/*++
Copyright (c) 2013 Microsoft Corporation

Module Name:

    realclosure.cpp

Abstract:

    Package for computing with elements of the realclosure of a field containing
       - all rationals
       - extended with computable transcendental real numbers (e.g., pi and e)
       - infinitesimals

Author:

    Leonardo (leonardo) 2013-01-02

Notes:

--*/
#include "math/realclosure/realclosure.h"
#include "math/realclosure/rcf_params.hpp"
#include "util/array.h"
#include "util/mpbq.h"
#include "math/realclosure/mpz_matrix.h"
#include "math/interval/interval_def.h"
#include "util/obj_ref.h"
#include "util/ref_vector.h"
#include "util/ref_buffer.h"
#include "util/common_msgs.h"

#ifndef REALCLOSURE_INI_BUFFER_SIZE
#define REALCLOSURE_INI_BUFFER_SIZE 32
#endif

#ifndef REALCLOSURE_INI_SEQ_SIZE
#define REALCLOSURE_INI_SEQ_SIZE 256
#endif

#ifndef REALCLOSURE_INI_DIV_PRECISION
#define REALCLOSURE_INI_DIV_PRECISION 24
#endif

namespace realclosure {

    // ---------------------------------
    //
    // Intervals with binary rational endpoints
    //
    // ---------------------------------

    struct mpbq_config {

        struct numeral_manager : public mpbq_manager {
            // division is not precise
            static bool precise() { return false; }
            static bool field() { return true; }
            unsigned m_div_precision;
            bool     m_to_plus_inf;

            numeral_manager(unsynch_mpq_manager & qm):mpbq_manager(qm), m_div_precision(REALCLOSURE_INI_DIV_PRECISION), m_to_plus_inf(true) {
            }

            void div(mpbq const & a, mpbq const & b, mpbq & c) {
                approx_div(a, b, c, m_div_precision, m_to_plus_inf);
            }

            void inv(mpbq & a) {
                mpbq one(1);
                scoped_mpbq r(*this);
                approx_div(one, a, r, m_div_precision, m_to_plus_inf);
                swap(a, r);
            }
        };

        typedef mpbq         numeral;
        numeral_manager &    m_manager;

        struct interval {
            numeral   m_lower;
            numeral   m_upper;
            unsigned char m_lower_inf;
            unsigned char m_upper_inf;
            unsigned char m_lower_open;
            unsigned char m_upper_open;
            interval():m_lower_inf(true), m_upper_inf(true), m_lower_open(true), m_upper_open(true) {}
            interval(numeral & l, numeral & u):m_lower_inf(false), m_upper_inf(false), m_lower_open(true), m_upper_open(true) {
                swap(m_lower, l);
                swap(m_upper, u);
            }
            numeral & lower() { return m_lower; }
            numeral & upper() { return m_upper; }
            void set_lower_is_inf(bool f) { m_lower_inf = f; }
            void set_upper_is_inf(bool f) { m_upper_inf = f; }
            void set_lower_is_open(bool f) { m_lower_open = f; }
            void set_upper_is_open(bool f) { m_upper_open = f; }
            numeral const & lower() const { return m_lower; }
            numeral const & upper() const { return m_upper; }
            bool lower_is_inf() const { return m_lower_inf != 0; }
            bool upper_is_inf() const { return m_upper_inf != 0; }
            bool lower_is_open() const { return m_lower_open != 0; }
            bool upper_is_open() const { return m_upper_open != 0; }
        };

        void set_rounding(bool to_plus_inf) { m_manager.m_to_plus_inf = to_plus_inf; }
        void round_to_minus_inf() { set_rounding(false); }
        void round_to_plus_inf() { set_rounding(true); }

        // Getters
        numeral const & lower(interval const & a) const { return a.m_lower; }
        numeral const & upper(interval const & a) const { return a.m_upper; }
        numeral & lower(interval & a) { return a.m_lower; }
        numeral & upper(interval & a) { return a.m_upper; }
        bool lower_is_open(interval const & a) const { return a.lower_is_open(); }
        bool upper_is_open(interval const & a) const { return a.upper_is_open(); }
        bool lower_is_inf(interval const & a) const { return a.lower_is_inf(); }
        bool upper_is_inf(interval const & a) const { return a.upper_is_inf(); }

        // Setters
        void set_lower(interval & a, numeral const & n) { m_manager.set(a.m_lower, n); }
        void set_upper(interval & a, numeral const & n) { m_manager.set(a.m_upper, n); }
        void set_lower_is_open(interval & a, bool v) { a.m_lower_open = v; }
        void set_upper_is_open(interval & a, bool v) { a.m_upper_open = v; }
        void set_lower_is_inf(interval & a, bool v) { a.m_lower_inf = v; }
        void set_upper_is_inf(interval & a, bool v) { a.m_upper_inf = v; }

        // Reference to numeral manager
        numeral_manager & m() const { return m_manager; }

        mpbq_config(numeral_manager & m):m_manager(m) {}
    };

    typedef interval_manager<mpbq_config> mpbqi_manager;
    typedef mpbqi_manager::interval       mpbqi;

    void swap(mpbqi & a, mpbqi & b) {
        swap(a.m_lower, b.m_lower);
        swap(a.m_upper, b.m_upper);
        std::swap(a.m_lower_inf, b.m_lower_inf);
        std::swap(a.m_upper_inf, b.m_upper_inf);
        std::swap(a.m_lower_open, b.m_lower_open);
        std::swap(a.m_upper_open, b.m_upper_open);
    }

    // ---------------------------------
    //
    // Values are represented as
    //   - arbitrary precision rationals (mpq)
    //   - rational functions on field extensions
    //
    // ---------------------------------

    struct value {
        unsigned m_ref_count;  //!< Reference counter
        bool     m_rational;   //!< True if the value is represented as an arbitrary precision rational value.
        mpbqi    m_interval;   //!< approximation as an interval with binary rational end-points
        // When performing an operation OP, we may have to make the width (upper - lower) of m_interval very small.
        // The precision (i.e., a small interval) needed for executing OP is usually unnecessary for subsequent operations,
        // This unnecessary precision will only slowdown the subsequent operations that do not need it.
        // To cope with this issue, we cache the value m_interval in m_old_interval whenever the width of m_interval is below
        // a give threshold. Then, after finishing OP, we restore the old_interval.
        mpbqi *  m_old_interval;
        value(bool rat):m_ref_count(0), m_rational(rat), m_old_interval(nullptr) {}
        bool is_rational() const { return m_rational; }
        mpbqi const & interval() const { return m_interval; }
        mpbqi & interval() { return m_interval; }
    };

    struct rational_value : public value {
        mpq      m_value;
        rational_value():value(true) {}
    };

    typedef ptr_array<value> polynomial;

    struct extension;
    bool rank_lt(extension * r1, extension * r2);

    struct rational_function_value : public value {
        polynomial   m_numerator;
        polynomial   m_denominator; // it is only needed if the extension is not algebraic.
        extension *  m_ext;
        bool         m_depends_on_infinitesimals; //!< True if the polynomial expression depends on infinitesimal values.
        rational_function_value(extension * ext):value(false), m_ext(ext), m_depends_on_infinitesimals(false) {}

        polynomial const & num() const { return m_numerator; }
        polynomial & num() { return m_numerator; }
        polynomial const & den() const { return m_denominator; }
        polynomial & den() { return m_denominator; }

        extension * ext() const { return m_ext; }
        bool depends_on_infinitesimals() const { return m_depends_on_infinitesimals; }
        void set_depends_on_infinitesimals(bool f) { m_depends_on_infinitesimals = f; }
    };

    // ---------------------------------
    //
    // Field Extensions
    //
    // ---------------------------------

    typedef int sign;

    typedef std::pair<polynomial, sign> p2s;

    typedef sarray<p2s> signs;

    struct extension {
        enum kind {
            TRANSCENDENTAL = 0,
            INFINITESIMAL  = 1,
            ALGEBRAIC      = 2
        };

        unsigned m_ref_count;
        unsigned m_kind:2;
        unsigned m_idx:30;
        mpbqi    m_interval;
        mpbqi *  m_old_interval;

        extension(kind k, unsigned idx):m_ref_count(0), m_kind(k), m_idx(idx), m_old_interval(nullptr) {}

        unsigned idx() const { return m_idx; }
        kind knd() const { return static_cast<kind>(m_kind); }

        bool is_algebraic() const { return knd() == ALGEBRAIC; }
        bool is_infinitesimal() const { return knd() == INFINITESIMAL; }
        bool is_transcendental() const { return knd() == TRANSCENDENTAL; }

        mpbqi const & interval() const { return m_interval; }
        mpbqi & interval() { return m_interval; }
    };

    bool rank_lt(extension * r1, extension * r2) {
        return r1->knd() < r2->knd() || (r1->knd() == r2->knd() && r1->idx() < r2->idx());
    }

    bool rank_eq(extension * r1, extension * r2) {
        return r1->knd() == r2->knd() && r1->idx() == r2->idx();
    }

    struct rank_lt_proc {
        bool operator()(extension * r1, extension * r2) const {
            return rank_lt(r1, r2);
        }
    };

    /**
       \brief Sign condition object, it encodes one conjunct of a sign assignment.
       If has to keep following m_prev to obtain the whole sign condition
    */
    struct sign_condition {
        unsigned         m_q_idx:31; // Sign condition for the polynomial at position m_q_idx in the field m_qs of sign_det structure
        unsigned         m_mark:1;   // auxiliary mark used during deletion
        int              m_sign;     // Sign of the polynomial associated with m_q_idx
        sign_condition * m_prev;     // Antecedent
        sign_condition():m_q_idx(0), m_mark(false), m_sign(0), m_prev(nullptr) {}
        sign_condition(unsigned qidx, int sign, sign_condition * prev):m_q_idx(qidx), m_mark(false), m_sign(sign), m_prev(prev) {}

        sign_condition * prev() const { return m_prev; }
        unsigned qidx() const { return m_q_idx; }
        int sign() const { return m_sign; }
    };

    struct sign_det {
        unsigned                m_ref_count;  // sign_det objects may be  shared between different roots of the same polynomial.
        mpz_matrix              M_s;          // Matrix used in the sign determination
        array<polynomial>       m_prs;        // Polynomials associated with the rows of M
        array<int>              m_taqrs;      // Result of the tarski query for each polynomial in m_prs
        array<sign_condition *> m_sign_conditions; // Sign conditions associated with the columns of M
        array<polynomial>       m_qs;     // Polynomials used in the sign conditions.
        sign_det():m_ref_count(0) {}

        array<polynomial> const & qs() const { return m_qs; }
        sign_condition * sc(unsigned idx) const { return m_sign_conditions[idx]; }
        unsigned num_roots() const { return m_prs.size(); }
        array<int> const & taqrs() const { return m_taqrs; }
        array<polynomial> const & prs() const { return m_prs; }
    };

    struct algebraic : public extension {
        polynomial   m_p;
        mpbqi        m_iso_interval;
        sign_det *   m_sign_det; //!< != 0         if m_iso_interval constrains more than one root of m_p.
        unsigned     m_sc_idx;   //!< != UINT_MAX  if m_sign_det != 0, in this case m_sc_idx < m_sign_det->m_sign_conditions.size()
        bool         m_depends_on_infinitesimals;  //!< True if the polynomial p depends on infinitesimal extensions.

        algebraic(unsigned idx):extension(ALGEBRAIC, idx), m_sign_det(nullptr), m_sc_idx(0), m_depends_on_infinitesimals(false) {}

        polynomial const & p() const { return m_p; }
        bool depends_on_infinitesimals() const { return m_depends_on_infinitesimals; }
        sign_det * sdt() const { return m_sign_det; }
        unsigned sc_idx() const { return m_sc_idx; }
        unsigned num_roots_inside_interval() const { return m_sign_det == nullptr ? 1 : m_sign_det->num_roots(); }
        mpbqi & iso_interval() { return m_iso_interval; }
    };

    struct transcendental : public extension {
        symbol        m_name;
        symbol        m_pp_name;
        unsigned      m_k;
        mk_interval & m_proc;

        transcendental(unsigned idx, symbol const & n, symbol const & pp_n, mk_interval & p):
            extension(TRANSCENDENTAL, idx), m_name(n), m_pp_name(pp_n), m_k(0), m_proc(p) {}

        void display(std::ostream & out, bool pp = false) const {
            if (pp)
                out << m_pp_name;
            else
                out << m_name;
        }
    };

    struct infinitesimal : public extension {
        symbol        m_name;
        symbol        m_pp_name;

        infinitesimal(unsigned idx, symbol const & n, symbol const & pp_n):extension(INFINITESIMAL, idx), m_name(n), m_pp_name(pp_n) {}

        void display(std::ostream & out, bool pp = false) const {
            if (pp) {
                if (m_pp_name.is_numerical())
                    out << "&epsilon;<sub>" << m_pp_name.get_num() << "</sub>";
                else
                    out << m_pp_name;

            }
            else {
                if (m_name.is_numerical())
                    out << "eps!" << m_name.get_num();
                else
                    out << m_name;
            }
        }
    };

    // ---------------------------------
    //
    // Predefined transcendental mk_interval procs
    //
    // ---------------------------------

    struct mk_pi_interval : public mk_interval {
        void operator()(unsigned k, mpqi_manager & im, mpqi_manager::interval & r) override {
            im.pi(k, r);
        }
    };

    struct mk_e_interval : public mk_interval {
        void operator()(unsigned k, mpqi_manager & im, mpqi_manager::interval & r) override {
            im.e(k, r);
        }
    };

    // ---------------------------------
    //
    // Manager
    //
    // ---------------------------------

    struct manager::imp {
        typedef ref_vector<value, imp> value_ref_vector;
        typedef ref_buffer<value, imp, REALCLOSURE_INI_BUFFER_SIZE> value_ref_buffer;
        typedef obj_ref<value, imp>    value_ref;
        typedef _scoped_interval<mpqi_manager>  scoped_mpqi;
        typedef _scoped_interval<mpbqi_manager> scoped_mpbqi;
        typedef sbuffer<int, REALCLOSURE_INI_BUFFER_SIZE> int_buffer;
        typedef sbuffer<unsigned, REALCLOSURE_INI_BUFFER_SIZE> unsigned_buffer;

        reslimit&                      m_limit;
        small_object_allocator *       m_allocator;
        bool                           m_own_allocator;
        unsynch_mpq_manager &          m_qm;
        mpz_matrix_manager             m_mm;
        mpbq_config::numeral_manager   m_bqm;
        mpqi_manager                   m_qim;
        mpbqi_manager                  m_bqim;
        ptr_vector<extension>          m_extensions[3];
        value *                        m_one;
        mk_pi_interval                 m_mk_pi_interval;
        value *                        m_pi;
        mk_e_interval                  m_mk_e_interval;
        value *                        m_e;
        ptr_vector<value>              m_to_restore; //!< Set of values v s.t. v->m_old_interval != 0
        ptr_vector<extension>          m_ex_to_restore;

        // Parameters
        bool                           m_use_prem; //!< use pseudo-remainder when computing sturm sequences
        bool                           m_clean_denominators;
        unsigned                       m_ini_precision; //!< initial precision for transcendentals, infinitesimals, etc.
        unsigned                       m_max_precision; //!< Maximum precision for interval arithmetic techniques, it switches to complete methods after that
        unsigned                       m_inf_precision; //!< 2^m_inf_precision is used as the lower bound of oo and -2^m_inf_precision is used as the upper_bound of -oo
        scoped_mpbq                    m_plus_inf_approx; // lower bound for binary rational intervals used to approximate an infinite positive value
        scoped_mpbq                    m_minus_inf_approx; // upper bound for binary rational intervals used to approximate an infinite negative value
        bool                           m_lazy_algebraic_normalization;

        // Tracing
        unsigned                       m_exec_depth;

        bool                           m_in_aux_values; // True if we are computing SquareFree polynomials or Sturm sequences. That is, the values being computed will be discarded.


        struct scoped_polynomial_seq {
            typedef ref_buffer<value, imp, REALCLOSURE_INI_SEQ_SIZE> value_seq;
            value_seq          m_seq_coeffs;
            sbuffer<unsigned>  m_begins;     // start position (in m_seq_coeffs) of each polynomial in the sequence
            sbuffer<unsigned>  m_szs;        // size of each polynomial in the sequence
        public:
            scoped_polynomial_seq(imp & m):m_seq_coeffs(m) {}
            ~scoped_polynomial_seq() {
            }

            /**
               \brief Add a new polynomial to the sequence.
               The contents of p is erased.
            */
            void push(unsigned sz, value * const * p) {
                m_begins.push_back(m_seq_coeffs.size());
                m_szs.push_back(sz);
                m_seq_coeffs.append(sz, p);
            }

            /**
               \brief Return the number of polynomials in the sequence.
            */
            unsigned size() const { return m_szs.size(); }

            /**
               \brief Return the vector of coefficients for the i-th polynomial in the sequence.
            */
            value * const * coeffs(unsigned i) const {
                return m_seq_coeffs.c_ptr() + m_begins[i];
            }

            /**
               \brief Return the size of the i-th polynomial in the sequence.
            */
            unsigned size(unsigned i) const { return m_szs[i]; }

            void reset() {
                m_seq_coeffs.reset();
                m_begins.reset();
                m_szs.reset();
            }

            scoped_polynomial_seq & operator=(scoped_polynomial_seq & s) {
                if (this == &s)
                    return *this;
                reset();
                m_seq_coeffs.append(s.m_seq_coeffs);
                m_begins.append(s.m_begins);
                m_szs.append(s.m_szs);
                return *this;
            }
        };

        struct scoped_sign_conditions {
            imp &                                                   m_imp;
            ptr_buffer<sign_condition, REALCLOSURE_INI_BUFFER_SIZE> m_scs;

            scoped_sign_conditions(imp & m):m_imp(m) {}
            ~scoped_sign_conditions() {
                m_imp.del_sign_conditions(m_scs.size(), m_scs.c_ptr());
            }

            sign_condition * & operator[](unsigned idx) { return m_scs[idx]; }
            unsigned size() const { return m_scs.size(); }
            bool empty() const { return m_scs.empty(); }
            void push_back(sign_condition * sc) { m_scs.push_back(sc); }
            void release() {
                // release ownership
                m_scs.reset();
            }
            void copy_from(scoped_sign_conditions & scs) {
                SASSERT(this != &scs);
                release();
                m_scs.append(scs.m_scs.size(), scs.m_scs.c_ptr());
                scs.release();
            }
            sign_condition * const * c_ptr() { return m_scs.c_ptr(); }
        };

        struct scoped_inc_depth {
            imp & m_imp;
            scoped_inc_depth(imp & m):m_imp(m) { m_imp.m_exec_depth++; }
            ~scoped_inc_depth() { m_imp.m_exec_depth--; }
        };

        #ifdef _TRACE
        #define INC_DEPTH() scoped_inc_depth __inc(*this)
        #else
        #define INC_DEPTH() ((void) 0)
        #endif

        imp(reslimit& lim, unsynch_mpq_manager & qm, params_ref const & p, small_object_allocator * a):
            m_limit(lim),
            m_allocator(a == nullptr ? alloc(small_object_allocator, "realclosure") : a),
            m_own_allocator(a == nullptr),
            m_qm(qm),
            m_mm(m_qm, *m_allocator),
            m_bqm(m_qm),
            m_qim(lim, m_qm),
            m_bqim(lim, m_bqm),
            m_plus_inf_approx(m_bqm),
            m_minus_inf_approx(m_bqm) {
            mpq one(1);
            m_one = mk_rational(one);
            inc_ref(m_one);
            m_pi = nullptr;
            m_e  = nullptr;

            m_exec_depth = 0;

            m_in_aux_values = false;


            updt_params(p);
        }

        ~imp() {
            restore_saved_intervals(); // to free memory
            dec_ref(m_one);
            dec_ref(m_pi);
            dec_ref(m_e);
            if (m_own_allocator)
                dealloc(m_allocator);
        }

        // Rational number manager
        unsynch_mpq_manager & qm() const { return m_qm; }

        // Binary rational number manager
        mpbq_config::numeral_manager & bqm() { return m_bqm; }

        // Rational interval manager
        mpqi_manager & qim() { return m_qim; }

        // Binary rational interval manager
        mpbqi_manager & bqim() { return m_bqim; }
        mpbqi_manager const & bqim() const { return m_bqim; }

        // Integer matrix manager
        mpz_matrix_manager & mm() { return m_mm; }

        small_object_allocator & allocator() { return *m_allocator; }

        void checkpoint() {
            if (!m_limit.inc())
                throw exception(Z3_CANCELED_MSG);
        }

        value * one() const {
            return m_one;
        }

        /**
           \brief Return the magnitude of the given interval.
           The magnitude is an approximation of the size of the interval.
        */
        int magnitude(mpbq const & l, mpbq const & u) {
            SASSERT(bqm().ge(u, l));
            scoped_mpbq w(bqm());
            bqm().sub(u, l, w);
            if (bqm().is_zero(w))
                return INT_MIN;
            SASSERT(bqm().is_pos(w));
            return bqm().magnitude_ub(w);
        }

        /**
           \brief Return the magnitude of the given interval.
           The magnitude is an approximation of the size of the interval.
        */
        int magnitude(mpbqi const & i) {
            if (i.lower_is_inf() || i.upper_is_inf())
                return INT_MAX;
            else
                return magnitude(i.lower(), i.upper());
        }

        /**
           \brief Return the magnitude of the given interval.
           The magnitude is an approximation of the size of the interval.
        */
        int magnitude(mpq const & l, mpq const & u) {
            SASSERT(qm().ge(u, l));
            scoped_mpq w(qm());
            qm().sub(u, l, w);
            if (qm().is_zero(w))
                return INT_MIN;
            SASSERT(qm().is_pos(w));
            return static_cast<int>(qm().log2(w.get().numerator())) + 1 - static_cast<int>(qm().log2(w.get().denominator()));
        }

        int magnitude(scoped_mpqi const & i) {
            SASSERT(!i->m_lower_inf && !i->m_upper_inf);
            return magnitude(i->m_lower, i->m_upper);
        }

        /**
           \brief Return true if the magnitude of the given interval is less than the parameter m_max_precision.
        */
        bool too_small(mpbqi const & i) {
            return magnitude(i) < -static_cast<int>(m_max_precision);
        }

#define SMALL_UNSIGNED 1 << 16
        static unsigned inc_precision(unsigned prec, unsigned inc) {
            if (prec < SMALL_UNSIGNED)
                return prec + inc;
            else
                return prec;
        }

        struct scoped_set_div_precision {
            mpbq_config::numeral_manager & m_bqm;
            unsigned                       m_old_precision;
            scoped_set_div_precision(mpbq_config::numeral_manager & bqm, unsigned prec):m_bqm(bqm) {
                m_old_precision = m_bqm.m_div_precision;
                m_bqm.m_div_precision = prec;
            }
            ~scoped_set_div_precision() {
                m_bqm.m_div_precision = m_old_precision;
            }
        };

        /**
           \brief c <- a/b with precision prec.
        */
        void div(mpbqi const & a, mpbqi const & b, unsigned prec, mpbqi & c) {
            SASSERT(!contains_zero(a));
            SASSERT(!contains_zero(b));
            scoped_set_div_precision set(bqm(), prec);
            bqim().div(a, b, c);
            SASSERT(!contains_zero(c));
        }

        /**
           \brief c <- a/b with precision prec.
        */
        void div(mpbqi const & a, mpz const & b, unsigned prec, mpbqi & c) {
            SASSERT(!contains_zero(a));
            SASSERT(!qm().is_zero(b));
            scoped_mpbqi bi(bqim());
            set_interval(bi, b);
            scoped_mpbqi r(bqim());
            div(a, bi, prec, r);
            swap(c, r);
        }

        /**
           \brief Save the current interval (i.e., approximation) of the given value or extension.
        */
        template<typename T>
        void save_interval(T * v, ptr_vector<T> & to_restore) {
            if (v->m_old_interval != nullptr)
                return; // interval was already saved.
            to_restore.push_back(v);
            inc_ref(v);
            v->m_old_interval = new (allocator()) mpbqi();
            set_interval(*(v->m_old_interval), v->m_interval);
        }
        void save_interval(value * v) {
            save_interval(v, m_to_restore);
        }
        void save_interval(extension * x) {
            save_interval(x, m_ex_to_restore);
        }

        /**
           \brief Save the current interval (i.e., approximation) of the given value IF it is too small.
        */
        void save_interval_if_too_small(value * v, unsigned new_prec) {
            if (new_prec > m_max_precision && !contains_zero(interval(v)))
                save_interval(v);
        }

        /**
           \brief Save the current interval (i.e., approximation) of the given value IF it is too small.
        */
        void save_interval_if_too_small(extension * x, unsigned new_prec) {
            if (new_prec > m_max_precision && !contains_zero(x->m_interval))
                save_interval(x);
        }

        /**
           \brief Restore the saved intervals (approximations) of RCF values and extensions
        */
        template<typename T>
        void restore_saved_intervals(ptr_vector<T> & to_restore) {
            unsigned sz = to_restore.size();
            for (unsigned i = 0; i < sz; i++) {
                T * v = to_restore[i];
                set_interval(v->m_interval, *(v->m_old_interval));
                bqim().del(*(v->m_old_interval));
                allocator().deallocate(sizeof(mpbqi), v->m_old_interval);
                v->m_old_interval = nullptr;
                dec_ref(v);
            }
            to_restore.reset();
        }
        void restore_saved_intervals() {
            restore_saved_intervals(m_to_restore);
            restore_saved_intervals(m_ex_to_restore);
        }

        void cleanup(extension::kind k) {
            ptr_vector<extension> & exts = m_extensions[k];
            // keep removing unused slots
            while (!exts.empty() && exts.back() == 0) {
                exts.pop_back();
            }
        }

        unsigned next_transcendental_idx() {
            cleanup(extension::TRANSCENDENTAL);
            return m_extensions[extension::TRANSCENDENTAL].size();
        }

        unsigned next_infinitesimal_idx() {
            cleanup(extension::INFINITESIMAL);
            return m_extensions[extension::INFINITESIMAL].size();
        }

        unsigned next_algebraic_idx() {
            cleanup(extension::ALGEBRAIC);
            return m_extensions[extension::ALGEBRAIC].size();
        }


        void updt_params(params_ref const & _p) {
            rcf_params p(_p);
            m_use_prem           = p.use_prem();
            m_clean_denominators = p.clean_denominators();
            m_ini_precision      = p.initial_precision();
            m_inf_precision      = p.inf_precision();
            m_max_precision      = p.max_precision();
            m_lazy_algebraic_normalization = p.lazy_algebraic_normalization();
            bqm().power(mpbq(2), m_inf_precision, m_plus_inf_approx);
            bqm().set(m_minus_inf_approx, m_plus_inf_approx);
            bqm().neg(m_minus_inf_approx);
        }

        /**
           \brief Reset the given polynomial.
           That is, after the call p is the 0 polynomial.
        */
        void reset_p(polynomial & p) {
            dec_ref(p.size(), p.c_ptr());
            p.finalize(allocator());
        }

        void del_rational(rational_value * v) {
            bqim().del(v->m_interval);
            qm().del(v->m_value);
            allocator().deallocate(sizeof(rational_value), v);
        }

        void del_rational_function(rational_function_value * v) {
            bqim().del(v->m_interval);
            reset_p(v->num());
            reset_p(v->den());
            dec_ref(v->ext());
            allocator().deallocate(sizeof(rational_function_value), v);
        }

        void del_value(value * v) {
            if (v->is_rational())
                del_rational(static_cast<rational_value*>(v));
            else
                del_rational_function(static_cast<rational_function_value*>(v));
        }

        void finalize(array<polynomial> & ps) {
            for (unsigned i = 0; i < ps.size(); i++)
                reset_p(ps[i]);
            ps.finalize(allocator());
        }

        void del_sign_condition(sign_condition * sc) {
            allocator().deallocate(sizeof(sign_condition), sc);
        }

        void del_sign_conditions(unsigned sz, sign_condition * const * to_delete) {
            ptr_buffer<sign_condition> all_to_delete;
            for (unsigned i = 0; i < sz; i++) {
                sign_condition * sc = to_delete[i];
                while (sc && sc->m_mark == false) {
                    sc->m_mark = true;
                    all_to_delete.push_back(sc);
                    sc = sc->m_prev;
                }
            }
            for (unsigned i = 0; i < all_to_delete.size(); i++) {
                del_sign_condition(all_to_delete[i]);
            }
        }

        void del_sign_det(sign_det * sd) {
            mm().del(sd->M_s);
            del_sign_conditions(sd->m_sign_conditions.size(), sd->m_sign_conditions.c_ptr());
            sd->m_sign_conditions.finalize(allocator());
            finalize(sd->m_prs);
            sd->m_taqrs.finalize(allocator());
            finalize(sd->m_qs);
            allocator().deallocate(sizeof(sign_det), sd);
        }

        void inc_ref_sign_det(sign_det * sd) {
            if (sd != nullptr)
                sd->m_ref_count++;
        }

        void dec_ref_sign_det(sign_det * sd) {
            if (sd != nullptr) {
                sd->m_ref_count--;
                if (sd->m_ref_count == 0) {
                    del_sign_det(sd);
                }
            }
        }

        void del_algebraic(algebraic * a) {
            reset_p(a->m_p);
            bqim().del(a->m_interval);
            bqim().del(a->m_iso_interval);
            dec_ref_sign_det(a->m_sign_det);
            allocator().deallocate(sizeof(algebraic), a);
        }

        void del_transcendental(transcendental * t) {
            bqim().del(t->m_interval);
            allocator().deallocate(sizeof(transcendental), t);
        }

        void del_infinitesimal(infinitesimal * i) {
            bqim().del(i->m_interval);
            allocator().deallocate(sizeof(infinitesimal), i);
        }

        void inc_ref(extension * ext) {
            SASSERT(ext != 0);
            ext->m_ref_count++;
        }

        void dec_ref(extension * ext) {
            SASSERT(m_extensions[ext->knd()][ext->idx()] == ext);
            SASSERT(ext->m_ref_count > 0);
            ext->m_ref_count--;
            if (ext->m_ref_count == 0) {
                m_extensions[ext->knd()][ext->idx()] = 0;
                switch (ext->knd()) {
                case extension::TRANSCENDENTAL: del_transcendental(static_cast<transcendental*>(ext)); break;
                case extension::INFINITESIMAL:  del_infinitesimal(static_cast<infinitesimal*>(ext)); break;
                case extension::ALGEBRAIC:      del_algebraic(static_cast<algebraic*>(ext)); break;
                }
            }
        }

        void inc_ref(value * v) {
            if (v)
                v->m_ref_count++;
        }

        void inc_ref(unsigned sz, value * const * p) {
            for (unsigned i = 0; i < sz; i++)
                inc_ref(p[i]);
        }

        void dec_ref(value * v) {
            if (v) {
                SASSERT(v->m_ref_count > 0);
                v->m_ref_count--;
                if (v->m_ref_count == 0)
                    del_value(v);
            }
        }

        void dec_ref(unsigned sz, value * const * p) {
            for (unsigned i = 0; i < sz; i++)
                dec_ref(p[i]);
        }

        void del(numeral & a) {
            dec_ref(a.m_value);
            a.m_value = nullptr;
        }

        void del(numeral_vector & v) {
            for (unsigned i = 0; i < v.size(); i++)
                del(v[i]);
        }

        /**
           \brief Return true if the given interval is smaller than 1/2^k
        */
        bool check_precision(mpbqi const & interval, unsigned k) {
            if (interval.lower_is_inf() || interval.upper_is_inf())
                return false;
            scoped_mpbq w(bqm());
            bqm().sub(interval.upper(), interval.lower(), w);
            return bqm().lt_1div2k(w, k);
        }

        /**
           \brief Return true if v is zero.
        */
        static bool is_zero(value * v) {
            return v == nullptr;
        }

        /**
           \brief Return true if v is represented using a nonzero arbitrary precision rational value.
        */
        static bool is_nz_rational(value * v) {
            SASSERT(v != 0);
            return v->is_rational();
        }

        /**
           \brief Return true if v is represented as rational value one.
        */
        bool is_rational_one(value * v) const {
            return !is_zero(v) && is_nz_rational(v) && qm().is_one(to_mpq(v));
        }

        /**
           \brief Return true if v is represented as rational value minus one.
        */
        bool is_rational_minus_one(value * v) const {
            return !is_zero(v) && is_nz_rational(v) && qm().is_minus_one(to_mpq(v));
        }

        /**
           \brief Return true if v is the value one;
         */
        bool is_one(value * v) const {
            return const_cast<imp*>(this)->compare(v, one()) == 0;
        }

        /**
           \brief Return true if p is the constant polynomial where the coefficient is
           the rational value 1.

           \remark This is NOT checking whether p is actually equal to 1.
           That is, it is just checking the representation.
        */
        bool is_rational_one(polynomial const & p) const {
            return p.size() == 1 && is_rational_one(p[0]);
        }
        bool is_rational_one(value_ref_buffer const & p) const {
            return p.size() == 1 && is_rational_one(p[0]);
        }

        bool is_denominator_one(rational_function_value * v) const {
            if (v->ext()->is_algebraic()) {
                SASSERT(v->den().size() == 0); // we do not use denominator for algebraic extensions
                return true;
            }
            else {
                return is_rational_one(v->den());
            }
        }

        template<typename T>
        bool is_one(polynomial const & p) const {
            return p.size() == 1 && is_one(p[0]);
        }

        /**
           \brief Return true if v is a represented as a rational function of the set of field extensions.
        */
        static bool is_rational_function(value * v) {
            SASSERT(v != 0);
            return !(v->is_rational());
        }

        static rational_value * to_nz_rational(value * v) {
            SASSERT(is_nz_rational(v));
            return static_cast<rational_value*>(v);
        }

        static rational_function_value * to_rational_function(value * v) {
            SASSERT(!is_nz_rational(v));
            return static_cast<rational_function_value*>(v);
        }

        static bool is_zero(numeral const & a) {
            return is_zero(a.m_value);
        }

        static bool is_nz_rational(numeral const & a) {
            SASSERT(!is_zero(a));
            return is_nz_rational(a.m_value);
        }

        /**
           \brief Return true if v is not a shared value. That is, we can perform
           destructive updates.
        */
        static bool is_unique(value * v) {
            SASSERT(v);
            return v->m_ref_count <= 1;
        }

        static bool is_unique(numeral const & a) {
            return is_unique(a.m_value);
        }

        static bool is_unique_nz_rational(value * v) {
            return is_nz_rational(v) && is_unique(v);
        }

        static bool is_unique_nz_rational(numeral const & a) {
            return is_unique_nz_rational(a.m_value);
        }

        static rational_value * to_nz_rational(numeral const & a) {
            SASSERT(is_nz_rational(a));
            return to_nz_rational(a.m_value);
        }

        static bool is_rational_function(numeral const & a) {
            return is_rational_function(a.m_value);
        }

        static rational_function_value * to_rational_function(numeral const & a) {
            SASSERT(is_rational_function(a));
            return to_rational_function(a.m_value);
        }

        static mpq & to_mpq(value * v) {
            SASSERT(is_nz_rational(v));
            return to_nz_rational(v)->m_value;
        }

        static mpq & to_mpq(numeral const & a) {
            SASSERT(is_nz_rational(a));
            return to_nz_rational(a)->m_value;
        }

        static int compare_rank(value * a, value * b) {
            SASSERT(a); SASSERT(b);
            if (is_nz_rational(a))
                return is_nz_rational(b) ? 0 : -1;
            else if (is_nz_rational(b)) {
                SASSERT(is_rational_function(a));
                return 1;
            }
            else if (rank_eq(to_rational_function(a)->ext(), to_rational_function(b)->ext()))
                return 0;
            else
                return rank_lt(to_rational_function(a)->ext(), to_rational_function(b)->ext()) ? -1 : 1;
        }

        static transcendental * to_transcendental(extension * ext) {
            SASSERT(ext->is_transcendental());
            return static_cast<transcendental*>(ext);
        }

        static infinitesimal * to_infinitesimal(extension * ext) {
            SASSERT(ext->is_infinitesimal());
            return static_cast<infinitesimal*>(ext);
        }

        static algebraic * to_algebraic(extension * ext) {
            SASSERT(ext->is_algebraic());
            return static_cast<algebraic*>(ext);
        }

        /**
           \brief Return True if the given extension depends on infinitesimal extensions.
           If it doesn't, then it is definitely a real value.

           If it does, then it may or may not be a real value.
                   Example: Assume eps is an infinitesimal, and pi is 3.14... .
                   Assume also that ext is the unique root between (3, 4) of the following polynomial:
                          x^2 - (pi + eps)*x + pi*ext
                   Thus, x is pi, but the system will return true, since its defining polynomial has infinitesimal
                   coefficients. In the future, we should be able to factor the polynomial
                   above as
                          (x - eps)*(x - pi)
                   and then detect that x is actually the root of (x - pi).
        */
        bool depends_on_infinitesimals(extension * ext) {
            switch (ext->knd()) {
            case extension::TRANSCENDENTAL: return false;
            case extension::INFINITESIMAL:  return true;
            case extension::ALGEBRAIC:      return to_algebraic(ext)->depends_on_infinitesimals();
            default:
                UNREACHABLE();
                return false;
            }
        }

        /**
           \brief Return true if v is definitely a real value.
        */
        bool depends_on_infinitesimals(value * v) const {
            if (is_zero(v) || is_nz_rational(v))
                return false;
            else
                return to_rational_function(v)->depends_on_infinitesimals();
        }

        bool depends_on_infinitesimals(unsigned sz, value * const * p) const {
            for (unsigned i = 0; i < sz; i++)
                if (depends_on_infinitesimals(p[i]))
                    return true;
            return false;
        }

        /**
           \brief Set the polynomial p with the given coefficients as[0], ..., as[n-1]
        */
        void set_p(polynomial & p, unsigned n, value * const * as) {
            SASSERT(n > 0);
            SASSERT(!is_zero(as[n - 1]));
            reset_p(p);
            p.set(allocator(), n, as);
            inc_ref(n, as);
        }

        /**
           \brief Return true if a is an open interval.
        */
        static bool is_open_interval(mpbqi const & a) {
            return a.lower_is_inf() && a.upper_is_inf();
        }

        /**
           \brief Return true if the interval contains zero.
        */
        bool contains_zero(mpbqi const & a) const {
            return bqim().contains_zero(a);
        }

        /**
           \brief Set the lower bound of the given interval.
        */
        void set_lower_core(mpbqi & a, mpbq const & k, bool open, bool inf) {
            bqm().set(a.lower(), k);
            a.set_lower_is_open(open);
            a.set_lower_is_inf(inf);
        }

        /**
           \brief a.lower <- k
        */
        void set_lower(mpbqi & a, mpbq const & k, bool open = true) {
            set_lower_core(a, k, open, false);
        }

        /**
           \brief a.lower <- -oo
        */
        void set_lower_inf(mpbqi & a) {
            bqm().reset(a.lower());
            a.set_lower_is_open(true);
            a.set_lower_is_inf(true);
        }

        /**
           \brief a.lower <- 0
        */
        void set_lower_zero(mpbqi & a) {
            bqm().reset(a.lower());
            a.set_lower_is_open(true);
            a.set_lower_is_inf(false);
        }

        /**
           \brief Set the upper bound of the given interval.
        */
        void set_upper_core(mpbqi & a, mpbq const & k, bool open, bool inf) {
            bqm().set(a.upper(), k);
            a.set_upper_is_open(open);
            a.set_upper_is_inf(inf);
        }

        /**
           \brief a.upper <- k
        */
        void set_upper(mpbqi & a, mpbq const & k, bool open = true) {
            set_upper_core(a, k, open, false);
        }

        /**
           \brief a.upper <- oo
        */
        void set_upper_inf(mpbqi & a) {
            bqm().reset(a.upper());
            a.set_upper_is_open(true);
            a.set_upper_is_inf(true);
        }

        /**
           \brief a.upper <- 0
        */
        void set_upper_zero(mpbqi & a) {
            bqm().reset(a.upper());
            a.set_upper_is_open(true);
            a.set_upper_is_inf(false);
        }

        /**
           \brief a <- b
        */
        void set_interval(mpbqi & a, mpbqi const & b) {
            set_lower_core(a, b.lower(), b.lower_is_open(), b.lower_is_inf());
            set_upper_core(a, b.upper(), b.upper_is_open(), b.upper_is_inf());
        }

        /**
           \brief a <- [b, b]
        */
        void set_interval(mpbqi & a, mpbq const & b) {
            set_lower_core(a, b, false, false);
            set_upper_core(a, b, false, false);
        }

        /**
           \brief a <- [b, b]
        */
        void set_interval(mpbqi & a, mpz const & b) {
            scoped_mpbq _b(bqm());
            bqm().set(_b, b);
            set_lower_core(a, _b, false, false);
            set_upper_core(a, _b, false, false);
        }

        sign_condition * mk_sign_condition(unsigned qidx, int sign, sign_condition * prev_sc) {
            return new (allocator()) sign_condition(qidx, sign, prev_sc);
        }

        /**
           \brief Make a rational_function_value using the given extension, numerator and denominator.
           This method does not set the interval. It remains (-oo, oo)
        */
        rational_function_value * mk_rational_function_value_core(extension * ext, unsigned num_sz, value * const * num, unsigned den_sz, value * const * den) {
            rational_function_value * r = new (allocator()) rational_function_value(ext);
            inc_ref(ext);
            set_p(r->num(), num_sz, num);
            if (ext->is_algebraic()) {
                // Avoiding wasteful allocation...
                // We do not use the denominator for algebraic extensions
                SASSERT(den_sz == 0 || (den_sz == 1 && is_rational_one(den[0])));

                SASSERT(r->den().size() == 0);
            }
            else {
                set_p(r->den(), den_sz, den);
            }
            r->set_depends_on_infinitesimals(depends_on_infinitesimals(ext) || depends_on_infinitesimals(num_sz, num) || depends_on_infinitesimals(den_sz, den));
            return r;
        }

        rational_function_value * mk_rational_function_value_core(algebraic * ext, unsigned num_sz, value * const * num) {
            return mk_rational_function_value_core(ext, num_sz, num, 0, nullptr);
        }

        /**
           \brief Create a value using the given extension.
        */
        rational_function_value * mk_rational_function_value(extension * ext) {
            value * num[2] = { nullptr, one() };
            value * den[1] = { one() };
            rational_function_value * v = mk_rational_function_value_core(ext, 2, num, 1, den);
            set_interval(v->interval(), ext->interval());
            return v;
        }

        /**
           \brief Create a new infinitesimal.
        */
        void mk_infinitesimal(symbol const & n, symbol const & pp_n, numeral & r) {
            unsigned idx = next_infinitesimal_idx();
            infinitesimal * eps = new (allocator()) infinitesimal(idx, n, pp_n);
            m_extensions[extension::INFINITESIMAL].push_back(eps);

            set_lower(eps->interval(), mpbq(0));
            set_upper(eps->interval(), mpbq(1, m_ini_precision));

            set(r, mk_rational_function_value(eps));

            SASSERT(sign(r) > 0);
            SASSERT(depends_on_infinitesimals(r));
        }

        void mk_infinitesimal(char const * n, char const * pp_n, numeral & r) {
            mk_infinitesimal(symbol(n), symbol(pp_n), r);
        }

        void mk_infinitesimal(numeral & r) {
            mk_infinitesimal(symbol(next_infinitesimal_idx()+1), symbol(next_infinitesimal_idx()+1), r);
        }

        void refine_transcendental_interval(transcendental * t) {
            scoped_mpqi i(qim());
            t->m_k++;
            t->m_proc(t->m_k, qim(), i);
            int m = magnitude(i);
            TRACE("rcf_transcendental",
                  tout << "refine_transcendental_interval rational: " << m << "\nrational interval: ";
                  qim().display(tout, i); tout << std::endl;);
            unsigned k;
            if (m >= 0)
                k = m_ini_precision;
            else
                k = inc_precision(-m, 8);
            scoped_mpbq l(bqm());
            mpq_to_mpbqi(i->m_lower, t->interval(), k);
            // save lower
            bqm().set(l, t->interval().lower());
            mpq_to_mpbqi(i->m_upper, t->interval(), k);
            bqm().set(t->interval().lower(), l);
        }

        void refine_transcendental_interval(transcendental * t, unsigned prec) {
            while (!check_precision(t->interval(), prec)) {
                TRACE("rcf_transcendental", tout << "refine_transcendental_interval: " << magnitude(t->interval()) << std::endl;);
                checkpoint();
                save_interval_if_too_small(t, prec);
                refine_transcendental_interval(t);
            }
        }

        void mk_transcendental(symbol const & n, symbol const & pp_n, mk_interval & proc, numeral & r) {
            unsigned idx = next_transcendental_idx();
            transcendental * t = new (allocator()) transcendental(idx, n, pp_n, proc);
            m_extensions[extension::TRANSCENDENTAL].push_back(t);

            while (contains_zero(t->interval())) {
                checkpoint();
                refine_transcendental_interval(t);
            }
            set(r, mk_rational_function_value(t));

            SASSERT(!depends_on_infinitesimals(r));
        }

        void mk_transcendental(char const * p, char const * pp_n, mk_interval & proc, numeral & r) {
            mk_transcendental(symbol(p), symbol(pp_n), proc, r);
        }

        void mk_transcendental(mk_interval & proc, numeral & r) {
            mk_transcendental(symbol(next_transcendental_idx()+1), symbol(next_transcendental_idx()+1), proc, r);
        }

        void mk_pi(numeral & r) {
            if (m_pi) {
                set(r, m_pi);
            }
            else {
                mk_transcendental(symbol("pi"), symbol("&pi;"), m_mk_pi_interval, r);
                m_pi = r.m_value;
                inc_ref(m_pi);
            }
        }

        void mk_e(numeral & r) {
            if (m_e) {
                set(r, m_e);
            }
            else {
                mk_transcendental(symbol("e"), symbol("e"), m_mk_e_interval, r);
                m_e = r.m_value;
                inc_ref(m_e);
            }
        }

        // ---------------------------------
        //
        // Root isolation
        //
        // ---------------------------------

        /**
           \brief r <- magnitude of the lower bound of |i|.
           That is, 2^r <= |i|.lower()
           Another way to view it is:
              2^r is smaller than the absolute value of any element in the interval i.

           Return true if succeeded, and false if i contains elements that are infinitely close to 0.

           \pre !contains_zero(i)
        */
        bool abs_lower_magnitude(mpbqi const & i, int & r) {
            SASSERT(!contains_zero(i));
            if (bqim().is_P(i)) {
                if (bqm().is_zero(i.lower()))
                    return false;
                r = bqm().magnitude_lb(i.lower());
                return true;
            }
            else {
                SASSERT(bqim().is_N(i));
                if (bqm().is_zero(i.upper()))
                    return false;
                scoped_mpbq tmp(bqm());
                tmp = i.upper();
                bqm().neg(tmp);
                r = bqm().magnitude_lb(tmp);
                return true;
            }
        }

        /**
           \brief r <- magnitude of the upper bound of |i|.
           That is, |i|.upper <= 2^r
           Another way to view it is:
              2^r is bigger than the absolute value of any element in the interval i.

           Return true if succeeded, and false if i is unbounded.

           \pre !contains_zero(i)
        */
        bool abs_upper_magnitude(mpbqi const & i, int & r) {
            SASSERT(!contains_zero(i));
            if (bqim().is_P(i)) {
                if (i.upper_is_inf())
                    return false;
                r = bqm().magnitude_ub(i.upper());
                return true;
            }
            else {
                SASSERT(bqim().is_N(i));
                if (i.lower_is_inf())
                    return false;
                scoped_mpbq tmp(bqm());
                tmp = i.lower();
                bqm().neg(tmp);
                r = bqm().magnitude_ub(tmp);
                return true;
            }
        }

        /**
           \brief Find positive root upper bound using Knuth's approach.

           Given p(x) = a_n * x^n + a_{n-1} * x^{n-1} + ... + a_0

           If a_n is positive,
           Let B = max({ (-a_{n-k}/a_n)^{1/k} |  1 <= k <= n,   a_{n-k} < 0 })
           Then, 2*B is a bound for the positive roots

           Similarly, if a_n is negative
           Let B = max({ (-a_{n-k}/a_n)^{1/k} |  1 <= k <= n,   a_{n-k} > 0 })
           Then, 2*B is a bound for the positive roots

           This procedure returns a N s.t.  2*B <= 2^N

           The computation is performed using the intervals associated with the coefficients of
           the polynomial.

           The procedure may fail if the interval for a_n is of the form  (l, 0) or (0, u).
           Similarly, the procedure will fail if one of the a_{n-k} has an interval of the form (l, oo) or (-oo, u).
           Both cases can only happen if the values of the coefficients depend on infinitesimal values.
        */
        bool pos_root_upper_bound(unsigned n, value * const * p, int & N) {
            SASSERT(n > 1);
            SASSERT(!is_zero(p[n-1]));
            int lc_sign = sign(p[n-1]);
            SASSERT(lc_sign != 0);
            int lc_mag;
            if (!abs_lower_magnitude(interval(p[n-1]), lc_mag))
                return false;
            N = -static_cast<int>(m_ini_precision);
            for (unsigned k = 2; k <= n; k++) {
                value * a = p[n - k];
                if (!is_zero(a) && sign(a) != lc_sign) {
                    int a_mag;
                    if (!abs_upper_magnitude(interval(a), a_mag))
                        return false;
                    int C = (a_mag - lc_mag)/static_cast<int>(k) + 1 /* compensate imprecision on division */ + 1 /* Knuth's bound is 2 x Max {... } */;
                    if (C > N)
                        N = C;
                }
            }
            return true;
        }


        /**
           \brief Auxiliary method for creating the intervals of the coefficients of the polynomials p(-x)
           without actually creating p(-x).

           'a' is the interval of the i-th coefficient of a polynomial
                a_n * x^n + ... + a_0
        */
        void neg_root_adjust(mpbqi const & a, unsigned i, mpbqi & r) {
            if (i % 2 == 0)
                bqim().neg(a, r);
            else
                bqim().set(r, a);
        }

        /**
           \brief Find negative root lower bound using Knuth's approach.

           This is similar to pos_root_upper_bound. In principle, we can use
           the same algorithm. We just have to adjust the coefficients by using
           the transformation p(-x).
        */
        bool neg_root_lower_bound(unsigned n, value * const * as, int & N) {
            SASSERT(n > 1);
            SASSERT(!is_zero(as[n-1]));
            scoped_mpbqi aux(bqim());
            neg_root_adjust(interval(as[n-1]), n-1, aux);
            int lc_sign = bqim().is_P(aux) ? 1 : -1;
            int lc_mag;
            if (!abs_lower_magnitude(aux, lc_mag))
                return false;
            N = -static_cast<int>(m_ini_precision);
            for (unsigned k = 2; k <= n; k++) {
                value * a = as[n - k];
                if (!is_zero(a)) {
                    neg_root_adjust(interval(as[n-k]), n-k, aux);
                    int a_sign = bqim().is_P(aux) ? 1 : -1;
                    if (a_sign != lc_sign) {
                        int a_mag;
                        if (!abs_upper_magnitude(aux, a_mag))
                            return false;
                        int C = (a_mag - lc_mag)/static_cast<int>(k) + 1 /* compensate imprecision on division */ + 1 /* Knuth's bound is 2 x Max {... } */;
                        if (C > N)
                            N = C;
                    }
                }
            }
            return true;
        }

        /**
           \brief q <- x^{n-1}*p(1/x)

           Given p(x) a_{n-1} * x^{n-1} + ... + a_0, this method stores
           a_0 * x^{n-1} + ... + a_{n-1} into q.
        */
        void reverse(unsigned n, value * const * p, value_ref_buffer & q) {
            unsigned i = n;
            while (i > 0) {
                --i;
                q.push_back(p[i]);
            }
        }

        /**
           \brief To compute the lower bound for positive roots we computer the upper bound for the polynomial q(x) = x^{n-1}*p(1/x).
           Assume U is an upper bound for roots of q(x), i.e., (r > 0 and q(r) = 0) implies r < U.
           Note that if r is a root for q(x), then 1/r is a root for p(x) and 1/U is a lower bound for positive roots of p(x).
           The polynomial q(x) is just p(x) "reversed".
        */
        bool pos_root_lower_bound(unsigned n, value * const * p, int & N) {
            value_ref_buffer q(*this);
            reverse(n, p, q);
            if (pos_root_upper_bound(n, q.c_ptr(), N)) {
                N = -N;
                return true;
            }
            else {
                return false;
            }
        }

        /**
           \brief See comment on pos_root_lower_bound.
        */
        bool neg_root_upper_bound(unsigned n, value * const * p, int & N) {
            value_ref_buffer q(*this);
            reverse(n, p, q);
            if (neg_root_lower_bound(n, q.c_ptr(), N)) {
                N = -N;
                return true;
            }
            else {
                return false;
            }
        }

        /**
           \brief Store in ds all (non-constant) derivatives of p.

           \post d.size() == n-2
        */
        void mk_derivatives(unsigned n, value * const * p, scoped_polynomial_seq & ds) {
            SASSERT(n >= 3); // p is at least quadratic
            SASSERT(!is_zero(p[0]));
            SASSERT(!is_zero(p[n-1]));
            value_ref_buffer p_prime(*this);
            derivative(n, p, p_prime);
            ds.push(p_prime.size(), p_prime.c_ptr());
            SASSERT(n >= 3);
            for (unsigned i = 0; i < n - 2; i++) {
                SASSERT(ds.size() > 0);
                unsigned prev = ds.size() - 1;
                n = ds.size(prev);
                p = ds.coeffs(prev);
                derivative(n, p, p_prime);
                ds.push(p_prime.size(), p_prime.c_ptr());
            }
        }


        /**
           \brief Auxiliary function for count_signs_at_zeros. (See comments at count_signs_at_zeros).

           - taq_p_q contains TaQ(p, q; interval)
        */
        void count_signs_at_zeros_core(// Input values
                                       int taq_p_q,
                                       unsigned p_sz, value * const * p,  // polynomial p
                                       unsigned q_sz, value * const * q,  // polynomial q
                                       mpbqi const & interval,
                                       int num_roots,                     // number of roots of p in the given interval
                                       // Output values
                                       int & q_eq_0, int & q_gt_0, int & q_lt_0,
                                       value_ref_buffer & q2) {
            if (taq_p_q == num_roots) {
                // q is positive in all roots of p
                q_eq_0 = 0;
                q_gt_0 = num_roots;
                q_lt_0 = 0;
            }
            else if (taq_p_q == -num_roots) {
                // q is negative in all roots of p
                q_eq_0 = 0;
                q_gt_0 = 0;
                q_lt_0 = num_roots;
            }
            if (taq_p_q == num_roots - 1) {
                // The following assignment is the only possibility
                q_eq_0 = 1;
                q_gt_0 = num_roots - 1;
                q_lt_0 = 0;
            }
            else if (taq_p_q == -(num_roots - 1)) {
                // The following assignment is the only possibility
                q_eq_0 = 1;
                q_gt_0 = 0;
                q_lt_0 = num_roots - 1;
            }
            else {
                // Expensive case
                // q2 <- q^2
                mul(q_sz, q, q_sz, q, q2);
                int taq_p_q2 = TaQ(p_sz, p, q2.size(), q2.c_ptr(), interval);
                SASSERT(0 <= taq_p_q2 && taq_p_q2 <= num_roots);
                // taq_p_q2 == q_gt_0 + q_lt_0
                SASSERT((taq_p_q2 + taq_p_q) % 2 == 0);
                SASSERT((taq_p_q2 - taq_p_q) % 2 == 0);
                q_eq_0 = num_roots - taq_p_q2;
                q_gt_0 = (taq_p_q2 + taq_p_q)/2;
                q_lt_0 = (taq_p_q2 - taq_p_q)/2;
            }
            SASSERT(q_eq_0 + q_gt_0 + q_lt_0 == num_roots);
        }


        /**
           \brief Given polynomials p and q, and an interval, compute the number of
           roots of p in the interval such that:
              - q is zero
              - q is positive
              - q is negative

           \pre num_roots is the number of roots of p in the given interval.

           \remark num_roots == q_eq_0 + q_gt_0 + q_lt_0
        */
        void count_signs_at_zeros(// Input values
                                  unsigned p_sz, value * const * p,  // polynomial p
                                  unsigned q_sz, value * const * q,  // polynomial q
                                  mpbqi const & interval,
                                  int num_roots,                     // number of roots of p in the given interval
                                  // Output values
                                  int & q_eq_0, int & q_gt_0, int & q_lt_0,
                                  value_ref_buffer & q2) {
            TRACE("rcf_count_signs",
                  tout << "p: "; display_poly(tout, p_sz, p); tout << "\n";
                  tout << "q: "; display_poly(tout, q_sz, q); tout << "\n";);
            SASSERT(num_roots > 0);
            int taq_p_q = TaQ(p_sz, p, q_sz, q, interval);
            count_signs_at_zeros_core(taq_p_q, p_sz, p, q_sz, q, interval, num_roots, q_eq_0, q_gt_0, q_lt_0, q2);
        }

        /**
           \brief Expand the set of Tarski Queries used in the sign determination algorithm.

           taqrs contains the results of TaQ(p, prs[i]; interval)
           We have that taqrs.size() == prs.size()

           We produce a new_taqrs and new_prs
           For each pr in new_prs we have
                pr      in new_prs,  TaQ(p, pr; interval) in new_taqrs
                pr*q    in new_prs,  TaQ(p, pr*q; interval) in new_taqrs
                if q2_sz != 0, we also have
                pr*q^2  in new_prs,  TaQ(p, pr*q^2; interval) in new_taqrs

        */
        void expand_taqrs(// Input values
                          int_buffer const &              taqrs,
                          scoped_polynomial_seq const &   prs,
                          unsigned p_sz, value * const *  p,
                          unsigned q_sz, value * const *  q,
                          bool use_q2, unsigned q2_sz, value * const * q2,
                          mpbqi const &                   interval,
                          // Output values
                          int_buffer &                    new_taqrs,
                          scoped_polynomial_seq &         new_prs
                          ) {
            SASSERT(taqrs.size() == prs.size());
            new_taqrs.reset(); new_prs.reset();
            for (unsigned i = 0; i < taqrs.size(); i++) {
                // Add prs * 1
                new_taqrs.push_back(taqrs[i]);
                new_prs.push(prs.size(i), prs.coeffs(i));
                // Add prs * q
                value_ref_buffer prq(*this);
                mul(prs.size(i), prs.coeffs(i), q_sz, q, prq);
                new_taqrs.push_back(TaQ(p_sz, p, prq.size(), prq.c_ptr(), interval));
                new_prs.push(prq.size(), prq.c_ptr());
                // If use_q2,
                // Add prs * q^2
                if (use_q2) {
                    value_ref_buffer prq2(*this);
                    mul(prs.size(i), prs.coeffs(i), q2_sz, q2, prq2);
                    new_taqrs.push_back(TaQ(p_sz, p, prq2.size(), prq2.c_ptr(), interval));
                    new_prs.push(prq2.size(), prq2.c_ptr());
                }
            }
            SASSERT(new_prs.size() == new_taqrs.size());
            SASSERT(use_q2  || new_prs.size() == 2*prs.size());
            SASSERT(!use_q2 || new_prs.size() == 3*prs.size());
        }

        /**
           \brief In the sign determination algorithm main loop, we keep processing polynomials q,
           and checking whether they discriminate the roots of the target polynomial.

           The vectors sc_cardinalities contains the cardinalities of the new realizable sign conditions.
           That is, we started we a sequence of sign conditions
                     sc_1, ..., sc_n,
           If q2_used is true, then we expanded this sequence as
                     sc1_1 and q == 0, sc_1 and q > 0, sc_1 and q < 0, ..., sc_n and q == 0, sc_n and q > 0, sc_n and q < 0
           If q2_used is false, then we considered only two possible signs of q.

           Thus, q is useful (i.e., it is a discriminator for the roots of p) IF
                If !q2_used,   then There is an i s.t. sc_cardinalities[2*i] > 0 && sc_cardinalities[2*i] > 0
                If q2_used,    then There is an i s.t. AtLeastTwo(sc_cardinalities[3*i] > 0, sc_cardinalities[3*i+1] > 0, sc_cardinalities[3*i+2] > 0)
        */
        bool keep_new_sc_assignment(unsigned sz, int const * sc_cardinalities, bool q2_used) {
            SASSERT(q2_used  || sz % 2 == 0);
            SASSERT(!q2_used || sz % 3 == 0);
            if (q2_used) {
                for (unsigned i = 0; i < sz; i += 3) {
                    unsigned c = 0;
                    if (sc_cardinalities[i] > 0)   c++;
                    if (sc_cardinalities[i+1] > 0) c++;
                    if (sc_cardinalities[i+2] > 0) c++;
                    if (c >= 2)
                        return true;
                }
            }
            else {
                for (unsigned i = 0; i < sz; i += 2) {
                    if (sc_cardinalities[i] > 0 && sc_cardinalities[i+1] > 0)
                        return true;
                }
            }
            return false;
        }

        /**
           \brief Store the polynomials in prs into the array of polynomials ps.
        */
        void set_array_p(array<polynomial> & ps, scoped_polynomial_seq const & prs) {
            unsigned sz = prs.size();
            ps.set(allocator(), sz, polynomial());
            for (unsigned i = 0; i < sz; i++) {
                unsigned pi_sz     = prs.size(i);
                value * const * pi = prs.coeffs(i);
                set_p(ps[i], pi_sz, pi);
            }
        }

        /**
           \brief Create a "sign determination" data-structure for an algebraic extension.

           The new object will assume the ownership of the elements stored in M and scs.
           M and scs will be empty after this operation.
        */
        sign_det * mk_sign_det(mpz_matrix & M_s, scoped_polynomial_seq const & prs, int_buffer const & taqrs, scoped_polynomial_seq const & qs, scoped_sign_conditions & scs) {
            sign_det * r = new (allocator()) sign_det();
            r->M_s.swap(M_s);
            set_array_p(r->m_prs, prs);
            r->m_taqrs.set(allocator(), taqrs.size(), taqrs.c_ptr());
            set_array_p(r->m_qs,  qs);
            r->m_sign_conditions.set(allocator(), scs.size(), scs.c_ptr());
            scs.release();
            return r;
        }

        /**
           \brief Create a new algebraic extension
         */
        algebraic * mk_algebraic(unsigned p_sz, value * const * p, mpbqi const & interval, mpbqi const & iso_interval, sign_det * sd, unsigned sc_idx) {
            unsigned idx = next_algebraic_idx();
            algebraic * r = new (allocator()) algebraic(idx);
            m_extensions[extension::ALGEBRAIC].push_back(r);

            set_p(r->m_p, p_sz, p);
            set_interval(r->m_interval, interval);
            set_interval(r->m_iso_interval, iso_interval);
            r->m_sign_det = sd;
            inc_ref_sign_det(sd);
            r->m_sc_idx   = sc_idx;
            r->m_depends_on_infinitesimals = depends_on_infinitesimals(p_sz, p);

            return r;
        }

        /**
           \brief Add a new root of p that is isolated by (interval, sd, sc_idx) to roots.
        */
        void add_root(unsigned p_sz, value * const * p, mpbqi const & interval, mpbqi const & iso_interval, sign_det * sd, unsigned sc_idx, numeral_vector & roots) {
            algebraic * a = mk_algebraic(p_sz, p, interval, iso_interval, sd, sc_idx);
            numeral r;
            set(r, mk_rational_function_value(a));
            roots.push_back(r);
        }

        /**
           \brief Simpler version of add_root that does not use sign_det data-structure. That is,
           interval contains only one root of p.
        */
        void add_root(unsigned p_sz, value * const * p, mpbqi const & interval, mpbqi const & iso_interval, numeral_vector & roots) {
            add_root(p_sz, p, interval, iso_interval, nullptr, UINT_MAX, roots);
        }

        /**
           \brief Create (the square) matrix for sign determination of q on the roots of p.
           It builds matrix based on the number of root of p where
                q is == 0, > 0 and < 0.
           The resultant matrix is stored in M.

           Return false if the sign of q is already determined, that is
           only one of the q_eq_0, q_gt_0, q_lt_0 is greater than zero.

           If the return value is true, then the resultant matrix M has size 2x2 or 3x3

           - q_eq_0 > 0, q_gt_0 > 0, q_lt_0 == 0
                 M <- {{1, 1},
                       {0, 1}}

                 Meaning:
                 M . [ #(q == 0), #(q > 0) ]^t == [ TaQ(p, 1), TaQ(p, q) ]^t

                 [ ... ]^t represents a column matrix.

           - q_eq_0 > 0, q_gt_0 == 0, q_lt_0 > 0
                 M <- {{1,  1},
                       {0, -1}}

                 Meaning:
                 M . [ #(q == 0), #(q < 0) ]^t == [ TaQ(p, 1), TaQ(p, q) ]^t

           - q_eq_0 == 0, q_gt_0 > 0, q_lt_0 > 0
                 M <- {{1, 1},
                       {1, -1}}

                 Meaning:
                 M . [ #(q > 0), #(q < 0) ]^t == [ TaQ(p, 1), TaQ(p, q) ]^t

           - q_eq_0 > 0,  q_gt_0 > 0, q_lt_0 > 0
                 M <- {{1, 1,  1},
                       {0, 1, -1},
                       {0, 1,  1}}

                 Meaning:
                 M . [ #(q == 0), #(q > 0), #(q < 0) ]^t == [ TaQ(p, 1), TaQ(p, q), TaQ(p, q^2) ]^t

        */
        bool mk_sign_det_matrix(int q_eq_0, int q_gt_0, int q_lt_0, scoped_mpz_matrix & M) {
            if      (q_eq_0 > 0  && q_gt_0 > 0  && q_lt_0 == 0) {
                // M <- {{1, 1},
                //       {0, 1}}
                mm().mk(2,2,M);
                M.set(0,0, 1); M.set(0,1, 1);
                M.set(1,0, 0); M.set(1,1, 1);
                return true;
            }
            else if (q_eq_0 > 0  && q_gt_0 == 0 && q_lt_0 > 0)  {
                // M <- {{1, 1},
                //       {0, -1}}
                mm().mk(2,2,M);
                M.set(0,0, 1); M.set(0,1,  1);
                M.set(1,0, 0); M.set(1,1, -1);
                return true;
            }
            else if (q_eq_0 == 0 && q_gt_0 > 0  && q_lt_0 > 0)  {
                // M <- {{1, 1},
                //       {1, -1}}
                mm().mk(2,2,M);
                M.set(0,0, 1); M.set(0,1,  1);
                M.set(1,0, 1); M.set(1,1, -1);
                return true;
            }
            else if (q_eq_0 > 0  && q_gt_0 > 0  && q_lt_0 > 0)  {
                // M <- {{1, 1,  1},
                //       {0, 1, -1},
                //       {0, 1,  1}}
                mm().mk(3,3,M);
                M.set(0,0, 1); M.set(0,1, 1); M.set(0,2,  1);
                M.set(1,0, 0); M.set(1,1, 1); M.set(1,2, -1);
                M.set(2,0, 0); M.set(2,1, 1); M.set(2,2,  1);
                return true;
            }
            else {
                // Sign of q is already determined.
                return false;
            }
        }


        /**
           \brief Isolate roots of p in the given interval using sign conditions to distinguish between them.
           We need this method when the polynomial contains roots that are infinitesimally close to each other.
           For example, given an infinitesimal eps, the polynomial (x - 1)(x - 1 - eps) == (1 + eps) + (- 2 - eps)*x + x^2
           has two roots 1 and 1+eps, we can't isolate these roots using intervals with binary rational end points.
           In this case, we use the signs of (some of the) derivatives in the roots.
           By Thom's lemma, we know we can always use the signs of the derivatives to distinguish between different roots.

           Remark: the polynomials do not need to be the derivatives of p. We use derivatives because a simple
           sequential search can be used to find the set of polynomials that can be used to distinguish between
           the different roots.

           \pre num_roots is the number of roots in the given interval
        */
        void sign_det_isolate_roots(unsigned p_sz, value * const * p, int num_roots, mpbqi const & interval, mpbqi const & iso_interval, numeral_vector & roots) {
            SASSERT(num_roots >= 2);
            scoped_polynomial_seq der_seq(*this);
            mk_derivatives(p_sz, p, der_seq);

            CASSERT("rcf_isolate_roots", TaQ_1(p_sz, p, interval) == num_roots);
            scoped_mpz_matrix M_s(mm());
            mm().mk(1, 1, M_s);
            M_s.set(0, 0, 1);

            // Sequence of polynomials associated with each row of M_s
            scoped_polynomial_seq prs(*this);
            value * one_p[] = { one() };
            prs.push(1, one_p); // start with the polynomial one

            // For i in [0, poly_rows.size()), taqrs[i] == TaQ(prs[i], p; interval)
            int_buffer taqrs;
            taqrs.push_back(num_roots);

            // Sequence of polynomials used to discriminate the roots of p
            scoped_polynomial_seq qs(*this);

            // Sequence of sign conditions associated with the columns of M_s
            // These are sign conditions on the polynomials in qs.
            scoped_sign_conditions scs(*this);
            scs.push_back(nullptr);

            // Starting configuration
            //
            // M_s   = {{1}}         Matrix of size 1x1 containing the value 1
            // prs   = [1]           Sequence of size 1 containing the constant polynomial 1 (one is always the first element of this sequence)
            // taqrs = [num_roots]   Sequence of size 1 containing the integer num_roots
            // scs   = [0]           Sequence of size 1 with the empty sign condition (i.e., NULL pointer)
            // qs    = {}            Empty set
            //

            scoped_mpz_matrix      new_M_s(mm());
            int_buffer             new_taqrs;
            scoped_polynomial_seq  new_prs(*this);
            scoped_sign_conditions new_scs(*this);

            int_buffer            sc_cardinalities;
            unsigned_buffer       cols_to_keep;
            unsigned_buffer       new_row_idxs;

            unsigned i = der_seq.size();
            // We perform the search backwards because the degrees are in decreasing order.
            while (i > 0) {
                --i;
                checkpoint();
                SASSERT(M_s.m() == M_s.n());
                SASSERT(M_s.m() == taqrs.size());
                SASSERT(M_s.m() == scs.size());
                TRACE("rcf_sign_det",
                      tout << M_s;
                      for (unsigned j = 0; j < scs.size(); j++) {
                          display_sign_conditions(tout, scs[j]);
                          tout << " = " << taqrs[j] << "\n";
                      }
                      tout << "qs:\n";
                      for (unsigned j = 0; j < qs.size(); j++) {
                          display_poly(tout, qs.size(j), qs.coeffs(j)); tout << "\n";
                      });
                // We keep executing this loop until we have only one root for each sign condition in scs.
                // When this happens the polynomials in qs are the ones used to discriminate the roots of p.
                unsigned q_sz     = der_seq.size(i);
                value * const * q = der_seq.coeffs(i);
                // q is a derivative of p.
                int q_eq_0, q_gt_0, q_lt_0;
                value_ref_buffer q2(*this);
                count_signs_at_zeros(p_sz, p, q_sz, q, iso_interval, num_roots, q_eq_0, q_gt_0, q_lt_0, q2);
                TRACE("rcf_sign_det",
                      tout << "q: "; display_poly(tout, q_sz, q); tout << "\n";
                      tout << "#(q == 0): " << q_eq_0 << ", #(q > 0): " << q_gt_0 << ", #(q < 0): " << q_lt_0 << "\n";);
                scoped_mpz_matrix M(mm());
                if (!mk_sign_det_matrix(q_eq_0, q_gt_0, q_lt_0, M)) {
                    // skip q since its sign does not discriminate the roots of p
                    continue;
                }
                bool use_q2 = M.n() == 3;
                mm().tensor_product(M_s, M, new_M_s);
                expand_taqrs(taqrs, prs, p_sz, p, q_sz, q, use_q2, q2.size(), q2.c_ptr(), iso_interval,
                             // --->
                             new_taqrs, new_prs);
                SASSERT(new_M_s.n() == new_M_s.m());           // it is a square matrix
                SASSERT(new_M_s.m() == new_taqrs.size());
                SASSERT(new_M_s.m() == new_prs.size());
                // The system must have a solution
                sc_cardinalities.resize(new_taqrs.size(), 0);
                // Solve
                //    new_M_s * sc_cardinalities = new_taqrs
                VERIFY(mm().solve(new_M_s, sc_cardinalities.c_ptr(), new_taqrs.c_ptr()));
                TRACE("rcf_sign_det", tout << "solution: "; for (unsigned i = 0; i < sc_cardinalities.size(); i++) { tout << sc_cardinalities[i] << " "; } tout << "\n";);
                // The solution must contain only positive values <= num_roots
                DEBUG_CODE(for (unsigned j = 0; j < sc_cardinalities.size(); j++) { SASSERT(0 <= sc_cardinalities[j] && sc_cardinalities[j] <= num_roots); });
                // We should keep q only if it discriminated something.
                // That is,
                //   If !use_q2,   then There is an i s.t. sc_cardinalities[2*i] > 0 && sc_cardinalities[2*i] > 0
                //   If use_q2,    then There is an i s.t. AtLeastTwo(sc_cardinalities[3*i] > 0, sc_cardinalities[3*i+1] > 0, sc_cardinalities[3*i+2] > 0)
                if (!keep_new_sc_assignment(sc_cardinalities.size(), sc_cardinalities.c_ptr(), use_q2)) {
                    // skip q since it did not reduced the cardinality of the existing sign conditions.
                    continue;
                }
                // keep q
                unsigned q_idx = qs.size();
                qs.push(q_sz, q);
                // We remove the columns associated with sign conditions that have cardinality zero,
                // and create new extended sign condition objects for the ones that have cardinality > 0.
                cols_to_keep.reset();
                unsigned j = 0; unsigned k = 0;
                unsigned step_sz = use_q2 ? 3 : 2;
                bool all_one = true;
                while (j < sc_cardinalities.size()) {
                    sign_condition * sc = scs[k];
                    k++;
                    for (unsigned s = 0; s < step_sz; s++) {
                        // Remark: the second row of M contains the sign for q
                        if (sc_cardinalities[j] > 0) {
                            new_scs.push_back(mk_sign_condition(q_idx, M.get_int(1, s), sc));
                            cols_to_keep.push_back(j);
                        }
                        if (sc_cardinalities[j] > 1)
                            all_one = false;
                        j++;
                    }
                }
                // Update scs with new_scs
                scs.copy_from(new_scs);
                SASSERT(new_scs.empty());
                // Update M_s
                mm().filter_cols(new_M_s, cols_to_keep.size(), cols_to_keep.c_ptr(), M_s);
                SASSERT(M_s.n() == cols_to_keep.size());
                new_row_idxs.resize(cols_to_keep.size(), 0);
                unsigned new_num_rows = mm().linear_independent_rows(M_s, new_row_idxs.c_ptr(), M_s);
                SASSERT(new_num_rows == cols_to_keep.size());
                // Update taqrs and prs
                prs.reset();
                taqrs.reset();
                for (unsigned j = 0; j < new_num_rows; j++) {
                    unsigned rid = new_row_idxs[j];
                    prs.push(new_prs.size(rid), new_prs.coeffs(rid));
                    taqrs.push_back(new_taqrs[rid]);
                }
                if (all_one)  {
                    // Stop each remaining sign condition in scs has cardinality one
                    // So, they are discriminating the roots of p.
                    break;
                }
            }
            TRACE("rcf_sign_det",
                  tout << "Final state\n";
                  display_poly(tout, p_sz, p); tout << "\n";
                  tout << M_s;
                  for (unsigned j = 0; j < scs.size(); j++) {
                      display_sign_conditions(tout, scs[j]);
                      tout << " = " << taqrs[j] << "\n";
                  }
                  tout << "qs:\n";
                  for (unsigned j = 0; j < qs.size(); j++) {
                      display_poly(tout, qs.size(j), qs.coeffs(j)); tout << "\n";
                  }
                  tout << "prs:\n";
                  for (unsigned j = 0; j < prs.size(); j++) {
                      display_poly(tout, prs.size(j), prs.coeffs(j)); tout << "\n";
                  });
            SASSERT(M_s.n() == M_s.m()); SASSERT(M_s.n() == static_cast<unsigned>(num_roots));
            sign_det * sd = mk_sign_det(M_s, prs, taqrs, qs, scs);
            for (unsigned idx = 0; idx < static_cast<unsigned>(num_roots); idx++) {
                add_root(p_sz, p, interval, iso_interval, sd, idx, roots);
            }
        }

        /**
           \brief Return true if p is a polynomial of the form a_{n-1}*x^{n-1} + a_0
        */
        bool is_nz_binomial(unsigned n, value * const * p) {
            SASSERT(n >= 2);
            SASSERT(!is_zero(p[0]));
            SASSERT(!is_zero(p[n-1]));
            for (unsigned i = 1; i < n - 1; i++) {
                if (!is_zero(p[i]))
                    return false;
            }
            return true;
        }

        /**
           \brief magnitude -> mpbq
        */
        void magnitude_to_mpbq(int mag, bool sign, mpbq & r) {
            if (mag < 0) {
                bqm().set(r, mpbq(1, -mag));
            }
            else {
                bqm().set(r, mpbq(2));
                bqm().power(r, mag);
            }
            if (sign)
                bqm().neg(r);
        }

        /**
           \brief Convert magnitudes for negative roots lower and upper bounds into an interval.
        */
        void mk_neg_interval(bool has_neg_lower, int neg_lower_N, bool has_neg_upper, int neg_upper_N, mpbqi & r) {
            scoped_mpbq aux(bqm());
            if (!has_neg_lower) {
                set_lower_inf(r);
            }
            else {
                magnitude_to_mpbq(neg_lower_N, true, aux);
                set_lower(r, aux);
            }
            if (!has_neg_upper) {
                set_upper_zero(r);
            }
            else {
                magnitude_to_mpbq(neg_upper_N, true, aux);
                set_upper(r, aux);
            }
        }

        /**
           \brief Convert magnitudes for negative roots lower and upper bounds into an interval.
        */
        void mk_pos_interval(bool has_pos_lower, int pos_lower_N, bool has_pos_upper, int pos_upper_N, mpbqi & r) {
            scoped_mpbq aux(bqm());
            if (!has_pos_lower) {
                set_lower_zero(r);
            }
            else {
                magnitude_to_mpbq(pos_lower_N, false, aux);
                set_lower(r, aux);
            }
            if (!has_pos_upper) {
                set_upper_inf(r);
            }
            else {
                magnitude_to_mpbq(pos_upper_N, false, aux);
                set_upper(r, aux);
            }
        }

        struct bisect_ctx {
            unsigned                m_p_sz;
            value * const *         m_p;
            bool                    m_depends_on_infinitesimals;
            scoped_polynomial_seq & m_sturm_seq;
            numeral_vector &        m_result_roots;
            bisect_ctx(unsigned p_sz, value * const * p, bool dinf, scoped_polynomial_seq & seq, numeral_vector & roots):
                m_p_sz(p_sz), m_p(p), m_depends_on_infinitesimals(dinf), m_sturm_seq(seq), m_result_roots(roots) {}
        };

        void bisect_isolate_roots(mpbqi & interval, mpbqi & iso_interval, int lower_sv, int upper_sv, bisect_ctx & ctx) {
            SASSERT(lower_sv >= upper_sv);
            int num_roots = lower_sv - upper_sv;
            if (num_roots == 0) {
                // interval does not contain roots
            }
            else if (num_roots == 1) {
                // Sturm sequences are for half-open intervals (a, b]
                // We must check if upper is the root
                if (eval_sign_at(ctx.m_p_sz, ctx.m_p, interval.upper()) == 0) {
                    // found precise root
                    numeral r;
                    set(r, mk_rational(interval.upper()));
                    ctx.m_result_roots.push_back(r);
                }
                else {
                    // interval is an isolating interval
                    add_root(ctx.m_p_sz, ctx.m_p, interval, iso_interval, ctx.m_result_roots);
                }
            }
            else if (ctx.m_depends_on_infinitesimals && check_precision(interval, m_max_precision)) {
                // IF
                //   - The polynomial depends on infinitesimals
                //   - The interval contains more than one root
                //   - The size of the interval is less than 1/2^m_max_precision
                // THEN
                //   - We switch to expensive sign determination procedure, since
                //     the roots may be infinitely close to each other.
                //
                sign_det_isolate_roots(ctx.m_p_sz, ctx.m_p, num_roots, interval, iso_interval, ctx.m_result_roots);
            }
            else {
                scoped_mpbq mid(bqm());
                bqm().add(interval.lower(), interval.upper(), mid);
                bqm().div2(mid);
                int mid_sv = sign_variations_at(ctx.m_sturm_seq, mid);
                int num_left_roots  = lower_sv - mid_sv;
                int num_right_roots = mid_sv - upper_sv;
                if (num_left_roots == 0) {
                    scoped_mpbqi right_interval(bqim());
                    set_lower(right_interval, mid);
                    set_upper(right_interval, interval.upper());
                    bisect_isolate_roots(right_interval, iso_interval, mid_sv, upper_sv, ctx);
                }
                else if (num_right_roots == 0) {
                    scoped_mpbqi left_interval(bqim());
                    set_lower(left_interval, interval.lower());
                    set_upper(left_interval, mid);
                    bisect_isolate_roots(left_interval, iso_interval,   lower_sv, mid_sv, ctx);
                }
                else {
                    scoped_mpbqi left_interval(bqim());
                    scoped_mpbqi right_interval(bqim());
                    set_lower(left_interval, interval.lower());
                    set_upper(left_interval, mid);
                    set_lower(right_interval, mid);
                    set_upper(right_interval, interval.upper());
                    bisect_isolate_roots(left_interval, left_interval,   lower_sv, mid_sv, ctx);
                    bisect_isolate_roots(right_interval, right_interval, mid_sv, upper_sv, ctx);
                }
            }
        }

        /**
           \brief Entry point for the root isolation procedure based on bisection.
        */
        void bisect_isolate_roots(// Input values
                                  unsigned p_sz, value * const * p, mpbqi & interval, mpbqi & iso_interval,
                                  // Extra Input values with already computed information
                                  scoped_polynomial_seq & sturm_seq, // sturm sequence for p
                                  int lower_sv,                      // number of sign variations at the lower bound of interval
                                  int upper_sv,                      // number of sign variations at the upper bound of interval
                                  // Output values
                                  numeral_vector & roots) {
            bool dinf = depends_on_infinitesimals(p_sz, p);
            bisect_ctx ctx(p_sz, p, dinf, sturm_seq, roots);
            bisect_isolate_roots(interval, iso_interval, lower_sv, upper_sv, ctx);
        }

        /**
           \brief Root isolation for polynomials that are
                   - nonlinear (degree > 2)
                   - zero is not a root
                   - square free
                   - nonconstant
        */
        void nl_nz_sqf_isolate_roots(unsigned n, value * const * p, numeral_vector & roots) {
            SASSERT(n > 2);
            SASSERT(!is_zero(p[0]));
            SASSERT(!is_zero(p[n-1]));
            int pos_lower_N, pos_upper_N, neg_lower_N, neg_upper_N;
            bool has_neg_lower = neg_root_lower_bound(n, p, neg_lower_N);
            bool has_neg_upper = neg_root_upper_bound(n, p, neg_upper_N);
            bool has_pos_lower = pos_root_lower_bound(n, p, pos_lower_N);
            bool has_pos_upper = pos_root_upper_bound(n, p, pos_upper_N);
            TRACE("rcf_isolate",
                  display_poly(tout, n, p); tout << "\n";
                  if (has_neg_lower) tout << "-2^" << neg_lower_N; else tout << "-oo";
                  tout << " < neg-roots < ";
                  if (has_neg_upper) tout << "-2^" << neg_upper_N; else tout << "0";
                  tout << "\n";
                  if (has_pos_lower) tout << "2^" << pos_lower_N; else tout << "0";
                  tout << " < pos-roots < ";
                  if (has_pos_upper) tout << "2^" << pos_upper_N; else tout << "oo";
                  tout << "\n";);
            // Compute the number of positive and negative roots
            scoped_polynomial_seq seq(*this);
            sturm_seq(n, p, seq);
            int num_sv_minus_inf = sign_variations_at_minus_inf(seq);
            int num_sv_zero      = sign_variations_at_zero(seq);
            int num_sv_plus_inf  = sign_variations_at_plus_inf(seq);
            int num_neg_roots    = num_sv_minus_inf - num_sv_zero;
            int num_pos_roots    = num_sv_zero      - num_sv_plus_inf;
            TRACE("rcf_isolate",
                  tout << "num_neg_roots: " << num_neg_roots << "\n";
                  tout << "num_pos_roots: " << num_pos_roots << "\n";);
            scoped_mpbqi pos_interval(bqim());
            scoped_mpbqi neg_interval(bqim());
            mk_neg_interval(has_neg_lower, neg_lower_N, has_neg_upper, neg_upper_N, neg_interval);
            mk_pos_interval(has_pos_lower, pos_lower_N, has_pos_upper, pos_upper_N, pos_interval);
            scoped_mpbqi minf_zero(bqim());
            set_lower_inf(minf_zero);
            set_upper_zero(minf_zero);
            scoped_mpbqi zero_inf(bqim());
            set_lower_zero(zero_inf);
            set_upper_inf(zero_inf);

            if (num_neg_roots > 0) {
                if (num_neg_roots == 1) {
                    add_root(n, p, neg_interval, minf_zero, nullptr, UINT_MAX, roots);
                }
                else {
                    if (has_neg_lower) {
                        bisect_isolate_roots(n, p, neg_interval, minf_zero, seq, num_sv_minus_inf, num_sv_zero, roots);
                    }
                    else {
                        sign_det_isolate_roots(n, p, num_neg_roots, minf_zero, minf_zero, roots);
                    }
                }
            }

            if (num_pos_roots > 0) {
                if (num_pos_roots == 1) {
                    add_root(n, p, pos_interval, zero_inf, nullptr, UINT_MAX, roots);
                }
                else {
                    if (has_pos_upper) {
                        bisect_isolate_roots(n, p, pos_interval, zero_inf, seq, num_sv_zero, num_sv_plus_inf, roots);
                    }
                    else {
                        sign_det_isolate_roots(n, p, num_pos_roots, zero_inf, zero_inf, roots);
                    }
                }
            }
        }

        /**
           \brief Root isolation for polynomials that are
                   - zero is not a root
                   - square free
                   - nonconstant
        */
        void nz_sqf_isolate_roots(unsigned n, value * const * p, numeral_vector & roots) {
            SASSERT(n > 1);
            SASSERT(!is_zero(p[0]));
            SASSERT(!is_zero(p[n-1]));
            if (n == 2) {
                // we don't need a field extension for linear polynomials.
                numeral r;
                value_ref v(*this);
                neg(p[0], v);
                div(v, p[1], v);
                set(r, v);
                roots.push_back(r);
            }
            else {
                nl_nz_sqf_isolate_roots(n, p, roots);
            }
        }

        /**
           \brief Root isolation for polynomials where 0 is not a root, and the denominators have been cleaned
           when m_clean_denominators == true

        */
        void nz_cd_isolate_roots(unsigned n, value * const * p, numeral_vector & roots) {
            SASSERT(n > 0);
            SASSERT(!is_zero(p[0]));
            SASSERT(!is_zero(p[n-1]));
            SASSERT(!m_clean_denominators || has_clean_denominators(n, p));
            if (n == 1) {
                // constant polynomial
                return;
            }
            value_ref_buffer sqf(*this);
            square_free(n, p, sqf);
            nz_sqf_isolate_roots(sqf.size(), sqf.c_ptr(), roots);
        }

        /**
           \brief Root isolation for polynomials where 0 is not a root.
        */
        void nz_isolate_roots(unsigned n, value * const * p, numeral_vector & roots) {
            TRACE("rcf_isolate",
                  tout << "nz_isolate_roots\n";
                  display_poly(tout, n, p); tout << "\n";);
            if (m_clean_denominators) {
                value_ref d(*this);
                value_ref_buffer norm_p(*this);
                clean_denominators(n, p, norm_p, d);
                nz_cd_isolate_roots(norm_p.size(), norm_p.c_ptr(), roots);
            }
            else {
                nz_cd_isolate_roots(n, p, roots);
            }
        }

        /**
           \brief Root isolation entry point.
        */
        void isolate_roots(unsigned n, numeral const * p, numeral_vector & roots) {
            TRACE("rcf_isolate_bug", tout << "isolate_roots: "; for (unsigned i = 0; i < n; i++) { display(tout, p[i]); tout << " "; } tout << "\n";);
            SASSERT(n > 0);
            SASSERT(!is_zero(p[n-1]));
            if (n == 1) {
                // constant polynomial
                return;
            }
            unsigned i = 0;
            for (; i < n; i++) {
                if (!is_zero(p[i]))
                    break;
            }
            SASSERT(i < n);
            SASSERT(!is_zero(p[i]));
            ptr_buffer<value> nz_p;
            for (; i < n; i++)
                nz_p.push_back(p[i].m_value);
            nz_isolate_roots(nz_p.size(), nz_p.c_ptr(), roots);
            if (nz_p.size() < n) {
                // zero is a root
                roots.push_back(numeral());
            }
        }

        // ---------------------------------
        //
        // Basic operations
        //
        // ---------------------------------

        void reset(numeral & a) {
            del(a);
            SASSERT(is_zero(a));
        }

        int sign(value * a) {
            if (is_zero(a))
                return 0;
            else if (is_nz_rational(a)) {
                return qm().is_pos(to_mpq(a)) ? 1 : -1;
            }
            else {
                SASSERT(!contains_zero(a->interval()));
                return bqim().is_P(a->interval()) ? 1 : -1;
            }
        }

        int sign(numeral const & a) {
            return sign(a.m_value);
        }

        /**
           \brief Return true the given rational function value is actually an integer.

           \pre a is a rational function (algebraic) extension.

           \remark If a is actually an integer, this method also updates its representation.
        */
        bool is_algebraic_int(numeral const & a) {
            SASSERT(is_rational_function(a));
            SASSERT(to_rational_function(a)->ext()->is_algebraic());
            // TODO
            return false;
        }

        /**
           \brief Return true if a is an integer
        */
        bool is_int(numeral const & a) {
            if (is_zero(a))
                return true;
            else if (is_nz_rational(a))
                return qm().is_int(to_mpq(a));
            else {
                rational_function_value * rf = to_rational_function(a);
                switch (rf->ext()->knd()) {
                case extension::TRANSCENDENTAL: return false;
                case extension::INFINITESIMAL:  return false;
                case extension::ALGEBRAIC: return is_algebraic_int(a);
                default:
                    UNREACHABLE();
                    return false;
                }
            }
        }

        /**
           \brief Return true if a depends on infinitesimal extensions.
        */
        bool depends_on_infinitesimals(numeral const & a) const {
            return depends_on_infinitesimals(a.m_value);
        }

        static void swap(mpbqi & a, mpbqi & b) {
            realclosure::swap(a, b);
        }

        /**
           \brief Store in interval an approximation of the rational number q with precision k.
           interval has binary rational end-points and the width is <= 1/2^k
        */
        void mpq_to_mpbqi(mpq const & q, mpbqi & interval, unsigned k) {
            interval.set_lower_is_inf(false);
            interval.set_upper_is_inf(false);
            if (bqm().to_mpbq(q, interval.lower())) {
                bqm().set(interval.upper(), interval.lower());
                interval.set_lower_is_open(false);
                interval.set_upper_is_open(false);
            }
            else {
                bqm().set(interval.upper(), interval.lower());
                bqm().mul2(interval.upper());
                interval.set_lower_is_open(true);
                interval.set_upper_is_open(true);
                if (qm().is_neg(q)) {
                    ::swap(interval.lower(), interval.upper());
                }
                while (contains_zero(interval) || !check_precision(interval, k) || bqm().is_zero(interval.lower()) || bqm().is_zero(interval.upper())) {
                    checkpoint();
                    bqm().refine_lower(q, interval.lower(), interval.upper());
                    bqm().refine_upper(q, interval.lower(), interval.upper());
                }
            }
        }

        void initialize_rational_value_interval(value * a) {
            // For rational values, we only compute the binary intervals if needed.
            SASSERT(is_nz_rational(a));
            mpq_to_mpbqi(to_mpq(a), a->m_interval, m_ini_precision);
        }

        mpbqi & interval(value * a) const {
            SASSERT(a != 0);
            if (contains_zero(a->m_interval)) {
                SASSERT(is_nz_rational(a));
                const_cast<imp*>(this)->initialize_rational_value_interval(a);
            }
            return a->m_interval;
        }

        rational_value * mk_rational() {
            return new (allocator()) rational_value();
        }

        /**
           \brief Make a rational and swap its value with v
        */
        rational_value * mk_rational_and_swap(mpq & v) {
            SASSERT(!qm().is_zero(v));
            rational_value * r = mk_rational();
            ::swap(r->m_value, v);
            return r;
        }

        rational_value * mk_rational(mpq const & v) {
            SASSERT(!qm().is_zero(v));
            rational_value * r = mk_rational();
            qm().set(r->m_value, v);
            return r;
        }

        rational_value * mk_rational(mpz const & v) {
            SASSERT(!qm().is_zero(v));
            rational_value * r = mk_rational();
            qm().set(r->m_value, v);
            return r;
        }

        rational_value * mk_rational(mpbq const & v) {
            SASSERT(!bqm().is_zero(v));
            scoped_mpq v_q(qm()); // v as a rational
            ::to_mpq(qm(), v, v_q);
            return mk_rational(v_q);
        }

        void reset_interval(value * a) {
            bqim().reset(a->m_interval);
        }

        template<typename T>
        void update_mpq_value(value * a, T & v) {
            SASSERT(is_nz_rational(a));
            qm().set(to_mpq(a), v);
            reset_interval(a);
        }

        template<typename T>
        void update_mpq_value(numeral & a, T & v) {
            update_mpq_value(a.m_value, v);
        }

        /**
           \brief a <- n
        */
        void set(numeral & a, int n) {
            if (n == 0) {
                reset(a);
                return;
            }

            del(a);
            a.m_value = mk_rational();
            inc_ref(a.m_value);
            update_mpq_value(a, n);
        }

        /**
           \brief a <- n
        */
        void set(numeral & a, mpz const & n) {
            if (qm().is_zero(n)) {
                reset(a);
                return;
            }

            del(a);
            a.m_value = mk_rational();
            inc_ref(a.m_value);
            update_mpq_value(a, n);
        }

        /**
           \brief a <- n
        */
        void set(numeral & a, mpq const & n) {
            if (qm().is_zero(n)) {
                reset(a);
                return;
            }
            del(a);
            a.m_value = mk_rational();
            inc_ref(a.m_value);
            update_mpq_value(a, n);
        }

        /**
           \brief a <- n
        */
        void set(numeral & a, numeral const & n) {
            inc_ref(n.m_value);
            dec_ref(a.m_value);
            a.m_value = n.m_value;
        }

        // ---------------------------------
        //
        // Polynomial arithmetic in RCF
        //
        // ---------------------------------

        /**
           \brief Remove 0s
        */
        void adjust_size(value_ref_buffer & r) {
            while (!r.empty() && r.back() == nullptr) {
                r.pop_back();
            }
        }

        /**
           \brief r <- p1 + p2
        */
        void add(unsigned sz1, value * const * p1, unsigned sz2, value * const * p2, value_ref_buffer & r) {
            SASSERT(p1 != r.c_ptr());
            SASSERT(p2 != r.c_ptr());
            r.reset();
            value_ref a_i(*this);
            unsigned min = std::min(sz1, sz2);
            unsigned i = 0;
            for (; i < min; i++) {
                add(p1[i], p2[i], a_i);
                r.push_back(a_i);
            }
            for (; i < sz1; i++)
                r.push_back(p1[i]);
            for (; i < sz2; i++)
                r.push_back(p2[i]);
            SASSERT(r.size() == std::max(sz1, sz2));
            adjust_size(r);
        }

        /**
           \brief r <- p + a
        */
        void add(unsigned sz, value * const * p, value * a, value_ref_buffer & r) {
            SASSERT(p != r.c_ptr());
            SASSERT(sz > 0);
            r.reset();
            value_ref a_0(*this);
            add(p[0], a, a_0);
            r.push_back(a_0);
            r.append(sz - 1, p + 1);
            adjust_size(r);
        }

        /**
           \brief r <- p1 - p2
        */
        void sub(unsigned sz1, value * const * p1, unsigned sz2, value * const * p2, value_ref_buffer & r) {
            SASSERT(p1 != r.c_ptr());
            SASSERT(p2 != r.c_ptr());
            r.reset();
            value_ref a_i(*this);
            unsigned min = std::min(sz1, sz2);
            unsigned i = 0;
            for (; i < min; i++) {
                sub(p1[i], p2[i], a_i);
                r.push_back(a_i);
            }
            for (; i < sz1; i++)
                r.push_back(p1[i]);
            for (; i < sz2; i++) {
                neg(p2[i], a_i);
                r.push_back(a_i);
            }
            SASSERT(r.size() == std::max(sz1, sz2));
            adjust_size(r);
        }

        /**
           \brief r <- p - a
        */
        void sub(unsigned sz, value * const * p, value * a, value_ref_buffer & r) {
            SASSERT(p != r.c_ptr());
            SASSERT(sz > 0);
            r.reset();
            value_ref a_0(*this);
            sub(p[0], a, a_0);
            r.push_back(a_0);
            r.append(sz - 1, p + 1);
            adjust_size(r);
        }

        /**
           \brief r <- a * p
        */
        void mul(value * a, unsigned sz, value * const * p, value_ref_buffer & r) {
            SASSERT(p != r.c_ptr());
            r.reset();
            if (a == nullptr)
                return;
            value_ref a_i(*this);
            for (unsigned i = 0; i < sz; i++) {
                mul(a, p[i], a_i);
                r.push_back(a_i);
            }
        }

        /**
           \brief r <- p1 * p2
        */
        void mul(unsigned sz1, value * const * p1, unsigned sz2, value * const * p2, value_ref_buffer & r) {
            SASSERT(p1 != r.c_ptr());
            SASSERT(p2 != r.c_ptr());
            r.reset();
            unsigned sz = sz1*sz2;
            r.resize(sz);
            if (sz1 < sz2) {
                std::swap(sz1, sz2);
                std::swap(p1, p2);
            }
            value_ref tmp(*this);
            for (unsigned i = 0; i < sz1; i++) {
                checkpoint();
                if (p1[i] == nullptr)
                    continue;
                for (unsigned j = 0; j < sz2; j++) {
                    // r[i+j] <- r[i+j] + p1[i]*p2[j]
                    mul(p1[i], p2[j], tmp);
                    add(r[i+j], tmp, tmp);
                    r.set(i+j, tmp);
                }
            }
            adjust_size(r);
        }

        /**
           \brief p <- p/a
        */
        void div(value_ref_buffer & p, value * a) {
            SASSERT(!is_zero(a));
            if (is_rational_one(a))
                return;
            value_ref a_i(*this);
            unsigned sz = p.size();
            for (unsigned i = 0; i < sz; i++) {
                div(p[i], a, a_i);
                p.set(i, a_i);
            }
        }

        /**
           \brief q <- quotient(p1, p2); r <- rem(p1, p2)
        */
        void div_rem(unsigned sz1, value * const * p1, unsigned sz2, value * const * p2,
                     value_ref_buffer & q, value_ref_buffer & r) {
            SASSERT(sz2 > 0);
            if (sz2 == 1) {
                q.reset(); q.append(sz1, p1);
                div(q, *p2);
                r.reset();
            }
            else {
                q.reset();
                r.reset(); r.append(sz1, p1);
                if (sz1 > 1) {
                    if (sz1 >= sz2) {
                        q.resize(sz1 - sz2 + 1);
                    }
                    else {
                        SASSERT(q.empty());
                    }
                    value * b_n = p2[sz2-1];
                    SASSERT(!is_zero(b_n));
                    value_ref ratio(*this);
                    value_ref aux(*this);
                    while (true) {
                        checkpoint();
                        sz1 = r.size();
                        if (sz1 < sz2) {
                            adjust_size(q);
                            break;
                        }
                        unsigned m_n = sz1 - sz2; // m-n
                        div(r[sz1 - 1], b_n, ratio);
                        // q[m_n] <- q[m_n] + r[sz1 - 1]/b_n
                        add(q[m_n], ratio, aux);
                        q.set(m_n, aux);
                        for (unsigned i = 0; i < sz2 - 1; i++) {
                            // r[i + m_n] <- r[i + m_n] - ratio * p2[i]
                            mul(ratio, p2[i], aux);
                            sub(r[i + m_n], aux, aux);
                            r.set(i + m_n, aux);
                        }
                        r.shrink(sz1 - 1);
                        adjust_size(r);
                    }
                }
            }
        }

        /**
           \brief q <- quotient(p1, p2)
        */
        void div(unsigned sz1, value * const * p1, unsigned sz2, value * const * p2, value_ref_buffer & q) {
            value_ref_buffer r(*this);
            div_rem(sz1, p1, sz2, p2, q, r);
        }

        /**
           \brief r <- p/a
        */
        void div(unsigned sz, value * const * p, value * a, value_ref_buffer & r) {
            r.reset();
            value_ref a_i(*this);
            for (unsigned i = 0; i < sz; i++) {
                div(p[i], a, a_i);
                r.push_back(a_i);
            }
        }

        /**
           \brief r <- rem(p1, p2)
        */
        void rem(unsigned sz1, value * const * p1, unsigned sz2, value * const * p2, value_ref_buffer & r) {
            SASSERT(p1 != r.c_ptr());
            SASSERT(p2 != r.c_ptr());
            TRACE("rcf_rem",
                  tout << "rem\n";
                  display_poly(tout, sz1, p1); tout << "\n";
                  display_poly(tout, sz2, p2); tout << "\n";);
            r.reset();
            SASSERT(sz2 > 0);
            if (sz2 == 1)
                return;
            r.append(sz1, p1);
            if (sz1 <= 1)
                return; // r is p1
            value * b_n = p2[sz2 - 1];
            value_ref ratio(*this);
            value_ref new_a(*this);
            SASSERT(b_n != 0);
            while (true) {
                checkpoint();
                sz1 = r.size();
                if (sz1 < sz2) {
                    TRACE("rcf_rem", tout << "rem result\n"; display_poly(tout, r.size(), r.c_ptr()); tout << "\n";);
                    return;
                }
                unsigned m_n = sz1 - sz2;
                div(r[sz1 - 1], b_n, ratio);
                for (unsigned i = 0; i < sz2 - 1; i++) {
                    mul(ratio, p2[i], new_a);
                    sub(r[i + m_n], new_a, new_a);
                    r.set(i + m_n, new_a);
                }
                r.shrink(sz1 - 1);
                adjust_size(r);
            }
        }

        /**
           \brief r <- prem(p1, p2) Pseudo-remainder

           We are working on a field, but it is useful to use the pseudo-remainder algorithm because
           it does not create rational function values.
           That is, if has_clean_denominators(p1) and has_clean_denominators(p2) then has_clean_denominators(r).
        */
        void prem(unsigned sz1, value * const * p1, unsigned sz2, value * const * p2, unsigned & d, value_ref_buffer & r) {
            SASSERT(p1 != r.c_ptr());
            SASSERT(p2 != r.c_ptr());
            TRACE("rcf_prem",
                  tout << "prem\n";
                  display_poly(tout, sz1, p1); tout << "\n";
                  display_poly(tout, sz2, p2); tout << "\n";);
            d = 0;
            r.reset();
            SASSERT(sz2 > 0);
            if (sz2 == 1)
                return;
            r.append(sz1, p1);
            if (sz1 <= 1)
                return; // r is p1
            value * b_n = p2[sz2 - 1];
            SASSERT(b_n != 0);
            value_ref a_m(*this);
            value_ref new_a(*this);
            while (true) {
                checkpoint();
                sz1 = r.size();
                if (sz1 < sz2) {
                    TRACE("rcf_prem", tout << "prem result\n"; display_poly(tout, r.size(), r.c_ptr()); tout << "\n";);
                    return;
                }
                unsigned m_n = sz1 - sz2;
                // r:  a_m * x^m + a_{m-1} * x^{m-1} + ... + a_0
                // p2: b_n * x^n + b_{n-1} * x^{n-1} + ... + b_0
                d++;
                a_m = r[sz1 - 1];
                // don't need to update position sz1 - 1, since it will become 0
                if (!is_rational_one(b_n)) {
                    for (unsigned i = 0; i < sz1 - 1; i++) {
                        mul(r[i], b_n, new_a);
                        r.set(i, new_a);
                    }
                }
                // buffer: a_m * x^m + b_n * a_{m-1} * x^{m-1} + ... + b_n * a_0
                // don't need to process i = sz2 - 1, because r[sz1 - 1] will become 0.
                for (unsigned i = 0; i < sz2 - 1; i++) {
                    mul(a_m, p2[i], new_a);
                    sub(r[i + m_n], new_a, new_a);
                    r.set(i + m_n, new_a);
                }
                r.shrink(sz1 - 1);
                adjust_size(r);
            }
        }

        void prem(unsigned sz1, value * const * p1, unsigned sz2, value * const * p2, value_ref_buffer & r) {
            unsigned d;
            prem(sz1, p1, sz2, p2, d, r);
        }

        /**
           \brief r <- -p
        */
        void neg(unsigned sz, value * const * p, value_ref_buffer & r) {
            SASSERT(p != r.c_ptr());
            r.reset();
            value_ref a_i(*this);
            for (unsigned i = 0; i < sz; i++) {
                neg(p[i], a_i);
                r.push_back(a_i);
            }
        }

        /**
           \brief r <- -r
        */
        void neg(value_ref_buffer & r) {
            value_ref a_i(*this);
            unsigned sz = r.size();
            for (unsigned i = 0; i < sz; i++) {
                neg(r[i], a_i);
                r.set(i, a_i);
            }
        }

        /**
           \brief p <- -p
        */
        void neg(polynomial & p) {
            value_ref a_i(*this);
            unsigned sz = p.size();
            for (unsigned i = 0; i < sz; i++) {
                neg(p[i], a_i);
                inc_ref(a_i.get());
                dec_ref(p[i]);
                p[i] = a_i.get();
            }
        }

        /**
           \brief r <- srem(p1, p2)
           Signed remainder
        */
        void srem(unsigned sz1, value * const * p1, unsigned sz2, value * const * p2, value_ref_buffer & r) {
            SASSERT(p1 != r.c_ptr());
            SASSERT(p2 != r.c_ptr());
            rem(sz1, p1, sz2, p2, r);
            neg(r);
        }

        /**
           \brief r <- sprem(p1, p2)
           Signed pseudo remainder
        */
        void sprem(unsigned sz1, value * const * p1, unsigned sz2, value * const * p2, value_ref_buffer & r) {
            SASSERT(p1 != r.c_ptr());
            SASSERT(p2 != r.c_ptr());
            unsigned d;
            prem(sz1, p1, sz2, p2, d, r);
            // We should not flip the sign if d is odd and leading coefficient of p2 is negative.
            if (d % 2 == 0 || (sz2 > 0 && sign(p2[sz2-1]) > 0))
                neg(r);
        }

        // ---------------------------------
        //
        // Structural equality
        //
        // ---------------------------------

        /**
           \brief Values a and b are said to be "structurally" equal if:
           - a and b are 0.
           - a and b are rationals and compare(a, b) == 0
           - a and b are rational function values p_a(x)/q_a(x) and p_b(y)/q_b(y) where x and y are field extensions, and
                  * x == y (pointer equality, i.e., they are the same field extension object).
                  * Every coefficient of p_a is structurally equal to every coefficient of p_b
                  * Every coefficient of q_a is structurally equal to every coefficient of q_b
           Clearly structural equality implies equality, but the reverse is not true.
        */
        bool struct_eq(value * a, value * b) const {
            if (a == b)
                return true;
            else if (a == nullptr || b == nullptr)
                return false;
            else if (is_nz_rational(a) && is_nz_rational(b))
                return qm().eq(to_mpq(a), to_mpq(b));
            else if (is_nz_rational(a) || is_nz_rational(b))
                return false;
            else {
                SASSERT(is_rational_function(a));
                SASSERT(is_rational_function(b));
                rational_function_value * rf_a = to_rational_function(a);
                rational_function_value * rf_b = to_rational_function(b);
                if (rf_a->ext() != rf_b->ext())
                    return false;
                return struct_eq(rf_a->num(), rf_b->num()) && struct_eq(rf_a->den(), rf_b->den());
            }
        }

        /**
           Auxiliary method for
                bool struct_eq(value * a, value * b)
        */
        bool struct_eq(unsigned sz_a, value * const * p_a, unsigned sz_b, value * const * p_b) const {
            if (sz_a != sz_b)
                return false;
            for (unsigned i = 0; i < sz_a; i++) {
                if (!struct_eq(p_a[i], p_b[i]))
                    return false;
            }
            return true;
        }

        /**
           Auxiliary method for
                bool struct_eq(value * a, value * b)
        */
        bool struct_eq(polynomial const & p_a, polynomial const & p_b) const {
            return struct_eq(p_a.size(), p_a.c_ptr(), p_b.size(), p_b.c_ptr());
        }

        // ---------------------------------
        //
        // Clean denominators
        //
        // ---------------------------------

        /**
           \brief We say 'a' has "clean" denominators if
           - a is 0
           - a is a rational_value that is an integer
           - a is a rational_function_value of the form p_a(x)/1 where the coefficients of p_a also have clean denominators.
        */
        bool has_clean_denominators(value * a) const {
            if (a == nullptr)
                return true;
            else if (is_nz_rational(a))
                return qm().is_int(to_mpq(a));
            else {
                rational_function_value * rf_a = to_rational_function(a);
                return is_denominator_one(rf_a) && has_clean_denominators(rf_a->num());
            }
        }

        /**
           \brief See comment at has_clean_denominators(value * a)
         */
        bool has_clean_denominators(unsigned sz, value * const * p) const {
            for (unsigned i = 0; i < sz; i++) {
                if (!has_clean_denominators(p[i]))
                    return false;
            }
            return true;
        }

        /**
           \brief See comment at has_clean_denominators(value * a)
         */
        bool has_clean_denominators(polynomial const & p) const {
            return has_clean_denominators(p.size(), p.c_ptr());
        }

        /**
           \brief "Clean" the denominators of 'a'. That is, return p and q s.t.
                     a == p/q
                  and
                     has_clean_denominators(p) and has_clean_denominators(q)
        */
        void clean_denominators_core(value * a, value_ref & p, value_ref & q) {
            INC_DEPTH();
            TRACE("rcf_clean", tout << "clean_denominators_core [" << m_exec_depth << "]\na: "; display(tout, a, false); tout << "\n";);
            p.reset(); q.reset();
            if (a == nullptr) {
                p = a;
                q = one();
            }
            else if (is_nz_rational(a)) {
                p = mk_rational(to_mpq(a).numerator());
                q = mk_rational(to_mpq(a).denominator());
            }
            else {
                rational_function_value * rf_a = to_rational_function(a);
                value_ref_buffer p_num(*this), p_den(*this);
                value_ref d_num(*this), d_den(*this);
                clean_denominators_core(rf_a->num(), p_num, d_num);
                if (is_denominator_one(rf_a)) {
                    p_den.push_back(one());
                    d_den = one();
                }
                else {
                    clean_denominators_core(rf_a->den(), p_den, d_den);
                }
                value_ref x(*this);
                x = mk_rational_function_value(rf_a->ext());
                mk_polynomial_value(p_num.size(), p_num.c_ptr(), x, p);
                mk_polynomial_value(p_den.size(), p_den.c_ptr(), x, q);
                if (!struct_eq(d_den, d_num)) {
                    mul(p, d_den, p);
                    mul(q, d_num, q);
                }
                if (sign(q) < 0) {
                    // make sure the denominator is positive
                    neg(p, p);
                    neg(q, q);
                }
            }
        }

        /**
           \brief Clean the denominators of the polynomial p, it returns clean_p and d s.t.
                      p = clean_p/d
                  and has_clean_denominators(clean_p) && has_clean_denominators(d)
        */
        void clean_denominators_core(unsigned p_sz, value * const * p, value_ref_buffer & norm_p, value_ref & d) {
            SASSERT(p_sz >= 1);
            value_ref_buffer nums(*this), dens(*this);
            value_ref a_n(*this), a_d(*this);
            bool all_one = true;
            for (unsigned i = 0; i < p_sz; i++) {
                if (p[i]) {
                    clean_denominators_core(p[i], a_n, a_d);
                    nums.push_back(a_n);
                    if (!is_rational_one(a_d))
                        all_one = false;
                    dens.push_back(a_d);
                }
                else {
                    nums.push_back(nullptr);
                    dens.push_back(nullptr);
                }
            }
            if (all_one) {
                norm_p = nums;
                d      = one();
            }
            else {
                // Compute lcm of the integer elements in dens.
                // This is a little trick to control the coefficient growth.
                // We don't compute lcm of the other elements of dens because it is too expensive.
                scoped_mpq lcm_z(qm());
                bool found_z = false;
                SASSERT(nums.size() == p_sz);
                SASSERT(dens.size() == p_sz);
                for (unsigned i = 0; i < p_sz; i++) {
                    if (!dens[i])
                        continue;
                    if (is_nz_rational(dens[i])) {
                        mpq const & _d = to_mpq(dens[i]);
                        SASSERT(qm().is_int(_d));
                        if (!found_z) {
                            found_z = true;
                            qm().set(lcm_z, _d);
                        }
                        else {
                            qm().lcm(lcm_z, _d, lcm_z);
                        }
                    }
                }

                value_ref lcm(*this);
                if (found_z) {
                    lcm = mk_rational(lcm_z);
                }
                else {
                    lcm = one();
                }

                // Compute the multipliers for nums.
                // Compute norm_p and d
                //
                // We do NOT use GCD to compute the LCM of the denominators of non-rational values.
                // However, we detect structurally equivalent denominators.
                //
                //   Thus   a/(b+1) + c/(b+1)  is converted into a*c/(b+1) instead of (a*(b+1) + c*(b+1))/(b+1)^2
                norm_p.reset();
                d = lcm;
                value_ref_buffer multipliers(*this);
                value_ref m(*this);
                for (unsigned i = 0; i < p_sz; i++) {
                    if (!nums[i]) {
                        norm_p.push_back(nullptr);
                    }
                    else {
                        SASSERT(dens[i]);
                        bool is_z;
                        if (!is_nz_rational(dens[i])) {
                            m = lcm;
                            is_z = false;
                        }
                        else {
                            scoped_mpq num_z(qm());
                            qm().div(lcm_z, to_mpq(dens[i]), num_z);
                            SASSERT(qm().is_int(num_z));
                            m = mk_rational_and_swap(num_z);
                            is_z = true;
                        }
                        bool found_lt_eq = false;
                        for (unsigned j = 0; j < p_sz; j++) {
                            TRACE("rcf_clean_bug", tout << "j: " << j << " "; display(tout, m, false); tout << "\n";);
                            if (!dens[j])
                                continue;
                            if (i != j && !is_nz_rational(dens[j])) {
                                if (struct_eq(dens[i], dens[j])) {
                                    if (j < i)
                                        found_lt_eq = true;
                                }
                                else {
                                    mul(m, dens[j], m);
                                }
                            }
                        }
                        if (!is_z && !found_lt_eq) {
                            mul(dens[i], d, d);
                        }
                        mul(m, nums[i], m);
                        norm_p.push_back(m);
                    }
                }
            }
            SASSERT(norm_p.size() == p_sz);
        }

        void clean_denominators_core(polynomial const & p, value_ref_buffer & norm_p, value_ref & d) {
            clean_denominators_core(p.size(), p.c_ptr(), norm_p, d);
        }

        void clean_denominators(value * a, value_ref & p, value_ref & q) {
            if (has_clean_denominators(a)) {
                p = a;
                q = one();
            }
            else {
                clean_denominators_core(a, p, q);
            }
        }

        void clean_denominators(unsigned sz, value * const * p, value_ref_buffer & norm_p, value_ref & d) {
            if (has_clean_denominators(sz, p)) {
                norm_p.append(sz, p);
                d = one();
            }
            else {
                clean_denominators_core(sz, p, norm_p, d);
            }
        }

        void clean_denominators(polynomial const & p, value_ref_buffer & norm_p, value_ref & d) {
            clean_denominators(p.size(), p.c_ptr(), norm_p, d);
        }

        void clean_denominators(numeral const & a, numeral & p, numeral & q) {
            value_ref _p(*this), _q(*this);
            clean_denominators(a.m_value, _p, _q);
            set(p, _p);
            set(q, _q);
        }

        // ---------------------------------
        //
        // GCD of integer coefficients
        //
        // ---------------------------------

        /**
           \brief If has_clean_denominators(a), then this method store the gcd of the integer coefficients in g.
           If !has_clean_denominators(a) it returns false.

           If g != 0, then it will compute the gcd of g and the coefficients in a.
        */
        bool gcd_int_coeffs(value * a, mpz & g) {
            if (a == nullptr) {
                return false;
            }
            else if (is_nz_rational(a)) {
                if (!qm().is_int(to_mpq(a)))
                    return false;
                else if (qm().is_zero(g)) {
                    qm().set(g, to_mpq(a).numerator());
                    qm().abs(g);
                }
                else {
                    qm().gcd(g, to_mpq(a).numerator(), g);
                }
                return true;
            }
            else {
                rational_function_value * rf_a = to_rational_function(a);
                if (!is_denominator_one(rf_a))
                    return false;
                else
                    return gcd_int_coeffs(rf_a->num(), g);
            }
        }

        /**
           \brief See comment in gcd_int_coeffs(value * a, mpz & g)
        */
        bool gcd_int_coeffs(unsigned p_sz, value * const * p, mpz & g) {
            if (p_sz == 0) {
                return false;
            }
            else {
                for (unsigned i = 0; i < p_sz; i++) {
                    if (p[i]) {
                        if (!gcd_int_coeffs(p[i], g))
                            return false;
                        if (qm().is_one(g))
                            return true;
                    }
                }
                return true;
            }
        }

        /**
           \brief See comment in gcd_int_coeffs(value * a, mpz & g)
        */
        bool gcd_int_coeffs(polynomial const & p, mpz & g) {
            return gcd_int_coeffs(p.size(), p.c_ptr(), g);
        }

        /**
           \brief Compute gcd_int_coeffs and divide p by it (if applicable).
        */
        void normalize_int_coeffs(value_ref_buffer & p) {
            scoped_mpz g(qm());
            if (gcd_int_coeffs(p.size(), p.c_ptr(), g) && !qm().is_one(g)) {
                SASSERT(qm().is_pos(g));
                value_ref a(*this);
                for (unsigned i = 0; i < p.size(); i++) {
                    if (p[i]) {
                        a = p[i];
                        p.set(i, nullptr);
                        exact_div_z(a, g);
                        p.set(i, a);
                    }
                }
            }
        }

        /**
           \brief a <- a/b where b > 0
           Auxiliary function for normalize_int_coeffs.
           It assumes has_clean_denominators(a), and that b divides all integer coefficients.

           FUTURE: perform the operation using destructive updates when a is not shared.
        */
        void exact_div_z(value_ref & a, mpz const & b) {
            if (a == 0) {
                return;
            }
            else if (is_nz_rational(a)) {
                scoped_mpq r(qm());
                SASSERT(qm().is_int(to_mpq(a)));
                qm().div(to_mpq(a), b, r);
                a = mk_rational_and_swap(r);
            }
            else {
                rational_function_value * rf = to_rational_function(a);
                SASSERT(is_denominator_one(rf));
                value_ref_buffer new_ais(*this);
                value_ref ai(*this);
                polynomial const & p = rf->num();
                for (unsigned i = 0; i < p.size(); i++) {
                    if (p[i]) {
                        ai = p[i];
                        exact_div_z(ai, b);
                        new_ais.push_back(ai);
                    }
                    else {
                        new_ais.push_back(nullptr);
                    }
                }
                rational_function_value * r = mk_rational_function_value_core(rf->ext(), new_ais.size(), new_ais.c_ptr(), 1, &m_one);
                set_interval(r->m_interval, rf->m_interval);
                a = r;
                // divide upper and lower by b
                div(r->m_interval, b, m_ini_precision, r->m_interval);
            }
        }

        // ---------------------------------
        //
        // GCD
        //
        // ---------------------------------

        bool is_monic(value_ref_buffer const & p) {
            return !p.empty() && is_rational_one(p[p.size() - 1]);
        }

        bool is_monic(polynomial const & p) {
            return !p.empty() && is_rational_one(p[p.size() - 1]);
        }

        /**
           \brief Force the leading coefficient of p to be 1.
        */
        void mk_monic(value_ref_buffer & p) {
            unsigned sz = p.size();
            if (sz > 0) {
                value_ref a_i(*this);
                SASSERT(p[sz-1] != 0);
                if (!is_rational_one(p[sz-1])) {
                    for (unsigned i = 0; i < sz - 1; i++) {
                        div(p[i], p[sz-1], a_i);
                        p.set(i, a_i);
                    }
                    p.set(sz-1, one());
                }
            }
        }

        /**
           \brief r <- gcd(p1, p2)
        */
        void gcd(unsigned sz1, value * const * p1, unsigned sz2, value * const * p2, value_ref_buffer & r) {
            INC_DEPTH();
            TRACE("rcf_gcd", tout << "GCD [" << m_exec_depth << "]\n";
                  display_poly(tout, sz1, p1); tout << "\n";
                  display_poly(tout, sz2, p2); tout << "\n";);
            if (sz1 == 0) {
                r.append(sz2, p2);
                mk_monic(r);
            }
            else if (sz2 == 0) {
                r.append(sz1, p1);
                mk_monic(r);
            }
            else {
                value_ref_buffer A(*this);
                value_ref_buffer B(*this);
                value_ref_buffer R(*this);
                A.append(sz1, p1);
                B.append(sz2, p2);
                while (true) {
                    TRACE("rcf_gcd",
                          tout << "A: "; display_poly(tout, A.size(), A.c_ptr()); tout << "\n";
                          tout << "B: "; display_poly(tout, B.size(), B.c_ptr()); tout << "\n";);
                    if (B.empty()) {
                        mk_monic(A);
                        r = A;
                        TRACE("rcf_gcd",
                              tout << "gcd result: "; display_poly(tout, r.size(), r.c_ptr()); tout << "\n";);
                        return;
                    }
                    rem(A.size(), A.c_ptr(), B.size(), B.c_ptr(), R);
                    A = B;
                    B = R;
                }
            }
        }

        void flip_sign_if_lc_neg(value_ref_buffer & r) {
            unsigned sz = r.size();
            if (sz == 0)
                return;
            if (sign(r[sz - 1]) < 0)
                neg(r);
        }

        void prem_gcd(unsigned sz1, value * const * p1, unsigned sz2, value * const * p2, value_ref_buffer & r) {
            INC_DEPTH();
            TRACE("rcf_gcd", tout << "prem-GCD [" << m_exec_depth << "]\n";
                  display_poly(tout, sz1, p1); tout << "\n";
                  display_poly(tout, sz2, p2); tout << "\n";);
            SASSERT(p1 != r.c_ptr());
            SASSERT(p2 != r.c_ptr());
            if (sz1 == 0) {
                r.append(sz2, p2);
                flip_sign_if_lc_neg(r);
            }
            else if (sz2 == 0) {
                r.append(sz1, p1);
                flip_sign_if_lc_neg(r);
            }
            else {
                value_ref_buffer A(*this);
                value_ref_buffer B(*this);
                value_ref_buffer R(*this);
                A.append(sz1, p1);
                B.append(sz2, p2);
                while (true) {
                    TRACE("rcf_gcd",
                          tout << "A: "; display_poly(tout, A.size(), A.c_ptr()); tout << "\n";
                          tout << "B: "; display_poly(tout, B.size(), B.c_ptr()); tout << "\n";);
                    if (B.empty()) {
                        normalize_int_coeffs(A);
                        flip_sign_if_lc_neg(A);
                        r = A;
                        TRACE("rcf_gcd",
                              tout << "gcd result: "; display_poly(tout, r.size(), r.c_ptr()); tout << "\n";);
                        return;
                    }
                    prem(A.size(), A.c_ptr(), B.size(), B.c_ptr(), R);
                    normalize_int_coeffs(R);
                    A = B;
                    B = R;
                }
            }
        }

        // ---------------------------------
        //
        // Derivatives and Sturm-Tarski Sequences
        //
        // ---------------------------------

        /**
           \brief r <- dp/dx
        */
        void derivative(unsigned sz, value * const * p, value_ref_buffer & r) {
            r.reset();
            if (sz > 1) {
                for (unsigned i = 1; i < sz; i++) {
                    mpq i_mpq(i);
                    value_ref a_i(*this);
                    a_i = mk_rational_and_swap(i_mpq);
                    mul(a_i, p[i], a_i);
                    r.push_back(a_i);
                }
                adjust_size(r);
            }
        }

        /**
           \brief r <- squarefree(p)
           Store in r the square free factors of p.
        */
        void square_free(unsigned sz, value * const * p, value_ref_buffer & r) {
            flet<bool> set(m_in_aux_values, true);
            if (sz <= 1) {
                r.append(sz, p);
            }
            else {
                value_ref_buffer p_prime(*this);
                value_ref_buffer g(*this);
                derivative(sz, p, p_prime);
                if (m_use_prem)
                    prem_gcd(sz, p, p_prime.size(), p_prime.c_ptr(), g);
                else
                    gcd(sz, p, p_prime.size(), p_prime.c_ptr(), g);
                if (g.size() <= 1) {
                    r.append(sz, p);
                }
                else {
                    div(sz, p, g.size(), g.c_ptr(), r);
                    if (m_use_prem)
                        normalize_int_coeffs(r);
                }
            }
        }

        /**
           \brief Keep expanding the Sturm sequence starting at seq
        */
        void sturm_seq_core(scoped_polynomial_seq & seq) {
            INC_DEPTH();
            flet<bool> set(m_in_aux_values, true);

            SASSERT(seq.size() >= 2);
            TRACE("rcf_sturm_seq",
                  unsigned sz = seq.size();
                  tout << "sturm_seq_core [" << m_exec_depth << "]\n";
                  display_poly(tout, seq.size(sz-2), seq.coeffs(sz-2)); tout << "\n";
                  display_poly(tout, seq.size(sz-1), seq.coeffs(sz-1)); tout << "\n";);
            value_ref_buffer r(*this);
            while (true) {
                unsigned sz = seq.size();
                if (m_use_prem) {
                    sprem(seq.size(sz-2), seq.coeffs(sz-2), seq.size(sz-1), seq.coeffs(sz-1), r);
                    normalize_int_coeffs(r);
                }
                else {
                    srem(seq.size(sz-2), seq.coeffs(sz-2), seq.size(sz-1), seq.coeffs(sz-1), r);
                }
                TRACE("rcf_sturm_seq",
                      tout << "sturm_seq_core [" << m_exec_depth << "], new polynomial\n";
                      display_poly(tout, r.size(), r.c_ptr()); tout << "\n";);
                if (r.empty())
                    return;
                seq.push(r.size(), r.c_ptr());
            }
        }

        /**
           \brief Store in seq the Sturm sequence for (p1; p2)
        */
        void sturm_seq(unsigned sz1, value * const * p1, unsigned sz2, value * const * p2, scoped_polynomial_seq & seq) {
            seq.reset();
            seq.push(sz1, p1);
            seq.push(sz2, p2);
            sturm_seq_core(seq);
        }

        /**
           \brief Store in seq the Sturm sequence for (p; p')
        */
        void sturm_seq(unsigned sz, value * const * p, scoped_polynomial_seq & seq) {
            seq.reset();
            value_ref_buffer p_prime(*this);
            seq.push(sz, p);
            derivative(sz, p, p_prime);
            seq.push(p_prime.size(), p_prime.c_ptr());
            sturm_seq_core(seq);
        }

        /**
           \brief Store in seq the Sturm sequence for (p1; p1' * p2)
        */
        void sturm_tarski_seq(unsigned sz1, value * const * p1, unsigned sz2, value * const * p2, scoped_polynomial_seq & seq) {
            seq.reset();
            value_ref_buffer p1_prime(*this);
            value_ref_buffer p1_prime_p2(*this);
            seq.push(sz1, p1);
            derivative(sz1, p1, p1_prime);
            mul(p1_prime.size(), p1_prime.c_ptr(), sz2, p2, p1_prime_p2);
            seq.push(p1_prime_p2.size(), p1_prime_p2.c_ptr());
            sturm_seq_core(seq);
        }

        // ---------------------------------
        //
        // Sign evaluation for polynomials
        // That is, sign of p(x) at b
        //
        // ---------------------------------

        /**
           \brief Return the sign of p(0)
        */
        int eval_sign_at_zero(unsigned n, value * const * p) {
            if (n == 0)
                return 0;
            return sign(p[0]);
        }

        /**
           \brief Return the sign of p(oo)
        */
        int eval_sign_at_plus_inf(unsigned n, value * const * p) {
            if (n == 0)
                return 0;
            SASSERT(!is_zero(p[n-1])); // p is well formed
            return sign(p[n-1]);
        }

        /**
           \brief Return the sign of p(-oo)
        */
        int eval_sign_at_minus_inf(unsigned n, value * const * p) {
            if (n == 0)
                return 0;
            SASSERT(!is_zero(p[n-1])); // p is well formed
            unsigned degree = n - 1;
            if (degree % 2 == 0)
                return sign(p[n - 1]);
            else
                return -sign(p[n - 1]);
        }

        /**
           \brief Store in r an approximation (as an interval) for the interval p(b).

           \pre n >= 2
        */
        void eval_sign_at_approx(unsigned n, value * const * p, mpbq const & b, mpbqi & r) {
            SASSERT(n >= 2);
            // We compute r using the Horner Sequence
            //  ((a_{n-1}*b + a_{n-2})*b + a_{n-3})*b + a_{n-4} ...
            // where a_i's are the intervals associated with coefficients of p.
            SASSERT(n > 0);
            SASSERT(p[n - 1] != 0);
            scoped_mpbqi bi(bqim());
            set_interval(bi, b);  // bi <- [b, b]
            // r <- a_n * bi
            bqim().mul(interval(p[n - 1]), bi, r);
            unsigned i = n - 1;
            while (i > 0) {
                checkpoint();
                --i;
                if (p[i] != nullptr)
                    bqim().add(r, interval(p[i]), r);
                if (i > 0)
                    bqim().mul(r, bi, r);
            }
        }

        /**
           \brief We say a polynomial has "refinable" approximated coefficients if the intervals
           approximating the coefficients do not have -oo or oo as lower/upper bounds.
        */
        bool has_refineable_approx_coeffs(unsigned n, value * const * p) {
            for (unsigned i = 0; i < n; i++) {
                if (p[i] != nullptr) {
                    mpbqi & a_i = interval(p[i]);
                    if (a_i.lower_is_inf() || a_i.upper_is_inf())
                        return false;
                }
            }
            return true;
        }

        /**
           \brief r <- p(b)
        */
        void mk_polynomial_value(unsigned n, value * const * p, value * b, value_ref & r) {
            SASSERT(n > 0);
            if (n == 1 || b == nullptr) {
                r = p[0];
            }
            else {
                SASSERT(n >= 2);
                // We compute the result using the Horner Sequence
                //  ((a_{n-1}*b + a_{n-2})*b + a_{n-3})*b + a_{n-4} ...
                // where a_i's are the coefficients of p.
                mul(p[n - 1], b, r); // r <- a_{n-1} * b
                unsigned i = n - 1;
                while (i > 0) {
                    --i;
                    if (p[i] != nullptr)
                        add(r, p[i], r); // r <- r + a_i
                    if (i > 0)
                        mul(r, b, r); // r <- r * b
                }
            }
        }

        /**
           \brief Evaluate the sign of p(b) by computing a value object.
        */
        int expensive_eval_sign_at(unsigned n, value * const * p, mpbq const & b) {
            flet<bool> set(m_in_aux_values, true);

            SASSERT(n > 1);
            SASSERT(p[n - 1] != 0);
            // Actually, given b = c/2^k, we compute the sign of (2^k)^n*p(c)
            // Original Horner Sequence
            //     ((a_n * b + a_{n-1})*b + a_{n-2})*b + a_{n-3} ...
            // Variation of the Horner Sequence for (2^k)^n*p(b)
            //     ((a_n * c + a_{n-1}*2_k)*c + a_{n-2}*(2_k)^2)*c + a_{n-3}*(2_k)^3 ... + a_0*(2_k)^n
            scoped_mpz mpz_twok(qm());
            qm().mul2k(mpz(1), b.k(), mpz_twok);
            value_ref twok(*this), twok_i(*this);
            twok = mk_rational(mpz_twok);
            twok_i = twok;
            value_ref c(*this);
            c = mk_rational(b.numerator());

            value_ref r(*this), ak(*this), rc(*this);

            r = p[n-1];
            unsigned i = n-1;
            while (i > 0) {
                --i;
                if (is_zero(p[i])) {
                    mul(r, c, r);
                }
                else {
                    // ak <- a_i * (2^k)^(n-i)
                    mul(p[i], twok_i, ak);
                    // r <- r * c + a_i * (2^k)^(n-i)
                    mul(r, c, rc);
                    add(ak, rc, r);
                }
                mul(twok_i, twok, twok_i);
            }
            return sign(r);
        }

        /**
           \brief Find the magnitude of the biggest interval use to approximate coefficients of p.

           \pre has_refineable_approx_coeffs(n, p)
        */
        int find_biggest_interval_magnitude(unsigned n, value * const * p) {
            int r = INT_MIN;
            for (unsigned i = 0; i < n; i++) {
                if (p[i] != nullptr) {
                    mpbqi & a_i = interval(p[i]);
                    SASSERT(!a_i.lower_is_inf() && !a_i.upper_is_inf());
                    int m = magnitude(a_i);
                    if (m > r)
                        r = m;
                }
            }
            return r;
        }

        /**
           \brief Return the sign of p(b)
        */
        int eval_sign_at(unsigned n, value * const * p, mpbq const & b) {
            if (n == 0)
                return 0;
            else if (n == 1)
                return sign(p[0]);
            else {
                scoped_mpbqi r(bqim());
                eval_sign_at_approx(n, p, b, r);
                if (!contains_zero(r)) {
                    // we are done
                    return bqim().is_P(r) ? 1 : -1;
                }
                else if (!has_refineable_approx_coeffs(n, p)) {
                    return expensive_eval_sign_at(n, p, b);
                }
                else {
                    int m = find_biggest_interval_magnitude(n, p);
                    unsigned prec;
                    if (m >= 0)
                        prec = 1;
                    else
                        prec = -m;
                    SASSERT(prec >= 1);
                    while (prec <= m_max_precision) {
                        checkpoint();
                        if (!refine_coeffs_interval(n, p, prec)) {
                            // Failed to refine intervals, p must depend on infinitesimal values.
                            // This can happen even if all intervals of coefficients of p are bounded.
                            return expensive_eval_sign_at(n, p, b);
                        }
                        eval_sign_at_approx(n, p, b, r);
                        if (!contains_zero(r)) {
                            // we are done
                            return bqim().is_P(r) ? 1 : -1;
                        }
                        prec++; // increase precision and try again.
                    }
                    return expensive_eval_sign_at(n, p, b);
                }
            }
        }

        // ---------------------------------
        //
        // Sign variations in polynomial sequences.
        //
        // ---------------------------------

        enum location {
            ZERO,
            MINUS_INF,
            PLUS_INF,
            MPBQ
        };

        /**
           \brief Compute the number of sign variations at position (loc, b) in the given polynomial sequence.
           The position (loc, b) should be interpreted in the following way:

           - (ZERO, *)      -> number of sign variations at 0.
           - (MINUS_INF, *) -> number of sign variations at -oo.
           - (PLUS_INF, *)  -> number of sign variations at oo.
           - (MPBQ, b)      -> number of sign variations at binary rational b.
        */
        unsigned sign_variations_at_core(scoped_polynomial_seq const & seq, location loc, mpbq const & b) {
            unsigned sz = seq.size();
            if (sz <= 1)
                return 0;
            unsigned r = 0;
            int sign, prev_sign;
            sign = 0;
            prev_sign = 0;
            unsigned i = 0;
            for (; i < sz; i++) {
                // find next nonzero
                unsigned psz      = seq.size(i);
                value * const * p = seq.coeffs(i);
                switch (loc) {
                case PLUS_INF:
                    sign = eval_sign_at_plus_inf(psz, p);
                    break;
                case MINUS_INF:
                    sign = eval_sign_at_minus_inf(psz, p);
                    break;
                case ZERO:
                    sign  = eval_sign_at_zero(psz, p);
                    break;
                case MPBQ:
                    sign = eval_sign_at(psz, p, b);
                    break;
                default:
                    UNREACHABLE();
                    break;
                }
                if (sign == 0)
                    continue;
                SASSERT(sign == 1 || sign == -1);
                // in the first iteration prev_sign == 0, then r is never incremented.
                if (sign != prev_sign && prev_sign != 0)
                    r++;
                // move to the next
                prev_sign = sign;
            }
            return r;
        }

        unsigned sign_variations_at_minus_inf(scoped_polynomial_seq const & seq) {
            mpbq dummy(0);
            return sign_variations_at_core(seq, MINUS_INF, dummy);
        }

        unsigned sign_variations_at_plus_inf(scoped_polynomial_seq const & seq) {
            mpbq dummy(0);
            return sign_variations_at_core(seq, PLUS_INF, dummy);
        }

        unsigned sign_variations_at_zero(scoped_polynomial_seq const & seq) {
            mpbq dummy(0);
            return sign_variations_at_core(seq, ZERO, dummy);
        }

        unsigned sign_variations_at(scoped_polynomial_seq const & seq, mpbq const & b) {
            return sign_variations_at_core(seq, MPBQ, b);
        }

        int sign_variations_at_lower(scoped_polynomial_seq & seq, mpbqi const & interval) {
            if (interval.lower_is_inf())
                return sign_variations_at_minus_inf(seq);
            else if (bqm().is_zero(interval.lower()))
                return sign_variations_at_zero(seq);
            else
                return sign_variations_at(seq, interval.lower());
        }

        int sign_variations_at_upper(scoped_polynomial_seq & seq, mpbqi const & interval) {
            if (interval.upper_is_inf())
                return sign_variations_at_plus_inf(seq);
            else if (bqm().is_zero(interval.upper()))
                return sign_variations_at_zero(seq);
            else
                return sign_variations_at(seq, interval.upper());
        }

        // ---------------------------------
        //
        // Tarski-Queries (see BPR book)
        //
        // ---------------------------------

        /**
           \brief Given a polynomial Sturm sequence seq for (P; P' * Q) and an interval (a, b], it returns
                 TaQ(Q, P; a, b) =
                    #{ x \in (a, b] | P(x) = 0 and Q(x) > 0 }
                    -
                    #{ x \in (a, b] | P(x) = 0 and Q(x) < 0 }

           \remark This method ignores whether the interval end-points are closed or open.
        */
        int TaQ(scoped_polynomial_seq & seq, mpbqi const & interval) {
            return sign_variations_at_lower(seq, interval) - sign_variations_at_upper(seq, interval);
        }

        /**
           \brief Return TaQ(Q, P; a, b) =
                    #{ x \in (a, b] | P(x) = 0 and Q(x) > 0 }
                    -
                    #{ x \in (a, b] | P(x) = 0 and Q(x) < 0 }

           \remark This method ignores whether the interval end-points are closed or open.
        */
        int TaQ(unsigned p_sz, value * const * p, unsigned q_sz, value * const * q, mpbqi const & interval) {
            INC_DEPTH();
            TRACE("rcf_TaQ", tout << "TaQ [" << m_exec_depth << "]\n";
                  display_poly(tout, p_sz, p); tout << "\n";
                  display_poly(tout, q_sz, q); tout << "\n";);
            scoped_polynomial_seq seq(*this);
            sturm_tarski_seq(p_sz, p, q_sz, q, seq);
            return TaQ(seq, interval);
        }

        /**
           \brief Return TaQ(1, P; a, b) =
                    #{ x \in (a, b] | P(x) = 0 }

           \remark This method ignores whether the interval end-points are closed or open.
        */
        int TaQ_1(unsigned p_sz, value * const * p, mpbqi const & interval) {
            INC_DEPTH();
            TRACE("rcf_TaQ", tout << "TaQ_1 [" << m_exec_depth << "]\n";
                  display_poly(tout, p_sz, p); tout << "\n";);
            scoped_polynomial_seq seq(*this);
            sturm_seq(p_sz, p, seq);
            return TaQ(seq, interval);
        }

        // ---------------------------------
        //
        // Interval refinement
        //
        // ---------------------------------

        void refine_rational_interval(rational_value * v, unsigned prec) {
            mpbqi & i = interval(v);
            if (!i.lower_is_open() && !i.upper_is_open()) {
                SASSERT(bqm().eq(i.lower(), i.upper()));
                return;
            }
            while (!check_precision(i, prec)) {
                checkpoint();
                bqm().refine_lower(to_mpq(v), i.lower(), i.upper());
                bqm().refine_upper(to_mpq(v), i.lower(), i.upper());
            }
        }

        /**
           \brief Refine the interval for each coefficient of in the polynomial p.
        */
        bool refine_coeffs_interval(unsigned n, value * const * p, unsigned prec) {
            for (unsigned i = 0; i < n; i++) {
                if (p[i] != nullptr && !refine_interval(p[i], prec))
                    return false;
            }
            return true;
        }

        /**
           \brief Refine the interval for each coefficient of in the polynomial p.
        */
        bool refine_coeffs_interval(polynomial const & p, unsigned prec) {
            return refine_coeffs_interval(p.size(), p.c_ptr(), prec);
        }

        /**
           \brief Store in r the interval p(v).
        */
        void polynomial_interval(polynomial const & p, mpbqi const & v, mpbqi & r) {
            // We compute r using the Horner Sequence
            //  ((a_n * v + a_{n-1})*v + a_{n-2})*v + a_{n-3} ...
            // where a_i's are the coefficients of p.
            unsigned sz = p.size();
            if (sz == 1) {
                bqim().set(r, interval(p[0]));
            }
            else {
                SASSERT(sz > 0);
                SASSERT(p[sz - 1] != 0);
                // r <- a_n * v
                bqim().mul(interval(p[sz-1]), v, r);
                unsigned i = sz - 1;
                while (i > 0) {
                    --i;
                    if (p[i] != 0)
                        bqim().add(r, interval(p[i]), r);
                    if (i > 0)
                        bqim().mul(r, v, r);
                }
            }
        }

        /**
           \brief Update the interval of v by using the intervals of
           extension and coefficients of the rational function.
        */
        void update_rf_interval(rational_function_value * v, unsigned prec) {
            if (is_denominator_one(v)) {
                polynomial_interval(v->num(), v->ext()->interval(), v->interval());
            }
            else  {
                scoped_mpbqi num_i(bqim()), den_i(bqim());
                polynomial_interval(v->num(), v->ext()->interval(), num_i);
                polynomial_interval(v->den(), v->ext()->interval(), den_i);
                if (!contains_zero(num_i) && !contains_zero(den_i)) {
                    div(num_i, den_i, inc_precision(prec, 2), v->interval());
                }
            }
        }

        void refine_transcendental_interval(rational_function_value * v, unsigned prec) {
            SASSERT(v->ext()->is_transcendental());
            polynomial const & n = v->num();
            polynomial const & d = v->den();
            unsigned _prec = prec;
            while (true) {
                VERIFY(refine_coeffs_interval(n, _prec)); // must return true because a transcendental never depends on an infinitesimal
                VERIFY(refine_coeffs_interval(d, _prec)); // must return true because a transcendental never depends on an infinitesimal
                refine_transcendental_interval(to_transcendental(v->ext()), _prec);
                update_rf_interval(v, prec);

                TRACE("rcf_transcendental", tout << "after update_rf_interval: " << magnitude(v->interval()) << " ";
                      bqim().display(tout, v->interval()); tout << std::endl;);

                if (check_precision(v->interval(), prec))
                    return;
                _prec++;
            }
        }

        bool refine_infinitesimal_interval(rational_function_value * v, unsigned prec) {
            SASSERT(v->ext()->is_infinitesimal());
            polynomial const & numerator   = v->num();
            polynomial const & denominator = v->den();
            unsigned num_idx = first_non_zero(numerator);
            unsigned den_idx = first_non_zero(denominator);
            if (num_idx == 0 && den_idx == 0) {
                unsigned _prec = prec;
                while (true) {
                    refine_interval(numerator[num_idx],   _prec);
                    refine_interval(denominator[num_idx], _prec);
                    mpbqi const & num_i = interval(numerator[num_idx]);
                    mpbqi const & den_i = interval(denominator[num_idx]);
                    SASSERT(!contains_zero(num_i));
                    SASSERT(!contains_zero(den_i));
                    if (is_open_interval(num_i) && is_open_interval(den_i)) {
                        // This case is simple because adding/subtracting infinitesimal quantities, will
                        // not change the interval.
                        div(num_i, den_i, inc_precision(prec, 2), v->interval());
                    }
                    else {
                        // The intervals num_i and den_i may not be open.
                        // Example: numerator[num_idx] or denominator[num_idx] are rationals
                        // that can be precisely represented as binary rationals.
                        scoped_mpbqi new_num_i(bqim());
                        scoped_mpbqi new_den_i(bqim());
                        mpbq tiny_value(1, _prec*2);
                        if (numerator.size() > 1)
                            add_infinitesimal(num_i, sign_of_first_non_zero(numerator, 1) > 0,   tiny_value, new_num_i);
                        else
                            bqim().set(new_num_i, num_i);
                        if (denominator.size() > 1)
                            add_infinitesimal(den_i, sign_of_first_non_zero(denominator, 1) > 0, tiny_value, new_den_i);
                        else
                            bqim().set(new_den_i, den_i);
                        div(new_num_i, new_den_i, inc_precision(prec, 2), v->interval());
                    }
                    if (check_precision(v->interval(), prec))
                        return true;
                    _prec++;
                }
            }
            else {
                // The following condition must hold because gcd(numerator, denominator) == 1
                // If num_idx > 0 and den_idx > 0, eps^{min(num_idx, den_idx)} is a factor of gcd(numerator, denominator)
                SASSERT(num_idx ==  0 || den_idx == 0);
                int s = sign(numerator[num_idx]) * sign(denominator[den_idx]);
                // The following must hold since numerator[num_idx] and denominator[den_idx] are not zero.
                SASSERT(s != 0);
                if (num_idx == 0) {
                    SASSERT(den_idx > 0);
                    // |v| is bigger than any binary rational
                    // Interval can't be refined. There is no way to isolate an infinity with an interval with binary rational end points.
                    return false;
                }
                else {
                    SASSERT(num_idx > 0);
                    SASSERT(den_idx == 0);
                    // |v| is infinitely close to zero.
                    if (s == 1) {
                        // it is close from above
                        set_lower(v->interval(), mpbq(0));
                        set_upper(v->interval(), mpbq(1, prec));
                    }
                    else {
                        // it is close from below
                        set_lower(v->interval(), mpbq(-1, prec));
                        set_upper(v->interval(), mpbq(0));
                    }
                    return true;
                }
            }
        }

        bool refine_algebraic_interval(algebraic * a, unsigned prec) {
            save_interval_if_too_small(a, prec);
            if (a->sdt() != nullptr) {
                // We don't bisect the interval, since it contains more than one root.
                // To bisect this kind of interval we would have to use Tarski queries.
                return false;
            }
            else {
                mpbqi & a_i = a->interval();
                if (a_i.lower_is_inf() || a_i.upper_is_inf()) {
                    // we can't bisect the infinite intervals
                    return false;
                }
                else {
                    mpbqi & a_i = a->interval();
                    SASSERT(!a_i.lower_is_inf() && !a_i.upper_is_inf());
                    int  lower_sign = INT_MIN;
                    while (!check_precision(a_i, prec)) {
                        checkpoint();
                        SASSERT(!bqm().eq(a_i.lower(), a_i.upper()));
                        scoped_mpbq m(bqm());
                        bqm().add(a_i.lower(), a_i.upper(), m);
                        bqm().div2(m);
                        int mid_sign   = eval_sign_at(a->p().size(), a->p().c_ptr(), m);
                        if (mid_sign == 0) {
                            // found the actual root
                            // set interval [m, m]
                            set_lower(a_i, m, false);
                            set_upper(a_i, m, false);
                            return true;
                        }
                        else {
                            SASSERT(mid_sign == 1 || mid_sign == -1);
                            if (lower_sign == INT_MIN) {
                                // initialize lower_sign
                                lower_sign = eval_sign_at(a->p().size(), a->p().c_ptr(), a_i.lower());
                            }
                            SASSERT(lower_sign == 1 || lower_sign == -1);
                            if (mid_sign == lower_sign) {
                                // improved lower bound
                                set_lower(a_i, m);
                            }
                            else {
                                // improved upper bound
                                set_upper(a_i, m);
                            }
                        }
                    }
                    return true;
                }
            }
        }

        bool refine_algebraic_interval(rational_function_value * v, unsigned prec) {
            SASSERT(v->ext()->is_algebraic());
            polynomial const & n = v->num();
            SASSERT(is_denominator_one(v));
            unsigned _prec = prec;
            while (true) {
                if (!refine_coeffs_interval(n, _prec) ||
                    !refine_algebraic_interval(to_algebraic(v->ext()), _prec))
                    return false;

                update_rf_interval(v, prec);
                TRACE("rcf_algebraic", tout << "after update_rf_interval: " << magnitude(v->interval()) << " "; bqim().display(tout, v->interval()); tout << std::endl;);
                if (check_precision(v->interval(), prec))
                    return true;
                _prec++;
            }
        }

        /**
           \brief Refine the interval of v to the desired precision (1/2^prec).
           Return false in case of failure. A failure can only happen if v depends on infinitesimal values.
        */
        bool refine_interval(value * v, unsigned prec) {
            checkpoint();
            SASSERT(!is_zero(v));
            int m = magnitude(interval(v));
            if (m == INT_MIN || (m < 0 && static_cast<unsigned>(-m) > prec))
                return true;
            save_interval_if_too_small(v, prec);
            if (is_nz_rational(v)) {
                refine_rational_interval(to_nz_rational(v), prec);
                return true;
            }
            else {
                rational_function_value * rf = to_rational_function(v);
                if (rf->ext()->is_transcendental()) {
                    refine_transcendental_interval(rf, prec);
                    return true;
                }
                else if (rf->ext()->is_infinitesimal())
                    return refine_infinitesimal_interval(rf, prec);
                else
                    return refine_algebraic_interval(rf, prec);
            }
        }

        // ---------------------------------
        //
        // Sign determination
        //
        // ---------------------------------

        /**
           \brief Return the position of the first non-zero coefficient of p.
         */
        static unsigned first_non_zero(polynomial const & p) {
            unsigned sz = p.size();
            for (unsigned i = 0; i < sz; i++) {
                if (p[i] != 0)
                    return i;
            }
            UNREACHABLE();
            return UINT_MAX;
        }

        /**
           \brief Return the sign of the first non zero coefficient starting at position start_idx
        */
        int sign_of_first_non_zero(polynomial const & p, unsigned start_idx) {
            unsigned sz = p.size();
            SASSERT(start_idx < sz);
            for (unsigned i = start_idx; i < sz; i++) {
                if (p[i] != 0)
                    return sign(p[i]);
            }
            UNREACHABLE();
            return 0;
        }

        /**
           out <- in + infinitesimal (if plus_eps == true)
           out <- in - infinitesimal (if plus_eps == false)

           We use the following rules for performing the assignment

           If plus_eps == True
               If lower(in) == v (closed or open), then lower(out) == v and open
               If upper(in) == v and open,         then upper(out) == v and open
               If upper(in) == v and closed,       then upper(out) == new_v and open
                        where new_v is v + tiny_value / 2^k, where k is the smallest natural such that sign(new_v) == sign(v)
           If plus_eps == False
               If lower(in) == v and open,         then lower(out) == v and open
               If lower(in) == v and closed,       then lower(out) == new_v and open
               If upper(in) == v (closed or open), then upper(out) == v and open
                        where new_v is v - tiny_value / 2^k, where k is the smallest natural such that sign(new_v) == sign(v)
        */
        void add_infinitesimal(mpbqi const & in, bool plus_eps, mpbq const & tiny_value, mpbqi & out) {
            set_interval(out, in);
            out.set_lower_is_open(true);
            out.set_upper_is_open(true);
            if (plus_eps) {
                if (!in.upper_is_open()) {
                    scoped_mpbq tval(bqm());
                    tval = tiny_value;
                    while (true) {
                        bqm().add(in.upper(), tval, out.upper());
                        if (bqm().is_pos(in.upper()) == bqm().is_pos(out.upper()))
                            return;
                        bqm().div2(tval);
                        checkpoint();
                    }
                }
            }
            else {
                if (!in.lower_is_open()) {
                    scoped_mpbq tval(bqm());
                    tval = tiny_value;
                    while (true) {
                        bqm().sub(in.lower(), tval, out.lower());
                        if (bqm().is_pos(in.lower()) == bqm().is_pos(out.lower()))
                            return;
                        bqm().div2(tval);
                        checkpoint();
                    }
                }
            }
        }

        /**
           \brief Determine the sign of an element of Q(trans_0, ..., trans_n)
        */
        void determine_transcendental_sign(rational_function_value * v) {
            // Remark: the sign of a rational function value on an transcendental is never zero.
            // Reason: The transcendental can be the root of a polynomial.
            SASSERT(v->ext()->is_transcendental());
            int m = magnitude(v->interval());
            unsigned prec = 1;
            if (m < 0)
                prec = static_cast<unsigned>(-m) + 1;
            while (contains_zero(v->interval())) {
                refine_transcendental_interval(v, prec);
                prec++;
            }
        }

        /**
           \brief Determine the sign of an element of Q(trans_0, ..., trans_n, eps_0, ..., eps_m)
        */
        void determine_infinitesimal_sign(rational_function_value * v) {
            // Remark: the sign of a rational function value on an infinitesimal is never zero.
            // Reason: An infinitesimal eps is transcendental in any field K. So, it can't be the root
            // of a polynomial.
            SASSERT(v->ext()->is_infinitesimal());
            polynomial const & numerator   = v->num();
            polynomial const & denominator = v->den();
            unsigned num_idx = first_non_zero(numerator);
            unsigned den_idx = first_non_zero(denominator);
            if (num_idx == 0 && den_idx == 0) {
                mpbqi const & num_i = interval(numerator[num_idx]);
                mpbqi const & den_i = interval(denominator[num_idx]);
                SASSERT(!contains_zero(num_i));
                SASSERT(!contains_zero(den_i));
                if (is_open_interval(num_i) && is_open_interval(den_i)) {
                    // This case is simple because adding/subtracting infinitesimal quantities, will
                    // not change the interval.
                    div(num_i, den_i, m_ini_precision, v->interval());
                }
                else {
                    // The intervals num_i and den_i may not be open.
                    // Example: numerator[num_idx] or denominator[num_idx] are rationals
                    // that can be precisely represented as binary rationals.
                    scoped_mpbqi new_num_i(bqim());
                    scoped_mpbqi new_den_i(bqim());
                    mpbq tiny_value(1, m_ini_precision); // 1/2^{m_ini_precision}
                    if (numerator.size() > 1)
                        add_infinitesimal(num_i, sign_of_first_non_zero(numerator, 1) > 0,   tiny_value, new_num_i);
                    else
                        bqim().set(new_num_i, num_i);
                    if (denominator.size() > 1)
                        add_infinitesimal(den_i, sign_of_first_non_zero(denominator, 1) > 0, tiny_value, new_den_i);
                    else
                        bqim().set(new_den_i, den_i);
                    div(new_num_i, new_den_i, m_ini_precision, v->interval());
                }
            }
            else {
                // The following condition must hold because gcd(numerator, denominator) == 1
                // If num_idx > 0 and den_idx > 0, eps^{min(num_idx, den_idx)} is a factor of gcd(numerator, denominator)
                SASSERT(num_idx ==  0 || den_idx == 0);
                int s = sign(numerator[num_idx]) * sign(denominator[den_idx]);
                // The following must hold since numerator[num_idx] and denominator[den_idx] are not zero.
                SASSERT(s != 0);
                if (num_idx == 0) {
                    SASSERT(den_idx > 0);
                    // |v| is bigger than any binary rational
                    if (s == 1) {
                        // it is "oo"
                        set_lower(v->interval(), m_plus_inf_approx);
                        set_upper_inf(v->interval());
                    }
                    else {
                        // it is "-oo"
                        set_lower_inf(v->interval());
                        set_upper(v->interval(), m_minus_inf_approx);
                    }
                }
                else {
                    SASSERT(num_idx > 0);
                    SASSERT(den_idx == 0);
                    // |v| is infinitely close to zero.
                    if (s == 1) {
                        // it is close from above
                        set_lower(v->interval(), mpbq(0));
                        set_upper(v->interval(), mpbq(1, m_ini_precision));
                    }
                    else {
                        // it is close from below
                        set_lower(v->interval(), mpbq(-1, m_ini_precision));
                        set_upper(v->interval(), mpbq(0));
                    }
                }
            }
            SASSERT(!contains_zero(v->interval()));
        }

        /**
           \brief Return true if x and q depend on infinitesimal values.
           That is, q(x) does not depend on infinitesimal values.
        */
        bool depends_on_infinitesimals(polynomial const & q, algebraic * x) {
            return x->depends_on_infinitesimals() || depends_on_infinitesimals(q.size(), q.c_ptr());
        }

        /**
           \brief This method is invoked when we know that q(x) is not zero and q(x) does not depend on infinitesimal values.
           The procedure will keep refining the intervals associated with x and coefficients of q until
           the interval of q(x) is of the form
              (l > 0, u)
              OR
              (l, u < 0)
        */
        void refine_until_sign_determined(polynomial const & q, algebraic * x, mpbqi & r) {
            SASSERT(!depends_on_infinitesimals(q, x));
            // If x and q do not depend on infinitesimals, must make sure that r satisfies our invariant
            // for intervals of polynomial values that do not depend on infinitesimals.
            // that is,
            //     Given r == (l, u), l != 0 and u != 0
            int m = magnitude(r);
            unsigned prec;
            if (m >= 0)
                prec = m_ini_precision;
            else
                prec = -m;
            while (true) {
                checkpoint();
                VERIFY(refine_coeffs_interval(q, prec)); // can't fail because q does not depend on infinitesimals
                VERIFY(refine_algebraic_interval(x, prec)); // can't fail because x does not depend on infinitesimals
                // Update r
                polynomial_interval(q, x->interval(), r);
                // Since q and r do not depend on infinitesimals -oo and +oo will never be end-points.
                SASSERT(!r.lower_is_inf());
                SASSERT(!r.upper_is_inf());
                if (!contains_zero(r) && !bqm().is_zero(r.lower()) && !bqm().is_zero(r.upper()))
                    return; // the interval r satisfies our requirements.
                prec++;
            }
        }

        /**
           \brief If q(x) != 0, return true and store in r an interval that contains the value q(x), but does not contain 0.
                  If q(x) == 0, return false
        */
        bool expensive_algebraic_poly_interval(polynomial const & q, algebraic * x, mpbqi & r) {
            polynomial_interval(q, x->interval(), r);
            if (!contains_zero(r)) {
                if (!depends_on_infinitesimals(q, x) && (bqm().is_zero(r.lower()) || bqm().is_zero(r.upper()))) {
                    // we don't want intervals of the form (l, 0) and (0, u) when
                    // q(x) does not depend on infinitesimals.
                    refine_until_sign_determined(q, x, r);
                }
                return true;
            }
            int num_roots = x->num_roots_inside_interval();
            SASSERT(x->sdt() != 0 || num_roots == 1);
            polynomial const & p = x->p();
            int taq_p_q = TaQ(p.size(), p.c_ptr(), q.size(), q.c_ptr(), x->iso_interval());
            if (num_roots == 1 && taq_p_q == 0)
                return false; // q(x) is zero
            if (taq_p_q == num_roots) {
                // q(x) is positive
                if (!depends_on_infinitesimals(q, x))
                    refine_until_sign_determined(q, x, r);
                else
                    set_lower_zero(r);
                SASSERT(!contains_zero(r));
                return true;
            }
            else if (taq_p_q == -num_roots) {
                // q(x) is negative
                if (!depends_on_infinitesimals(q, x))
                    refine_until_sign_determined(q, x, r);
                else
                    set_upper_zero(r);
                SASSERT(!contains_zero(r));
                return true;
            }
            else {
                SASSERT(num_roots > 1);
                SASSERT(x->sdt() != 0);
                int q_eq_0, q_gt_0, q_lt_0;
                value_ref_buffer q2(*this);
                count_signs_at_zeros_core(taq_p_q, p.size(), p.c_ptr(), q.size(), q.c_ptr(), x->iso_interval(), num_roots, q_eq_0, q_gt_0, q_lt_0, q2);
                if (q_eq_0 > 0 && q_gt_0 == 0 && q_lt_0 == 0) {
                    // q(x) is zero
                    return false;
                }
                else if (q_eq_0 == 0 && q_gt_0 > 0 && q_lt_0 == 0) {
                    // q(x) is positive
                    set_lower_zero(r);
                    return true;
                }
                else if (q_eq_0 == 0 && q_gt_0 == 0 && q_lt_0 > 0) {
                    // q(x) is negative
                    set_upper_zero(r);
                    return true;
                }
                else {
                    sign_det & sdt = *(x->sdt());
                    // Remark:
                    // By definition of algebraic and sign_det, we know that
                    //   sdt.M_s * [1, ..., 1]^t = sdt.taqrs()^t
                    // That is,
                    //   [1, ..., 1]^t = sdt.M_s^-1 * sdt.taqrs()^t
                    // Moreover the number of roots in x->iso_interval() is equal to the number of rows and columns in sdt.M_s.
                    // The column j of std.M_s is associated with the sign condition sdt.m_scs[j].
                    // The row i of sdt.M_s is associated with the polynomial sdt.prs()[i].
                    //
                    // The extension x is encoded using the sign condition x->sc_idx() of std.m_scs
                    //
                    scoped_mpz_matrix M(mm());
                    VERIFY(mk_sign_det_matrix(q_eq_0, q_gt_0, q_lt_0, M));
                    bool use_q2 = M.n() == 3;
                    scoped_mpz_matrix new_M_s(mm());
                    mm().tensor_product(sdt.M_s, M, new_M_s);
                    array<polynomial> const & prs = sdt.prs(); // polynomials associated with the rows of M_s
                    array<int> const & taqrs      = sdt.taqrs(); // For each i in [0, taqrs.size())  TaQ(p, prs[i]; x->iso_interval()) == taqrs[i]
                    SASSERT(prs.size() == taqrs.size());
                    int_buffer   new_taqrs;
                    value_ref_buffer prq(*this);
                    // fill new_taqrs using taqrs and the new tarski queries containing q (and q^2 when use_q2 == true).
                    for (unsigned i = 0; i < taqrs.size(); i++) {
                        // Add TaQ(p, prs[i] * 1; x->iso_interval())
                        new_taqrs.push_back(taqrs[i]);
                        // Add TaQ(p, prs[i] * q; x->iso_interval())
                        mul(prs[i].size(), prs[i].c_ptr(), q.size(), q.c_ptr(), prq);
                        new_taqrs.push_back(TaQ(p.size(), p.c_ptr(), prq.size(), prq.c_ptr(), x->iso_interval()));
                        if (use_q2) {
                            // Add TaQ(p, prs[i] * q^2; x->iso_interval())
                            mul(prs[i].size(), prs[i].c_ptr(), q2.size(), q2.c_ptr(), prq);
                            new_taqrs.push_back(TaQ(p.size(), p.c_ptr(), prq.size(), prq.c_ptr(), x->iso_interval()));
                        }
                    }
                    int_buffer   sc_cardinalities;
                    sc_cardinalities.resize(new_taqrs.size(), 0);
                    // Solve
                    //    new_M_s * sc_cardinalities = new_taqrs
                    VERIFY(mm().solve(new_M_s, sc_cardinalities.c_ptr(), new_taqrs.c_ptr()));
                    DEBUG_CODE({
                            // check if sc_cardinalities has the expected structure
                            //   - contains only 0 or 1
                            //   - !use_q2 IMPLIES for all i in [0, taqrs.size())   (sc_cardinalities[2*i] == 1) + (sc_cardinalities[2*i + 1] == 1) == 1
                            //   -  use_q2 IMPLIES for all i in [0, taqrs.size())   (sc_cardinalities[3*i] == 1) + (sc_cardinalities[3*i + 1] == 1) + (sc_cardinalities[3*i + 2] == 1) == 1
                            for (unsigned i = 0; i < sc_cardinalities.size(); i++) {
                                SASSERT(sc_cardinalities[i] == 0 || sc_cardinalities[i] == 1);
                            }
                            if (!use_q2) {
                                for (unsigned i = 0; i < taqrs.size(); i++) {
                                    SASSERT((sc_cardinalities[2*i] == 1) + (sc_cardinalities[2*i + 1] == 1) == 1);
                                }
                            }
                            else {
                                for (unsigned i = 0; i < taqrs.size(); i++) {
                                    SASSERT((sc_cardinalities[3*i] == 1) + (sc_cardinalities[3*i + 1] == 1) + (sc_cardinalities[3*i + 2] == 1) == 1);
                                }
                            }
                        });
                    // Remark:
                    //   Note that we found the sign of q for every root of p in the interval x->iso_interval() :)
                    unsigned sc_idx = x->sc_idx();
                    if (use_q2) {
                        if (sc_cardinalities[3*sc_idx] == 1) {
                            // q(x) is zero
                            return false;
                        }
                        else if (sc_cardinalities[3*sc_idx + 1] == 1) {
                            // q(x) is positive
                            set_lower_zero(r);
                            return true;
                        }
                        else {
                            SASSERT(sc_cardinalities[3*sc_idx + 2] == 1);
                            // q(x) is negative
                            set_upper_zero(r);
                            return true;
                        }
                    }
                    else {
                        if (q_eq_0 == 0) {
                            if (sc_cardinalities[2*sc_idx] == 1) {
                                // q(x) is positive
                                set_lower_zero(r);
                                return true;
                            }
                            else {
                                SASSERT(sc_cardinalities[2*sc_idx + 1] == 1);
                                // q(x) is negative
                                set_upper_zero(r);
                                return true;
                            }
                        }
                        else if (q_gt_0 == 0) {
                            if (sc_cardinalities[2*sc_idx] == 1) {
                                // q(x) is zero
                                return false;
                            }
                            else {
                                SASSERT(sc_cardinalities[2*sc_idx + 1] == 1);
                                // q(x) is negative
                                set_upper_zero(r);
                                return true;
                            }
                        }
                        else {
                            SASSERT(q_lt_0 == 0);
                            if (sc_cardinalities[2*sc_idx] == 1) {
                                // q(x) is zero
                                return false;
                            }
                            else {
                                SASSERT(sc_cardinalities[2*sc_idx + 1] == 1);
                                // q(x) is positive
                                set_lower_zero(r);
                                return true;
                            }
                        }
                    }
                }
            }
        }

        bool expensive_determine_algebraic_sign(rational_function_value * v) {
            SASSERT(contains_zero(v->interval()));
            SASSERT(v->ext()->is_algebraic());
            TRACE("rcf_algebraic_sign",
                  tout << "expensive_determine_algebraic_sign\n"; display(tout, v, false);
                  tout << "\ninterval: ";  bqim().display(tout, v->interval()); tout << "\n";);
            algebraic * x = to_algebraic(v->ext());
            scoped_mpbqi num_interval(bqim());
            SASSERT(is_denominator_one(v));
            if (!expensive_algebraic_poly_interval(v->num(), x, num_interval))
                return false; // it is zero
            SASSERT(!contains_zero(num_interval));
            set_interval(v->interval(), num_interval);
            SASSERT(!contains_zero(v->interval()));
            return true; // it is not zero
        }

        /**
           \brief Determine the sign of an rational function value
           p(x)/q(x) when x is an algebraic extension.
        */
        bool determine_algebraic_sign(rational_function_value * v) {
            SASSERT(v->ext()->is_algebraic());
            mpbqi & interval = v->interval();
            if (interval.lower_is_inf() || interval.upper_is_inf()) {
                return expensive_determine_algebraic_sign(v);
            }
            else {
                int m = magnitude(v->interval());
                unsigned prec = 1;
                if (m < 0)
                    prec = static_cast<unsigned>(-m) + 1;
                while (contains_zero(v->interval())) {
                    if (!refine_algebraic_interval(v, prec))
                        return expensive_determine_algebraic_sign(v);
                    prec++;
                    if (prec > m_max_precision)
                        return expensive_determine_algebraic_sign(v);
                }
                SASSERT(!contains_zero(v->interval()));
                return true;
            }
        }

        /**
           \brief Determine the sign of the new rational function value.
           The idea is to keep refining the interval until interval of v does not contain 0.
           After a couple of steps we switch to expensive sign determination procedure.

           Return false if v is actually zero.
        */
        bool determine_sign(rational_function_value * v) {
            if (!contains_zero(v->interval()))
                return true;
            bool r;
            switch (v->ext()->knd()) {
            case extension::TRANSCENDENTAL: determine_transcendental_sign(v); r = true; break; // it is never zero
            case extension::INFINITESIMAL:  determine_infinitesimal_sign(v);  r = true; break; // it is never zero
            case extension::ALGEBRAIC:      r = determine_algebraic_sign(v); break;
            default:
                UNREACHABLE();
                r = false;
            }
            TRACE("rcf_determine_sign_bug",
                  tout << "result: " << r << "\n";
                  display_compact(tout, v); tout << "\n";
                  tout << "sign: " << sign(v) << "\n";);
            return r;
        }

        bool determine_sign(value_ref & r) {
            SASSERT(is_rational_function(r.get()));
            return determine_sign(to_rational_function(r.get()));
        }

        // ---------------------------------
        //
        // Arithmetic operations
        //
        // ---------------------------------

        /**
           \brief Compute polynomials new_p1 and new_p2 s.t.
                  - p1/p2 == new_p1/new_p2, AND
                  - new_p2 is a Monic polynomial, AND
                  - gcd(new_p1, new_p2) == 1
        */
        void normalize_fraction(unsigned sz1, value * const * p1, unsigned sz2, value * const * p2, value_ref_buffer & new_p1, value_ref_buffer & new_p2) {
            INC_DEPTH();
            TRACE("rcf_arith", tout << "normalize [" << m_exec_depth << "]\n";
                  display_poly(tout, sz1, p1); tout << "\n";
                  display_poly(tout, sz2, p2); tout << "\n";);
            SASSERT(sz1 > 0 && sz2 > 0);
            if (sz2 == 1) {
                // - new_p1 <- p1/p2[0];       new_p2 <- one              IF  sz2 == 1
                div(sz1, p1, p2[0], new_p1);
                new_p2.reset(); new_p2.push_back(one());
            }
            else {
                value * lc = p2[sz2 - 1];
                if (is_rational_one(lc)) {
                    // p2 is monic
                    normalize_num_monic_den(sz1, p1, sz2, p2, new_p1, new_p2);
                }
                else {
                    // p2 is not monic
                    value_ref_buffer tmp1(*this);
                    value_ref_buffer tmp2(*this);
                    div(sz1, p1, lc, tmp1);
                    div(sz2, p2, lc, tmp2);
                    normalize_num_monic_den(tmp1.size(), tmp1.c_ptr(), tmp2.size(), tmp2.c_ptr(), new_p1, new_p2);
                }
            }
            TRACE("normalize_fraction_bug",
                  display_poly(tout, sz1, p1); tout << "\n";
                  display_poly(tout, sz2, p2); tout << "\n";
                  tout << "====>\n";
                  display_poly(tout, new_p1.size(), new_p1.c_ptr()); tout << "\n";
                  display_poly(tout, new_p2.size(), new_p2.c_ptr()); tout << "\n";);
        }

        /**
           \brief Auxiliary function for normalize_fraction.
           It produces new_p1 and new_p2 s.t.
                 new_p1/new_p2 == p1/p2
                 gcd(new_p1, new_p2) == 1

           Assumptions:
           \pre p2 is monic
           \pre sz2 > 1
        */
        void normalize_num_monic_den(unsigned sz1, value * const * p1, unsigned sz2, value * const * p2,
                                     value_ref_buffer & new_p1, value_ref_buffer & new_p2) {
            SASSERT(sz2 > 1);
            SASSERT(is_rational_one(p2[sz2-1]));

            value_ref_buffer g(*this);

            gcd(sz1, p1, sz2, p2, g);
            SASSERT(is_monic(g));

            if (is_rational_one(g)) {
                new_p1.append(sz1, p1);
                new_p2.append(sz2, p2);
            }
            else {
                div(sz1, p1, g.size(), g.c_ptr(), new_p1);
                div(sz2, p2, g.size(), g.c_ptr(), new_p2);
                SASSERT(is_monic(new_p2));
            }
        }

        /**
           \brief Simplify p1(x) using x's defining polynomial.

           By definition of polynomial division, we have:

           new_p1(x) == quotient(p1,p)(x) * p(x) + rem(p1,p)(x)

           Since p(x) == 0, we have that

           new_p1(x) = rem(p1,p)(x)
        */
        void normalize_algebraic(algebraic * x, unsigned sz1, value * const * p1, value_ref_buffer & new_p1) {
            polynomial const & p = x->p();
            if (!m_lazy_algebraic_normalization || !m_in_aux_values || is_monic(p)) {
                rem(sz1, p1, p.size(), p.c_ptr(), new_p1);
            }
            else {
                new_p1.reset();
                new_p1.append(sz1, p1);
            }
        }

        /**
           \brief Create a new value using the a->ext(), and the given numerator and denominator.
           Use interval(a) + interval(b) as an initial approximation for the interval of the result, and invoke determine_sign()
        */
        void mk_add_value(rational_function_value * a, value * b, unsigned num_sz, value * const * num, unsigned den_sz, value * const * den, value_ref & r) {
            SASSERT(num_sz > 0);
            // den_sz may be zero for algebraic extensions.
            // We do not use denominators for algebraic extensions.
            if (num_sz == 1 && den_sz <= 1) {
                // In this case, the normalization rules guarantee that den is one.
                SASSERT(den_sz == 0 || is_rational_one(den[0]));
                r = num[0];
            }
            else {
                scoped_mpbqi ri(bqim());
                bqim().add(interval(a), interval(b), ri);
                r = mk_rational_function_value_core(a->ext(), num_sz, num, den_sz, den);
                swap(r->interval(), ri);
                if (determine_sign(r)) {
                    SASSERT(!contains_zero(r->interval()));
                }
                else {
                    // The new value is 0
                    r = nullptr;
                }
            }
        }

        /**
           \brief Add a value of 'a' the form n/1 with b where rank(a) > rank(b)
        */
        void add_p_v(rational_function_value * a, value * b, value_ref & r) {
            SASSERT(is_denominator_one(a));
            SASSERT(compare_rank(a, b) > 0);
            polynomial const & an  = a->num();
            polynomial const & one = a->den();
            SASSERT(an.size() > 1);
            value_ref_buffer new_num(*this);
            add(an.size(), an.c_ptr(), b, new_num);
            SASSERT(new_num.size() == an.size());
            mk_add_value(a, b, new_num.size(), new_num.c_ptr(), one.size(), one.c_ptr(), r);
        }

        /**
           \brief Add a value 'a' of the form n/d with b where rank(a) > rank(b)
        */
        void add_rf_v(rational_function_value * a, value * b, value_ref & r) {
            value_ref_buffer b_ad(*this);
            value_ref_buffer num(*this);
            polynomial const & an = a->num();
            if (is_denominator_one(a)) {
                add_p_v(a, b, r);
            }
            else {
                SASSERT(!a->ext()->is_algebraic());
                polynomial const & ad = a->den();
                // b_ad <- b * ad
                mul(b, ad.size(), ad.c_ptr(), b_ad);
                // num <- a + b * ad
                add(an.size(), an.c_ptr(), b_ad.size(), b_ad.c_ptr(), num);
                if (num.empty())
                    r = nullptr;
                else {
                    value_ref_buffer new_num(*this);
                    value_ref_buffer new_den(*this);
                    normalize_fraction(num.size(), num.c_ptr(), ad.size(), ad.c_ptr(), new_num, new_den);
                    SASSERT(!new_num.empty());
                    mk_add_value(a, b, new_num.size(), new_num.c_ptr(), new_den.size(), new_den.c_ptr(), r);
                }
            }
        }

        /**
           \brief Add values 'a' and 'b' of the form n/1 and rank(a) == rank(b)
        */
        void add_p_p(rational_function_value * a, rational_function_value * b, value_ref & r) {
            SASSERT(is_denominator_one(a));
            SASSERT(is_denominator_one(b));
            SASSERT(compare_rank(a, b) == 0);
            polynomial const & an  = a->num();
            polynomial const & one = a->den();
            polynomial const & bn  = b->num();
            value_ref_buffer new_num(*this);
            add(an.size(), an.c_ptr(), bn.size(), bn.c_ptr(), new_num);
            if (new_num.empty())
                r = nullptr;
            else {
                // We don't need to invoke normalize_algebraic even if x (== a->ext()) is algebraic.
                // Reason: by construction the polynomials a->num() and b->num() are "normalized".
                // That is, their degrees are < degree of the polynomial defining x.
                // Moreover, when we add polynomials, the degree can only decrease.
                // So, degree of new_num must be < degree of x's defining polynomial.
                mk_add_value(a, b, new_num.size(), new_num.c_ptr(), one.size(), one.c_ptr(), r);
            }
        }

        /**
           \brief Add values 'a' and 'b' of the form n/d and rank(a) == rank(b)
        */
        void add_rf_rf(rational_function_value * a, rational_function_value * b, value_ref & r) {
            SASSERT(compare_rank(a, b) == 0);
            polynomial const & an = a->num();
            polynomial const & bn = b->num();
            if (is_denominator_one(a) && is_denominator_one(b)) {
                add_p_p(a, b, r);
            }
            else {
                SASSERT(!a->ext()->is_algebraic());
                polynomial const & ad = a->den();
                polynomial const & bd = b->den();
                value_ref_buffer an_bd(*this);
                value_ref_buffer bn_ad(*this);
                mul(an.size(), an.c_ptr(), bd.size(), bd.c_ptr(), an_bd);
                mul(bn.size(), bn.c_ptr(), ad.size(), ad.c_ptr(), bn_ad);
                value_ref_buffer num(*this);
                add(an_bd.size(), an_bd.c_ptr(), bn_ad.size(), bn_ad.c_ptr(), num);
                if (num.empty()) {
                    r = nullptr;
                }
                else {
                    value_ref_buffer den(*this);
                    mul(ad.size(), ad.c_ptr(), bd.size(), bd.c_ptr(), den);
                    value_ref_buffer new_num(*this);
                    value_ref_buffer new_den(*this);
                    normalize_fraction(num.size(), num.c_ptr(), den.size(), den.c_ptr(), new_num, new_den);
                    SASSERT(!new_num.empty());
                    mk_add_value(a, b, new_num.size(), new_num.c_ptr(), new_den.size(), new_den.c_ptr(), r);
                }
            }
        }

        void add(value * a, value * b, value_ref & r) {
            if (a == nullptr) {
                r = b;
            }
            else if (b == nullptr) {
                r = a;
            }
            else if (is_nz_rational(a) && is_nz_rational(b)) {
                scoped_mpq v(qm());
                qm().add(to_mpq(a), to_mpq(b), v);
                if (qm().is_zero(v))
                    r = nullptr;
                else
                    r = mk_rational_and_swap(v);
            }
            else {
                INC_DEPTH();
                TRACE("rcf_arith", tout << "add [" << m_exec_depth << "]\n";
                      display(tout, a, false); tout << "\n";
                      display(tout, b, false); tout << "\n";);
                switch (compare_rank(a, b)) {
                case -1: add_rf_v(to_rational_function(b), a, r); break;
                case 0:  add_rf_rf(to_rational_function(a), to_rational_function(b), r); break;
                case 1:  add_rf_v(to_rational_function(a), b, r); break;
                default: UNREACHABLE();
                }
            }
        }

        void sub(value * a, value * b, value_ref & r) {
            if (a == nullptr) {
                neg(b, r);
            }
            else if (b == nullptr) {
                r = a;
            }
            else if (is_nz_rational(a) && is_nz_rational(b)) {
                scoped_mpq v(qm());
                qm().sub(to_mpq(a), to_mpq(b), v);
                if (qm().is_zero(v))
                    r = nullptr;
                else
                    r = mk_rational_and_swap(v);
            }
            else {
                value_ref neg_b(*this);
                neg(b, neg_b);
                switch (compare_rank(a, neg_b)) {
                case -1: add_rf_v(to_rational_function(neg_b), a, r); break;
                case 0:  add_rf_rf(to_rational_function(a), to_rational_function(neg_b), r); break;
                case 1:  add_rf_v(to_rational_function(a), neg_b, r); break;
                default: UNREACHABLE();
                }
            }
        }

        void neg_rf(rational_function_value * a, value_ref & r) {
            polynomial const & an = a->num();
            polynomial const & ad = a->den();
            value_ref_buffer new_num(*this);
            neg(an.size(), an.c_ptr(), new_num);
            scoped_mpbqi ri(bqim());
            bqim().neg(interval(a), ri);
            r = mk_rational_function_value_core(a->ext(), new_num.size(), new_num.c_ptr(), ad.size(), ad.c_ptr());
            swap(r->interval(), ri);
            SASSERT(!contains_zero(r->interval()));
        }

        void neg(value * a, value_ref & r) {
            if (a == nullptr) {
                r = nullptr;
            }
            else if (is_nz_rational(a)) {
                scoped_mpq v(qm());
                qm().set(v, to_mpq(a));
                qm().neg(v);
                r = mk_rational_and_swap(v);
            }
            else {
                neg_rf(to_rational_function(a), r);
            }
        }

        /**
           \brief Create a new value using the a->ext(), and the given numerator and denominator.
           Use interval(a) * interval(b) as an initial approximation for the interval of the result, and invoke determine_sign()
        */
        void mk_mul_value(rational_function_value * a, value * b, unsigned num_sz, value * const * num, unsigned den_sz, value * const * den, value_ref & r) {
            SASSERT(num_sz > 0);
            if (num_sz == 1 && den_sz <= 1) {
                // den_sz may be zero for algebraic extensions.
                // We do not use denominators for algebraic extensions.
                // In this case, the normalization rules guarantee that den is one.
                SASSERT(den_sz == 0 || is_rational_one(den[0]));
                r = num[0];
            }
            else {
                scoped_mpbqi ri(bqim());
                bqim().mul(interval(a), interval(b), ri);
                r = mk_rational_function_value_core(a->ext(), num_sz, num, den_sz, den);
                swap(ri, r->interval());
                if (determine_sign(r)) {
                    SASSERT(!contains_zero(r->interval()));
                }
                else {
                    // The new value is 0
                    r = nullptr;
                }
            }
        }

        /**
           \brief Multiply a value of 'a' the form n/1 with b where rank(a) > rank(b)
        */
        void mul_p_v(rational_function_value * a, value * b, value_ref & r) {
            SASSERT(is_denominator_one(a));
            SASSERT(b != 0);
            SASSERT(compare_rank(a, b) > 0);
            polynomial const & an  = a->num();
            polynomial const & one = a->den();
            SASSERT(an.size() > 1);
            value_ref_buffer new_num(*this);
            mul(b, an.size(), an.c_ptr(), new_num);
            SASSERT(new_num.size() == an.size());
            mk_mul_value(a, b, new_num.size(), new_num.c_ptr(), one.size(), one.c_ptr(), r);
        }

        /**
           \brief Multiply a value 'a' of the form n/d with b where rank(a) > rank(b)
        */
        void mul_rf_v(rational_function_value * a, value * b, value_ref & r) {
            polynomial const & an = a->num();
            if (is_denominator_one(a)) {
                mul_p_v(a, b, r);
            }
            else {
                SASSERT(!a->ext()->is_algebraic());
                polynomial const & ad = a->den();
                value_ref_buffer num(*this);
                // num <- b * an
                mul(b, an.size(), an.c_ptr(), num);
                SASSERT(num.size() == an.size());
                value_ref_buffer new_num(*this);
                value_ref_buffer new_den(*this);
                normalize_fraction(num.size(), num.c_ptr(), ad.size(), ad.c_ptr(), new_num, new_den);
                SASSERT(!new_num.empty());
                mk_mul_value(a, b, new_num.size(), new_num.c_ptr(), new_den.size(), new_den.c_ptr(), r);
            }
        }

        /**
           \brief Multiply values 'a' and 'b' of the form n/1 and rank(a) == rank(b)
        */
        void mul_p_p(rational_function_value * a, rational_function_value * b, value_ref & r) {
            SASSERT(is_denominator_one(a));
            SASSERT(is_denominator_one(b));
            SASSERT(compare_rank(a, b) == 0);
            polynomial const & an  = a->num();
            polynomial const & one = a->den();
            polynomial const & bn  = b->num();
            value_ref_buffer new_num(*this);
            mul(an.size(), an.c_ptr(), bn.size(), bn.c_ptr(), new_num);
            SASSERT(!new_num.empty());
            extension * x = a->ext();
            if (x->is_algebraic()) {
                value_ref_buffer new_num2(*this);
                normalize_algebraic(to_algebraic(x), new_num.size(), new_num.c_ptr(), new_num2);
                SASSERT(!new_num.empty());
                mk_mul_value(a, b, new_num2.size(), new_num2.c_ptr(), one.size(), one.c_ptr(), r);
            }
            else {
                mk_mul_value(a, b, new_num.size(), new_num.c_ptr(), one.size(), one.c_ptr(), r);
            }
        }

        /**
           \brief Multiply values 'a' and 'b' of the form n/d and rank(a) == rank(b)
        */
        void mul_rf_rf(rational_function_value * a, rational_function_value * b, value_ref & r) {
            SASSERT(compare_rank(a, b) == 0);
            polynomial const & an = a->num();
            polynomial const & bn = b->num();
            if (is_denominator_one(a) && is_denominator_one(b)) {
                mul_p_p(a, b, r);
            }
            else {
                SASSERT(!a->ext()->is_algebraic());
                polynomial const & ad = a->den();
                polynomial const & bd = b->den();
                value_ref_buffer num(*this);
                value_ref_buffer den(*this);
                mul(an.size(), an.c_ptr(), bn.size(), bn.c_ptr(), num);
                mul(ad.size(), ad.c_ptr(), bd.size(), bd.c_ptr(), den);
                SASSERT(!num.empty()); SASSERT(!den.empty());
                value_ref_buffer new_num(*this);
                value_ref_buffer new_den(*this);
                normalize_fraction(num.size(), num.c_ptr(), den.size(), den.c_ptr(), new_num, new_den);
                SASSERT(!new_num.empty());
                mk_mul_value(a, b, new_num.size(), new_num.c_ptr(), new_den.size(), new_den.c_ptr(), r);
            }
        }

        void mul(value * a, value * b, value_ref & r) {
            if (a == nullptr || b == nullptr) {
                r = nullptr;
            }
            else if (is_rational_one(a)) {
                r = b;
            }
            else if (is_rational_one(b)) {
                r = a;
            }
            else if (is_rational_minus_one(a)) {
                neg(b, r);
            }
            else if (is_rational_minus_one(b)) {
                neg(a, r);
            }
            else if (is_nz_rational(a) && is_nz_rational(b)) {
                scoped_mpq v(qm());
                qm().mul(to_mpq(a), to_mpq(b), v);
                r = mk_rational_and_swap(v);
            }
            else {
                INC_DEPTH();
                TRACE("rcf_arith", tout << "mul [" << m_exec_depth << "]\n";
                      display(tout, a, false); tout << "\n";
                      display(tout, b, false); tout << "\n";);
                switch (compare_rank(a, b)) {
                case -1: mul_rf_v(to_rational_function(b), a, r); break;
                case 0:  mul_rf_rf(to_rational_function(a), to_rational_function(b), r); break;
                case 1:  mul_rf_v(to_rational_function(a), b, r); break;
                default: UNREACHABLE();
                }
            }
        }

        void div(value * a, value * b, value_ref & r) {
            if (a == nullptr) {
                r = nullptr;
            }
            else if (b == nullptr) {
                throw exception("division by zero");
            }
            else if (is_rational_one(b)) {
                r = a;
            }
            else if (is_rational_one(a)) {
                inv(b, r);
            }
            else if (is_rational_minus_one(b)) {
                neg(a, r);
            }
            else if (is_nz_rational(a) && is_nz_rational(b)) {
                scoped_mpq v(qm());
                qm().div(to_mpq(a), to_mpq(b), v);
                r = mk_rational_and_swap(v);
            }
            else {
                value_ref inv_b(*this);
                inv(b, inv_b);
                switch (compare_rank(a, inv_b)) {
                case -1: mul_rf_v(to_rational_function(inv_b), a, r);  break;
                case 0:  mul_rf_rf(to_rational_function(a), to_rational_function(inv_b), r); break;
                case 1:  mul_rf_v(to_rational_function(a), inv_b, r); break;
                default: UNREACHABLE();
                }
            }
        }

        /**
           \brief Invert 1/q(alpha) given that p(alpha) = 0. That is, we find h s.t.

           q(alpha) * h(alpha) = 1

           The procedure succeeds (and returns true) if the GCD(q, p) = 1.

           If the GCD(q, p) != 1, then it returns false, and store the GCD in g.

           The following procedure is essentially a special case of the extended polynomial GCD algorithm.
        */
        bool inv_algebraic(unsigned q_sz, value * const * q, unsigned p_sz, value * const * p, value_ref_buffer & g, value_ref_buffer & h) {
            TRACE("inv_algebraic",
                  tout << "q: "; display_poly(tout, q_sz, q); tout << "\n";
                  tout << "p: "; display_poly(tout, p_sz, p); tout << "\n";);
            SASSERT(q_sz > 0);
            SASSERT(q_sz < p_sz);
            // Q <- q
            value_ref_buffer Q(*this);
            Q.append(q_sz, q);
            // R <- 1
            value_ref_buffer R(*this);
            R.push_back(one());
            value_ref_buffer Quo(*this), Rem(*this), aux(*this);

            // We find h(alpha), by rewriting the equation
            //    q(alpha) * h(alpha) = 1
            // until we have
            //    1 * h(alpha) = R(alpha)
            while (true) {
                // In every iteration of the loop we have
                //   Q(alpha) * h(alpha) = R(alpha)
                TRACE("inv_algebraic",
                      tout << "Q: "; display_poly(tout, Q.size(), Q.c_ptr()); tout << "\n";
                      tout << "R: "; display_poly(tout, R.size(), R.c_ptr()); tout << "\n";);
                if (Q.size() == 1) {
                    // If the new Q is the constant polynomial, they we are done.
                    // We just divide R by Q[0].
                    // h(alpha) = R(alpha) / Q[0]
                    div(R.size(), R.c_ptr(), Q[0], h);
                    TRACE("inv_algebraic", tout << "h: "; display_poly(tout, h.size(), h.c_ptr()); tout << "\n";);
                    // g <- 1
                    g.reset(); g.push_back(one());
                    return true;
                }
                else {
                    div_rem(p_sz, p, Q.size(), Q.c_ptr(), Quo, Rem);
                    if (Rem.empty()) {
                        // failed
                        // GCD(q, p) != 1
                        g = Q;
                        mk_monic(g);
                        return false;
                    }
                    else {
                        // By the definition of polynomial division, we have
                        //     p == Quo * Q + Rem
                        // Since, we have p(alpha) = 0
                        //     Quo(alpha) * Q(alpha) = -Rem(alpha)    (*)
                        // Now, if we multiply the equation
                        //     Q(alpha) * h(alpha) = R(alpha)
                        // by Quo(alpha) and apply (*), we get
                        //     -Rem(alpha) * h(alpha) = R(alpha) * Quo(alpha)
                        // Thus, we update Q, and R for the next iteration, as
                        //    Q <- -REM
                        //    R <- R * Quo
                        // Q <- -Rem
                        neg(Rem.size(), Rem.c_ptr(), Q);
                        mul(R.size(), R.c_ptr(), Quo.size(), Quo.c_ptr(), aux);
                        // Moreover since p(alpha) = 0, we can simplify Q, by using
                        // Q(alpha) = REM(Q, p)(alpha)
                        rem(aux.size(), aux.c_ptr(), p_sz, p, R);
                        SASSERT(R.size() < p_sz);
                        //
                    }
                }
            }
        }

        /**
           \brief r <- 1/a  specialized version when a->ext() is algebraic.
           It avoids the use of rational functions.
        */
        void inv_algebraic(rational_function_value * a, value_ref & r) {
            SASSERT(a->ext()->is_algebraic());
            SASSERT(is_denominator_one(a));
            scoped_mpbqi ri(bqim());
            bqim().inv(interval(a), ri);
            algebraic * alpha = to_algebraic(a->ext());
            polynomial const & q = a->num();
            polynomial const & p = alpha->p();
            value_ref_buffer norm_q(*this);
            // since p(alpha) = 0, we have that q(alpha) = rem(q, p)(alpha)
            rem(q.size(), q.c_ptr(), p.size(), p.c_ptr(), norm_q);
            SASSERT(norm_q.size() < p.size());
            value_ref_buffer new_num(*this), g(*this);
            if (inv_algebraic(norm_q.size(), norm_q.c_ptr(), p.size(), p.c_ptr(), g, new_num)) {
                if (new_num.size() == 1) {
                    r = new_num[0];
                }
                else {
                    r = mk_rational_function_value_core(alpha, new_num.size(), new_num.c_ptr());
                    swap(r->interval(), ri);
                    SASSERT(!contains_zero(r->interval()));
                }
            }
            else {
                // We failed to compute 1/a
                // because q and p are not co-prime
                // This can happen because we don't use minimal
                // polynomials to represent algebraic extensions such
                // as alpha.

                // We recover from the failure by refining the defining polynomial of alpha
                // with p/gcd(p, q)
                // Remark: g contains the gcd of p, q
                // And try again :)

                value_ref_buffer new_p(*this);
                div(p.size(), p.c_ptr(), g.size(), g.c_ptr(), new_p);
                if (m_clean_denominators) {
                    value_ref_buffer tmp(*this);
                    value_ref d(*this);
                    clean_denominators(new_p.size(), new_p.c_ptr(), tmp, d);
                    new_p = tmp;
                }
                SASSERT(new_p.size() >= 2);

                if (new_p.size() == 2) {
                    // Easy case: alpha is actually equal to
                    //    -new_p[0]/new_p[1]
                    value_ref alpha_val(*this);
                    alpha_val = new_p[0];
                    neg(alpha_val, alpha_val);
                    div(alpha_val, new_p[1], alpha_val);
                    // Thus, a is equal to q(alpha_val)
                    value_ref new_a(*this);
                    mk_polynomial_value(q.size(), q.c_ptr(), alpha_val, new_a);
                    // Remark new_a does not depend on alpha anymore
                    // r == 1/inv(new_a)
                    inv(new_a, r);
                }
                else if (alpha->sdt() == nullptr) {
                    // Another easy case: we just have to replace
                    //    alpha->p() with new_p.
                    // The m_iso_interval for p() is also an isolating interval for new_p,
                    // since the roots of new_p() are a subset of the roots of p
                    reset_p(alpha->m_p);
                    set_p(alpha->m_p, new_p.size(), new_p.c_ptr());

                    // The new call will succeed because q and new_p are co-prime
                    inv_algebraic(a, r);
                }
                else {
                    // Let sdt be alpha->sdt();
                    // In principal, the signs of the polynomials sdt->qs can be used
                    // to discriminate the roots of new_p. The signs of this polynomials
                    // depend only on alpha, and not on the polynomial used to define alpha
                    // So, in principle, we can reuse m_qs and m_sign_conditions.
                    // However, we have to recompute the tarski queries with respect to new_p.
                    // This values will be different, since new_p has less roots than p.
                    //
                    // Instead of trying to reuse the information in sdt, we simply
                    // isolate the roots of new_p, and check the one that is equal to alpha.
                    // and copy all the information from them.
                    SASSERT(new_p.size() > 2);
                    // we can invoke nl_nz_sqf_isolate_roots, because we know
                    //   - new_p is not linear
                    //   - new_p is square free (it is a factor of the square free polynomial p)
                    //   - 0 is not a root of new_p (it is a factor of p, and 0 is not a root of p)
                    numeral_vector roots;
                    nl_nz_sqf_isolate_roots(new_p.size(), new_p.c_ptr(), roots);
                    SASSERT(roots.size() > 0);
                    algebraic * new_alpha;
                    if (roots.size() == 1) {
                        new_alpha = to_algebraic(to_rational_function(roots[0].m_value)->ext());
                    }
                    else {
                        value_ref alpha_val(*this);
                        alpha_val = mk_rational_function_value(alpha);
                        // search for the root that is equal to alpha
                        unsigned i = 0;
                        for (i = 0; i < roots.size(); i++) {
                            if (compare(alpha_val, roots[i].m_value) == 0) {
                                // found it;
                                break;
                            }
                        }
                        new_alpha = to_algebraic(to_rational_function(roots[i].m_value)->ext());
                    }
                    SASSERT(new_alpha->p().size() == new_p.size());
                    // We now that alpha and new_alpha represent the same value.
                    // Thus, we update alpha fields with the fields from new_alpha.

                    // copy new_alpha->m_p
                    reset_p(alpha->m_p);
                    set_p(alpha->m_p, new_alpha->m_p.size(), new_alpha->m_p.c_ptr());
                    // copy new_alpha->m_sign_det
                    inc_ref_sign_det(new_alpha->m_sign_det);
                    dec_ref_sign_det(alpha->m_sign_det);
                    alpha->m_sign_det = new_alpha->m_sign_det;
                    // copy remaining fields
                    set_interval(alpha->m_iso_interval, new_alpha->m_iso_interval);
                    alpha->m_sc_idx = new_alpha->m_sc_idx;
                    alpha->m_depends_on_infinitesimals = new_alpha->m_depends_on_infinitesimals;

                    // The new call will succeed because q and new_p are co-prime
                    inv_algebraic(a, r);
                }
            }
        }

        void inv_rf(rational_function_value * a, value_ref & r) {
            if (a->ext()->is_algebraic()) {
                inv_algebraic(a, r);
            }
            else {
                SASSERT(!a->ext()->is_algebraic());
                polynomial const & an = a->num();
                polynomial const & ad = a->den();
                scoped_mpbqi ri(bqim());
                bqim().inv(interval(a), ri);
                // The GCD of an and ad is one, we may use a simpler version of normalize
                value_ref_buffer new_num(*this);
                value_ref_buffer new_den(*this);
                normalize_fraction(ad.size(), ad.c_ptr(), an.size(), an.c_ptr(), new_num, new_den);
                r = mk_rational_function_value_core(a->ext(), new_num.size(), new_num.c_ptr(), new_den.size(), new_den.c_ptr());
                swap(r->interval(), ri);
                SASSERT(!contains_zero(r->interval()));
            }
        }

        void inv(value * a, value_ref & r) {
            if (a == nullptr) {
                throw exception("division by zero");
            }
            if (is_nz_rational(a)) {
                scoped_mpq v(qm());
                qm().inv(to_mpq(a), v);
                r = mk_rational_and_swap(v);
            }
            else {
                inv_rf(to_rational_function(a), r);
            }
        }

        void set(numeral & n, value * v) {
            inc_ref(v);
            dec_ref(n.m_value);
            n.m_value = v;
        }

        void set(numeral & n, value_ref const & v) {
            set(n, v.get());
        }

        void neg(numeral & a) {
            value_ref r(*this);
            neg(a.m_value, r);
            set(a, r);
        }

        void neg(numeral const & a, numeral & b) {
            value_ref r(*this);
            neg(a.m_value, r);
            set(b, r);
        }

        void inv(numeral & a) {
            value_ref r(*this);
            inv(a.m_value, r);
            set(a, r);
        }

        void inv(numeral const & a, numeral & b) {
            value_ref r(*this);
            inv(a.m_value, r);
            set(b, r);
        }

        void add(numeral const & a, numeral const & b, numeral & c) {
            value_ref r(*this);
            add(a.m_value, b.m_value, r);
            set(c, r);
        }

        void sub(numeral const & a, numeral const & b, numeral & c) {
            value_ref r(*this);
            sub(a.m_value, b.m_value, r);
            set(c, r);
        }

        void mul(numeral const & a, numeral const & b, numeral & c) {
            value_ref r(*this);
            mul(a.m_value, b.m_value, r);
            set(c, r);
        }

        void div(numeral const & a, numeral const & b, numeral & c) {
            value_ref r(*this);
            div(a.m_value, b.m_value, r);
            set(c, r);
        }

        /**
           \brief a <- b^{1/k}
        */
        void root(numeral const & a, unsigned k, numeral & b) {
            if (k == 0)
                throw exception("0-th root is indeterminate");

            if (k == 1 || is_zero(a)) {
                set(b, a);
                return;
            }

            if (sign(a) < 0 && k % 2 == 0)
                throw exception("even root of negative number");

            // create the polynomial p of the form x^k - a
            value_ref_buffer p(*this);
            value_ref neg_a(*this);
            neg(a.m_value, neg_a);
            p.push_back(neg_a);
            for (unsigned i = 0; i < k - 1; i++)
                p.push_back(nullptr);
            p.push_back(one());

            numeral_vector roots;
            nz_isolate_roots(p.size(), p.c_ptr(), roots);
            SASSERT(roots.size() == 1 || roots.size() == 2);
            if (roots.size() == 1 || sign(roots[0].m_value) > 0) {
                set(b, roots[0]);
            }
            else {
                SASSERT(roots.size() == 2);
                SASSERT(sign(roots[1].m_value) > 0);
                set(b, roots[1]);
            }
            del(roots);
        }

        /**
           \brief a <- b^k
        */
        void power(numeral const & a, unsigned k, numeral & b) {
            unsigned mask = 1;
            value_ref power(*this);
            value_ref _b(*this);
            power = a.m_value;
            _b = one();
            while (mask <= k) {
                checkpoint();
                if (mask & k)
                    mul(_b, power, _b);
                mul(power, power, power);
                mask = mask << 1;
            }
            set(b, _b);
        }

        // ---------------------------------
        //
        // Comparison
        //
        // ---------------------------------

        int compare(value * a, value * b) {
            if (a == nullptr)
                return -sign(b);
            else if (b == nullptr)
                return sign(a);
            else if (is_nz_rational(a) && is_nz_rational(b)) {
                if (qm().eq(to_mpq(a), to_mpq(b)))
                    return 0;
                else
                    return qm().lt(to_mpq(a), to_mpq(b)) ? -1 : 1;
            }
            else {
                // FUTURE: try to refine interval before switching to sub+sign approach
                if (bqim().before(interval(a), interval(b)))
                    return -1;
                else if (bqim().before(interval(b), interval(a)))
                    return 1;
                else {
                    value_ref diff(*this);
                    sub(a, b, diff);
                    return sign(diff);
                }
            }
        }

        int compare(numeral const & a, numeral const & b) {
            return compare(a.m_value, b.m_value);
        }

        // ---------------------------------
        //
        // "Pretty printing"
        //
        // ---------------------------------

        struct collect_algebraic_refs {
            char_vector            m_visited; // Set of visited algebraic extensions.
            ptr_vector<algebraic>  m_found;   // vector/list of visited algebraic extensions.

            void mark(extension * ext) {
                if (ext->is_algebraic()) {
                    m_visited.reserve(ext->idx() + 1, false);
                    if (!m_visited[ext->idx()]) {
                        m_visited[ext->idx()] = true;
                        algebraic * a = to_algebraic(ext);
                        m_found.push_back(a);
                        mark(a->p());
                    }
                }
            }

            void mark(polynomial const & p) {
                for (unsigned i = 0; i < p.size(); i++) {
                    mark(p[i]);
                }
            }

            void mark(value * v) {
                if (v == nullptr || is_nz_rational(v))
                    return;
                rational_function_value * rf = to_rational_function(v);
                mark(rf->ext());
                mark(rf->num());
                mark(rf->den());
            }
        };

        static unsigned num_nz_coeffs(polynomial const & p) {
            unsigned r = 0;
            for (unsigned i = 0; i < p.size(); i++) {
                if (p[i])
                    r++;
            }
            return r;
        }

        bool use_parenthesis(value * v) const {
            if (is_zero(v) || is_nz_rational(v))
                return false;
            rational_function_value * rf = to_rational_function(v);
            return num_nz_coeffs(rf->num()) > 1 || !is_denominator_one(rf);
        }

        template<typename DisplayVar>
        void display_polynomial(std::ostream & out, unsigned sz, value * const * p, DisplayVar const & display_var, bool compact, bool pp) const {
            if (sz == 0) {
                out << "0";
                return;
            }
            unsigned i = sz;
            bool first = true;
            while (i > 0) {
                --i;
                if (p[i] == nullptr)
                    continue;
                if (first)
                    first = false;
                else
                    out << " + ";
                if (i == 0)
                    display(out, p[i], compact, pp);
                else {
                    if (!is_rational_one(p[i])) {
                        if (use_parenthesis(p[i])) {
                            out << "(";
                            display(out, p[i], compact, pp);
                            out << ")";
                            if (pp)
                                out << " ";
                            else
                                out << "*";
                        }
                        else {
                            display(out, p[i], compact, pp);
                            if (pp)
                                out << " ";
                            else
                                out << "*";
                        }
                    }
                    display_var(out, compact, pp);
                    if (i > 1) {
                        if (pp)
                            out << "<sup>" << i << "</sup>";
                        else
                            out << "^" << i;
                    }
                }
            }
        }

        template<typename DisplayVar>
        void display_polynomial(std::ostream & out, polynomial const & p, DisplayVar const & display_var, bool compact, bool pp) const {
            display_polynomial(out, p.size(), p.c_ptr(), display_var, compact, pp);
        }

        struct display_free_var_proc {
            void operator()(std::ostream & out, bool compact, bool pp) const {
                out << "x";
            }
        };

        struct display_ext_proc {
            imp const &  m;
            extension *  m_ref;
            display_ext_proc(imp const & _m, extension * r):m(_m), m_ref(r) {}
            void operator()(std::ostream & out, bool compact, bool pp) const {
                m.display_ext(out, m_ref, compact, pp);
            }
        };

        void display_polynomial_expr(std::ostream & out, polynomial const & p, extension * ext, bool compact, bool pp) const {
            display_polynomial(out, p, display_ext_proc(*this, ext), compact, pp);
        }

        static void display_poly_sign(std::ostream & out, int s) {
            if (s < 0)
                out << " < 0";
            else if (s == 0)
                out << " = 0";
            else
                out << " > 0";
        }

        void display_sign_conditions(std::ostream & out, sign_condition * sc) const {
            bool first = true;
            out << "{";
            while (sc) {
                if (first)
                    first = false;
                else
                    out << ", ";
                out << "q(" << sc->qidx() << ")";
                display_poly_sign(out, sc->sign());
                sc = sc->prev();
            }
            out << "}";
        }

        void display_sign_conditions(std::ostream & out, sign_condition * sc, array<polynomial> const & qs, bool compact, bool pp) const {
            bool first = true;
            out << "{";
            while (sc) {
                if (first)
                    first = false;
                else
                    out << ", ";
                display_polynomial(out, qs[sc->qidx()], display_free_var_proc(), compact, pp);
                display_poly_sign(out, sc->sign());
                sc = sc->prev();
            }
            out << "}";
        }

        void display_interval(std::ostream & out, mpbqi const & i, bool pp) const {
            if (pp)
                bqim().display_pp(out, i);
            else
                bqim().display(out, i);
        }

        void display_algebraic_def(std::ostream & out, algebraic * a, bool compact, bool pp) const {
            out << "root(";
            display_polynomial(out, a->p(), display_free_var_proc(), compact, pp);
            out << ", ";
            display_interval(out, a->iso_interval(), pp);
            out << ", ";
            if (a->sdt() != nullptr)
                display_sign_conditions(out, a->sdt()->sc(a->sc_idx()), a->sdt()->qs(), compact, pp);
            else
                out << "{}";
            out << ")";
        }

        void display_poly(std::ostream & out, unsigned n, value * const * p) const {
            collect_algebraic_refs c;
            for (unsigned i = 0; i < n; i++)
                c.mark(p[i]);
            display_polynomial(out, n, p, display_free_var_proc(), true, false);
            std::sort(c.m_found.begin(), c.m_found.end(), rank_lt_proc());
            for (unsigned i = 0; i < c.m_found.size(); i++) {
                algebraic * ext = c.m_found[i];
                out << "\n   r!" << ext->idx() << " := ";
                display_algebraic_def(out, ext, true, false);
            }
        }

        void display_ext(std::ostream & out, extension * r, bool compact, bool pp) const {
            switch (r->knd()) {
            case extension::TRANSCENDENTAL: to_transcendental(r)->display(out, pp); break;
            case extension::INFINITESIMAL:  to_infinitesimal(r)->display(out, pp); break;
            case extension::ALGEBRAIC:
                if (compact) {
                    if (pp)
                        out << "&alpha;<sub>" << r->idx() << "</sub>";
                    else
                        out << "r!" << r->idx();
                }
                else {
                    display_algebraic_def(out, to_algebraic(r), compact, pp);
                }
            }
        }

        void display(std::ostream & out, value * v, bool compact, bool pp=false) const {
            if (v == nullptr)
                out << "0";
            else if (is_nz_rational(v))
                qm().display(out, to_mpq(v));
            else {
                rational_function_value * rf = to_rational_function(v);
                if (is_denominator_one(rf)) {
                    display_polynomial_expr(out, rf->num(), rf->ext(), compact, pp);
                }
                else if (is_rational_one(rf->num())) {
                    out << "1/(";
                    display_polynomial_expr(out, rf->den(), rf->ext(), compact, pp);
                    out << ")";
                }
                else {
                    out << "(";
                    display_polynomial_expr(out, rf->num(), rf->ext(), compact, pp);
                    out << ")/(";
                    display_polynomial_expr(out, rf->den(), rf->ext(), compact, pp);
                    out << ")";
                }
            }
        }

        void display_compact(std::ostream & out, value * a, bool pp=false) const {
            collect_algebraic_refs c;
            c.mark(a);
            if (c.m_found.empty()) {
                display(out, a, true, pp);
            }
            else {
                std::sort(c.m_found.begin(), c.m_found.end(), rank_lt_proc());
                out << "[";
                display(out, a, true, pp);
                for (unsigned i = 0; i < c.m_found.size(); i++) {
                    algebraic * ext = c.m_found[i];
                    if (pp)
                        out << "; &alpha;<sub>" << ext->idx() << "</sub> := ";
                    else
                        out << "; r!" << ext->idx() << " := ";
                    display_algebraic_def(out, ext, true, pp);
                }
                out << "]";
            }
        }

        void display(std::ostream & out, numeral const & a, bool compact=false, bool pp=false) const {
            if (compact)
                display_compact(out, a.m_value, pp);
            else
                display(out, a.m_value, false, pp);
        }

        void display_non_rational_in_decimal(std::ostream & out, numeral const & a, unsigned precision) {
            SASSERT(!is_zero(a));
            SASSERT(!is_nz_rational(a));
            mpbqi const & i = interval(a.m_value);
            if (refine_interval(a.m_value, precision*4)) {
                // hack
                if (bqm().is_int(i.lower()))
                    bqm().display_decimal(out, i.upper(), precision);
                else
                    bqm().display_decimal(out, i.lower(), precision);
            }
            else {
                if (sign(a.m_value) > 0)
                    out << "?";
                else
                    out << "-?";
            }
        }

        void display_decimal(std::ostream & out, numeral const & a, unsigned precision) const {
            if (is_zero(a)) {
                out << "0";
            }
            else if (is_nz_rational(a)) {
                qm().display_decimal(out, to_mpq(a), precision);
            }
            else {
                const_cast<imp*>(this)->display_non_rational_in_decimal(out, a, precision);
            }
        }

        void display_interval(std::ostream & out, numeral const & a) const {
            if (is_zero(a))
                out << "[0, 0]";
            else
                display_interval(out, interval(a.m_value), false);
        }
    };

    // Helper object for restoring the value intervals.
    class save_interval_ctx {
        manager::imp * m;
    public:
        save_interval_ctx(manager const * _this):m(_this->m_imp) { SASSERT (m); }
        ~save_interval_ctx() { m->restore_saved_intervals(); }
    };

    manager::manager(reslimit& lim, unsynch_mpq_manager & m, params_ref const & p, small_object_allocator * a) {
        m_imp = alloc(imp, lim, m, p, a);
    }

    manager::~manager() {
        dealloc(m_imp);
    }

    void manager::get_param_descrs(param_descrs & r) {
        rcf_params::collect_param_descrs(r);
    }

    void manager::updt_params(params_ref const & p) {
        m_imp->updt_params(p);
    }

    unsynch_mpq_manager & manager::qm() const {
        return m_imp->m_qm;
    }

    void manager::del(numeral & a) {
        m_imp->del(a);
    }

    void manager::mk_infinitesimal(char const * n, char const * pp_n, numeral & r) {
        m_imp->mk_infinitesimal(n, pp_n, r);
    }

    void manager::mk_infinitesimal(numeral & r) {
        m_imp->mk_infinitesimal(r);
    }

    void manager::mk_transcendental(char const * n, char const * pp_n, mk_interval & proc, numeral & r) {
        m_imp->mk_transcendental(n, pp_n, proc, r);
    }

    void manager::mk_transcendental(mk_interval & proc, numeral & r) {
        m_imp->mk_transcendental(proc, r);
    }

    void manager::mk_pi(numeral & r) {
        m_imp->mk_pi(r);
    }

    void manager::mk_e(numeral & r) {
        m_imp->mk_e(r);
    }

    void manager::isolate_roots(unsigned n, numeral const * as, numeral_vector & roots) {
        save_interval_ctx ctx(this);
        m_imp->isolate_roots(n, as, roots);
    }

    void manager::reset(numeral & a) {
        m_imp->reset(a);
    }

    int manager::sign(numeral const & a) {
        save_interval_ctx ctx(this);
        return m_imp->sign(a);
    }

    bool manager::is_zero(numeral const & a) {
        return sign(a) == 0;
    }

    bool manager::is_pos(numeral const & a) {
        return sign(a) > 0;
    }

    bool manager::is_neg(numeral const & a) {
        return sign(a) < 0;
    }

    bool manager::is_int(numeral const & a) {
        return m_imp->is_int(a);
    }

    bool manager::depends_on_infinitesimals(numeral const & a) {
        return m_imp->depends_on_infinitesimals(a);
    }

    void manager::set(numeral & a, int n) {
        m_imp->set(a, n);
    }

    void manager::set(numeral & a, mpz const & n) {
        m_imp->set(a, n);
    }

    void manager::set(numeral & a, mpq const & n) {
        m_imp->set(a, n);
    }

    void manager::set(numeral & a, numeral const & n) {
        m_imp->set(a, n);
    }

    void manager::swap(numeral & a, numeral & b) {
        std::swap(a.m_value, b.m_value);
    }

    void manager::root(numeral const & a, unsigned k, numeral & b) {
        save_interval_ctx ctx(this);
        m_imp->root(a, k, b);
    }

    void manager::power(numeral const & a, unsigned k, numeral & b) {
        save_interval_ctx ctx(this);
        m_imp->power(a, k, b);
    }

    void manager::add(numeral const & a, numeral const & b, numeral & c) {
        save_interval_ctx ctx(this);
        m_imp->add(a, b, c);
    }

    void manager::add(numeral const & a, mpz const & b, numeral & c) {
        scoped_numeral _b(*this);
        set(_b, b);
        add(a, _b, c);
    }

    void manager::sub(numeral const & a, numeral const & b, numeral & c) {
        save_interval_ctx ctx(this);
        m_imp->sub(a, b, c);
    }

    void manager::mul(numeral const & a, numeral const & b, numeral & c) {
        save_interval_ctx ctx(this);
        m_imp->mul(a, b, c);
    }

    void manager::neg(numeral & a) {
        save_interval_ctx ctx(this);
        m_imp->neg(a);
    }

    void manager::neg(numeral const & a, numeral & b) {
        save_interval_ctx ctx(this);
        m_imp->neg(a, b);
    }

    void manager::inv(numeral & a) {
        save_interval_ctx ctx(this);
        m_imp->inv(a);
    }

    void manager::inv(numeral const & a, numeral & b) {
        save_interval_ctx ctx(this);
        m_imp->inv(a, b);
    }

    void manager::div(numeral const & a, numeral const & b, numeral & c) {
        save_interval_ctx ctx(this);
        m_imp->div(a, b, c);
    }

    int manager::compare(numeral const & a, numeral const & b) {
        save_interval_ctx ctx(this);
        return m_imp->compare(a, b);
    }

    bool manager::eq(numeral const & a, numeral const & b) {
        return compare(a, b) == 0;
    }

    bool manager::eq(numeral const & a, mpq const & b) {
        scoped_numeral _b(*this);
        set(_b, b);
        return eq(a, _b);
    }

    bool manager::eq(numeral const & a, mpz const & b) {
        scoped_numeral _b(*this);
        set(_b, b);
        return eq(a, _b);
    }

    bool manager::lt(numeral const & a, numeral const & b) {
        return compare(a, b) < 0;
    }

    bool manager::lt(numeral const & a, mpq const & b) {
        scoped_numeral _b(*this);
        set(_b, b);
        return lt(a, _b);
    }

    bool manager::lt(numeral const & a, mpz const & b) {
        scoped_numeral _b(*this);
        set(_b, b);
        return lt(a, _b);
    }

    bool manager::gt(numeral const & a, mpq const & b) {
        scoped_numeral _b(*this);
        set(_b, b);
        return gt(a, _b);
    }

    bool manager::gt(numeral const & a, mpz const & b) {
        scoped_numeral _b(*this);
        set(_b, b);
        return gt(a, _b);
    }

    void manager::display(std::ostream & out, numeral const & a, bool compact, bool pp) const {
        save_interval_ctx ctx(this);
        m_imp->display(out, a, compact, pp);
    }

    void manager::display_decimal(std::ostream & out, numeral const & a, unsigned precision) const {
        save_interval_ctx ctx(this);
        m_imp->display_decimal(out, a, precision);
    }

    void manager::display_interval(std::ostream & out, numeral const & a) const {
        save_interval_ctx ctx(this);
        m_imp->display_interval(out, a);
    }

    void manager::clean_denominators(numeral const & a, numeral & p, numeral & q) {
        save_interval_ctx ctx(this);
        m_imp->clean_denominators(a, p, q);
    }
};

void pp(realclosure::manager::imp * imp, realclosure::polynomial const & p, realclosure::extension * ext) {
    imp->display_polynomial_expr(std::cout, p, ext, false, false);
    std::cout << std::endl;
}

void pp(realclosure::manager::imp * imp, realclosure::value * v) {
    imp->display(std::cout, v, false);
    std::cout << std::endl;
}

void pp(realclosure::manager::imp * imp, unsigned sz, realclosure::value * const * p) {
    for (unsigned i = 0; i < sz; i++)
        pp(imp, p[i]);
}

void pp(realclosure::manager::imp * imp, realclosure::manager::imp::value_ref_buffer const & p) {
    for (unsigned i = 0; i < p.size(); i++)
        pp(imp, p[i]);
}

void pp(realclosure::manager::imp * imp, realclosure::manager::imp::value_ref const & v) {
    pp(imp, v.get());
}

void pp(realclosure::manager::imp * imp, realclosure::mpbqi const & i) {
    imp->bqim().display(std::cout, i);
    std::cout << std::endl;
}

void pp(realclosure::manager::imp * imp, realclosure::manager::imp::scoped_mpqi const & i) {
    imp->qim().display(std::cout, i);
    std::cout << std::endl;
}

void pp(realclosure::manager::imp * imp, mpbq const & n) {
    imp->bqm().display(std::cout, n);
    std::cout << std::endl;
}

void pp(realclosure::manager::imp * imp, mpq const & n) {
    imp->qm().display(std::cout, n);
    std::cout << std::endl;
}

void pp(realclosure::manager::imp * imp, realclosure::extension * x) {
    imp->display_ext(std::cout, x, false, false);
    std::cout << std::endl;
}
