G = U;
                    484:                else {
                    485:                        D += dp_hm(G); G = dp_rest(G);
1.1       noro      486:                }
                    487:        }
1.6       noro      488:        return D;
1.1       noro      489: }
                    490:
                    491: def remove_zero(L)
                    492: {
                    493:        for ( R = []; L != []; L = cdr(L) )
                    494:                if ( car(L) )
                    495:                        R = cons(car(L),R);
                    496:        return R;
                    497: }
                    498:
                    499: def z_subst(F,V)
                    500: {
                    501:        for ( ; V != []; V = cdr(V) )
                    502:                F = subst(F,car(V),0);
                    503:        return F;
                    504: }
                    505:
                    506: def flatmf(L) {
                    507:     for ( S = []; L != []; L = cdr(L) )
                    508:                if ( type(F=car(car(L))) != NUM )
                    509:                        S = append(S,[F]);
                    510:        return S;
                    511: }
                    512:
                    513: def member(A,L) {
                    514:     for ( ; L != []; L = cdr(L) )
                    515:                if ( A == car(L) )
                    516:                        return 1;
                    517:        return 0;
                    518: }
                    519:
                    520: def intersection(A,B)
                    521: {
                    522:        for ( L = []; A != []; A = cdr(A) )
                    523:        if ( member(car(A),B) )
                    524:                L = cons(car(A),L);
                    525:        return L;
                    526: }
                    527:
                    528: def b_subst(F,V)
                    529: {
                    530:        D = deg(F,V);
                    531:        C = newvect(D+1);
                    532:        for ( I = D; I >= 0; I-- )
                    533:                C[I] = coef(F,I,V);
                    534:        for ( I = 0, R = 0; I <= D; I++ )
                    535:                if ( C[I] )
                    536:                        R += C[I]*v_factorial(V,I);
                    537:        return R;
                    538: }
                    539:
                    540: def v_factorial(V,N)
                    541: {
                    542:        for ( J = N-1, R = 1; J >= 0; J-- )
                    543:                R *= V-J;
                    544:        return R;
                    545: }
                    546: end$
                    547:

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