/** @file normal.cpp
 *
 *  This file implements several functions that work on univariate and
 *  multivariate polynomials and rational functions.
 *  These functions include polynomial quotient and remainder, GCD and LCM
 *  computation, square-free factorization and rational function normalization. */

/*
 *  GiNaC Copyright (C) 1999-2001 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., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
 */

#include <algorithm>
#include <map>

#include "normal.h"
#include "basic.h"
#include "ex.h"
#include "add.h"
#include "constant.h"
#include "expairseq.h"
#include "fail.h"
#include "inifcns.h"
#include "lst.h"
#include "mul.h"
#include "numeric.h"
#include "power.h"
#include "relational.h"
#include "pseries.h"
#include "symbol.h"
#include "utils.h"

namespace GiNaC {

// If comparing expressions (ex::compare()) is fast, you can set this to 1.
// Some routines like quo(), rem() and gcd() will then return a quick answer
// when they are called with two identical arguments.
#define FAST_COMPARE 1

// Set this if you want divide_in_z() to use remembering
#define USE_REMEMBER 0

// Set this if you want divide_in_z() to use trial division followed by
// polynomial interpolation (always slower except for completely dense
// polynomials)
#define USE_TRIAL_DIVISION 0

// Set this to enable some statistical output for the GCD routines
#define STATISTICS 0


#if STATISTICS
// Statistics variables
static int gcd_called = 0;
static int sr_gcd_called = 0;
static int heur_gcd_called = 0;
static int heur_gcd_failed = 0;

// Print statistics at end of program
static struct _stat_print {
	_stat_print() {}
	~_stat_print() {
		cout << "gcd() called " << gcd_called << " times\n";
		cout << "sr_gcd() called " << sr_gcd_called << " times\n";
		cout << "heur_gcd() called " << heur_gcd_called << " times\n";
		cout << "heur_gcd() failed " << heur_gcd_failed << " times\n";
	}
} stat_print;
#endif


/** Return pointer to first symbol found in expression.  Due to GiNaCīs
 *  internal ordering of terms, it may not be obvious which symbol this
 *  function returns for a given expression.
 *
 *  @param e  expression to search
 *  @param x  pointer to first symbol found (returned)
 *  @return "false" if no symbol was found, "true" otherwise */
static bool get_first_symbol(const ex &e, const symbol *&x)
{
	if (is_ex_exactly_of_type(e, symbol)) {
		x = static_cast<symbol *>(e.bp);
		return true;
	} else if (is_ex_exactly_of_type(e, add) || is_ex_exactly_of_type(e, mul)) {
		for (unsigned i=0; i<e.nops(); i++)
			if (get_first_symbol(e.op(i), x))
				return true;
	} else if (is_ex_exactly_of_type(e, power)) {
		if (get_first_symbol(e.op(0), x))
			return true;
	}
	return false;
}


/*
 *  Statistical information about symbols in polynomials
 */

/** This structure holds information about the highest and lowest degrees
 *  in which a symbol appears in two multivariate polynomials "a" and "b".
 *  A vector of these structures with information about all symbols in
 *  two polynomials can be created with the function get_symbol_stats().
 *
 *  @see get_symbol_stats */
struct sym_desc {
	/** Pointer to symbol */
	const symbol *sym;

	/** Highest degree of symbol in polynomial "a" */
	int deg_a;

	/** Highest degree of symbol in polynomial "b" */
	int deg_b;

	/** Lowest degree of symbol in polynomial "a" */
	int ldeg_a;

	/** Lowest degree of symbol in polynomial "b" */
	int ldeg_b;

	/** Maximum of deg_a and deg_b (Used for sorting) */
	int max_deg;

	/** Maximum number of terms of leading coefficient of symbol in both polynomials */
	int max_lcnops;

	/** Commparison operator for sorting */
	bool operator<(const sym_desc &x) const
	{
		if (max_deg == x.max_deg)
			return max_lcnops < x.max_lcnops;
		else
			return max_deg < x.max_deg;
	}
};

// Vector of sym_desc structures
typedef std::vector<sym_desc> sym_desc_vec;

// Add symbol the sym_desc_vec (used internally by get_symbol_stats())
static void add_symbol(const symbol *s, sym_desc_vec &v)
{
	sym_desc_vec::iterator it = v.begin(), itend = v.end();
	while (it != itend) {
		if (it->sym->compare(*s) == 0)  // If it's already in there, don't add it a second time
			return;
		it++;
	}
	sym_desc d;
	d.sym = s;
	v.push_back(d);
}

// Collect all symbols of an expression (used internally by get_symbol_stats())
static void collect_symbols(const ex &e, sym_desc_vec &v)
{
	if (is_ex_exactly_of_type(e, symbol)) {
		add_symbol(static_cast<symbol *>(e.bp), v);
	} else if (is_ex_exactly_of_type(e, add) || is_ex_exactly_of_type(e, mul)) {
		for (unsigned i=0; i<e.nops(); i++)
			collect_symbols(e.op(i), v);
	} else if (is_ex_exactly_of_type(e, power)) {
		collect_symbols(e.op(0), v);
	}
}

/** Collect statistical information about symbols in polynomials.
 *  This function fills in a vector of "sym_desc" structs which contain
 *  information about the highest and lowest degrees of all symbols that
 *  appear in two polynomials. The vector is then sorted by minimum
 *  degree (lowest to highest). The information gathered by this
 *  function is used by the GCD routines to identify trivial factors
 *  and to determine which variable to choose as the main variable
 *  for GCD computation.
 *
 *  @param a  first multivariate polynomial
 *  @param b  second multivariate polynomial
 *  @param v  vector of sym_desc structs (filled in) */
static void get_symbol_stats(const ex &a, const ex &b, sym_desc_vec &v)
{
	collect_symbols(a.eval(), v);   // eval() to expand assigned symbols
	collect_symbols(b.eval(), v);
	sym_desc_vec::iterator it = v.begin(), itend = v.end();
	while (it != itend) {
		int deg_a = a.degree(*(it->sym));
		int deg_b = b.degree(*(it->sym));
		it->deg_a = deg_a;
		it->deg_b = deg_b;
		it->max_deg = std::max(deg_a, deg_b);
		it->max_lcnops = std::max(a.lcoeff(*(it->sym)).nops(), b.lcoeff(*(it->sym)).nops());
		it->ldeg_a = a.ldegree(*(it->sym));
		it->ldeg_b = b.ldegree(*(it->sym));
		it++;
	}
	sort(v.begin(), v.end());
#if 0
	std::clog << "Symbols:\n";
	it = v.begin(); itend = v.end();
	while (it != itend) {
		std::clog << " " << *it->sym << ": deg_a=" << it->deg_a << ", deg_b=" << it->deg_b << ", ldeg_a=" << it->ldeg_a << ", ldeg_b=" << it->ldeg_b << ", max_deg=" << it->max_deg << ", max_lcnops=" << it->max_lcnops << endl;
		std::clog << "  lcoeff_a=" << a.lcoeff(*(it->sym)) << ", lcoeff_b=" << b.lcoeff(*(it->sym)) << endl;
		it++;
	}
#endif
}


/*
 *  Computation of LCM of denominators of coefficients of a polynomial
 */

// Compute LCM of denominators of coefficients by going through the
// expression recursively (used internally by lcm_of_coefficients_denominators())
static numeric lcmcoeff(const ex &e, const numeric &l)
{
	if (e.info(info_flags::rational))
		return lcm(ex_to<numeric>(e).denom(), l);
	else if (is_ex_exactly_of_type(e, add)) {
		numeric c = _num1();
		for (unsigned i=0; i<e.nops(); i++)
			c = lcmcoeff(e.op(i), c);
		return lcm(c, l);
	} else if (is_ex_exactly_of_type(e, mul)) {
		numeric c = _num1();
		for (unsigned i=0; i<e.nops(); i++)
			c *= lcmcoeff(e.op(i), _num1());
		return lcm(c, l);
	} else if (is_ex_exactly_of_type(e, power)) {
		if (is_ex_exactly_of_type(e.op(0), symbol))
			return l;
		else
			return pow(lcmcoeff(e.op(0), l), ex_to<numeric>(e.op(1)));
	}
	return l;
}

/** Compute LCM of denominators of coefficients of a polynomial.
 *  Given a polynomial with rational coefficients, this function computes
 *  the LCM of the denominators of all coefficients. This can be used
 *  to bring a polynomial from Q[X] to Z[X].
 *
 *  @param e  multivariate polynomial (need not be expanded)
 *  @return LCM of denominators of coefficients */
static numeric lcm_of_coefficients_denominators(const ex &e)
{
	return lcmcoeff(e, _num1());
}

/** Bring polynomial from Q[X] to Z[X] by multiplying in the previously
 *  determined LCM of the coefficient's denominators.
 *
 *  @param e  multivariate polynomial (need not be expanded)
 *  @param lcm  LCM to multiply in */
static ex multiply_lcm(const ex &e, const numeric &lcm)
{
	if (is_ex_exactly_of_type(e, mul)) {
		ex c = _ex1();
		numeric lcm_accum = _num1();
		for (unsigned i=0; i<e.nops(); i++) {
			numeric op_lcm = lcmcoeff(e.op(i), _num1());
			c *= multiply_lcm(e.op(i), op_lcm);
			lcm_accum *= op_lcm;
		}
		c *= lcm / lcm_accum;
		return c;
	} else if (is_ex_exactly_of_type(e, add)) {
		ex c = _ex0();
		for (unsigned i=0; i<e.nops(); i++)
			c += multiply_lcm(e.op(i), lcm);
		return c;
	} else if (is_ex_exactly_of_type(e, power)) {
		if (is_ex_exactly_of_type(e.op(0), symbol))
			return e * lcm;
		else
			return pow(multiply_lcm(e.op(0), lcm.power(ex_to<numeric>(e.op(1)).inverse())), e.op(1));
	} else
		return e * lcm;
}


/** Compute the integer content (= GCD of all numeric coefficients) of an
 *  expanded polynomial.
 *
 *  @param e  expanded polynomial
 *  @return integer content */
numeric ex::integer_content(void) const
{
	GINAC_ASSERT(bp!=0);
	return bp->integer_content();
}

numeric basic::integer_content(void) const
{
	return _num1();
}

numeric numeric::integer_content(void) const
{
	return abs(*this);
}

numeric add::integer_content(void) const
{
	epvector::const_iterator it = seq.begin();
	epvector::const_iterator itend = seq.end();
	numeric c = _num0();
	while (it != itend) {
		GINAC_ASSERT(!is_ex_exactly_of_type(it->rest,numeric));
		GINAC_ASSERT(is_ex_exactly_of_type(it->coeff,numeric));
		c = gcd(ex_to<numeric>(it->coeff), c);
		it++;
	}
	GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
	c = gcd(ex_to<numeric>(overall_coeff),c);
	return c;
}

numeric mul::integer_content(void) const
{
#ifdef DO_GINAC_ASSERT
	epvector::const_iterator it = seq.begin();
	epvector::const_iterator itend = seq.end();
	while (it != itend) {
		GINAC_ASSERT(!is_ex_exactly_of_type(recombine_pair_to_ex(*it),numeric));
		++it;
	}
#endif // def DO_GINAC_ASSERT
	GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
	return abs(ex_to<numeric>(overall_coeff));
}


/*
 *  Polynomial quotients and remainders
 */

/** Quotient q(x) of polynomials a(x) and b(x) in Q[x].
 *  It satisfies a(x)=b(x)*q(x)+r(x).
 *
 *  @param a  first polynomial in x (dividend)
 *  @param b  second polynomial in x (divisor)
 *  @param x  a and b are polynomials in x
 *  @param check_args  check whether a and b are polynomials with rational
 *         coefficients (defaults to "true")
 *  @return quotient of a and b in Q[x] */
ex quo(const ex &a, const ex &b, const symbol &x, bool check_args)
{
	if (b.is_zero())
		throw(std::overflow_error("quo: division by zero"));
	if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric))
		return a / b;
#if FAST_COMPARE
	if (a.is_equal(b))
		return _ex1();
#endif
	if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
		throw(std::invalid_argument("quo: arguments must be polynomials over the rationals"));

	// Polynomial long division
	ex q = _ex0();
	ex r = a.expand();
	if (r.is_zero())
		return r;
	int bdeg = b.degree(x);
	int rdeg = r.degree(x);
	ex blcoeff = b.expand().coeff(x, bdeg);
	bool blcoeff_is_numeric = is_ex_exactly_of_type(blcoeff, numeric);
	while (rdeg >= bdeg) {
		ex term, rcoeff = r.coeff(x, rdeg);
		if (blcoeff_is_numeric)
			term = rcoeff / blcoeff;
		else {
			if (!divide(rcoeff, blcoeff, term, false))
				return (new fail())->setflag(status_flags::dynallocated);
		}
		term *= power(x, rdeg - bdeg);
		q += term;
		r -= (term * b).expand();
		if (r.is_zero())
			break;
		rdeg = r.degree(x);
	}
	return q;
}


/** Remainder r(x) of polynomials a(x) and b(x) in Q[x].
 *  It satisfies a(x)=b(x)*q(x)+r(x).
 *
 *  @param a  first polynomial in x (dividend)
 *  @param b  second polynomial in x (divisor)
 *  @param x  a and b are polynomials in x
 *  @param check_args  check whether a and b are polynomials with rational
 *         coefficients (defaults to "true")
 *  @return remainder of a(x) and b(x) in Q[x] */
ex rem(const ex &a, const ex &b, const symbol &x, bool check_args)
{
	if (b.is_zero())
		throw(std::overflow_error("rem: division by zero"));
	if (is_ex_exactly_of_type(a, numeric)) {
		if  (is_ex_exactly_of_type(b, numeric))
			return _ex0();
		else
			return b;
	}
#if FAST_COMPARE
	if (a.is_equal(b))
		return _ex0();
#endif
	if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
		throw(std::invalid_argument("rem: arguments must be polynomials over the rationals"));

	// Polynomial long division
	ex r = a.expand();
	if (r.is_zero())
		return r;
	int bdeg = b.degree(x);
	int rdeg = r.degree(x);
	ex blcoeff = b.expand().coeff(x, bdeg);
	bool blcoeff_is_numeric = is_ex_exactly_of_type(blcoeff, numeric);
	while (rdeg >= bdeg) {
		ex term, rcoeff = r.coeff(x, rdeg);
		if (blcoeff_is_numeric)
			term = rcoeff / blcoeff;
		else {
			if (!divide(rcoeff, blcoeff, term, false))
				return (new fail())->setflag(status_flags::dynallocated);
		}
		term *= power(x, rdeg - bdeg);
		r -= (term * b).expand();
		if (r.is_zero())
			break;
		rdeg = r.degree(x);
	}
	return r;
}


/** Pseudo-remainder of polynomials a(x) and b(x) in Z[x].
 *
 *  @param a  first polynomial in x (dividend)
 *  @param b  second polynomial in x (divisor)
 *  @param x  a and b are polynomials in x
 *  @param check_args  check whether a and b are polynomials with rational
 *         coefficients (defaults to "true")
 *  @return pseudo-remainder of a(x) and b(x) in Z[x] */
ex prem(const ex &a, const ex &b, const symbol &x, bool check_args)
{
	if (b.is_zero())
		throw(std::overflow_error("prem: division by zero"));
	if (is_ex_exactly_of_type(a, numeric)) {
		if (is_ex_exactly_of_type(b, numeric))
			return _ex0();
		else
			return b;
	}
	if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
		throw(std::invalid_argument("prem: arguments must be polynomials over the rationals"));

	// Polynomial long division
	ex r = a.expand();
	ex eb = b.expand();
	int rdeg = r.degree(x);
	int bdeg = eb.degree(x);
	ex blcoeff;
	if (bdeg <= rdeg) {
		blcoeff = eb.coeff(x, bdeg);
		if (bdeg == 0)
			eb = _ex0();
		else
			eb -= blcoeff * power(x, bdeg);
	} else
		blcoeff = _ex1();

	int delta = rdeg - bdeg + 1, i = 0;
	while (rdeg >= bdeg && !r.is_zero()) {
		ex rlcoeff = r.coeff(x, rdeg);
		ex term = (power(x, rdeg - bdeg) * eb * rlcoeff).expand();
		if (rdeg == 0)
			r = _ex0();
		else
			r -= rlcoeff * power(x, rdeg);
		r = (blcoeff * r).expand() - term;
		rdeg = r.degree(x);
		i++;
	}
	return power(blcoeff, delta - i) * r;
}


/** Sparse pseudo-remainder of polynomials a(x) and b(x) in Z[x].
 *
 *  @param a  first polynomial in x (dividend)
 *  @param b  second polynomial in x (divisor)
 *  @param x  a and b are polynomials in x
 *  @param check_args  check whether a and b are polynomials with rational
 *         coefficients (defaults to "true")
 *  @return sparse pseudo-remainder of a(x) and b(x) in Z[x] */

ex sprem(const ex &a, const ex &b, const symbol &x, bool check_args)
{
	if (b.is_zero())
		throw(std::overflow_error("prem: division by zero"));
	if (is_ex_exactly_of_type(a, numeric)) {
		if (is_ex_exactly_of_type(b, numeric))
			return _ex0();
		else
			return b;
	}
	if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
		throw(std::invalid_argument("prem: arguments must be polynomials over the rationals"));

	// Polynomial long division
	ex r = a.expand();
	ex eb = b.expand();
	int rdeg = r.degree(x);
	int bdeg = eb.degree(x);
	ex blcoeff;
	if (bdeg <= rdeg) {
		blcoeff = eb.coeff(x, bdeg);
		if (bdeg == 0)
			eb = _ex0();
		else
			eb -= blcoeff * power(x, bdeg);
	} else
		blcoeff = _ex1();

	while (rdeg >= bdeg && !r.is_zero()) {
		ex rlcoeff = r.coeff(x, rdeg);
		ex term = (power(x, rdeg - bdeg) * eb * rlcoeff).expand();
		if (rdeg == 0)
			r = _ex0();
		else
			r -= rlcoeff * power(x, rdeg);
		r = (blcoeff * r).expand() - term;
		rdeg = r.degree(x);
	}
	return r;
}


/** Exact polynomial division of a(X) by b(X) in Q[X].
 *  
 *  @param a  first multivariate polynomial (dividend)
 *  @param b  second multivariate polynomial (divisor)
 *  @param q  quotient (returned)
 *  @param check_args  check whether a and b are polynomials with rational
 *         coefficients (defaults to "true")
 *  @return "true" when exact division succeeds (quotient returned in q),
 *          "false" otherwise */
bool divide(const ex &a, const ex &b, ex &q, bool check_args)
{
	q = _ex0();
	if (b.is_zero())
		throw(std::overflow_error("divide: division by zero"));
	if (a.is_zero())
		return true;
	if (is_ex_exactly_of_type(b, numeric)) {
		q = a / b;
		return true;
	} else if (is_ex_exactly_of_type(a, numeric))
		return false;
#if FAST_COMPARE
	if (a.is_equal(b)) {
		q = _ex1();
		return true;
	}
#endif
	if (check_args && (!a.info(info_flags::rational_polynomial) ||
	                   !b.info(info_flags::rational_polynomial)))
		throw(std::invalid_argument("divide: arguments must be polynomials over the rationals"));

	// Find first symbol
	const symbol *x;
	if (!get_first_symbol(a, x) && !get_first_symbol(b, x))
		throw(std::invalid_argument("invalid expression in divide()"));

	// Polynomial long division (recursive)
	ex r = a.expand();
	if (r.is_zero())
		return true;
	int bdeg = b.degree(*x);
	int rdeg = r.degree(*x);
	ex blcoeff = b.expand().coeff(*x, bdeg);
	bool blcoeff_is_numeric = is_ex_exactly_of_type(blcoeff, numeric);
	while (rdeg >= bdeg) {
		ex term, rcoeff = r.coeff(*x, rdeg);
		if (blcoeff_is_numeric)
			term = rcoeff / blcoeff;
		else
			if (!divide(rcoeff, blcoeff, term, false))
				return false;
		term *= power(*x, rdeg - bdeg);
		q += term;
		r -= (term * b).expand();
		if (r.is_zero())
			return true;
		rdeg = r.degree(*x);
	}
	return false;
}


#if USE_REMEMBER
/*
 *  Remembering
 */

typedef std::pair<ex, ex> ex2;
typedef std::pair<ex, bool> exbool;

struct ex2_less {
	bool operator() (const ex2 &p, const ex2 &q) const 
	{
		int cmp = p.first.compare(q.first);
		return ((cmp<0) || (!(cmp>0) && p.second.compare(q.second)<0));
	}
};

typedef std::map<ex2, exbool, ex2_less> ex2_exbool_remember;
#endif


/** Exact polynomial division of a(X) by b(X) in Z[X].
 *  This functions works like divide() but the input and output polynomials are
 *  in Z[X] instead of Q[X] (i.e. they have integer coefficients). Unlike
 *  divide(), it doesnīt check whether the input polynomials really are integer
 *  polynomials, so be careful of what you pass in. Also, you have to run
 *  get_symbol_stats() over the input polynomials before calling this function
 *  and pass an iterator to the first element of the sym_desc vector. This
 *  function is used internally by the heur_gcd().
 *  
 *  @param a  first multivariate polynomial (dividend)
 *  @param b  second multivariate polynomial (divisor)
 *  @param q  quotient (returned)
 *  @param var  iterator to first element of vector of sym_desc structs
 *  @return "true" when exact division succeeds (the quotient is returned in
 *          q), "false" otherwise.
 *  @see get_symbol_stats, heur_gcd */
static bool divide_in_z(const ex &a, const ex &b, ex &q, sym_desc_vec::const_iterator var)
{
	q = _ex0();
	if (b.is_zero())
		throw(std::overflow_error("divide_in_z: division by zero"));
	if (b.is_equal(_ex1())) {
		q = a;
		return true;
	}
	if (is_ex_exactly_of_type(a, numeric)) {
		if (is_ex_exactly_of_type(b, numeric)) {
			q = a / b;
			return q.info(info_flags::integer);
		} else
			return false;
	}
#if FAST_COMPARE
	if (a.is_equal(b)) {
		q = _ex1();
		return true;
	}
#endif

#if USE_REMEMBER
	// Remembering
	static ex2_exbool_remember dr_remember;
	ex2_exbool_remember::const_iterator remembered = dr_remember.find(ex2(a, b));
	if (remembered != dr_remember.end()) {
		q = remembered->second.first;
		return remembered->second.second;
	}
#endif

	// Main symbol
	const symbol *x = var->sym;

	// Compare degrees
	int adeg = a.degree(*x), bdeg = b.degree(*x);
	if (bdeg > adeg)
		return false;

#if USE_TRIAL_DIVISION

	// Trial division with polynomial interpolation
	int i, k;

	// Compute values at evaluation points 0..adeg
	vector<numeric> alpha; alpha.reserve(adeg + 1);
	exvector u; u.reserve(adeg + 1);
	numeric point = _num0();
	ex c;
	for (i=0; i<=adeg; i++) {
		ex bs = b.subs(*x == point);
		while (bs.is_zero()) {
			point += _num1();
			bs = b.subs(*x == point);
		}
		if (!divide_in_z(a.subs(*x == point), bs, c, var+1))
			return false;
		alpha.push_back(point);
		u.push_back(c);
		point += _num1();
	}

	// Compute inverses
	vector<numeric> rcp; rcp.reserve(adeg + 1);
	rcp.push_back(_num0());
	for (k=1; k<=adeg; k++) {
		numeric product = alpha[k] - alpha[0];
		for (i=1; i<k; i++)
			product *= alpha[k] - alpha[i];
		rcp.push_back(product.inverse());
	}

	// Compute Newton coefficients
	exvector v; v.reserve(adeg + 1);
	v.push_back(u[0]);
	for (k=1; k<=adeg; k++) {
		ex temp = v[k - 1];
		for (i=k-2; i>=0; i--)
			temp = temp * (alpha[k] - alpha[i]) + v[i];
		v.push_back((u[k] - temp) * rcp[k]);
	}

	// Convert from Newton form to standard form
	c = v[adeg];
	for (k=adeg-1; k>=0; k--)
		c = c * (*x - alpha[k]) + v[k];

	if (c.degree(*x) == (adeg - bdeg)) {
		q = c.expand();
		return true;
	} else
		return false;

#else

	// Polynomial long division (recursive)
	ex r = a.expand();
	if (r.is_zero())
		return true;
	int rdeg = adeg;
	ex eb = b.expand();
	ex blcoeff = eb.coeff(*x, bdeg);
	while (rdeg >= bdeg) {
		ex term, rcoeff = r.coeff(*x, rdeg);
		if (!divide_in_z(rcoeff, blcoeff, term, var+1))
			break;
		term = (term * power(*x, rdeg - bdeg)).expand();
		q += term;
		r -= (term * eb).expand();
		if (r.is_zero()) {
#if USE_REMEMBER
			dr_remember[ex2(a, b)] = exbool(q, true);
#endif
			return true;
		}
		rdeg = r.degree(*x);
	}
#if USE_REMEMBER
	dr_remember[ex2(a, b)] = exbool(q, false);
#endif
	return false;

#endif
}


/*
 *  Separation of unit part, content part and primitive part of polynomials
 */

/** Compute unit part (= sign of leading coefficient) of a multivariate
 *  polynomial in Z[x]. The product of unit part, content part, and primitive
 *  part is the polynomial itself.
 *
 *  @param x  variable in which to compute the unit part
 *  @return unit part
 *  @see ex::content, ex::primpart */
ex ex::unit(const symbol &x) const
{
	ex c = expand().lcoeff(x);
	if (is_ex_exactly_of_type(c, numeric))
		return c < _ex0() ? _ex_1() : _ex1();
	else {
		const symbol *y;
		if (get_first_symbol(c, y))
			return c.unit(*y);
		else
			throw(std::invalid_argument("invalid expression in unit()"));
	}
}


/** Compute content part (= unit normal GCD of all coefficients) of a
 *  multivariate polynomial in Z[x].  The product of unit part, content part,
 *  and primitive part is the polynomial itself.
 *
 *  @param x  variable in which to compute the content part
 *  @return content part
 *  @see ex::unit, ex::primpart */
ex ex::content(const symbol &x) const
{
	if (is_zero())
		return _ex0();
	if (is_ex_exactly_of_type(*this, numeric))
		return info(info_flags::negative) ? -*this : *this;
	ex e = expand();
	if (e.is_zero())
		return _ex0();

	// First, try the integer content
	ex c = e.integer_content();
	ex r = e / c;
	ex lcoeff = r.lcoeff(x);
	if (lcoeff.info(info_flags::integer))
		return c;

	// GCD of all coefficients
	int deg = e.degree(x);
	int ldeg = e.ldegree(x);
	if (deg == ldeg)
		return e.lcoeff(x) / e.unit(x);
	c = _ex0();
	for (int i=ldeg; i<=deg; i++)
		c = gcd(e.coeff(x, i), c, NULL, NULL, false);
	return c;
}


/** Compute primitive part of a multivariate polynomial in Z[x].
 *  The product of unit part, content part, and primitive part is the
 *  polynomial itself.
 *
 *  @param x  variable in which to compute the primitive part
 *  @return primitive part
 *  @see ex::unit, ex::content */
ex ex::primpart(const symbol &x) const
{
	if (is_zero())
		return _ex0();
	if (is_ex_exactly_of_type(*this, numeric))
		return _ex1();

	ex c = content(x);
	if (c.is_zero())
		return _ex0();
	ex u = unit(x);
	if (is_ex_exactly_of_type(c, numeric))
		return *this / (c * u);
	else
		return quo(*this, c * u, x, false);
}


/** Compute primitive part of a multivariate polynomial in Z[x] when the
 *  content part is already known. This function is faster in computing the
 *  primitive part than the previous function.
 *
 *  @param x  variable in which to compute the primitive part
 *  @param c  previously computed content part
 *  @return primitive part */
ex ex::primpart(const symbol &x, const ex &c) const
{
	if (is_zero())
		return _ex0();
	if (c.is_zero())
		return _ex0();
	if (is_ex_exactly_of_type(*this, numeric))
		return _ex1();

	ex u = unit(x);
	if (is_ex_exactly_of_type(c, numeric))
		return *this / (c * u);
	else
		return quo(*this, c * u, x, false);
}


/*
 *  GCD of multivariate polynomials
 */

/** Compute GCD of polynomials in Q[X] using the Euclidean algorithm (not
 *  really suited for multivariate GCDs). This function is only provided for
 *  testing purposes.
 *
 *  @param a  first multivariate polynomial
 *  @param b  second multivariate polynomial
 *  @param x  pointer to symbol (main variable) in which to compute the GCD in
 *  @return the GCD as a new expression
 *  @see gcd */

static ex eu_gcd(const ex &a, const ex &b, const symbol *x)
{
//std::clog << "eu_gcd(" << a << "," << b << ")\n";

	// Sort c and d so that c has higher degree
	ex c, d;
	int adeg = a.degree(*x), bdeg = b.degree(*x);
	if (adeg >= bdeg) {
		c = a;
		d = b;
	} else {
		c = b;
		d = a;
	}

	// Normalize in Q[x]
	c = c / c.lcoeff(*x);
	d = d / d.lcoeff(*x);

	// Euclidean algorithm
	ex r;
	for (;;) {
//std::clog << " d = " << d << endl;
		r = rem(c, d, *x, false);
		if (r.is_zero())
			return d / d.lcoeff(*x);
		c = d;
		d = r;
	}
}


/** Compute GCD of multivariate polynomials using the Euclidean PRS algorithm
 *  with pseudo-remainders ("World's Worst GCD Algorithm", staying in Z[X]).
 *  This function is only provided for testing purposes.
 *
 *  @param a  first multivariate polynomial
 *  @param b  second multivariate polynomial
 *  @param x  pointer to symbol (main variable) in which to compute the GCD in
 *  @return the GCD as a new expression
 *  @see gcd */

static ex euprem_gcd(const ex &a, const ex &b, const symbol *x)
{
//std::clog << "euprem_gcd(" << a << "," << b << ")\n";

	// Sort c and d so that c has higher degree
	ex c, d;
	int adeg = a.degree(*x), bdeg = b.degree(*x);
	if (adeg >= bdeg) {
		c = a;
		d = b;
	} else {
		c = b;
		d = a;
	}

	// Calculate GCD of contents
	ex gamma = gcd(c.content(*x), d.content(*x), NULL, NULL, false);

	// Euclidean algorithm with pseudo-remainders
	ex r;
	for (;;) {
//std::clog << " d = " << d << endl;
		r = prem(c, d, *x, false);
		if (r.is_zero())
			return d.primpart(*x) * gamma;
		c = d;
		d = r;
	}
}


/** Compute GCD of multivariate polynomials using the primitive Euclidean
 *  PRS algorithm (complete content removal at each step). This function is
 *  only provided for testing purposes.
 *
 *  @param a  first multivariate polynomial
 *  @param b  second multivariate polynomial
 *  @param x  pointer to symbol (main variable) in which to compute the GCD in
 *  @return the GCD as a new expression
 *  @see gcd */

static ex peu_gcd(const ex &a, const ex &b, const symbol *x)
{
//std::clog << "peu_gcd(" << a << "," << b << ")\n";

	// Sort c and d so that c has higher degree
	ex c, d;
	int adeg = a.degree(*x), bdeg = b.degree(*x);
	int ddeg;
	if (adeg >= bdeg) {
		c = a;
		d = b;
		ddeg = bdeg;
	} else {
		c = b;
		d = a;
		ddeg = adeg;
	}

	// Remove content from c and d, to be attached to GCD later
	ex cont_c = c.content(*x);
	ex cont_d = d.content(*x);
	ex gamma = gcd(cont_c, cont_d, NULL, NULL, false);
	if (ddeg == 0)
		return gamma;
	c = c.primpart(*x, cont_c);
	d = d.primpart(*x, cont_d);

	// Euclidean algorithm with content removal
	ex r;
	for (;;) {
//std::clog << " d = " << d << endl;
		r = prem(c, d, *x, false);
		if (r.is_zero())
			return gamma * d;
		c = d;
		d = r.primpart(*x);
	}
}


/** Compute GCD of multivariate polynomials using the reduced PRS algorithm.
 *  This function is only provided for testing purposes.
 *
 *  @param a  first multivariate polynomial
 *  @param b  second multivariate polynomial
 *  @param x  pointer to symbol (main variable) in which to compute the GCD in
 *  @return the GCD as a new expression
 *  @see gcd */

static ex red_gcd(const ex &a, const ex &b, const symbol *x)
{
//std::clog << "red_gcd(" << a << "," << b << ")\n";

	// Sort c and d so that c has higher degree
	ex c, d;
	int adeg = a.degree(*x), bdeg = b.degree(*x);
	int cdeg, ddeg;
	if (adeg >= bdeg) {
		c = a;
		d = b;
		cdeg = adeg;
		ddeg = bdeg;
	} else {
		c = b;
		d = a;
		cdeg = bdeg;
		ddeg = adeg;
	}

	// Remove content from c and d, to be attached to GCD later
	ex cont_c = c.content(*x);
	ex cont_d = d.content(*x);
	ex gamma = gcd(cont_c, cont_d, NULL, NULL, false);
	if (ddeg == 0)
		return gamma;
	c = c.primpart(*x, cont_c);
	d = d.primpart(*x, cont_d);

	// First element of divisor sequence
	ex r, ri = _ex1();
	int delta = cdeg - ddeg;

	for (;;) {
		// Calculate polynomial pseudo-remainder
//std::clog << " d = " << d << endl;
		r = prem(c, d, *x, false);
		if (r.is_zero())
			return gamma * d.primpart(*x);
		c = d;
		cdeg = ddeg;

		if (!divide(r, pow(ri, delta), d, false))
			throw(std::runtime_error("invalid expression in red_gcd(), division failed"));
		ddeg = d.degree(*x);
		if (ddeg == 0) {
			if (is_ex_exactly_of_type(r, numeric))
				return gamma;
			else
				return gamma * r.primpart(*x);
		}

		ri = c.expand().lcoeff(*x);
		delta = cdeg - ddeg;
	}
}


/** Compute GCD of multivariate polynomials using the subresultant PRS
 *  algorithm. This function is used internally by gcd().
 *
 *  @param a   first multivariate polynomial
 *  @param b   second multivariate polynomial
 *  @param var iterator to first element of vector of sym_desc structs
 *  @return the GCD as a new expression
 *  @see gcd */

static ex sr_gcd(const ex &a, const ex &b, sym_desc_vec::const_iterator var)
{
//std::clog << "sr_gcd(" << a << "," << b << ")\n";
#if STATISTICS
	sr_gcd_called++;
#endif

	// The first symbol is our main variable
	const symbol &x = *(var->sym);

	// Sort c and d so that c has higher degree
	ex c, d;
	int adeg = a.degree(x), bdeg = b.degree(x);
	int cdeg, ddeg;
	if (adeg >= bdeg) {
		c = a;
		d = b;
		cdeg = adeg;
		ddeg = bdeg;
	} else {
		c = b;
		d = a;
		cdeg = bdeg;
		ddeg = adeg;
	}

	// Remove content from c and d, to be attached to GCD later
	ex cont_c = c.content(x);
	ex cont_d = d.content(x);
	ex gamma = gcd(cont_c, cont_d, NULL, NULL, false);
	if (ddeg == 0)
		return gamma;
	c = c.primpart(x, cont_c);
	d = d.primpart(x, cont_d);
//std::clog << " content " << gamma << " removed, continuing with sr_gcd(" << c << "," << d << ")\n";

	// First element of subresultant sequence
	ex r = _ex0(), ri = _ex1(), psi = _ex1();
	int delta = cdeg - ddeg;

	for (;;) {
		// Calculate polynomial pseudo-remainder
//std::clog << " start of loop, psi = " << psi << ", calculating pseudo-remainder...\n";
//std::clog << " d = " << d << endl;
		r = prem(c, d, x, false);
		if (r.is_zero())
			return gamma * d.primpart(x);
		c = d;
		cdeg = ddeg;
//std::clog << " dividing...\n";
		if (!divide_in_z(r, ri * pow(psi, delta), d, var))
			throw(std::runtime_error("invalid expression in sr_gcd(), division failed"));
		ddeg = d.degree(x);
		if (ddeg == 0) {
			if (is_ex_exactly_of_type(r, numeric))
				return gamma;
			else
				return gamma * r.primpart(x);
		}

		// Next element of subresultant sequence
//std::clog << " calculating next subresultant...\n";
		ri = c.expand().lcoeff(x);
		if (delta == 1)
			psi = ri;
		else if (delta)
			divide_in_z(pow(ri, delta), pow(psi, delta-1), psi, var+1);
		delta = cdeg - ddeg;
	}
}


/** Return maximum (absolute value) coefficient of a polynomial.
 *  This function is used internally by heur_gcd().
 *
 *  @param e  expanded multivariate polynomial
 *  @return maximum coefficient
 *  @see heur_gcd */
numeric ex::max_coefficient(void) const
{
	GINAC_ASSERT(bp!=0);
	return bp->max_coefficient();
}

/** Implementation ex::max_coefficient().
 *  @see heur_gcd */
numeric basic::max_coefficient(void) const
{
	return _num1();
}

numeric numeric::max_coefficient(void) const
{
	return abs(*this);
}

numeric add::max_coefficient(void) const
{
	epvector::const_iterator it = seq.begin();
	epvector::const_iterator itend = seq.end();
	GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
	numeric cur_max = abs(ex_to<numeric>(overall_coeff));
	while (it != itend) {
		numeric a;
		GINAC_ASSERT(!is_ex_exactly_of_type(it->rest,numeric));
		a = abs(ex_to<numeric>(it->coeff));
		if (a > cur_max)
			cur_max = a;
		it++;
	}
	return cur_max;
}

numeric mul::max_coefficient(void) const
{
#ifdef DO_GINAC_ASSERT
	epvector::const_iterator it = seq.begin();
	epvector::const_iterator itend = seq.end();
	while (it != itend) {
		GINAC_ASSERT(!is_ex_exactly_of_type(recombine_pair_to_ex(*it),numeric));
		it++;
	}
#endif // def DO_GINAC_ASSERT
	GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
	return abs(ex_to<numeric>(overall_coeff));
}


/** Apply symmetric modular homomorphism to a multivariate polynomial.
 *  This function is used internally by heur_gcd().
 *
 *  @param e  expanded multivariate polynomial
 *  @param xi  modulus
 *  @return mapped polynomial
 *  @see heur_gcd */
ex ex::smod(const numeric &xi) const
{
	GINAC_ASSERT(bp!=0);
	return bp->smod(xi);
}

ex basic::smod(const numeric &xi) const
{
	return *this;
}

ex numeric::smod(const numeric &xi) const
{
	return GiNaC::smod(*this, xi);
}

ex add::smod(const numeric &xi) const
{
	epvector newseq;
	newseq.reserve(seq.size()+1);
	epvector::const_iterator it = seq.begin();
	epvector::const_iterator itend = seq.end();
	while (it != itend) {
		GINAC_ASSERT(!is_ex_exactly_of_type(it->rest,numeric));
		numeric coeff = GiNaC::smod(ex_to<numeric>(it->coeff), xi);
		if (!coeff.is_zero())
			newseq.push_back(expair(it->rest, coeff));
		it++;
	}
	GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
	numeric coeff = GiNaC::smod(ex_to<numeric>(overall_coeff), xi);
	return (new add(newseq,coeff))->setflag(status_flags::dynallocated);
}

ex mul::smod(const numeric &xi) const
{
#ifdef DO_GINAC_ASSERT
	epvector::const_iterator it = seq.begin();
	epvector::const_iterator itend = seq.end();
	while (it != itend) {
		GINAC_ASSERT(!is_ex_exactly_of_type(recombine_pair_to_ex(*it),numeric));
		it++;
	}
#endif // def DO_GINAC_ASSERT
	mul * mulcopyp = new mul(*this);
	GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
	mulcopyp->overall_coeff = GiNaC::smod(ex_to<numeric>(overall_coeff),xi);
	mulcopyp->clearflag(status_flags::evaluated);
	mulcopyp->clearflag(status_flags::hash_calculated);
	return mulcopyp->setflag(status_flags::dynallocated);
}


/** xi-adic polynomial interpolation */
static ex interpolate(const ex &gamma, const numeric &xi, const symbol &x)
{
	ex g = _ex0();
	ex e = gamma;
	numeric rxi = xi.inverse();
	for (int i=0; !e.is_zero(); i++) {
		ex gi = e.smod(xi);
		g += gi * power(x, i);
		e = (e - gi) * rxi;
	}
	return g;
}

/** Exception thrown by heur_gcd() to signal failure. */
class gcdheu_failed {};

/** Compute GCD of multivariate polynomials using the heuristic GCD algorithm.
 *  get_symbol_stats() must have been called previously with the input
 *  polynomials and an iterator to the first element of the sym_desc vector
 *  passed in. This function is used internally by gcd().
 *
 *  @param a  first multivariate polynomial (expanded)
 *  @param b  second multivariate polynomial (expanded)
 *  @param ca  cofactor of polynomial a (returned), NULL to suppress
 *             calculation of cofactor
 *  @param cb  cofactor of polynomial b (returned), NULL to suppress
 *             calculation of cofactor
 *  @param var iterator to first element of vector of sym_desc structs
 *  @return the GCD as a new expression
 *  @see gcd
 *  @exception gcdheu_failed() */
static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const_iterator var)
{
//std::clog << "heur_gcd(" << a << "," << b << ")\n";
#if STATISTICS
	heur_gcd_called++;
#endif

	// Algorithms only works for non-vanishing input polynomials
	if (a.is_zero() || b.is_zero())
		return (new fail())->setflag(status_flags::dynallocated);

	// GCD of two numeric values -> CLN
	if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric)) {
		numeric g = gcd(ex_to<numeric>(a), ex_to<numeric>(b));
		if (ca)
			*ca = ex_to<numeric>(a) / g;
		if (cb)
			*cb = ex_to<numeric>(b) / g;
		return g;
	}

	// The first symbol is our main variable
	const symbol &x = *(var->sym);

	// Remove integer content
	numeric gc = gcd(a.integer_content(), b.integer_content());
	numeric rgc = gc.inverse();
	ex p = a * rgc;
	ex q = b * rgc;
	int maxdeg =  std::max(p.degree(x),q.degree(x));
	
	// Find evaluation point
	numeric mp = p.max_coefficient();
	numeric mq = q.max_coefficient();
	numeric xi;
	if (mp > mq)
		xi = mq * _num2() + _num2();
	else
		xi = mp * _num2() + _num2();

	// 6 tries maximum
	for (int t=0; t<6; t++) {
		if (xi.int_length() * maxdeg > 100000) {
//std::clog << "giving up heur_gcd, xi.int_length = " << xi.int_length() << ", maxdeg = " << maxdeg << std::endl;
			throw gcdheu_failed();
		}

		// Apply evaluation homomorphism and calculate GCD
		ex cp, cq;
		ex gamma = heur_gcd(p.subs(x == xi), q.subs(x == xi), &cp, &cq, var+1).expand();
		if (!is_ex_exactly_of_type(gamma, fail)) {

			// Reconstruct polynomial from GCD of mapped polynomials
			ex g = interpolate(gamma, xi, x);

			// Remove integer content
			g /= g.integer_content();

			// If the calculated polynomial divides both p and q, this is the GCD
			ex dummy;
			if (divide_in_z(p, g, ca ? *ca : dummy, var) && divide_in_z(q, g, cb ? *cb : dummy, var)) {
				g *= gc;
				ex lc = g.lcoeff(x);
				if (is_ex_exactly_of_type(lc, numeric) && ex_to<numeric>(lc).is_negative())
					return -g;
				else
					return g;
			}
#if 0
			cp = interpolate(cp, xi, x);
			if (divide_in_z(cp, p, g, var)) {
				if (divide_in_z(g, q, cb ? *cb : dummy, var)) {
					g *= gc;
					if (ca)
						*ca = cp;
					ex lc = g.lcoeff(x);
					if (is_ex_exactly_of_type(lc, numeric) && ex_to<numeric>(lc).is_negative())
						return -g;
					else
						return g;
				}
			}
			cq = interpolate(cq, xi, x);
			if (divide_in_z(cq, q, g, var)) {
				if (divide_in_z(g, p, ca ? *ca : dummy, var)) {
					g *= gc;
					if (cb)
						*cb = cq;
					ex lc = g.lcoeff(x);
					if (is_ex_exactly_of_type(lc, numeric) && ex_to<numeric>(lc).is_negative())
						return -g;
					else
						return g;
				}
			}
#endif
		}

		// Next evaluation point
		xi = iquo(xi * isqrt(isqrt(xi)) * numeric(73794), numeric(27011));
	}
	return (new fail())->setflag(status_flags::dynallocated);
}


/** Compute GCD (Greatest Common Divisor) of multivariate polynomials a(X)
 *  and b(X) in Z[X].
 *
 *  @param a  first multivariate polynomial
 *  @param b  second multivariate polynomial
 *  @param check_args  check whether a and b are polynomials with rational
 *         coefficients (defaults to "true")
 *  @return the GCD as a new expression */
ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args)
{
//std::clog << "gcd(" << a << "," << b << ")\n";
#if STATISTICS
	gcd_called++;
#endif

	// GCD of numerics -> CLN
	if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric)) {
		numeric g = gcd(ex_to<numeric>(a), ex_to<numeric>(b));
		if (ca || cb) {
			if (g.is_zero()) {
				if (ca)
					*ca = _ex0();
				if (cb)
					*cb = _ex0();
			} else {
				if (ca)
					*ca = ex_to<numeric>(a) / g;
				if (cb)
					*cb = ex_to<numeric>(b) / g;
			}
		}
		return g;
	}

	// Check arguments
	if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial))) {
		throw(std::invalid_argument("gcd: arguments must be polynomials over the rationals"));
	}

	// Partially factored cases (to avoid expanding large expressions)
	if (is_ex_exactly_of_type(a, mul)) {
		if (is_ex_exactly_of_type(b, mul) && b.nops() > a.nops())
			goto factored_b;
factored_a:
		ex g = _ex1();
		ex acc_ca = _ex1();
		ex part_b = b;
		for (unsigned i=0; i<a.nops(); i++) {
			ex part_ca, part_cb;
			g *= gcd(a.op(i), part_b, &part_ca, &part_cb, check_args);
			acc_ca *= part_ca;
			part_b = part_cb;
		}
		if (ca)
			*ca = acc_ca;
		if (cb)
			*cb = part_b;
		return g;
	} else if (is_ex_exactly_of_type(b, mul)) {
		if (is_ex_exactly_of_type(a, mul) && a.nops() > b.nops())
			goto factored_a;
factored_b:
		ex g = _ex1();
		ex acc_cb = _ex1();
		ex part_a = a;
		for (unsigned i=0; i<b.nops(); i++) {
			ex part_ca, part_cb;
			g *= gcd(part_a, b.op(i), &part_ca, &part_cb, check_args);
			acc_cb *= part_cb;
			part_a = part_ca;
		}
		if (ca)
			*ca = part_a;
		if (cb)
			*cb = acc_cb;
		return g;
	}

#if FAST_COMPARE
	// Input polynomials of the form poly^n are sometimes also trivial
	if (is_ex_exactly_of_type(a, power)) {
		ex p = a.op(0);
		if (is_ex_exactly_of_type(b, power)) {
			if (p.is_equal(b.op(0))) {
				// a = p^n, b = p^m, gcd = p^min(n, m)
				ex exp_a = a.op(1), exp_b = b.op(1);
				if (exp_a < exp_b) {
					if (ca)
						*ca = _ex1();
					if (cb)
						*cb = power(p, exp_b - exp_a);
					return power(p, exp_a);
				} else {
					if (ca)
						*ca = power(p, exp_a - exp_b);
					if (cb)
						*cb = _ex1();
					return power(p, exp_b);
				}
			}
		} else {
			if (p.is_equal(b)) {
				// a = p^n, b = p, gcd = p
				if (ca)
					*ca = power(p, a.op(1) - 1);
				if (cb)
					*cb = _ex1();
				return p;
			}
		}
	} else if (is_ex_exactly_of_type(b, power)) {
		ex p = b.op(0);
		if (p.is_equal(a)) {
			// a = p, b = p^n, gcd = p
			if (ca)
				*ca = _ex1();
			if (cb)
				*cb = power(p, b.op(1) - 1);
			return p;
		}
	}
#endif

	// Some trivial cases
	ex aex = a.expand(), bex = b.expand();
	if (aex.is_zero()) {
		if (ca)
			*ca = _ex0();
		if (cb)
			*cb = _ex1();
		return b;
	}
	if (bex.is_zero()) {
		if (ca)
			*ca = _ex1();
		if (cb)
			*cb = _ex0();
		return a;
	}
	if (aex.is_equal(_ex1()) || bex.is_equal(_ex1())) {
		if (ca)
			*ca = a;
		if (cb)
			*cb = b;
		return _ex1();
	}
#if FAST_COMPARE
	if (a.is_equal(b)) {
		if (ca)
			*ca = _ex1();
		if (cb)
			*cb = _ex1();
		return a;
	}
#endif

	// Gather symbol statistics
	sym_desc_vec sym_stats;
	get_symbol_stats(a, b, sym_stats);

	// The symbol with least degree is our main variable
	sym_desc_vec::const_iterator var = sym_stats.begin();
	const symbol &x = *(var->sym);

	// Cancel trivial common factor
	int ldeg_a = var->ldeg_a;
	int ldeg_b = var->ldeg_b;
	int min_ldeg = std::min(ldeg_a,ldeg_b);
	if (min_ldeg > 0) {
		ex common = power(x, min_ldeg);
//std::clog << "trivial common factor " << common << std::endl;
		return gcd((aex / common).expand(), (bex / common).expand(), ca, cb, false) * common;
	}

	// Try to eliminate variables
	if (var->deg_a == 0) {
//std::clog << "eliminating variable " << x << " from b" << std::endl;
		ex c = bex.content(x);
		ex g = gcd(aex, c, ca, cb, false);
		if (cb)
			*cb *= bex.unit(x) * bex.primpart(x, c);
		return g;
	} else if (var->deg_b == 0) {
//std::clog << "eliminating variable " << x << " from a" << std::endl;
		ex c = aex.content(x);
		ex g = gcd(c, bex, ca, cb, false);
		if (ca)
			*ca *= aex.unit(x) * aex.primpart(x, c);
		return g;
	}

	ex g;
#if 1
	// Try heuristic algorithm first, fall back to PRS if that failed
	try {
		g = heur_gcd(aex, bex, ca, cb, var);
	} catch (gcdheu_failed) {
		g = fail();
	}
	if (is_ex_exactly_of_type(g, fail)) {
//std::clog << "heuristics failed" << std::endl;
#if STATISTICS
		heur_gcd_failed++;
#endif
#endif
//		g = heur_gcd(aex, bex, ca, cb, var);
//		g = eu_gcd(aex, bex, &x);
//		g = euprem_gcd(aex, bex, &x);
//		g = peu_gcd(aex, bex, &x);
//		g = red_gcd(aex, bex, &x);
		g = sr_gcd(aex, bex, var);
		if (g.is_equal(_ex1())) {
			// Keep cofactors factored if possible
			if (ca)
				*ca = a;
			if (cb)
				*cb = b;
		} else {
			if (ca)
				divide(aex, g, *ca, false);
			if (cb)
				divide(bex, g, *cb, false);
		}
#if 1
	} else {
		if (g.is_equal(_ex1())) {
			// Keep cofactors factored if possible
			if (ca)
				*ca = a;
			if (cb)
				*cb = b;
		}
	}
#endif
	return g;
}


/** Compute LCM (Least Common Multiple) of multivariate polynomials in Z[X].
 *
 *  @param a  first multivariate polynomial
 *  @param b  second multivariate polynomial
 *  @param check_args  check whether a and b are polynomials with rational
 *         coefficients (defaults to "true")
 *  @return the LCM as a new expression */
ex lcm(const ex &a, const ex &b, bool check_args)
{
	if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric))
		return lcm(ex_to<numeric>(a), ex_to<numeric>(b));
	if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
		throw(std::invalid_argument("lcm: arguments must be polynomials over the rationals"));
	
	ex ca, cb;
	ex g = gcd(a, b, &ca, &cb, false);
	return ca * cb * g;
}


/*
 *  Square-free factorization
 */

/** Compute square-free factorization of multivariate polynomial a(x) using
 *  Yunīs algorithm.  Used internally by sqrfree().
 *
 *  @param a  multivariate polynomial over Z[X], treated here as univariate
 *            polynomial in x.
 *  @param x  variable to factor in
 *  @return   vector of factors sorted in ascending degree */
static exvector sqrfree_yun(const ex &a, const symbol &x)
{
	exvector res;
	ex w = a;
	ex z = w.diff(x);
	ex g = gcd(w, z);
	if (g.is_equal(_ex1())) {
		res.push_back(a);
		return res;
	}
	ex y;
	do {
		w = quo(w, g, x);
		y = quo(z, g, x);
		z = y - w.diff(x);
		g = gcd(w, z);
		res.push_back(g);
	} while (!z.is_zero());
	return res;
}
/** Compute square-free factorization of multivariate polynomial in Q[X].
 *
 *  @param a  multivariate polynomial over Q[X]
 *  @param x  lst of variables to factor in, may be left empty for autodetection
 *  @return   polynomail a in square-free factored form. */
ex sqrfree(const ex &a, const lst &l)
{
	if (is_ex_of_type(a,numeric) ||     // algorithm does not trap a==0
	    is_ex_of_type(a,symbol))        // shortcut
		return a;
	// If no lst of variables to factorize in was specified we have to
	// invent one now.  Maybe one can optimize here by reversing the order
	// or so, I don't know.
	lst args;
	if (l.nops()==0) {
		sym_desc_vec sdv;
		get_symbol_stats(a, _ex0(), sdv);
		for (sym_desc_vec::iterator it=sdv.begin(); it!=sdv.end(); ++it)
			args.append(*it->sym);
	} else {
		args = l;
	}
	// Find the symbol to factor in at this stage
	if (!is_ex_of_type(args.op(0), symbol))
		throw (std::runtime_error("sqrfree(): invalid factorization variable"));
	const symbol x = ex_to<symbol>(args.op(0));
	// convert the argument from something in Q[X] to something in Z[X]
	numeric lcm = lcm_of_coefficients_denominators(a);
	ex tmp = multiply_lcm(a,lcm);
	// find the factors
	exvector factors = sqrfree_yun(tmp,x);
	// construct the next list of symbols with the first element popped
	lst newargs;
	for (int i=1; i<args.nops(); ++i)
		newargs.append(args.op(i));
	// recurse down the factors in remaining vars
	if (newargs.nops()>0) {
		for (exvector::iterator i=factors.begin(); i!=factors.end(); ++i)
			*i = sqrfree(*i, newargs);
	}
	// Done with recursion, now construct the final result
	ex result = _ex1();
	exvector::iterator it = factors.begin();
	for (int p = 1; it!=factors.end(); ++it, ++p)
		result *= power(*it, p);
	// Yun's algorithm does not account for constant factors.  (For
	// univariate polynomials it works only in the monic case.)  We can
	// correct this by inserting what has been lost back into the result:
	result = result * quo(tmp, result, x);
	return result *	lcm.inverse();
}


/*
 *  Normal form of rational functions
 */

/*
 *  Note: The internal normal() functions (= basic::normal() and overloaded
 *  functions) all return lists of the form {numerator, denominator}. This
 *  is to get around mul::eval()'s automatic expansion of numeric coefficients.
 *  E.g. (a+b)/3 is automatically converted to a/3+b/3 but we want to keep
 *  the information that (a+b) is the numerator and 3 is the denominator.
 */

/** Create a symbol for replacing the expression "e" (or return a previously
 *  assigned symbol). The symbol is appended to sym_lst and returned, the
 *  expression is appended to repl_lst.
 *  @see ex::normal */
static ex replace_with_symbol(const ex &e, lst &sym_lst, lst &repl_lst)
{
	// Expression already in repl_lst? Then return the assigned symbol
	for (unsigned i=0; i<repl_lst.nops(); i++)
		if (repl_lst.op(i).is_equal(e))
			return sym_lst.op(i);
	
	// Otherwise create new symbol and add to list, taking care that the
	// replacement expression doesn't contain symbols from the sym_lst
	// because subs() is not recursive
	symbol s;
	ex es(s);
	ex e_replaced = e.subs(sym_lst, repl_lst);
	sym_lst.append(es);
	repl_lst.append(e_replaced);
	return es;
}

/** Create a symbol for replacing the expression "e" (or return a previously
 *  assigned symbol). An expression of the form "symbol == expression" is added
 *  to repl_lst and the symbol is returned.
 *  @see ex::to_rational */
static ex replace_with_symbol(const ex &e, lst &repl_lst)
{
	// Expression already in repl_lst? Then return the assigned symbol
	for (unsigned i=0; i<repl_lst.nops(); i++)
		if (repl_lst.op(i).op(1).is_equal(e))
			return repl_lst.op(i).op(0);
	
	// Otherwise create new symbol and add to list, taking care that the
	// replacement expression doesn't contain symbols from the sym_lst
	// because subs() is not recursive
	symbol s;
	ex es(s);
	ex e_replaced = e.subs(repl_lst);
	repl_lst.append(es == e_replaced);
	return es;
}

/** Default implementation of ex::normal(). It replaces the object with a
 *  temporary symbol.
 *  @see ex::normal */
ex basic::normal(lst &sym_lst, lst &repl_lst, int level) const
{
	return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
}


/** Implementation of ex::normal() for symbols. This returns the unmodified symbol.
 *  @see ex::normal */
ex symbol::normal(lst &sym_lst, lst &repl_lst, int level) const
{
	return (new lst(*this, _ex1()))->setflag(status_flags::dynallocated);
}


/** Implementation of ex::normal() for a numeric. It splits complex numbers
 *  into re+I*im and replaces I and non-rational real numbers with a temporary
 *  symbol.
 *  @see ex::normal */
ex numeric::normal(lst &sym_lst, lst &repl_lst, int level) const
{
	numeric num = numer();
	ex numex = num;

	if (num.is_real()) {
		if (!num.is_integer())
			numex = replace_with_symbol(numex, sym_lst, repl_lst);
	} else { // complex
		numeric re = num.real(), im = num.imag();
		ex re_ex = re.is_rational() ? re : replace_with_symbol(re, sym_lst, repl_lst);
		ex im_ex = im.is_rational() ? im : replace_with_symbol(im, sym_lst, repl_lst);
		numex = re_ex + im_ex * replace_with_symbol(I, sym_lst, repl_lst);
	}

	// Denominator is always a real integer (see numeric::denom())
	return (new lst(numex, denom()))->setflag(status_flags::dynallocated);
}


/** Fraction cancellation.
 *  @param n  numerator
 *  @param d  denominator
 *  @return cancelled fraction {n, d} as a list */
static ex frac_cancel(const ex &n, const ex &d)
{
	ex num = n;
	ex den = d;
	numeric pre_factor = _num1();

//std::clog << "frac_cancel num = " << num << ", den = " << den << std::endl;

	// Handle trivial case where denominator is 1
	if (den.is_equal(_ex1()))
		return (new lst(num, den))->setflag(status_flags::dynallocated);

	// Handle special cases where numerator or denominator is 0
	if (num.is_zero())
		return (new lst(num, _ex1()))->setflag(status_flags::dynallocated);
	if (den.expand().is_zero())
		throw(std::overflow_error("frac_cancel: division by zero in frac_cancel"));

	// Bring numerator and denominator to Z[X] by multiplying with
	// LCM of all coefficients' denominators
	numeric num_lcm = lcm_of_coefficients_denominators(num);
	numeric den_lcm = lcm_of_coefficients_denominators(den);
	num = multiply_lcm(num, num_lcm);
	den = multiply_lcm(den, den_lcm);
	pre_factor = den_lcm / num_lcm;

	// Cancel GCD from numerator and denominator
	ex cnum, cden;
	if (gcd(num, den, &cnum, &cden, false) != _ex1()) {
		num = cnum;
		den = cden;
	}

	// Make denominator unit normal (i.e. coefficient of first symbol
	// as defined by get_first_symbol() is made positive)
	const symbol *x;
	if (get_first_symbol(den, x)) {
		GINAC_ASSERT(is_ex_exactly_of_type(den.unit(*x),numeric));
		if (ex_to<numeric>(den.unit(*x)).is_negative()) {
			num *= _ex_1();
			den *= _ex_1();
		}
	}

	// Return result as list
//std::clog << " returns num = " << num << ", den = " << den << ", pre_factor = " << pre_factor << std::endl;
	return (new lst(num * pre_factor.numer(), den * pre_factor.denom()))->setflag(status_flags::dynallocated);
}


/** Implementation of ex::normal() for a sum. It expands terms and performs
 *  fractional addition.
 *  @see ex::normal */
ex add::normal(lst &sym_lst, lst &repl_lst, int level) const
{
	if (level == 1)
		return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
	else if (level == -max_recursion_level)
		throw(std::runtime_error("max recursion level reached"));

	// Normalize children and split each one into numerator and denominator
	exvector nums, dens;
	nums.reserve(seq.size()+1);
	dens.reserve(seq.size()+1);
	epvector::const_iterator it = seq.begin(), itend = seq.end();
	while (it != itend) {
		ex n = recombine_pair_to_ex(*it).bp->normal(sym_lst, repl_lst, level-1);
		nums.push_back(n.op(0));
		dens.push_back(n.op(1));
		it++;
	}
	ex n = overall_coeff.bp->normal(sym_lst, repl_lst, level-1);
	nums.push_back(n.op(0));
	dens.push_back(n.op(1));
	GINAC_ASSERT(nums.size() == dens.size());

	// Now, nums is a vector of all numerators and dens is a vector of
	// all denominators
//std::clog << "add::normal uses " << nums.size() << " summands:\n";

	// Add fractions sequentially
	exvector::const_iterator num_it = nums.begin(), num_itend = nums.end();
	exvector::const_iterator den_it = dens.begin(), den_itend = dens.end();
//std::clog << " num = " << *num_it << ", den = " << *den_it << std::endl;
	ex num = *num_it++, den = *den_it++;
	while (num_it != num_itend) {
//std::clog << " num = " << *num_it << ", den = " << *den_it << std::endl;
		ex next_num = *num_it++, next_den = *den_it++;

		// Trivially add sequences of fractions with identical denominators
		while ((den_it != den_itend) && next_den.is_equal(*den_it)) {
			next_num += *num_it;
			num_it++; den_it++;
		}

		// Additiion of two fractions, taking advantage of the fact that
		// the heuristic GCD algorithm computes the cofactors at no extra cost
		ex co_den1, co_den2;
		ex g = gcd(den, next_den, &co_den1, &co_den2, false);
		num = ((num * co_den2) + (next_num * co_den1)).expand();
		den *= co_den2;		// this is the lcm(den, next_den)
	}
//std::clog << " common denominator = " << den << std::endl;

	// Cancel common factors from num/den
	return frac_cancel(num, den);
}


/** Implementation of ex::normal() for a product. It cancels common factors
 *  from fractions.
 *  @see ex::normal() */
ex mul::normal(lst &sym_lst, lst &repl_lst, int level) const
{
	if (level == 1)
		return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
	else if (level == -max_recursion_level)
		throw(std::runtime_error("max recursion level reached"));

	// Normalize children, separate into numerator and denominator
	ex num = _ex1();
	ex den = _ex1(); 
	ex n;
	epvector::const_iterator it = seq.begin(), itend = seq.end();
	while (it != itend) {
		n = recombine_pair_to_ex(*it).bp->normal(sym_lst, repl_lst, level-1);
		num *= n.op(0);
		den *= n.op(1);
		it++;
	}
	n = overall_coeff.bp->normal(sym_lst, repl_lst, level-1);
	num *= n.op(0);
	den *= n.op(1);

	// Perform fraction cancellation
	return frac_cancel(num, den);
}


/** Implementation of ex::normal() for powers. It normalizes the basis,
 *  distributes integer exponents to numerator and denominator, and replaces
 *  non-integer powers by temporary symbols.
 *  @see ex::normal */
ex power::normal(lst &sym_lst, lst &repl_lst, int level) const
{
	if (level == 1)
		return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
	else if (level == -max_recursion_level)
		throw(std::runtime_error("max recursion level reached"));

	// Normalize basis and exponent (exponent gets reassembled)
	ex n_basis = basis.bp->normal(sym_lst, repl_lst, level-1);
	ex n_exponent = exponent.bp->normal(sym_lst, repl_lst, level-1);
	n_exponent = n_exponent.op(0) / n_exponent.op(1);

	if (n_exponent.info(info_flags::integer)) {

		if (n_exponent.info(info_flags::positive)) {

			// (a/b)^n -> {a^n, b^n}
			return (new lst(power(n_basis.op(0), n_exponent), power(n_basis.op(1), n_exponent)))->setflag(status_flags::dynallocated);

		} else if (n_exponent.info(info_flags::negative)) {

			// (a/b)^-n -> {b^n, a^n}
			return (new lst(power(n_basis.op(1), -n_exponent), power(n_basis.op(0), -n_exponent)))->setflag(status_flags::dynallocated);
		}

	} else {

		if (n_exponent.info(info_flags::positive)) {

			// (a/b)^x -> {sym((a/b)^x), 1}
			return (new lst(replace_with_symbol(power(n_basis.op(0) / n_basis.op(1), n_exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);

		} else if (n_exponent.info(info_flags::negative)) {

			if (n_basis.op(1).is_equal(_ex1())) {

				// a^-x -> {1, sym(a^x)}
				return (new lst(_ex1(), replace_with_symbol(power(n_basis.op(0), -n_exponent), sym_lst, repl_lst)))->setflag(status_flags::dynallocated);

			} else {

				// (a/b)^-x -> {sym((b/a)^x), 1}
				return (new lst(replace_with_symbol(power(n_basis.op(1) / n_basis.op(0), -n_exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
			}

		} else {	// n_exponent not numeric

			// (a/b)^x -> {sym((a/b)^x, 1}
			return (new lst(replace_with_symbol(power(n_basis.op(0) / n_basis.op(1), n_exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
		}
	}
}


/** Implementation of ex::normal() for pseries. It normalizes each coefficient
 *  and replaces the series by a temporary symbol.
 *  @see ex::normal */
ex pseries::normal(lst &sym_lst, lst &repl_lst, int level) const
{
	epvector newseq;
	for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
		ex restexp = i->rest.normal();
		if (!restexp.is_zero())
			newseq.push_back(expair(restexp, i->coeff));
	}
	ex n = pseries(relational(var,point), newseq);
	return (new lst(replace_with_symbol(n, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
}


/** Implementation of ex::normal() for relationals. It normalizes both sides.
 *  @see ex::normal */
ex relational::normal(lst &sym_lst, lst &repl_lst, int level) const
{
	return (new lst(relational(lh.normal(), rh.normal(), o), _ex1()))->setflag(status_flags::dynallocated);
}


/** Normalization of rational functions.
 *  This function converts an expression to its normal form
 *  "numerator/denominator", where numerator and denominator are (relatively
 *  prime) polynomials. Any subexpressions which are not rational functions
 *  (like non-rational numbers, non-integer powers or functions like sin(),
 *  cos() etc.) are replaced by temporary symbols which are re-substituted by
 *  the (normalized) subexpressions before normal() returns (this way, any
 *  expression can be treated as a rational function). normal() is applied
 *  recursively to arguments of functions etc.
 *
 *  @param level maximum depth of recursion
 *  @return normalized expression */
ex ex::normal(int level) const
{
	lst sym_lst, repl_lst;

	ex e = bp->normal(sym_lst, repl_lst, level);
	GINAC_ASSERT(is_ex_of_type(e, lst));

	// Re-insert replaced symbols
	if (sym_lst.nops() > 0)
		e = e.subs(sym_lst, repl_lst);

	// Convert {numerator, denominator} form back to fraction
	return e.op(0) / e.op(1);
}

/** Get numerator of an expression. If the expression is not of the normal
 *  form "numerator/denominator", it is first converted to this form and
 *  then the numerator is returned.
 *
 *  @see ex::normal
 *  @return numerator */
ex ex::numer(void) const
{
	lst sym_lst, repl_lst;

	ex e = bp->normal(sym_lst, repl_lst, 0);
	GINAC_ASSERT(is_ex_of_type(e, lst));

	// Re-insert replaced symbols
	if (sym_lst.nops() > 0)
		return e.op(0).subs(sym_lst, repl_lst);
	else
		return e.op(0);
}

/** Get denominator of an expression. If the expression is not of the normal
 *  form "numerator/denominator", it is first converted to this form and
 *  then the denominator is returned.
 *
 *  @see ex::normal
 *  @return denominator */
ex ex::denom(void) const
{
	lst sym_lst, repl_lst;

	ex e = bp->normal(sym_lst, repl_lst, 0);
	GINAC_ASSERT(is_ex_of_type(e, lst));

	// Re-insert replaced symbols
	if (sym_lst.nops() > 0)
		return e.op(1).subs(sym_lst, repl_lst);
	else
		return e.op(1);
}

/** Get numerator and denominator of an expression. If the expresison is not
 *  of the normal form "numerator/denominator", it is first converted to this
 *  form and then a list [numerator, denominator] is returned.
 *
 *  @see ex::normal
 *  @return a list [numerator, denominator] */
ex ex::numer_denom(void) const
{
	lst sym_lst, repl_lst;

	ex e = bp->normal(sym_lst, repl_lst, 0);
	GINAC_ASSERT(is_ex_of_type(e, lst));

	// Re-insert replaced symbols
	if (sym_lst.nops() > 0)
		return e.subs(sym_lst, repl_lst);
	else
		return e;
}


/** Default implementation of ex::to_rational(). It replaces the object with a
 *  temporary symbol.
 *  @see ex::to_rational */
ex basic::to_rational(lst &repl_lst) const
{
	return replace_with_symbol(*this, repl_lst);
}


/** Implementation of ex::to_rational() for symbols. This returns the
 *  unmodified symbol.
 *  @see ex::to_rational */
ex symbol::to_rational(lst &repl_lst) const
{
	return *this;
}


/** Implementation of ex::to_rational() for a numeric. It splits complex
 *  numbers into re+I*im and replaces I and non-rational real numbers with a
 *  temporary symbol.
 *  @see ex::to_rational */
ex numeric::to_rational(lst &repl_lst) const
{
	if (is_real()) {
		if (!is_rational())
			return replace_with_symbol(*this, repl_lst);
	} else { // complex
		numeric re = real();
		numeric im = imag();
		ex re_ex = re.is_rational() ? re : replace_with_symbol(re, repl_lst);
		ex im_ex = im.is_rational() ? im : replace_with_symbol(im, repl_lst);
		return re_ex + im_ex * replace_with_symbol(I, repl_lst);
	}
	return *this;
}


/** Implementation of ex::to_rational() for powers. It replaces non-integer
 *  powers by temporary symbols.
 *  @see ex::to_rational */
ex power::to_rational(lst &repl_lst) const
{
	if (exponent.info(info_flags::integer))
		return power(basis.to_rational(repl_lst), exponent);
	else
		return replace_with_symbol(*this, repl_lst);
}


/** Implementation of ex::to_rational() for expairseqs.
 *  @see ex::to_rational */
ex expairseq::to_rational(lst &repl_lst) const
{
	epvector s;
	s.reserve(seq.size());
	for (epvector::const_iterator it=seq.begin(); it!=seq.end(); ++it) {
		s.push_back(split_ex_to_pair(recombine_pair_to_ex(*it).to_rational(repl_lst)));
		// s.push_back(combine_ex_with_coeff_to_pair((*it).rest.to_rational(repl_lst),
	}
	ex oc = overall_coeff.to_rational(repl_lst);
	if (oc.info(info_flags::numeric))
		return thisexpairseq(s, overall_coeff);
	else s.push_back(combine_ex_with_coeff_to_pair(oc,_ex1()));
	return thisexpairseq(s, default_overall_coeff());
}


/** Rationalization of non-rational functions.
 *  This function converts a general expression to a rational polynomial
 *  by replacing all non-rational subexpressions (like non-rational numbers,
 *  non-integer powers or functions like sin(), cos() etc.) to temporary
 *  symbols. This makes it possible to use functions like gcd() and divide()
 *  on non-rational functions by applying to_rational() on the arguments,
 *  calling the desired function and re-substituting the temporary symbols
 *  in the result. To make the last step possible, all temporary symbols and
 *  their associated expressions are collected in the list specified by the
 *  repl_lst parameter in the form {symbol == expression}, ready to be passed
 *  as an argument to ex::subs().
 *
 *  @param repl_lst collects a list of all temporary symbols and their replacements
 *  @return rationalized expression */
ex ex::to_rational(lst &repl_lst) const
{
	return bp->to_rational(repl_lst);
}


} // namespace GiNaC