fixed a bug in the raising/lowering of dummy indices and extended it to work
[ginac.git] / ginac / indexed.cpp
index 6e36eb305fc157d4d109f3464ce19be919dc572d..dcd6d9510d6d0ac688802373f30f7d93e0f07362 100644 (file)
@@ -1,9 +1,9 @@
 /** @file indexed.cpp
  *
- *  Implementation of GiNaC's index carrying objects. */
+ *  Implementation of GiNaC's indexed expressions. */
 
 /*
- *  GiNaC Copyright (C) 1999 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
  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  */
 
-#include <string>
+#include <iostream>
+#include <stdexcept>
 
 #include "indexed.h"
-#include "ex.h"
 #include "idx.h"
-#include "debugmsg.h"
+#include "add.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"
 
 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
 //////////
 
-// public
+indexed::indexed() : symtree(sy_none())
+{
+       tinfo_key = TINFO_indexed;
+}
 
-indexed::indexed()
+void indexed::copy(const indexed & other)
 {
-    debugmsg("indexed default constructor",LOGLEVEL_CONSTRUCT);
-    tinfo_key=TINFO_indexed;
+       inherited::copy(other);
+       symtree = other.symtree;
 }
 
-indexed::~indexed()
+DEFAULT_DESTROY(indexed)
+
+//////////
+// other constructors
+//////////
+
+indexed::indexed(const ex & b) : inherited(b), symtree(sy_none())
 {
-    debugmsg("indexed destructor",LOGLEVEL_DESTRUCT);
-    destroy(0);
+       tinfo_key = TINFO_indexed;
+       validate();
 }
 
-indexed::indexed(indexed const & other)
+indexed::indexed(const ex & b, const ex & i1) : inherited(b, i1), symtree(sy_none())
 {
-    debugmsg("indexed copy constructor",LOGLEVEL_CONSTRUCT);
-    copy (other);
+       tinfo_key = TINFO_indexed;
+       validate();
 }
 
-indexed const & indexed::operator=(indexed const & other)
+indexed::indexed(const ex & b, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(sy_none())
 {
-    debugmsg("indexed operator=",LOGLEVEL_ASSIGNMENT);
-    if (this != &other) {
-        destroy(1);
-        copy(other);
-    }
-    return *this;
+       tinfo_key = TINFO_indexed;
+       validate();
 }
 
-// protected
+indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(sy_none())
+{
+       tinfo_key = TINFO_indexed;
+       validate();
+}
 
-void indexed::copy(indexed const & other)
+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())
 {
-    exprseq::copy(other);
+       tinfo_key = TINFO_indexed;
+       validate();
 }
 
-void indexed::destroy(bool call_parent)
+indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(symm)
 {
-    if (call_parent) {
-        exprseq::destroy(call_parent);
-    }
+       tinfo_key = TINFO_indexed;
+       validate();
+}
+
+indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(symm)
+{
+       tinfo_key = TINFO_indexed;
+       validate();
+}
+
+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)
+{
+       tinfo_key = TINFO_indexed;
+       validate();
+}
+
+indexed::indexed(const ex & b, const exvector & v) : inherited(b), symtree(sy_none())
+{
+       seq.insert(seq.end(), v.begin(), v.end());
+       tinfo_key = TINFO_indexed;
+       validate();
+}
+
+indexed::indexed(const ex & b, const symmetry & symm, const exvector & v) : inherited(b), symtree(symm)
+{
+       seq.insert(seq.end(), v.begin(), v.end());
+       tinfo_key = TINFO_indexed;
+       validate();
+}
+
+indexed::indexed(const symmetry & symm, const exprseq & es) : inherited(es), symtree(symm)
+{
+       tinfo_key = TINFO_indexed;
+}
+
+indexed::indexed(const symmetry & symm, const exvector & v, bool discardable) : inherited(v, discardable), symtree(symm)
+{
+       tinfo_key = TINFO_indexed;
+}
+
+indexed::indexed(const symmetry & symm, exvector * vp) : inherited(vp), symtree(symm)
+{
+       tinfo_key = TINFO_indexed;
 }
 
 //////////
-// other constructors
+// archiving
+//////////
+
+indexed::indexed(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
+{
+       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_ex("symmetry", symtree);
+}
+
+DEFAULT_UNARCHIVE(indexed)
+
+//////////
+// functions overriding virtual functions from base classes
 //////////
 
-// public
+void indexed::print(const print_context & c, unsigned level) const
+{
+       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"
+                   << ", 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);
+
+       } else {
+
+               bool is_tex = is_of_type(c, print_latex);
+               const ex & base = seq[0];
+               bool need_parens = is_ex_exactly_of_type(base, add) || is_ex_exactly_of_type(base, mul)
+                               || is_ex_exactly_of_type(base, ncmul) || is_ex_exactly_of_type(base, power)
+                               || is_ex_of_type(base, indexed);
+               if (is_tex)
+                       c.s << "{";
+               if (need_parens)
+                       c.s << "(";
+               base.print(c);
+               if (need_parens)
+                       c.s << ")";
+               if (is_tex)
+                       c.s << "}";
+               printindices(c, level);
+       }
+}
+
+bool indexed::info(unsigned inf) const
+{
+       if (inf == info_flags::indexed) return true;
+       if (inf == info_flags::has_indices) return seq.size() > 1;
+       return inherited::info(inf);
+}
+
+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));
+       }
+};
+
+bool indexed::all_index_values_are(unsigned inf) const
+{
+       // No indices? Then no property can be fulfilled
+       if (seq.size() < 2)
+               return false;
+
+       // Check all indices
+       return find_if(seq.begin() + 1, seq.end(), bind2nd(idx_is_not(), inf)) == seq.end();
+}
 
-indexed::indexed(ex const & i1) : exprseq(i1)
+int indexed::compare_same_type(const basic & other) const
 {
-    debugmsg("indexed constructor from ex",LOGLEVEL_CONSTRUCT);
-    tinfo_key=TINFO_indexed;
-    GINAC_ASSERT(all_of_type_idx());
+       GINAC_ASSERT(is_a<indexed>(other));
+       return inherited::compare_same_type(other);
 }
 
-indexed::indexed(ex const & i1, ex const & i2) : exprseq(i1,i2)
+ex indexed::eval(int level) const
 {
-    debugmsg("indexed constructor from ex,ex",LOGLEVEL_CONSTRUCT);
-    tinfo_key=TINFO_indexed;
-    GINAC_ASSERT(all_of_type_idx());
+       // First evaluate children, then we will end up here again
+       if (level > 1)
+               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;
+
+       // 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));
+               v[0] = seq[0] / f;
+               return f * thisexprseq(v);
+       }
+
+       // Canonicalize indices according to the symmetry properties
+       if (seq.size() > 2) {
+               exvector v = seq;
+               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 ex(sig) * thisexprseq(v);
+               }
+       }
+
+       // Let the class of the base object perform additional evaluations
+       return ex_to<basic>(base).eval_indexed(*this);
 }
 
-indexed::indexed(ex const & i1, ex const & i2, ex const & i3)
-    : exprseq(i1,i2,i3)
+ex indexed::thisexprseq(const exvector & v) const
 {
-    debugmsg("indexed constructor from ex,ex,ex",LOGLEVEL_CONSTRUCT);
-    tinfo_key=TINFO_indexed;
-    GINAC_ASSERT(all_of_type_idx());
+       return indexed(ex_to<symmetry>(symtree), v);
 }
 
-indexed::indexed(exvector const & iv) : exprseq(iv)
+ex indexed::thisexprseq(exvector * vp) const
 {
-    debugmsg("indexed constructor from exvector",LOGLEVEL_CONSTRUCT);
-    tinfo_key=TINFO_indexed;
-    GINAC_ASSERT(all_of_type_idx());
+       return indexed(ex_to<symmetry>(symtree), vp);
 }
 
-indexed::indexed(exvector * ivp) : exprseq(ivp)
+ex indexed::expand(unsigned options) const
 {
-    debugmsg("indexed constructor from exvector *",LOGLEVEL_CONSTRUCT);
-    tinfo_key=TINFO_indexed;
-    GINAC_ASSERT(all_of_type_idx());
+       GINAC_ASSERT(seq.size() > 0);
+
+       if ((options & expand_options::expand_indexed) && is_ex_exactly_of_type(seq[0], add)) {
+
+               // expand_indexed expands (a+b).i -> a.i + b.i
+               const ex & base = seq[0];
+               ex sum = _ex0;
+               for (unsigned i=0; i<base.nops(); i++) {
+                       exvector s = seq;
+                       s[0] = base.op(i);
+                       sum += thisexprseq(s).expand();
+               }
+               return sum;
+
+       } else
+               return inherited::expand(options);
 }
 
 //////////
-// functions overriding virtual functions from bases classes
+// virtual functions which can be overridden by derived classes
 //////////
 
-// public
+// none
+
+//////////
+// non-virtual functions in this class
+//////////
 
-basic * indexed::duplicate() const
+void indexed::printindices(const print_context & c, unsigned level) const
 {
-    debugmsg("indexed duplicate",LOGLEVEL_DUPLICATE);
-    return new indexed(*this);
+       if (seq.size() > 1) {
+
+               exvector::const_iterator it=seq.begin() + 1, itend = seq.end();
+
+               if (is_of_type(c, print_latex)) {
+
+                       // TeX output: group by variance
+                       bool first = true;
+                       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) { // Variance changed
+                                       // The empty {} prevents indices from ending up on top of each other
+                                       if (!first)
+                                               c.s << "}{}";
+                                       covariant = cur_covariant;
+                                       if (covariant)
+                                               c.s << "_{";
+                                       else
+                                               c.s << "^{";
+                               }
+                               it->print(c, level);
+                               c.s << " ";
+                               first = false;
+                               it++;
+                       }
+                       c.s << "}";
+
+               } else {
+
+                       // Ordinary output
+                       while (it != itend) {
+                               it->print(c, level);
+                               it++;
+                       }
+               }
+       }
 }
 
-void indexed::printraw(ostream & os) 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
 {
-    debugmsg("indexed printraw",LOGLEVEL_PRINT);
-    os << "indexed(indices=";
-    printrawindices(os);
-    os << ",hash=" << hashvalue << ",flags=" << flags << ")";
+       GINAC_ASSERT(seq.size() > 0);
+       exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
+       while (it != itend) {
+               if (!is_ex_of_type(*it, idx))
+                       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);
+       }
 }
 
-void indexed::printtree(ostream & os, unsigned indent) const
+/** Implementation of ex::diff() for an indexed object always returns 0.
+ *
+ *  @see ex::diff */
+ex indexed::derivative(const symbol & s) const
 {
-    debugmsg("indexed printtree",LOGLEVEL_PRINT);
-    os << string(indent,' ') << "indexed: " << seq.size() << " indices";
-    os << ",hash=" << hashvalue << ",flags=" << flags << endl;
-    printtreeindices(os,indent);
+       return _ex0;
 }
 
-void indexed::print(ostream & os, unsigned upper_precedence) const
+//////////
+// global functions
+//////////
+
+/** Check whether two sorted index vectors are consistent (i.e. equal). */
+static bool indices_consistent(const exvector & v1, const exvector & v2)
 {
-    debugmsg("indexed print",LOGLEVEL_PRINT);
-    os << "UNNAMEDINDEX";
-    printindices(os);
+       // Number of indices must be the same
+       if (v1.size() != v2.size())
+               return false;
+
+       return equal(v1.begin(), v1.end(), v2.begin(), ex_is_equal());
 }
 
-void indexed::printcsrc(ostream & os, unsigned type,
-                        unsigned upper_precedence) const
+exvector indexed::get_indices(void) const
 {
-    debugmsg("indexed print csrc",LOGLEVEL_PRINT);
-    print(os,upper_precedence);
+       GINAC_ASSERT(seq.size() >= 1);
+       return exvector(seq.begin() + 1, seq.end());
 }
 
-bool indexed::info(unsigned inf) const
+exvector indexed::get_dummy_indices(void) const
 {
-    if (inf==info_flags::indexed) return true;
-    if (inf==info_flags::has_indices) return seq.size()!=0;
-    return exprseq::info(inf);
+       exvector free_indices, dummy_indices;
+       find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
+       return dummy_indices;
 }
 
-exvector indexed::get_indices(void) const
+exvector indexed::get_dummy_indices(const indexed & other) const
 {
-    return seq;
+       exvector indices = get_free_indices();
+       exvector other_indices = other.get_free_indices();
+       indices.insert(indices.end(), other_indices.begin(), other_indices.end());
+       exvector dummy_indices;
+       find_dummy_indices(indices, dummy_indices);
+       return dummy_indices;
+}
 
-    /*
-    idxvector filtered_indices;
-    filtered_indices.reserve(indices.size());
-    for (idxvector::const_iterator cit=indices.begin(); cit!=indices.end(); ++cit) {
-        if ((*cit).get_type()==t) {
-            filtered_indices.push_back(*cit);
-        }
-    }
-    return filtered_indices;
-    */
+bool indexed::has_dummy_index_for(const ex & i) const
+{
+       exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
+       while (it != itend) {
+               if (is_dummy_pair(*it, i))
+                       return true;
+               it++;
+       }
+       return false;
 }
 
-// protected
+exvector indexed::get_free_indices(void) const
+{
+       exvector free_indices, dummy_indices;
+       find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
+       return free_indices;
+}
 
-int indexed::compare_same_type(basic const & other) const
+exvector add::get_free_indices(void) const
 {
-    GINAC_ASSERT(is_of_type(other,indexed));
-    return exprseq::compare_same_type(other);
+       exvector free_indices;
+       for (unsigned i=0; i<nops(); i++) {
+               if (i == 0)
+                       free_indices = op(i).get_free_indices();
+               else {
+                       exvector free_indices_of_term = op(i).get_free_indices();
+                       if (!indices_consistent(free_indices, free_indices_of_term))
+                               throw (std::runtime_error("add::get_free_indices: inconsistent indices in sum"));
+               }
+       }
+       return free_indices;
 }
 
-bool indexed::is_equal_same_type(basic const & other) const
+exvector mul::get_free_indices(void) const
 {
-    GINAC_ASSERT(is_of_type(other,indexed));
-    return exprseq::is_equal_same_type(other);
+       // Concatenate free indices of all factors
+       exvector un;
+       for (unsigned i=0; i<nops(); i++) {
+               exvector free_indices_of_factor = op(i).get_free_indices();
+               un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
+       }
+
+       // And remove the dummy indices
+       exvector free_indices, dummy_indices;
+       find_free_and_dummy(un, free_indices, dummy_indices);
+       return free_indices;
 }
 
-unsigned indexed::return_type(void) const
+exvector ncmul::get_free_indices(void) const
 {
-    return return_types::noncommutative;
+       // Concatenate free indices of all factors
+       exvector un;
+       for (unsigned i=0; i<nops(); i++) {
+               exvector free_indices_of_factor = op(i).get_free_indices();
+               un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
+       }
+
+       // And remove the dummy indices
+       exvector free_indices, dummy_indices;
+       find_free_and_dummy(un, free_indices, dummy_indices);
+       return free_indices;
 }
-   
-unsigned indexed::return_type_tinfo(void) const
+
+exvector power::get_free_indices(void) const
 {
-    return tinfo_key;
+       // Return free indices of basis
+       return basis.get_free_indices();
 }
 
-ex indexed::thisexprseq(exvector const & v) const
+/** Rename dummy indices in an expression.
+ *
+ *  @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
+ *    before and which we would like to use in "e", too. This gets updated
+ *    by the function */
+static ex rename_dummy_indices(const ex & e, exvector & global_dummy_indices, exvector & local_dummy_indices)
 {
-    return indexed(v);
+       unsigned global_size = global_dummy_indices.size(),
+                local_size = local_dummy_indices.size();
+
+       // Any local dummy indices at all?
+       if (local_size == 0)
+               return e;
+
+       if (global_size < local_size) {
+
+               // 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) {
+                       if (find_if(global_dummy_indices.begin(), global_dummy_indices.end(), bind2nd(ex_is_equal(), *it)) == global_dummy_indices.end()) {
+                               global_dummy_indices.push_back(*it);
+                               global_size++;
+                               remaining--;
+                       }
+                       it++;
+               }
+
+               // If this is the first set of local indices, do nothing
+               if (old_global_size == 0)
+                       return e;
+       }
+       GINAC_ASSERT(local_size <= global_size);
+
+       // 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));
+       }
 }
 
-ex indexed::thisexprseq(exvector * vp) const
+/** Given a set of indices, extract those of class varidx. */
+static void find_variant_indices(const exvector & v, exvector & variant_indices)
 {
-    return indexed(vp);
+       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);
+       }
 }
 
-//////////
-// virtual functions which can be overridden by derived classes
-//////////
+/** 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: ;
+       }
+
+       return something_changed;
+}
 
-// none
+/* 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)
+{
+       // Remember whether the product was commutative or noncommutative
+       // (because we chop it into factors and need to reassemble later)
+       bool non_commutative = is_ex_exactly_of_type(e, ncmul);
+
+       // Collect factors in an exvector, store squares twice
+       exvector v;
+       v.reserve(e.nops() * 2);
+
+       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));
+               v.push_back(e.op(0));
+               v.push_back(e.op(0));
+       } else {
+               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)) {
+                               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 (unsigned j=0; j<f.nops(); j++)
+                                       v.push_back(f.op(j));
+                       } else
+                               v.push_back(f);
+               }
+       }
+
+       // Perform contractions
+       bool something_changed = false;
+       GINAC_ASSERT(v.size() > 1);
+       exvector::iterator it1, itend = v.end(), next_to_last = itend - 1;
+       for (it1 = v.begin(); it1 != next_to_last; it1++) {
+
+try_again:
+               if (!is_ex_of_type(*it1, indexed))
+                       continue;
+
+               bool first_noncommutative = (it1->return_type() != return_types::commutative);
+
+               // 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);
+
+               exvector::iterator it2;
+               for (it2 = it1 + 1; it2 != itend; it2++) {
+
+                       if (!is_ex_of_type(*it2, indexed))
+                               continue;
+
+                       bool second_noncommutative = (it2->return_type() != return_types::commutative);
+
+                       // 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);
+                       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);
+                       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.empty()) {
+                               if (sp.is_defined(*it1, *it2)) {
+                                       *it1 = sp.evaluate(*it1, *it2);
+                                       *it2 = _ex1;
+                                       goto contraction_done;
+                               }
+                       }
+
+                       // Try to contract the first one with the second one
+                       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 = ex_to<basic>(it2->op(0)).contract_with(it2, it1, v);
+                       }
+                       if (contracted) {
+contraction_done:
+                               if (first_noncommutative || second_noncommutative
+                                || is_ex_exactly_of_type(*it1, add) || is_ex_exactly_of_type(*it2, add)
+                                || is_ex_exactly_of_type(*it1, mul) || is_ex_exactly_of_type(*it2, mul)
+                                || is_ex_exactly_of_type(*it1, ncmul) || is_ex_exactly_of_type(*it2, ncmul)) {
+
+                                       // One of the factors became a sum or product:
+                                       // re-expand expression and run again
+                                       // 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, true)) : ex(mul(v)));
+                                       return simplify_indexed(r, free_indices, dummy_indices, sp);
+                               }
+
+                               // Both objects may have new indices now or they might
+                               // even not be indexed objects any more, so we have to
+                               // start over
+                               something_changed = true;
+                               goto try_again;
+                       }
+               }
+       }
+
+       // Find free indices (concatenate them all and call find_free_and_dummy())
+       // and all dummy indices that appear
+       exvector un, individual_dummy_indices;
+       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);
+                       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());
+       }
+       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, 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 ex_to<basic>(r.op(0).op(0)).scalar_mul_indexed(r.op(0), ex_to<numeric>(r.op(1)));
+       else
+               return r;
+}
 
-//////////
-// non-virtual functions in this class
-//////////
+/** Simplify indexed expression, return list of free indices. */
+ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
+{
+       // Expand the expression
+       ex e_expanded = e.expand();
+
+       // Simplification of single indexed object: just find the free indices
+       // and perform dummy index renaming/repositioning
+       if (is_ex_of_type(e_expanded, indexed)) {
+
+               // 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);
+       }
+
+       // Simplification of sum = sum of simplifications, check consistency of
+       // free indices in each term
+       if (is_ex_exactly_of_type(e_expanded, add)) {
+               bool first = true;
+               ex sum = _ex0;
+               free_indices.clear();
+
+               for (unsigned i=0; i<e_expanded.nops(); i++) {
+                       exvector free_indices_of_term;
+                       ex term = simplify_indexed(e_expanded.op(i), free_indices_of_term, dummy_indices, sp);
+                       if (!term.is_zero()) {
+                               if (first) {
+                                       free_indices = free_indices_of_term;
+                                       sum = term;
+                                       first = false;
+                               } else {
+                                       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 = ex_to<basic>(sum.op(0)).add_indexed(sum, term);
+                                       else
+                                               sum += term;
+                               }
+                       }
+               }
+
+               return sum;
+       }
+
+       // 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)))
+               return simplify_indexed_product(e_expanded, free_indices, dummy_indices, sp);
+
+       // Cannot do anything
+       free_indices.clear();
+       return e_expanded;
+}
 
-// protected
+/** 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 GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
+}
 
-void indexed::printrawindices(ostream & os) const
+/** 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
 {
-    if (seq.size()!=0) {
-        for (exvector::const_iterator cit=seq.begin(); cit!=seq.end(); ++cit) {
-            (*cit).printraw(os);
-            os << ",";
-        }
-    }
+       exvector free_indices, dummy_indices;
+       return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
 }
 
-void indexed::printtreeindices(ostream & os, unsigned indent) const
+/** Symmetrize expression over its free indices. */
+ex ex::symmetrize(void) const
 {
-    if (seq.size()!=0) {
-        for (exvector::const_iterator cit=seq.begin(); cit!=seq.end(); ++cit) {
-            os << string(indent+delta_indent,' ');
-            (*cit).printraw(os);
-            os << endl;
-        }
-    }
+       return GiNaC::symmetrize(*this, get_free_indices());
 }
 
-void indexed::printindices(ostream & os) const
+/** Antisymmetrize expression over its free indices. */
+ex ex::antisymmetrize(void) const
 {
-    if (seq.size()!=0) {
-        if (seq.size()>1) {
-            os << "{";
-        }
-        exvector::const_iterator last=seq.end()-1;
-        exvector::const_iterator cit=seq.begin();
-        for (; cit!=last; ++cit) {
-            (*cit).print(os);
-            os << ",";
-        }
-        (*cit).print(os);
-        if (seq.size()>1) {
-            os << "}";
-        }
-    }
+       return GiNaC::antisymmetrize(*this, get_free_indices());
 }
 
-bool indexed::all_of_type_idx(void) const
+/** Symmetrize expression by cyclic permutation over its free indices. */
+ex ex::symmetrize_cyclic(void) const
 {
-    // used only inside of ASSERTs
-    for (exvector::const_iterator cit=seq.begin(); cit!=seq.end(); ++cit) {
-        if (!is_ex_of_type(*cit,idx)) return false;
-    }
-    return true;
+       return GiNaC::symmetrize_cyclic(*this, get_free_indices());
 }
 
 //////////
-// static member variables
+// helper classes
 //////////
 
-// none
+void scalar_products::add(const ex & v1, const ex & v2, const ex & sp)
+{
+       spm[make_key(v1, v2)] = sp;
+}
 
-//////////
-// global constants
-//////////
+void scalar_products::add_vectors(const lst & l)
+{
+       // Add all possible pairs of products
+       unsigned num = l.nops();
+       for (unsigned i=0; i<num; i++) {
+               ex a = l.op(i);
+               for (unsigned j=0; j<num; j++) {
+                       ex b = l.op(j);
+                       add(a, b, a*b);
+               }
+       }
+}
 
-const indexed some_indexed;
-type_info const & typeid_indexed=typeid(some_indexed);
+void scalar_products::clear(void)
+{
+       spm.clear();
+}
+
+/** Check whether scalar product pair is defined. */
+bool scalar_products::is_defined(const ex & v1, const ex & v2) const
+{
+       return spm.find(make_key(v1, v2)) != spm.end();
+}
+
+/** Return value of defined scalar product pair. */
+ex scalar_products::evaluate(const ex & v1, const ex & v2) const
+{
+       return spm.find(make_key(v1, v2))->second;
+}
+
+void scalar_products::debugprint(void) const
+{
+       std::cerr << "map size=" << spm.size() << std::endl;
+       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=" << i->second << std::endl;
+               ++i;
+       }
+}
+
+/** Make key from object pair. */
+spmapkey scalar_products::make_key(const ex & v1, const ex & v2)
+{
+       // If indexed, extract base objects
+       ex s1 = is_ex_of_type(v1, indexed) ? v1.op(0) : v1;
+       ex s2 = is_ex_of_type(v2, indexed) ? v2.op(0) : v2;
+
+       // Enforce canonical order in pair
+       if (s1.compare(s2) > 0)
+               return spmapkey(s2, s1);
+       else
+               return spmapkey(s1, s2);
+}
 
 } // namespace GiNaC