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fix #2615

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Signed-off-by: Nikolaj Bjorner <nbjorner@microsoft.com>
This commit is contained in:
Nikolaj Bjorner 2019-10-06 19:00:14 -07:00
parent f9b6e4e247
commit 7c10fb83a0
4 changed files with 90 additions and 76 deletions
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87
src/math/polynomial/algebraic_numbers.cpp
View file

@ -16,16 +16,16 @@ Author:
Notes: Notes:
--*/ --*/
#include "math/polynomial/algebraic_numbers.h"
#include "math/polynomial/upolynomial.h"
#include "util/mpbq.h" #include "util/mpbq.h"
#include "util/basic_interval.h" #include "util/basic_interval.h"
#include "math/polynomial/sexpr2upolynomial.h"
#include "util/scoped_ptr_vector.h" #include "util/scoped_ptr_vector.h"
#include "util/mpbqi.h" #include "util/mpbqi.h"
#include "util/timeit.h" #include "util/timeit.h"
#include "math/polynomial/algebraic_params.hpp"
#include "util/common_msgs.h" #include "util/common_msgs.h"
#include "math/polynomial/algebraic_numbers.h"
#include "math/polynomial/upolynomial.h"
#include "math/polynomial/sexpr2upolynomial.h"
#include "math/polynomial/algebraic_params.hpp"
namespace algebraic_numbers { namespace algebraic_numbers {
@ -124,6 +124,10 @@ namespace algebraic_numbers {
~imp() { ~imp() {
} }
bool acell_inv(algebraic_cell const& c) {
return c.m_sign_lower == (upm().eval_sign_at(c.m_p_sz, c.m_p, lower(&c)) == polynomial::sign_neg);
}
void checkpoint() { void checkpoint() {
if (!m_limit.inc()) if (!m_limit.inc())
throw algebraic_exception(Z3_CANCELED_MSG); throw algebraic_exception(Z3_CANCELED_MSG);
@ -366,17 +370,17 @@ namespace algebraic_numbers {
return c; return c;
} }
int sign_lower(algebraic_cell * c) { polynomial::sign sign_lower(algebraic_cell * c) const {
return c->m_sign_lower == 0 ? 1 : -1; return c->m_sign_lower == 0 ? polynomial::sign_pos : polynomial::sign_neg;
} }
mpbq & lower(algebraic_cell * c) { mpbq const & lower(algebraic_cell const * c) const { return c->m_interval.lower(); }
return c->m_interval.lower();
}
mpbq & upper(algebraic_cell * c) { mpbq const & upper(algebraic_cell const * c) const { return c->m_interval.upper(); }
return c->m_interval.upper();
} mpbq & lower(algebraic_cell * c) { return c->m_interval.lower(); }
mpbq & upper(algebraic_cell * c) { return c->m_interval.upper(); }
void update_sign_lower(algebraic_cell * c) { void update_sign_lower(algebraic_cell * c) {
polynomial::sign sl = upm().eval_sign_at(c->m_p_sz, c->m_p, lower(c)); polynomial::sign sl = upm().eval_sign_at(c->m_p_sz, c->m_p, lower(c));
@ -384,6 +388,7 @@ namespace algebraic_numbers {
SASSERT(sl != polynomial::sign_zero); SASSERT(sl != polynomial::sign_zero);
SASSERT(upm().eval_sign_at(c->m_p_sz, c->m_p, upper(c)) == -sl); SASSERT(upm().eval_sign_at(c->m_p_sz, c->m_p, upper(c)) == -sl);
c->m_sign_lower = sl == polynomial::sign_neg; c->m_sign_lower = sl == polynomial::sign_neg;
SASSERT(acell_inv(*c));
} }
// Make sure the GCD of the coefficients is one and the leading coefficient is positive // Make sure the GCD of the coefficients is one and the leading coefficient is positive
@ -393,6 +398,7 @@ namespace algebraic_numbers {
if (upm().m().is_neg(c->m_p[c->m_p_sz-1])) { if (upm().m().is_neg(c->m_p[c->m_p_sz-1])) {
upm().neg(c->m_p_sz, c->m_p); upm().neg(c->m_p_sz, c->m_p);
c->m_sign_lower = !(c->m_sign_lower); c->m_sign_lower = !(c->m_sign_lower);
SASSERT(acell_inv(*c));
} }
} }
@ -469,6 +475,8 @@ namespace algebraic_numbers {
target->m_sign_lower = source->m_sign_lower; target->m_sign_lower = source->m_sign_lower;
target->m_not_rational = source->m_not_rational; target->m_not_rational = source->m_not_rational;
target->m_i = source->m_i; target->m_i = source->m_i;
SASSERT(acell_inv(*source));
SASSERT(acell_inv(*target));
} }
void set(numeral & a, unsigned sz, mpz const * p, mpbq const & lower, mpbq const & upper, bool minimal) { void set(numeral & a, unsigned sz, mpz const * p, mpbq const & lower, mpbq const & upper, bool minimal) {
@ -499,7 +507,7 @@ namespace algebraic_numbers {
update_sign_lower(c); update_sign_lower(c);
normalize_coeffs(c); normalize_coeffs(c);
} }
SASSERT(sign_lower(a.to_algebraic()) == upm().eval_sign_at(a.to_algebraic()->m_p_sz, a.to_algebraic()->m_p, a.to_algebraic()->m_interval.lower())); SASSERT(acell_inv(*a.to_algebraic()));
} }
TRACE("algebraic", tout << "a: "; display_root(tout, a); tout << "\n";); TRACE("algebraic", tout << "a: "; display_root(tout, a); tout << "\n";);
} }
@ -519,6 +527,7 @@ namespace algebraic_numbers {
algebraic_cell * c = new (mem) algebraic_cell(); algebraic_cell * c = new (mem) algebraic_cell();
a.m_cell = TAG(void *, c, ROOT); a.m_cell = TAG(void *, c, ROOT);
copy(c, b.to_algebraic()); copy(c, b.to_algebraic());
SASSERT(acell_inv(*c));
} }
} }
else { else {
@ -532,6 +541,7 @@ namespace algebraic_numbers {
del_poly(a.to_algebraic()); del_poly(a.to_algebraic());
del_interval(a.to_algebraic()); del_interval(a.to_algebraic());
copy(a.to_algebraic(), b.to_algebraic()); copy(a.to_algebraic(), b.to_algebraic());
SASSERT(acell_inv(*a.to_algebraic()));
} }
} }
} }
@ -693,6 +703,7 @@ namespace algebraic_numbers {
algebraic_cell * c = a.to_algebraic(); algebraic_cell * c = a.to_algebraic();
if (!upm().normalize_interval_core(c->m_p_sz, c->m_p, sign_lower(c), bqm(), lower(c), upper(c))) if (!upm().normalize_interval_core(c->m_p_sz, c->m_p, sign_lower(c), bqm(), lower(c), upper(c)))
reset(a); reset(a);
SASSERT(acell_inv(*c));
} }
} }
@ -727,7 +738,9 @@ namespace algebraic_numbers {
Return FALSE, if actual root was found. Return FALSE, if actual root was found.
*/ */
bool refine_core(algebraic_cell * c) { bool refine_core(algebraic_cell * c) {
return upm().refine_core(c->m_p_sz, c->m_p, sign_lower(c), bqm(), lower(c), upper(c)); bool r = upm().refine_core(c->m_p_sz, c->m_p, sign_lower(c), bqm(), lower(c), upper(c));
SASSERT(acell_inv(*c));
return r;
} }
/** /**
@ -746,7 +759,10 @@ namespace algebraic_numbers {
if (a.is_basic()) if (a.is_basic())
return false; return false;
algebraic_cell * c = a.to_algebraic(); algebraic_cell * c = a.to_algebraic();
if (!refine_core(c)) { if (refine_core(c)) {
return true;
}
else {
// root was found // root was found
scoped_mpq r(qm()); scoped_mpq r(qm());
to_mpq(qm(), lower(c), r); to_mpq(qm(), lower(c), r);
@ -754,7 +770,6 @@ namespace algebraic_numbers {
a.m_cell = mk_basic_cell(r); a.m_cell = mk_basic_cell(r);
return false; return false;
} }
return true;
} }
bool refine(numeral & a, unsigned k) { bool refine(numeral & a, unsigned k) {
@ -776,6 +791,7 @@ namespace algebraic_numbers {
a.m_cell = mk_basic_cell(r); a.m_cell = mk_basic_cell(r);
return false; return false;
} }
SASSERT(acell_inv(*c));
return true; return true;
} }
@ -1573,6 +1589,7 @@ namespace algebraic_numbers {
upm().p_minus_x(c->m_p_sz, c->m_p); upm().p_minus_x(c->m_p_sz, c->m_p);
bqim().neg(c->m_interval); bqim().neg(c->m_interval);
update_sign_lower(c); update_sign_lower(c);
SASSERT(acell_inv(*c));
} }
} }
@ -1583,13 +1600,14 @@ namespace algebraic_numbers {
if (a.is_basic()) if (a.is_basic())
return; return;
algebraic_cell * cell_a = a.to_algebraic(); algebraic_cell * cell_a = a.to_algebraic();
mpbq & lower = cell_a->m_interval.lower(); SASSERT(acell_inv(*cell_a));
mpbq & upper = cell_a->m_interval.upper(); mpbq & _lower = cell_a->m_interval.lower();
if (!bqm().is_zero(lower) && !bqm().is_zero(upper)) mpbq & _upper = cell_a->m_interval.upper();
if (!bqm().is_zero(_lower) && !bqm().is_zero(_upper))
return; return;
int sign_l = sign_lower(cell_a); auto sign_l = sign_lower(cell_a);
SASSERT(sign_l != 0); SASSERT(!polynomial::is_zero(sign_l));
int sign_u = -sign_l; auto sign_u = -sign_l;
#define REFINE_LOOP(BOUND, TARGET_SIGN) \ #define REFINE_LOOP(BOUND, TARGET_SIGN) \
while (true) { \ while (true) { \
@ -1600,21 +1618,23 @@ namespace algebraic_numbers {
scoped_mpq r(qm()); \ scoped_mpq r(qm()); \
to_mpq(qm(), BOUND, r); \ to_mpq(qm(), BOUND, r); \
set(a, r); \ set(a, r); \
return; \ break; \
} \
if (new_sign == TARGET_SIGN) { \
break; \
} \ } \
if (new_sign == TARGET_SIGN) \
return; \
} }
if (bqm().is_zero(lower)) { if (bqm().is_zero(_lower)) {
bqm().set(lower, upper); bqm().set(_lower, _upper);
REFINE_LOOP(lower, sign_l); REFINE_LOOP(_lower, sign_l);
} }
else { else {
SASSERT(bqm().is_zero(upper)); SASSERT(bqm().is_zero(_upper));
bqm().set(upper, lower); bqm().set(_upper, _lower);
REFINE_LOOP(upper, sign_u); REFINE_LOOP(_upper, sign_u);
} }
SASSERT(acell_inv(*cell_a));
} }
void inv(numeral & a) { void inv(numeral & a) {
@ -1642,6 +1662,8 @@ namespace algebraic_numbers {
// convert isolating interval back as a binary rational bound // convert isolating interval back as a binary rational bound
upm().convert_q2bq_interval(cell_a->m_p_sz, cell_a->m_p, inv_lower, inv_upper, bqm(), lower(cell_a), upper(cell_a)); upm().convert_q2bq_interval(cell_a->m_p_sz, cell_a->m_p, inv_lower, inv_upper, bqm(), lower(cell_a), upper(cell_a));
TRACE("algebraic_bug", tout << "after inv: "; display_root(tout, a); tout << "\n"; display_interval(tout, a); tout << "\n";); TRACE("algebraic_bug", tout << "after inv: "; display_root(tout, a); tout << "\n"; display_interval(tout, a); tout << "\n";);
update_sign_lower(cell_a);
SASSERT(acell_inv(*cell_a));
} }
} }
@ -2118,7 +2140,6 @@ namespace algebraic_numbers {
// compute the resultants // compute the resultants
polynomial_ref q_i(pm()); polynomial_ref q_i(pm());
std::stable_sort(xs.begin(), xs.end(), var_degree_lt(*this, x2v)); std::stable_sort(xs.begin(), xs.end(), var_degree_lt(*this, x2v));
// std::cout << "R: " << R << "\n";
for (unsigned i = 0; i < xs.size(); i++) { for (unsigned i = 0; i < xs.size(); i++) {
checkpoint(); checkpoint();
polynomial::var x_i = xs[i]; polynomial::var x_i = xs[i];
@ -2127,7 +2148,6 @@ namespace algebraic_numbers {
SASSERT(!v_i.is_basic()); SASSERT(!v_i.is_basic());
algebraic_cell * c = v_i.to_algebraic(); algebraic_cell * c = v_i.to_algebraic();
q_i = pm().to_polynomial(c->m_p_sz, c->m_p, x_i); q_i = pm().to_polynomial(c->m_p_sz, c->m_p, x_i);
// std::cout << "q_i: " << q_i << std::endl;
pm().resultant(R, q_i, x_i, R); pm().resultant(R, q_i, x_i, R);
SASSERT(!pm().is_zero(R)); SASSERT(!pm().is_zero(R));
} }
@ -2136,7 +2156,6 @@ namespace algebraic_numbers {
upm().to_numeral_vector(R, _R); upm().to_numeral_vector(R, _R);
unsigned k = upm().nonzero_root_lower_bound(_R.size(), _R.c_ptr()); unsigned k = upm().nonzero_root_lower_bound(_R.size(), _R.c_ptr());
TRACE("anum_eval_sign", tout << "R: " << R << "\nk: " << k << "\nri: "<< ri << "\n";); TRACE("anum_eval_sign", tout << "R: " << R << "\nk: " << k << "\nri: "<< ri << "\n";);
// std::cout << "R: " << R << "\n";
scoped_mpbq mL(bqm()), L(bqm()); scoped_mpbq mL(bqm()), L(bqm());
bqm().set(mL, -1); bqm().set(mL, -1);
bqm().set(L, 1); bqm().set(L, 1);

1
src/math/polynomial/polynomial.h
View file

@ -48,6 +48,7 @@ namespace polynomial {
inline sign operator-(sign s) { switch (s) { case sign_neg: return sign_pos; case sign_pos: return sign_neg; default: return sign_zero; } }; inline sign operator-(sign s) { switch (s) { case sign_neg: return sign_pos; case sign_pos: return sign_neg; default: return sign_zero; } };
inline sign to_sign(int s) { return s == 0 ? sign_zero : (s > 0 ? sign_pos : sign_neg); } inline sign to_sign(int s) { return s == 0 ? sign_zero : (s > 0 ? sign_pos : sign_neg); }
inline sign operator*(sign a, sign b) { return to_sign((int)a * (int)b); } inline sign operator*(sign a, sign b) { return to_sign((int)a * (int)b); }
inline bool is_zero(sign s) { return s == sign_zero; }
int lex_compare(monomial const * m1, monomial const * m2); int lex_compare(monomial const * m1, monomial const * m2);
int lex_compare2(monomial const * m1, monomial const * m2, var min_var); int lex_compare2(monomial const * m1, monomial const * m2, var min_var);

56
src/math/polynomial/upolynomial.cpp
View file

@ -1375,7 +1375,7 @@ namespace upolynomial {
} }
return sign_changes(Q.size(), Q.c_ptr()); return sign_changes(Q.size(), Q.c_ptr());
#endif #endif
int prev_sign = 0; polynomial::sign prev_sign = polynomial::sign_zero;
unsigned num_vars = 0; unsigned num_vars = 0;
// a0 a1 a2 a3 // a0 a1 a2 a3
// a0 a0+a1 a0+a1+a2 a0+a1+a2+a3 // a0 a0+a1 a0+a1+a2 a0+a1+a2+a3
@ -1388,10 +1388,10 @@ namespace upolynomial {
for (k = 1; k < sz - i; k++) { for (k = 1; k < sz - i; k++) {
m().add(Q[k], Q[k-1], Q[k]); m().add(Q[k], Q[k-1], Q[k]);
} }
int sign = sign_of(Q[k-1]); auto sign = sign_of(Q[k-1]);
if (sign == 0) if (polynomial::is_zero(sign))
continue; continue;
if (sign != prev_sign && prev_sign != 0) { if (sign != prev_sign && !polynomial::is_zero(prev_sign)) {
num_vars++; num_vars++;
if (num_vars > 1) if (num_vars > 1)
return num_vars; return num_vars;
@ -2342,7 +2342,6 @@ namespace upolynomial {
#else #else
scoped_numeral U(m()); scoped_numeral U(m());
root_upper_bound(p1.size(), p1.c_ptr(), U); root_upper_bound(p1.size(), p1.c_ptr(), U);
std::cout << "U: " << U << "\n";
unsigned pos_k = m().log2(U) + 1; unsigned pos_k = m().log2(U) + 1;
unsigned neg_k = pos_k; unsigned neg_k = pos_k;
#endif #endif
@ -2752,18 +2751,15 @@ namespace upolynomial {
The arguments sign_a and sign_b must contain the values returned by The arguments sign_a and sign_b must contain the values returned by
eval_sign_at(sz, p, a) and eval_sign_at(sz, p, b). eval_sign_at(sz, p, a) and eval_sign_at(sz, p, b).
*/ */
bool manager::refine_core(unsigned sz, numeral const * p, int sign_a, mpbq_manager & bqm, mpbq & a, mpbq & b) { bool manager::refine_core(unsigned sz, numeral const * p, polynomial::sign sign_a, mpbq_manager & bqm, mpbq & a, mpbq & b) {
SASSERT(sign_a == eval_sign_at(sz, p, a)); SASSERT(sign_a == eval_sign_at(sz, p, a));
int sign_b = -sign_a; SASSERT(-sign_a == eval_sign_at(sz, p, b));
(void)sign_b; SASSERT(sign_a != 0);
SASSERT(sign_b == eval_sign_at(sz, p, b));
SASSERT(sign_a == -sign_b);
SASSERT(sign_a != 0 && sign_b != 0);
scoped_mpbq mid(bqm); scoped_mpbq mid(bqm);
bqm.add(a, b, mid); bqm.add(a, b, mid);
bqm.div2(mid); bqm.div2(mid);
int sign_mid = eval_sign_at(sz, p, mid); auto sign_mid = eval_sign_at(sz, p, mid);
if (sign_mid == 0) { if (polynomial::is_zero(sign_mid)) {
swap(mid, a); swap(mid, a);
return false; return false;
} }
@ -2771,15 +2767,15 @@ namespace upolynomial {
swap(mid, a); swap(mid, a);
return true; return true;
} }
SASSERT(sign_mid == sign_b); SASSERT(sign_mid == -sign_a);
swap(mid, b); swap(mid, b);
return true; return true;
} }
// See refine_core // See refine_core
bool manager::refine(unsigned sz, numeral const * p, mpbq_manager & bqm, mpbq & a, mpbq & b) { bool manager::refine(unsigned sz, numeral const * p, mpbq_manager & bqm, mpbq & a, mpbq & b) {
int sign_a = eval_sign_at(sz, p, a); polynomial::sign sign_a = eval_sign_at(sz, p, a);
SASSERT(sign_a != 0); SASSERT(!polynomial::is_zero(sign_a));
return refine_core(sz, p, sign_a, bqm, a, b); return refine_core(sz, p, sign_a, bqm, a, b);
} }
@ -2788,8 +2784,8 @@ namespace upolynomial {
// //
// Return TRUE, if interval was squeezed, and new interval is stored in (a,b). // Return TRUE, if interval was squeezed, and new interval is stored in (a,b).
// Return FALSE, if the actual root was found, it is stored in a. // Return FALSE, if the actual root was found, it is stored in a.
bool manager::refine_core(unsigned sz, numeral const * p, int sign_a, mpbq_manager & bqm, mpbq & a, mpbq & b, unsigned prec_k) { bool manager::refine_core(unsigned sz, numeral const * p, polynomial::sign sign_a, mpbq_manager & bqm, mpbq & a, mpbq & b, unsigned prec_k) {
SASSERT(sign_a != 0); SASSERT(sign_a != polynomial::sign_zero);
SASSERT(sign_a == eval_sign_at(sz, p, a)); SASSERT(sign_a == eval_sign_at(sz, p, a));
SASSERT(-sign_a == eval_sign_at(sz, p, b)); SASSERT(-sign_a == eval_sign_at(sz, p, b));
scoped_mpbq w(bqm); scoped_mpbq w(bqm);
@ -2806,16 +2802,16 @@ namespace upolynomial {
} }
bool manager::refine(unsigned sz, numeral const * p, mpbq_manager & bqm, mpbq & a, mpbq & b, unsigned prec_k) { bool manager::refine(unsigned sz, numeral const * p, mpbq_manager & bqm, mpbq & a, mpbq & b, unsigned prec_k) {
int sign_a = eval_sign_at(sz, p, a); polynomial::sign sign_a = eval_sign_at(sz, p, a);
SASSERT(eval_sign_at(sz, p, b) == -sign_a); SASSERT(eval_sign_at(sz, p, b) == -sign_a);
SASSERT(sign_a != 0); SASSERT(sign_a != 0);
return refine_core(sz, p, sign_a, bqm, a, b, prec_k); return refine_core(sz, p, sign_a, bqm, a, b, prec_k);
} }
bool manager::convert_q2bq_interval(unsigned sz, numeral const * p, mpq const & a, mpq const & b, mpbq_manager & bqm, mpbq & c, mpbq & d) { bool manager::convert_q2bq_interval(unsigned sz, numeral const * p, mpq const & a, mpq const & b, mpbq_manager & bqm, mpbq & c, mpbq & d) {
int sign_a = eval_sign_at(sz, p, a); polynomial::sign sign_a = eval_sign_at(sz, p, a);
int sign_b = eval_sign_at(sz, p, b); polynomial::sign sign_b = eval_sign_at(sz, p, b);
SASSERT(sign_a != 0 && sign_b != 0); SASSERT(!polynomial::is_zero(sign_a) && !polynomial::is_zero(sign_b));
SASSERT(sign_a == -sign_b); SASSERT(sign_a == -sign_b);
bool found_d = false; bool found_d = false;
TRACE("convert_bug", TRACE("convert_bug",
@ -2846,8 +2842,8 @@ namespace upolynomial {
} }
SASSERT(bqm.lt(upper, b)); SASSERT(bqm.lt(upper, b));
while (true) { while (true) {
int sign_upper = eval_sign_at(sz, p, upper); auto sign_upper = eval_sign_at(sz, p, upper);
if (sign_upper == 0) { if (polynomial::is_zero(sign_upper)) {
// found root // found root
bqm.swap(c, upper); bqm.swap(c, upper);
bqm.del(lower); bqm.del(upper); bqm.del(lower); bqm.del(upper);
@ -2891,8 +2887,8 @@ namespace upolynomial {
SASSERT(bqm.lt(lower, upper)); SASSERT(bqm.lt(lower, upper));
SASSERT(bqm.lt(lower, b)); SASSERT(bqm.lt(lower, b));
while (true) { while (true) {
int sign_lower = eval_sign_at(sz, p, lower); polynomial::sign sign_lower = eval_sign_at(sz, p, lower);
if (sign_lower == 0) { if (polynomial::is_zero(sign_lower)) {
// found root // found root
bqm.swap(c, lower); bqm.swap(c, lower);
bqm.del(lower); bqm.del(upper); bqm.del(lower); bqm.del(upper);
@ -2923,14 +2919,12 @@ namespace upolynomial {
else { else {
SASSERT(sign_a == eval_sign_at(sz, p, a)); SASSERT(sign_a == eval_sign_at(sz, p, a));
} }
int sign_b = -sign_a; SASSERT(-sign_a == eval_sign_at(sz, p, b));
(void) sign_b; SASSERT(sign_a != 0);
SASSERT(sign_b == eval_sign_at(sz, p, b));
SASSERT(sign_a != 0 && sign_b != 0);
if (has_zero_roots(sz, p)) { if (has_zero_roots(sz, p)) {
return false; // zero is the root return false; // zero is the root
} }
int sign_zero = eval_sign_at_zero(sz, p); auto sign_zero = eval_sign_at_zero(sz, p);
if (sign_a == sign_zero) { if (sign_a == sign_zero) {
m.reset(a); m.reset(a);
} }

4
src/math/polynomial/upolynomial.h
View file

@ -863,11 +863,11 @@ namespace upolynomial {
// Return FALSE, if the actual root was found, it is stored in a. // Return FALSE, if the actual root was found, it is stored in a.
// //
// See upolynomial.cpp for additional comments // See upolynomial.cpp for additional comments
bool refine_core(unsigned sz, numeral const * p, int sign_a, mpbq_manager & bqm, mpbq & a, mpbq & b); bool refine_core(unsigned sz, numeral const * p, polynomial::sign sign_a, mpbq_manager & bqm, mpbq & a, mpbq & b);
bool refine(unsigned sz, numeral const * p, mpbq_manager & bqm, mpbq & a, mpbq & b); bool refine(unsigned sz, numeral const * p, mpbq_manager & bqm, mpbq & a, mpbq & b);
bool refine_core(unsigned sz, numeral const * p, int sign_a, mpbq_manager & bqm, mpbq & a, mpbq & b, unsigned prec_k); bool refine_core(unsigned sz, numeral const * p, polynomial::sign sign_a, mpbq_manager & bqm, mpbq & a, mpbq & b, unsigned prec_k);
bool refine(unsigned sz, numeral const * p, mpbq_manager & bqm, mpbq & a, mpbq & b, unsigned prec_k); bool refine(unsigned sz, numeral const * p, mpbq_manager & bqm, mpbq & a, mpbq & b, unsigned prec_k);
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