source: git/Singular/LIB/schreyer.lib @ fdde6ce

//////////////////////////////////////////////////////////////////////////////
version="$Id$";
category="General purpose";
info="
LIBRARY: schreyer.lib     Helpers for working with the Schreyer induced ordering
AUTHOR:  Oleksandr Motsak <U@D>, where U={motsak}, D={mathematik.uni-kl.de}

PROCEDURES:
 Sres(M,l)      Schreyer resolution of module M of maximal length l
 Ssyz(M)        Schreyer resolution of module M of length 1
 Scontinue(l)   continue the resolution computation by most l steps

KEYWORDS:  syzygy; Schreyer induced ordering; Schreyer free resolution
NOTE:  requires the dynamic module: syzextra
";

static proc prepareSyz( module I, list # )
{
  int i;
  int k = 0;
  int r = nrows(I);
  int c = ncols(I);


  if( size(#) > 0 )
  {
    if( typeof(#[1]) == "int" || typeof(#[1]) == "bigint" )
    {
      k = #[1];
    }
  }

  if( k < r )
  {
    "// *** Wrong k: ", k, " < nrows: ", r, " => setting k = r = ", r;
    k = r;
  }

//   "k: ", k;  "c: ", c;   "I: ", I;

  for( i = c; i > 0; i-- )
  {
    I[i] = I[i] + gen(k + i);
  }

//  DetailedPrint(I);

  return(I);
}

static proc separateSyzGB( module J, int c )
{
  module II, G; vector v; int i;

  J = simplify(J, 2);

  for( i = ncols(J); i > 0; i-- )
  {
    v = J[i];
    if( leadcomp(v) > c )
    {
      II[i] = v;
    } else
    {
      G[i] = v; // leave only gen(i): i <= c
    }
  }

  II = simplify(II, 2);
  G = simplify(G, 2);

  return (list(G, II));
}

static proc splitSyzGB( module J, int c )
{
  module JJ; vector v, vv; int i;

  for( i = ncols(J); i > 0; i-- )
  {
    v = J[i];

    vv = 0;
    
    while( leadcomp(v) <= c )
    {
      vv = vv + lead(v);
      v  = v  - lead(v);
    }

    J[i] = vv;
    JJ[i] = v;
  }

  J = simplify(J, 2);
  JJ = simplify(JJ, 2);

  return (list(J, JJ));
}


static proc Sinit(module M)
{
  def @save = basering;
  
  int @DEBUG = !system("with", "ndebug");
  if( @DEBUG )
  {
    "Sinit::Input";
    type(M);
    DetailedPrint(M);
    attrib(M);
  }

  int @RANK = nrows(M); int @SIZE = ncols(M);

  int @IS_A_SB = attrib(M, "isSB"); // ??? only if all weights were zero?!

  if( !@IS_A_SB ) 
  {
    M = std(M); // this should be faster than computing std in S (later on)
  }

  def S = MakeInducedSchreyerOrdering(1); // 1 puts history terms to the back
  // TODO: NOTE: +1 causes trouble to Singular interpreter!!!???
  setring S; // a new ring with a Schreyer ordering

  if( @DEBUG )
  {
    "Sinit::StartingISRing";
    basering; 
//    DetailedPrint(basering);
  }

  // Setup the leading syzygy^{-1} module to zero:
  module Z = 0; Z[@RANK] = 0; attrib(Z, "isHomog", intvec(0));  

  module MRES = Z;
  
  list RES; RES[1] = Z;

  module F = freemodule(@RANK);
  intvec @V = deg(F[1..@RANK]);
  
  module M = imap(@save, M);
  
  attrib(M, "isHomog", @V);
  attrib(M, "isSB", 1);

  
  if( @DEBUG )
  {
    "Sinit::SB_Input: ";
    type(M);
    attrib(M);
    attrib(M, "isHomog");
    DetailedPrint(M);
  }

  if( @DEBUG )
  {
    // 0^th syz. property
    if( size(module(transpose( transpose(M) * transpose(MRES) ))) > 0 )
    {
      transpose( transpose(M) * transpose(MRES) );
      "transpose( transpose(M) * transpose(MRES) ) != 0!!!";
      $
    }
  }

  RES[size(RES)+1] = M; // list of all syzygy modules
  MRES = MRES, M;

  attrib(MRES, "isHomog", @V);  

  attrib(S, "InducionLeads", lead(M));
  attrib(S, "InducionStart", @RANK);  
  
  if( @DEBUG )
  {
    "Sinit::MRES";
    DetailedPrint(MRES);
    attrib(MRES, "isHomog");
    attrib(S);
  }

  export RES;
  export MRES;
  return (S);
}

static proc Sstep()
{
  int @DEBUG = !system("with", "ndebug");

  if( @DEBUG )
  {
    "Sstep::NextInducedRing";
    DetailedPrint(basering);

    attrib(basering, "InducionLeads");
    attrib(basering, "InducionStart");

    GetInducedData();
  }

  // syzygy step:

/*
  // is initial weights are all zeroes!
  def L =  lead(M);
  intvec @V = deg(M[1..ncols(M)]);  @W;  @V;  @W = @V;  attrib(L, "isHomog", @W);  
  SetInducedReferrence(L, @RANK, 0);
*/

//  def L =  lead(MRES);
//  @W = @W, @V;
//  attrib(L, "isHomog", @W);  


  // General setting:
//  SetInducedReferrence(MRES, 0, 0); // limit: 0!
  int @l = size(RES);

  module M = RES[@l];

  module L = attrib(basering, "InducionLeads");
  int limit = attrib(basering, "InducionStart");

//  L;  limit;
  
  int @RANK = ncols(MRES) - ncols(M); // nrows(M); // what if M is zero?!

/*
  if( @RANK !=  nrows(M) )
  {
    type(MRES);
    @RANK;
    type(M);
    pause();
  }
*/
  
  intvec @W = attrib(M, "isHomog");
  intvec @V = deg(M[1..ncols(M)]);
  @V = @W, @V;
   
  if( @DEBUG )
  {
    "Sstep::NextInput: ";
    M;
    @V;
    @RANK;   
    DetailedPrint(MRES);
    attrib(MRES, "isHomog");
  }

  
      
  SetInducedReferrence(L, limit, 0);
  
  def K = prepareSyz(M, @RANK);
//  K;
  
//   attrib(K, "isHomog", @V);   DetailedPrint(K, 1000);

//  pause();
  
  K = idPrepare(K, @RANK); // std(K); // ?
  K = simplify(K, 2);

//  K;

  module N = separateSyzGB(K, @RANK)[2]; // 1^st syz. module: vectors which start in lower part (comp >= @RANK)
  attrib(N, "isHomog", @V);

// "N_0: "; N; DetailedPrint(N, 10);
  
  N = std(N); // TODO: fix "wrong weights"!!!?
  attrib(N, "isHomog", @V);

//  N;
  
  if( @DEBUG )
  {
    if( size(N) > 0 )
    {
      // next syz. property
      if( size(module(transpose( transpose(N) * transpose(MRES) ))) > 0 )
      {
        MRES;

        "N: "; N; DetailedPrint(N, 10);

        "K:"; K; DetailedPrint(K, 10);

        "RANKS: ", @RANK;

        "transpose( transpose(N) * transpose(MRES) ) != 0!!!";
        transpose( transpose(N) * transpose(MRES) );

        "transpose(N) * transpose(MRES): ";
        transpose(N) * transpose(MRES);
        DetailedPrint(module(_), 2);
        $
      }
    }
  }
  
  RES[@l + 1] = N; // list of all syzygy modules
  
  MRES = MRES, N;
  attrib(MRES, "isHomog", @V);


  L = L, lead(N);
  attrib(basering, "InducionLeads", L);

  if( @DEBUG )
  {
    "Sstep::NextSyzOutput: ";
    DetailedPrint(N);
    attrib(N, "isHomog");
  }

}

proc Scontinue(int l)
"USAGE:  Scontinue(l)
RETURN:  nothing, instead it changes RES and MRES variables in the current ring
PURPOSE: computes further (at most l) syzygies 
NOTE:    must be used within a ring returned by Sres or Ssyz. RES and MRES are
         explained in Sres
EXAMPLE: example Scontinue; shows an example
"
{
  def data = GetInducedData();
           
  if( (!defined(RES)) || (!defined(MRES)) || (typeof(data) != "list") || (size(data) != 2) )
  {
    ERROR("Sorry, but basering does not seem to be returned by Sres or Ssyz");
  }
  for (;  (l != 0) && (size(RES[size(RES)]) > 0); l-- )
  {
    Sstep(); 
  }
}
example
{ "EXAMPLE:"; echo = 2;
  ring r;
  module M = maxideal(1); M;
  def S = Ssyz(M); setring S; S;
  "Only the first syzygy: ";
  RES; MRES;
  "More syzygies: ";
  Scontinue(10);
  RES; MRES;
}

proc Ssyz(module M)
"USAGE:  Ssyz(M)
RETURN:  ring, containing a list of modules RES and a module MRES
PURPOSE: computes the first syzygy module of M (wrt some Schreyer ordering)
NOTE:    The output is explained in Sres
EXAMPLE: example Ssyz; shows an example
"
{
  def S = Sinit(M); setring S;
  
  Sstep(); // NOTE: what if M is zero?

  return (S);
}
example
{ "EXAMPLE:"; echo = 2;
  ring r;
  module M = maxideal(1); M;
  def S = Ssyz(M); setring S; S;
  "Only the first syzygy: ";
  RES;
  MRES; // Note gen(i)
  kill S;
  setring r; kill M;

  module M = 0;
  def S = Ssyz(M); setring S; S;
  "Only the first syzygy: ";
  RES;
  MRES;
}

proc Sres(module M, int l)
"USAGE:  Sres(M, l)
RETURN:  ring, containing a list of modules RES and a module MRES
PURPOSE: computes (at most l) syzygy modules of M wrt the classical Schreyer
         induced ordering with gen(i) > gen(j) if i > j, provided both gens
         are from the same syzygy level.
NOTE:    RES contains the images of maps subsituting the beginning of the
         Schreyer free resolution of baseRing^r/M, while MRES is a sum of
         these images in a big free sum, containing all the syzygy modules.
         The syzygy modules are shifted so that gen(i) correspons to MRES[i].
         The leading zero module RES[0] indicates the fact that coker of the
         first map is zero. The number of zeroes inducates the rank of input.
NOTE:    If l == 0 then l is set to be nvars(basering) + 1
EXAMPLE: example Sres; shows an example
"
{
  def S = Sinit(M); setring S;

  if (l == 0)
  {
    l = nvars(basering) + 1; // not really an estimate...?!
  }
  
  Sstep(); l = l - 1;
  
  Scontinue(l);
  
  return (S);
}
example
{ "EXAMPLE:"; echo = 2;
  ring r;
  module M = maxideal(1); M;
  def S = Sres(M, 0); setring S; S;
  RES;
  MRES;
  kill S;
  setring r; kill M;

  def A = nc_algebra(-1,0); setring A;
  ideal Q = var(1)^2, var(2)^2, var(3)^2;
  qring SCA = twostd(Q);
  basering;

  module M = maxideal(1);
  def S = Sres(M, 2); setring S; S;
  RES;
  MRES;
}



// ================================================================== //


LIB "general.lib"; // for sort

// TODO: in C++!
static proc Tail(def M)
{
  int i = ncols(M); def m;
  while (i > 0)
  {
    m = M[i];

    m = m - lead(m); // m = tail(m)
    
    M[i] = m;
    
    i--;
  }
  return (M);
}

/* static */ proc SSinit(def M)
{
  if( (typeof(M) != "module") && (typeof(M) != "ideal") )
  {
    ERROR("Sorry: need an ideal or a module for input");
  }

  // TODO! DONE?
  def @save = basering;
  
  int @DEBUG = !system("with", "ndebug");
  
  if( @DEBUG )
  {
    "SSinit::Input";
    type(M);
//    DetailedPrint(M);
    attrib(M);
  }

  int @RANK = nrows(M); int @SIZE = ncols(M);

  int @IS_A_SB = attrib(M, "isSB"); // ??? only if all weights were zero?!

  if( !@IS_A_SB ) 
  {
    def opts = option(get);
    option(redSB); option(redTail); 
    M = std(M);
    option(set, opts);
    kill opts;
  } else
  {
    M = simplify(M, 2 + 4 + 32);
  }

  def LEAD = lead(M);

  // sort wrt neg.deg.rev.lex!
  intvec iv_ds = sort(LEAD, "ds", 1)[2]; // ,1 => reversed!

  M = M[iv_ds]; // sort M wrt ds on current leading terms
  LEAD = LEAD[iv_ds];
  
  def TAIL = Tail(M);
  
  intvec @DEGS = deg(M[1..@SIZE]); // store actuall degrees of input elements
  
  // TODO: what about real modules? weighted ones?
  
  list @l = ringlist(@save);

  int @z = 0; ideal @m = maxideal(1); intvec @wdeg = deg(@m[1..ncols(@m)]);

  // NOTE: @wdeg will be ignored anyway :(
  @l[3] = list(list("C", @z), list("lp", @wdeg)); 

  kill @z, @wdeg; // since these vars are ring independent!

  def S = ring(@l); // --MakeInducedSchreyerOrdering(1);

  module F = freemodule(@RANK);
  intvec @V = deg(F[1..@RANK]);
  
  setring S; // ring with an easy divisibility test ("C, lex")

  if( @DEBUG )
  {
    "SSinit::NewRing(C, lex)";
    basering; 
//    DetailedPrint(basering);
  }

  // Setup the leading syzygy^{-1} module to zero:
  module Z = 0; Z[@RANK] = 0; attrib(Z, "isHomog", intvec(0));  

  module MRES = Z;
  
  list RES;  RES[1] = Z;
  list LRES; LRES[1] = Z;
  list TRES; TRES[1] = Z;
  
  def M = imap(@save, M);

  attrib(M, "isHomog", @V);
  attrib(M, "isSB", 1);
  attrib(M, "degrees", @DEGS);  
  
  def LEAD = imap(@save, LEAD);
  
  attrib(LEAD, "isHomog", @V);
  attrib(LEAD, "isSB", 1);  
  
  def TAIL = imap(@save, TAIL);

  if( @DEBUG )
  {
    "SSinit::(sorted) SB_Input: ";
    type(M);
    attrib(M);
    attrib(M, "isHomog");
//    DetailedPrint(M);
  }

  if( @DEBUG )
  {
    // 0^th syz. property
    if( size(module(transpose( transpose(M) * transpose(MRES) ))) > 0 )
    {
      transpose( transpose(M) * transpose(MRES) );
      "ERROR: transpose( transpose(M) * transpose(MRES) ) != 0!!!";
      $
    }
  }

  RES[size(RES)+1] = M; // list of all syzygy modules
  LRES[size(LRES)+1] = LEAD; // list of all syzygy modules
  TRES[size(TRES)+1] = TAIL; // list of all syzygy modules
  
  MRES = MRES, M; //?

  attrib(MRES, "isHomog", @V);
  
  attrib(S, "InducionStart", @RANK);
  attrib(S, "LEAD2SYZ", 1);
  attrib(S, "TAILREDSYZ", 0);
  attrib(S, "DEBUG", @DEBUG);
  
  if( @DEBUG )
  {
    "SSinit::MRES";
    MRES;
//    DetailedPrint(MRES);
    attrib(MRES, "isHomog");
    attrib(S);
  }

  export RES;
  export MRES;
  export LRES;
  export TRES;
  return (S);
}
example
{ "EXAMPLE:"; echo = 2;
  ring R = 0, (w, x, y, z), dp;

  def M = maxideal(1);
  def S = SSinit(M); setring S; S;
  
  "Only the first initialization: ";
  RES; LRES; TRES;
  MRES; 

  kill S; setring R; kill M;
  
  ideal M = w^2 - x*z,  w*x - y*z,  x^2 - w*y, x*y - z^2, y^2 - w*z;
  def S = SSinit(M); setring S; S;

  "Only the first initialization: ";
  RES; LRES; TRES;
  MRES; 

  kill S; setring R; kill M;
}


LIB "poly.lib"; // for lcm



/// Compute L(Syz(L))
proc SSComputeLeadingSyzygyTerms(def L)
{
  if( typeof( attrib(basering, "DEBUG") ) == "int" )
  {
    int @DEBUG = attrib(basering, "DEBUG");
  } else
  {
    int @DEBUG = !system("with", "ndebug");
  }

  if( @DEBUG )
  {
    "SSComputeLeadingSyzygyTerms::Input: ";
    L;
  }

  int i, j, r; intvec iv_ds;
  int N = ncols(L);
  def a, b;
  poly aa, bb;

  bigint c;

  ideal M;

  module S = 0;
  
  for(i = 1; i <= N; i++)
  {
    a = L[i];
//    "a: ", a;
    c = leadcomp(a);
    r = int(c);

    if(r > 0)
    {
      aa = a[r];
    } else
    {
      aa = a;
    }

    M = 0;
    
    for(j = i-1; j > 0; j--)
    {
      b = L[j];
//      "b: ", b;

      if( leadcomp(b) == c )
      {
        if(r > 0)
        {
          bb = b[r];
        } else
        {
          bb = b;
        }

        M[j] = (lcm(aa, bb) / aa);
      }
    }
    
    // TODO: add quotient relations here...

    M = simplify(M, 1 + 2 + 32);

    iv_ds = sort(M, "ds", 1)[2]; // ,1 => reversed!

    M = M[iv_ds];
    
    S = S, M * gen(i);
  }

  S = simplify(S, 2);

  S = sort(S, "ds", 1)[1]; // ,1 => reversed! // TODO: not needed?
  
  if( @DEBUG )
  {
    "SSComputeLeadingSyzygyTerms::Output: ";
    S;
  }  

  attrib(S, "isSB", 1); 

  return (S);
}

/// Compute Syz(L), where L is a monomial (leading) module
proc SSCompute2LeadingSyzygyTerms(def L, int @TAILREDSYZ)
{
  if( typeof( attrib(basering, "DEBUG") ) == "int" )
  {
    int @DEBUG = attrib(basering, "DEBUG");
  } else
  {
    int @DEBUG = !system("with", "ndebug");
  }

  if( @DEBUG )
  {
    "SSCompute2LeadingSyzygyTerms::Input: ";
    L;
    "@TAILREDSYZ: ", @TAILREDSYZ;
  }

  int i, j, r; 
  int N = ncols(L);
  def a, b;

  poly aa, bb, @lcm;

  bigint c;

  module M;

  module S = 0;

  for(i = 1; i <= N; i++)
  {
    a = L[i];
//    "a: ", a;
    c = leadcomp(a);
    r = int(c);

    aa = leadmonomial(a);

    M = 0;

    for(j = i-1; j > 0; j--)
    {
      b = L[j];
//      "b: ", b;

      if( leadcomp(b) == c )
      {
        bb = leadmonomial(b);
        @lcm = lcm(aa, bb);

        M[j] = (@lcm / aa)* gen(i) - (@lcm / bb)* gen(j);
      }
    }
    
    M = simplify(M, 2);

    // TODO: add quotient relations here...
    S = S, M;
  }

  if( @TAILREDSYZ )
  {
    // Make sure that 2nd syzygy terms are not reducible by 1st
    def opts = option(get);
    option(redSB); option(redTail); 
    S = std(S); // binomial module
    option(set, opts);
    //  kill opts;
  } else
  {
    S = simplify(S, 2 + 32);
  }

  S = sort(S, "ds", 1)[1]; // ,1 => reversed!

  if( @DEBUG )
  {
    "SSCompute2LeadingSyzygyTerms::Syz(LEAD): "; S;

    if( size(S) > 0 and size(L) > 0 )
    {
      if( size(module(transpose( transpose(S) * transpose(L) ))) > 0 )
      {
        transpose( transpose(S) * transpose(L) );
        "ERROR: transpose( transpose(S) * transpose(L) ) != 0!!!";
        $
      }
    }
  }

  module S2 = Tail(S);
  S = lead(S); // (C,lp) on base ring!
              
  if( @DEBUG )
  {
    "SSCompute2LeadingSyzygyTerms::Output: "; S; S2;
  }  
  
  attrib(S, "isSB", 1); 

  return (S, S2);
}

// -------------------------------------------------------- //

/// TODO: save shortcut LM(m) * "t" -> ?
proc SSReduceTerm(poly m, def t, def L, def T, list #)
{
  if( typeof( attrib(basering, "DEBUG") ) == "int" )
  {
    int @DEBUG = attrib(basering, "DEBUG");
  } else
  {
    int @DEBUG = !system("with", "ndebug");
  }

  if( @DEBUG )
  {
    "SSReduce::Input: ";

    "mult: ", m;
    "term: ", t;
    "L: ", L;
    "T: ", T;
    if( size(#) > 0 )
    {
      "LSyz: ", #;
    }
//    "attrib(LS, 'isSB')", attrib(LS, "isSB");
  }
  
  vector s = 0;

  if( t == 0 )
  {
    return (s);
  }

  def product = m * t;

  bigint c = leadcomp(t);
  int r = int(c);
  
  def a, b, nf, bb;

  // looking for an appropriate reducer
  for( int k = ncols(L); k > 0; k-- )
  {
    a = L[k];
    // with the same mod. component
    if( leadcomp(a) == c )
    {
      b = - (leadmonomial(product) / leadmonomial(L[k]));
      
      // which divides the product
      if( b != 0 )
      {
//        "b: ", b;
        bb = b * gen(k);
        nf = bb;

        if( size(#) > 0 )
        {
          if( typeof(#[1]) == "module" )
          {
            nf = NF(bb, #[1]);
//        "NF: ", nf;
          }
        }

        // while the complement (the fraction) is not reducible by leading syzygies 
        if( nf != 0 )
        {
          /// TODO: save shortcut LM(m) * T[i] -> ?

          // choose ANY such reduction... (with the biggest index?)
          s = bb + SSTraverseTail(b, T[k], L, T, #);
          break;
        }
      }
    }
  }  
  if( @DEBUG )
  {
    "SSReduceTerm::Output: ", s;
  }
  return (s);
}

// TODO: store m * @tail -.-^-.-^-.--> ?
proc SSTraverseTail(poly m, def @tail, def L, def T, list #)
{
  if( typeof( attrib(basering, "DEBUG") ) == "int" )
  {
    int @DEBUG = attrib(basering, "DEBUG");
  } else
  {
    int @DEBUG = !system("with", "ndebug");
  }

  if( @DEBUG )
  {
    "SSTraverse::Input: ";

    "mult: ", m;
    "tail: ", @tail; // T[i];

    if( size(#) > 0 )
    {
      "LSyz: "; #[1];
    }
  }

  vector s = 0;

  def @l;

  // iterate tail-terms in ANY order!
  while( size(@tail) > 0 )
  {
    @l = lead(@tail);
    s = s + SSReduceTerm(m, @l, L, T, #);
    @tail = @tail - @l;
  }

  if( @DEBUG )
  {
    "SSTraverseTail::Output: ", s;
  }
  return (s);
}

// -------------------------------------------------------- //

// module (N, LL, TT) = SSComputeSyzygy(L, T);
// Compute Syz(L ++ T) = N = LL ++ TT
proc SSComputeSyzygy(def L, def T)
{
  if( typeof( attrib(basering, "DEBUG") ) == "int" )
  {
    int @DEBUG = attrib(basering, "DEBUG");
  } else
  {
    int @DEBUG = !system("with", "ndebug");
  }
  
  if( @DEBUG )
  {
    "SSComputeSyzygy::Input";
    "basering: ", basering; attrib(basering);
//    DetailedPrint(basering);

//    "iCompShift: ", iCompShift;

    "L: "; L;
    "T: "; T;
  }

  def a; bigint c; int r, k; poly aa;

  int @LEAD2SYZ = 0;
  if( typeof( attrib(basering, "LEAD2SYZ") ) == "int" )
  {
    @LEAD2SYZ = attrib(basering, "LEAD2SYZ");
  }

  int @TAILREDSYZ = 1;
  if( typeof( attrib(basering, "TAILREDSYZ") ) == "int" )
  {
    @TAILREDSYZ = attrib(basering, "TAILREDSYZ");
//    @TAILREDSYZ;
  }
  
  /// Get the critical leading syzygy terms
  if( @LEAD2SYZ ) // & 2nd syz. term
  {
    def a2; int r2; poly aa2;  
    module LL, LL2;
    (LL, LL2) = SSCompute2LeadingSyzygyTerms(L, @TAILREDSYZ); // ++ 
  } else
  {
    module LL = SSComputeLeadingSyzygyTerms(L);
  }

  module TT, SYZ;

  if( size(LL) > 0 )
  {
    list LS;

    if( @TAILREDSYZ )
    {
      LS = list(LL);
    }
    
    vector @tail;

    for(k = ncols(LL); k > 0; k-- )
    {
      // leading syz. term:
      a = LL[k]; c = leadcomp(a); r = int(c); aa = leadmonomial(a);
      //    "A: ", a, " --->>>> ", aa, " **** [", r, "]: ";

      /// TODO: save shortcut (aa) * T[r] -> ?
      @tail = SSTraverseTail(aa, T[r], L, T, LS);
             
      // get the 2nd syzygy term...
      
      if( @LEAD2SYZ ) // with the 2nd syz. term:
      {      
        a2 = LL2[k]; c = leadcomp(a2); r2 = int(c); aa2 = leadmonomial(a2);
        @tail = @tail + 
             /// TODO: save shortcut (aa2) * T[r2] -> ?
             a2 + SSTraverseTail(aa2, T[r2], L, T, LS);
      } else
      {
        @tail = @tail + SSReduceTerm(aa, L[r], L, T, LS);
      }
      
      
      TT[k] = @tail;
      SYZ[k] = a + @tail;
    }
  }

/*
  def opts = option(get); option(redSB); option(redTail); 
  module SYZ = std(syz(M)); 
  option(set, opts); kill opts;
  
  module LL = lead(SYZ); // TODO: WRONG ORDERING!!!!!!!!
  module TT = Tail(SYZ);
*/
  
  if( @DEBUG )
  {
    "SSComputeSyzygy::Output";

    "SYZ: "; SYZ;
    "LL: "; LL;
    "TT: "; TT;
  }

  return (SYZ, LL, TT);
}

// resolution/syzygy step:
static proc SSstep()
{
  /// TODO!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
  int @DEBUG = !system("with", "ndebug");

  if( @DEBUG )
  {
    "SSstep::NextInducedRing"; 
    "basering: ", basering; attrib(basering);
  }

/*
  // is initial weights are all zeroes!
  def L =  lead(M);
  intvec @V = deg(M[1..ncols(M)]);  @W;  @V;  @W = @V;  attrib(L, "isHomog", @W);  
  SetInducedReferrence(L, @RANK, 0);
*/

//  def L =  lead(MRES);
//  @W = @W, @V;
//  attrib(L, "isHomog", @W);  


  // General setting:
//  SetInducedReferrence(MRES, 0, 0); // limit: 0!
  int @l = size(RES);

  def M =  RES[@l];

  def L = LRES[@l];
  def T = TRES[@l];


  //// TODO: wrong !!!!!
  int @RANK = ncols(MRES) - ncols(M); // nrows(M); // what if M is zero?!

  

/*
  if( @RANK !=  nrows(M) )
  {
    type(MRES);
    @RANK;
    type(M);
    pause();
  }
*/
  
  intvec @W = attrib(M, "isHomog"); intvec @V = attrib(M, "degrees"); @V = @W, @V;
   
  if( @DEBUG )
  {
    "Sstep::NextInput: ";
    M;
    L;
    @V;
    @RANK;
//    DetailedPrint(MRES);
    attrib(MRES, "isHomog");
  }

      
  // TODO: N  = SYZ( M )!!!
  module N, LL, TT;
  (N, LL, TT) = SSComputeSyzygy(/*M, */L, T/*, @RANK*/); 

  // shift syz.comp by @RANK:
  module Z;
  Z = 0; Z[@RANK] = 0; Z = Z, transpose(LL);   LL = transpose(Z);
  Z = 0; Z[@RANK] = 0; Z = Z, transpose(TT);   TT = transpose(Z);
  Z = 0; Z[@RANK] = 0; Z = Z, transpose(N);     N = transpose(Z);


  if( size(N) > 0 )
  {
    if( @DEBUG )
    {
      // next syz. property
      if( size(module(transpose( transpose(N) * transpose(MRES) ))) > 0 )
      {
        "MRES", MRES;

        "N: "; N; // DetailedPrint(N, 2);

        "LL:"; LL; // DetailedPrint(LL, 1);
        "TT:"; TT; // DetailedPrint(TT, 10);

        "RANKS: ", @RANK;

        "transpose( transpose(N) * transpose(MRES) ) != 0!!!";
        transpose( transpose(N) * transpose(MRES) );

        "transpose(N) * transpose(MRES): ";
        transpose(N) * transpose(MRES);
        // DetailedPrint(module(_), 2);
        $
      }
    }
  }

  attrib(N, "isHomog", @V);

  // TODO: correct the following: 
  intvec @DEGS = deg(N[1..ncols(N)]); // no mod. comp. weights :(

  
  attrib(N, "degrees", @DEGS);
  
   RES[@l + 1] = N; // list of all syzygy modules
  LRES[@l + 1] = LL; // list of all syzygy modules
  TRES[@l + 1] = TT; // list of all syzygy modules

  MRES = MRES, N;
  
  attrib(MRES, "isHomog", @V);

//  L = L, lead(N);  attrib(basering, "InducionLeads", L);

  if( @DEBUG )
  {
    "SSstep::NextSyzOutput: ";
    N;
//    DetailedPrint(N);
    attrib(N);
  }

}

proc SScontinue(int l)
"USAGE:  SScontinue(l)
RETURN:  nothing, instead it changes RES and MRES variables in the current ring
PURPOSE: computes further (at most l) syzygies 
NOTE:    must be used within a ring returned by Sres or Ssyz. RES and MRES are
         explained in Sres
EXAMPLE: example Scontinue; shows an example
"
{

  /// TODO!
//  def data = GetInducedData();

  if( (!defined(RES)) || (!defined(MRES)) ) /* || (typeof(data) != "list") || (size(data) != 2) */
  {
    ERROR("Sorry, but basering does not seem to be returned by Sres or Ssyz");
  }
  for (;  (l != 0) && (size(RES[size(RES)]) > 0); l-- )
  {
    SSstep(); 
  }
}
example
{ "EXAMPLE:"; echo = 2;
  ring r;
  module M = maxideal(1); M;
  def S = SSsyz(M); setring S; S;
  "Only the first syzygy: ";
  RES; MRES;
  "More syzygies: ";
  SScontinue(10);
  RES; MRES;
}

proc SSsyz(def M)
"USAGE:  SSsyz(M)
RETURN:  ring, containing a list of modules RES and a module MRES
PURPOSE: computes the first syzygy module of M (wrt some Schreyer ordering)?
NOTE:    The output is explained in Sres
EXAMPLE: example Ssyz; shows an example
"
{
  if( (typeof(M) != "module") && (typeof(M) != "ideal") )
  {
    ERROR("Sorry: need an ideal or a module for input");
  }

  def SS = SSinit(M); setring SS;
  
  SSstep(); // NOTE: what if M is zero?

  return (SS);
}
example
{ "EXAMPLE:"; echo = 2;
  ring r;

/*  ideal M = 0;
  def S = SSsyz(M); setring S; S;
  "Only the first syzygy: ";
  RES; LRES; TRES;
  MRES;
  
  kill S; setring r; kill M;
*/  

  module M = maxideal(1); M;
  def S = SSres(M, 0); setring S; S;
  MRES;
  RES;
  "";
  LRES;
  "";
  TRES;

  kill S; setring r; kill M;

  kill r;
  
  ring R = 0, (w, x, y, z), dp;
  ideal M = w^2 - x*z,  w*x - y*z,  x^2 - w*y, x*y - z^2, y^2 - w*z;
  
  def S = SSres(M, 0); setring S; S;
  MRES;
  RES;
  "";
  LRES;
  "";
  TRES;
}

proc SSres(def M, int l)
"USAGE:  SSres(I, l)
RETURN:  ring, containing a list of modules RES and a module MRES
PURPOSE: computes (at most l) syzygy modules of M wrt the classical Schreyer
         induced ordering with gen(i) > gen(j) if i > j, provided both gens
         are from the same syzygy level.???
NOTE:    RES contains the images of maps subsituting the beginning of the
         Schreyer free resolution of baseRing^r/M, while MRES is a sum of
         these images in a big free sum, containing all the syzygy modules.
         The syzygy modules are shifted so that gen(i) correspons to MRES[i].
         The leading zero module RES[0] indicates the fact that coker of the
         first map is zero. The number of zeroes inducates the rank of input.
NOTE:    If l == 0 then l is set to be nvars(basering) + 1
EXAMPLE: example SSres; shows an example
"
{
  if( (typeof(M) != "module") && (typeof(M) != "ideal") )
  {
    ERROR("Sorry: need an ideal or a module for input");
  }

  def SS = SSinit(M); setring SS;

  if (l == 0)
  {
    l = nvars(basering) + 1; // not really an estimate...?!
  }

  SSstep(); l = l - 1;

  SScontinue(l);

  return (SS);
}
example
{ "EXAMPLE:"; echo = 2;
  ring r;
  module M = maxideal(1); M;
  def S = SSres(M, 0); setring S; S;
  RES;
  MRES;
  kill S;
  setring r; kill M;

  def A = nc_algebra(-1,0); setring A;
  ideal Q = var(1)^2, var(2)^2, var(3)^2;
  qring SCA = twostd(Q);
  basering;

  module M = maxideal(1);
  def S = SSres(M, 2); setring S; S;
  RES;
  MRES;
}



static proc loadme()
{
  int @DEBUG = !system("with", "ndebug");

  if( @DEBUG )
  {
    
    "ndebug?: ", system("with", "ndebug");
    "om_ndebug?: ", system("with", "om_ndebug");

    listvar(Top);
    listvar(Schreyer);
  }
//  listvar(Syzextra); listvar(Syzextra_g);

  if( !defined(DetailedPrint) )
  {
    if( !@DEBUG )
    {

      if( @DEBUG )
      {
        "Loading the Release version!";
      }
      load("syzextra.so");

      if( @DEBUG )
      {
        listvar(Syzextra);
      }
      
//      export Syzextra;

//      exportto(Schreyer, Syzextra::noop);
      exportto(Schreyer, Syzextra::DetailedPrint);
      exportto(Schreyer, Syzextra::leadmonomial);
      exportto(Schreyer, Syzextra::leadcomp);
//      exportto(Schreyer, Syzextra::leadrawexp);
//      exportto(Schreyer, Syzextra::ISUpdateComponents);
      exportto(Schreyer, Syzextra::SetInducedReferrence);
      exportto(Schreyer, Syzextra::GetInducedData);
//      exportto(Schreyer, Syzextra::GetAMData);
//      exportto(Schreyer, Syzextra::SetSyzComp);
      exportto(Schreyer, Syzextra::MakeInducedSchreyerOrdering);
//      exportto(Schreyer, Syzextra::MakeSyzCompOrdering);
      exportto(Schreyer, Syzextra::idPrepare);
//      exportto(Schreyer, Syzextra::reduce_syz);
//      exportto(Schreyer, Syzextra::p_Content);

    }
    else
    {
      if( @DEBUG )
      {
        "Loading the Debug version!";
      }

      load("syzextra_g.so");

      if( @DEBUG )
      {      
        listvar(Syzextra_g);
      }
      
//      export Syzextra_g;
//      exportto(Schreyer, Syzextra_g::noop);
      exportto(Schreyer, Syzextra_g::DetailedPrint);
      exportto(Schreyer, Syzextra_g::leadmonomial);
      exportto(Schreyer, Syzextra_g::leadcomp);
//      exportto(Schreyer, Syzextra_g::leadrawexp);
//      exportto(Schreyer, Syzextra_g::ISUpdateComponents);
      exportto(Schreyer, Syzextra_g::SetInducedReferrence);
      exportto(Schreyer, Syzextra_g::GetInducedData);
//      exportto(Schreyer, Syzextra_g::GetAMData);
//      exportto(Schreyer, Syzextra_g::SetSyzComp);
      exportto(Schreyer, Syzextra_g::MakeInducedSchreyerOrdering);
//      exportto(Schreyer, Syzextra_g::MakeSyzCompOrdering);
      exportto(Schreyer, Syzextra_g::idPrepare);
//      exportto(Schreyer, Syzextra_g::reduce_syz);
//      exportto(Schreyer, Syzextra_g::p_Content);

      
    }

    exportto(Top, DetailedPrint);
    exportto(Top, GetInducedData);

    if( @DEBUG )
    {
      listvar(Top);
      listvar(Schreyer);
    }
  }
  
  if( !defined(GetInducedData) )
  {
    ERROR("Sorry but we are missing the dynamic module (syzextra(_g)?.so)...");
  }

}

static proc mod_init()
{
  loadme();
}


proc testallSexamples()
{
  example Ssyz;
  example Scontinue;
  example Sres;  
}

proc testallSSexamples()
{
  example SSsyz;
  example SScontinue;
  example SSres;  
}

example
{ "EXAMPLE:"; echo = 2;
  testallSexamples();
  testallSSexamples();
}