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Signed-off-by: Lev Nachmanson <levnach@hotmail.com>
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Lev Nachmanson
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7cfafc133f
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5e1c104667
6 changed files with 601 additions and 367 deletions
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@ -1164,7 +1164,288 @@ static void tst_unsound_lws2380() {
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run_test(true); // Levelwise projection
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}
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// Test case for unsound lemma - levelwise produces cell that's too large
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// Input: 5 polynomials with max_var=x3, sample x0=-7, x1=-1, x2=1
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// Counterexample: x0=-4, x1=-8, x2=5, x3=6
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static void tst_unsound_lws_x3() {
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std::cout << "=== tst_unsound_lws_x3 ===\n";
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params_ref ps;
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ps.set_bool("lws", true);
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reslimit rlim;
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nlsat::solver s(rlim, ps, false);
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anum_manager & am = s.am();
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nlsat::pmanager & pm = s.pm();
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nlsat::assignment sample_as(am);
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nlsat::assignment counter_as(am);
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polynomial::cache cache(pm);
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// Create 4 variables x0, x1, x2, x3
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nlsat::var x0 = s.mk_var(false);
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nlsat::var x1 = s.mk_var(false);
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nlsat::var x2 = s.mk_var(false);
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nlsat::var x3 = s.mk_var(false);
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polynomial_ref _x0(pm), _x1(pm), _x2(pm), _x3(pm);
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_x0 = pm.mk_polynomial(x0);
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_x1 = pm.mk_polynomial(x1);
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_x2 = pm.mk_polynomial(x2);
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_x3 = pm.mk_polynomial(x3);
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// p[0]: x3 + x0
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polynomial_ref p0(pm);
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p0 = _x3 + _x0;
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// p[1]: x1
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polynomial_ref p1(pm);
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p1 = _x1;
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// p[2]: 9 x1^2 x3^2 - 6 x0 x1 x2 x3 + 3 x0 x1^2 x3 - 12 x1^2 x3 + x0^2 x2^2 - x0^2 x1 x2 - 4 x0 x1 x2 + 2 x0 x1^2 + 4 x1^2
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polynomial_ref p2(pm);
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p2 = 9 * (_x1^2) * (_x3^2)
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- 6 * _x0 * _x1 * _x2 * _x3
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+ 3 * _x0 * (_x1^2) * _x3
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- 12 * (_x1^2) * _x3
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+ (_x0^2) * (_x2^2)
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- (_x0^2) * _x1 * _x2
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- 4 * _x0 * _x1 * _x2
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+ 2 * _x0 * (_x1^2)
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+ 4 * (_x1^2);
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// p[3]: 9 x1^2 x3^2 - 6 x0 x1 x2 x3 + 6 x0 x1^2 x3 - 12 x1^2 x3 + x0^2 x2^2 - 2 x0^2 x1 x2 - 4 x0 x1 x2 + x0^2 x1^2 + 4 x0 x1^2 + 4 x1^2
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polynomial_ref p3(pm);
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p3 = 9 * (_x1^2) * (_x3^2)
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- 6 * _x0 * _x1 * _x2 * _x3
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+ 6 * _x0 * (_x1^2) * _x3
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- 12 * (_x1^2) * _x3
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+ (_x0^2) * (_x2^2)
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- 2 * (_x0^2) * _x1 * _x2
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- 4 * _x0 * _x1 * _x2
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+ (_x0^2) * (_x1^2)
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+ 4 * _x0 * (_x1^2)
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+ 4 * (_x1^2);
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// p[4]: x3 + x1 + x0
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polynomial_ref p4(pm);
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p4 = _x3 + _x1 + _x0;
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std::cout << "p0: " << p0 << "\n";
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std::cout << "p1: " << p1 << "\n";
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std::cout << "p2: " << p2 << "\n";
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std::cout << "p3: " << p3 << "\n";
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std::cout << "p4: " << p4 << "\n\n";
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// Set sample: x0=-7, x1=-1, x2=1, x3=? (need to pick a value in the cell)
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// For the sample, we need an x3 value. Let's use x3=8 (which is > -x0 = 7, so p0 > 0)
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scoped_anum val(am);
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am.set(val, -7); sample_as.set(x0, val);
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am.set(val, -1); sample_as.set(x1, val);
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am.set(val, 1); sample_as.set(x2, val);
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am.set(val, 8); sample_as.set(x3, val);
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// Counterexample: x0=-4, x1=-8, x2=5, x3=6
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am.set(val, -4); counter_as.set(x0, val);
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am.set(val, -8); counter_as.set(x1, val);
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am.set(val, 5); counter_as.set(x2, val);
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am.set(val, 6); counter_as.set(x3, val);
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std::cout << "Sample point: x0=-7, x1=-1, x2=1, x3=8\n";
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std::cout << "Counterexample: x0=-4, x1=-8, x2=5, x3=6\n\n";
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// Evaluate polynomials at sample
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std::cout << "Polynomial signs at SAMPLE:\n";
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std::cout << " p0 sign: " << am.eval_sign_at(p0, sample_as) << "\n";
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std::cout << " p1 sign: " << am.eval_sign_at(p1, sample_as) << "\n";
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std::cout << " p2 sign: " << am.eval_sign_at(p2, sample_as) << "\n";
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std::cout << " p3 sign: " << am.eval_sign_at(p3, sample_as) << "\n";
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std::cout << " p4 sign: " << am.eval_sign_at(p4, sample_as) << "\n\n";
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// Evaluate polynomials at counterexample
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std::cout << "Polynomial signs at COUNTEREXAMPLE:\n";
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std::cout << " p0 sign: " << am.eval_sign_at(p0, counter_as) << "\n";
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std::cout << " p1 sign: " << am.eval_sign_at(p1, counter_as) << "\n";
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std::cout << " p2 sign: " << am.eval_sign_at(p2, counter_as) << "\n";
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std::cout << " p3 sign: " << am.eval_sign_at(p3, counter_as) << "\n";
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std::cout << " p4 sign: " << am.eval_sign_at(p4, counter_as) << "\n\n";
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// Set solver assignment for levelwise (without x3)
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am.set(val, -7); sample_as.set(x0, val);
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am.set(val, -1); sample_as.set(x1, val);
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am.set(val, 1); sample_as.set(x2, val);
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s.set_rvalues(sample_as);
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// Build polynomial vector
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polynomial_ref_vector polys(pm);
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polys.push_back(p0);
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polys.push_back(p1);
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polys.push_back(p2);
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polys.push_back(p3);
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polys.push_back(p4);
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unsigned max_x = x3;
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// Print roots of each polynomial at sample
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std::cout << "Roots of polynomials at sample (in x3):\n";
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for (unsigned i = 0; i < polys.size(); ++i) {
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polynomial_ref p(polys.get(i), pm);
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if (pm.max_var(p) != x3) {
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std::cout << " p" << i << ": max_var is not x3, skipping\n";
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continue;
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}
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scoped_anum_vector roots(am);
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am.isolate_roots(p, nlsat::undef_var_assignment(sample_as, x3), roots);
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std::cout << " p" << i << " roots: ";
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if (roots.empty()) {
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std::cout << "(none)";
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} else {
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for (unsigned j = 0; j < roots.size(); ++j) {
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if (j > 0) std::cout << ", ";
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am.display_decimal(std::cout, roots[j], 5);
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}
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}
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std::cout << "\n";
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}
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std::cout << "\n";
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// Compute and evaluate resultant of p3 and p4
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std::cout << "Resultant of p3 and p4 (in x3):\n";
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polynomial_ref res_p3_p4(pm);
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{
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pm.resultant(p3, p4, x3, res_p3_p4);
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std::cout << " Res(p3, p4) = ";
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pm.display(std::cout, res_p3_p4);
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std::cout << "\n";
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std::cout << " Sign at sample (x0=-7, x1=-1, x2=1): " << am.eval_sign_at(res_p3_p4, sample_as) << "\n";
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std::cout << " Sign at counter (x0=-4, x1=-8, x2=5): " << am.eval_sign_at(res_p3_p4, counter_as) << "\n";
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// Check roots of the resultant at x2 level (parametric in x0, x1)
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std::cout << " Roots at sample x0,x1 (in x2): ";
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scoped_anum_vector res_roots(am);
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nlsat::assignment partial_sample(am);
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scoped_anum val(am);
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am.set(val, -7); partial_sample.set(x0, val);
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am.set(val, -1); partial_sample.set(x1, val);
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am.isolate_roots(res_p3_p4, nlsat::undef_var_assignment(partial_sample, x2), res_roots);
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for (unsigned j = 0; j < res_roots.size(); ++j) {
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if (j > 0) std::cout << ", ";
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am.display_decimal(std::cout, res_roots[j], 5);
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}
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std::cout << "\n";
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// Check roots at counterexample x0,x1
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std::cout << " Roots at counter x0,x1 (in x2): ";
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nlsat::assignment partial_counter(am);
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am.set(val, -4); partial_counter.set(x0, val);
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am.set(val, -8); partial_counter.set(x1, val);
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scoped_anum_vector res_roots_counter(am);
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am.isolate_roots(res_p3_p4, nlsat::undef_var_assignment(partial_counter, x2), res_roots_counter);
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for (unsigned j = 0; j < res_roots_counter.size(); ++j) {
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if (j > 0) std::cout << ", ";
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am.display_decimal(std::cout, res_roots_counter[j], 5);
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}
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std::cout << "\n";
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// Compute and check discriminant of Res(p3,p4) in x2
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std::cout << "\n Discriminant of Res(p3,p4) in x2:\n";
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polynomial_ref disc_res(pm);
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pm.discriminant(res_p3_p4, x2, disc_res);
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std::cout << " Disc = ";
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pm.display(std::cout, disc_res);
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std::cout << "\n";
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std::cout << " Sign at sample (x0=-7, x1=-1): " << am.eval_sign_at(disc_res, sample_as) << "\n";
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std::cout << " Sign at counter (x0=-4, x1=-8): " << am.eval_sign_at(disc_res, counter_as) << "\n";
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}
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std::cout << "\n";
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std::cout << "Running levelwise with max_x = x3\n";
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// Run levelwise
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nlsat::levelwise lws(s, polys, max_x, s.sample(), pm, am, cache);
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auto cell = lws.single_cell();
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std::cout << "Levelwise " << (lws.failed() ? "FAILED" : "succeeded") << "\n";
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std::cout << "Cell intervals (count=" << cell.size() << "):\n";
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for (auto const& interval : cell) {
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nlsat::display(std::cout << " ", s, interval) << "\n";
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}
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// Evaluate cell bounds at counterexample to check if counterexample is in cell
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std::cout << "\n--- Checking if counterexample is in cell ---\n";
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std::cout << "For a SECTOR (lower_root, upper_root), variable x satisfies:\n";
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std::cout << " x > lower_root AND x < upper_root\n\n";
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// For univariate evaluation, we need to substitute lower vars
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// Level 0: x0 interval, evaluate at x0=-4
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// Level 1: x1 interval (parametric in x0), evaluate at (x0=-4, x1=-8)
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// Level 2: x2 interval (parametric in x0,x1), evaluate at (x0=-4,x1=-8,x2=5)
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bool counterexample_outside_cell = false;
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for (unsigned i = 0; i < cell.size(); ++i) {
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auto const& interval = cell[i];
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nlsat::var level = i;
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std::cout << "Level " << level << ":\n";
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// Build assignment up to this level (exclusive) for root isolation
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nlsat::assignment partial_as(am);
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scoped_anum val(am);
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if (level > 0) { am.set(val, -4); partial_as.set(x0, val); }
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if (level > 1) { am.set(val, -8); partial_as.set(x1, val); }
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if (level > 2) { am.set(val, 5); partial_as.set(x2, val); }
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scoped_anum counter_val(am);
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if (level == 0) am.set(counter_val, -4);
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else if (level == 1) am.set(counter_val, -8);
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else if (level == 2) am.set(counter_val, 5);
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if (interval.is_section()) {
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std::cout << " Section case\n";
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} else {
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// Isolate roots and check bounds
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if (!interval.l_inf()) {
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polynomial_ref lower_p(interval.l, pm);
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scoped_anum_vector lower_roots(am);
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am.isolate_roots(lower_p, nlsat::undef_var_assignment(partial_as, level), lower_roots);
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if (lower_roots.size() >= interval.l_index) {
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std::cout << " Lower root (root[" << interval.l_index << "]): ";
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am.display_decimal(std::cout, lower_roots[interval.l_index - 1], 10);
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std::cout << "\n";
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std::cout << " Counter x" << level << " = ";
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am.display_decimal(std::cout, counter_val, 10);
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int cmp = am.compare(counter_val, lower_roots[interval.l_index - 1]);
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std::cout << " -> " << (cmp > 0 ? "ABOVE" : (cmp < 0 ? "BELOW" : "EQUAL")) << " lower bound\n";
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if (cmp <= 0) counterexample_outside_cell = true;
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}
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}
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if (!interval.u_inf()) {
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polynomial_ref upper_p(interval.u, pm);
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scoped_anum_vector upper_roots(am);
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am.isolate_roots(upper_p, nlsat::undef_var_assignment(partial_as, level), upper_roots);
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if (upper_roots.size() >= interval.u_index) {
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std::cout << " Upper root (root[" << interval.u_index << "]): ";
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am.display_decimal(std::cout, upper_roots[interval.u_index - 1], 10);
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std::cout << "\n";
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std::cout << " Counter x" << level << " = ";
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am.display_decimal(std::cout, counter_val, 10);
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int cmp = am.compare(counter_val, upper_roots[interval.u_index - 1]);
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std::cout << " -> " << (cmp > 0 ? "ABOVE" : (cmp < 0 ? "BELOW" : "EQUAL")) << " upper bound\n";
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if (cmp >= 0) counterexample_outside_cell = true;
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}
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}
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}
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std::cout << "\n";
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}
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// The counterexample has different polynomial signs than the sample.
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// For a sound cell, the counterexample must be OUTSIDE the cell.
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ENSURE(counterexample_outside_cell);
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std::cout << "SUCCESS: Counterexample is OUTSIDE the cell (cell is sound)\n";
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std::cout << "=== END tst_unsound_lws_x3 ===\n\n";
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}
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void tst_nlsat() {
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tst_unsound_lws_x3();
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std::cout << "------------------\n";
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tst_unsound_lws2380();
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std::cout << "------------------\n";
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tst_polynomial_cache_mk_unique();
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