Ticket #54: freegb.lib

//////////////////////////////////////////////////////////////////////////////
version="$Id: freegb.lib,v 1.12 2008/10/06 17:04:27 Singular Exp $";
category="Noncommutative";
info="
LIBRARY: freegb.lib   Twosided Noncommutative Groebner bases in Free Algebras
AUTHOR: Viktor Levandovskyy,     levandov@math.rwth-aachen.de

BACKGROUND: 
TODO: WRITE DOWN SOME THEORY BEHIND (DIVISIBILITY, ORDERING ON FREE ASSOC. ALG., GB) 
TODO: ADD REFERENCES
TODO: EXPLAIN YOUR ASSUMPTIONS/IMPLICITE AGREEMENTS/INPUT, OUTPUT FORMATS ETC.
TODO: BAD NAMES, SEE 3.9.1 Procedures in a library, 5^th rule. PLEASE RENAME!


MAIN PROCEDURES: 

freegbRing(int d);    creates a ring with d blocks of shifted original variables
freegbasis(list L, int n); computes two-sided GB of an ideal up to degree n

AUXILIARY PROCEDURES:

lp2lstr(ideal K, def save); converts letter-place ideal to a list of modules
lst2str(list L[,int n]);    converts a list of modules into polys in free alg.
mod2str(module M[,int n]);  converts a module into a polynomial in free algebra
vct2str(module M[,int n]);  converts a vector into a word in free algebra
"

// TODO: LOTS OF GLOBAL UNLISTED PROCS WITHOUT HELP/EXAMPLES... ->
// -> MAKE THEM STATIC OR DOCUMENT THEM!
// TODO: AT LEAST SOME (E.G. Serre MUST BE LISTED IN THE HEADER)

// TODO: MOVE THE FOLLOWING TO THE ABOVE BACKGROUND...

// this library computes two-sided GB of an ideal
// in a free associative algebra

// a monomial is encoded via a vector V
// where V[1] = coefficient
// V[1+i] = the corresponding symbol





LIB "discretize.lib"; // for replace
LIB "qhmoduli.lib"; // for Max


// obsolete?

proc lshift(module M, int s, string varing, def lpring)
{
  // FINALLY IMPLEMENTED AS A PART OT THE CODE
  // shifts a poly from the ring @R to s positions
  // M lives in varing, the result in lpring
  // to be run from varing
  int i, j, k, sm, sv;
  vector v;
  //  execute("setring "+lpring);
  setring lpring;
  poly @@p;
  ideal I;
  execute("setring "+varing);
  sm = ncols(M);
  for (i=1; i<=s; i++)
  {
    // modules, e.g. free polynomials
    for (j=1; j<=sm; j++)
    {
      //vectors, e.g. free monomials
      v  = M[j];
      sv = size(v);
      sp = "@@p = @@p + ";
      for (k=2; k<=sv; k++)
      {
        sp = sp + string(v[k])+"("+string(k-1+s)+")*";
      }
      sp = sp + string(v[1])+";"; // coef;
      setring lpring;
      //      execute("setring "+lpring);
      execute(sp);
      execute("setring "+varing);
    }
    setring lpring;
    //    execute("setring "+lpring);
    I = I,@@p;
    @@p = 0;
  }
  setring lpring;
  //execute("setring "+lpring);
  export(I);
  //  setring varing;
  execute("setring "+varing);
}

proc skip0(vector v)
{
  // skips zeros in a vector, producing another vector
  int sv = nrows(v);
  int sw = size(v);
  if (sv == sw)
  {
    return(v);
  }
  int i;
  int j=1;
  vector w;
  for (i=1; i<=sv; i++)
  {
    if (v[i] != 0)
    {
      w = w + v[i]*gen(j);
      j++;
    }
  }
  return(w);
}


// TODO: ANY ASSUMPTIONS ON THE BASERING? 
// TODO: EXPLAIN WHICH FREE ALGEBRA IS MEANT?

proc lst2str(list L, list #)
"USAGE:  lst2str(L[,n]);  L a list of modules, n an optional integer
RETURN:  list of strings
PURPOSE: converts a list of modules into a list of strings representing 
         corresponding module as an element in the free algebra
EXAMPLE: example lst2str; shows examples
NOTE:    if an optional integer is not 0, stars signs are used in multiplication
"
{
  // returns a list of strings
  // being sentences in words built from L
  // if #[1] = 1, use * between generators
  int useStar = 0;
  if ( size(#)>0 )
  {
    if (typeof(#[1]) == "int")
    {
      useStar = 1;
    }
  }
  int i;
  int s = size(L);
  list N;
  for(i=1; i<=s; i++)
  {
    if ((typeof(L[i]) == "module") || (typeof(L[i]) == "matrix") )
    {
      N[i] = mod2str(L[i],useStar);
    }
    else
    {
      ERROR("module or matrix expected in the list");
      brake;
//      return(N);
    }
  }
  return(N);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y,z),(dp(1),dp(2));
  module M = [-1,x,y],[-7,y,y],[3,x,x];
  module N = [1,x,y,x,y],[-2,y,x,y,x],[6,x,y,y,x,y];
  list L; L[1] = M; L[2] = N;
  lst2str(L);
  lst2str(L[1],1);
}


// TODO: ANY ASSUMPTIONS ON THE BASERING/INPUT!? 
// TODO: EXPLAIN WHICH FREE ALGEBRA IS MEANT?

proc mod2str(module M, list #)
"USAGE:  mod2str(M[,n]);  M a module, n an optional integer
RETURN:  string
PURPOSE: Converts the module M into a string representing M
         as an element in the free algebra
EXAMPLE: example mod2str; shows examples
NOTE: if an optional integer is not 0, stars signs are used in multiplication
"
{
  // returns a string
  // a sentence in words built from M
  // if #[1] = 1, use * between generators
  int useStar = 0;
  if ( size(#)>0 )
  {
    if (typeof(#[1]) == "int")
    {
      useStar = 1;
    }
  }
  int i;
  int s    = ncols(M);
  string t;
  string mp;
  for(i=1; i<=s; i++)
  {
    mp = vct2str(M[i],useStar);
    if (mp[1] == "-")
    {
      t = t + mp;
    }
    else
    {
      t = t + "+" + mp;
    }
  }
  if (t[1]=="+")
  {
    t = t[2..size(t)]; // remove first "+"
  }
  return(t);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y,z),(dp);
  module M = [1,x,y,x,y],[-2,y,x,y,x],[6,x,y,y,x,y];
  mod2str(M);
  mod2str(M,1);
  // TODO: WHAT ABOUT ZERO?
  M = 0;
  mod2str(M); // TODO: FIX BUG: MUST BE JUST A ZERO!
  M = [0], [0], [1], [0, x];
  mod2str(M); // TODO: FIX THIS BUGS!
  M = [1, x, y, z], [0, x], [2, y], [0];
  mod2str(M);
}

// TODO: ANY ASSUMPTIONS ON THE BASERING/INPUT!? 
// TODO: EXPLAIN WHICH FREE ALGEBRA IS MEANT?

proc vct2str(vector v, list #)
"USAGE:  vct2str(v[,n]);  v a vector, n an optional integer
RETURN:  string
PURPOSE: Converts a vector into a string representing a word in the free alg.
EXAMPLE: example vct2str; shows examples
NOTE: if an optional integer is not 0, stars signs are used in multiplication
"
{
  // if #[1] = 1, use * between generators
  int useStar = 0;
  if ( size(#)>0 )
  {
    if (typeof(#[1]) == "int")
    {
      useStar = 1;
    }
  }
  int ppl = printlevel-voice+2;
  // for a word, encoded by v
  // produces a string for it
  v = skip0(v);
  number cf = leadcoef(v[1]);
  int s = size(v);
  string vs,vv,vp,err;
  int i,j,p,q;
  for (i=1; i<=s-1; i++)
  {
    p     = IsVar(v[i+1]);
    if (p==0)
    {
      err = "Error: monomial expected at" + string(i+1);
      dbprint(ppl,err);
      return("_");
    }
    if (p==1)
    {
      if (useStar && (size(vs) >0))       {   vs = vs + "*"; }
        vs = vs + string(v[i+1]);
    }
    else //power
    {
      vv = string(v[i+1]);
      q = find(vv,"^");
      if (q==0)
      {
        q = find(vv,string(p));
        if (q==0)
        {
          err = "error in find for string "+vv;
          dbprint(ppl,err);
          return("_");
        }
      }
      // q>0
      vp = vv[1..q-1];
      for(j=1;j<=p;j++)
      {
         if (useStar && (size(vs) >0))       {   vs = vs + "*"; }
         vs = vs + vp;
      }
    }
  }
  string scf;
  if (cf == -1)
  {
    scf = "-";
  }
  else
  {
    scf = string(cf);
    if (cf == 1)
    {
      scf = "";
    }
  }
  if (useStar && (size(scf) >0) && (scf!="-") )       {   scf = scf + "*"; }
  vs = scf + vs;
  return(vs);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = (0,a),(x,y3,z(1)),dp;
  vector v = [-7,x,y3^4,x2,z(1)^3];
  vct2str(v);
  vct2str(v,1);
  vector w = [-7a^5+6a,x,y3,y3,x,z(1),z(1)];
  vct2str(w);
  vct2str(w,1);
  // TODO: WHAT ABOUT ZERO?
  vector M = 0;
  vct2str(M); 
  M = [0];
  vct2str(M);
  M = [0, x, y3, z(1)];
  vct2str(M,1); // TODO: FIX BUG: MUST BE ZERO!
  M = [1, x, 0, z(1)];
  vct2str(M,1);
  M = [1, 0, z(1), 0, 0, x*y3, 0, 666];
  vct2str(M,1); // TODO: FIX BUG: WHAT IS THIS???

}

proc IsVar(poly p)
{
  // checks whether p is a variable indeed
  // if it's a power of a variable, returns the power
  if (p==0) {  return(0); } //"p=0";
  poly q   = leadmonom(p);
  if ( (p-lead(p)) !=0 ) { return(0); } // "p-lm(p)>0";
  intvec v = leadexp(p);
  int s = size(v);
  int i=1;
  int cnt = 0;
  int pwr = 0;
  for (i=1; i<=s; i++)
  {
    if (v[i] != 0)
    {
      cnt++;
      pwr = v[i];
    }
  }
  //  "cnt:";  cnt;
  if (cnt==1) { return(pwr); }
  else { return(0); }
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y),dp;
  poly f = xy+1;
  IsVar(f);
  poly g = xy;
  IsVar(g);
  poly h = y^3;
  IsVar(h);
  poly i = 1;
  IsVar(i);
}

// new conversion routines

proc id2words(ideal I, int d)
{
  // NOT FINISHED
  // input: ideal I of polys in letter-place notation
  // in the ring with d real vars
  // output: the list of strings: associative words
  // extract names of vars
  int i,m,n; string s; string place = "(1)";
  list lv;
  for(i=1; i<=d; i++)
  {
    s = string(var(i));
    // get rid of place
    n = find(s, place);
    if (n>0)
    {
      s = s[1..n-1];
    }
    lv[i] = s;
  }
  poly p,q;
  for (i=1; i<=ncols(I); i++)
  {
    if (I[i] != 0)
    {
      p = I[i];
      while (p!=0)
      {
         q = leadmonom(p);
      }
    }
  }

  return(lv);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x(1),y(1),z(1),x(2),y(2),z(2)),dp;
  ideal I = x(1)*y(2) -z(1)*x(2);
  id2words(I,3);
}

proc mono2word(poly p, int d)
{
}

// given the element -7xy^2x, it is represented as [-7,x,y^2,x] or as [-7,x,y,y,x]
// use the orig ord on (x,y,z) and expand it blockwise to (x(i),y(i),z(i))

// the correspondences:
// monomial in K<x,y,z>    <<--->> vector in R
// polynomial in K<x,y,z>  <<--->> list of vectors (matrix/module) in R
// ideal in K<x,y,z>       <<--->> list of matrices/modules in R


// 1. form a new ring
// 2. NOP
// 3. compute GB -> with the kernel stuff
// 4. skip shifted elts (check that no such exist?)
// 5. go back to orig vars, produce strings/modules
// 6. return the result


// TODO: BAD NAME -> RENAME!
// TODO: NO ASSUMPTIONS? WHAT ABOUT NON-COMM. INPUT RING? NON-HOMOG. INPUT?
// TODO: ADD STEP-BY-STEP COMMENTS.

proc freegbasis(list LM, int d)
"USAGE:  freegbasis(L, d);  L a list of modules, d an integer
RETURN:  ring, 
  TODO: EXPLAIN OUTPUT FORMAT
PURPOSE: compute the two-sided Groebner basis of an ideal, encoded by L 
         (TODO: EXPLAIN FORMAT OF L)
         in the free associative algebra,
         (TODO: WHICH free associative algebra? SYMBOLS? GROUND FIELD?)
         up to degree d
BACKGROUND: 
  TODO: EXPLAIN ALGORITHM, GIVE REFERENCES!
  TODO: WHAT ABOUT TERMINATION?
DISPLAY: 
  TODO: EXPLAIN!
EXAMPLE: example freegbasis; shows examples
"
{
  // d = up to degree, will be shifted to d+1
  if (d<1) {"bad d"; return(0);}

  int ppl = printlevel-voice+2;
  string err = "";

  int i,j,s;
  def save = basering;
  // determine max no of places in the input
  int slm = size(LM); // numbers of polys in the ideal
  int sm;
  intvec iv;
  module M;
  for (i=1; i<=slm; i++)
  {
    // modules, e.g. free polynomials
    M  = LM[i];
    sm = ncols(M);
    for (j=1; j<=sm; j++)
    {
      //vectors, e.g. free monomials
      iv = iv, size(M[j])-1; // 1 place is reserved by the coeff
    }
  }

  // TODO: THE FOLLOWING SEEMS TO BE THE CONTENT OF 'freegbRing'!
  // AND THUS IS OVERSIMPLIFIED: WHAT IF VARIABLES/PARAMETERS HAVE BRACKETS?
  // SEE EXAMPLE FOR 'freegbRing'!

  int D = Max(iv); // max size of input words
  if (d<D) {"bad d"; return(LM);}
  D = D + d-1;
  //  D = d;
  list LR  = ringlist(save);
  list L, tmp;
  L[1] = LR[1]; // ground field
  L[4] = LR[4]; // quotient ideal
  tmp  = LR[2]; // varnames
  s = size(LR[2]);
  for (i=1; i<=D; i++)
  {
    for (j=1; j<=s; j++)
    {
      tmp[i*s+j] = string(tmp[j])+"("+string(i+1)+")";
    }
  }
  for (i=1; i<=s; i++)
  {
    tmp[i] = string(tmp[i])+"("+string(1)+")";
  }
  L[2] = tmp;
  list OrigNames = LR[2];
  // ordering: d blocks of the ord on r
  // try to get whether the ord on r is blockord itself
  s = size(LR[3]);
  if (s==2)
  {
    // not a blockord, 1 block + module ord
    tmp = LR[3][s]; // module ord
    for (i=1; i<=D; i++)
    {
      LR[3][s-1+i] = LR[3][1];
    }
    LR[3][s+D] = tmp;
  }
  if (s>2)
  {
    // there are s-1 blocks
    int nb = s-1;
    tmp = LR[3][s]; // module ord
    for (i=1; i<=D; i++)
    {
      for (j=1; j<=nb; j++)
      {
        LR[3][i*nb+j] = LR[3][j];
      }
    }
    //    size(LR[3]);
    LR[3][nb*(D+1)+1] = tmp;
  }
  L[3] = LR[3];
  def @R = ring(L); 

  // HERE END 'freegbRing'! ;-)


  // TODO: THE FOLLOWING CONVERSION DESERVES TO BE AN AUXILIARY PROCEDURE!

  setring @R; // !
  ideal I;
  poly @p;
  s = size(OrigNames);
  //  "s:";s;
  // convert LM to canonical vectors (no powers)
  setring save; // !!
  kill M; // M was defined earlier
  module M;
  slm = size(LM); // numbers of polys in the ideal
  int sv,k,l;
  vector v;
  //  poly p;
  string sp;
  setring @R; // !!!
  poly @@p=0;
  setring save; // !!!! 
  for (l=1; l<=slm; l++)
  {
    // modules, e.g. free polynomials
    M  = LM[l];
    sm = ncols(M); // in intvec iv the sizes are stored
    // modules, e.g. free polynomials
    for (j=1; j<=sm; j++)
    {
      //vectors, e.g. free monomials
      v  = M[j];
      sv = size(v);
      //        "sv:";sv;
      sp = "@@p = @@p + ";
      for (k=2; k<=sv; k++)
      {
        sp = sp + string(v[k])+"("+string(k-1)+")*";
      }
      sp = sp + string(v[1])+";"; // coef;
      setring @R; // !!!!!
      execute(sp);
      setring save; // !!!!! !
    }
    setring @R; // !!!!! !!
    //      "@@p:"; @@p;
    I = I,@@p;
    @@p = 0;
    setring save; // !!!!! !!!
  }
  kill sp;
  // 3. compute GB
  setring @R; // !!!!! !!!! 

  // TODO: UNBELIEVABLE PLENITUDE OF SETRING ABOVE!
  // YOU CAN DO BETTER AS FOLLOWS: 
  //  1. GENERATE ONLY 1 (ONE!!!!) STRING sp FOR THE WHOLE(!!!) L 
  //              BEFORE(!!!!) SWITCHING TO @R 
  //  2. EXECUTE IT IN @R 

  dbprint(ppl,"computing GB");
  ideal J = system("freegb",I,d,nvars(save));
  //  ideal J = slimgb(I);
  dbprint(ppl,J);
  // 4. skip shifted elts
  ideal K = select1(J,1..s); // s = size(OrigNames)
  dbprint(ppl,K);
  dbprint(ppl, "done with GB");

  // TODO: THE FOLLOWING DESERVES TO BE AN AUXILIARY PROCEDURE AS WELL! IS IT lp2lstr?

  // K contains vars x(1),...z(1) = images of originals
  // 5. go back to orig vars, produce strings/modules
  if (K[1] == 0)
  {
    "no reasonable output, GB gives 0";
    return(0);
  }

  // YOU HAVE A LOT OF SETRINGS BELOW, ARE YOU SURE THAT YOU CANNOT ELIMINATE SOME...?

  int sk = size(K);
  int sp, sx, a, b;
  intvec x;
  poly p,q;
  poly pn;
  // vars in 'save'
  setring save; // !
  module N;
  list LN;
  vector V;
  poly pn;
  // test and skip exponents >=2
  setring @R; // !!
  for(i=1; i<=sk; i++)
  {
    p  = K[i];
    while (p!=0)
    {
      q  = lead(p);
      //      "processing q:";q;
      x  = leadexp(q);
      sx = size(x);
      for(k=1; k<=sx; k++)
      {
        if ( x[k] >= 2 )
        {
          err = "skip: the value x[k] is " + string(x[k]);
          dbprint(ppl,err);
          //            return(0);
          K[i] = 0;
          p    = 0;
          q    = 0;
          break;
        }
      }
      p  = p - q;
    }
  }
  K  = simplify(K,2);
  sk = size(K);

  for(i=1; i<=sk; i++)
  {
    //    setring save;
    //    V  = 0;
    setring @R; // !!!
    p  = K[i];
    while (p!=0)
    {
      q  = lead(p);
      err =  "processing q:" + string(q);
      dbprint(ppl,err);
      x  = leadexp(q);
      sx = size(x);
      pn = leadcoef(q);
      setring save; // !!!!
      pn = imap(@R,pn);
      V  = V + leadcoef(pn)*gen(1);
      for(k=1; k<=sx; k++)
      {
        if (x[k] ==1)
        {
          a = k / s; // block number=a+1, a!=0
          b = k % s; // remainder
          //          printf("a: %s, b: %s",a,b);
          if (b == 0)
          {
            // that is it's the last var in the block
            b = s;
            a = a-1;
          }
          V = V + var(b)*gen(a+2);
        }
//         else
//         {
//           printf("error: the value x[k] is %s", x[k]);
//           return(0);
//         }
      }
      err = "V: " + string(V);
      dbprint(ppl,err);
      //      printf("V: %s", string(V));
      N = N,V;
      V  = 0;
      setring @R; // !!!!!
      p  = p - q;
      pn = 0;
    }
    setring save; // !!!!! !
    LN[i] = simplify(N,2);
    N     = 0;
  }
  setring save; // !!!!! !!
  return(LN);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y,z),(dp(1),dp(2));
  module M = [-1,x,y],[-7,y,y],[3,x,x];
  module N = [1,x,y,x],[-1,y,x,y];
  list L; L[1] = M; L[2] = N;
  lst2str(L);
  def U = freegbasis(L,5);
  lst2str(U);
  kill U;
  setring r; // non-homog. input:
  M = [-1,x,y],[-7,y,y,z],[3,x,x,x,z,z];
  N = [1,x,y,z,x],[-1,y,x,y];
  L[1] = M; L[2] = N;
  lst2str(L);
  def U = freegbasis(L,5);
  lst2str(U); // OK
  kill U,r;
  ring r = 0,(x,y,z),(dp(1),dp(2));
  def R = nc_algebra(1,0); setring R; // should be the same as 1st!
  module M = [-1,x,y],[-7,y,y],[3,x,x];
  module N = [1,x,y,x],[-1,y,x,y];
  list L; L[1] = M; L[2] = N;
  lst2str(L);
  def U = freegbasis(L,5);
  lst2str(U); // OK
  kill U, R;
  setring r;
  def R = nc_algebra(-1,1); setring R; // some non-commutativity
  R;
  module M = [-1,x,y],[-7,y,y],[3,x,x];
  module N = [1,x,y,x],[-1,y,x,y];
  list L; L[1] = M; L[2] = N;
  lst2str(L);
  def U = freegbasis(L,5);  
  lst2str(U);


}

proc crs(list LM, int d)
"USAGE:  crs(L, d);  L a list of modules, d an integer
RETURN:  ring
PURPOSE: create a ring and shift the ideal
EXAMPLE: example crs; shows examples
"
{
  // d = up to degree, will be shifted to d+1
  if (d<1) {"bad d"; return(0);}

  int ppl = printlevel-voice+2;
  string err = "";

  int i,j,s;
  def save = basering;
  // determine max no of places in the input
  int slm = size(LM); // numbers of polys in the ideal
  int sm;
  intvec iv;
  module M;
  for (i=1; i<=slm; i++)
  {
    // modules, e.g. free polynomials
    M  = LM[i];
    sm = ncols(M);
    for (j=1; j<=sm; j++)
    {
      //vectors, e.g. free monomials
      iv = iv, size(M[j])-1; // 1 place is reserved by the coeff
    }
  }
  int D = Max(iv); // max size of input words
  if (d<D) {"bad d"; return(LM);}
  D = D + d-1;
  //  D = d;
  list LR  = ringlist(save);
  list L, tmp;
  L[1] = LR[1]; // ground field
  L[4] = LR[4]; // quotient ideal
  tmp  = LR[2]; // varnames
  s = size(LR[2]);
  for (i=1; i<=D; i++)
  {
    for (j=1; j<=s; j++)
    {
      tmp[i*s+j] = string(tmp[j])+"("+string(i)+")";
    }
  }
  for (i=1; i<=s; i++)
  {
    tmp[i] = string(tmp[i])+"("+string(0)+")";
  }
  L[2] = tmp;
  list OrigNames = LR[2];
  // ordering: d blocks of the ord on r
  // try to get whether the ord on r is blockord itself
  s = size(LR[3]);
  if (s==2)
  {
    // not a blockord, 1 block + module ord
    tmp = LR[3][s]; // module ord
    for (i=1; i<=D; i++)
    {
      LR[3][s-1+i] = LR[3][1];
    }
    LR[3][s+D] = tmp;
  }
  if (s>2)
  {
    // there are s-1 blocks
    int nb = s-1;
    tmp = LR[3][s]; // module ord
    for (i=1; i<=D; i++)
    {
      for (j=1; j<=nb; j++)
      {
        LR[3][i*nb+j] = LR[3][j];
      }
    }
    //    size(LR[3]);
    LR[3][nb*(D+1)+1] = tmp;
  }
  L[3] = LR[3];
  def @R = ring(L);
  setring @R;
  ideal I;
  poly @p;
  s = size(OrigNames);
  //  "s:";s;
  // convert LM to canonical vectors (no powers)
  setring save;
  kill M; // M was defined earlier
  module M;
  slm = size(LM); // numbers of polys in the ideal
  int sv,k,l;
  vector v;
  //  poly p;
  string sp;
  setring @R;
  poly @@p=0;
  setring save;
  for (l=1; l<=slm; l++)
  {
    // modules, e.g. free polynomials
    M  = LM[l];
    sm = ncols(M); // in intvec iv the sizes are stored
    for (i=0; i<=d-iv[l]; i++)
    {
      // modules, e.g. free polynomials
      for (j=1; j<=sm; j++)
      {
        //vectors, e.g. free monomials
        v  = M[j];
        sv = size(v);
        //        "sv:";sv;
        sp = "@@p = @@p + ";
        for (k=2; k<=sv; k++)
        {
          sp = sp + string(v[k])+"("+string(k-2+i)+")*";
        }
        sp = sp + string(v[1])+";"; // coef;
        setring @R;
        execute(sp);
        setring save;
      }
      setring @R;
      //      "@@p:"; @@p;
      I = I,@@p;
      @@p = 0;
      setring save;
    }
  }
  setring @R;
  export I;
  return(@R);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y,z),(dp(1),dp(2));
  module M = [-1,x,y],[-7,y,y],[3,x,x];
  module N = [1,x,y,x],[-1,y,x,y];
  list L; L[1] = M; L[2] = N;
  lst2str(L);
  def U = crs(L,5);
  setring U; U;
  I;
}

proc polylen(ideal I)
{
  // returns the ideal of length of polys
  int i;
  intvec J;
  number s = 0;
  for(i=1;i<=ncols(I);i++)
  {
    J[i] = size(I[i]);
    s = s + J[i];
  }
  printf("the sum of length %s",s);
  //  print(s);
  return(J);
}


// TODO: ASSUMPTIONS? 
// TODO: PROC FORGETS ABOUT NON-COMM. RELATIONS IN THE 
//// BASERING! DOCUMENT IT OR FIX...
// TODO: WRITE DOWN WHAT IS THE ORDERING IN THE OUTPUT RING?
// TODO: OVERSIMPLIFIED: WHAT IF PARAMETERS HAVE BRACKETS
//// AND COINCIDE WITH GENERATED VARIABLES? SEE EXAMPLE


proc freegbRing(int d)
"USAGE:  freegbRing(d); d an integer
RETURN:  ring
PURPOSE: creates a ring with d blocks of shifted original variables
EXAMPLE: example freegbRing; shows examples
"
{
  // d = up to degree, will be shifted to d+1
  if (d<1) {"bad d"; return(0);}

  int ppl = printlevel-voice+2; 
  string err = "";

  int i,j,s;
  def save = basering;
  int D = d-1;
  list LR  = ringlist(save);
  list L, tmp;
  L[1] = LR[1]; // ground field
  L[4] = LR[4]; // quotient ideal
  tmp  = LR[2]; // varnames
  s = size(LR[2]);
  for (i=1; i<=D; i++)
  {
    for (j=1; j<=s; j++)
    {
      tmp[i*s+j] = string(tmp[j])+"("+string(i+1)+")";
    }
  }
  for (i=1; i<=s; i++)
  {
    tmp[i] = string(tmp[i])+"("+string(1)+")";
  }
  L[2] = tmp;
  list OrigNames = LR[2];
  // ordering: d blocks of the ord on r
  // try to get whether the ord on r is blockord itself
  // TODO: make L(2) ordering! exponent is maximally 2
  s = size(LR[3]);
  if (s==2)
  {
    // not a blockord, 1 block + module ord
    tmp = LR[3][s]; // module ord
    for (i=1; i<=D; i++)
    {
      LR[3][s-1+i] = LR[3][1];
    }
    LR[3][s+D] = tmp;
  }
  if (s>2)
  {
    // there are s-1 blocks
    int nb = s-1;
    tmp = LR[3][s]; // module ord
    for (i=1; i<=D; i++)
    {
      for (j=1; j<=nb; j++)
      {
        LR[3][i*nb+j] = LR[3][j];
      }
    }
    //    size(LR[3]);
    LR[3][nb*(D+1)+1] = tmp;
  }
  L[3] = LR[3];
  def @R = ring(L);
  //  setring @R;
  return (@R);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y,z),(dp(1),dp(2));
  r;
  def A = freegbRing(2); setring A;
  A;
  kill A, r;
  ring r = 0,(x(1..3)),(dp(1),lp(2));
  r;
  def A = freegbRing(2); setring A;
  A; // OK
  kill A, r;
  ring r = (0,a(1),b(1)),(a, b),(lp(1),dp(1));
  r;
  def A = freegbRing(2); setring A; 
  A; // BUG: parameter should not be named as a variable and vice verse!
  a(1); typeof(a(1));
  kill A, r;
  ring r = 0,(x,y,z),dp;  
  def R = nc_algebra(-1, 1); setring R; 
  R;  
  def A = freegbRing(2); setring A;
  A; // NOTE: the putput is a purely commutative ring!
  kill A, R, r;
}

proc ex_shift()
{
  LIB "freegb.lib";
  ring r = 0,(x,y,z),(dp(1),dp(2));
  module M = [-1,x,y],[-7,y,y],[3,x,x];
  module N = [1,x,y,x],[-1,y,x,y];
  list L; L[1] = M; L[2] = N;
  lst2str(L);
  def U = crs(L,5);
  setring U; U;
  I;
  poly p = I[2]; // I[8];
  p;
  system("stest",p,7,7,3); // error -> the world is ok
  poly q1 = system("stest",p,1,7,3); //ok
  poly q6 = system("stest",p,6,7,3); //ok
  system("btest",p,3); //ok
  system("btest",q1,3); //ok
  system("btest",q6,3); //ok
}

proc test_shrink()
{
  LIB "freegb.lib";
  ring r =0,(x,y,z),dp;
  int d = 5;
  def R = freegbRing(d);
  setring R;
  poly p1 = x(1)*y(2)*z(3);
  poly p2 = x(1)*y(4)*z(5);
  poly p3 = x(1)*y(1)*z(3);
  poly p4 = x(1)*y(2)*z(2);
  poly p5 = x(3)*z(5);
  poly p6 = x(1)*y(1)*x(3)*z(5);
  poly p7 = x(1)*y(2)*x(3)*y(4)*z(5);
  poly p8 = p1+p2+p3+p4+p5 + p6 + p7;
  p1; system("shrinktest",p1,3);
  p2; system("shrinktest",p2,3);
  p3; system("shrinktest",p3,3);
  p4; system("shrinktest",p4,3);
  p5; system("shrinktest",p5,3);
  p6; system("shrinktest",p6,3);
  p7; system("shrinktest",p7,3);
  p8; system("shrinktest",p8,3);
  poly p9 = p1 + 2*p2 + 5*p5 + 7*p7;
  p9; system("shrinktest",p9,3);
}

proc ex2()
{
  option(prot);
  LIB "freegb.lib";
  ring r = 0,(x,y),dp;
  module M = [-1,x,y],[3,x,x]; // 3x^2 - xy
  def U = freegb(M,7);
  lst2str(U);
}

proc ex_nonhomog()
{
  option(prot);
  LIB "freegb.lib";
  ring r = 0,(x,y,h),dp;
  list L;
  module M;
  M = [-1,y,y],[1,x,x,x];  // x3-y2
  L[1] = M;
  M = [1,x,h],[-1,h,x];  // xh-hx
  L[2] = M;
  M = [1,y,h],[-1,h,y];  // yh-hy
  L[3] = M;
  def U = freegb(L,4);
  lst2str(U);
  // strange elements in the basis
}

proc ex_nonhomog_comm()
{
  option(prot);
  LIB "freegb.lib";
  ring r = 0,(x,y),dp;
  module M = [-1,y,y],[1,x,x,x];
  def U = freegb(M,5);
  lst2str(U);
}

proc ex_nonhomog_h()
{
  option(prot);
  LIB "freegb.lib";
  ring r = 0,(x,y,h),(a(1,1),dp);
  module M = [-1,y,y,h],[1,x,x,x]; // x3 - y2h
  def U = freegb(M,6);
  lst2str(U);
}

proc ex_nonhomog_h2()
{
  option(prot);
  LIB "freegb.lib";
  ring r = 0,(x,y,h),(dp);
  list L;
  module M;
  M = [-1,y,y,h],[1,x,x,x]; // x3 - y2h
  L[1] = M;
  M = [1,x,h],[-1,h,x]; // xh - hx
  L[2] = M;
  M = [1,y,h],[-1,h,y]; // yh - hy
  L[3] = M;
  def U = freegbasis(L,3);
  lst2str(U);
  // strange answer CHECK
}


proc ex_nonhomog_3()
{
  option(prot);
  LIB "./freegb.lib";
  ring r = 0,(x,y,z),(dp);
  list L;
  module M;
  M = [1,z,y],[-1,x]; // zy - x
  L[1] = M;
  M = [1,z,x],[-1,y]; // zx - y
  L[2] = M;
  M = [1,y,x],[-1,z]; // yx - z
  L[3] = M;
  lst2str(L);
  list U = freegb(L,4);
  lst2str(U);
  // strange answer CHECK
}

proc ex_densep_2()
{
  option(prot);
  LIB "freegb.lib";
  ring r = (0,a,b,c),(x,y),(Dp); // deglex
  module M = [1,x,x], [a,x,y], [b,y,x], [c,y,y];
  lst2str(M);
  list U = freegb(M,5);
  lst2str(U);
  // a=b is important -> finite basis!!!
  module M = [1,x,x], [a,x,y], [a,y,x], [c,y,y];
  lst2str(M);
  list U = freegb(M,5);
  lst2str(U);
}


// 1. form a new ring
// 2. produce shifted generators
// 3. compute GB
// 4. skip shifted elts
// 5. go back to orig vars, produce strings/modules
// 6. return the result

proc freegbold(list LM, int d)
"USAGE:  freegbold(L, d);  L a list of modules, d an integer
RETURN:  ring
PURPOSE: compute the two-sided Groebner basis of an ideal, encoded by L in
the free associative algebra, up to degree d
EXAMPLE: example freegbold; shows examples
"
{
  // d = up to degree, will be shifted to d+1
  if (d<1) {"bad d"; return(0);}

  int ppl = printlevel-voice+2;
  string err = "";

  int i,j,s;
  def save = basering;
  // determine max no of places in the input
  int slm = size(LM); // numbers of polys in the ideal
  int sm;
  intvec iv;
  module M;
  for (i=1; i<=slm; i++)
  {
    // modules, e.g. free polynomials
    M  = LM[i];
    sm = ncols(M);
    for (j=1; j<=sm; j++)
    {
      //vectors, e.g. free monomials
      iv = iv, size(M[j])-1; // 1 place is reserved by the coeff
    }
  }
  int D = Max(iv); // max size of input words
  if (d<D) {"bad d"; return(LM);}
  D = D + d-1;
  //  D = d;
  list LR  = ringlist(save);
  list L, tmp;
  L[1] = LR[1]; // ground field
  L[4] = LR[4]; // quotient ideal
  tmp  = LR[2]; // varnames
  s = size(LR[2]);
  for (i=1; i<=D; i++)
  {
    for (j=1; j<=s; j++)
    {
      tmp[i*s+j] = string(tmp[j])+"("+string(i+1)+")";
    }
  }
  for (i=1; i<=s; i++)
  {
    tmp[i] = string(tmp[i])+"("+string(1)+")";
  }
  L[2] = tmp;
  list OrigNames = LR[2];
  // ordering: d blocks of the ord on r
  // try to get whether the ord on r is blockord itself
  // TODO: make L(2) ordering! exponent is maximally 2
  s = size(LR[3]);
  if (s==2)
  {
    // not a blockord, 1 block + module ord
    tmp = LR[3][s]; // module ord
    for (i=1; i<=D; i++)
    {
      LR[3][s-1+i] = LR[3][1];
    }
    LR[3][s+D] = tmp;
  }
  if (s>2)
  {
    // there are s-1 blocks
    int nb = s-1;
    tmp = LR[3][s]; // module ord
    for (i=1; i<=D; i++)
    {
      for (j=1; j<=nb; j++)
      {
        LR[3][i*nb+j] = LR[3][j];
      }
    }
    //    size(LR[3]);
    LR[3][nb*(D+1)+1] = tmp;
  }
  L[3] = LR[3];
  def @R = ring(L);
  setring @R;
  ideal I;
  poly @p;
  s = size(OrigNames);
  //  "s:";s;
  // convert LM to canonical vectors (no powers)
  setring save;
  kill M; // M was defined earlier
  module M;
  slm = size(LM); // numbers of polys in the ideal
  int sv,k,l;
  vector v;
  //  poly p;
  string sp;
  setring @R;
  poly @@p=0;
  setring save;
  for (l=1; l<=slm; l++)
  {
    // modules, e.g. free polynomials
    M  = LM[l];
    sm = ncols(M); // in intvec iv the sizes are stored
    for (i=0; i<=d-iv[l]; i++)
    {
      // modules, e.g. free polynomials
      for (j=1; j<=sm; j++)
      {
        //vectors, e.g. free monomials
        v  = M[j];
        sv = size(v);
        //        "sv:";sv;
        sp = "@@p = @@p + ";
        for (k=2; k<=sv; k++)
        {
          sp = sp + string(v[k])+"("+string(k-1+i)+")*";
        }
        sp = sp + string(v[1])+";"; // coef;
        setring @R;
        execute(sp);
        setring save;
      }
      setring @R;
      //      "@@p:"; @@p;
      I = I,@@p;
      @@p = 0;
      setring save;
    }
  }
  kill sp;
  // 3. compute GB
  setring @R;
  dbprint(ppl,"computing GB");
  //  ideal J = groebner(I);
  ideal J = slimgb(I);
  dbprint(ppl,J);
  // 4. skip shifted elts
  ideal K = select1(J,1..s); // s = size(OrigNames)
  dbprint(ppl,K);
  dbprint(ppl, "done with GB");
  // K contains vars x(1),...z(1) = images of originals
  // 5. go back to orig vars, produce strings/modules
  if (K[1] == 0)
  {
    "no reasonable output, GB gives 0";
    return(0);
  }
  int sk = size(K);
  int sp, sx, a, b;
  intvec x;
  poly p,q;
  poly pn;
  // vars in 'save'
  setring save;
  module N;
  list LN;
  vector V;
  poly pn;
  // test and skip exponents >=2
  setring @R;
  for(i=1; i<=sk; i++)
  {
    p  = K[i];
    while (p!=0)
    {
      q  = lead(p);
      //      "processing q:";q;
      x  = leadexp(q);
      sx = size(x);
      for(k=1; k<=sx; k++)
      {
        if ( x[k] >= 2 )
        {
          err = "skip: the value x[k] is " + string(x[k]);
          dbprint(ppl,err);
          //            return(0);
          K[i] = 0;
          p    = 0;
          q    = 0;
          break;
        }
      }
      p  = p - q;
    }
  }
  K  = simplify(K,2);
  sk = size(K);
  for(i=1; i<=sk; i++)
  {
    //    setring save;
    //    V  = 0;
    setring @R;
    p  = K[i];
    while (p!=0)
    {
      q  = lead(p);
      err =  "processing q:" + string(q);
      dbprint(ppl,err);
      x  = leadexp(q);
      sx = size(x);
      pn = leadcoef(q);
      setring save;
      pn = imap(@R,pn);
      V  = V + leadcoef(pn)*gen(1);
      for(k=1; k<=sx; k++)
      {
        if (x[k] ==1)
        {
          a = k / s; // block number=a+1, a!=0
          b = k % s; // remainder
          //          printf("a: %s, b: %s",a,b);
          if (b == 0)
          {
            // that is it's the last var in the block
            b = s;
            a = a-1;
          }
          V = V + var(b)*gen(a+2);
        }
//         else
//         {
//           printf("error: the value x[k] is %s", x[k]);
//           return(0);
//         }
      }
      err = "V: " + string(V);
      dbprint(ppl,err);
      //      printf("V: %s", string(V));
      N = N,V;
      V  = 0;
      setring @R;
      p  = p - q;
      pn = 0;
    }
    setring save;
    LN[i] = simplify(N,2);
    N     = 0;
  }
  setring save;
  return(LN);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y,z),(dp(1),dp(2));
  module M = [-1,x,y],[-7,y,y],[3,x,x];
  module N = [1,x,y,x],[-1,y,x,y];
  list L; L[1] = M; L[2] = N;
  lst2str(L);
  def U = freegbold(L,5);
  lst2str(U);
}

proc sgb(ideal I, int d)
{
  // new code
  // map x_i to x_i(1) via map()
  //LIB "freegb.lib";
  def save = basering;
  //int d =7;// degree
  int nv = nvars(save);
  def R = freegbRing(d);
  setring R;
  int i;
  ideal Imap;
  for (i=1; i<=nv; i++)
  {
    Imap[i] = var(i);
  }
  //ideal I = x(1)*y(2), y(1)*x(2)+z(1)*z(2);
  ideal I = x(1)*x(2),x(1)*y(2) + z(1)*x(2);
  option(prot);
  //option(teach);
  ideal J = system("freegb",I,d,nv);
}



static proc checkCeq()
{
  ring r = 0,(x,y),Dp;
  def A = freegbRing(4);
  setring A;
  A;
  // I = x2-xy
  ideal I = x(1)*x(2) - x(1)*y(2), x(2)*x(3) - x(2)*y(3), x(3)*x(4) - x(3)*y(4);
  ideal C = x(2)-x(1),x(3)-x(2),x(4)-x(3),y(2)-y(1),y(3)-y(2),y(4)-y(3);
  ideal K = I,C;
  groebner(K);
}


proc exHom1()
{
  // we start with
  // z*y - x, z*x - y, y*x - z
  LIB "freegb.lib";
  LIB "elim.lib";
  ring r = 0,(x,y,z,h),dp;
  list L;
  module M;
  M = [1,z,y],[-1,x,h]; // zy - xh
  L[1] = M;
  M = [1,z,x],[-1,y,h]; // zx - yh
  L[2] = M;
  M = [1,y,x],[-1,z,h]; // yx - zh
  L[3] = M;
  lst2str(L);
  def U = crs(L,4);
  setring U;
  I = I,
    y(2)*h(3)+z(2)*x(3),     y(3)*h(4)+z(3)*x(4),
    y(2)*x(3)-z(2)*h(3),     y(3)*x(4)-z(3)*h(4);
  I = simplify(I,2);
  ring r2 = 0,(x(0..4),y(0..4),z(0..4),h(0..4)),dp;
  ideal J = imap(U,I);
  //  ideal K = homog(J,h);
  option(redSB);
  option(redTail);
  ideal L = groebner(J); //(K);
  ideal LL = sat(L,ideal(h))[1];
  ideal M = subst(LL,h,1);
  M = simplify(M,2);
  setring U;
  ideal M = imap(r2,M);
  lst2str(U);
}

static proc test1()
{
  LIB "freegb.lib";
  ring r = 0,(x,y),Dp;
  int d = 10; // degree
  def R = freegbRing(d);
  setring R;
  ideal I = x(1)*x(2) - y(1)*y(2);
  option(prot);
  option(teach);
  ideal J = system("freegb",I,d,2);
  J;
}

static proc test2()
{
  LIB "freegb.lib";
  ring r = 0,(x,y),Dp;
  int d = 10; // degree
  def R = freegbRing(d);
  setring R;
  ideal I = x(1)*x(2) - x(1)*y(2);
  option(prot);
  option(teach);
  ideal J = system("freegb",I,d,2);
  J;
}

static proc test3()
{
  LIB "freegb.lib";
  ring r = 0,(x,y,z),dp;
  int d =5; // degree
  def R = freegbRing(d);
  setring R;
  ideal I = x(1)*y(2), y(1)*x(2)+z(1)*z(2);
  option(prot);
  option(teach);
  ideal J = system("freegb",I,d,3);
}

proc schur2-3()
{
  // nonhomog:
  //  h^4-10*h^2+9,f*e-e*f+h, h*2-e*h-2*e,h*f-f*h+2*f
  // homogenized with t
  //  h^4-10*h^2*t^2+9*t^4,f*e-e*f+h*t, h*2-e*h-2*e*t,h*f-f*h+2*f*t,
  // t*h - h*t, t*f - f*t, t*e - e*t
}

proc adem(int i, int j)
{
  // produces Adem relations for i<2j in char 0
  // assume: 0<i<2j
  // requires presence of vars up to i+j
  if ( (i<0) || (i >= 2*j) )
  {
    ERROR("arguments out of range"); return(0);
  }
  ring @r = 0,(s(i+j..0)),lp;
  poly p,q;
  number n;
  int ii = i div 2; int k;
  // k=0 => s(0)=1
  n = binomial(j-1,i);
  q = n*s(i+j)*s(0);
  printf("k=0, term=%s",q);
  p = p + q;
  for (k=1; k<= ii; k++)
  {
    n = binomial(j-k-1,i-2*k);
    q = n*s(i+j-k)*s(k);;
    printf("k=%s, term=%s",k,q);
    p = p + q;
  }
  poly AdemRel = p;
  export AdemRel;
  return(@r);
}
example
{
  "EXAMPLE:"; echo = 2;
  def A = adem(2,5);
  setring A;
  AdemRel;
}

/*
1,1: 0
1,2: s(3)*s(0) == s(3) -> def for s(3):=s(1)s(2)
2,1: adm
2,2: s(3)*s(1) == s(1)s(2)s(1)
1,3: 0 ( since 2*s(4)*s(0) = 0 mod 2)
3,1: adm
2,3: s(5)*s(0)+s(4)*s(1) == s(5)+s(4)*s(1)
3,2: 0
3,3: s(5)*s(1)
1,4: 3*s(5)*s(0) == s(5)  -> def for s(5):=s(1)*s(4)
4,1: adm
2,4: 3*s(6)*s(0)+s(5)*s(1) == s(6) + s(5)*s(1) == s(6) + s(1)*s(4)*s(1)
4,2: adm
4,3: s(5)*s(2)
3,4: s(7)*s(0)+2*s(6)*s(1) == s(7) -> def for s(7):=s(3)*s(4)
4,4: s(7)*s(1)+s(6)*s(2)
*/

/* s1,s2:
s1*s1 =0, s2*s2 = s1*s2*s1
*/

/*
try char 0:
s1,s2:
s1*s1 =0, s2*s2 = s1*s2*s1, s(1)*s(3)== s(1)*s(1)*s(3) == 0 = 2*s(4) ->def for s(4)
hence 2==0! only in char 2
 */

proc adem2mod(int n)
{
  // Adem rels modulo 2
}

proc stringpoly2lplace(string s)
{
  // decomposes sentence into terms
  s = replace(s,newline,""); // get rid of newlines
  s = replace(s," ",""); // get rid of empties
  //arith symbols: +,-
  // decompose into words with coeffs
  list LS;
  int i,j,ie,je,k,cnt;
  // s[1]="-" situation
  if (s[1]=="-")
  {
    LS = stringpoly2lplace(string(s[2..size(s)]));
    LS[1] = string("-"+string(LS[1]));
    return(LS);
  }
  i = find(s,"-",2);
  // i==1 might happen if the 1st symbol coeff is negative
  j = find(s,"+");
  list LL;
  if (i==j)
  {
    "return a monomial";
    // that is both are 0 -> s is a monomial
    LS[1] = s;
    return(LS);
  }
  if (i==0)
  {
    "i==0 situation";
    // no minuses at all => pluses only
    cnt++;
    LS[cnt] = string(s[1..j-1]);
    s = s[j+1..size(s)];
    while (s!= "")
    {
      j = find(s,"+");
      cnt++;
      if (j==0)
      {
        LS[cnt] = string(s);
        s = "";
      }
      else
      {
        LS[cnt] = string(s[1..j-1]);
        s = s[j+1..size(s)];
      }
    }
    return(LS);
  }
  if (j==0)
  {
    "j==0 situation";
    // no pluses at all except the lead coef => the rest are minuses only
    cnt++;
    LS[cnt] = string(s[1..i-1]);
    s = s[i..size(s)];
    while (s!= "")
    {
      i = find(s,"-",2);
      cnt++;
      if (i==0)
      {
        LS[cnt] = string(s);
        s = "";
      }
      else
      {
        LS[cnt] = string(s[1..i-1]);
        s = s[i..size(s)];
      }
    }
    return(LS);
  }
  // now i, j are nonzero
  if (i>j)
  {
    "i>j situation";
    // + comes first, at place j
    cnt++;
    //    "cnt:"; cnt; "j:"; j;
    LS[cnt] = string(s[1..j-1]);
    s = s[j+1..size(s)];
    LL = stringpoly2lplace(s);
    LS = LS + LL;
    kill LL;
    return(LS);
  }
  else
  {
    "j>i situation";
    // - might come first, at place i
    if (i>1)
    {
      cnt++;
      LS[cnt] = string(s[1..i-1]);
      s = s[i..size(s)];
    }
    else
    {
      // i==1->  minus at leadcoef
      ie = find(s,"-",i+1);
      je = find(s,"+",i+1);
      if (je == ie)
      {
         "ie=je situation";
        //monomial
        cnt++;
        LS[cnt] = s;
        return(LS);
      }
      if (je < ie)
      {
         "je<ie situation";
        // + comes first
        cnt++;
        LS[cnt] = s[1..je-1];
        s = s[je+1..size(s)];
      }
      else
      {
        // ie < je
         "ie<je situation";
        cnt++;
        LS[cnt] = s[1..ie-1];
        s = s[ie..size(s)];
      }
    }
    "going into recursion with "+s;
    LL = stringpoly2lplace(s);
    LS = LS + LL;
    return(LS);
  }
}
example
{
  "EXAMPLE:"; echo = 2;
  string s = "x*y+y*z+z*t"; // + only
  stringpoly2lplace(s);
  string s2 = "x*y - y*z-z*t*w*w"; // +1, - only
  stringpoly2lplace(s2);
  string s3 = "-x*y + y - 2*x +7*w*w*w";
  stringpoly2lplace(s3);
}

proc addplaces(list L)
{
  // adds places to the list of strings
  // according to their order in the list
  int s = size(L);
  int i;
  for (i=1; i<=s; i++)
  {
    if (typeof(L[i]) == "string")
    {
      L[i] = L[i] + "(" + string(i) + ")";
    }
    else
    {
      ERROR("entry of type string expected");
      return(0);
    }
  }
  return(L);
}
example
{
  "EXAMPLE:"; echo = 2;
  string a = "f1";   string b = "f2";
  list L = a,b,a;
  addplaces(L);
}

proc sent2lplace(string s)
{
  // SENTence of words TO LetterPLACE
  list L =   stringpoly2lplace(s);
  int i; int ss = size(L);
  for(i=1; i<=ss; i++)
  {
    L[i] = str2lplace(L[i]);
  }
  return(L);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(f2,f1),dp;
  string s = "f2*f1*f1 - 2*f1*f2*f1+ f1*f1*f2";
  sent2lplace(s);
}

proc testnumber(string s)
{
  string t;
  if (s[1]=="-")
  {
    // two situations: either there's a negative number
    t = s[2..size(s)];
    if (testnumber(t))
    {
      //a negative number
    }
    else
    {
      // a variable times -1
    }
    // or just a "-" for -1
  }
  t = "ring @r=(";
  t = t + charstr(basering)+"),";
  t = t + string(var(1))+",dp;";
  //  write(":w tstnum.tst",t);
  t = t+ "number @@Nn = " + s + ";"+"$";
  write(":w tstnum.tst",t);
  string runsing = system("Singular");
  int k;
  t = runsing+ " -teq <tstnum.tst >tstnum.out";
  k = system("sh",t);
  if (k!=0)
  {
    ERROR("Problems running Singular");
  }
  int i = system("sh", "grep error tstnum.out > /dev/NULL");
  if (i!=0)
  {
    // no error: s is a number
    i = 1;
  }
  k = system("sh","rm tstnum.tst tstnum.out > /dev/NULL");
  return(i);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = (0,a),x,dp;
  string s = "a^2+7*a-2";
  testnumber(s);
  s = "b+a";
  testnumber(s);
}

proc str2lplace(string s)
{
  // converts a word (monomial) with coeff into letter-place
  // string: coef*var1^exp1*var2^exp2*...varN^expN
  s = strpower2rep(s); // expand powers
  if (size(s)==0) { return(0); }
  int i,j,k,insC;
  string a,b,c,d,t;
  // 1. get coeff
  i = find(s,"*");
  if (i==0) { return(s); }
  list VN;
  c = s[1..i-1]; // incl. the case like (-a^2+1)
  int tn = testnumber(c);
  if (tn == 0)
  {
    // failed test
    if (c[1]=="-")
    {
      // two situations: either there's a negative number
      t = c[2..size(c)];
      if (testnumber(t))
      {
         //a negative number
        // nop here
      }
      else
      {
         // a variable times -1
          c = "-1";
          j++; VN[j] = t; //string(c[2..size(c)]);
          insC = 1;
      }
    }
    else
    {
      // just a variable with coeff 1
          j++; VN[j] = string(c);
          c = "1";
          insC = 1;
    }
  }
 // get vars
  t = s;
  //  t = s[i+1..size(s)];
  k = size(t); //j = 0;
  while (k>0)
  {
    t = t[i+1..size(t)]; //next part
    i = find(t,"*"); // next *
    if (i==0)
    {
      // last monomial
      j++;
      VN[j] = t;
      k = size(t);
      break;
    }
    b = t[1..i-1];
    //    print(b);
    j++;
    VN[j] = b;
    k = size(t);
  }
  VN = addplaces(VN);
  VN[size(VN)+1] = string(c);
  return(VN);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = (0,a),(f2,f1),dp;
  str2lplace("-2*f2^2*f1^2*f2");
  str2lplace("-f1*f2");
  str2lplace("(-a^2+7a)*f1*f2");
}

proc strpower2rep(string s)
{
  // makes x*x*x*x out of x^4 ., rep statys for repetitions
  // looks for "-" problem
  // exception: "-" as coeff
  string ex,t;
  int i,j,k;

  i = find(s,"^"); // first ^
  if (i==0) { return(s); } // no ^ signs

  if (s[1] == "-")
  {
    // either -coef or -1
    // got the coeff:
    i = find(s,"*");
    if (i==0)
    {
      // no *'s   => coef == -1 or s == -23
      i = size(s)+1;
    }
    t = string(s[2..i-1]); // without "-"
    if ( testnumber(t) )
    {
      // a good number
      t = strpower2rep(string(s[2..size(s)]));
      t = "-"+t;
      return(t);
    }
    else
    {
      // a variable
      t = strpower2rep(string(s[2..size(s)]));
      t = "-1*"+ t;
      return(t);
    }
  }
  // the case when leadcoef is a number in ()
  if (s[1] == "(")
  {
    i = find(s,")",2);    // must be nonzero
    t = s[2..i-1];
    if ( testnumber(t) )
    {
      // a good number
    }
    else {"strpower2rep: bad number as coef";}
    ex = string(s[i+2..size(s)]); // 2 because of *
    ex =  strpower2rep(ex);
    t = "("+t+")*"+ex;
    return(t);
  }

  i = find(s,"^"); // first ^
  j = find(s,"*",i+1); // next * == end of ^
  if (j==0)
  {
    ex = s[i+1..size(s)];
  }
  else
  {
    ex = s[i+1..j-1];
  }
  execute("int @exp = " + ex + ";"); //@exp = exponent
  // got varname
  for (k=i-1; k>0; k--)
  {
    if (s[k] == "*") break;
  }
  string varn = s[k+1..i-1];
  //  "varn:";  varn;
  string pref;
  if (k>0)
  {
    pref = s[1..k]; // with * on the k-th place
  }
  //  "pref:";  pref;
  string suf;
  if ( (j>0) && (j+1 <= size(s)) )
  {
    suf = s[j+1..size(s)]; // without * on the 1st place
  }
  //  "suf:"; suf;
  string toins;
  for (k=1; k<=@exp; k++)
  {
    toins = toins + varn+"*";
  }
  //  "toins: ";  toins;
  if (size(suf) == 0)
  {
    toins = toins[1..size(toins)-1]; // get rid of trailing *
  }
  else
  {
    suf = strpower2rep(suf);
  }
  ex = pref + toins + suf;
  return(ex);
  //  return(strpower2rep(ex));
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = (0,a),(x,y,z,t),dp;
  strpower2rep("-x^4");
  strpower2rep("-2*x^4*y^3*z*t^2");
  strpower2rep("-a^2*x^4");
}

proc Liebr(poly a, poly b, list #)
{
  // alias ppLiebr;
  //if int N is given compute [a,[...[a,b]]]] left normed bracket
  poly q;
  int N=1;
  if (size(#)>0)
  {
    if (typeof(#[1])=="int")
    {
      N = int(#[1]);
    }
  }
  if (N<=0) { return(q); }
  while (b!=0)
  {
    q = q + pmLiebr(a,lead(b));
    b = b - lead(b);
  }
  int i;
  if (N >1)
  {
    for(i=1; i<=N; i++)
    {
      q = Liebr(a,q);
    }
  }
  return(q);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x(1),y(1),x(2),y(2),x(3),y(3),x(4),y(4)),dp;
  poly a = x(1)*y(2); poly b = y(1);
  int uptodeg=4; int lV=2;
  export uptodeg; export lV;
  Liebr(a,b);
  Liebr(x(1),y(1),2);
}

proc pmLiebr(poly a, poly b)
{
  //  a poly, b mono
  poly s;
  while (a!=0)
  {
    s = s + mmLiebr(lead(a),lead(b));
    a = a - lead(a);
  }
  return(s);
}

//proc pshift(poly a, int i, int uptodeg, int lV)
proc pshift(poly a, int i)
{
  // shifts a monomial a by i
  // calls pLPshift(p,sh,uptodeg,lVblock);
  return(system("stest",a,i,uptodeg,lV));
}

proc mmLiebr(poly a, poly b)
{
  // a,b, monomials
  a = lead(a);
  b = lead(b);
  int sa = deg(a);
  int sb = deg(b);
  poly v = a*pshift(b,sa) - b*pshift(a,sb);
  return(v);
}

static proc test_shift()
{
  LIB "freegb.lib";
  ring r = 0,(a,b),dp;
  int d =5;
  def R = freegbRing(d);
  setring R;
  int uptodeg = d; export uptodeg;
  int lV = 2; export lV;
  poly p = mmLiebr(a(1),b(1));
  poly p = Liebr(a(1),b(1));
}

proc Serre(intmat A, int zu)
{
  // zu = 1 -> with commutators [f_i,f_j]; zu == 0 without them
  // suppose that A is cartan matrix
  // then Serre's relations are
  // (ad f_j)^{1-A_{ij}} ( f_i)
  int ppl = printlevel-voice+2;
  int n = ncols(A); // hence n variables
  int i,j,k,l;
  poly p,q;
  ideal I;
  for (i=1; i<=n; i++)
  {
    for (j=1; j<=n; j++)
    {
      l = 1 - A[i,j];
      //     printf("i:%s, j: %s, l: %s",i,j,l);
      dbprint(ppl,"i, j, l: ",i,j,l);
      //      if ((i!=j) && (l >0))
      //      if ( (i!=j) &&  ( ((zu ==0) &&  (l >=2)) || ((zu ==1) &&  (l >=1)) ) )
      if ((i!=j) && (l >0))
      {
        q = Liebr(var(j),var(i));
        //        printf("first bracket: %s",q);
        dbprint(ppl,"first bracket: ",q);
        //        if (l >=2)
        //        {
          for (k=1; k<=l-1; k++)
          {
            q = Liebr(var(j),q);
            //            printf("further bracket: %s",q);
            dbprint(ppl,"further bracket:",q);
          }
          //        }
      }
      if (q!=0) { I = I,q; q=0;}
    }
  }
  I = simplify(I,2);
  return(I);
}
example
{
  "EXAMPLE:"; echo = 2;
  intmat A[2][2] = 2, -1, -1, 2; // sl_3 == A_2
  ring r = 0,(f1,f2),dp;
  int uptodeg = 3; 
  int lV = 2;
  export uptodeg; export lV;
  def R = freegbRing(uptodeg);
  setring R;
  ideal I = Serre(A,1);
  I;
  Serre(A,0);
}

// TODO: EXPLAIN CONVERTION RULE, ENCODING OF 'LN'!
// THIS SEEMS TO BE THE LAST PART OF FREEGBASIS?
// TODO: ASSUMPTIONS! WHY NO SQUARES IN ELEMENTS FROM K?

proc lp2lstr(ideal K, def save)
"USAGE:  lp2lstr(K,save); K an ideal, save a ring
RETURN:  nothing (exports object LN into save)
PURPOSE: converts letter-place ideal to list of modules
EXAMPLE: example lp2lstr; shows examples
"
{
  def @R = basering;
  string err;
  int s = nvars(save);
  int i,j,k;
    // K contains vars x(1),...z(1) = images of originals
  // 5. go back to orig vars, produce strings/modules
  int sk = size(K);
  int sp, sx, a, b;
  intvec x;
  poly p,q;
  poly pn;
  // vars in 'save'
  setring save;
  module N;
  list LN;
  vector V;
  poly pn;
  // test and skip exponents >=2
  setring @R;

  // TODO: YOU CAN FOLD TWO FOLLOWING LOOPS TOGETHER! 
  // THE FOLLOWING CHECK/PREPROCESSING MIGHT BE EXPENSIVE!
  for(i=1; i<=sk; i++)
  {
    p  = K[i];
    while (p!=0)
    {
      q  = lead(p);
      //      "processing q:";q;
      x  = leadexp(q);
      sx = size(x);
      for(k=1; k<=sx; k++)
      {
        if ( x[k] >= 2 )
        {
          err = "skip: the value x[k] is " + string(x[k]);
          dbprint(ppl,err);
          //            return(0);
          K[i] = 0;
          p    = 0;
          q    = 0;
          break;
        }
      }
      p  = p - q;
    }
  }
  K  = simplify(K,2);
  sk = size(K);
  for(i=1; i<=sk; i++)
  {
    //    setring save;
    //    V  = 0;
    setring @R;
    p  = K[i];
    while (p!=0)
    {
      q  = lead(p);
      err =  "processing q:" + string(q);
      dbprint(ppl,err);
      x  = leadexp(q);
      sx = size(x);
      pn = leadcoef(q);
      setring save;
      pn = imap(@R,pn);
      V  = V + leadcoef(pn)*gen(1);
      for(k=1; k<=sx; k++)
      {
        if (x[k] ==1)
        {
          a = k / s; // block number=a+1, a!=0
          b = k % s; // remainder
          //          printf("a: %s, b: %s",a,b);
          if (b == 0)
          {
            // that is it's the last var in the block
            b = s;
            a = a-1;
          }
          V = V + var(b)*gen(a+2);
        }
      }
      err = "V: " + string(V);
      dbprint(ppl,err);
      //      printf("V: %s", string(V));
      N = N,V;
      V  = 0;
      setring @R;
      p  = p - q;
      pn = 0;
    }
    setring save;
    LN[i] = simplify(N,2);
    N     = 0;
  }
  setring save;
  export LN;
  //  return(LN);
}
example
{
  "EXAMPLE:"; echo = 2;
  intmat A[2][2] = 2, -1, -1, 2; // sl_3 == A_2
  ring r = 0,(f1,f2),dp;
  int uptodeg = 3; 
  def R = freegbRing(uptodeg);
  setring R;
  ideal I = Serre(A,1); // TODO: BUG: WHY IS THIS FAILED??? HIDING ARGUMENTS IS A BAD STYLE!!!

  int lV = 2; // NOT CLEAR WHERE DO YOU NEED THIS?
  export uptodeg; export lV; // TODO: BUG: THEY WHERE NOR DEFINED BEFORE IN CODE!!!
  I = Serre(A,1); 

  lp2lstr(I,r);
  setring r;
  lst2str(LN,1);


  kill uptodeg; 
  kill lV;
}

proc strList2poly(list L)
{
  //  list L comes from sent2lplace (which takes a poly on the input)
  // each entry of L is a sublist with the coef on the last place
  int s = size(L); int t;
  int i,j;
  list M;
  poly p,q;
  string Q;
  for(i=1; i<=s; i++)
  {
    M = L[i];
    t = size(M);
    //    q = M[t]; // a constant
    Q = string(M[t]);
    for(j=1; j<t; j++)
    {
      //      q = q*M[j];
      Q = Q+"*"+string(M[j]);
    }
    execute("q="+Q+";");
    //    q;
    p = p + q;
  }
  kill Q;
  return(p);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r =0,(x,y,z,t),Dp;
  def A = freegbRing(4);
  setring A;
  string t = "-2*y*z*y*z + y*t*z*z - z*x*x*y  + 2*z*y*z*y";
  list L = sent2lplace(t);
  L;
  poly p = strList2poly(L);
  p;
}

proc file2lplace(string fname)
{
  // format: from the usual string to letterplace
  string s = read(fname);
  // assume: file is a comma-sep list of polys
  // the vars are declared before
  // the file ends with ";"
  string t; int i;
  ideal I;
  list tst;
  while (s!="")
  {
    i = find(s,",");
    "i"; i;
    if (i==0)
    {
      i = find(s,";");
      if (i==0)
      {
        // no ; ??
         "no colon or semicolon found anymore";
         return(I);
      }
      // no "," but ";" on the i-th place
      t = s[1..i-1];
      s = "";
      "processing: "; t;
      tst = sent2lplace(t);
      tst;
      I = I, strList2poly(tst);
      return(I);
    }
    // here i !=0
    t = s[1..i-1];
    s = s[i+1..size(s)];
    "processing: "; t;
    tst = sent2lplace(t);
    tst;
    I = I, strList2poly(tst);
  }
  return(I);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r =0,(x,y,z,t),dp;
  def A = freegbRing(4);
  setring A;
  string fn = "myfile";
  string s1 = "z*y*y*y - 3*y*z*x*y  + 3*y*y*z*y - y*x*y*z,";
  string s2 = "-2*y*x*y*z + y*y*z*z - z*z*y*y + 2*z*y*z*y,";
  string s3 = "z*y*x*t - 2*y*z*y*t + y*y*z*t - t*z*y*y + 2*t*y*z*y - t*x*y*z;";
  write(":w "+fn,s1);  write(":a "+fn,s2);   write(":a "+fn,s3);
  read(fn);
  ideal I = file2lplace(fn);
  I;
}

static proc get_ls3nilp()
{
//first app of file2lplace
  ring r =0,(x,y,z,t),dp;
  int d = 10;
  def A = freegbRing(d);
  setring A;
  ideal I = file2lplace("./ls3nilp.bg");
  // and now test the correctness: go back from lplace to strings
  lp2lstr(I,r);
  setring r;
  lst2str(LN,1); // agree!
}

static proc doc_example
{
  LIB "freegb.lib";
  ring r = 0,(x,y,z),dp;
  int d =4; // degree bound
  def R = freegbRing(d);
  setring R;
  ideal I = x(1)*y(2) + y(1)*z(2), x(1)*x(2) + x(1)*y(2) - y(1)*x(2) - y(1)*y(2);
  option(redSB);option(redTail);
  ideal J = system("freegb",I,d,nvars(r));
  J;
  // visualization:
  lp2lstr(J,r); // export an object called LN to the ring r
  setring r;  // change to the ring r
  lst2str(LN,1); // output the strings
}



// TODO:
// multiply two letterplace polynomials, lpMult
// reduction/ Normalform? needs kernel stuff