Quelle dt.c Sprache: C

/****************************************************************************
**
** This file is part of GAP, a system for computational discrete algebra.
**
** Copyright of GAP belongs to its developers, whose names are too numerous
** to list here. Please refer to the COPYRIGHT file for details.
**
** SPDX-License-Identifier: GPL-2.0-or-later
**
** This file implements the part of the deep thought package which deals
** with computing the deep thought polynomials.
**
** Deep Thought deals with trees. A tree <tree> is a concatenation of
** several nodes where each node is a 5-tuple of immediate integers. If
** <tree> is an atom it contains only one node, thus it is itself a
** 5-tuple. If <tree> is not an atom we obtain its list representation by
**
** <tree> := topnode(<tree>) concat left(<tree>) concat right(<tree>) .
**
** Let us denote the i-th node of <tree> by (<tree>, i) and the tree rooted
** at (<tree>, i) by tree(<tree>, i). Let <a> be tree(<tree>, i)
** The first entry of (<tree>, i) is pos(a),
** and the second entry is num(a). The third entry of (<tree>, i) gives a
** mark.(<tree>, i)[3] = 1 means that (<tree>, i) is marked,
** (<tree>, i)[3] = 0 means that (<tree>, i) is not marked. The fourth entry
** of (<tree>, i) contains the number of knodes of tree(<tree>, i). The
** fifth entry of (<tree>, i) finally contains a boundary for
** pos( tree(<tree>, i) ). (<tree>, i)[5] <= 0 means that
** pos( tree(<tree>, i) ) is unbounded. If tree(<tree>, i) is an atom we
** already know that pos( tree(<tree>, i) ) is unbound. Thus we then can
** use the fifth component of (<tree>, i) to store the side. In this case
** (<tree>, i)[5] = -1 means that tree(<tree>, i) is an atom from the
** right hand word, and (<tree>, i)[5] = -2 means that tree(<tree>, i) is
** an atom from the left hand word.
**
** A second important data structure deep thought deals with is a deep
** thought monomial. A deep thought monomial g_<tree> is a product of
** binomial coefficients with a coefficient c. Deep thought monomials
** are represented in this implementation by formula
** vectors, which are lists of integers. The first entry of a formula
** vector is 0, to distinguish formula vectors from trees. The second
** entry is the coefficient c, and the third and fourth entries are
** num( left(tree) ) and num( right(tree) ). The remaining part of the
** formula vector is a concatenation of pairs of integers. A pair (i, j)
** with i > 0 represents binomial(x_i, j). A pair (0, j) represents
** binomial(y_gen, j) when word*gen^power is calculated.
**
** Finally deep thought has to deal with pseudorepresentatives. A
** pseudorepresentative <a> is stored in list of length 4. The first entry
** stores left( <a> ), the second entry contains right( <a> ), the third
** entry contains num( <a> ) and the last entry finally gives a boundary
** for pos( ) for all trees which are represented by <a>.
*/

#include "dt.h"

#include "calls.h"
#include "error.h"
#include "gvars.h"
#include "integer.h"
#include "modules.h"
#include "plist.h"

static void UnmarkTree(Obj z);
static UInt Mark(Obj tree, Obj reftree, Int indexx);
static Int AlmostEqual(Obj tree1, Int index1, Obj tree2, Int index2);
static Int Equal(Obj tree1, Int index1, Obj tree2, Int index2);
static Obj Mark2(Obj tree, Int index1, Obj reftree, Int index2);
static UInt FindTree(Obj tree, Int indexx);
static Obj MakeFormulaVector(Obj tree, Obj pr);
static Int Leftof(Obj tree1, Int index1, Obj tree2, Int index2);
static Int Leftof2(Obj tree1, Int index1, Obj tree2, Int index2);
static Int Earlier(Obj tree1, Int index1, Obj tree2, Int index2);
static void FindNewReps(Obj tree, Obj reps, Obj pr, Obj max);
static void FindSubs(Obj tree,
 Int x,
 Obj list1,
 Obj list2,
 Obj a,
 Obj b,
 Int al,
 Int ar,
 Int bl,
 Int br,
 Obj reps,
 Obj pr,
 Obj max);
static void SetSubs(Obj list, Obj a, Obj tree);
static void UnmarkAEClass(Obj tree, Obj list);

#ifdef TEST_TREE
static void TestTree(Obj tree);
static Obj Part(Obj list, Int pos1, Int pos2);
#endif

/****************************************************************************
**
*F DT_POS(tree, index) . . . . . . . . . . . . . position of (<tree>, index)
**
** 'DT_POS' returns pos(<a>) where <a> is the subtree of <tree> rooted at
** (<tree>, index). <index> has to be a positive integer less or equal than
** the number of nodes of <tree>.
*/
#define DT_POS(tree, index) \
 (ELM_PLIST(tree, (index-1)*5 + 1 ) )

/***************************************************************************
**
*F SET_DT_POS(tree, index, obj) . . . assign the position of(<tree>, index)
**
** 'SET_DT_POS sets pos(<a>) to the object <obj>, where <a> is the subtree
** of <tree>, rooted at (<tree>, index). <index> has to be a positive
** integer less or equal to the number of nodes of <tree>
*/
#define SET_DT_POS(tree, index, obj) \
 SET_ELM_PLIST(tree, (index-1)*5 + 1, obj)

/***************************************************************************
**
*F DT_GEN(tree, index) . . . . . . . . . . . . . generator of (<tree>, index)
**
** 'DT_GEN' returns num(<a>) where <a> is the subtree of <tree> rooted at
** (<tree>, index). <index> has to be a positive integer less or equal than
** the number of nodes of <tree>.
*/
#define DT_GEN(tree, index) \
 (ELM_PLIST(tree, (index-1)*5 + 2) )

/**************************************************************************
**
*F DT_IS_MARKED(tree, index) . . . . . . tests if (<tree>, index) is marked
**
** 'DT_IS_MARKED' returns 1 (as C integer) if (<tree>, index) is marked, and
** 0 otherwise. <index> has to be a positive integer less or equal to the
** number of nodes of <tree>.
*/
#define DT_IS_MARKED(tree, index) \
 (INT_INTOBJ (ELM_PLIST(tree, (index-1)*5 + 3) ) )

/**************************************************************************
**
*F DT_MARK(tree, index) . . . . . . . . . . . . . . . . . . . . mark a node
**
** 'DT_MARK' marks the node (<tree>, index). <index> has to be a positive
** integer less or equal to the number of nodes of <tree>.
*/
#define DT_MARK(tree, index) \
 SET_ELM_PLIST(tree, (index-1)*5 + 3, INTOBJ_INT(1) )

/**************************************************************************
**
*F DT_UNMARK(tree, index) . . . . . . . . . . . remove the mark from a node
**
** 'DT_UNMARK' removes the mark from the node (<tree>, index). <index> has
** has to be a positive integer less or equal to the number of nodes of
** <tree>.
*/
#define DT_UNMARK(tree, index) \
 SET_ELM_PLIST(tree, (index-1)*5 + 3, INTOBJ_INT(0) )

/****************************************************************************
**
*F DT_RIGHT(tree, index) . . . .determine the right subnode of (<tree>, index)
*F DT_LEFT(tree, index) . . . . determine the left subnode of (<tree>, index)
**
** 'DT_RIGHT' returns the right subnode of (<tree>, index). That means if
** DT_RIGHT(tree, index) = index2, then (<tree>, index2) is the right
** subnode of (<tree>, index).
**
** 'DT_LEFT' returns the left subnode of (<tree>, index). That means if
** DT_LEFT(tree, index) = index2, then (<tree>, index2) is the left
** subnode of (<tree>, index).
**
** Before calling 'DT_RIGHT' or 'DT_LEFT' it should be ensured, that
** (<tree>, index) is not an atom. <index> has to be a positive integer
** less or equal to the number of nodes of <tree>.
*/
#define DT_RIGHT(tree, index) \
 ( INT_INTOBJ(ELM_PLIST(tree, index*5 + 4) ) + index + 1)
#define DT_LEFT(tree, index) \
 ( index + 1 )

/****************************************************************************
**
*F DT_SIDE(tree, index) . . . . . . . determine the side of (<tree>, index)
*V RIGHT. . . . . . . . . . . . . . . integer describing "right"
*V LEFT . . . . . . . . . . . . . . . integer describing "left"
**
** 'DT_SIDE' returns 'LEFT' if (<tree>, index) is an atom from the Left-hand
** word, and 'RIGHT' if (<tree>, index) is an atom of the Right-hand word.
** Otherwise 'DT_SIDE' returns an integer bigger than 1. <index> has to be
** a positive integer less or equal to the number of nodes of <tree>.
*/
#define RIGHT -1
#define LEFT -2
#define DT_SIDE(tree, index) \
 (INT_INTOBJ( ELM_PLIST(tree, (index-1)*5 + 5 ) ) )

/****************************************************************************
**
*F DT_LENGTH(tree, index) . . . . . . . . number of nodes of (<tree>, index)
**
** 'DT_LENGTH' returns the number of nodes of (<tree>, index). <index> has
** to be a positive integer less or equal to the number of nodes of <tree>.
*/
#define DT_LENGTH(tree, index) \
 ( INT_INTOBJ(ELM_PLIST(tree, (index-1)*5 + 4) ) )

/***************************************************************************
**
*F DT_MAX(tree, index) . . . . . . . . . . . . . . . . boundary of a node
**
** 'DT_MAX(tree, index)' returns a boundary for 'DT_POS(tree, index)'.
** 'DT_MAX(tree, index) = 0 ' means that 'DT_POS(tree, index)' is unbound.
** <index> has to be a positive integer less or equal to the number of nodes
** of tree.
*/
#define DT_MAX(tree, index) \
 (ELM_PLIST(tree, (index-1)*5 + 5 ) )

/****************************************************************************
**
*F CELM(list, pos) . . . . . . . . . . element of a plain list as C integer
**
** 'CELM' returns the <pos>-th element of the plain list <list>. <pos> has
** to be a positive integer less or equal to the physical length of <list>.
** Before calling 'CELM' it should be ensured that the <pos>-th entry of
** <list> is an immediate integer object.
*/
#define CELM(list, pos) ( INT_INTOBJ(ELM_PLIST(list, pos) ) )

/****************************************************************************
**
*V Dt_add
**
** Dt_add is used to store the library function dt_add.
*/

static Obj Dt_add;

/****************************************************************************
**
*F UnmarkTree( <tree> ) . . . . . . . remove the marks of all nodes of <tree>
**
** 'UnmarkTree' removes all marks of all nodes of the tree <tree>.
*/
static void UnmarkTree(Obj tree)
{
 UInt i, len; // loop variable

 len = DT_LENGTH(tree, 1);
 for (i=1; i <= len; i++ )
 DT_UNMARK(tree, i);
}

/****************************************************************************
**
*F FuncUnmarkTree(<self>, <tree>) . . remove the marks of all nodes of <tree>
**
** 'FuncUnmarkTree' implements the internal function 'UnmarkTree'.
**
** 'UnmarkTree( <tree> )'
**
** 'UnmarkTree' removes all marks of all nodes of the tree <tree>.
*/
static Obj FuncUnmarkTree(Obj self, Obj tree)
{
 UnmarkTree(tree);
 return 0;
}

/*****************************************************************************
**
*F Mark(<tree>, <reftree>, <index>) . . . . . . . . find all nodes of <tree>
** which are almost equal
** to (<reftree>, index)
**
** 'Mark' determines all nodes of the tree <tree>, rooting subtrees almost
** equal to the tree rooted at (<reftree>, index). 'Mark' marks these nodes
** and returns the number of different nodes among these nodes. Since it
** is assumed that the set {pos(a) | a almost equal to (<reftree>, index) }
** is equal to {1,...,n} for a positive integer n, 'Mark' actually returns
** the Maximum of {pos(a) | a almost equal to (<reftree>, index)}.
*/
static UInt Mark(Obj tree, Obj reftree, Int indexx)
{
 UInt i, // loop variable
 m, // integer to return
 len;
 Obj refgen;

 m = 0;
 i = 1;
 len = DT_LENGTH(tree, 1);
 refgen = DT_GEN(reftree, indexx);
 while ( i <= len )
 {
 /* skip all nodes (<tree>, i) with
 ** num(<tree>, i) > num(<reftree>, indexx) */
 while( i < len &&
 DT_GEN(tree, i) > refgen )
 i++;
 if ( AlmostEqual(tree, i, reftree, indexx) )
 {
 DT_MARK(tree, i);
 if ( m < INT_INTOBJ( DT_POS(tree, i) ) )
 m = INT_INTOBJ( DT_POS(tree, i) );
 }
 /* Since num(a) < num(b) holds for all subtrees <a> of an arbitrary
 ** tree we can now skip the whole tree rooted at (<tree>, i).
 ** If (<tree>, i) is the left subnode of another node we can even
 ** skip the tree rooted at that node, because of
 ** num( right(a) ) < num( left(a) ) for all trees <a>.
 ** Note that (<tree>, i) is the left subnode of another node, if and
 ** only if the previous node (<tree>, i-1) is not an atom. in this
 ** case (<tree>, i) is the left subnode of (<tree>, i-1). */
 if ( DT_LENGTH(tree, i-1) == 1 )
 // skip the tree rooted at (<tree>, i).
 i = i + DT_LENGTH(tree, i);
 else
 // skip the tree rooted at (<tree>, i-1)
 i = i - 1 + DT_LENGTH(tree, i-1);
 }
 return m;
}

/****************************************************************************
**
*F AmostEqual(<tree1>,<index1>,<tree2>,<index2>) . . test of almost equality
**
** 'AlmostEqual' tests if tree(<tree1>, index1) is almost equal to
** tree(<tree2>, index2). 'AlmostEqual' returns 1
** if these trees are almost equal, and 0 otherwise. <index1> has to be
** a positive integer less or equal to the number of nodes of <tree1>,
** and <index2> has to be a positive integer less or equal to the number of
** nodes of <tree2>.
*/
static Int AlmostEqual(Obj tree1, Int index1, Obj tree2, Int index2)
{
 UInt k, schranke; // loop variable
 /* First the two top nodes of tree(<tree1>, index1) and
 ** tree(<tree2>, index2) (that are (<tree1>, index1) and
 ** (<tree2, index2) ) are compared by testing the equality of the 2-nd,
 ** 5-th and 6-th entries the nodes. */
 if ( DT_GEN(tree1, index1) != DT_GEN(tree2, index2) )
 return 0;
 if ( DT_SIDE(tree1, index1) != DT_SIDE(tree2, index2) )
 return 0;
 if ( DT_LENGTH(tree1, index1) != DT_LENGTH(tree2, index2) )
 return 0;
 /* For the comparison of the remaining nodes of tree(<tree1>, index1)
 ** and tree(<tree2>, index2) it is also necessary to compare the first
 ** entries of the nodes. Note that we know at this point, that
 ** tree(<tree1>, index1) and tree(<tree2>, index2) have the same number
 ** of nodes */
 schranke = index1 + DT_LENGTH(tree1, index1);
 for (k = index1 + 1; k < schranke; k++ )
 {
 if ( DT_GEN(tree1, k) != DT_GEN(tree2, k + index2 - index1 ) )
 return 0;
 if ( DT_POS(tree1, k) != DT_POS(tree2, k + index2 - index1 ) )
 return 0;
 if ( DT_SIDE(tree1, k) !=
 DT_SIDE(tree2, k + index2 - index1) )
 return 0;
 if ( DT_LENGTH(tree1, k) != DT_LENGTH(tree2, k + index2 - index1) )
 return 0;
 }
 return 1;
}

/*****************************************************************************
**
*F Equal(<tree1>,<index1>,<tree2>,<index2>) . . . . . . . . test of equality
**
** 'Equal' tests if tree(<tree1>, index1) is equal to
** tree(<tree2>, index2). 'Equal' returns 1
** if these trees are equal, and 0 otherwise. <index1> has to be
** a positive integer less or equal to the number of nodes of <tree1>,
** and <index2> has to be a positive integer less or equal to the number of
** nodes of <tree2>.
*/
static Int Equal(Obj tree1, Int index1, Obj tree2, Int index2)
{
 UInt k, schranke; // loop variable

 /* Each node of tree(<tree1>, index1) is compared to the corresponding
 ** node of tree(<tree2>, index2) by testing the equality of the 1-st,
 ** 2-nd, 5-th and 6-th nodes. */
 schranke = index1 + DT_LENGTH(tree1, index1);
 for (k=index1; k < schranke; k++)
 {
 if ( DT_GEN(tree1, k) != DT_GEN(tree2, k + index2 - index1 ) )
 return 0;
 if ( DT_POS(tree1, k) != DT_POS(tree2, k + index2 - index1 ) )
 return 0;
 if ( DT_SIDE(tree1, k) !=
 DT_SIDE(tree2, k + index2 - index1) )
 return 0;
 if ( DT_LENGTH(tree1, k) != DT_LENGTH(tree2, k + index2 - index1) )
 return 0;
 }
 return 1;
}

/****************************************************************************
**
*F Mark2(<tree>,<index1>,<reftree>,<index2>) . . find all subtrees of
** tree(<tree>, index1) which
** are almost equal to
** tree(<reftree>, index2)
**
** 'Mark2' determines all subtrees of tree(<tree>, index1) that are almost
** equal to tree(<reftree>, index2). 'Mark2' marks the top nodes of these
** trees and returns a list of lists <list> such that <list>[i]
** for each subtree <a> of <tree> which is almost equal to
** tree(<reftree>, index2) and for which pos(<a>) = i holds contains an
** integer describing the position of the top node of <a> in <tree>.
** For example <list>[i] = [j, k] means that tree(<tree>, j) and
** tree(<tree>, k) are almost equal to tree(<reftree>, index2) and
** that pos(tree(<tree>, j) = pos(tree(<tree>, k) = i holds.
**
** <index1> has to be a positive integer less or equal to the number of nodes
** of <tree>, and <index2> has to be a positive integer less or equal to
** the number of nodes of <reftree>.
*/
static Obj Mark2(Obj tree, Int index1, Obj reftree, Int index2)
{
 UInt i, // loop variable
 len;
 Obj new,
 list, // list to return
 refgen;

 // initialize <list>
 list = NEW_PLIST(T_PLIST, 0);
 i = index1;
 len = index1 + DT_LENGTH(tree, index1) - 1;
 refgen = DT_GEN(reftree, index2);
 while( i <= len )
 {
 /* skip all nodes (<tree>, i) with
 ** num(<tree>, i) > num(<reftree>, index) */
 while( i < len &&
 DT_GEN(tree, i) > refgen )
 i++;
 if ( AlmostEqual(tree, i, reftree, index2) )
 {
 DT_MARK(tree, i);
 // if <list> is too small grow it appropriately
 if ( LEN_PLIST(list) < INT_INTOBJ( DT_POS(tree, i) ) )
 {
 GROW_PLIST(list, INT_INTOBJ( DT_POS(tree, i) ) );
 SET_LEN_PLIST(list, INT_INTOBJ( DT_POS(tree, i) ) );
 }
 /* if <list> has no entry at position pos(tree(<tree>, i))
 ** create a new list <new>, assign it to list at position
 ** pos(tree(<tree>, i)), and add i to <new> */
 if ( ELM_PLIST(list, INT_INTOBJ( DT_POS(tree, i) ) ) == 0)
 {
 new = NewPlistFromArgs(INTOBJ_INT(i));
 SET_ELM_PLIST(list, INT_INTOBJ( DT_POS(tree, i) ), new);
 // tell gasman that list has changed
 CHANGED_BAG(list);
 }
 // add i to <list>[ pos(tree(<tree>, i)) ]
 else
 {
 new = ELM_PLIST(list, INT_INTOBJ( DT_POS(tree, i) ) );
 PushPlist(new, INTOBJ_INT(i) );
 }
 }
 /* Since num(a) < num(b) holds for all subtrees <a> of an arbitrary
 ** tree we can now skip the whole tree rooted at (<tree>, i).
 ** If (<tree>, i) is the left subnode of another node we can even
 ** skip the tree rooted at that node, because of
 ** num( right(a) ) < num( left(a) ) for all trees <a>.
 ** Note that (<tree>, i) is the left subnode of another node, if and
 ** only if the previous node (<tree>, i-1) is not an atom. In this
 ** case (<tree>, i) is the left subnode of (<tree>, i-1). */
 if ( DT_LENGTH(tree, i-1) == 1 )
 // skip tree(<tree>, i)
 i = i + DT_LENGTH(tree, i);
 else
 // skip tree(<tree>, i-1)
 i = i - 1 + DT_LENGTH(tree, i-1);
 }
 return list;
}

/*****************************************************************************
**
*F FindTree(<tree>, <index>)
**
** 'FindTree' looks for a subtree <a> of tree(<tree>, index) such that
** the top node of
** <a> is not marked but all the other nodes of <a> are marked. It is
** assumed that if the top node of a subtree of tree(<tree>, index)
** is marked, all
** nodes of are marked. Hence it suffices to look for a subtree <a>
** of <tree> such that the top node of <a> is unmarked and the left and the
** right node of <a> are marked. 'FindTree' returns an integer such
** that tree(<tree> ,i) has the properties mentioned above. If such a tree
** does not exist 'Findtree' returns 0 (as C integer). Note that this holds
** if and only if tree(<tree>, index) is marked.
*/
static UInt FindTree(Obj tree, Int indexx)
{
 UInt i; // loop variable

 // return 0 if (<tree>, indexx) is marked
 if ( DT_IS_MARKED(tree, indexx) )
 return 0;
 i = indexx;
 /* loop over all nodes of tree(<tree>, indexx) to find a tree with the
 ** properties described above. */
 while( i < indexx + DT_LENGTH(tree, indexx) )
 {
 // skip all nodes that are unmarked and rooting non-atoms
 while( !( DT_IS_MARKED(tree, i) ) && DT_LENGTH(tree, i) > 1 )
 i++;
 /* if (<tree>, i) is unmarked we now know that tree(<tree>, i) is
 ** an atom and we can return i. Note that an unmarked atom has the
 ** desired properties. */
 if ( !( DT_IS_MARKED(tree, i) ) )
 return i;
 // go to the previous node
 i--;
 /* If the right node of tree(<tree>, i) is marked return i.
 ** Else go to the right node of tree(<tree>, i). */
 if ( DT_IS_MARKED(tree, DT_RIGHT(tree, i) ) )
 return i;
 i = DT_RIGHT(tree, i);
 }
 return 0;
}

/****************************************************************************
**
*F MakeFormulaVector(<tree>, <pr>) . . . . . . . . . compute the polynomial
** g_<tree> for <tree>
**
** 'MakeFormulaVector' returns the polynomial g_<tree> for a tree <tree>
** and a pc-presentation <pr> of a nilpotent group. This polynomial g_<tree>
** is a product of binomial coefficients with a coefficient c ( see the
** header of this file ).
**
** For the calculation of the coefficient c the top node of <tree> is ignored
** because it can happen that trees are equal except for the top node.
** Hence it suffices to compute the formula vector for one of these trees.
** Then we get the "correct" coefficient for the polynomial for each <tree'>
** of those trees by multiplying the coefficient given by the formula vector
** with c_( num(left(<tree'>)), num(right(<tree'>)); num(<tree'>) ). This
** is also the reason for storing num(left(<tree>)) and num(right(<tree>))
** in the formula vector.
**
** 'MakeFormulaVector' only returns correct results if all nodes of <tree>
** are unmarked.
*/
static Obj MakeFormulaVector(Obj tree, Obj pr)
{
 UInt i, // denominator of a binomial coefficient
 j, // loop variable
 u; // node index
 Obj rel, // stores relations of <pr>
 vec, // stores formula vector to return
 prod,// stores the product of two integers
 gen;

 // initialize <vec> and set the first four elements
 vec = NewPlistFromArgs(INTOBJ_INT(0), INTOBJ_INT(1),
 DT_GEN(tree, DT_LEFT(tree, 1)),
 DT_GEN(tree, DT_RIGHT(tree, 1)));
 /* loop over all almost equal classes of subtrees of <tree> except for
 ** <tree> itself. */
 u = FindTree(tree, 1);
 while( u > 1 )
 {
 /* mark all subtrees of <tree> almost equal to tree(<tree>, u) and
 ** get the number of different trees in this almost equal class */
 i = Mark(tree, tree, u);
 /* if tree(<tree>, u) is an atom from the Right-hand word append
 ** [ 0, i ] to <vec> */
 if ( DT_SIDE(tree, u) == RIGHT )
 {
 GROW_PLIST(vec, LEN_PLIST(vec)+2);
 SET_LEN_PLIST(vec, LEN_PLIST(vec)+2);
 SET_ELM_PLIST(vec, LEN_PLIST(vec)-1, INTOBJ_INT(0) );
 SET_ELM_PLIST(vec, LEN_PLIST(vec), INTOBJ_INT(i) );
 }
 /* if tree(<tree>, u) is an atom from the Left-hand word append
 ** [ num(tree(<tree>, u)), i ] to <vec> */
 else if ( DT_SIDE(tree, u) == LEFT)
 {
 GROW_PLIST(vec, LEN_PLIST(vec)+2);
 SET_LEN_PLIST(vec, LEN_PLIST(vec)+2);
 SET_ELM_PLIST(vec, LEN_PLIST(vec)-1, DT_GEN(tree, u) );
 SET_ELM_PLIST(vec, LEN_PLIST(vec), INTOBJ_INT(i) );
 }
 /* if tree(<tree>, u) is not an atom multiply
 ** <vec>[2] with binomial(d, i) where
 ** d = c_(num(left(<tree>,u)), num(right(<tree>,u)); num(<tree>,u)) */
 else
 {
 j = 3;
 rel = ELM_PLIST( ELM_PLIST(pr, INT_INTOBJ( DT_GEN(tree,
 DT_LEFT(tree, u) ) ) ),
 INT_INTOBJ( DT_GEN(tree, DT_RIGHT(tree, u) ) ) );
 gen = DT_GEN(tree, u);
 while ( 1 )
 {
 if ( ELM_PLIST(rel, j) == gen )
 {
 prod = ProdInt(ELM_PLIST(vec, 2),
 BinomialInt(ELM_PLIST(rel, j+1),
 INTOBJ_INT(i) ) );
 SET_ELM_PLIST(vec, 2, prod);
 // tell gasman that vec has changed
 CHANGED_BAG(vec);
 break;
 }
 j+=2;
 }
 }
 u = FindTree(tree, 1);
 }
 return vec;
}

/**************************************************************************
**
*F FuncMakeFormulaVector(<self>,<tree>,<pr>) . . . . . compute the formula
** vector for <tree>
**
** 'FuncMakeFormulaVector' implements the internal function
** 'MakeFormulaVector(<tree>, <pr>)'.
**
** 'MakeFormulaVector(<tree>, <pr>)'
**
** 'MakeFormulaVector' returns the formula vector for the tree <tree> and
** the pc-presentation <pr>.
*/
static Obj FuncMakeFormulaVector(Obj self, Obj tree, Obj pr)
{
 if (LEN_PLIST(tree) == 5)
 ErrorMayQuit(" has to be a non-atom", 0, 0);
 return MakeFormulaVector(tree, pr);
}

/****************************************************************************
**
*F Leftof(<tree1>,<index1>,<tree2>,<index2>) . . . . test if one tree is left
** of another tree
**
** 'Leftof' returns 1 if tree(<tree1>, index1) is left of tree(<tree2>,index2)
** in the word being collected at the first instance, that
** tree(<tree1>, index1) and tree(<tree2>, index2) both occur. It is assumed
** that tree(<tree1>, index1) is not equal to tree(<tree2>, index2).
*/
static Int Leftof(Obj tree1, Int index1, Obj tree2, Int index2)
{
 if ( DT_LENGTH(tree1, index1) == 1 && DT_LENGTH(tree2, index2) == 1 ) {
 if (DT_SIDE(tree1, index1) == LEFT && DT_SIDE(tree2, index2) == RIGHT)
 return 1;
 else if (DT_SIDE(tree1, index1) == RIGHT &&
 DT_SIDE(tree2, index2) == LEFT )
 return 0;
 else if (DT_GEN(tree1, index1) == DT_GEN(tree2, index2) )
 return ( DT_POS(tree1, index1) < DT_POS(tree2, index2) );
 else
 return ( DT_GEN(tree1, index1) < DT_GEN(tree2, index2) );
 }
 if ( DT_LENGTH(tree1, index1) > 1 &&
 DT_LENGTH(tree2, index2) > 1 &&
 Equal( tree1, DT_RIGHT(tree1, index1) ,
 tree2, DT_RIGHT(tree2, index2) ) )
 {
 if ( Equal( tree1, DT_LEFT(tree1, index1),
 tree2, DT_LEFT(tree2, index2) ) ) {
 if ( DT_GEN(tree1, index1) == DT_GEN(tree2, index2) )
 return ( DT_POS(tree1, index1) < DT_POS(tree2, index2) );
 else
 return ( DT_GEN(tree1, index1) < DT_GEN(tree2, index2) );
 }
 }
 if( Earlier(tree1, index1, tree2, index2) )
 return !Leftof2( tree2, index2, tree1, index1);
 else
 return Leftof2( tree1, index1, tree2, index2);
}

/*****************************************************************************
**
*F Leftof2(<tree1>,<index1>,<tree2>,<index2>) . . . . . test if one tree is
** left of another tree
**
** 'Leftof2' returns 1 if tree(<tree1>, index1) is left of
** tree(<tree2>,index2)in the word being collected at the first instance,
** that tree(<tree1>, index1) and tree(<tree2>, index2) both occur. It is
** assumed that tree(<tree2>, index2) occurs earlier than
** tree(<tree1>,index1). Furthermore it is assumed that if both
** tree(<tree1>, index1) and tree(<tree2>, index2) are non-atoms, then their
** right trees and their left trees are not equal.
*/
static Int Leftof2(Obj tree1, Int index1, Obj tree2, Int index2)
{
 if ( DT_GEN(tree2, index2) < DT_GEN(tree1, DT_RIGHT(tree1, index1) ) )
 return 0;
 else if (Equal(tree1, DT_RIGHT(tree1, index1), tree2, index2 ) )
 return 0;
 else if (DT_GEN(tree2, index2) == DT_GEN(tree1, DT_RIGHT(tree1, index1)) )
 return Leftof(tree1, DT_RIGHT(tree1, index1), tree2, index2 );
 else if (Equal(tree1, DT_LEFT(tree1, index1), tree2, index2) )
 return 0;
 else
 return Leftof(tree1, DT_LEFT(tree1, index1), tree2, index2);
}

/****************************************************************************
**
*F Earlier(<tree1>,<index1>,<tree2>,<index2>) . . . test if one tree occurs
** earlier than another
**
** 'Earlier' returns 1 if tree(<tree1>, index1) occurs strictly earlier than
** tree(<tree2>, index2). It is assumed that at least one of these trees
** is a non-atom. Furthermore it is assumed that if both of these trees are
** non-atoms, right(tree(<tree1>, index1) ) does not equal
** right(tree(<tree2>, index2) ) or left(tree(<tree1>, index1) ) does not
** equal left(tree(<tree2>, index2) ).
*/
static Int Earlier(Obj tree1, Int index1, Obj tree2, Int index2)
{
 if ( DT_LENGTH(tree1, index1) == 1 )
 return 1;
 if ( DT_LENGTH(tree2, index2) == 1 )
 return 0;
 if ( Equal(tree1, DT_RIGHT(tree1, index1),
 tree2, DT_RIGHT(tree2, index2) ) )
 return Leftof(tree1, DT_LEFT(tree2, index2),
 tree2, DT_LEFT(tree1, index1) );
 if ( DT_GEN(tree1, DT_RIGHT(tree1, index1) ) ==
 DT_GEN(tree2, DT_RIGHT(tree2, index2) ) )
 return Leftof( tree1, DT_RIGHT(tree1, index1) ,
 tree2, DT_RIGHT(tree2, index2) );
 return (DT_GEN(tree1, DT_RIGHT(tree1, index1) ) <
 DT_GEN(tree2, DT_RIGHT(tree2, index2) ) );
}

/****************************************************************************
**
** GetPols( <list>, <pr>, <pols> )
**
** GetPols computes all representatives which are represented by the
** pseudorepresentative <list>, converts them all into the corresponding
** deep thought monomial and stores all these monomials in the list <pols>.
*/

// See below:
static void GetReps(Obj list, Obj reps);
static void FindNewReps2(Obj tree, Obj reps, Obj pr);

static void GetPols(Obj list, Obj pr, Obj pols)
{
 Obj lreps,
 rreps,
 tree,
 tree1;
 UInt i,j,k,l, lenr, lenl, len;

 lreps = NEW_PLIST(T_PLIST, 2);
 rreps = NEW_PLIST(T_PLIST, 2);
 /* get the representatives that are represented by <list>[1] and those
 ** which are represented by <list>[2]. */
 GetReps( ELM_PLIST(list, 1), lreps );
 GetReps( ELM_PLIST(list, 2), rreps );
 lenr = LEN_PLIST(rreps);
 lenl = LEN_PLIST(lreps);
 for (i=1; i<=lenl; i++)
 for (j=1; j<=lenr; j++)
 {
 /* now get all representatives, which can be constructed from
 ** <lreps>[] and <rreps>[<j>] and add the corresponding
 ** deep thought monomials to <pols> */
 k = LEN_PLIST( ELM_PLIST(lreps, i) )
 + LEN_PLIST( ELM_PLIST(rreps, j) ) + 5;/* m"ogliche Inkom-*/
 tree = NEW_PLIST(T_PLIST, k); /* patibilit"at nach*/
 SET_LEN_PLIST(tree, k); /*"Anderung der Datenstruktur */
 SET_ELM_PLIST(tree, 1, INTOBJ_INT(1) );
 SET_ELM_PLIST(tree, 2, ELM_PLIST( list, 3) );
 SET_ELM_PLIST(tree, 3, INTOBJ_INT(0) );
 SET_ELM_PLIST(tree, 4, INTOBJ_INT((int)(k/5)) );
 SET_ELM_PLIST(tree, 5, INTOBJ_INT(0) );
 tree1 = ELM_PLIST(lreps, i);
 len = LEN_PLIST( tree1 );
 for (l=1; l<=len; l++)
 SET_ELM_PLIST(tree, l+5, ELM_PLIST(tree1, l) );
 k = LEN_PLIST(tree1) + 5;
 tree1 = ELM_PLIST(rreps, j);
 len = LEN_PLIST( tree1 );
 for (l=1; l<=len; l++)
 SET_ELM_PLIST(tree, l+k, ELM_PLIST(tree1, l) );
 UnmarkTree(tree);
 FindNewReps2(tree, pols, pr);
 }
}

/****************************************************************************
**
*F FuncGetPols( <self>, <list>, <pr>, <pols> )
**
** FuncGetPols implements the internal function GetPols.
*/

static Obj FuncGetPols(Obj self, Obj list, Obj pr, Obj pols)
{
 if (LEN_PLIST(list) != 4)
 ErrorMayQuit(" must be a generalised representative not a tree",
 0, 0);
 GetPols(list, pr, pols);
 return (Obj) 0;
}

/****************************************************************************
**
*F GetReps( <list>, <reps> )
**
** GetReps computes all representatives which are represented by the
** pseudorepresentative <list> and adds them to the list <reps>.
*/

// See below:
static void FindNewReps1(Obj tree, Obj reps);

static void GetReps(Obj list, Obj reps)
{
 Obj lreps,
 rreps,
 tree,
 tree1;
 UInt i,j,k,l, lenr, lenl, len;;

 if ( LEN_PLIST(list) != 4 )
 {
 SET_ELM_PLIST(reps, 1, list);
 SET_LEN_PLIST(reps, 1);
 return;
 }
 lreps = NEW_PLIST(T_PLIST, 2);
 rreps = NEW_PLIST(T_PLIST, 2);
 /* now get all representatives which are represented by <list>[1] and
 ** all representatives which are represented by <list>[2]. */
 GetReps( ELM_PLIST(list, 1), lreps );
 GetReps( ELM_PLIST(list, 2), rreps );
 lenl = LEN_PLIST( lreps );
 lenr = LEN_PLIST( rreps );
 for (i=1; i<=lenl; i++)
 for (j=1; j<=lenr; j++)
 {
 /* compute all representatives which can be constructed from
 ** <lreps>[] and <rreps>[<j>] and add them to <reps>. */
 k = LEN_PLIST( ELM_PLIST(lreps, i) )
 + LEN_PLIST( ELM_PLIST(rreps, j) ) + 5;/* m"ogliche Inkom-*/
 tree = NEW_PLIST(T_PLIST, k); /* patibilit"at nach*/
 SET_LEN_PLIST(tree, k); /*"Anderung der Datenstruktur */
 SET_ELM_PLIST(tree, 1, INTOBJ_INT(1) );
 SET_ELM_PLIST(tree, 2, ELM_PLIST( list, 3) );
 SET_ELM_PLIST(tree, 3, INTOBJ_INT(0) );
 SET_ELM_PLIST(tree, 4, INTOBJ_INT((int)(k/5)) );
 if ( IS_INTOBJ( ELM_PLIST(list, 4) ) &&
 CELM(list, 4) < 100 && CELM(list, 4) > 0 )
 SET_ELM_PLIST(tree, 5, ELM_PLIST(list, 4) );
 else
 SET_ELM_PLIST(tree, 5, INTOBJ_INT(0) );
 tree1 = ELM_PLIST(lreps, i);
 len = LEN_PLIST( tree1 );
 for (l=1; l<=len; l++)
 SET_ELM_PLIST(tree, l+5, ELM_PLIST(tree1, l) );
 k = LEN_PLIST(tree1) + 5;
 tree1 = ELM_PLIST(rreps, j);
 len = LEN_PLIST( tree1 );
 for (l=1; l<=len; l++)
 SET_ELM_PLIST(tree, l+k, ELM_PLIST(tree1, l) );
 UnmarkTree(tree);
 FindNewReps1(tree, reps);
 }
}

/**************************************************************************
**
*F FindNewReps(<tree>,<reps>,<pr>,<max>) . . construct new representatives
**
** 'FindNewReps' constructs all trees <tree'> with the following properties.
** 1) left(<tree'>) is equivalent to left(<tree>).
** right(<tree'>) is equivalent to right(<tree>).
** num(<tree'>) = num(<tree>)
** 2) <tree'> is the least tree in its equivalence class.
** 3) for each marked node of (<tree>, i) of <tree> tree(<tree>, i) is equal
** to tree(<tree'>, i).
** There are three versions of FindNewReps. FindNewReps1 adds all found
** trees to the list <reps>. This version is called by GetReps.
** FindNewReps2 computes for each found tree the corresponding deep thought
** monomial adds these deep thought monomials to <reps>. This version
** is called from GetPols.
** The third version FindNewReps finally assumes that <reps> is the list of
** pseudorepresentatives. This Version adds all found trees to <reps> and
** additionally all trees, that fulfill 1), 2) and 3) except for
** num(<tree'>) = num(<tree>). This version is called from the library
** function calrepsn.
** It is assumed that both left(<tree>) and right(<tree>) are the least
** elements in their equivalence class.
*/

// See below:
static void FindSubs1(Obj tree,
 Int x,
 Obj list1,
 Obj list2,
 Obj a,
 Obj b,
 Int al,
 Int ar,
 Int bl,
 Int br,
 Obj reps);

static void FindNewReps1(Obj tree, Obj reps)
{
 Obj y, // stores a copy of <tree>
 lsubs, /* stores pos(<subtree>) for all subtrees of
 ** left(<tree>) in a given almost equal class */

 rsubs, /* stores pos(<subtree>) for all subtrees of
 ** right(<tree>) in the same almost equal class */

 llist, /* stores all elements of an almost equal class
 ** of subtrees of left(<tree>) */

 rlist; /* stores all elements of the same almost equal
 ** class of subtrees of right(<tree>) */
 Int a, // stores a subtree of right((<tree>)
 n, // Length of lsubs
 m, // Length of rsubs
 i; // loop variable

 /* get a subtree of right(<tree>) which is unmarked but whose
 ** subtrees are all marked */
 a = FindTree(tree, DT_RIGHT(tree, 1) );
 /* If we do not find such a tree we at the bottom of the recursion.
 ** If leftof(left(<tree>), right(<tree>) ) holds we add all <tree>
 ** to <reps>. */
 if ( a == 0 )
 {
 if ( Leftof(tree, DT_LEFT(tree, 1), tree, DT_RIGHT(tree, 1) ) )
 {
 y = ShallowCopyPlist(tree);
 AssPlist(reps, LEN_PLIST(reps) + 1, y);
 }
 return;
 }
 /* get all subtrees of left(<tree>) which are almost equal to
 ** tree(<tree>, a) and mark them */
 llist = Mark2(tree, DT_LEFT(tree, 1), tree, a);
 /* get all subtrees of right(<tree>) which are almost equal to
 ** tree(<tree>, a) and mark them */
 rlist = Mark2(tree, DT_RIGHT(tree, 1), tree, a);
 n = LEN_PLIST(llist);
 m = LEN_PLIST(rlist);
 /* if no subtrees of left(<tree>) almost equal to
 ** tree(<tree>, a) have been found there is no possibility
 ** to change the pos-argument in the trees stored in llist and
 ** rlist, so call FindNewReps without changing any pos-arguments.
 */
 if ( n == 0 )
 {
 FindNewReps1(tree, reps);
 // unmark all top nodes of the trees stored in rlist
 UnmarkAEClass(tree, rlist);
 return;
 }
 /* store all pos-arguments that occur in the trees of llist.
 ** Note that the set of the pos-arguments in llist actually
 ** equals {1,...,n}. */
 lsubs = NEW_PLIST( T_PLIST, n );
 SET_LEN_PLIST(lsubs, n);
 for (i=1; i<=n; i++)
 SET_ELM_PLIST(lsubs, i, INTOBJ_INT(i) );
 /* store all pos-arguments that occur in the trees of rlist.
 ** Note that the set of the pos-arguments in rlist actually
 ** equals {1,...,m}. */
 rsubs = NEW_PLIST( T_PLIST, m );
 SET_LEN_PLIST(rsubs, m);
 for (i=1; i<=m; i++)
 SET_ELM_PLIST(rsubs, i, INTOBJ_INT(i) );
 /* find all possibilities for lsubs and rsubs such that
 ** lsubs[1] < lsubs[2] <...<lsubs[n],
 ** rsubs[1] < rsubs[2] <...<rsubs[n],
 ** and set(lsubs concat rsubs) equals {1,...,k} for a positive
 ** integer k. For each found lsubs and rsubs 'FindSubs' changes
 ** pos-arguments of the subtrees in llist and rlist accordingly
 ** and then calls 'FindNewReps' with the changed tree <tree>.
 */
 FindSubs1(tree, a, llist, rlist, lsubs, rsubs, 1, n, 1, m, reps);
 /* Unmark the subtrees of <tree> in llist and rlist and reset
 ** pos-arguments to the original state. */
 UnmarkAEClass(tree, rlist);
 UnmarkAEClass(tree, llist);
}

// See below:
static void FindSubs2(Obj tree,
 Int x,
 Obj list1,
 Obj list2,
 Obj a,
 Obj b,
 Int al,
 Int ar,
 Int bl,
 Int br,
 Obj reps,
 Obj pr);

static void
FindNewReps2(Obj tree, Obj reps, Obj pr /* pc-presentation for a
 ** nilpotent group <G> */
)
{
 Obj lsubs, /* stores pos(<subtree>) for all subtrees of
 ** left(<tree>) in a given almost equal class */

 rsubs, /* stores pos(<subtree>) for all subtrees of
 ** right(<tree>) in the same almost equal class */

 llist, /* stores all elements of an almost equal class
 ** of subtrees of left(<tree>) */

 rlist; /* stores all elements of the same almost equal
 ** class of subtrees of right(<tree>) */
 Int a, // stores a subtree of right((<tree>)
 n, // Length of lsubs
 m, // Length of rsubs
 i; // loop variable

 /* get a subtree of right(<tree>) which is unmarked but whose
 ** subtrees are all marked */
 a = FindTree(tree, DT_RIGHT(tree, 1) );
 /* If we do not find such a tree we at the bottom of the recursion.
 ** If leftof(left(<tree>), right(<tree>) ) holds we convert <tree>
 ** into the corresponding deep thought monomial and add that to
 ** <reps>. */
 if ( a == 0 )
 {
 if ( Leftof(tree, DT_LEFT(tree, 1), tree, DT_RIGHT(tree, 1) ) )
 {
 /* get the formula vector of tree and add it to
 ** reps[ rel[1] ]. */
 UnmarkTree(tree);
 tree = MakeFormulaVector( tree, pr);
 CALL_3ARGS(Dt_add, tree, reps, pr);
 }
 return;
 }
 /* get all subtrees of left(<tree>) which are almost equal to
 ** tree(<tree>, a) and mark them */
 llist = Mark2(tree, DT_LEFT(tree, 1), tree, a);
 /* get all subtrees of right(<tree>) which are almost equal to
 ** tree(<tree>, a) and mark them */
 rlist = Mark2(tree, DT_RIGHT(tree, 1), tree, a);
 n = LEN_PLIST(llist);
 m = LEN_PLIST(rlist);
 /* if no subtrees of left(<tree>) almost equal to
 ** tree(<tree>, a) have been found there is no possibility
 ** to change the pos-argument in the trees stored in llist and
 ** rlist, so call FindNewReps without changing any pos-arguments.
 */
 if ( n == 0 )
 {
 FindNewReps2(tree, reps, pr);
 // unmark all top nodes of the trees stored in rlist
 UnmarkAEClass(tree, rlist);
 return;
 }
 /* store all pos-arguments that occur in the trees of llist.
 ** Note that the set of the pos-arguments in llist actually
 ** equals {1,...,n}. */
 lsubs = NEW_PLIST( T_PLIST, n );
 SET_LEN_PLIST(lsubs, n);
 for (i=1; i<=n; i++)
 SET_ELM_PLIST(lsubs, i, INTOBJ_INT(i) );
 /* store all pos-arguments that occur in the trees of rlist.
 ** Note that the set of the pos-arguments in rlist actually
 ** equals {1,...,m}. */
 rsubs = NEW_PLIST( T_PLIST, m );
 SET_LEN_PLIST(rsubs, m);
 for (i=1; i<=m; i++)
 SET_ELM_PLIST(rsubs, i, INTOBJ_INT(i) );
 /* find all possibilities for lsubs and rsubs such that
 ** lsubs[1] < lsubs[2] <...<lsubs[n],
 ** rsubs[1] < rsubs[2] <...<rsubs[n],
 ** and set(lsubs concat rsubs) equals {1,...,k} for a positive
 ** integer k. For each found lsubs and rsubs 'FindSubs' changes
 ** pos-arguments of the subtrees in llist and rlist accordingly
 ** and then calls 'FindNewReps' with the changed tree <tree>.
 */
 FindSubs2(tree, a, llist, rlist, lsubs, rsubs, 1, n, 1, m, reps, pr);
 /* Unmark the subtrees of <tree> in llist and rlist and reset
 ** pos-arguments to the original state. */
 UnmarkAEClass(tree, rlist);
 UnmarkAEClass(tree, llist);
}

static void FindNewReps(Obj tree,
 Obj reps,
 Obj pr, /* pc-presentation for a
 ** nilpotent group <G> */

 Obj max /* every generator <g_i> of <G> with
 ** i > max lies in the center of <G> */
)
{
 Obj y, // stores a copy of <tree>
 lsubs, /* stores pos(<subtree>) for all subtrees of
 ** left(<tree>) in a given almost equal class */

 rsubs, /* stores pos(<subtree>) for all subtrees of
 ** right(<tree>) in the same almost equal class */

 llist, /* stores all elements of an almost equal class
 ** of subtrees of left(<tree>) */

 rlist, /* stores all elements of the same almost equal
 ** class of subtrees of right(<tree>) */
 list1, // stores a sublist of <reps>
 rel; // stores a commutator relation from <pr>
 Int a; // stores a subtree of right((<tree>)
 UInt n, // Length of lsubs
 m, // Length of rsubs
 i, lenrel; // loop variable

 /* get a subtree of right(<tree>) which is unmarked but whose
 ** subtrees are all marked */
 a = FindTree(tree, DT_RIGHT(tree, 1) );
 /* If we do not find such a tree we at the bottom of the recursion.
 ** If leftof(left(<tree>), right(<tree>) ) holds we add all trees
 ** <tree'> with left(<tree'>) = left(<tree>),
 ** right(<tree'>) = right(<tree>) to <reps>, and <tree'> is the
 ** least element in its equivalence class. Note that for such a
 ** tree we have pos(<tree'>) = 1 and num(<tree'>) = j where j is a
 ** positive integer for which
 ** c_( num(left(<tree>), num(right(<tree>)), j ) does not equal
 ** 0. These integers are contained in the list
 ** pr[ num(left(<tree>)) ][ num(right(<tree>)) ]. */
 if ( a == 0 )
 {
 if ( Leftof(tree, DT_LEFT(tree, 1), tree, DT_RIGHT(tree, 1) ) )
 {
 // get pr[ num(left(<tree>)) ][ num(right(<tree>)) ]
 rel = ELM_PLIST( ELM_PLIST(pr, INT_INTOBJ( DT_GEN(tree,
 DT_LEFT(tree, 1)))) ,
 INT_INTOBJ( DT_GEN(tree, DT_RIGHT(tree, 1) ) ) );
 if ( ELM_PLIST(rel, 3) > max )
 {
 UnmarkTree(tree);
 tree = MakeFormulaVector(tree, pr);
 list1 = ELM_PLIST(reps, CELM(rel, 3) );
 PushPlist(list1, tree);
 }
 else
 {
 y = ShallowCopyPlist(tree);
 lenrel = LEN_PLIST(rel);
 for ( i=3;
 i < lenrel &&
 ELM_PLIST(rel, i) <= max;
 i+=2 )
 {
 list1 = ELM_PLIST(reps, CELM(rel, i) );
 PushPlist(list1, y);
 }
 }
 }
 return;
 }
 /* get all subtrees of left(<tree>) which are almost equal to
 ** tree(<tree>, a) and mark them */
 llist = Mark2(tree, DT_LEFT(tree, 1), tree, a);
 /* get all subtrees of right(<tree>) which are almost equal to
 ** tree(<tree>, a) and mark them */
 rlist = Mark2(tree, DT_RIGHT(tree, 1), tree, a);
 n = LEN_PLIST(llist);
 m = LEN_PLIST(rlist);
 /* if no subtrees of left(<tree>) almost equal to
 ** tree(<tree>, a) have been found there is no possibility
 ** to change the pos-argument in the trees stored in llist and
 ** rlist, so call FindNewReps without changing any pos-arguments.
 */
 if ( n == 0 )
 {
 FindNewReps(tree, reps, pr, max);
 // unmark all top nodes of the trees stored in rlist
 UnmarkAEClass(tree, rlist);
 return;
 }
 /* store all pos-arguments that occur in the trees of llist.
 ** Note that the set of the pos-arguments in llist actually
 ** equals {1,...,n}. */
 lsubs = NEW_PLIST( T_PLIST, n );
 SET_LEN_PLIST(lsubs, n);
 for (i=1; i<=n; i++)
 SET_ELM_PLIST(lsubs, i, INTOBJ_INT(i) );
 /* store all pos-arguments that occur in the trees of rlist.
 ** Note that the set of the pos-arguments in rlist actually
 ** equals {1,...,m}. */
 rsubs = NEW_PLIST( T_PLIST, m );
 SET_LEN_PLIST(rsubs, m);
 for (i=1; i<=m; i++)
 SET_ELM_PLIST(rsubs, i, INTOBJ_INT(i) );
 /* find all possibilities for lsubs and rsubs such that
 ** lsubs[1] < lsubs[2] <...<lsubs[n],
 ** rsubs[1] < rsubs[2] <...<rsubs[n],
 ** and set(lsubs concat rsubs) equals {1,...,k} for a positive
 ** integer k. For each found lsubs and rsubs 'FindSubs' changes
 ** pos-arguments of the subtrees in llist and rlist accordingly
 ** and then calls 'FindNewReps' with the changed tree <tree>.
 */
 FindSubs(tree, a, llist, rlist, lsubs, rsubs, 1, n, 1, m, reps, pr, max);
 /* Unmark the subtrees of <tree> in llist and rlist and reset
 ** pos-arguments to the original state. */
 UnmarkAEClass(tree, rlist);
 UnmarkAEClass(tree, llist);
}

/***************************************************************************
**
*F FuncFindNewReps(<self>,<args>) . . . . . . construct new representatives
**
** 'FuncFindNewReps' implements the internal function 'FindNewReps'.
*/

static Obj FuncFindNewReps(Obj self, Obj tree, Obj reps, Obj pr, Obj max)
{

#ifdef TEST_TREE
 // test if <tree> is really a tree
 TestTree(tree);
#endif
 if (LEN_PLIST(tree) < 15)
 ErrorMayQuit(" must be a tree not a plain list", 0, 0);
 FindNewReps(tree, reps, pr, max);
 return 0;
}

/***************************************************************************
**
*F TestTree(<obj>) . . . . . . . . . . . . . . . . . . . . . . test a tree
**
** 'TestTree' tests if <tree> is a tree. If <tree> is not a tree 'TestTree'
** signals an error.
*/
#ifdef TEST_TREE
static void TestTree(
 Obj tree)
{
 if ( TNUM_OBJ(tree) != T_PLIST || LEN_PLIST(tree) % 7 != 0)
 ErrorMayQuit(
 " must be a plain list, whose length is a multiple of 7",
 0, 0);
 if ( DT_LENGTH(tree, 1) != LEN_PLIST(tree)/7 )
 ErrorMayQuit(" must be a tree, not a plain list", 0, 0);
 if ( DT_SIDE(tree, 1) >= DT_LENGTH(tree, 1) )
 ErrorMayQuit(" must be a tree, not a plain list", 0, 0);
 if ( DT_LENGTH(tree, 1) == 1)
 {
 if ( DT_SIDE(tree, 1) != LEFT && DT_SIDE(tree, 1) != RIGHT )
 ErrorMayQuit(" must be a tree, not a plain list", 0, 0);
 return;
 }
 if ( DT_SIDE(tree, 1) <= 1 )
 ErrorMayQuit(" must be a tree, not a plain list", 0, 0);
 if (DT_LENGTH(tree, 1) !=
 DT_LENGTH(tree, DT_LEFT(tree, 1)) +
 DT_LENGTH(tree, DT_RIGHT(tree, 1)) +
 1 )
 ErrorMayQuit(" must be a tree, not a plain list", 0, 0);
 if ( DT_SIDE(tree, 1) != DT_LENGTH(tree, DT_LEFT(tree, 1) ) + 1 )
 ErrorMayQuit(" must be a tree, not a plain list", 0, 0);
 TestTree( Part(tree, (DT_LEFT(tree, 1) - 1)*7,
 (DT_RIGHT(tree, 1) - 1)*7 ) );
 TestTree( Part(tree, (DT_RIGHT(tree, 1) - 1)*7, LEN_PLIST(tree) ) );
}
#endif

/****************************************************************************
**
*F Part(<list>, <pos1>, <pos2>) . . . . . . . . . . . return a part of list
**
** 'Part' returns <list>{ [<pos1>+1 .. <pos2>] }.
*/
#ifdef TEST_TREE
static Obj Part(
 Obj list,
 Int pos1,
 Int pos2 )
{
 Int i, length;
 Obj part;

 length = pos2 - pos1;
 part = NEW_PLIST(T_PLIST, length);
 SET_LEN_PLIST(part, length);
 for (i=1; i <= length; i++)
 {
 SET_ELM_PLIST(part, i, ELM_PLIST(list, pos1+i) );
 }
 return part;
}
#endif

/***************************************************************************
**
*F FindSubs(<tree>,<x>,<list1>,<list2>,<a>,,<al>,<ar>,<bl>, ,<reps>,
** <pr>,<max> ) . . . . . . . . . find possible pos-arguments for
** the trees in <list1> and <list2>
**
** 'FindSubs' finds all possibilities for a and b such that
** 1) a[1] < a[2] <..< a[ ar ]
** b[1] < b[2] <..< b[ br ]
** 2) set( a concat b ) = {1,..,k} for a positive integer k.
** 3) a[1],...,a[ al-1 ] and b[1],..,b[ bl-1 ] remain unchanged.
** For each found possibility 'FindSubs' sets the pos-arguments in the
** trees of <list1> and <list2> according to the entries of <a> and
** . Then it calls 'FindNewReps' with the changed tree <tree> as
v** argument.
**
** It is assumed that the conditions 1) and 2) hold for a{ [1..al-1] } and
** b{ [1..bl-1] }.
**
** There are three versions of FindSubs according to the three versions of
** FindNewReps. FindSubs1 is called from FindNewReps1 and calls
** FindNewReps1. FindSubs2 is called from FindNewReps2 and calls
** FindNewReps2. FindSubs is called from FindNewReps and calls FindNewReps.
*/

static void FindSubs1(Obj tree,
 Int x, // subtree of <tree>
 Obj list1, /* list containing all subtrees of
 ** left(<tree>) almost equal to
 ** tree(<tree>, x) */

 Obj list2, /* list containing all subtrees of
 ** right(<tree>) almost equal to
 ** tree(<tree>, x) */

 Obj a, /* list to change, containing the
 ** pos-arguments of the trees in list1 */

 Obj b, /* list to change, containing the
 ** pos-arguments of the trees in list2 */
 Int al,
 Int ar,
 Int bl,
 Int br,
 Obj reps // list of representatives for all trees
)
{
 Int i; // loop variable

 // if <al> > <ar> or <bl> > nothing remains to change.
 if ( al > ar || bl > br )
 {
 /* Set the pos-arguments of the trees in <list1> and <list2>
 ** according to the entries of <a> and . */
 SetSubs( list1, a, tree);
 SetSubs( list2, b, tree);
 FindNewReps1(tree, reps);
 return;
 }
 /* If a[ ar] is bigger or equal to the boundary of pos(tree(<tree>, x)
 ** the execution of the statements in the body of this if-statement
 ** would have the consequence that some subtrees of <tree> in <list1>
 ** would get a pos-argument bigger than the boundary of
 ** pos(tree<tree>, x). But since the trees in <list1> are almost
 ** equal to tree(<tree>, x) they have all the same boundary for their
 ** pos-argument as tree(<tree>, x). So these statements are only
 ** executed when <a>[ar] is less than the boundary of
 ** pos(tree(<tree>, x).
 */
 if ( INT_INTOBJ( DT_MAX(tree, x) ) <= 0 ||
 ELM_PLIST(a, ar) < DT_MAX(tree, x) )
 {
 for (i=al; i<=ar; i++)
 SET_ELM_PLIST(a, i, INTOBJ_INT( CELM(a,i) + 1 ) );
 FindSubs1(tree, x, list1, list2, a, b, al, ar, bl+1, br, reps);
 for (i=al; i<=ar; i++)
 SET_ELM_PLIST(a, i, INTOBJ_INT( CELM(a, i) - 1 ) );
 }
 FindSubs1(tree, x, list1, list2, a, b, al+1, ar, bl+1, br, reps);
 /* If b[ br] is bigger or equal to the boundary of pos(tree(<tree>, x)
 ** the execution of the statements in the body of this if-statement
 ** would have the consequence that some subtrees of <tree> in <list2>
 ** would get a pos-argument bigger than the boundary of
 ** pos(tree<tree>, x). But since the trees in <list2> are almost
 ** equal to tree(<tree>, x) they have all the same boundary for their
 ** pos-argument as tree(<tree>, x). So these statements are only
 ** executed when [br] is less than the boundary of
 ** pos(tree(<tree>, x).
 */
 if ( INT_INTOBJ( DT_MAX(tree, x) ) <= 0 ||
 ELM_PLIST(b, br) < DT_MAX(tree, x) )
 {
 for (i=bl; i<=br; i++)
 SET_ELM_PLIST(b, i, INTOBJ_INT( CELM(b, i) + 1 ) );
 FindSubs1(tree, x, list1, list2, a, b, al+1, ar, bl, br, reps);
 for (i=bl; i<=br; i++)
 SET_ELM_PLIST(b, i, INTOBJ_INT( CELM(b, i) - 1 ) );
 }
}

static void FindSubs2(Obj tree,
 Int x, // subtree of <tree>
 Obj list1, /* list containing all subtrees of
 ** left(<tree>) almost equal to
 ** tree(<tree>, x) */

 Obj list2, /* list containing all subtrees of
 ** right(<tree>) almost equal to
 ** tree(<tree>, x) */

 Obj a, /* list to change, containing the
 ** pos-arguments of the trees in list1 */

 Obj b, /* list to change, containing the
 ** pos-arguments of the trees in list2 */
 Int al,
 Int ar,
 Int bl,
 Int br,
 Obj reps, // list of representatives for all trees
 Obj pr // pc-presentation
)
{
 Int i; // loop variable

 // if <al> > <ar> or <bl> > nothing remains to change.
 if ( al > ar || bl > br )
 {
 /* Set the pos-arguments of the trees in <list1> and <list2>
 ** according to the entries of <a> and . */
 SetSubs( list1, a, tree);
 SetSubs( list2, b, tree);
 FindNewReps2(tree, reps, pr);
 return;
 }
 /* If a[ ar] is bigger or equal to the boundary of pos(tree(<tree>, x)
 ** the execution of the statements in the body of this if-statement
 ** would have the consequence that some subtrees of <tree> in <list1>
 ** would get a pos-argument bigger than the boundary of
 ** pos(tree<tree>, x). But since the trees in <list1> are almost
 ** equal to tree(<tree>, x) they have all the same boundary for their
 ** pos-argument as tree(<tree>, x). So these statements are only
 ** executed when <a>[ar] is less than the boundary of
 ** pos(tree(<tree>, x).
 */
 if ( INT_INTOBJ( DT_MAX(tree, x) ) <= 0 ||
 ELM_PLIST(a, ar) < DT_MAX(tree, x) )
 {
 for (i=al; i<=ar; i++)
 SET_ELM_PLIST(a, i, INTOBJ_INT( CELM(a,i) + 1 ) );
 FindSubs2(tree, x, list1, list2, a, b, al, ar, bl+1, br, reps, pr);
 for (i=al; i<=ar; i++)
 SET_ELM_PLIST(a, i, INTOBJ_INT( CELM(a, i) - 1 ) );
 }
 FindSubs2(tree, x, list1, list2, a, b, al+1, ar, bl+1, br, reps, pr);
 /* If b[ br] is bigger or equal to the boundary of pos(tree(<tree>, x)
 ** the execution of the statements in the body of this if-statement
 ** would have the consequence that some subtrees of <tree> in <list2>
 ** would get a pos-argument bigger than the boundary of
 ** pos(tree<tree>, x). But since the trees in <list2> are almost
 ** equal to tree(<tree>, x) they have all the same boundary for their
 ** pos-argument as tree(<tree>, x). So these statements are only
 ** executed when [br] is less than the boundary of
 ** pos(tree(<tree>, x).
 */
 if ( INT_INTOBJ( DT_MAX(tree, x) ) <= 0 ||
 ELM_PLIST(b, br) < DT_MAX(tree, x) )
 {
 for (i=bl; i<=br; i++)
 SET_ELM_PLIST(b, i, INTOBJ_INT( CELM(b, i) + 1 ) );
 FindSubs2(tree, x, list1, list2, a, b, al+1, ar, bl, br, reps, pr);
 for (i=bl; i<=br; i++)
 SET_ELM_PLIST(b, i, INTOBJ_INT( CELM(b, i) - 1 ) );
 }
}

static void FindSubs(Obj tree,
 Int x, // subtree of <tree>
 Obj list1, /* list containing all subtrees of
 ** left(<tree>) almost equal to
 ** tree(<tree>, x) */

 Obj list2, /* list containing all subtrees of
 ** right(<tree>) almost equal to
 ** tree(<tree>, x) */

 Obj a, /* list to change, containing the
 ** pos-arguments of the trees in list1 */

 Obj b, /* list to change, containing the
 ** pos-arguments of the trees in list2 */
 Int al,
 Int ar,
 Int bl,
 Int br,
 Obj reps, // list of representatives for all trees
 Obj pr, // pc-presentation
 Obj max // needed to call 'FindNewReps'
)
{
 Int i; // loop variable

 // if <al> > <ar> or <bl> > nothing remains to change.
 if ( al > ar || bl > br )
 {
 /* Set the pos-arguments of the trees in <list1> and <list2>
 ** according to the entries of <a> and . */
 SetSubs( list1, a, tree);
 SetSubs( list2, b, tree);
 FindNewReps(tree, reps, pr, max);
 return;
 }
 /* If a[ ar] is bigger or equal to the boundary of pos(tree(<tree>, x)
 ** the execution of the statements in the body of this if-statement
 ** would have the consequence that some subtrees of <tree> in <list1>
 ** would get a pos-argument bigger than the boundary of
 ** pos(tree<tree>, x). But since the trees in <list1> are almost
 ** equal to tree(<tree>, x) they have all the same boundary for their
 ** pos-argument as tree(<tree>, x). So these statements are only
--> --------------------

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--> --------------------

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