indexed.cpp

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/*
 *  GiNaC Copyright (C) 1999-2023 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
 *  the Free Software Foundation; either version 2 of the License, or
 *  (at your option) any later version.
 *
 *  This program is distributed in the hope that it will be useful,
 *  but WITHOUT ANY WARRANTY; without even the implied warranty of
 *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *  GNU General Public License for more details.
 *
 *  You should have received a copy of the GNU General Public License
 *  along with this program; if not, write to the Free Software
 *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
 */
 
#include "indexed.h"
#include "idx.h"
#include "add.h"
#include "mul.h"
#include "ncmul.h"
#include "power.h"
#include "relational.h"
#include "symmetry.h"
#include "operators.h"
#include "lst.h"
#include "archive.h"
#include "symbol.h"
#include "utils.h"
#include "integral.h"
#include "matrix.h"
#include "inifcns.h"
 
#include <iostream>
#include <limits>
#include <sstream>
#include <stdexcept>
 
namespace GiNaC {
 
GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(indexed, exprseq,
  print_func<print_context>(&indexed::do_print).
  print_func<print_latex>(&indexed::do_print_latex).
  print_func<print_tree>(&indexed::do_print_tree))
 
 
// default constructor
 
indexed::indexed() : symtree(not_symmetric())
{
}
 
// other constructors
 
indexed::indexed(const ex & b) : inherited{b}, symtree(not_symmetric())
{
    validate();
}
 
indexed::indexed(const ex & b, const ex & i1) : inherited{b, i1}, symtree(not_symmetric())
{
    validate();
}
 
indexed::indexed(const ex & b, const ex & i1, const ex & i2) : inherited{b, i1, i2}, symtree(not_symmetric())
{
    validate();
}
 
indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3) : inherited{b, i1, i2, i3}, symtree(not_symmetric())
{
    validate();
}
 
indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3, const ex & i4) : inherited{b, i1, i2, i3, i4}, symtree(not_symmetric())
{
    validate();
}
 
indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2) : inherited{b, i1, i2}, symtree(symm)
{
    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)
{
    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)
{
    validate();
}
 
indexed::indexed(const ex & b, const exvector & v) : inherited{b}, symtree(not_symmetric())
{
    seq.insert(seq.end(), v.begin(), v.end());
    validate();
}
 
indexed::indexed(const ex & b, const symmetry & symm, const exvector & v) : inherited{b}, symtree(symm)
{
    seq.insert(seq.end(), v.begin(), v.end());
    validate();
}
 
indexed::indexed(const symmetry & symm, const exprseq & es) : inherited(es), symtree(symm)
{
}
 
indexed::indexed(const symmetry & symm, const exvector & v) : inherited(v), symtree(symm)
{
}
 
indexed::indexed(const symmetry & symm, exvector && v) : inherited(std::move(v)), symtree(symm)
{
}
 
// archiving
 
void indexed::read_archive(const archive_node &n, lst &sym_lst)
{
    inherited::read_archive(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 = not_symmetric();
                break;
        }
        const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
    }
}
GINAC_BIND_UNARCHIVER(indexed);
 
void indexed::archive(archive_node &n) const
{
    inherited::archive(n);
    n.add_ex("symmetry", symtree);
}
 
// functions overriding virtual functions from base classes
 
void indexed::printindices(const print_context & c, unsigned level) const
{
    if (seq.size() > 1) {
 
        auto it = seq.begin() + 1, itend = seq.end();
 
        if (is_a<print_latex>(c)) {
 
            // TeX output: group by variance
            bool first = true;
            bool covariant = true;
 
            while (it != itend) {
                bool cur_covariant = (is_a<varidx>(*it) ? 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::print_indexed(const print_context & c, const char *openbrace, const char *closebrace, unsigned level) const
{
    if (precedence() <= level)
        c.s << openbrace << '(';
    c.s << openbrace;
    seq[0].print(c, precedence());
    c.s << closebrace;
    printindices(c, level);
    if (precedence() <= level)
        c.s << ')' << closebrace;
}
 
void indexed::do_print(const print_context & c, unsigned level) const
{
    print_indexed(c, "", "", level);
}
 
void indexed::do_print_latex(const print_latex & c, unsigned level) const
{
    print_indexed(c, "{", "}", level);
}
 
void indexed::do_print_tree(const print_tree & c, unsigned level) const
{
    c.s << std::string(level, ' ') << class_name() << " @" << this
        << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
        << ", " << seq.size()-1 << " indices"
        << ", symmetry=" << symtree << std::endl;
    seq[0].print(c, level + c.delta_indent);
    printindices(c, level + c.delta_indent);
}
 
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);
}
 
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(),
                   [inf](const ex & e) { return !(ex_to<idx>(e).get_value().info(inf)); }) == seq.end();
}
 
int indexed::compare_same_type(const basic & other) const
{
    GINAC_ASSERT(is_a<indexed>(other));
    return inherited::compare_same_type(other);
}
 
ex indexed::eval() const
{
    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_exactly_a<mul>(base) && is_exactly_a<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 * thiscontainer(v);
    }
 
    if((typeid(*this) == typeid(indexed)) && seq.size()==1)
        return base;
 
    // 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 != std::numeric_limits<int>::max()) {
            // Something has changed while sorting indices, more evaluations later
            if (sig == 0)
                return _ex0;
            return ex(sig) * thiscontainer(v);
        }
    }
 
    // Let the class of the base object perform additional evaluations
    return ex_to<basic>(base).eval_indexed(*this);
}
 
ex indexed::real_part() const
{
    if(op(0).info(info_flags::real))
        return *this;
    return real_part_function(*this).hold();
}
 
ex indexed::imag_part() const
{
    if(op(0).info(info_flags::real))
        return 0;
    return imag_part_function(*this).hold();
}
 
ex indexed::thiscontainer(const exvector & v) const
{
    return indexed(ex_to<symmetry>(symtree), v);
}
 
ex indexed::thiscontainer(exvector && v) const
{
    return indexed(ex_to<symmetry>(symtree), std::move(v));
}
 
unsigned indexed::return_type() const
{
    if(is_a<matrix>(op(0)))
        return return_types::commutative;
    else
        return op(0).return_type();
}
 
ex indexed::expand(unsigned options) const
{
    GINAC_ASSERT(seq.size() > 0);
 
    if (options & expand_options::expand_indexed) {
        ex newbase = seq[0].expand(options);
        if (is_exactly_a<add>(newbase)) {
            ex sum = _ex0;
            for (size_t i=0; i<newbase.nops(); i++) {
                exvector s = seq;
                s[0] = newbase.op(i);
                sum += thiscontainer(s).expand(options);
            }
            return sum;
        }
        if (!are_ex_trivially_equal(newbase, seq[0])) {
            exvector s = seq;
            s[0] = newbase;
            return ex_to<indexed>(thiscontainer(s)).inherited::expand(options);
        }
    }
    return inherited::expand(options);
}
 
// virtual functions which can be overridden by derived classes
 
// none
 
// non-virtual functions in this class
 
void indexed::validate() const
{
    GINAC_ASSERT(seq.size() > 0);
    auto it = seq.begin() + 1, itend = seq.end();
    while (it != itend) {
        if (!is_a<idx>(*it))
            throw(std::invalid_argument("indices of indexed object must be of type idx"));
        it++;
    }
 
    if (!symtree.is_zero()) {
        if (!is_exactly_a<symmetry>(symtree))
            throw(std::invalid_argument("symmetry of indexed object must be of type symmetry"));
        const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
    }
}
 
ex indexed::derivative(const symbol & s) const
{
    return _ex0;
}
 
// global functions
 
struct idx_is_equal_ignore_dim {
    bool operator() (const ex &lh, const ex &rh) const
    {
        if (lh.is_equal(rh))
            return true;
        else
            try {
                // Replacing the dimension might cause an error (e.g. with
                // index classes that only work in a fixed number of dimensions)
                return lh.is_equal(ex_to<idx>(rh).replace_dim(ex_to<idx>(lh).get_dim()));
            } catch (...) {
                return false;
            }
    }
};
 
static bool indices_consistent(const exvector & v1, const exvector & v2)
{
    // Number of indices must be the same
    if (v1.size() != v2.size())
        return false;
 
    return equal(v1.begin(), v1.end(), v2.begin(), idx_is_equal_ignore_dim());
}
 
exvector indexed::get_indices() const
{
    GINAC_ASSERT(seq.size() >= 1);
    return exvector(seq.begin() + 1, seq.end());
}
 
exvector indexed::get_dummy_indices() const
{
    exvector free_indices, dummy_indices;
    find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
    return dummy_indices;
}
 
exvector indexed::get_dummy_indices(const indexed & other) const
{
    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;
}
 
bool indexed::has_dummy_index_for(const ex & i) const
{
    auto it = seq.begin() + 1, itend = seq.end();
    while (it != itend) {
        if (is_dummy_pair(*it, i))
            return true;
        it++;
    }
    return false;
}
 
exvector indexed::get_free_indices() const
{
    exvector free_indices, dummy_indices;
    find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
    return free_indices;
}
 
exvector add::get_free_indices() const
{
    exvector free_indices;
    for (size_t 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;
}
 
exvector mul::get_free_indices() const
{
    // Concatenate free indices of all factors
    exvector un;
    for (size_t 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;
}
 
exvector ncmul::get_free_indices() const
{
    // Concatenate free indices of all factors
    exvector un;
    for (size_t 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;
}
 
struct is_summation_idx {
    bool operator()(const ex & e)
    {
        return is_dummy_pair(e, e);
    }
};
 
exvector integral::get_free_indices() const
{
    if (a.get_free_indices().size() || b.get_free_indices().size())
        throw (std::runtime_error("integral::get_free_indices: boundary values should not have free indices"));
    return f.get_free_indices();
}
 
template<class T> size_t number_of_type(const exvector&v)
{
    size_t number = 0;
    for (auto & it : v)
        if (is_exactly_a<T>(it))
            ++number;
    return number;
}
 
template<class T> static ex rename_dummy_indices(const ex & e, exvector & global_dummy_indices, exvector & local_dummy_indices)
{
    size_t global_size = number_of_type<T>(global_dummy_indices),
           local_size = number_of_type<T>(local_dummy_indices);
 
    // 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
        size_t old_global_size = global_size;
        int remaining = local_size - global_size;
        auto it = local_dummy_indices.begin(), itend = local_dummy_indices.end();
        while (it != itend && remaining > 0) {
            if (is_exactly_a<T>(*it) &&
                find_if(global_dummy_indices.begin(), global_dummy_indices.end(),
                        [it](const ex &lh) { return idx_is_equal_ignore_dim()(lh, *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 vectors of index symbols
    exvector local_syms, global_syms;
    local_syms.reserve(local_size);
    global_syms.reserve(local_size);
    for (size_t i=0; local_syms.size()!=local_size; i++)
        if(is_exactly_a<T>(local_dummy_indices[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 (size_t i=0; global_syms.size()!=local_size; i++) // don't use more global symbols than necessary
        if(is_exactly_a<T>(global_dummy_indices[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
    exvector local_uniq, global_uniq;
    set_difference(local_syms.begin(), local_syms.end(), global_syms.begin(), global_syms.end(), std::back_insert_iterator<exvector>(local_uniq), ex_is_less());
    set_difference(global_syms.begin(), global_syms.end(), local_syms.begin(), local_syms.end(), std::back_insert_iterator<exvector>(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.begin(), local_uniq.end()), lst(global_uniq.begin(), global_uniq.end()), subs_options::no_pattern);
    }
}
 
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);
    }
}
 
bool reposition_dummy_indices(ex & e, exvector & variant_dummy_indices, exvector & moved_indices)
{
    bool something_changed = false;
 
    // Find dummy symbols that occur twice in the same indexed object.
    exvector local_var_dummies;
    local_var_dummies.reserve(e.nops()/2);
    for (size_t i=1; i<e.nops(); ++i) {
        if (!is_a<varidx>(e.op(i)))
            continue;
        for (size_t j=i+1; j<e.nops(); ++j) {
            if (is_dummy_pair(e.op(i), e.op(j))) {
                local_var_dummies.push_back(e.op(i));
                for (auto k = variant_dummy_indices.begin(); k!=variant_dummy_indices.end(); ++k) {
                    if (e.op(i).op(0) == k->op(0)) {
                        variant_dummy_indices.erase(k);
                        break;
                    }
                }
                break;
            }
        }
    }
 
    // In the case where a dummy symbol occurs twice in the same indexed object
    // we try all possibilities of raising/lowering and keep the least one in
    // the sense of ex_is_less.
    ex optimal_e = e;
    size_t numpossibs = 1 << local_var_dummies.size();
    for (size_t i=0; i<numpossibs; ++i) {
        ex try_e = e;
        for (size_t j=0; j<local_var_dummies.size(); ++j) {
            exmap m;
            if (1<<j & i) {
                ex curr_idx = local_var_dummies[j];
                ex curr_toggle = ex_to<varidx>(curr_idx).toggle_variance();
                m[curr_idx] = curr_toggle;
                m[curr_toggle] = curr_idx;
            }
            try_e = e.subs(m, subs_options::no_pattern);
        }
        if(ex_is_less()(try_e, optimal_e))
        {   optimal_e = try_e;
            something_changed = true;
        }
    }
    e = optimal_e;
 
    if (!is_a<indexed>(e))
        return true;
 
    exvector seq = ex_to<indexed>(e).seq;
 
    // If a dummy index is encountered for the first time in the
    // product, pull it up, otherwise, pull it down
    for (auto it2 = seq.begin()+1, it2end = seq.end(); 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()) {
                    /*
                     * N.B. we don't want to use
                     *
                     *  e = e.subs(lst{
                     *  *it2 == ex_to<varidx>(*it2).toggle_variance(),
                     *  ex_to<varidx>(*it2).toggle_variance() == *it2
                     *  }, subs_options::no_pattern);
                     *
                     * since this can trigger non-trivial repositioning of indices,
                     * e.g. due to non-trivial symmetry properties of e, thus
                     * invalidating iterators
                     */
                    *it2 = ex_to<varidx>(*it2).toggle_variance();
                    something_changed = true;
                }
                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()) {
                    *it2 = ex_to<varidx>(*it2).toggle_variance();
                    something_changed = true;
                }
                goto next_index;
            }
        }
 
next_index: ;
    }
 
    if (something_changed)
        e = ex_to<indexed>(e).thiscontainer(seq);
 
    return something_changed;
}
 
/* Ordering that only compares the base expressions of indexed objects. */
struct ex_base_is_less {
    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;
    }
};
 
/* An auxiliary function used by simplify_indexed() and expand_dummy_sum() 
 * It returns an exvector of factors from the supplied product */
static void product_to_exvector(const ex & e, exvector & v, bool & non_commutative)
{
    // Remember whether the product was commutative or noncommutative
    // (because we chop it into factors and need to reassemble later)
    non_commutative = is_exactly_a<ncmul>(e);
 
    // Collect factors in an exvector, store squares twice
    v.reserve(e.nops() * 2);
 
    if (is_exactly_a<power>(e)) {
        // 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 (size_t i=0; i<e.nops(); i++) {
            ex f = e.op(i);
            if (is_exactly_a<power>(f) && f.op(1).is_equal(_ex2)) {
                v.push_back(f.op(0));
                v.push_back(f.op(0));
            } else if (is_exactly_a<ncmul>(f)) {
                // Noncommutative factor found, split it as well
                non_commutative = true; // everything becomes noncommutative, ncmul will sort out the commutative factors later
                for (size_t j=0; j<f.nops(); j++)
                    v.push_back(f.op(j));
            } else
                v.push_back(f);
        }
    }
}
 
template<class T> ex idx_symmetrization(const ex& r,const exvector& local_dummy_indices)
{   exvector dummy_syms;
    dummy_syms.reserve(r.nops());
    for (auto & it : local_dummy_indices)
            if(is_exactly_a<T>(it))
                dummy_syms.push_back(it.op(0));
    if(dummy_syms.size() < 2)
        return r;
    ex q=symmetrize(r, dummy_syms);
    return q;
}
 
// Forward declaration needed in absence of friend injection, C.f. [namespace.memdef]:
ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp);
 
ex simplify_indexed_product(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
{
    // Collect factors in an exvector
    exvector v;
 
    // Remember whether the product was commutative or noncommutative
    // (because we chop it into factors and need to reassemble later)
    bool non_commutative;
    product_to_exvector(e, v, non_commutative);
 
    // Perform contractions
    bool something_changed = false;
    bool has_nonsymmetric = 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_a<indexed>(*it1))
            continue;
 
        bool first_noncommutative = (it1->return_type() != return_types::commutative);
        bool first_nonsymmetric = ex_to<symmetry>(ex_to<indexed>(*it1).get_symmetry()).has_nonsymmetric();
 
        // 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_a<indexed>(*it2))
                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);
            size_t 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() && it1->nops()==2 && it2->nops()==2) {
 
                ex dim = minimal_dim(
                    ex_to<idx>(it1->op(1)).get_dim(),
                    ex_to<idx>(it2->op(1)).get_dim()
                );
 
                // User-defined scalar product?
                if (sp.is_defined(*it1, *it2, dim)) {
 
                    // Yes, substitute it
                    *it1 = sp.evaluate(*it1, *it2, dim);
                    *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_exactly_a<add>(*it1) || is_exactly_a<add>(*it2)
                 || is_exactly_a<mul>(*it1) || is_exactly_a<mul>(*it2)
                 || is_exactly_a<ncmul>(*it1) || is_exactly_a<ncmul>(*it2)) {
 
                    // One of the factors became a sum or product:
                    // re-expand expression and run again
                    // Non-commutative products are always re-expanded to give
                    // eval_ncmul() the chance to re-order and canonicalize
                    // the product
                    bool is_a_product = (is_exactly_a<mul>(*it1) || is_exactly_a<ncmul>(*it1)) &&
                                        (is_exactly_a<mul>(*it2) || is_exactly_a<ncmul>(*it2));
                    ex r = (non_commutative ? ex(ncmul(std::move(v))) : ex(mul(std::move(v))));
 
                    // If new expression is a product we can call this function again,
                    // otherwise we need to pass argument to simplify_indexed() to be expanded
                    if (is_a_product)
                        return simplify_indexed_product(r, free_indices, dummy_indices, sp);
                    else
                        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;
            }
            else if (!has_nonsymmetric &&
                    (first_nonsymmetric ||
                     ex_to<symmetry>(ex_to<indexed>(*it2).get_symmetry()).has_nonsymmetric())) {
                has_nonsymmetric = true;
            }
        }
    }
 
    // 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_a<indexed>(*it1)) {
            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_a<indexed>(*it1))
                continue;
 
            if (reposition_dummy_indices(*it1, variant_dummy_indices, moved_indices))
                something_changed = true;
        }
    }
 
    ex r;
    if (something_changed)
        r = non_commutative ? ex(ncmul(std::move(v))) : ex(mul(std::move(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 (has_nonsymmetric) {
        ex q = idx_symmetrization<idx>(r, local_dummy_indices);
        if (q.is_zero()) {
            free_indices.clear();
            return _ex0;
        }
        q = idx_symmetrization<varidx>(q, local_dummy_indices);
        if (q.is_zero()) {
            free_indices.clear();
            return _ex0;
        }
        q = idx_symmetrization<spinidx>(q, local_dummy_indices);
        if (q.is_zero()) {
            free_indices.clear();
            return _ex0;
        }
    }
 
    // Dummy index renaming
    r = rename_dummy_indices<idx>(r, dummy_indices, local_dummy_indices);
    r = rename_dummy_indices<varidx>(r, dummy_indices, local_dummy_indices);
    r = rename_dummy_indices<spinidx>(r, dummy_indices, local_dummy_indices);
 
    // Product of indexed object with a scalar?
    if (is_exactly_a<mul>(r) && r.nops() == 2
     && is_exactly_a<numeric>(r.op(1)) && is_a<indexed>(r.op(0)))
        return ex_to<basic>(r.op(0).op(0)).scalar_mul_indexed(r.op(0), ex_to<numeric>(r.op(1)));
    else
        return r;
}
 
class terminfo {
public:
    terminfo(const ex & orig_, const ex & symm_) : orig(orig_), symm(symm_) {}
 
    ex orig; 
    ex symm; 
};
 
class terminfo_is_less {
public:
    bool operator() (const terminfo & ti1, const terminfo & ti2) const
    {
        return (ti1.symm.compare(ti2.symm) < 0);
    }
};
 
class symminfo {
public:
    symminfo() : num(0) {}
 
    symminfo(const ex & symmterm_, const ex & orig_, size_t num_) : orig(orig_), num(num_)
    {
        if (is_exactly_a<mul>(symmterm_) && is_exactly_a<numeric>(symmterm_.op(symmterm_.nops()-1))) {
            coeff = symmterm_.op(symmterm_.nops()-1);
            symmterm = symmterm_ / coeff;
        } else {
            coeff = 1;
            symmterm = symmterm_;
        }
    }
 
    ex symmterm;  
    ex coeff;     
    ex orig;      
    size_t num; 
};
 
class symminfo_is_less_by_symmterm {
public:
    bool operator() (const symminfo & si1, const symminfo & si2) const
    {
        return (si1.symmterm.compare(si2.symmterm) < 0);
    }
};
 
class symminfo_is_less_by_orig {
public:
    bool operator() (const symminfo & si1, const symminfo & si2) const
    {
        return (si1.orig.compare(si2.orig) < 0);
    }
};
 
bool hasindex(const ex &x, const ex &sym)
{   
    if(is_a<idx>(x) && x.op(0)==sym)
        return true;
    else
        for(size_t i=0; i<x.nops(); ++i)
            if(hasindex(x.op(i), sym))
                return true;
    return false;
}
 
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_a<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
        e_expanded = rename_dummy_indices<idx>(e_expanded, dummy_indices, local_dummy_indices);
        e_expanded = rename_dummy_indices<varidx>(e_expanded, dummy_indices, local_dummy_indices);
        e_expanded = rename_dummy_indices<spinidx>(e_expanded, dummy_indices, local_dummy_indices);
        return e_expanded;
    }
 
    // Simplification of sum = sum of simplifications, check consistency of
    // free indices in each term
    if (is_exactly_a<add>(e_expanded)) {
        bool first = true;
        ex sum;
        free_indices.clear();
 
        for (size_t 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)) {
                        std::ostringstream s;
                        s << "simplify_indexed: inconsistent indices in sum: ";
                        s << exprseq(free_indices) << " vs. " << exprseq(free_indices_of_term);
                        throw (std::runtime_error(s.str()));
                    }
                    if (is_a<indexed>(sum) && is_a<indexed>(term))
                        sum = ex_to<basic>(sum.op(0)).add_indexed(sum, term);
                    else
                        sum += term;
                }
            }
        }
 
        // If the sum turns out to be zero, we are finished
        if (sum.is_zero()) {
            free_indices.clear();
            return sum;
        }
 
        // More than one term and more than one dummy index?
        size_t num_terms_orig = (is_exactly_a<add>(sum) ? sum.nops() : 1);
        if (num_terms_orig < 2 || dummy_indices.size() < 2)
            return sum;
 
        // Chop the sum into terms and symmetrize each one over the dummy
        // indices
        std::vector<terminfo> terms;
        for (size_t i=0; i<sum.nops(); i++) {
            const ex & term = sum.op(i);
            exvector dummy_indices_of_term;
            dummy_indices_of_term.reserve(dummy_indices.size());
            for (auto & i : dummy_indices)
                if (hasindex(term,i.op(0)))
                    dummy_indices_of_term.push_back(i);
            ex term_symm = idx_symmetrization<idx>(term, dummy_indices_of_term);
            term_symm = idx_symmetrization<varidx>(term_symm, dummy_indices_of_term);
            term_symm = idx_symmetrization<spinidx>(term_symm, dummy_indices_of_term);
            if (term_symm.is_zero())
                continue;
            terms.push_back(terminfo(term, term_symm));
        }
 
        // Sort by symmetrized terms
        std::sort(terms.begin(), terms.end(), terminfo_is_less());
 
        // Combine equal symmetrized terms
        std::vector<terminfo> terms_pass2;
        for (std::vector<terminfo>::const_iterator i=terms.begin(); i!=terms.end(); ) {
            size_t num = 1;
            auto j = i + 1;
            while (j != terms.end() && j->symm == i->symm) {
                num++;
                j++;
            }
            terms_pass2.push_back(terminfo(i->orig * num, i->symm * num));
            i = j;
        }
 
        // If there is only one term left, we are finished
        if (terms_pass2.size() == 1)
            return terms_pass2[0].orig;
 
        // Chop the symmetrized terms into subterms
        std::vector<symminfo> sy;
        for (auto & i : terms_pass2) {
            if (is_exactly_a<add>(i.symm)) {
                size_t num = i.symm.nops();
                for (size_t j=0; j<num; j++)
                    sy.push_back(symminfo(i.symm.op(j), i.orig, num));
            } else
                sy.push_back(symminfo(i.symm, i.orig, 1));
        }
 
        // Sort by symmetrized subterms
        std::sort(sy.begin(), sy.end(), symminfo_is_less_by_symmterm());
 
        // Combine equal symmetrized subterms
        std::vector<symminfo> sy_pass2;
        exvector result;
        for (auto i=sy.begin(); i!=sy.end(); ) {
 
            // Combine equal terms
            auto j = i + 1;
            if (j != sy.end() && j->symmterm == i->symmterm) {
 
                // More than one term, collect the coefficients
                ex coeff = i->coeff;
                while (j != sy.end() && j->symmterm == i->symmterm) {
                    coeff += j->coeff;
                    j++;
                }
 
                // Add combined term to result
                if (!coeff.is_zero())
                    result.push_back(coeff * i->symmterm);
 
            } else {
 
                // Single term, store for second pass
                sy_pass2.push_back(*i);
            }
 
            i = j;
        }
 
        // Were there any remaining terms that didn't get combined?
        if (sy_pass2.size() > 0) {
 
            // Yes, sort by their original terms
            std::sort(sy_pass2.begin(), sy_pass2.end(), symminfo_is_less_by_orig());
 
            for (std::vector<symminfo>::const_iterator i=sy_pass2.begin(); i!=sy_pass2.end(); ) {
 
                // How many symmetrized terms of this original term are left?
                size_t num = 1;
                auto j = i + 1;
                while (j != sy_pass2.end() && j->orig == i->orig) {
                    num++;
                    j++;
                }
 
                if (num == i->num) {
 
                    // All terms left, then add the original term to the result
                    result.push_back(i->orig);
 
                } else {
 
                    // Some terms were combined with others, add up the remaining symmetrized terms
                    std::vector<symminfo>::const_iterator k;
                    for (k=i; k!=j; k++)
                        result.push_back(k->coeff * k->symmterm);
                }
 
                i = j;
            }
        }
 
        // Add all resulting terms
        ex sum_symm = dynallocate<add>(result);
        if (sum_symm.is_zero())
            free_indices.clear();
        return sum_symm;
    }
 
    // Simplification of products
    if (is_exactly_a<mul>(e_expanded)
     || is_exactly_a<ncmul>(e_expanded)
     || (is_exactly_a<power>(e_expanded) && is_a<indexed>(e_expanded.op(0)) && 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;
}
 
ex ex::simplify_indexed(unsigned options) const
{
    exvector free_indices, dummy_indices;
    scalar_products sp;
    return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
}
 
ex ex::simplify_indexed(const scalar_products & sp, unsigned options) const
{
    exvector free_indices, dummy_indices;
    return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
}
 
ex ex::symmetrize() const
{
    return GiNaC::symmetrize(*this, get_free_indices());
}
 
ex ex::antisymmetrize() const
{
    return GiNaC::antisymmetrize(*this, get_free_indices());
}
 
ex ex::symmetrize_cyclic() const
{
    return GiNaC::symmetrize_cyclic(*this, get_free_indices());
}
 
// helper classes
 
spmapkey::spmapkey(const ex & v1_, const ex & v2_, const ex & dim_) : dim(dim_)
{
    // If indexed, extract base objects
    ex s1 = is_a<indexed>(v1_) ? v1_.op(0) : v1_;
    ex s2 = is_a<indexed>(v2_) ? v2_.op(0) : v2_;
 
    // Enforce canonical order in pair
    if (s1.compare(s2) > 0) {
        v1 = s2;
        v2 = s1;
    } else {
        v1 = s1;
        v2 = s2;
    }
}
 
bool spmapkey::operator==(const spmapkey &other) const
{
    if (!v1.is_equal(other.v1))
        return false;
    if (!v2.is_equal(other.v2))
        return false;
    if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
        return true;
    else
        return dim.is_equal(other.dim);
}
 
bool spmapkey::operator<(const spmapkey &other) const
{
    int cmp = v1.compare(other.v1);
    if (cmp)
        return cmp < 0;
    cmp = v2.compare(other.v2);
    if (cmp)
        return cmp < 0;
 
    // Objects are equal, now check dimensions
    if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
        return false;
    else
        return dim.compare(other.dim) < 0;
}
 
void spmapkey::debugprint() const
{
    std::cerr << "(" << v1 << "," << v2 << "," << dim << ")";
}
 
void scalar_products::add(const ex & v1, const ex & v2, const ex & sp)
{
    spm[spmapkey(v1, v2)] = sp;
}
 
void scalar_products::add(const ex & v1, const ex & v2, const ex & dim, const ex & sp)
{
    spm[spmapkey(v1, v2, dim)] = sp;
}
 
void scalar_products::add_vectors(const lst & l, const ex & dim)
{
    // Add all possible pairs of products
    for (auto & it1 : l)
        for (auto & it2 : l)
            add(it1, it2, it1 * it2);
}
 
void scalar_products::clear()
{
    spm.clear();
}
 
bool scalar_products::is_defined(const ex & v1, const ex & v2, const ex & dim) const
{
    return spm.find(spmapkey(v1, v2, dim)) != spm.end();
}
 
ex scalar_products::evaluate(const ex & v1, const ex & v2, const ex & dim) const
{
    return spm.find(spmapkey(v1, v2, dim))->second;
}
 
void scalar_products::debugprint() const
{
    std::cerr << "map size=" << spm.size() << std::endl;
    for (auto & it : spm) {
        const spmapkey & k = it.first;
        std::cerr << "item key=";
        k.debugprint();
        std::cerr << ", value=" << it.second << std::endl;
    }
}
 
exvector get_all_dummy_indices_safely(const ex & e)
{
    if (is_a<indexed>(e))
        return ex_to<indexed>(e).get_dummy_indices();
    else if (is_a<power>(e) && e.op(1)==2) {
        return e.op(0).get_free_indices();
    }   
    else if (is_a<mul>(e) || is_a<ncmul>(e)) {
        exvector dummies;
        exvector free_indices;
        for (std::size_t i = 0; i < e.nops(); ++i) {
            exvector dummies_of_factor = get_all_dummy_indices_safely(e.op(i));
            dummies.insert(dummies.end(), dummies_of_factor.begin(),
                dummies_of_factor.end());
            exvector free_of_factor = e.op(i).get_free_indices();
            free_indices.insert(free_indices.begin(), free_of_factor.begin(),
                free_of_factor.end());
        }
        exvector free_out, dummy_out;
        find_free_and_dummy(free_indices.begin(), free_indices.end(), free_out,
            dummy_out);
        dummies.insert(dummies.end(), dummy_out.begin(), dummy_out.end());
        return dummies;
    }
    else if(is_a<add>(e)) {
        exvector result;
        for(std::size_t i = 0; i < e.nops(); ++i) {
            exvector dummies_of_term = get_all_dummy_indices_safely(e.op(i));
            sort(dummies_of_term.begin(), dummies_of_term.end());
            exvector new_vec;
            set_union(result.begin(), result.end(), dummies_of_term.begin(),
                dummies_of_term.end(), std::back_inserter<exvector>(new_vec),
                ex_is_less());
            result.swap(new_vec);
        }
        return result;
    }
    return exvector();
}
 
exvector get_all_dummy_indices(const ex & e)
{
    exvector p;
    bool nc;
    product_to_exvector(e, p, nc);
    auto ip = p.begin(), ipend = p.end();
    exvector v, v1;
    while (ip != ipend) {
        if (is_a<indexed>(*ip)) {
            v1 = ex_to<indexed>(*ip).get_dummy_indices();
            v.insert(v.end(), v1.begin(), v1.end());
            auto ip1 = ip + 1;
            while (ip1 != ipend) {
                if (is_a<indexed>(*ip1)) {
                    v1 = ex_to<indexed>(*ip).get_dummy_indices(ex_to<indexed>(*ip1));
                    v.insert(v.end(), v1.begin(), v1.end());
                }
                ++ip1;
            }
        }
        ++ip;
    }
    return v;
}
 
lst rename_dummy_indices_uniquely(const exvector & va, const exvector & vb)
{
    exvector common_indices;
    set_intersection(va.begin(), va.end(), vb.begin(), vb.end(), std::back_insert_iterator<exvector>(common_indices), ex_is_less());
    if (common_indices.empty()) {
        return lst{lst{}, lst{}};
    } else {
        exvector new_indices, old_indices;
        old_indices.reserve(2*common_indices.size());
        new_indices.reserve(2*common_indices.size());
        exvector::const_iterator ip = common_indices.begin(), ipend = common_indices.end();
        while (ip != ipend) {
            ex newsym = dynallocate<symbol>();
            ex newidx;
            if(is_exactly_a<spinidx>(*ip))
                newidx = dynallocate<spinidx>(newsym, ex_to<spinidx>(*ip).get_dim(),
                                              ex_to<spinidx>(*ip).is_covariant(),
                                              ex_to<spinidx>(*ip).is_dotted());
            else if (is_exactly_a<varidx>(*ip))
                newidx = dynallocate<varidx>(newsym, ex_to<varidx>(*ip).get_dim(),
                                             ex_to<varidx>(*ip).is_covariant());
            else
                newidx = dynallocate<idx>(newsym, ex_to<idx>(*ip).get_dim());
            old_indices.push_back(*ip);
            new_indices.push_back(newidx);
            if(is_a<varidx>(*ip)) {
                old_indices.push_back(ex_to<varidx>(*ip).toggle_variance());
                new_indices.push_back(ex_to<varidx>(newidx).toggle_variance());
            }
            ++ip;
        }
        return lst{lst(old_indices.begin(), old_indices.end()), lst(new_indices.begin(), new_indices.end())};
    }
}
 
ex rename_dummy_indices_uniquely(const exvector & va, const exvector & vb, const ex & b)
{
    lst indices_subs = rename_dummy_indices_uniquely(va, vb);
    return (indices_subs.op(0).nops()>0 ? b.subs(ex_to<lst>(indices_subs.op(0)), ex_to<lst>(indices_subs.op(1)), subs_options::no_pattern|subs_options::no_index_renaming) : b);
}
 
ex rename_dummy_indices_uniquely(const ex & a, const ex & b)
{
    exvector va = get_all_dummy_indices_safely(a);
    if (va.size() > 0) {
        exvector vb = get_all_dummy_indices_safely(b);
        if (vb.size() > 0) {
            sort(va.begin(), va.end(), ex_is_less());
            sort(vb.begin(), vb.end(), ex_is_less());
            lst indices_subs = rename_dummy_indices_uniquely(va, vb);
            if (indices_subs.op(0).nops() > 0)
                return b.subs(ex_to<lst>(indices_subs.op(0)), ex_to<lst>(indices_subs.op(1)), subs_options::no_pattern|subs_options::no_index_renaming);
        }
    }
    return b;
}
 
ex rename_dummy_indices_uniquely(exvector & va, const ex & b, bool modify_va)
{
    if (va.size() > 0) {
        exvector vb = get_all_dummy_indices_safely(b);
        if (vb.size() > 0) {
            sort(vb.begin(), vb.end(), ex_is_less());
            lst indices_subs = rename_dummy_indices_uniquely(va, vb);
            if (indices_subs.op(0).nops() > 0) {
                if (modify_va) {
                    for (auto & i : ex_to<lst>(indices_subs.op(1)))
                        va.push_back(i);
                    exvector uncommon_indices;
                    set_difference(vb.begin(), vb.end(), indices_subs.op(0).begin(), indices_subs.op(0).end(), std::back_insert_iterator<exvector>(uncommon_indices), ex_is_less());
                    for (auto & ip : uncommon_indices)
                        va.push_back(ip);
                    sort(va.begin(), va.end(), ex_is_less());
                }
                return b.subs(ex_to<lst>(indices_subs.op(0)), ex_to<lst>(indices_subs.op(1)), subs_options::no_pattern|subs_options::no_index_renaming);
            }
        }
    }
    return b;
}
 
ex expand_dummy_sum(const ex & e, bool subs_idx)
{
    ex e_expanded = e.expand();
    pointer_to_map_function_1arg<bool> fcn(expand_dummy_sum, subs_idx);
    if (is_a<add>(e_expanded) || is_a<lst>(e_expanded) || is_a<matrix>(e_expanded)) {
        return e_expanded.map(fcn);
    } else if (is_a<ncmul>(e_expanded) || is_a<mul>(e_expanded) || is_a<power>(e_expanded) || is_a<indexed>(e_expanded)) {
        exvector v;
        if (is_a<indexed>(e_expanded))
            v = ex_to<indexed>(e_expanded).get_dummy_indices();
        else
            v = get_all_dummy_indices(e_expanded);
        ex result = e_expanded;
        for (const auto & nu : v) {
            if (ex_to<idx>(nu).get_dim().info(info_flags::nonnegint)) {
                int idim = ex_to<numeric>(ex_to<idx>(nu).get_dim()).to_int();
                ex en = 0;
                for (int i=0; i < idim; i++) {
                    if (subs_idx && is_a<varidx>(nu)) {
                        ex other = ex_to<varidx>(nu).toggle_variance();
                        en += result.subs(lst{
                            nu == idx(i, idim),
                            other == idx(i, idim)
                        });
                    } else {
                        en += result.subs( nu.op(0) == i );
                    }
                }
                result = en;
            }
        }
        return result;
    } else {
        return e;
    }
}
 
} // namespace GiNaC