fixed a bug in the raising/lowering of dummy indices and extended it to work
[ginac.git] / ginac / indexed.cpp
index acbf7d489967445a4d1e2d52f79cf5d91f06d850..dcd6d9510d6d0ac688802373f30f7d93e0f07362 100644 (file)
@@ -3,7 +3,7 @@
  *  Implementation of GiNaC's indexed expressions. */
 
 /*
- *  GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
+ *  GiNaC Copyright (C) 1999-2002 Johannes Gutenberg University Mainz, Germany
  *
  *  This program is free software; you can redistribute it and/or modify
  *  it under the terms of the GNU General Public License as published by
@@ -20,8 +20,8 @@
  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  */
 
+#include <iostream>
 #include <stdexcept>
-#include <algorithm>
 
 #include "indexed.h"
 #include "idx.h"
 #include "mul.h"
 #include "ncmul.h"
 #include "power.h"
+#include "relational.h"
+#include "symmetry.h"
 #include "lst.h"
 #include "print.h"
 #include "archive.h"
 #include "utils.h"
-#include "debugmsg.h"
 
 namespace GiNaC {
 
 GINAC_IMPLEMENT_REGISTERED_CLASS(indexed, exprseq)
 
 //////////
-// default constructor, destructor, copy constructor assignment operator and helpers
+// default ctor, dtor, copy ctor, assignment operator and helpers
 //////////
 
-indexed::indexed() : symmetry(unknown)
+indexed::indexed() : symtree(sy_none())
 {
-       debugmsg("indexed default constructor", LOGLEVEL_CONSTRUCT);
        tinfo_key = TINFO_indexed;
 }
 
 void indexed::copy(const indexed & other)
 {
        inherited::copy(other);
-       symmetry = other.symmetry;
+       symtree = other.symtree;
 }
 
 DEFAULT_DESTROY(indexed)
@@ -61,97 +61,81 @@ DEFAULT_DESTROY(indexed)
 // other constructors
 //////////
 
-indexed::indexed(const ex & b) : inherited(b), symmetry(unknown)
+indexed::indexed(const ex & b) : inherited(b), symtree(sy_none())
 {
-       debugmsg("indexed constructor from ex", LOGLEVEL_CONSTRUCT);
        tinfo_key = TINFO_indexed;
-       assert_all_indices_of_type_idx();
+       validate();
 }
 
-indexed::indexed(const ex & b, const ex & i1) : inherited(b, i1), symmetry(unknown)
+indexed::indexed(const ex & b, const ex & i1) : inherited(b, i1), symtree(sy_none())
 {
-       debugmsg("indexed constructor from ex,ex", LOGLEVEL_CONSTRUCT);
        tinfo_key = TINFO_indexed;
-       assert_all_indices_of_type_idx();
+       validate();
 }
 
-indexed::indexed(const ex & b, const ex & i1, const ex & i2) : inherited(b, i1, i2), symmetry(unknown)
+indexed::indexed(const ex & b, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(sy_none())
 {
-       debugmsg("indexed constructor from ex,ex,ex", LOGLEVEL_CONSTRUCT);
        tinfo_key = TINFO_indexed;
-       assert_all_indices_of_type_idx();
+       validate();
 }
 
-indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symmetry(unknown)
+indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(sy_none())
 {
-       debugmsg("indexed constructor from ex,ex,ex,ex", LOGLEVEL_CONSTRUCT);
        tinfo_key = TINFO_indexed;
-       assert_all_indices_of_type_idx();
+       validate();
 }
 
-indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3, const ex & i4) : inherited(b, i1, i2, i3, i4), symmetry(unknown)
+indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3, const ex & i4) : inherited(b, i1, i2, i3, i4), symtree(sy_none())
 {
-       debugmsg("indexed constructor from ex,ex,ex,ex,ex", LOGLEVEL_CONSTRUCT);
        tinfo_key = TINFO_indexed;
-       assert_all_indices_of_type_idx();
+       validate();
 }
 
-indexed::indexed(const ex & b, symmetry_type symm, const ex & i1, const ex & i2) : inherited(b, i1, i2), symmetry(symm)
+indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(symm)
 {
-       debugmsg("indexed constructor from ex,symmetry,ex,ex", LOGLEVEL_CONSTRUCT);
        tinfo_key = TINFO_indexed;
-       assert_all_indices_of_type_idx();
+       validate();
 }
 
-indexed::indexed(const ex & b, symmetry_type symm, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symmetry(symm)
+indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(symm)
 {
-       debugmsg("indexed constructor from ex,symmetry,ex,ex,ex", LOGLEVEL_CONSTRUCT);
        tinfo_key = TINFO_indexed;
-       assert_all_indices_of_type_idx();
+       validate();
 }
 
-indexed::indexed(const ex & b, symmetry_type symm, const ex & i1, const ex & i2, const ex & i3, const ex & i4) : inherited(b, i1, i2, i3, i4), symmetry(symm)
+indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2, const ex & i3, const ex & i4) : inherited(b, i1, i2, i3, i4), symtree(symm)
 {
-       debugmsg("indexed constructor from ex,symmetry,ex,ex,ex,ex", LOGLEVEL_CONSTRUCT);
        tinfo_key = TINFO_indexed;
-       assert_all_indices_of_type_idx();
+       validate();
 }
 
-indexed::indexed(const ex & b, const exvector & v) : inherited(b), symmetry(unknown)
+indexed::indexed(const ex & b, const exvector & v) : inherited(b), symtree(sy_none())
 {
-       debugmsg("indexed constructor from ex,exvector", LOGLEVEL_CONSTRUCT);
        seq.insert(seq.end(), v.begin(), v.end());
        tinfo_key = TINFO_indexed;
-       assert_all_indices_of_type_idx();
+       validate();
 }
 
-indexed::indexed(const ex & b, symmetry_type symm, const exvector & v) : inherited(b), symmetry(symm)
+indexed::indexed(const ex & b, const symmetry & symm, const exvector & v) : inherited(b), symtree(symm)
 {
-       debugmsg("indexed constructor from ex,symmetry,exvector", LOGLEVEL_CONSTRUCT);
        seq.insert(seq.end(), v.begin(), v.end());
        tinfo_key = TINFO_indexed;
-       assert_all_indices_of_type_idx();
+       validate();
 }
 
-indexed::indexed(symmetry_type symm, const exprseq & es) : inherited(es), symmetry(symm)
+indexed::indexed(const symmetry & symm, const exprseq & es) : inherited(es), symtree(symm)
 {
-       debugmsg("indexed constructor from symmetry,exprseq", LOGLEVEL_CONSTRUCT);
        tinfo_key = TINFO_indexed;
-       assert_all_indices_of_type_idx();
 }
 
-indexed::indexed(symmetry_type symm, const exvector & v, bool discardable) : inherited(v, discardable), symmetry(symm)
+indexed::indexed(const symmetry & symm, const exvector & v, bool discardable) : inherited(v, discardable), symtree(symm)
 {
-       debugmsg("indexed constructor from symmetry,exvector", LOGLEVEL_CONSTRUCT);
        tinfo_key = TINFO_indexed;
-       assert_all_indices_of_type_idx();
 }
 
-indexed::indexed(symmetry_type symm, exvector * vp) : inherited(vp), symmetry(symm)
+indexed::indexed(const symmetry & symm, exvector * vp) : inherited(vp), symtree(symm)
 {
-       debugmsg("indexed constructor from symmetry,exvector *", LOGLEVEL_CONSTRUCT);
        tinfo_key = TINFO_indexed;
-       assert_all_indices_of_type_idx();
 }
 
 //////////
@@ -160,40 +144,47 @@ indexed::indexed(symmetry_type symm, exvector * vp) : inherited(vp), symmetry(sy
 
 indexed::indexed(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 {
-       debugmsg("indexed constructor from archive_node", LOGLEVEL_CONSTRUCT);
-       unsigned int symm;
-       if (!(n.find_unsigned("symmetry", symm)))
-               throw (std::runtime_error("unknown indexed symmetry type in archive"));
+       if (!n.find_ex("symmetry", symtree, sym_lst)) {
+               // GiNaC versions <= 0.9.0 had an unsigned "symmetry" property
+               unsigned symm = 0;
+               n.find_unsigned("symmetry", symm);
+               switch (symm) {
+                       case 1:
+                               symtree = sy_symm();
+                               break;
+                       case 2:
+                               symtree = sy_anti();
+                               break;
+                       default:
+                               symtree = sy_none();
+                               break;
+               }
+               const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
+       }
 }
 
 void indexed::archive(archive_node &n) const
 {
        inherited::archive(n);
-       n.add_unsigned("symmetry", symmetry);
+       n.add_ex("symmetry", symtree);
 }
 
 DEFAULT_UNARCHIVE(indexed)
 
 //////////
-// functions overriding virtual functions from bases classes
+// functions overriding virtual functions from base classes
 //////////
 
 void indexed::print(const print_context & c, unsigned level) const
 {
-       debugmsg("indexed print", LOGLEVEL_PRINT);
        GINAC_ASSERT(seq.size() > 0);
 
        if (is_of_type(c, print_tree)) {
 
                c.s << std::string(level, ' ') << class_name()
                    << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
-                   << ", " << seq.size()-1 << " indices";
-               switch (symmetry) {
-                       case symmetric: c.s << ", symmetric"; break;
-                       case antisymmetric: c.s << ", antisymmetric"; break;
-                       default: break;
-               }
-               c.s << std::endl;
+                   << ", " << seq.size()-1 << " indices"
+                   << ", symmetry=" << symtree << std::endl;
                unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
                seq[0].print(c, level + delta_indent);
                printindices(c, level + delta_indent);
@@ -225,9 +216,9 @@ bool indexed::info(unsigned inf) const
        return inherited::info(inf);
 }
 
-struct idx_is_not : public binary_function<ex, unsigned, bool> {
+struct idx_is_not : public std::binary_function<ex, unsigned, bool> {
        bool operator() (const ex & e, unsigned inf) const {
-               return !(ex_to_idx(e).get_value().info(inf));
+               return !(ex_to<idx>(e).get_value().info(inf));
        }
 };
 
@@ -243,118 +234,55 @@ bool indexed::all_index_values_are(unsigned inf) const
 
 int indexed::compare_same_type(const basic & other) const
 {
-       GINAC_ASSERT(is_of_type(other, indexed));
+       GINAC_ASSERT(is_a<indexed>(other));
        return inherited::compare_same_type(other);
 }
 
-// The main difference between sort_index_vector() and canonicalize_indices()
-// is that the latter takes the symmetry of the object into account. Once we
-// implement mixed symmetries, canonicalize_indices() will only be able to
-// reorder index pairs with known symmetry properties, while sort_index_vector()
-// always sorts the whole vector.
-
-/** Bring a vector of indices into a canonic order. This operation only makes
- *  sense if the object carrying these indices is either symmetric or totally
- *  antisymmetric with respect to the indices.
- *
- *  @param itbegin Start of index vector
- *  @param itend End of index vector
- *  @param antisymm Whether the object is antisymmetric
- *  @return the sign introduced by the reordering of the indices if the object
- *          is antisymmetric (or 0 if two equal indices are encountered). For
- *          symmetric objects, this is always +1. If the index vector was
- *          already in a canonic order this function returns INT_MAX. */
-static int canonicalize_indices(exvector::iterator itbegin, exvector::iterator itend, bool antisymm)
-{
-       bool something_changed = false;
-       int sig = 1;
-
-       // Simple bubble sort algorithm should be sufficient for the small
-       // number of indices expected
-       exvector::iterator it1 = itbegin, next_to_last_idx = itend - 1;
-       while (it1 != next_to_last_idx) {
-               exvector::iterator it2 = it1 + 1;
-               while (it2 != itend) {
-                       int cmpval = it1->compare(*it2);
-                       if (cmpval == 1) {
-                               it1->swap(*it2);
-                               something_changed = true;
-                               if (antisymm)
-                                       sig = -sig;
-                       } else if (cmpval == 0 && antisymm) {
-                               something_changed = true;
-                               sig = 0;
-                       }
-                       it2++;
-               }
-               it1++;
-       }
-
-       return something_changed ? sig : INT_MAX;
-}
-
 ex indexed::eval(int level) const
 {
        // First evaluate children, then we will end up here again
        if (level > 1)
-               return indexed(symmetry, evalchildren(level));
+               return indexed(ex_to<symmetry>(symtree), evalchildren(level));
 
        const ex &base = seq[0];
 
        // If the base object is 0, the whole object is 0
        if (base.is_zero())
-               return _ex0();
+               return _ex0;
 
        // If the base object is a product, pull out the numeric factor
        if (is_ex_exactly_of_type(base, mul) && is_ex_exactly_of_type(base.op(base.nops() - 1), numeric)) {
-               exvector v = seq;
-               ex f = ex_to_numeric(base.op(base.nops() - 1));
+               exvector v(seq);
+               ex f = ex_to<numeric>(base.op(base.nops() - 1));
                v[0] = seq[0] / f;
                return f * thisexprseq(v);
        }
 
        // Canonicalize indices according to the symmetry properties
-       if (seq.size() > 2 && (symmetry != unknown && symmetry != mixed)) {
+       if (seq.size() > 2) {
                exvector v = seq;
-               int sig = canonicalize_indices(v.begin() + 1, v.end(), symmetry == antisymmetric);
+               GINAC_ASSERT(is_exactly_a<symmetry>(symtree));
+               int sig = canonicalize(v.begin() + 1, ex_to<symmetry>(symtree));
                if (sig != INT_MAX) {
                        // Something has changed while sorting indices, more evaluations later
                        if (sig == 0)
-                               return _ex0();
+                               return _ex0;
                        return ex(sig) * thisexprseq(v);
                }
        }
 
        // Let the class of the base object perform additional evaluations
-       return base.bp->eval_indexed(*this);
-}
-
-int indexed::degree(const ex & s) const
-{
-       return is_equal(*s.bp) ? 1 : 0;
-}
-
-int indexed::ldegree(const ex & s) const
-{
-       return is_equal(*s.bp) ? 1 : 0;
-}
-
-ex indexed::coeff(const ex & s, int n) const
-{
-       if (is_equal(*s.bp))
-               return n==1 ? _ex1() : _ex0();
-       else
-               return n==0 ? ex(*this) : _ex0();
+       return ex_to<basic>(base).eval_indexed(*this);
 }
 
 ex indexed::thisexprseq(const exvector & v) const
 {
-       return indexed(symmetry, v);
+       return indexed(ex_to<symmetry>(symtree), v);
 }
 
 ex indexed::thisexprseq(exvector * vp) const
 {
-       return indexed(symmetry, vp);
+       return indexed(ex_to<symmetry>(symtree), vp);
 }
 
 ex indexed::expand(unsigned options) const
@@ -365,7 +293,7 @@ ex indexed::expand(unsigned options) const
 
                // expand_indexed expands (a+b).i -> a.i + b.i
                const ex & base = seq[0];
-               ex sum = _ex0();
+               ex sum = _ex0;
                for (unsigned i=0; i<base.nops(); i++) {
                        exvector s = seq;
                        s[0] = base.op(i);
@@ -400,10 +328,11 @@ void indexed::printindices(const print_context & c, unsigned level) const
                        bool covariant = true;
 
                        while (it != itend) {
-                               bool cur_covariant = (is_ex_of_type(*it, varidx) ? ex_to_varidx(*it).is_covariant() : true);
-                               if (first || cur_covariant != covariant) {
+                               bool cur_covariant = (is_ex_of_type(*it, varidx) ? ex_to<varidx>(*it).is_covariant() : true);
+                               if (first || cur_covariant != covariant) { // Variance changed
+                                       // The empty {} prevents indices from ending up on top of each other
                                        if (!first)
-                                               c.s << "}";
+                                               c.s << "}{}";
                                        covariant = cur_covariant;
                                        if (covariant)
                                                c.s << "_{";
@@ -428,10 +357,10 @@ void indexed::printindices(const print_context & c, unsigned level) const
        }
 }
 
-/** Check whether all indices are of class idx. This function is used
- *  internally to make sure that all constructed indexed objects really
- *  carry indices and not some other classes. */
-void indexed::assert_all_indices_of_type_idx(void) const
+/** Check whether all indices are of class idx and validate the symmetry
+ *  tree. This function is used internally to make sure that all constructed
+ *  indexed objects really carry indices and not some other classes. */
+void indexed::validate(void) const
 {
        GINAC_ASSERT(seq.size() > 0);
        exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
@@ -440,6 +369,20 @@ void indexed::assert_all_indices_of_type_idx(void) const
                        throw(std::invalid_argument("indices of indexed object must be of type idx"));
                it++;
        }
+
+       if (!symtree.is_zero()) {
+               if (!is_ex_exactly_of_type(symtree, symmetry))
+                       throw(std::invalid_argument("symmetry of indexed object must be of type symmetry"));
+               const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
+       }
+}
+
+/** Implementation of ex::diff() for an indexed object always returns 0.
+ *
+ *  @see ex::diff */
+ex indexed::derivative(const symbol & s) const
+{
+       return _ex0;
 }
 
 //////////
@@ -550,7 +493,7 @@ exvector power::get_free_indices(void) const
 
 /** Rename dummy indices in an expression.
  *
- *  @param e Expression to be worked on
+ *  @param e Expression to work on
  *  @param local_dummy_indices The set of dummy indices that appear in the
  *    expression "e"
  *  @param global_dummy_indices The set of dummy indices that have appeared
@@ -558,8 +501,8 @@ exvector power::get_free_indices(void) const
  *    by the function */
 static ex rename_dummy_indices(const ex & e, exvector & global_dummy_indices, exvector & local_dummy_indices)
 {
-       int global_size = global_dummy_indices.size(),
-           local_size = local_dummy_indices.size();
+       unsigned global_size = global_dummy_indices.size(),
+                local_size = local_dummy_indices.size();
 
        // Any local dummy indices at all?
        if (local_size == 0)
@@ -569,6 +512,7 @@ static ex rename_dummy_indices(const ex & e, exvector & global_dummy_indices, ex
 
                // More local indices than we encountered before, add the new ones
                // to the global set
+               int old_global_size = global_size;
                int remaining = local_size - global_size;
                exvector::const_iterator it = local_dummy_indices.begin(), itend = local_dummy_indices.end();
                while (it != itend && remaining > 0) {
@@ -579,27 +523,111 @@ static ex rename_dummy_indices(const ex & e, exvector & global_dummy_indices, ex
                        }
                        it++;
                }
-       }
 
-       // Replace index symbols in expression
+               // If this is the first set of local indices, do nothing
+               if (old_global_size == 0)
+                       return e;
+       }
        GINAC_ASSERT(local_size <= global_size);
-       bool all_equal = true;
-       lst local_syms, global_syms;
-       for (unsigned i=0; i<local_size; i++) {
-               ex loc_sym = local_dummy_indices[i].op(0);
-               ex glob_sym = global_dummy_indices[i].op(0);
-               if (!loc_sym.is_equal(glob_sym)) {
-                       all_equal = false;
-                       local_syms.append(loc_sym);
-                       global_syms.append(glob_sym);
+
+       // Construct lists of index symbols
+       exlist local_syms, global_syms;
+       for (unsigned i=0; i<local_size; i++)
+               local_syms.push_back(local_dummy_indices[i].op(0));
+       shaker_sort(local_syms.begin(), local_syms.end(), ex_is_less(), ex_swap());
+       for (unsigned i=0; i<global_size; i++)
+               global_syms.push_back(global_dummy_indices[i].op(0));
+       shaker_sort(global_syms.begin(), global_syms.end(), ex_is_less(), ex_swap());
+
+       // Remove common indices
+       exlist local_uniq, global_uniq;
+       set_difference(local_syms.begin(), local_syms.end(), global_syms.begin(), global_syms.end(), std::back_insert_iterator<exlist>(local_uniq), ex_is_less());
+       set_difference(global_syms.begin(), global_syms.end(), local_syms.begin(), local_syms.end(), std::back_insert_iterator<exlist>(global_uniq), ex_is_less());
+
+       // Replace remaining non-common local index symbols by global ones
+       if (local_uniq.empty())
+               return e;
+       else {
+               while (global_uniq.size() > local_uniq.size())
+                       global_uniq.pop_back();
+               return e.subs(lst(local_uniq), lst(global_uniq));
+       }
+}
+
+/** Given a set of indices, extract those of class varidx. */
+static void find_variant_indices(const exvector & v, exvector & variant_indices)
+{
+       exvector::const_iterator it1, itend;
+       for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
+               if (is_exactly_a<varidx>(*it1))
+                       variant_indices.push_back(*it1);
+       }
+}
+
+/** Raise/lower dummy indices in a single indexed objects to canonicalize their
+ *  variance.
+ *
+ *  @param e Object to work on
+ *  @param variant_dummy_indices The set of indices that might need repositioning (will be changed by this function)
+ *  @param moved_indices The set of indices that have been repositioned (will be changed by this function)
+ *  @return true if 'e' was changed */
+bool reposition_dummy_indices(ex & e, exvector & variant_dummy_indices, exvector & moved_indices)
+{
+       bool something_changed = false;
+
+       // If a dummy index is encountered for the first time in the
+       // product, pull it up, otherwise, pull it down
+       exvector::const_iterator it2, it2start, it2end;
+       for (it2start = ex_to<indexed>(e).seq.begin(), it2end = ex_to<indexed>(e).seq.end(), it2 = it2start + 1; it2 != it2end; ++it2) {
+               if (!is_exactly_a<varidx>(*it2))
+                       continue;
+
+               exvector::iterator vit, vitend;
+               for (vit = variant_dummy_indices.begin(), vitend = variant_dummy_indices.end(); vit != vitend; ++vit) {
+                       if (it2->op(0).is_equal(vit->op(0))) {
+                               if (ex_to<varidx>(*it2).is_covariant()) {
+                                       e = e.subs(lst(
+                                               *it2 == ex_to<varidx>(*it2).toggle_variance(),
+                                               ex_to<varidx>(*it2).toggle_variance() == *it2
+                                       ));
+                                       something_changed = true;
+                                       it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
+                                       it2start = ex_to<indexed>(e).seq.begin();
+                                       it2end = ex_to<indexed>(e).seq.end();
+                               }
+                               moved_indices.push_back(*vit);
+                               variant_dummy_indices.erase(vit);
+                               goto next_index;
+                       }
+               }
+
+               for (vit = moved_indices.begin(), vitend = moved_indices.end(); vit != vitend; ++vit) {
+                       if (it2->op(0).is_equal(vit->op(0))) {
+                               if (ex_to<varidx>(*it2).is_contravariant()) {
+                                       e = e.subs(*it2 == ex_to<varidx>(*it2).toggle_variance());
+                                       something_changed = true;
+                                       it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
+                                       it2start = ex_to<indexed>(e).seq.begin();
+                                       it2end = ex_to<indexed>(e).seq.end();
+                               }
+                               goto next_index;
+                       }
                }
+
+next_index: ;
        }
-       if (all_equal)
-               return e;
-       else
-               return e.subs(local_syms, global_syms);
+
+       return something_changed;
 }
 
+/* Ordering that only compares the base expressions of indexed objects. */
+struct ex_base_is_less : public std::binary_function<ex, ex, bool> {
+       bool operator() (const ex &lh, const ex &rh) const
+       {
+               return (is_a<indexed>(lh) ? lh.op(0) : lh).compare(is_a<indexed>(rh) ? rh.op(0) : rh) < 0;
+       }
+};
+
 /** Simplify product of indexed expressions (commutative, noncommutative and
  *  simple squares), return list of free indices. */
 ex simplify_indexed_product(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
@@ -614,19 +642,19 @@ ex simplify_indexed_product(const ex & e, exvector & free_indices, exvector & du
 
        if (is_ex_exactly_of_type(e, power)) {
                // We only get called for simple squares, split a^2 -> a*a
-               GINAC_ASSERT(e.op(1).is_equal(_ex2()));
+               GINAC_ASSERT(e.op(1).is_equal(_ex2));
                v.push_back(e.op(0));
                v.push_back(e.op(0));
        } else {
-               for (int i=0; i<e.nops(); i++) {
+               for (unsigned i=0; i<e.nops(); i++) {
                        ex f = e.op(i);
-                       if (is_ex_exactly_of_type(f, power) && f.op(1).is_equal(_ex2())) {
+                       if (is_ex_exactly_of_type(f, power) && f.op(1).is_equal(_ex2)) {
                                v.push_back(f.op(0));
                    v.push_back(f.op(0));
                        } else if (is_ex_exactly_of_type(f, ncmul)) {
                                // Noncommutative factor found, split it as well
                                non_commutative = true; // everything becomes noncommutative, ncmul will sort out the commutative factors later
-                               for (int j=0; j<f.nops(); j++)
+                               for (unsigned j=0; j<f.nops(); j++)
                                        v.push_back(f.op(j));
                        } else
                                v.push_back(f);
@@ -648,7 +676,7 @@ try_again:
                // Indexed factor found, get free indices and look for contraction
                // candidates
                exvector free1, dummy1;
-               find_free_and_dummy(ex_to_indexed(*it1).seq.begin() + 1, ex_to_indexed(*it1).seq.end(), free1, dummy1);
+               find_free_and_dummy(ex_to<indexed>(*it1).seq.begin() + 1, ex_to<indexed>(*it1).seq.end(), free1, dummy1);
 
                exvector::iterator it2;
                for (it2 = it1 + 1; it2 != itend; it2++) {
@@ -661,42 +689,33 @@ try_again:
                        // Find free indices of second factor and merge them with free
                        // indices of first factor
                        exvector un;
-                       find_free_and_dummy(ex_to_indexed(*it2).seq.begin() + 1, ex_to_indexed(*it2).seq.end(), un, dummy1);
+                       find_free_and_dummy(ex_to<indexed>(*it2).seq.begin() + 1, ex_to<indexed>(*it2).seq.end(), un, dummy1);
                        un.insert(un.end(), free1.begin(), free1.end());
 
                        // Check whether the two factors share dummy indices
                        exvector free, dummy;
                        find_free_and_dummy(un, free, dummy);
-                       if (dummy.size() == 0)
+                       unsigned num_dummies = dummy.size();
+                       if (num_dummies == 0)
                                continue;
 
                        // At least one dummy index, is it a defined scalar product?
                        bool contracted = false;
-                       if (free.size() == 0) {
+                       if (free.empty()) {
                                if (sp.is_defined(*it1, *it2)) {
                                        *it1 = sp.evaluate(*it1, *it2);
-                                       *it2 = _ex1();
+                                       *it2 = _ex1;
                                        goto contraction_done;
                                }
                        }
 
-                       // Contraction of symmetric with antisymmetric object is zero
-                       if ((ex_to_indexed(*it1).symmetry == indexed::symmetric &&
-                            ex_to_indexed(*it2).symmetry == indexed::antisymmetric
-                         || ex_to_indexed(*it1).symmetry == indexed::antisymmetric &&
-                            ex_to_indexed(*it2).symmetry == indexed::symmetric)
-                        && dummy.size() > 1) {
-                               free_indices.clear();
-                               return _ex0();
-                       }
-
                        // Try to contract the first one with the second one
-                       contracted = it1->op(0).bp->contract_with(it1, it2, v);
+                       contracted = ex_to<basic>(it1->op(0)).contract_with(it1, it2, v);
                        if (!contracted) {
 
                                // That didn't work; maybe the second object knows how to
                                // contract itself with the first one
-                               contracted = it2->op(0).bp->contract_with(it2, it1, v);
+                               contracted = ex_to<basic>(it2->op(0)).contract_with(it2, it1, v);
                        }
                        if (contracted) {
 contraction_done:
@@ -710,7 +729,7 @@ contraction_done:
                                        // Non-commutative products are always re-expanded to give
                                        // simplify_ncmul() the chance to re-order and canonicalize
                                        // the product
-                                       ex r = (non_commutative ? ex(ncmul(v)) : ex(mul(v)));
+                                       ex r = (non_commutative ? ex(ncmul(v, true)) : ex(mul(v)));
                                        return simplify_indexed(r, free_indices, dummy_indices, sp);
                                }
 
@@ -726,35 +745,70 @@ contraction_done:
        // Find free indices (concatenate them all and call find_free_and_dummy())
        // and all dummy indices that appear
        exvector un, individual_dummy_indices;
-       it1 = v.begin(); itend = v.end();
-       while (it1 != itend) {
+       for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
                exvector free_indices_of_factor;
                if (is_ex_of_type(*it1, indexed)) {
                        exvector dummy_indices_of_factor;
-                       find_free_and_dummy(ex_to_indexed(*it1).seq.begin() + 1, ex_to_indexed(*it1).seq.end(), free_indices_of_factor, dummy_indices_of_factor);
+                       find_free_and_dummy(ex_to<indexed>(*it1).seq.begin() + 1, ex_to<indexed>(*it1).seq.end(), free_indices_of_factor, dummy_indices_of_factor);
                        individual_dummy_indices.insert(individual_dummy_indices.end(), dummy_indices_of_factor.begin(), dummy_indices_of_factor.end());
                } else
                        free_indices_of_factor = it1->get_free_indices();
                un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
-               it1++;
        }
        exvector local_dummy_indices;
        find_free_and_dummy(un, free_indices, local_dummy_indices);
        local_dummy_indices.insert(local_dummy_indices.end(), individual_dummy_indices.begin(), individual_dummy_indices.end());
 
+       // Filter out the dummy indices with variance
+       exvector variant_dummy_indices;
+       find_variant_indices(local_dummy_indices, variant_dummy_indices);
+
+       // Any indices with variance present at all?
+       if (!variant_dummy_indices.empty()) {
+
+               // Yes, bring the product into a canonical order that only depends on
+               // the base expressions of indexed objects
+               if (!non_commutative)
+                       std::sort(v.begin(), v.end(), ex_base_is_less());
+
+               exvector moved_indices;
+
+               // Iterate over all indexed objects in the product
+               for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
+                       if (!is_ex_of_type(*it1, indexed))
+                               continue;
+
+                       if (reposition_dummy_indices(*it1, variant_dummy_indices, moved_indices))
+                               something_changed = true;
+               }
+       }
+
        ex r;
        if (something_changed)
-               r = non_commutative ? ex(ncmul(v)) : ex(mul(v));
+               r = non_commutative ? ex(ncmul(v, true)) : ex(mul(v));
        else
                r = e;
 
+       // The result should be symmetric with respect to exchange of dummy
+       // indices, so if the symmetrization vanishes, the whole expression is
+       // zero. This detects things like eps.i.j.k * p.j * p.k = 0.
+       if (local_dummy_indices.size() >= 2) {
+               lst dummy_syms;
+               for (int i=0; i<local_dummy_indices.size(); i++)
+                       dummy_syms.append(local_dummy_indices[i].op(0));
+               if (r.symmetrize(dummy_syms).is_zero()) {
+                       free_indices.clear();
+                       return _ex0;
+               }
+       }
+
        // Dummy index renaming
        r = rename_dummy_indices(r, dummy_indices, local_dummy_indices);
 
        // Product of indexed object with a scalar?
        if (is_ex_exactly_of_type(r, mul) && r.nops() == 2
         && is_ex_exactly_of_type(r.op(1), numeric) && is_ex_of_type(r.op(0), indexed))
-               return r.op(0).op(0).bp->scalar_mul_indexed(r.op(0), ex_to_numeric(r.op(1)));
+               return ex_to<basic>(r.op(0).op(0)).scalar_mul_indexed(r.op(0), ex_to<numeric>(r.op(1)));
        else
                return r;
 }
@@ -766,11 +820,27 @@ ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indi
        ex e_expanded = e.expand();
 
        // Simplification of single indexed object: just find the free indices
-       // and perform dummy index renaming
+       // and perform dummy index renaming/repositioning
        if (is_ex_of_type(e_expanded, indexed)) {
-               const indexed &i = ex_to_indexed(e_expanded);
+
+               // Find the dummy indices
+               const indexed &i = ex_to<indexed>(e_expanded);
                exvector local_dummy_indices;
                find_free_and_dummy(i.seq.begin() + 1, i.seq.end(), free_indices, local_dummy_indices);
+
+               // Filter out the dummy indices with variance
+               exvector variant_dummy_indices;
+               find_variant_indices(local_dummy_indices, variant_dummy_indices);
+
+               // Any indices with variance present at all?
+               if (!variant_dummy_indices.empty()) {
+
+                       // Yes, reposition them
+                       exvector moved_indices;
+                       reposition_dummy_indices(e_expanded, variant_dummy_indices, moved_indices);
+               }
+
+               // Rename the dummy indices
                return rename_dummy_indices(e_expanded, dummy_indices, local_dummy_indices);
        }
 
@@ -778,7 +848,7 @@ ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indi
        // free indices in each term
        if (is_ex_exactly_of_type(e_expanded, add)) {
                bool first = true;
-               ex sum = _ex0();
+               ex sum = _ex0;
                free_indices.clear();
 
                for (unsigned i=0; i<e_expanded.nops(); i++) {
@@ -793,7 +863,7 @@ ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indi
                                        if (!indices_consistent(free_indices, free_indices_of_term))
                                                throw (std::runtime_error("simplify_indexed: inconsistent indices in sum"));
                                        if (is_ex_of_type(sum, indexed) && is_ex_of_type(term, indexed))
-                                               sum = sum.op(0).bp->add_indexed(sum, term);
+                                               sum = ex_to<basic>(sum.op(0)).add_indexed(sum, term);
                                        else
                                                sum += term;
                                }
@@ -806,7 +876,7 @@ ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indi
        // Simplification of products
        if (is_ex_exactly_of_type(e_expanded, mul)
         || is_ex_exactly_of_type(e_expanded, ncmul)
-        || (is_ex_exactly_of_type(e_expanded, power) && is_ex_of_type(e_expanded.op(0), indexed) && e_expanded.op(1).is_equal(_ex2())))
+        || (is_ex_exactly_of_type(e_expanded, power) && is_ex_of_type(e_expanded.op(0), indexed) && e_expanded.op(1).is_equal(_ex2)))
                return simplify_indexed_product(e_expanded, free_indices, dummy_indices, sp);
 
        // Cannot do anything
@@ -814,17 +884,47 @@ ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indi
        return e_expanded;
 }
 
-ex simplify_indexed(const ex & e)
+/** Simplify/canonicalize expression containing indexed objects. This
+ *  performs contraction of dummy indices where possible and checks whether
+ *  the free indices in sums are consistent.
+ *
+ *  @return simplified expression */
+ex ex::simplify_indexed(void) const
 {
        exvector free_indices, dummy_indices;
        scalar_products sp;
-       return simplify_indexed(e, free_indices, dummy_indices, sp);
+       return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
 }
 
-ex simplify_indexed(const ex & e, const scalar_products & sp)
+/** Simplify/canonicalize expression containing indexed objects. This
+ *  performs contraction of dummy indices where possible, checks whether
+ *  the free indices in sums are consistent, and automatically replaces
+ *  scalar products by known values if desired.
+ *
+ *  @param sp Scalar products to be replaced automatically
+ *  @return simplified expression */
+ex ex::simplify_indexed(const scalar_products & sp) const
 {
        exvector free_indices, dummy_indices;
-       return simplify_indexed(e, free_indices, dummy_indices, sp);
+       return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
+}
+
+/** Symmetrize expression over its free indices. */
+ex ex::symmetrize(void) const
+{
+       return GiNaC::symmetrize(*this, get_free_indices());
+}
+
+/** Antisymmetrize expression over its free indices. */
+ex ex::antisymmetrize(void) const
+{
+       return GiNaC::antisymmetrize(*this, get_free_indices());
+}
+
+/** Symmetrize expression by cyclic permutation over its free indices. */
+ex ex::symmetrize_cyclic(void) const
+{
+       return GiNaC::symmetrize_cyclic(*this, get_free_indices());
 }
 
 //////////
@@ -869,10 +969,12 @@ ex scalar_products::evaluate(const ex & v1, const ex & v2) const
 void scalar_products::debugprint(void) const
 {
        std::cerr << "map size=" << spm.size() << std::endl;
-       for (spmap::const_iterator cit=spm.begin(); cit!=spm.end(); ++cit) {
-               const spmapkey & k = cit->first;
+       spmap::const_iterator i = spm.begin(), end = spm.end();
+       while (i != end) {
+               const spmapkey & k = i->first;
                std::cerr << "item key=(" << k.first << "," << k.second;
-               std::cerr << "), value=" << cit->second << std::endl;
+               std::cerr << "), value=" << i->second << std::endl;
+               ++i;
        }
 }