mul.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 "mul.h"
#include "add.h"
#include "power.h"
#include "operators.h"
#include "matrix.h"
#include "indexed.h"
#include "lst.h"
#include "archive.h"
#include "utils.h"
#include "symbol.h"
#include "compiler.h"
 
#include <iostream>
#include <limits>
#include <stdexcept>
#include <vector>
 
namespace GiNaC {
 
GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mul, expairseq,
  print_func<print_context>(&mul::do_print).
  print_func<print_latex>(&mul::do_print_latex).
  print_func<print_csrc>(&mul::do_print_csrc).
  print_func<print_tree>(&mul::do_print_tree).
  print_func<print_python_repr>(&mul::do_print_python_repr))
 
 
 
// default constructor
 
mul::mul()
{
}
 
// other constructors
 
// public
 
mul::mul(const ex & lh, const ex & rh)
{
    overall_coeff = _ex1;
    construct_from_2_ex(lh,rh);
    GINAC_ASSERT(is_canonical());
}
 
mul::mul(const exvector & v)
{
    overall_coeff = _ex1;
    construct_from_exvector(v);
    GINAC_ASSERT(is_canonical());
}
 
mul::mul(const epvector & v)
{
    overall_coeff = _ex1;
    construct_from_epvector(v);
    GINAC_ASSERT(is_canonical());
}
 
mul::mul(const epvector & v, const ex & oc, bool do_index_renaming)
{
    overall_coeff = oc;
    construct_from_epvector(v, do_index_renaming);
    GINAC_ASSERT(is_canonical());
}
 
mul::mul(epvector && vp)
{
    overall_coeff = _ex1;
    construct_from_epvector(std::move(vp));
    GINAC_ASSERT(is_canonical());
}
 
mul::mul(epvector && vp, const ex & oc, bool do_index_renaming)
{
    overall_coeff = oc;
    construct_from_epvector(std::move(vp), do_index_renaming);
    GINAC_ASSERT(is_canonical());
}
 
mul::mul(const ex & lh, const ex & mh, const ex & rh)
{
    exvector factors;
    factors.reserve(3);
    factors.push_back(lh);
    factors.push_back(mh);
    factors.push_back(rh);
    overall_coeff = _ex1;
    construct_from_exvector(factors);
    GINAC_ASSERT(is_canonical());
}
 
// archiving
 
// functions overriding virtual functions from base classes
 
void mul::print_overall_coeff(const print_context & c, const char *mul_sym) const
{
    const numeric &coeff = ex_to<numeric>(overall_coeff);
    if (coeff.csgn() == -1)
        c.s << '-';
    if (!coeff.is_equal(*_num1_p) &&
        !coeff.is_equal(*_num_1_p)) {
        if (coeff.is_rational()) {
            if (coeff.is_negative())
                (-coeff).print(c);
            else
                coeff.print(c);
        } else {
            if (coeff.csgn() == -1)
                (-coeff).print(c, precedence());
            else
                coeff.print(c, precedence());
        }
        c.s << mul_sym;
    }
}
 
void mul::do_print(const print_context & c, unsigned level) const
{
    if (precedence() <= level)
        c.s << '(';
 
    print_overall_coeff(c, "*");
 
    bool first = true;
    for (auto & it : seq) {
        if (!first)
            c.s << '*';
        else
            first = false;
        recombine_pair_to_ex(it).print(c, precedence());
    }
 
    if (precedence() <= level)
        c.s << ')';
}
 
void mul::do_print_latex(const print_latex & c, unsigned level) const
{
    if (precedence() <= level)
        c.s << "{(";
 
    print_overall_coeff(c, " ");
 
    // Separate factors into those with negative numeric exponent
    // and all others
    exvector neg_powers, others;
    for (auto & it : seq) {
        GINAC_ASSERT(is_exactly_a<numeric>(it.coeff));
        if (ex_to<numeric>(it.coeff).is_negative())
            neg_powers.push_back(recombine_pair_to_ex(expair(it.rest, -it.coeff)));
        else
            others.push_back(recombine_pair_to_ex(it));
    }
 
    if (!neg_powers.empty()) {
 
        // Factors with negative exponent are printed as a fraction
        c.s << "\\frac{";
        mul(others).eval().print(c);
        c.s << "}{";
        mul(neg_powers).eval().print(c);
        c.s << "}";
 
    } else {
 
        // All other factors are printed in the ordinary way
        for (auto & vit : others) {
            c.s << ' ';
            vit.print(c, precedence());
        }
    }
 
    if (precedence() <= level)
        c.s << ")}";
}
 
void mul::do_print_csrc(const print_csrc & c, unsigned level) const
{
    if (precedence() <= level)
        c.s << "(";
 
    if (!overall_coeff.is_equal(_ex1)) {
        if (overall_coeff.is_equal(_ex_1))
            c.s << "-";
        else {
            overall_coeff.print(c, precedence());
            c.s << "*";
        }
    }
 
    // Print arguments, separated by "*" or "/"
    auto it = seq.begin(), itend = seq.end();
    while (it != itend) {
 
        // If the first argument is a negative integer power, it gets printed as "1.0/<expr>"
        bool needclosingparenthesis = false;
        if (it == seq.begin() && it->coeff.info(info_flags::negint)) {
            if (is_a<print_csrc_cl_N>(c)) {
                c.s << "recip(";
                needclosingparenthesis = true;
            } else
                c.s << "1.0/";
        }
 
        // If the exponent is 1 or -1, it is left out
        if (it->coeff.is_equal(_ex1) || it->coeff.is_equal(_ex_1))
            it->rest.print(c, precedence());
        else if (it->coeff.info(info_flags::negint))
            ex(power(it->rest, -ex_to<numeric>(it->coeff))).print(c, level);
        else
            ex(power(it->rest, ex_to<numeric>(it->coeff))).print(c, level);
 
        if (needclosingparenthesis)
            c.s << ")";
 
        // Separator is "/" for negative integer powers, "*" otherwise
        ++it;
        if (it != itend) {
            if (it->coeff.info(info_flags::negint))
                c.s << "/";
            else
                c.s << "*";
        }
    }
 
    if (precedence() <= level)
        c.s << ")";
}
 
void mul::do_print_python_repr(const print_python_repr & c, unsigned level) const
{
    c.s << class_name() << '(';
    op(0).print(c);
    for (size_t i=1; i<nops(); ++i) {
        c.s << ',';
        op(i).print(c);
    }
    c.s << ')';
}
 
bool mul::info(unsigned inf) const
{
    switch (inf) {
        case info_flags::polynomial:
        case info_flags::integer_polynomial:
        case info_flags::cinteger_polynomial:
        case info_flags::rational_polynomial:
        case info_flags::real:
        case info_flags::rational:
        case info_flags::integer:
        case info_flags::crational:
        case info_flags::cinteger:
        case info_flags::even:
        case info_flags::crational_polynomial:
        case info_flags::rational_function: {
            for (auto & it : seq) {
                if (!recombine_pair_to_ex(it).info(inf))
                    return false;
            }
            if (overall_coeff.is_equal(*_num1_p) && inf == info_flags::even)
                return true;
            return overall_coeff.info(inf);
        }
        case info_flags::positive:
        case info_flags::negative: {
            if ((inf==info_flags::positive) && (flags & status_flags::is_positive))
                return true;
            else if ((inf==info_flags::negative) && (flags & status_flags::is_negative))
                return true;
            if (flags & status_flags::purely_indefinite)
                return false;
 
            bool pos = true;
            for (auto & it : seq) {
                const ex& factor = recombine_pair_to_ex(it);
                if (factor.info(info_flags::positive))
                    continue;
                else if (factor.info(info_flags::negative))
                    pos = !pos;
                else
                    return false;
            }
            if (overall_coeff.info(info_flags::negative))
                pos = !pos;
            setflag(pos ? status_flags::is_positive : status_flags::is_negative);
            return (inf == info_flags::positive? pos : !pos);
        }
        case info_flags::nonnegative: {
            if  (flags & status_flags::is_positive)
                return true;
            bool pos = true;
            for (auto & it : seq) {
                const ex& factor = recombine_pair_to_ex(it);
                if (factor.info(info_flags::nonnegative) || factor.info(info_flags::positive))
                    continue;
                else if (factor.info(info_flags::negative))
                    pos = !pos;
                else
                    return false;
            }
            return (overall_coeff.info(info_flags::negative)? !pos : pos);
        }
        case info_flags::posint:
        case info_flags::negint: {
            bool pos = true;
            for (auto & it : seq) {
                const ex& factor = recombine_pair_to_ex(it);
                if (factor.info(info_flags::posint))
                    continue;
                else if (factor.info(info_flags::negint))
                    pos = !pos;
                else
                    return false;
            }
            if (overall_coeff.info(info_flags::negint))
                pos = !pos;
            else if (!overall_coeff.info(info_flags::posint))
                return false;
            return (inf ==info_flags::posint? pos : !pos); 
        }
        case info_flags::nonnegint: {
            bool pos = true;
            for (auto & it : seq) {
                const ex& factor = recombine_pair_to_ex(it);
                if (factor.info(info_flags::nonnegint) || factor.info(info_flags::posint))
                    continue;
                else if (factor.info(info_flags::negint))
                    pos = !pos;
                else
                    return false;
            }
            if (overall_coeff.info(info_flags::negint))
                pos = !pos;
            else if (!overall_coeff.info(info_flags::posint))
                return false;
            return pos; 
        }
        case info_flags::indefinite: {
            if (flags & status_flags::purely_indefinite)
                return true;
            if (flags & (status_flags::is_positive | status_flags::is_negative))
                return false;
            for (auto & it : seq) {
                const ex& term = recombine_pair_to_ex(it);
                if (term.info(info_flags::positive) || term.info(info_flags::negative))
                    return false;
            }
            setflag(status_flags::purely_indefinite);
            return true;
        }
    }
    return inherited::info(inf);
}
 
bool mul::is_polynomial(const ex & var) const
{
    for (auto & it : seq) {
        if (!it.rest.is_polynomial(var) ||
            (it.rest.has(var) && !it.coeff.info(info_flags::nonnegint))) {
            return false;
        }
    }
    return true;
}
 
int mul::degree(const ex & s) const
{
    // Sum up degrees of factors
    int deg_sum = 0;
    for (auto & it : seq) {
        if (ex_to<numeric>(it.coeff).is_integer())
            deg_sum += recombine_pair_to_ex(it).degree(s);
        else {
            if (it.rest.has(s))
                throw std::runtime_error("mul::degree() undefined degree because of non-integer exponent");
        }
    }
    return deg_sum;
}
 
int mul::ldegree(const ex & s) const
{
    // Sum up degrees of factors
    int deg_sum = 0;
    for (auto & it : seq) {
        if (ex_to<numeric>(it.coeff).is_integer())
            deg_sum += recombine_pair_to_ex(it).ldegree(s);
        else {
            if (it.rest.has(s))
                throw std::runtime_error("mul::ldegree() undefined degree because of non-integer exponent");
        }
    }
    return deg_sum;
}
 
ex mul::coeff(const ex & s, int n) const
{
    exvector coeffseq;
    coeffseq.reserve(seq.size()+1);
    
    if (n==0) {
        // product of individual coeffs
        // if a non-zero power of s is found, the resulting product will be 0
        for (auto & it : seq)
            coeffseq.push_back(recombine_pair_to_ex(it).coeff(s,n));
        coeffseq.push_back(overall_coeff);
        return dynallocate<mul>(coeffseq);
    }
    
    bool coeff_found = false;
    for (auto & it : seq) {
        ex t = recombine_pair_to_ex(it);
        ex c = t.coeff(s, n);
        if (!c.is_zero()) {
            coeffseq.push_back(c);
            coeff_found = 1;
        } else {
            coeffseq.push_back(t);
        }
    }
    if (coeff_found) {
        coeffseq.push_back(overall_coeff);
        return dynallocate<mul>(coeffseq);
    }
    
    return _ex0;
}
 
ex mul::eval() const
{
    if (flags & status_flags::evaluated) {
        GINAC_ASSERT(seq.size()>0);
        GINAC_ASSERT(seq.size()>1 || !overall_coeff.is_equal(_ex1));
        return *this;
    }
 
    const epvector evaled = evalchildren();
    if (unlikely(!evaled.empty())) {
        // start over evaluating a new object
        return dynallocate<mul>(std::move(evaled), overall_coeff);
    }
 
    size_t seq_size = seq.size();
    if (overall_coeff.is_zero()) {
        // *(...,x;0) -> 0
        return _ex0;
    } else if (seq_size==0) {
        // *(;c) -> c
        return overall_coeff;
    } else if (seq_size==1 && overall_coeff.is_equal(_ex1)) {
        // *(x;1) -> x
        return recombine_pair_to_ex(*(seq.begin()));
    } else if ((seq_size==1) &&
               is_exactly_a<add>((*seq.begin()).rest) &&
               ex_to<numeric>((*seq.begin()).coeff).is_equal(*_num1_p)) {
        // *(+(x,y,...);c) -> +(*(x,c),*(y,c),...) (c numeric(), no powers of +())
        const add & addref = ex_to<add>((*seq.begin()).rest);
        epvector distrseq;
        distrseq.reserve(addref.seq.size());
        for (auto & it : addref.seq) {
            distrseq.push_back(addref.combine_pair_with_coeff_to_pair(it, overall_coeff));
        }
        return dynallocate<add>(std::move(distrseq),
                                ex_to<numeric>(addref.overall_coeff).mul_dyn(ex_to<numeric>(overall_coeff)))
            .setflag(status_flags::evaluated);
    } else if ((seq_size >= 2) && (! (flags & status_flags::expanded))) {
        // Strip the content and the unit part from each term. Thus
        // things like (-x+a)*(3*x-3*a) automagically turn into - 3*(x-a)^2
 
        auto i = seq.begin(), last = seq.end();
        auto j = seq.begin();
        epvector s;
        numeric oc = *_num1_p;
        bool something_changed = false;
        while (i!=last) {
            if (likely(! (is_a<add>(i->rest) && i->coeff.is_equal(_ex1)))) {
                // power::eval has such a rule, no need to handle powers here
                ++i;
                continue;
            }
 
            // XXX: What is the best way to check if the polynomial is a primitive? 
            numeric c = i->rest.integer_content();
            const numeric lead_coeff =
                ex_to<numeric>(ex_to<add>(i->rest).seq.begin()->coeff).div(c);
            const bool canonicalizable = lead_coeff.is_integer();
 
            // XXX: The main variable is chosen in a random way, so this code 
            // does NOT transform the term into the canonical form (thus, in some
            // very unlucky event it can even loop forever). Hopefully the main
            // variable will be the same for all terms in *this
            const bool unit_normal = lead_coeff.is_pos_integer();
            if (likely((c == *_num1_p) && ((! canonicalizable) || unit_normal))) {
                ++i;
                continue;
            }
 
            if (! something_changed) {
                s.reserve(seq_size);
                something_changed = true;
            }
 
            while ((j!=i) && (j!=last)) {
                s.push_back(*j);
                ++j;
            }
 
            if (! unit_normal)
                c = c.mul(*_num_1_p);
 
            oc = oc.mul(c);
 
            // divide add by the number in place to save at least 2 .eval() calls
            const add& addref = ex_to<add>(i->rest);
            add & primitive = dynallocate<add>(addref);
            primitive.clearflag(status_flags::hash_calculated);
            primitive.overall_coeff = ex_to<numeric>(primitive.overall_coeff).div_dyn(c);
            for (auto & ai : primitive.seq)
                ai.coeff = ex_to<numeric>(ai.coeff).div_dyn(c);
 
            s.push_back(expair(primitive, _ex1));
 
            ++i;
            ++j;
        }
        if (something_changed) {
            while (j!=last) {
                s.push_back(*j);
                ++j;
            }
            return dynallocate<mul>(std::move(s), ex_to<numeric>(overall_coeff).mul_dyn(oc));
        }
    }
 
    return this->hold();
}
 
ex mul::evalf() const
{
    epvector s;
    s.reserve(seq.size());
 
    for (auto & it : seq)
        s.push_back(expair(it.rest.evalf(), it.coeff));
    return dynallocate<mul>(std::move(s), overall_coeff.evalf());
}
 
void mul::find_real_imag(ex & rp, ex & ip) const
{
    rp = overall_coeff.real_part();
    ip = overall_coeff.imag_part();
    for (auto & it : seq) {
        ex factor = recombine_pair_to_ex(it);
        ex new_rp = factor.real_part();
        ex new_ip = factor.imag_part();
        if (new_ip.is_zero()) {
            rp *= new_rp;
            ip *= new_rp;
        } else {
            ex temp = rp*new_rp - ip*new_ip;
            ip = ip*new_rp + rp*new_ip;
            rp = temp;
        }
    }
    rp = rp.expand();
    ip = ip.expand();
}
 
ex mul::real_part() const
{
    ex rp, ip;
    find_real_imag(rp, ip);
    return rp;
}
 
ex mul::imag_part() const
{
    ex rp, ip;
    find_real_imag(rp, ip);
    return ip;
}
 
ex mul::evalm() const
{
    // numeric*matrix
    if (seq.size() == 1 && seq[0].coeff.is_equal(_ex1)
     && is_a<matrix>(seq[0].rest))
        return ex_to<matrix>(seq[0].rest).mul(ex_to<numeric>(overall_coeff));
 
    // Evaluate children first, look whether there are any matrices at all
    // (there can be either no matrices or one matrix; if there were more
    // than one matrix, it would be a non-commutative product)
    epvector s;
    s.reserve(seq.size());
 
    bool have_matrix = false;
    epvector::iterator the_matrix;
 
    for (auto & it : seq) {
        const ex &m = recombine_pair_to_ex(it).evalm();
        s.push_back(split_ex_to_pair(m));
        if (is_a<matrix>(m)) {
            have_matrix = true;
            the_matrix = s.end() - 1;
        }
    }
 
    if (have_matrix) {
 
        // The product contained a matrix. We will multiply all other factors
        // into that matrix.
        matrix m = ex_to<matrix>(the_matrix->rest);
        s.erase(the_matrix);
        ex scalar = dynallocate<mul>(std::move(s), overall_coeff);
        return m.mul_scalar(scalar);
 
    } else
        return dynallocate<mul>(std::move(s), overall_coeff);
}
 
ex mul::eval_ncmul(const exvector & v) const
{
    if (seq.empty())
        return inherited::eval_ncmul(v);
 
    // Find first noncommutative element and call its eval_ncmul()
    for (auto & it : seq)
        if (it.rest.return_type() == return_types::noncommutative)
            return it.rest.eval_ncmul(v);
    return inherited::eval_ncmul(v);
}
 
bool tryfactsubs(const ex & origfactor, const ex & patternfactor, int & nummatches, exmap& repls)
{   
    ex origbase;
    int origexponent;
    int origexpsign;
 
    if (is_exactly_a<power>(origfactor) && origfactor.op(1).info(info_flags::integer)) {
        origbase = origfactor.op(0);
        int expon = ex_to<numeric>(origfactor.op(1)).to_int();
        origexponent = expon > 0 ? expon : -expon;
        origexpsign = expon > 0 ? 1 : -1;
    } else {
        origbase = origfactor;
        origexponent = 1;
        origexpsign = 1;
    }
 
    ex patternbase;
    int patternexponent;
    int patternexpsign;
 
    if (is_exactly_a<power>(patternfactor) && patternfactor.op(1).info(info_flags::integer)) {
        patternbase = patternfactor.op(0);
        int expon = ex_to<numeric>(patternfactor.op(1)).to_int();
        patternexponent = expon > 0 ? expon : -expon;
        patternexpsign = expon > 0 ? 1 : -1;
    } else {
        patternbase = patternfactor;
        patternexponent = 1;
        patternexpsign = 1;
    }
 
    exmap saverepls = repls;
    if (origexponent < patternexponent || origexpsign != patternexpsign || !origbase.match(patternbase,saverepls))
        return false;
    repls = saverepls;
 
    int newnummatches = origexponent / patternexponent;
    if (newnummatches < nummatches)
        nummatches = newnummatches;
    return true;
}
 
bool algebraic_match_mul_with_mul(const mul &e, const ex &pat, exmap& repls,
                                  int factor, int &nummatches, const std::vector<bool> &subsed,
                                  std::vector<bool> &matched)
{
    GINAC_ASSERT(subsed.size() == e.nops());
    GINAC_ASSERT(matched.size() == e.nops());
 
    if (factor == (int)pat.nops())
        return true;
 
    for (size_t i=0; i<e.nops(); ++i) {
        if(subsed[i] || matched[i])
            continue;
        exmap newrepls = repls;
        int newnummatches = nummatches;
        if (tryfactsubs(e.op(i), pat.op(factor), newnummatches, newrepls)) {
            matched[i] = true;
            if (algebraic_match_mul_with_mul(e, pat, newrepls, factor+1,
                    newnummatches, subsed, matched)) {
                repls = newrepls;
                nummatches = newnummatches;
                return true;
            }
            else
                matched[i] = false;
        }
    }
 
    return false;
}
 
bool mul::has(const ex & pattern, unsigned options) const
{
    if(!(options & has_options::algebraic))
        return basic::has(pattern,options);
    if(is_a<mul>(pattern)) {
        exmap repls;
        int nummatches = std::numeric_limits<int>::max();
        std::vector<bool> subsed(nops(), false);
        std::vector<bool> matched(nops(), false);
        if(algebraic_match_mul_with_mul(*this, pattern, repls, 0, nummatches,
                subsed, matched))
            return true;
    }
    return basic::has(pattern, options);
}
 
ex mul::algebraic_subs_mul(const exmap & m, unsigned options) const
{   
    std::vector<bool> subsed(nops(), false);
    ex divide_by = 1;
    ex multiply_by = 1;
 
    for (auto & it : m) {
 
        if (is_exactly_a<mul>(it.first)) {
retry1:
            int nummatches = std::numeric_limits<int>::max();
            std::vector<bool> currsubsed(nops(), false);
            exmap repls;
            
            if (!algebraic_match_mul_with_mul(*this, it.first, repls, 0, nummatches, subsed, currsubsed))
                continue;
 
            for (size_t j=0; j<subsed.size(); j++)
                if (currsubsed[j])
                    subsed[j] = true;
            ex subsed_pattern
                = it.first.subs(repls, subs_options::no_pattern);
            divide_by *= pow(subsed_pattern, nummatches);
            ex subsed_result
                = it.second.subs(repls, subs_options::no_pattern);
            multiply_by *= pow(subsed_result, nummatches);
            goto retry1;
 
        } else {
 
            for (size_t j=0; j<this->nops(); j++) {
                int nummatches = std::numeric_limits<int>::max();
                exmap repls;
                if (!subsed[j] && tryfactsubs(op(j), it.first, nummatches, repls)){
                    subsed[j] = true;
                    ex subsed_pattern
                        = it.first.subs(repls, subs_options::no_pattern);
                    divide_by *= pow(subsed_pattern, nummatches);
                    ex subsed_result
                        = it.second.subs(repls, subs_options::no_pattern);
                    multiply_by *= pow(subsed_result, nummatches);
                }
            }
        }
    }
 
    bool subsfound = false;
    for (size_t i=0; i<subsed.size(); i++) {
        if (subsed[i]) {
            subsfound = true;
            break;
        }
    }
    if (!subsfound)
        return subs_one_level(m, options | subs_options::algebraic);
 
    return ((*this)/divide_by)*multiply_by;
}
 
ex mul::conjugate() const
{
    // The base class' method is wrong here because we have to be careful at
    // branch cuts. power::conjugate takes care of that already, so use it.
    std::unique_ptr<epvector> newepv(nullptr);
    for (auto i=seq.begin(); i!=seq.end(); ++i) {
        if (newepv) {
            newepv->push_back(split_ex_to_pair(recombine_pair_to_ex(*i).conjugate()));
            continue;
        }
        ex x = recombine_pair_to_ex(*i);
        ex c = x.conjugate();
        if (c.is_equal(x)) {
            continue;
        }
        newepv.reset(new epvector);
        newepv->reserve(seq.size());
        for (auto j=seq.begin(); j!=i; ++j) {
            newepv->push_back(*j);
        }
        newepv->push_back(split_ex_to_pair(c));
    }
    ex x = overall_coeff.conjugate();
    if (!newepv && are_ex_trivially_equal(x, overall_coeff)) {
        return *this;
    }
    return thisexpairseq(newepv ? std::move(*newepv) : seq, x);
}
 
 
// protected
 
ex mul::derivative(const symbol & s) const
{
    size_t num = seq.size();
    exvector addseq;
    addseq.reserve(num);
    
    // D(a*b*c) = D(a)*b*c + a*D(b)*c + a*b*D(c)
    epvector mulseq = seq;
    auto i = seq.begin(), end = seq.end();
    auto i2 = mulseq.begin();
    while (i != end) {
        expair ep = split_ex_to_pair(pow(i->rest, i->coeff - _ex1) *
                                     i->rest.diff(s));
        ep.swap(*i2);
        addseq.push_back(dynallocate<mul>(mulseq, overall_coeff * i->coeff));
        ep.swap(*i2);
        ++i; ++i2;
    }
    return dynallocate<add>(addseq);
}
 
int mul::compare_same_type(const basic & other) const
{
    return inherited::compare_same_type(other);
}
 
unsigned mul::return_type() const
{
    if (seq.empty()) {
        // mul without factors: should not happen, but commutates
        return return_types::commutative;
    }
    
    bool all_commutative = true;
    epvector::const_iterator noncommutative_element; // point to first found nc element
    
    epvector::const_iterator i = seq.begin(), end = seq.end();
    while (i != end) {
        unsigned rt = i->rest.return_type();
        if (rt == return_types::noncommutative_composite)
            return rt; // one ncc -> mul also ncc
        if ((rt == return_types::noncommutative) && (all_commutative)) {
            // first nc element found, remember position
            noncommutative_element = i;
            all_commutative = false;
        }
        if ((rt == return_types::noncommutative) && (!all_commutative)) {
            // another nc element found, compare type_infos
            if (noncommutative_element->rest.return_type_tinfo() != i->rest.return_type_tinfo()) {
                    // different types -> mul is ncc
                    return return_types::noncommutative_composite;
            }
        }
        ++i;
    }
    // all factors checked
    return all_commutative ? return_types::commutative : return_types::noncommutative;
}
 
return_type_t mul::return_type_tinfo() const
{
    if (seq.empty())
        return make_return_type_t<mul>(); // mul without factors: should not happen
    
    // return type_info of first noncommutative element
    for (auto & it : seq)
        if (it.rest.return_type() == return_types::noncommutative)
            return it.rest.return_type_tinfo();
 
    // no noncommutative element found, should not happen
    return make_return_type_t<mul>();
}
 
ex mul::thisexpairseq(const epvector & v, const ex & oc, bool do_index_renaming) const
{
    return dynallocate<mul>(v, oc, do_index_renaming);
}
 
ex mul::thisexpairseq(epvector && vp, const ex & oc, bool do_index_renaming) const
{
    return dynallocate<mul>(std::move(vp), oc, do_index_renaming);
}
 
expair mul::split_ex_to_pair(const ex & e) const
{
    if (is_exactly_a<power>(e)) {
        const power & powerref = ex_to<power>(e);
        if (is_exactly_a<numeric>(powerref.exponent))
            return expair(powerref.basis,powerref.exponent);
    }
    return expair(e,_ex1);
}
 
expair mul::combine_ex_with_coeff_to_pair(const ex & e,
                                          const ex & c) const
{
    GINAC_ASSERT(is_exactly_a<numeric>(c));
 
    // First, try a common shortcut:
    if (is_exactly_a<symbol>(e))
        return expair(e, c);
 
    // trivial case: exponent 1
    if (c.is_equal(_ex1))
        return split_ex_to_pair(e);
 
    // to avoid duplication of power simplification rules,
    // we create a temporary power object
    // otherwise it would be hard to correctly evaluate
    // expression like (4^(1/3))^(3/2)
    return split_ex_to_pair(pow(e,c));
}
 
expair mul::combine_pair_with_coeff_to_pair(const expair & p,
                                            const ex & c) const
{
    GINAC_ASSERT(is_exactly_a<numeric>(p.coeff));
    GINAC_ASSERT(is_exactly_a<numeric>(c));
 
    // First, try a common shortcut:
    if (is_exactly_a<symbol>(p.rest))
        return expair(p.rest, p.coeff * c);
 
    // trivial case: exponent 1
    if (c.is_equal(_ex1))
        return p;
    if (p.coeff.is_equal(_ex1))
        return expair(p.rest, c);
 
    // to avoid duplication of power simplification rules,
    // we create a temporary power object
    // otherwise it would be hard to correctly evaluate
    // expression like (4^(1/3))^(3/2)
    return split_ex_to_pair(pow(recombine_pair_to_ex(p),c));
}
 
ex mul::recombine_pair_to_ex(const expair & p) const
{
    if (p.coeff.is_equal(_ex1))
        return p.rest;
    else
        return dynallocate<power>(p.rest, p.coeff);
}
 
bool mul::expair_needs_further_processing(epp it)
{
    if (is_exactly_a<mul>(it->rest) &&
        ex_to<numeric>(it->coeff).is_integer()) {
        // combined pair is product with integer power -> expand it
        *it = split_ex_to_pair(recombine_pair_to_ex(*it));
        return true;
    }
    if (is_exactly_a<numeric>(it->rest)) {
        if (it->coeff.is_equal(_ex1)) {
            // pair has coeff 1 and must be moved to the end
            return true;
        }
        expair ep = split_ex_to_pair(recombine_pair_to_ex(*it));
        if (!ep.is_equal(*it)) {
            // combined pair is a numeric power which can be simplified
            *it = ep;
            return true;
        }
    }
    return false;
}       
 
ex mul::default_overall_coeff() const
{
    return _ex1;
}
 
void mul::combine_overall_coeff(const ex & c)
{
    GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
    GINAC_ASSERT(is_exactly_a<numeric>(c));
    overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c));
}
 
void mul::combine_overall_coeff(const ex & c1, const ex & c2)
{
    GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
    GINAC_ASSERT(is_exactly_a<numeric>(c1));
    GINAC_ASSERT(is_exactly_a<numeric>(c2));
    overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c1).power(ex_to<numeric>(c2)));
}
 
bool mul::can_make_flat(const expair & p) const
{
    GINAC_ASSERT(is_exactly_a<numeric>(p.coeff));
 
    // (x*y)^c == x^c*y^c  if c ∈ ℤ
    return p.coeff.info(info_flags::integer);
}
 
bool mul::can_be_further_expanded(const ex & e)
{
    if (is_exactly_a<mul>(e)) {
        for (auto & it : ex_to<mul>(e).seq) {
            if (is_exactly_a<add>(it.rest) && it.coeff.info(info_flags::posint))
                return true;
        }
    } else if (is_exactly_a<power>(e)) {
        if (is_exactly_a<add>(e.op(0)) && e.op(1).info(info_flags::posint))
            return true;
    }
    return false;
}
 
ex mul::expand(unsigned options) const
{
    // Check for trivial case: expanding the monomial (~ 30% of all calls)
    bool monomial_case = true;
    for (const auto & i : seq) {
        if (!is_a<symbol>(i.rest) || !i.coeff.info(info_flags::integer)) {
            monomial_case = false;
            break;
        }
    }
    if (monomial_case) {
        setflag(status_flags::expanded);
        return *this;
    }
 
    // do not rename indices if the object has no indices at all
    if ((!(options & expand_options::expand_rename_idx)) && 
        this->info(info_flags::has_indices))
        options |= expand_options::expand_rename_idx;
 
    const bool skip_idx_rename = !(options & expand_options::expand_rename_idx);
 
    // First, expand the children
    epvector expanded = expandchildren(options);
    const epvector & expanded_seq = (expanded.empty() ? seq : expanded);
 
    // Now, look for all the factors that are sums and multiply each one out
    // with the next one that is found while collecting the factors which are
    // not sums
    ex last_expanded = _ex1;
 
    epvector non_adds;
    non_adds.reserve(expanded_seq.size());
 
    for (const auto & cit : expanded_seq) {
        if (is_exactly_a<add>(cit.rest) &&
            (cit.coeff.is_equal(_ex1))) {
            if (is_exactly_a<add>(last_expanded)) {
 
                // Expand a product of two sums, aggressive version.
                // Caring for the overall coefficients in separate loops can
                // sometimes give a performance gain of up to 15%!
 
                const int sizedifference = ex_to<add>(last_expanded).seq.size()-ex_to<add>(cit.rest).seq.size();
                // add2 is for the inner loop and should be the bigger of the two sums
                // in the presence of asymptotically good sorting:
                const add& add1 = (sizedifference<0 ? ex_to<add>(last_expanded) : ex_to<add>(cit.rest));
                const add& add2 = (sizedifference<0 ? ex_to<add>(cit.rest) : ex_to<add>(last_expanded));
                epvector distrseq;
                distrseq.reserve(add1.seq.size()+add2.seq.size());
 
                // Multiply add2 with the overall coefficient of add1 and append it to distrseq:
                if (!add1.overall_coeff.is_zero()) {
                    if (add1.overall_coeff.is_equal(_ex1))
                        distrseq.insert(distrseq.end(), add2.seq.begin(), add2.seq.end());
                    else
                        for (const auto & i : add2.seq)
                            distrseq.push_back(expair(i.rest, ex_to<numeric>(i.coeff).mul_dyn(ex_to<numeric>(add1.overall_coeff))));
                }
 
                // Multiply add1 with the overall coefficient of add2 and append it to distrseq:
                if (!add2.overall_coeff.is_zero()) {
                    if (add2.overall_coeff.is_equal(_ex1))
                        distrseq.insert(distrseq.end(), add1.seq.begin(), add1.seq.end());
                    else
                        for (const auto & i : add1.seq)
                            distrseq.push_back(expair(i.rest, ex_to<numeric>(i.coeff).mul_dyn(ex_to<numeric>(add2.overall_coeff))));
                }
 
                // Compute the new overall coefficient and put it together:
                ex tmp_accu = dynallocate<add>(distrseq, add1.overall_coeff*add2.overall_coeff);
 
                exvector add1_dummy_indices, add2_dummy_indices, add_indices;
                lst dummy_subs;
 
                if (!skip_idx_rename) {
                    for (const auto & i : add1.seq) {
                        add_indices = get_all_dummy_indices_safely(i.rest);
                        add1_dummy_indices.insert(add1_dummy_indices.end(), add_indices.begin(), add_indices.end());
                    }
                    for (const auto & i : add2.seq) {
                        add_indices = get_all_dummy_indices_safely(i.rest);
                        add2_dummy_indices.insert(add2_dummy_indices.end(), add_indices.begin(), add_indices.end());
                    }
 
                    sort(add1_dummy_indices.begin(), add1_dummy_indices.end(), ex_is_less());
                    sort(add2_dummy_indices.begin(), add2_dummy_indices.end(), ex_is_less());
                    dummy_subs = rename_dummy_indices_uniquely(add1_dummy_indices, add2_dummy_indices);
                }
 
                // Multiply explicitly all non-numeric terms of add1 and add2:
                for (const auto & i2 : add2.seq) {
                    // We really have to combine terms here in order to compactify
                    // the result.  Otherwise it would become waayy tooo bigg.
                    numeric oc(*_num0_p);
                    epvector distrseq2;
                    distrseq2.reserve(add1.seq.size());
                    const ex i2_new = (skip_idx_rename || (dummy_subs.op(0).nops() == 0) ?
                            i2.rest :
                            i2.rest.subs(ex_to<lst>(dummy_subs.op(0)),
                                         ex_to<lst>(dummy_subs.op(1)), subs_options::no_pattern));
                    for (const auto & i1 : add1.seq) {
                        // Don't push_back expairs which might have a rest that evaluates to a numeric,
                        // since that would violate an invariant of expairseq:
                        const ex rest = dynallocate<mul>(i1.rest, i2_new);
                        if (is_exactly_a<numeric>(rest)) {
                            oc += ex_to<numeric>(rest).mul(ex_to<numeric>(i1.coeff).mul(ex_to<numeric>(i2.coeff)));
                        } else {
                            distrseq2.push_back(expair(rest, ex_to<numeric>(i1.coeff).mul_dyn(ex_to<numeric>(i2.coeff))));
                        }
                    }
                    tmp_accu += dynallocate<add>(std::move(distrseq2), oc);
                }
                last_expanded = tmp_accu;
            } else {
                if (!last_expanded.is_equal(_ex1))
                    non_adds.push_back(split_ex_to_pair(last_expanded));
                last_expanded = cit.rest;
            }
 
        } else {
            non_adds.push_back(cit);
        }
    }
 
    // Now the only remaining thing to do is to multiply the factors which
    // were not sums into the "last_expanded" sum
    if (is_exactly_a<add>(last_expanded)) {
        size_t n = last_expanded.nops();
        exvector distrseq;
        distrseq.reserve(n);
        exvector va;
        if (! skip_idx_rename) {
            va = get_all_dummy_indices_safely(mul(non_adds));
            sort(va.begin(), va.end(), ex_is_less());
        }
 
        for (size_t i=0; i<n; ++i) {
            epvector factors = non_adds;
            if (skip_idx_rename)
                factors.push_back(split_ex_to_pair(last_expanded.op(i)));
            else
                factors.push_back(split_ex_to_pair(rename_dummy_indices_uniquely(va, last_expanded.op(i))));
            ex term = dynallocate<mul>(factors, overall_coeff);
            if (can_be_further_expanded(term)) {
                distrseq.push_back(term.expand());
            } else {
                if (options == 0)
                    ex_to<basic>(term).setflag(status_flags::expanded);
                distrseq.push_back(term);
            }
        }
 
        return dynallocate<add>(distrseq).setflag(options == 0 ? status_flags::expanded : 0);
    }
 
    non_adds.push_back(split_ex_to_pair(last_expanded));
    ex result = dynallocate<mul>(non_adds, overall_coeff);
    if (can_be_further_expanded(result)) {
        return result.expand();
    } else {
        if (options == 0)
            ex_to<basic>(result).setflag(status_flags::expanded);
        return result;
    }
}
 
  
// new virtual functions which can be overridden by derived classes
 
// none
 
// non-virtual functions in this class
 
 
epvector mul::expandchildren(unsigned options) const
{
    auto cit = seq.begin(), last = seq.end();
    while (cit!=last) {
        const ex & factor = recombine_pair_to_ex(*cit);
        const ex & expanded_factor = factor.expand(options);
        if (!are_ex_trivially_equal(factor,expanded_factor)) {
            
            // something changed, copy seq, eval and return it
            epvector s;
            s.reserve(seq.size());
            
            // copy parts of seq which are known not to have changed
            auto cit2 = seq.begin();
            while (cit2!=cit) {
                s.push_back(*cit2);
                ++cit2;
            }
 
            // copy first changed element
            s.push_back(split_ex_to_pair(expanded_factor));
            ++cit2;
 
            // copy rest
            while (cit2!=last) {
                s.push_back(split_ex_to_pair(recombine_pair_to_ex(*cit2).expand(options)));
                ++cit2;
            }
            return s;
        }
        ++cit;
    }
 
    return epvector(); // nothing has changed
}
 
GINAC_BIND_UNARCHIVER(mul);
 
} // namespace GiNaC