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https://github.com/Z3Prover/z3
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Computing missing coeff for assign-bounds lemma
This commit is contained in:
Arie Gurfinkel
parent
1764bb8785
commit
8e57ab5d97
1 changed files with 218 additions and 4 deletions
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@ -73,6 +73,129 @@ namespace spacer {
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class linear_combinator {
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struct scaled_lit {
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bool is_pos;
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app *lit;
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rational coeff;
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scaled_lit(bool is_pos, app *lit, const rational &coeff) :
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is_pos(is_pos), lit(lit), coeff(coeff) {}
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};
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ast_manager &m;
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th_rewriter m_rw;
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arith_util m_arith;
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expr_ref m_sum;
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bool m_is_strict;
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rational m_lc;
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vector<scaled_lit> m_lits;
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public:
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linear_combinator(ast_manager &m) : m(m), m_rw(m), m_arith(m),
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m_sum(m), m_is_strict(false),
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m_lc(1) {}
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void add_lit(app* lit, rational const &coeff, bool is_pos = true) {
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m_lits.push_back(scaled_lit(is_pos, lit, coeff));
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}
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void normalize_coeff() {
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for (auto &lit : m_lits)
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m_lc = lcm(m_lc, denominator(lit.coeff));
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if (!m_lc.is_one()) {
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for (auto &lit : m_lits)
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lit.coeff *= m_lc;
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}
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}
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rational const &lc() const {return m_lc;}
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bool process_lit(scaled_lit &lit0) {
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arith_util a(m);
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app* lit = lit0.lit;
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rational &coeff = lit0.coeff;
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bool is_pos = lit0.is_pos;
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if (m.is_not(lit)) {
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lit = to_app(lit->get_arg(0));
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is_pos = !is_pos;
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}
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if (!m_arith.is_le(lit) && !m_arith.is_lt(lit) &&
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!m_arith.is_ge(lit) && !m_arith.is_gt(lit) && !m.is_eq(lit)) {
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return false;
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}
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SASSERT(lit->get_num_args() == 2);
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sort* s = m.get_sort(lit->get_arg(0));
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bool is_int = m_arith.is_int(s);
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if (!is_int && m_arith.is_int_expr(lit->get_arg(0))) {
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is_int = true;
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s = m_arith.mk_int();
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}
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if (!is_int && is_pos && (m_arith.is_gt(lit) || m_arith.is_lt(lit))) {
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m_is_strict = true;
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}
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if (!is_int && !is_pos && (m_arith.is_ge(lit) || m_arith.is_le(lit))) {
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m_is_strict = true;
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}
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SASSERT(m_arith.is_int(s) || m_arith.is_real(s));
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expr_ref sign1(m), sign2(m), term(m);
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sign1 = m_arith.mk_numeral(m.is_eq(lit)?coeff:abs(coeff), s);
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sign2 = m_arith.mk_numeral(m.is_eq(lit)?-coeff:-abs(coeff), s);
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if (!m_sum.get()) {
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m_sum = m_arith.mk_numeral(rational(0), s);
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}
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expr* a0 = lit->get_arg(0);
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expr* a1 = lit->get_arg(1);
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if (is_pos && (m_arith.is_ge(lit) || m_arith.is_gt(lit))) {
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std::swap(a0, a1);
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}
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if (!is_pos && (m_arith.is_le(lit) || m_arith.is_lt(lit))) {
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std::swap(a0, a1);
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}
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//
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// Multiplying by coefficients over strict
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// and non-strict inequalities:
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//
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// (a <= b) * 2
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// (a - b <= 0) * 2
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// (2a - 2b <= 0)
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// (a < b) * 2 <=>
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// (a +1 <= b) * 2 <=>
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// 2a + 2 <= 2b <=>
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// 2a+2-2b <= 0
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bool strict_ineq =
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is_pos?(m_arith.is_gt(lit) || m_arith.is_lt(lit)):(m_arith.is_ge(lit) || m_arith.is_le(lit));
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if (is_int && strict_ineq) {
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m_sum = m_arith.mk_add(m_sum, sign1);
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}
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term = m_arith.mk_mul(sign1, a0);
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m_sum = m_arith.mk_add(m_sum, term);
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term = m_arith.mk_mul(sign2, a1);
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m_sum = m_arith.mk_add(m_sum, term);
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m_rw(m_sum);
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return true;
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}
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expr_ref operator()(){
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if (!m_sum) normalize_coeff();
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m_sum.reset();
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for (auto &lit : m_lits) {
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if (!process_lit(lit)) return expr_ref(m);
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}
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return m_sum;
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}
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};
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/*
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* ====================================
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* methods for transforming proofs
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@ -101,6 +224,60 @@ namespace spacer {
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return pf;
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}
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static bool match_mul(expr *e, expr_ref &var, expr_ref &val, arith_util &a) {
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expr *e1 = nullptr, *e2 = nullptr;
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if (!a.is_mul(e, e1, e2)) {
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if (a.is_numeral(e)) return false;
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if (!var || var == e) {
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var = e;
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val = a.mk_numeral(rational(1), get_sort(e));
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return true;
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}
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return false;
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}
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if (!a.is_numeral(e1)) std::swap(e1, e2);
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if (!a.is_numeral(e1)) return false;
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// if variable is given, match it as well
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if (!var || var == e2) {
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var = e2;
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val = e1;
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return true;
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}
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return false;
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}
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static expr_ref get_coeff(expr *lit0, expr_ref &var) {
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ast_manager &m = var.m();
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arith_util a(m);
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expr *lit = nullptr;
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if (!m.is_not(lit0, lit)) lit = lit0;
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expr *e1 = nullptr, *e2 = nullptr;
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// assume e2 is numeral and ignore it
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if ((a.is_le(lit, e1, e2) || a.is_lt(lit, e1, e2) ||
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a.is_ge(lit, e1, e2) || a.is_gt(lit, e1, e2) ||
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m.is_eq(lit, e1, e2))) {
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if (a.is_numeral(e1)) std::swap(e1, e2);
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}
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else {
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e1 = lit;
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}
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expr_ref val(m);
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if (!a.is_add(e1)) {
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if (match_mul(e1, var, val, a)) return val;
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}
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else {
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for (auto *arg : *to_app(e1)) {
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if (match_mul(arg, var, val, a)) return val;
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}
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}
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return expr_ref(m);
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}
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// convert assign-bounds lemma to a farkas lemma by adding missing coeff
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// assume that missing coeff is for premise at position 0
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static proof_ref mk_fk_from_ab(ast_manager &m,
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@ -108,9 +285,44 @@ namespace spacer {
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unsigned num_params,
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parameter const *params) {
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SASSERT(num_params == parents.size() + 1 /* one param is missing */);
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// compute missing coefficient
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linear_combinator lcb(m);
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for (unsigned i = 1, sz = parents.size(); i < sz; ++i) {
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app *p = to_app(m.get_fact(parents.get(i)));
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rational const &r = params[i+1].get_rational();
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lcb.add_lit(p, r);
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}
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TRACE("spacer.fkab",
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tout << "lit0 is: " << mk_pp(m.get_fact(parents.get(0)), m) << "\n"
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<< "LCB is: " << lcb() << "\n";);
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expr_ref var(m), val1(m), val2(m);
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val1 = get_coeff(m.get_fact(parents.get(0)), var);
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val2 = get_coeff(lcb(), var);
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TRACE("spacer.fkab",
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tout << "var: " << var
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<< " val1: " << val1 << " val2: " << val2 << "\n";);
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rational rat1, rat2, coeff0;
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arith_util a(m);
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if (a.is_numeral(val1, rat1) && a.is_numeral(val2, rat2)) {
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coeff0 = abs(rat2/rat1);
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coeff0 = coeff0 / lcb.lc();
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TRACE("spacer.fkab", tout << "coeff0: " << coeff0 << "\n";);
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}
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else {
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IF_VERBOSE(1, verbose_stream()
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<< "\n\n\nFAILED TO FIND COEFFICIENT\n\n\n";);
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// failed to find a coefficient
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return proof_ref(m);
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}
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buffer<parameter> v;
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v.push_back(parameter(symbol("farkas")));
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v.push_back(parameter(rational(1)));
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v.push_back(parameter(coeff0));
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for (unsigned i = 2; i < num_params; ++i)
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v.push_back(params[i]);
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@ -124,11 +336,11 @@ namespace spacer {
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v.size(), v.c_ptr());
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SASSERT(is_arith_lemma(m, pf));
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DEBUG_CODE(
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proof_checker pc(m);
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expr_ref_vector side(m);
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SASSERT(pc.check(pf, side));
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);
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ENSURE(pc.check(pf, side)););
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return pf;
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}
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@ -179,7 +391,9 @@ namespace spacer {
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d->get_num_parameters(),
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d->get_parameters());
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}
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else {
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// fall back to th-lemma
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if (!th_lemma) {
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th_lemma = mk_th_lemma(m, hyps,
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d->get_num_parameters(),
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d->get_parameters());
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