z3/src/sat/smt/ba_solver.cpp

/*++
Copyright (c) 2017 Microsoft Corporation

Module Name:

    ba_solver.cpp

Abstract:

    Extension for cardinality and xor reasoning.

Author:

    Nikolaj Bjorner (nbjorner) 2017-01-30

--*/

#include <cmath>
#include "util/mpz.h"
#include "sat/sat_types.h"
#include "sat/smt/ba_solver.h"
#include "sat/sat_simplifier_params.hpp"
#include "sat/sat_xor_finder.h"


namespace sat {

    static unsigned _bad_id = 11111111; // 2759; //
#define BADLOG(_cmd_) if (p.id() == _bad_id) { _cmd_; }

    ba_solver::card& ba_solver::constraint::to_card() {
        SASSERT(is_card());
        return static_cast<card&>(*this);
    }

    ba_solver::card const& ba_solver::constraint::to_card() const{
        SASSERT(is_card());
        return static_cast<card const&>(*this);
    }

    ba_solver::pb& ba_solver::constraint::to_pb() {
        SASSERT(is_pb());
        return static_cast<pb&>(*this);
    }

    ba_solver::pb const& ba_solver::constraint::to_pb() const{
        SASSERT(is_pb());
        return static_cast<pb const&>(*this);
    }

    ba_solver::pb_base const& ba_solver::constraint::to_pb_base() const{
        SASSERT(is_pb() || is_card());
        return static_cast<pb_base const&>(*this);
    }


    unsigned ba_solver::constraint::fold_max_var(unsigned w) const {
        if (lit() != null_literal) w = std::max(w, lit().var());
        for (unsigned i = 0; i < size(); ++i) w = std::max(w, get_lit(i).var());
        return w;
    }


    std::ostream& operator<<(std::ostream& out, ba_solver::constraint const& cnstr) {
        if (cnstr.lit() != null_literal) out << cnstr.lit() << " == ";
        switch (cnstr.tag()) {
        case ba_solver::card_t: {
            ba_solver::card const& c = cnstr.to_card();
            for (literal l : c) {
                out << l << " ";
            }
            out << " >= " << c.k();
            break;
        }
        case ba_solver::pb_t: {
            ba_solver::pb const& p = cnstr.to_pb();
            bool first = true;
            for (ba_solver::wliteral wl : p) {
                if (!first) out << "+ ";
                if (wl.first != 1) out << wl.first << " * ";
                out << wl.second << " ";
                first = false;
            }
            out << " >= " << p.k();
            break;
        }
        case ba_solver::xr_t: {
            ba_solver::xr const& x = cnstr.to_xr();
            for (unsigned i = 0; i < x.size(); ++i) {
                out << x[i] << " ";
                if (i + 1 < x.size()) out << "x ";
            }            
            break;
        }
        default:
            UNREACHABLE();
        }
        return out;
    }


    // -----------------------
    // pb_base

    bool ba_solver::pb_base::well_formed() const {
        uint_set vars;        
        if (lit() != null_literal) vars.insert(lit().var());
        for (unsigned i = 0; i < size(); ++i) {
            bool_var v = get_lit(i).var();
            if (vars.contains(v)) return false;
            if (get_coeff(i) > k()) return false;
            vars.insert(v);
        }
        return true;
    }

    // ----------------------
    // card

    ba_solver::card::card(unsigned id, literal lit, literal_vector const& lits, unsigned k):
        pb_base(card_t, id, lit, lits.size(), get_obj_size(lits.size()), k) {        
        for (unsigned i = 0; i < size(); ++i) {
            m_lits[i] = lits[i];
        }
    }

    void ba_solver::card::negate() {
        m_lit.neg();
        for (unsigned i = 0; i < m_size; ++i) {
            m_lits[i].neg();
        }
        m_k = m_size - m_k + 1;
        SASSERT(m_size >= m_k && m_k > 0);
    }

    bool ba_solver::card::is_watching(literal l) const {
        unsigned sz = std::min(k() + 1, size());
        for (unsigned i = 0; i < sz; ++i) {
            if ((*this)[i] == l) return true;
        }
        return false;
    }

    // -----------------------------------
    // pb

    ba_solver::pb::pb(unsigned id, literal lit, svector<ba_solver::wliteral> const& wlits, unsigned k):
        pb_base(pb_t, id, lit, wlits.size(), get_obj_size(wlits.size()), k),
        m_slack(0),
        m_num_watch(0),
        m_max_sum(0) {
        for (unsigned i = 0; i < size(); ++i) {
            m_wlits[i] = wlits[i];
        }
        update_max_sum();
    }

    void ba_solver::pb::update_max_sum() {
        m_max_sum = 0;
        for (unsigned i = 0; i < size(); ++i) {
            m_wlits[i].first = std::min(k(), m_wlits[i].first);
            if (m_max_sum + m_wlits[i].first < m_max_sum) {
                throw default_exception("addition of pb coefficients overflows");
            }
            m_max_sum += m_wlits[i].first;
        }
    }

    void ba_solver::pb::negate() {
        m_lit.neg();
        unsigned w = 0;
        for (unsigned i = 0; i < m_size; ++i) {
            m_wlits[i].second.neg();
            VERIFY(w + m_wlits[i].first >= w);
            w += m_wlits[i].first;
        }        
        m_k = w - m_k + 1;
        VERIFY(w >= m_k && m_k > 0);
    }

    bool ba_solver::pb::is_watching(literal l) const {
        for (unsigned i = 0; i < m_num_watch; ++i) {
            if ((*this)[i].second == l) return true;
        }
        return false;
    }


    bool ba_solver::pb::is_cardinality() const {
        if (size() == 0) return false;
        unsigned w = (*this)[0].first;
        for (wliteral wl : *this) if (w != wl.first) return false;
        return true;
    }


    

    // ----------------------------            
    // card

    bool ba_solver::init_watch(card& c) {
        literal root = c.lit();
        if (root != null_literal && value(root) == l_false) {
            clear_watch(c);
            c.negate();
            root.neg();
        }
        if (root != null_literal) {
            if (!is_watched(root, c)) watch_literal(root, c);
            if (!c.is_pure() && !is_watched(~root, c)) watch_literal(~root, c);
        }
        TRACE("ba", display(tout << "init watch: ", c, true););
        SASSERT(root == null_literal || value(root) == l_true);
        unsigned j = 0, sz = c.size(), bound = c.k();
        // put the non-false literals into the head.

        if (bound == sz) {
            for (literal l : c) assign(c, l);
            return false;
        }

        for (unsigned i = 0; i < sz; ++i) {
            if (value(c[i]) != l_false) {
                if (j != i) {
                    if (c.is_watched() && j <= bound && i > bound) {
                        unwatch_literal(c[j], c);
                        watch_literal(c[i], c);
                    }
                    c.swap(i, j);
                }
                ++j;
            }
        }
        DEBUG_CODE(
            bool is_false = false;
            for (literal l : c) {
                SASSERT(!is_false || value(l) == l_false);
                is_false = value(l) == l_false;
            });

        // j is the number of non-false, sz - j the number of false.

        if (j < bound) {
            if (c.is_watched()) clear_watch(c);
            SASSERT(0 < bound && bound < sz);
            literal alit = c[j];
            
            //
            // we need the assignment level of the asserting literal to be maximal.
            // such that conflict resolution can use the asserting literal as a starting
            // point.
            //

            for (unsigned i = bound; i < sz; ++i) {                
                if (lvl(alit) < lvl(c[i])) {
                    c.swap(i, j);
                    alit = c[j];
                }
            }
            set_conflict(c, alit);
            return false;
        }
        else if (j == bound) {
            for (unsigned i = 0; i < bound; ++i) {
                assign(c, c[i]);                
            }
            return false;
        }
        else {
            if (c.is_watched()) return true;
            clear_watch(c);
            for (unsigned i = 0; i <= bound; ++i) {
                watch_literal(c[i], c);
            }
            c.set_watch();
            return true;
        }
    }

    void ba_solver::clear_watch(card& c) {
        if (c.is_clear()) return;
        c.clear_watch();
        unsigned sz = std::min(c.k() + 1, c.size());
        for (unsigned i = 0; i < sz; ++i) {
            unwatch_literal(c[i], c);            
        }
    }

    // -----------------------
    // constraint

    void ba_solver::set_conflict(constraint& c, literal lit) {
        m_stats.m_num_conflicts++;
        TRACE("ba", display(tout, c, true); );
        if (!validate_conflict(c)) {
            IF_VERBOSE(0, display(verbose_stream(), c, true));
            UNREACHABLE();
        }
        SASSERT(validate_conflict(c));
        if (c.is_xr() && value(lit) == l_true) lit.neg();
        SASSERT(value(lit) == l_false);
        set_conflict(justification::mk_ext_justification(s().scope_lvl(), c.cindex()), ~lit);
        SASSERT(inconsistent());
    }

    void ba_solver::assign(constraint& c, literal lit) {
        if (inconsistent()) return;
        switch (value(lit)) {
        case l_true: 
            break;
        case l_false: 
            set_conflict(c, lit); 
            break;
        default:
            m_stats.m_num_propagations++;
            m_num_propagations_since_pop++;
            //TRACE("ba", tout << "#prop: " << m_stats.m_num_propagations << " - " << c.lit() << " => " << lit << "\n";);
            SASSERT(validate_unit_propagation(c, lit));
            if (get_config().m_drat) {
                svector<drat::premise> ps;
                literal_vector lits;
                get_antecedents(lit, c, lits);
                lits.push_back(lit);
                ps.push_back(drat::premise(drat::s_ext(), c.lit())); // null_literal case.
                drat_add(lits, ps);
            }
            assign(lit, justification::mk_ext_justification(s().scope_lvl(), c.cindex()));
            break;
        }
    }

    // -------------------
    // pb_base

    void ba_solver::simplify(pb_base& p) {
        SASSERT(s().at_base_lvl());
        if (p.lit() != null_literal && value(p.lit()) == l_false) {
            TRACE("ba", tout << "pb: flip sign " << p << "\n";);
            IF_VERBOSE(1, verbose_stream() << "sign is flipped " << p << "\n";);
            return;
        }
        bool nullify = p.lit() != null_literal && value(p.lit()) == l_true;
        if (nullify) {
            IF_VERBOSE(100, display(verbose_stream() << "nullify tracking literal\n", p, true););
            SASSERT(lvl(p.lit()) == 0);
            nullify_tracking_literal(p);
            init_watch(p);
        }

        SASSERT(p.lit() == null_literal || value(p.lit()) != l_false);

        unsigned true_val = 0, slack = 0, num_false = 0;
        for (unsigned i = 0; i < p.size(); ++i) {
            literal l = p.get_lit(i);
            bool_var v = l.var();
            if (s().was_eliminated(v)) {
                VERIFY(p.learned());
                remove_constraint(p, "contains eliminated");
                return;
            }
            switch (value(l)) {
            case l_true: true_val += p.get_coeff(i); break;
            case l_false: ++num_false; break;
            default: slack += p.get_coeff(i); break;
            }
        }
        if (p.k() == 1 && p.lit() == null_literal) {
            literal_vector lits(p.literals());
            s().mk_clause(lits.size(), lits.c_ptr(), p.learned());
            IF_VERBOSE(100, display(verbose_stream() << "add clause: " << lits << "\n", p, true););
            remove_constraint(p, "implies clause");
        }
        else if (true_val == 0 && num_false == 0) {
            if (p.lit() == null_literal || value(p.lit()) == l_true) {
                init_watch(p);
            }
        }
        else if (true_val >= p.k()) {
            if (p.lit() != null_literal) {
                IF_VERBOSE(100, display(verbose_stream() << "assign true literal ", p, true););
                s().assign_scoped(p.lit());
            }        
            remove_constraint(p, "is true");
        }
        else if (slack + true_val < p.k()) {
            if (p.lit() != null_literal) {
                IF_VERBOSE(100, display(verbose_stream() << "assign false literal ", p, true););
                s().assign_scoped(~p.lit());
            }
            else {
                IF_VERBOSE(1, verbose_stream() << "unsat during simplification\n";);
                s().set_conflict(justification(0));
            }
            remove_constraint(p, "is false");
        }
        else if (slack + true_val == p.k()) {
            literal_vector lits(p.literals());    
            assert_unconstrained(p.lit(), lits);
            remove_constraint(p, "is tight");
        }
        else {
            
            unsigned sz = p.size();
            clear_watch(p);
            unsigned j = 0;
            for (unsigned i = 0; i < sz; ++i) {
                literal l = p.get_lit(i);
                if (value(l) == l_undef) {
                    if (i != j) p.swap(i, j);
                    ++j;
                }
            }
            sz = j;
            // _bad_id = p.id();
            BADLOG(display(verbose_stream() << "simplify ", p, true));

            unsigned k = p.k() - true_val;

            if (k == 1 && p.lit() == null_literal) {
                literal_vector lits(sz, p.literals().c_ptr());
                s().mk_clause(sz, lits.c_ptr(), p.learned());
                remove_constraint(p, "is clause");
                return;
            }
            p.set_size(sz);
            p.set_k(k);
            if (p.lit() == null_literal || value(p.lit()) == l_true) {
                init_watch(p);
            }
            else {
                SASSERT(value(p.lit()) == l_undef);
            }
            BADLOG(display(verbose_stream() << "simplified ", p, true); verbose_stream() << "\n");
            // display(verbose_stream(), c, true);
            _bad_id = 11111111;
            SASSERT(p.well_formed());
            m_simplify_change = true;
        }
    }

    /*
      \brief split PB constraint into two because root is reused in arguments.
    
      x <=> a*x + B*y >= k
     
      x => a*x + By >= k
      ~x => a*x + By < k
     
      k*~x + a*x + By >= k
      (B+a-k + 1)*x + a*~x + B*~y >= B + a - k + 1

      (k - a) * ~x + By >= k - a
      k' * x + B'y >= k'
       
    */
    
    void ba_solver::split_root(pb_base& p) {
        SASSERT(p.lit() != null_literal);
        SASSERT(!p.learned());
        m_weights.resize(2*s().num_vars(), 0);
        unsigned k = p.k();
        unsigned w, w1, w2;
        literal root = p.lit();
        m_weights[(~root).index()] = k;
        for (unsigned i = 0; i < p.size(); ++i) {
            m_weights[p.get_lit(i).index()] += p.get_coeff(i);
        }
        literal_vector lits(p.literals());
        lits.push_back(~root);

        for (literal l : lits) {
            w1 = m_weights[l.index()];
            w2 = m_weights[(~l).index()];
            if (w1 >= w2) {
                if (w2 >= k) {
                    for (literal l2 : lits) m_weights[l2.index()] = 0;
                    // constraint is true
                    return;
                }
                k -= w2;
                m_weights[(~l).index()] = 0;
                m_weights[l.index()] = w1 - w2;
            }
        }
        SASSERT(k > 0);

        // ~root * (k - a) + p >= k - a

        m_wlits.reset();
        for (literal l : lits) {
            w = m_weights[l.index()];
            if (w != 0) {
                m_wlits.push_back(wliteral(w, l));
            } 
            m_weights[l.index()] = 0;
        }
        
        add_pb_ge(null_literal, m_wlits, k, false);        
    }


    // -------------------
    // pb


    // watch a prefix of literals, such that the slack of these is >= k
    bool ba_solver::init_watch(pb& p) {
        clear_watch(p);        
        if (p.lit() != null_literal && value(p.lit()) == l_false) {
            p.negate();           
        }
        
        VERIFY(p.lit() == null_literal || value(p.lit()) == l_true);
        unsigned sz = p.size(), bound = p.k();
     
        // put the non-false literals into the head.
        unsigned slack = 0, slack1 = 0, num_watch = 0, j = 0;
        for (unsigned i = 0; i < sz; ++i) {
            if (value(p[i].second) != l_false) {
                if (j != i) {
                    p.swap(i, j);
                }
                if (slack <= bound) {
                    slack += p[j].first;
                    ++num_watch;
                }
                else {
                    slack1 += p[j].first;
                }
                ++j;
            }
        }
        BADLOG(verbose_stream() << "watch " << num_watch << " out of " << sz << "\n");

        DEBUG_CODE(
            bool is_false = false;
            for (unsigned k = 0; k < sz; ++k) {
                SASSERT(!is_false || value(p[k].second) == l_false);
                SASSERT((k < j) == (value(p[k].second) != l_false));
                is_false = value(p[k].second) == l_false;
            });

        if (slack < bound) {
            literal lit = p[j].second;
            VERIFY(value(lit) == l_false);
            for (unsigned i = j + 1; i < sz; ++i) {
                if (lvl(lit) < lvl(p[i].second)) {
                    lit = p[i].second;
                }
            }
            set_conflict(p, lit);
            return false;
        }
        else {            
            for (unsigned i = 0; i < num_watch; ++i) {
                watch_literal(p[i], p);
            }
            p.set_slack(slack);
            p.set_num_watch(num_watch);

            SASSERT(validate_watch(p, null_literal));

            TRACE("ba", display(tout << "init watch: ", p, true););

            // slack is tight:
            if (slack + slack1 == bound) {
                SASSERT(slack1 == 0);
                SASSERT(j == num_watch);                
                for (unsigned i = 0; i < j; ++i) {
                    assign(p, p[i].second);
                }
            }
            return true;
        }
    }

    /*
      Chai Kuhlmann:
      Lw - set of watched literals
      Lu - set of unwatched literals that are not false
      
      Lw = Lw \ { alit }
      Sw -= value
      a_max = max { a | l in Lw u Lu, l = undef }
      while (Sw < k + a_max & Lu != 0) {
        a_s = max { a | l in Lu }
        Sw += a_s
        Lw = Lw u {l_s}
        Lu = Lu \ {l_s}
      }
      if (Sw < k) return conflict
      for (li in Lw | Sw < k + ai) 
         assign li
      return no-conflict

     a_max index: index of non-false literal with maximal weight.
    */

    void ba_solver::add_index(pb& p, unsigned index, literal lit) {
        if (value(lit) == l_undef) {
            m_pb_undef.push_back(index);
            if (p[index].first > m_a_max) {
                m_a_max = p[index].first;
            }
        }
    }    
    
    /*
      \brief propagate assignment to alit in constraint p.
      
      TBD: 
      - consider reordering literals in watch list so that the search for watched literal takes average shorter time.
      - combine with caching literals that are assigned to 'true' to a cold store where they are not being revisited.
        Since 'true' literals may be unassigned (unless they are assigned at level 0) the cache has to be backtrack
        friendly (and the overhead of backtracking has to be taken into account).
     */
    lbool ba_solver::add_assign(pb& p, literal alit) {
        BADLOG(display(verbose_stream() << "assign: " << alit << " watch: " << p.num_watch() << " size: " << p.size(), p, true));
        TRACE("ba", display(tout << "assign: " << alit << "\n", p, true););
        SASSERT(!inconsistent());
        unsigned sz = p.size();
        unsigned bound = p.k();
        unsigned num_watch = p.num_watch();
        unsigned slack = p.slack();
        SASSERT(value(alit) == l_false);
        SASSERT(p.lit() == null_literal || value(p.lit()) == l_true);
        SASSERT(num_watch <= sz);
        SASSERT(num_watch > 0);
        unsigned index = 0;
        m_a_max = 0;
        m_pb_undef.reset();
        for (; index < num_watch; ++index) {
            literal lit = p[index].second;
            if (lit == alit) {
                break;
            }
            add_index(p, index, lit);
        }
        if (index == num_watch || num_watch == 0) {
            _bad_id = p.id();
            BADLOG(
                verbose_stream() << "BAD: " << p.id() << "\n";
                display(verbose_stream(), p, true);
                verbose_stream() << "alit: " << alit << "\n";
                verbose_stream() << "num watch " << num_watch << "\n");
            UNREACHABLE();
            return l_undef;
        }
        
        SASSERT(validate_watch(p, null_literal));
        // SASSERT(validate_watch(p, null_literal));
        
        SASSERT(index < num_watch);
        unsigned index1 = index + 1;
        for (; m_a_max == 0 && index1 < num_watch; ++index1) {
            add_index(p, index1, p[index1].second);
        }
        
        unsigned val = p[index].first;
        SASSERT(value(p[index].second) == l_false);
        SASSERT(val <= slack);
        slack -= val;

        // find literals to swap with:            
        for (unsigned j = num_watch; j < sz && slack < bound + m_a_max; ++j) {
            literal lit = p[j].second;
            if (value(lit) != l_false) {
                slack += p[j].first;
                SASSERT(!is_watched(p[j].second, p));
                watch_literal(p[j], p);
                p.swap(num_watch, j);
                add_index(p, num_watch, lit);
                BADLOG(verbose_stream() << "add watch: " << lit << " num watch: " << num_watch << " max: " << m_a_max << " slack " << slack << "\n");                
                ++num_watch;
            }
        }
        
        SASSERT(!inconsistent());
        DEBUG_CODE(for (auto idx : m_pb_undef) { SASSERT(value(p[idx].second) == l_undef); });
        
        if (slack < bound) {
            // maintain watching the literal
            slack += val;
            p.set_slack(slack);
            p.set_num_watch(num_watch);
            SASSERT(validate_watch(p, null_literal));
            BADLOG(display(verbose_stream() << "conflict: " << alit << " watch: " << p.num_watch() << " size: " << p.size(), p, true));            
            SASSERT(bound <= slack);
            TRACE("ba", tout << "conflict " << alit << "\n";);
            set_conflict(p, alit);
            return l_false;
        }

        if (num_watch == 1) { _bad_id = p.id(); }

        BADLOG(verbose_stream() << "size: " << p.size() << " index: " << index << " num watch: " << num_watch << "\n");

        // swap out the watched literal.
        --num_watch;
        SASSERT(num_watch > 0);
        p.set_slack(slack);
        p.set_num_watch(num_watch);
        p.swap(num_watch, index);


        // 
        // slack >= bound, but slack - w(l) < bound 
        // l must be true.
        // 
        if (slack < bound + m_a_max) {            
            BADLOG(verbose_stream() << "slack " << slack << " " << bound << " " << m_a_max << "\n";);
            TRACE("ba", tout << p << "\n"; for(auto j : m_pb_undef) tout << j << " "; tout << "\n";);
            for (unsigned index1 : m_pb_undef) {
                if (index1 == num_watch) {
                    index1 = index;
                }
                wliteral wl = p[index1];
                literal lit = wl.second;
                SASSERT(value(lit) == l_undef);
                if (slack < bound + wl.first) {
                    BADLOG(verbose_stream() << "Assign " << lit << " " << wl.first << "\n");                
                    assign(p, lit);
                }
            }
        }

        SASSERT(validate_watch(p, alit)); // except that alit is still watched.

        TRACE("ba", display(tout << "assign: " << alit << "\n", p, true););

        BADLOG(verbose_stream() << "unwatch " << alit << " watch: " << p.num_watch() << " size: " << p.size() << " slack: " << p.slack() << " " << inconsistent() << "\n");

        return l_undef;
    }

    void ba_solver::watch_literal(wliteral l, pb& p) {
        watch_literal(l.second, p);
    }

    void ba_solver::clear_watch(pb& p) {
        p.clear_watch();
        for (unsigned i = 0; i < p.num_watch(); ++i) {
            unwatch_literal(p[i].second, p);          
        }  
        p.set_num_watch(0);
        DEBUG_CODE(for (wliteral wl : p) VERIFY(!is_watched(wl.second, p)););
    }

    void ba_solver::recompile(pb& p) {
        // IF_VERBOSE(2, verbose_stream() << "re: " << p << "\n";);
        SASSERT(p.num_watch() == 0);
        m_weights.resize(2*s().num_vars(), 0);
        for (wliteral wl : p) {
            m_weights[wl.second.index()] += wl.first;
        }
        unsigned k = p.k();
        unsigned sz = p.size();
        bool all_units = true;
        unsigned j = 0;
        for (unsigned i = 0; i < sz && 0 < k; ++i) {
            literal l = p[i].second;
            unsigned w1 = m_weights[l.index()];
            unsigned w2 = m_weights[(~l).index()];
            if (w1 == 0 || w1 < w2) {
                continue;
            }
            else if (k <= w2) {
                k = 0;
                break;
            }
            else {
                SASSERT(w2 <= w1 && w2 < k);
                k -= w2;
                w1 -= w2;
                m_weights[l.index()] = 0;
                m_weights[(~l).index()] = 0;        
                if (w1 == 0) {
                    continue;
                }    
                else {
                    p[j] = wliteral(w1, l);            
                    all_units &= w1 == 1;
                    ++j;
                }
            }
        }
        sz = j;
        // clear weights
        for (wliteral wl : p) {
            m_weights[wl.second.index()] = 0;
            m_weights[(~wl.second).index()] = 0;
        }

        if (k == 0) {
            if (p.lit() != null_literal) {
                s().assign_scoped(p.lit());
            }
            remove_constraint(p, "recompiled to true");
            return;
        }

        else if (k == 1 && p.lit() == null_literal) {
            literal_vector lits(sz, p.literals().c_ptr());
            s().mk_clause(sz, lits.c_ptr(), p.learned());
            remove_constraint(p, "recompiled to clause");
            return;
        }

        else if (all_units) {
            literal_vector lits(sz, p.literals().c_ptr());
            add_at_least(p.lit(), lits, k, p.learned());
            remove_constraint(p, "recompiled to cardinality");
            return;
        }
        else {
            p.set_size(sz);
            p.update_max_sum();
            if (p.max_sum() < k) {
                if (p.lit() == null_literal) {
                    s().set_conflict(justification(0));
                }
                else {
                    s().assign_scoped(~p.lit());
                }
                remove_constraint(p, "recompiled to false");
                return;
            }
            p.set_k(k);
            SASSERT(p.well_formed());
            
            if (clausify(p)) {
                return;
            }
            if (p.lit() == null_literal || value(p.lit()) == l_true) {
                init_watch(p);
            }
        }
    }

    void ba_solver::display(std::ostream& out, pb const& p, bool values) const {
        if (p.lit() != null_literal) out << p.lit() << " == ";
        if (values) {
            out << "[watch: " << p.num_watch() << ", slack: " << p.slack() << "]";
        }
        if (p.lit() != null_literal && values) {
            out << "@(" << value(p.lit());
            if (value(p.lit()) != l_undef) {
                out << ":" << lvl(p.lit());
            }
            out << "): ";
        }
        unsigned i = 0;
        for (wliteral wl : p) {
            literal l = wl.second;
            unsigned w = wl.first;
            if (i > 0) out << "+ ";
            if (i++ == p.num_watch()) out << " | ";
            if (w > 1) out << w << " * ";
            out << l;
            if (values) {
                out << "@(" << value(l);
                if (value(l) != l_undef) {
                    out << ":" << lvl(l);
                }
                out << ") ";
            }
            else {
                out << " ";
            }
        }
        out << ">= " << p.k()  << "\n";
    }


    // ---------------------------
    // conflict resolution
    

    void ba_solver::inc_coeff(literal l, unsigned offset) {
        SASSERT(offset > 0);
        bool_var v = l.var();
        SASSERT(v != null_bool_var);
        m_coeffs.reserve(v + 1, 0);
        TRACE("ba_verbose", tout << l << " " << offset << "\n";);

        int64_t coeff0 = m_coeffs[v];
        if (coeff0 == 0) {
            m_active_vars.push_back(v);
        }
        
        int64_t loffset = static_cast<int64_t>(offset);
        int64_t inc = l.sign() ? -loffset : loffset;
        int64_t coeff1 = inc + coeff0;
        m_coeffs[v] = coeff1;
        if (coeff1 > INT_MAX || coeff1 < INT_MIN) {
            m_overflow = true;
            return;
        }

        if (coeff0 > 0 && inc < 0) {
            inc_bound(std::max((int64_t)0, coeff1) - coeff0);
        }
        else if (coeff0 < 0 && inc > 0) {
            inc_bound(coeff0 - std::min((int64_t)0, coeff1));
        }
        int64_t lbound = static_cast<int64_t>(m_bound);

        // reduce coefficient to be no larger than bound.
        if (coeff1 > lbound) {
            m_coeffs[v] = lbound;
        }
        else if (coeff1 < 0 && -coeff1 > lbound) {
            m_coeffs[v] = -lbound;
        }        
    }

    int64_t ba_solver::get_coeff(bool_var v) const {
        return m_coeffs.get(v, 0);
    }

    uint64_t ba_solver::get_coeff(literal lit) const {
        int64_t c1 = get_coeff(lit.var());
        SASSERT((c1 < 0) == lit.sign());
        int64_t c = std::abs(c1);
        m_overflow |= (c != c1);
        return static_cast<uint64_t>(c);
    }

    ba_solver::wliteral ba_solver::get_wliteral(bool_var v) {
        int64_t c1 = get_coeff(v);
        literal l = literal(v, c1 < 0);
        c1 = std::abs(c1);
        unsigned c = static_cast<unsigned>(c1);
        // TRACE("ba", tout << l << " " << c << "\n";);
        m_overflow |= c != c1;
        return wliteral(c, l);
    }

    unsigned ba_solver::get_abs_coeff(bool_var v) const {
        int64_t c1 = std::abs(get_coeff(v));
        unsigned c = static_cast<unsigned>(c1);
        m_overflow |= c != c1;
        return c;
    }

    int ba_solver::get_int_coeff(bool_var v) const {
        int64_t c1 = m_coeffs.get(v, 0);
        int c = static_cast<int>(c1);
        m_overflow |= c != c1;
        return c;
    }

    void ba_solver::inc_bound(int64_t i) {
        int64_t new_bound = m_bound;
        new_bound += i;
        unsigned nb = static_cast<unsigned>(new_bound);
        m_overflow |= new_bound < 0 || nb != new_bound;
        m_bound = nb;        
    }

    void ba_solver::reset_coeffs() {
        for (unsigned i = m_active_vars.size(); i-- > 0; ) {
            m_coeffs[m_active_vars[i]] = 0;
        }
        m_active_vars.reset();
    }


    static bool _debug_conflict = false;
    static literal _debug_consequent = null_literal;
    static unsigned_vector _debug_var2position;

// #define DEBUG_CODE(_x_) _x_

    void ba_solver::bail_resolve_conflict(unsigned idx) {
        literal_vector const& lits = s().m_trail;
        while (m_num_marks > 0) {
            bool_var v = lits[idx].var();
            if (s().is_marked(v)) {
                s().reset_mark(v);
                --m_num_marks;
            }
            if (idx == 0 && !_debug_conflict) {
                _debug_conflict = true;
                _debug_var2position.reserve(s().num_vars());
                for (unsigned i = 0; i < lits.size(); ++i) {
                    _debug_var2position[lits[i].var()] = i;
                }
                IF_VERBOSE(0, 
                           active2pb(m_A);
                           uint64_t c = 0;
                           for (wliteral l : m_A.m_wlits) c += l.first;
                           verbose_stream() << "sum of coefficients: " << c << "\n";
                           display(verbose_stream(), m_A, true);
                           verbose_stream() << "conflicting literal: " << s().m_not_l << "\n";);

                for (literal l : lits) {
                    if (s().is_marked(l.var())) {
                        IF_VERBOSE(0, verbose_stream() << "missing mark: " << l << "\n";);
                        s().reset_mark(l.var());
                    }
                }
                m_num_marks = 0;
                resolve_conflict();                
            }
            --idx;
        }
    }

    lbool ba_solver::resolve_conflict() { 
        if (0 == m_num_propagations_since_pop) {
            return l_undef;
        }

        if (s().m_config.m_pb_resolve == PB_ROUNDING) {
            return resolve_conflict_rs();
        }
       
        m_overflow = false;
        reset_coeffs();
        m_num_marks = 0;
        m_bound = 0;
        literal consequent = s().m_not_l;
        justification js = s().m_conflict;
        TRACE("ba", tout << consequent << " " << js << "\n";);
        bool unique_max;
        m_conflict_lvl = s().get_max_lvl(consequent, js, unique_max);
        if (m_conflict_lvl == 0) {
            return l_undef;
        }
        if (consequent != null_literal) {
            consequent.neg();
            process_antecedent(consequent, 1);
        }
        literal_vector const& lits = s().m_trail;
        unsigned idx = lits.size() - 1;
        unsigned offset = 1;
        DEBUG_CODE(active2pb(m_A););

        do {


            if (m_overflow || offset > (1 << 12)) {
                IF_VERBOSE(20, verbose_stream() << "offset: " << offset << "\n";
                           DEBUG_CODE(active2pb(m_A); display(verbose_stream(), m_A);););
                goto bail_out;
            }

            if (offset == 0) {
                goto process_next_resolvent;            
            }

            DEBUG_CODE(TRACE("sat_verbose", display(tout, m_A);););
            TRACE("ba", tout << "process consequent: " << consequent << " : "; s().display_justification(tout, js) << "\n";);
            SASSERT(offset > 0);

            DEBUG_CODE(justification2pb(js, consequent, offset, m_B););
            
            if (_debug_conflict) {
                IF_VERBOSE(0, 
                           verbose_stream() << consequent << "\n";
                           s().display_justification(verbose_stream(), js);
                           verbose_stream() << "\n";);
                _debug_consequent = consequent;
            }
            switch(js.get_kind()) {
            case justification::NONE:
                SASSERT (consequent != null_literal);
                inc_bound(offset);
                break;
            case justification::BINARY:
                inc_bound(offset);
                SASSERT (consequent != null_literal);
                inc_coeff(consequent, offset);
                process_antecedent(js.get_literal(), offset);
                break;
            case justification::TERNARY:
                inc_bound(offset); 
                SASSERT (consequent != null_literal);
                inc_coeff(consequent, offset);				
                process_antecedent(js.get_literal1(), offset);
                process_antecedent(js.get_literal2(), offset);
                break;
            case justification::CLAUSE: {
                inc_bound(offset); 
                clause & c = s().get_clause(js);
                unsigned i = 0;
                if (consequent != null_literal) {
                    inc_coeff(consequent, offset);
                    if (c[0] == consequent) {
                        i = 1;
                    }
                    else {
                        SASSERT(c[1] == consequent);
                        process_antecedent(c[0], offset);
                        i = 2;
                    }
                }
                unsigned sz = c.size();
                for (; i < sz; i++)
                    process_antecedent(c[i], offset);
                break;
            }
            case justification::EXT_JUSTIFICATION: {
                constraint& cnstr = index2constraint(js.get_ext_justification_idx());
                ++m_stats.m_num_resolves;
                switch (cnstr.tag()) {
                case card_t: {
                    card& c = cnstr.to_card();
                    inc_bound(static_cast<int64_t>(offset) * c.k());
                    process_card(c, offset);
                    break;
                }
                case pb_t: {
                    pb& p = cnstr.to_pb();
                    m_lemma.reset();
                    inc_bound(offset);
                    inc_coeff(consequent, offset);
                    get_antecedents(consequent, p, m_lemma);
                    TRACE("ba", display(tout, p, true); tout << m_lemma << "\n";);
                    if (_debug_conflict) {
                        verbose_stream() << consequent << " ";
                        verbose_stream() << "antecedents: " << m_lemma << "\n";
                    }
                    for (literal l : m_lemma) process_antecedent(~l, offset);
                    break;
                }
                case xr_t: {
                    // jus.push_back(js);
                    m_lemma.reset();
                    inc_bound(offset);
                    inc_coeff(consequent, offset);
                    get_xr_antecedents(consequent, idx, js, m_lemma);
                    for (literal l : m_lemma) process_antecedent(~l, offset);
                    break;
                }
                default:
                    UNREACHABLE();
                    break;
                }
                break;
            }
            default:
                UNREACHABLE();
                break;
            }
            
            SASSERT(validate_lemma());            

            DEBUG_CODE(
                active2pb(m_C);
                VERIFY(validate_resolvent());
                m_A = m_C;
                TRACE("ba", display(tout << "conflict: ", m_A);););

            cut();

        process_next_resolvent:
            
            // find the next marked variable in the assignment stack
            bool_var v;
            while (true) {
                consequent = lits[idx];
                v = consequent.var();
                if (s().is_marked(v)) {
                    if (s().lvl(v) == m_conflict_lvl) {
                        break;
                    }
                }
                if (idx == 0) {
                    IF_VERBOSE(2, verbose_stream() << "did not find marked literal\n";);
                    goto bail_out;
                }
                SASSERT(idx > 0);
                --idx;
            }
            
            SASSERT(lvl(v) == m_conflict_lvl);
            s().reset_mark(v);
            --idx;
            TRACE("sat_verbose", tout << "Unmark: v" << v << "\n";);
            --m_num_marks;
            js = s().m_justification[v];
            offset = get_abs_coeff(v);
            if (offset > m_bound) {
                int64_t bound64 = static_cast<int64_t>(m_bound);
                m_coeffs[v] = (get_coeff(v) < 0) ? -bound64 : bound64;
                offset = m_bound;
                DEBUG_CODE(active2pb(m_A););
            }
            SASSERT(value(consequent) == l_true);
        }        
        while (m_num_marks > 0);
        
        DEBUG_CODE(for (bool_var i = 0; i < static_cast<bool_var>(s().num_vars()); ++i) SASSERT(!s().is_marked(i)););
        SASSERT(validate_lemma());

        if (!create_asserting_lemma()) {
            goto bail_out;
        }
        active2lemma();


        DEBUG_CODE(VERIFY(validate_conflict(m_lemma, m_A)););
        
        return l_true;

    bail_out:
        if (m_overflow) {
            ++m_stats.m_num_overflow;
            m_overflow = false;
        }
        bail_resolve_conflict(idx);
        return l_undef;
    }


    unsigned ba_solver::ineq::bv_coeff(bool_var v) const {
        for (unsigned i = size(); i-- > 0; ) {
            if (lit(i).var() == v) return coeff(i);
        }
        UNREACHABLE();
        return 0;
    }

    void ba_solver::ineq::divide(unsigned c) {
        if (c == 1) return;
        for (unsigned i = size(); i-- > 0; ) {
            m_wlits[i].first = (coeff(i) + c - 1) / c;
        }
        m_k = (m_k + c - 1) / c;
    }

    /**
     * Remove literal at position i, subtract coefficient from bound.
     */
    void ba_solver::ineq::weaken(unsigned i) {
        unsigned ci = coeff(i);
        SASSERT(m_k >= ci);
        m_k -= ci;
        m_wlits[i] = m_wlits.back();
        m_wlits.pop_back();
    }

    /** 
     * Round coefficient of inequality to 1.
     */
    void ba_solver::round_to_one(ineq& ineq, bool_var v) {
        unsigned c = ineq.bv_coeff(v);
        if (c == 1) return;
        unsigned sz = ineq.size();
        for (unsigned i = 0; i < sz; ++i) {
            unsigned ci = ineq.coeff(i); 
            unsigned q = ci % c;
            if (q != 0 && !is_false(ineq.lit(i))) {
#if 1
                // code review by Elffers:
                ineq.weaken(i);
                --i;
                --sz;                
#else
                if (q == ci) {
                    ineq.weaken(i);
                    --i;
                    --sz;
                }
                else {
                    ineq.m_wlits[i].first -= q;
                    ineq.m_k -= q;
                }
#endif
            }
        }
        ineq.divide(c);
        TRACE("ba", display(tout << "var: " << v << " " << c << ": ", ineq, true););
    }

    void ba_solver::round_to_one(bool_var w) {
        unsigned c = get_abs_coeff(w);
        if (c == 1 || c == 0) return;
        for (bool_var v : m_active_vars) {
            wliteral wl = get_wliteral(v);
            unsigned q = wl.first % c;
            if (q != 0 && !is_false(wl.second)) {
                m_coeffs[v] = wl.first - q;
                m_bound -= q;
                SASSERT(m_bound > 0);
            }
        }        
        SASSERT(validate_lemma());
        divide(c);
        SASSERT(validate_lemma());
        TRACE("ba", active2pb(m_B); display(tout, m_B, true););
    }

    void ba_solver::divide(unsigned c) {
        SASSERT(c != 0);
        if (c == 1) return;
        reset_active_var_set();
        unsigned j = 0, sz = m_active_vars.size();
        for (unsigned i = 0; i < sz; ++i) {
            bool_var v = m_active_vars[i];
            int ci = get_int_coeff(v);
            if (!test_and_set_active(v) || ci == 0) continue;
            if (ci > 0) {
                m_coeffs[v] = (ci + c - 1) / c;
            }
            else {
                m_coeffs[v] = -static_cast<int64_t>((-ci + c - 1) / c);
            }
            m_active_vars[j++] = v;
        }
        m_active_vars.shrink(j);
        m_bound = static_cast<unsigned>((m_bound + c - 1) / c);                    
    }

    void ba_solver::resolve_on(literal consequent) {
        round_to_one(consequent.var());
        m_coeffs[consequent.var()] = 0;
    }

    void ba_solver::resolve_with(ineq const& ineq) {
        TRACE("ba", display(tout, ineq, true););
        inc_bound(ineq.m_k);        
        TRACE("ba", tout << "bound: " << m_bound << "\n";);

        for (unsigned i = ineq.size(); i-- > 0; ) {
            literal l = ineq.lit(i);
            inc_coeff(l, static_cast<unsigned>(ineq.coeff(i)));
            TRACE("ba", tout << "bound: " << m_bound << " lit: " << l << " coeff: " << ineq.coeff(i) << "\n";);
        }
    }

    void ba_solver::reset_marks(unsigned idx) {
        while (m_num_marks > 0) {
            SASSERT(idx > 0);
            bool_var v = s().m_trail[idx].var();
            if (s().is_marked(v)) {
                s().reset_mark(v);
                --m_num_marks;
            }
            --idx;
        }
    }
    
    /**
     * \brief mark variables that are on the assignment stack but 
     * below the current processing level.
     */
    void ba_solver::mark_variables(ineq const& ineq) {
        for (wliteral wl : ineq.m_wlits) {
            literal l = wl.second;
            if (!is_false(l)) continue;
            bool_var v = l.var();
            unsigned level = lvl(v);
            if (!s().is_marked(v) && !is_visited(v) && level == m_conflict_lvl) {
                s().mark(v);
                ++m_num_marks;
            }
        }
    }

    lbool ba_solver::resolve_conflict_rs() {
        if (0 == m_num_propagations_since_pop) {
            return l_undef;
        }
        m_overflow = false;
        reset_coeffs();
        init_visited();
        m_num_marks = 0;
        m_bound = 0;
        literal consequent = s().m_not_l;
        justification js = s().m_conflict;
        bool unique_max;
        m_conflict_lvl = s().get_max_lvl(consequent, js, unique_max);
        if (m_conflict_lvl == 0) {
            return l_undef;
        }
        if (consequent != null_literal) {
            consequent.neg();
            process_antecedent(consequent, 1);
        }
        TRACE("ba", tout << consequent << " " << js << "\n";);
        unsigned idx = s().m_trail.size() - 1;
        
        do {
            TRACE("ba", s().display_justification(tout << "process consequent: " << consequent << " : ", js) << "\n";
                  if (consequent != null_literal) { active2pb(m_A); display(tout, m_A, true); }
                  );

            switch (js.get_kind()) {
            case justification::NONE:
                SASSERT(consequent != null_literal);
                round_to_one(consequent.var());
                inc_bound(1);
                inc_coeff(consequent, 1);
                break;
            case justification::BINARY:
                SASSERT(consequent != null_literal);
                round_to_one(consequent.var());
                inc_bound(1);
                inc_coeff(consequent, 1);
                process_antecedent(js.get_literal());
                break;
            case justification::TERNARY:
                SASSERT(consequent != null_literal);
                round_to_one(consequent.var());
                inc_bound(1);
                inc_coeff(consequent, 1);
                process_antecedent(js.get_literal1());
                process_antecedent(js.get_literal2());
                break;
            case justification::CLAUSE: {
                clause & c = s().get_clause(js);
                unsigned i = 0;
                if (consequent != null_literal) {
                    round_to_one(consequent.var());
                    inc_coeff(consequent, 1);
                    if (c[0] == consequent) {
                        i = 1;
                    }
                    else {
                        SASSERT(c[1] == consequent);
                        process_antecedent(c[0]);
                        i = 2;
                    }
                }
                inc_bound(1);
                unsigned sz = c.size();
                for (; i < sz; i++)
                    process_antecedent(c[i]);
                break;
            }
            case justification::EXT_JUSTIFICATION: {
                ++m_stats.m_num_resolves;
                ext_justification_idx index = js.get_ext_justification_idx();
                constraint& cnstr = index2constraint(index);
                SASSERT(!cnstr.was_removed());
                switch (cnstr.tag()) {
                case card_t: 
                case pb_t: {
                    pb_base const& p = cnstr.to_pb_base();
                    unsigned k = p.k(), sz = p.size();
                    m_A.reset(0);
                    for (unsigned i = 0; i < sz; ++i) {
                        literal l = p.get_lit(i);
                        unsigned c = p.get_coeff(i);
                        if (l == consequent || !is_visited(l.var())) {
                            m_A.push(l, c);
                        }
                        else {
                            SASSERT(k > c);
                            TRACE("ba", tout << "visited: " << l << "\n";);
                            k -= c;
                        }
                    }
                    SASSERT(k > 0);
                    if (p.lit() != null_literal) m_A.push(~p.lit(), k);
                    m_A.m_k = k;
                    break;                 
                }
                default:
                    constraint2pb(cnstr, consequent, 1, m_A);
                    break;
                }
                mark_variables(m_A);
                if (consequent == null_literal) {
                    SASSERT(validate_ineq(m_A)); 
                    m_bound = static_cast<unsigned>(m_A.m_k);
                    for (wliteral wl : m_A.m_wlits) {
                        process_antecedent(wl.second, wl.first);
                    } 
                }
                else {
                    round_to_one(consequent.var());
                    if (cnstr.is_pb()) round_to_one(m_A, consequent.var()); 
                    SASSERT(validate_ineq(m_A)); 
                    resolve_with(m_A);
                }
                break;
            }
            default:
                UNREACHABLE();
                break;
            }            

            SASSERT(validate_lemma());
            cut();

            // find the next marked variable in the assignment stack
            bool_var v;
            while (true) {
                consequent = s().m_trail[idx];
                v = consequent.var();
                mark_visited(v);
                if (s().is_marked(v)) {
                    int64_t c = get_coeff(v);
                    if (c == 0 || ((c < 0) == consequent.sign())) {
                        s().reset_mark(v);
                        --m_num_marks;
                    }
                    else {
                        break;
                    }
                }
                if (idx == 0) {
                    TRACE("ba", tout << "there is no consequent\n";);
                    goto bail_out;
                }
                --idx;
            }
            
            SASSERT(lvl(v) == m_conflict_lvl);
            s().reset_mark(v);
            --idx;
            --m_num_marks;
            js = s().m_justification[v];
        }        
        while (m_num_marks > 0 && !m_overflow);
        TRACE("ba", active2pb(m_A); display(tout, m_A, true););

        // TBD: check if this is useful
        if (!m_overflow && consequent != null_literal) {
            round_to_one(consequent.var());
        }
        if (!m_overflow && create_asserting_lemma()) {
            active2lemma();
            return l_true;
        }
        
    bail_out:
        TRACE("ba", tout << "bail " << m_overflow << "\n";);
        if (m_overflow) {
            ++m_stats.m_num_overflow;
            m_overflow = false;
        }
        return l_undef;
    }


    bool ba_solver::create_asserting_lemma() {
        int64_t bound64 = m_bound;
        int64_t slack = -bound64;
        reset_active_var_set();
        unsigned j = 0, sz = m_active_vars.size();
        for (unsigned i = 0; i < sz; ++i) {
            bool_var v = m_active_vars[i];
            unsigned c = get_abs_coeff(v);
            if (!test_and_set_active(v) || c == 0) continue;
            slack += c;
            m_active_vars[j++] = v;
        }
        m_active_vars.shrink(j);
        m_lemma.reset();        
        m_lemma.push_back(null_literal);
        unsigned num_skipped = 0;
        int64_t asserting_coeff = 0;
        for (unsigned i = 0; 0 <= slack && i < m_active_vars.size(); ++i) { 
            bool_var v = m_active_vars[i];
            int64_t coeff = get_coeff(v);
            lbool val = value(v);
            bool is_true = val == l_true;
            bool append = coeff != 0 && val != l_undef && ((coeff < 0) == is_true);
            if (append) {
                literal lit(v, !is_true);
                if (lvl(lit) == m_conflict_lvl) {
                    if (m_lemma[0] == null_literal) {
                        asserting_coeff = std::abs(coeff);
                        slack -= asserting_coeff;
                        m_lemma[0] = ~lit;
                    }
                    else {
                        ++num_skipped;
                        if (asserting_coeff < std::abs(coeff)) {
                            m_lemma[0] = ~lit;
                            slack -= (std::abs(coeff) - asserting_coeff);
                            asserting_coeff = std::abs(coeff);
                        }
                    }
                }
                else if (lvl(lit) < m_conflict_lvl) {
                    slack -= std::abs(coeff);
                    m_lemma.push_back(~lit);
                }
            }
        }
        if (slack >= 0) {
            TRACE("ba", tout << "slack is non-negative\n";);
            IF_VERBOSE(20, verbose_stream() << "(sat.card slack: " << slack << " skipped: " << num_skipped << ")\n";);
            return false;
        }
        if (m_overflow) {
            TRACE("ba", tout << "overflow\n";);
            return false;
        }        
        if (m_lemma[0] == null_literal) {
            if (m_lemma.size() == 1) {
                s().set_conflict(justification(0));
            }
            TRACE("ba", tout << "no asserting literal\n";);
            return false;
        }

        TRACE("ba", tout << m_lemma << "\n";);

        if (get_config().m_drat) {
            svector<drat::premise> ps; // TBD fill in
            drat_add(m_lemma, ps);
        }

        s().m_lemma.reset();
        s().m_lemma.append(m_lemma);
        for (unsigned i = 1; i < m_lemma.size(); ++i) {
            CTRACE("ba", s().is_marked(m_lemma[i].var()), tout << "marked: " << m_lemma[i] << "\n";);
            s().mark(m_lemma[i].var());
        }
        return true;
    }

    /*
      \brief compute a cut for current resolvent.
     */

    void ba_solver::cut() {

        // bypass cut if there is a unit coefficient
        for (bool_var v : m_active_vars) {
            if (1 == get_abs_coeff(v)) return;
        }

        unsigned g = 0;

        for (unsigned i = 0; g != 1 && i < m_active_vars.size(); ++i) {
            bool_var v = m_active_vars[i];
            unsigned coeff = get_abs_coeff(v);
            if (coeff == 0) {
                continue;
            }
            if (m_bound < coeff) {
                int64_t bound64 = m_bound;
                if (get_coeff(v) > 0) {
                    m_coeffs[v] = bound64;
                }
                else {
                    m_coeffs[v] = -bound64;
                }
                coeff = m_bound;
            }
            SASSERT(0 < coeff && coeff <= m_bound);
            if (g == 0) {
                g = coeff;
            }
            else {
                g = u_gcd(g, coeff);
            }
        }

        if (g >= 2) {
            reset_active_var_set();
            unsigned j = 0, sz = m_active_vars.size();
            for (unsigned i = 0; i < sz; ++i) {
                bool_var v = m_active_vars[i];
                int64_t c = m_coeffs[v];
                if (!test_and_set_active(v) || c == 0) continue;
                m_coeffs[v] /= static_cast<int>(g);
                m_active_vars[j++] = v;
            }
            m_active_vars.shrink(j);
            m_bound = (m_bound + g - 1) / g;
            ++m_stats.m_num_cut;
        }        
    }

    void ba_solver::process_card(card& c, unsigned offset) {
        literal lit = c.lit();
        SASSERT(c.k() <= c.size());       
        SASSERT(lit == null_literal || value(lit) != l_undef);
        SASSERT(0 < offset);
        for (unsigned i = c.k(); i < c.size(); ++i) {
            process_antecedent(c[i], offset);
        }
        for (unsigned i = 0; i < c.k(); ++i) {
            inc_coeff(c[i], offset);                        
        }
        if (lit != null_literal) {
            uint64_t offset1 = static_cast<uint64_t>(offset) * c.k();
            if (offset1 > UINT_MAX) {
                m_overflow = true;
            }
            if (value(lit) == l_true) {
                process_antecedent(~lit, static_cast<unsigned>(offset1));
            }
            else {
                process_antecedent(lit, static_cast<unsigned>(offset1));
            }
        }
    }

    void ba_solver::process_antecedent(literal l, unsigned offset) {
        SASSERT(value(l) == l_false);
        bool_var v = l.var();
        unsigned level = lvl(v);

        if (!s().is_marked(v) && level == m_conflict_lvl) {
            s().mark(v);
            ++m_num_marks;
            if (_debug_conflict && _debug_consequent != null_literal && _debug_var2position[_debug_consequent.var()] < _debug_var2position[l.var()]) {
                IF_VERBOSE(0, verbose_stream() << "antecedent " << l << " is above consequent in stack\n";);
            }
        }
        inc_coeff(l, offset);                
    }   

    literal ba_solver::get_asserting_literal(literal p) {
        if (get_abs_coeff(p.var()) != 0) {
            return p;
        }
        unsigned level = 0;        
        for (unsigned i = 0; i < m_active_vars.size(); ++i) { 
            bool_var v = m_active_vars[i];
            literal lit(v, get_coeff(v) < 0);
            if (value(lit) == l_false && lvl(lit) > level) {
                p = lit;
                level = lvl(lit);
            }
        }
        return p;        
    }

    ba_solver::ba_solver(ast_manager& m, sat_internalizer& si)
        : m(m), si(si), m_pb(m),
          m_solver(nullptr), m_lookahead(nullptr), 
          m_constraint_id(0), m_ba(*this), m_sort(m_ba) {
        TRACE("ba", tout << this << "\n";);
        m_num_propagations_since_pop = 0;
    }

    ba_solver::~ba_solver() {
        m_stats.reset();
        for (constraint* c : m_constraints) {
            c->deallocate(m_allocator);
        }
        for (constraint* c : m_learned) {
            c->deallocate(m_allocator);
        }
    }

    void ba_solver::add_at_least(bool_var v, literal_vector const& lits, unsigned k) {
        literal lit = v == null_bool_var ? null_literal : literal(v, false);
        add_at_least(lit, lits, k, m_is_redundant);
    }

    ba_solver::constraint* ba_solver::add_at_least(literal lit, literal_vector const& lits, unsigned k, bool learned) {
        if (k == 1 && lit == null_literal) {
            literal_vector _lits(lits);
            s().mk_clause(_lits.size(), _lits.c_ptr(), learned);
            return nullptr;
        }
        if (!learned && clausify(lit, lits.size(), lits.c_ptr(), k)) {
            return nullptr;
        }
        void * mem = m_allocator.allocate(card::get_obj_size(lits.size()));
        constraint_base::initialize(mem, this);
        card* c = new (constraint_base::ptr2mem(mem)) card(next_id(), lit, lits, k);
        c->set_learned(learned);
        add_constraint(c);
        return c;
    }

    void ba_solver::add_constraint(constraint* c) {
        literal_vector lits(c->literals());
        if (c->learned()) {
            m_learned.push_back(c);
        }
        else {
            SASSERT(!m_solver || s().at_base_lvl());
            m_constraints.push_back(c);
        }
        literal lit = c->lit();
        if (c->learned() && m_solver && !s().at_base_lvl()) {
            SASSERT(lit == null_literal);
            // gets initialized after backjump.
            m_constraint_to_reinit.push_back(c);
        }
        else if (lit == null_literal) {
            init_watch(*c);
        }
        else {
            if (m_solver) m_solver->set_external(lit.var());
            watch_literal(lit, *c);
            watch_literal(~lit, *c);
        }        
        SASSERT(c->well_formed());
    }


    bool ba_solver::init_watch(constraint& c) {
        if (inconsistent()) return false;
        switch (c.tag()) {
        case card_t: return init_watch(c.to_card()); 
        case pb_t: return init_watch(c.to_pb());
        case xr_t: return init_watch(c.to_xr()); 
        }
        UNREACHABLE();
        return false;
    }   

    lbool ba_solver::add_assign(constraint& c, literal l) {
        switch (c.tag()) {
        case card_t: return add_assign(c.to_card(), l); 
        case pb_t: return add_assign(c.to_pb(), l); 
        case xr_t: return add_assign(c.to_xr(), l);
        }
        UNREACHABLE();
        return l_undef;
    }

    ba_solver::constraint* ba_solver::add_pb_ge(literal lit, svector<wliteral> const& wlits, unsigned k, bool learned) {
        bool units = true;
        for (wliteral wl : wlits) units &= wl.first == 1;
        if (k == 0 && lit == null_literal) {
            return nullptr;
        }
        if (!learned) {
            for (wliteral wl : wlits) s().set_external(wl.second.var()); 
        }
        if (units || k == 1) {
            literal_vector lits;
            for (wliteral wl : wlits) lits.push_back(wl.second);
            return add_at_least(lit, lits, k, learned);
        }
        void * mem = m_allocator.allocate(pb::get_obj_size(wlits.size()));
        constraint_base::initialize(mem, this);
        pb* p = new (constraint_base::ptr2mem(mem)) pb(next_id(), lit, wlits, k);
        p->set_learned(learned);
        add_constraint(p);
        return p;
    }

    void ba_solver::add_pb_ge(bool_var v, svector<wliteral> const& wlits, unsigned k) {
        literal lit = v == null_bool_var ? null_literal : literal(v, false);
        add_pb_ge(lit, wlits, k, m_is_redundant);
    }

    /*
      \brief return true to keep watching literal.
    */
    bool ba_solver::propagate(literal l, ext_constraint_idx idx) {
        SASSERT(value(l) == l_true);
        constraint& c = index2constraint(idx);
        if (c.lit() != null_literal && l.var() == c.lit().var()) {
            init_watch(c);
            return true;
        }
        else if (c.lit() != null_literal && value(c.lit()) != l_true) {
        // else if (c.lit() != null_literal && value(c.lit()) == l_false) {
            return true;
        }
        else {
            return l_undef != add_assign(c, ~l);
        }
    }

    double ba_solver::get_reward(card const& c, literal_occs_fun& literal_occs) const {
        unsigned k = c.k(), slack = 0;
        bool do_add = get_config().m_lookahead_reward == heule_schur_reward;
        double to_add = do_add ? 0: 1;
        for (literal l : c) {
            switch (value(l)) {
            case l_true:  --k; if (k == 0) return 0; 
            case l_undef: 
                if (do_add) to_add += literal_occs(l); 
                ++slack; break;
            case l_false: break;
            }
        }
        if (k >= slack) return 1;
        return pow(0.5, slack - k + 1) * to_add;
    }

    double ba_solver::get_reward(pb const& c, literal_occs_fun& occs) const {
        unsigned k = c.k(), slack = 0;
        bool do_add = get_config().m_lookahead_reward == heule_schur_reward;
        double to_add = do_add ? 0 : 1;
        double undefs = 0;
        for (wliteral wl : c) {
            literal l = wl.second;
            unsigned w = wl.first;
            switch (value(l)) {
            case l_true:  if (k <= w) return 0; 
            case l_undef: 
                if (do_add) to_add += occs(l);
                ++undefs; 
                slack += w; 
                break; // TBD multiplier factor on this
            case l_false: break;
            }
        }
        if (k >= slack || 0 == undefs) return 0;
        double avg = slack / undefs;
        return pow(0.5, (slack - k + 1)/avg) * to_add;
    }

    double ba_solver::get_reward(literal l, ext_justification_idx idx, literal_occs_fun& occs) const {
        constraint const& c = index2constraint(idx);
        switch (c.tag()) {
        case card_t: return get_reward(c.to_card(), occs);
        case pb_t: return get_reward(c.to_pb(), occs); 
        case xr_t: return 0;
        default: UNREACHABLE(); return 0;
        }
    }


    void ba_solver::ensure_parity_size(bool_var v) {
        if (m_parity_marks.size() <= static_cast<unsigned>(v)) {
            m_parity_marks.resize(static_cast<unsigned>(v) + 1, 0);
        }
    }
    
    unsigned ba_solver::get_parity(bool_var v) {
        return m_parity_marks.get(v, 0);
    }

    void ba_solver::inc_parity(bool_var v) {
        ensure_parity_size(v);
        m_parity_marks[v]++;
    }

    void ba_solver::reset_parity(bool_var v) {
        ensure_parity_size(v);
        m_parity_marks[v] = 0;
    }
    

    /**
       \brief retrieve a sufficient set of literals from p that imply l.
       
       Find partition: 

         - Ax + coeff*l + B*y >= k
         - all literals in x are false.
         - B < k

       Then x is an explanation for l

     */

    bool ba_solver::assigned_above(literal above, literal below) {
        unsigned l = lvl(above);
        SASSERT(l == lvl(below));
        if (l == 0) return false;
        unsigned start = s().m_scopes[l-1].m_trail_lim;
        literal_vector const& lits = s().m_trail;

#if 0
        IF_VERBOSE(10, verbose_stream() << "level " << l << " scope level " << s().scope_lvl() << " tail lim start: " 
                   << start << " size of lits: " << lits.size() << " num scopes " << s().m_scopes.size() << "\n";);
#endif

        for (unsigned sz = lits.size(); sz-- > start; ) {
            if (lits[sz] == above) return true;
            if (lits[sz] == below) return false;
        }
        UNREACHABLE();
        return false;
    }

    void ba_solver::get_antecedents(literal l, pb const& p, literal_vector& r) {
        TRACE("ba", display(tout << l << " level: " << s().scope_lvl() << " ", p, true););
        SASSERT(p.lit() == null_literal || value(p.lit()) == l_true);

        if (p.lit() != null_literal) {
            r.push_back(p.lit());
        }

        unsigned k = p.k();

        if (_debug_conflict) {
            IF_VERBOSE(0, display(verbose_stream(), p, true);
                       verbose_stream() << "literal: " << l << " value: " << value(l) << " num-watch: " << p.num_watch() << " slack: " << p.slack() << "\n";);
        }

        if (value(l) == l_false) {
            // The literal comes from a conflict.
            // it is forced true, but assigned to false.
            unsigned slack = 0;
            for (wliteral wl : p) {
                if (value(wl.second) != l_false) {
                    slack += wl.first;
                }
            }
            SASSERT(slack < k);
            for (wliteral wl : p) {
                literal lit = wl.second;
                if (lit != l && value(lit) == l_false) {
                    unsigned w = wl.first;
                    if (slack + w < k) {
                        slack += w;
                    }
                    else {
                        r.push_back(~lit);
                    }
                } 
            }
        }
        else {
            // comes from a unit propagation
            SASSERT(value(l) == l_true);
            unsigned coeff = 0, j = 0;
            for (; j < p.size(); ++j) {
                if (p[j].second == l) {
                    coeff = p[j].first;
                    break;
                }
            }
            
            ++j;
            if (j < p.num_watch()) {
                j = p.num_watch();
            }
            CTRACE("ba", coeff == 0, display(tout << l << " coeff: " << coeff << "\n", p, true);); 
            
            if (_debug_conflict) {
                std::cout << "coeff " << coeff << "\n";
            }

            SASSERT(coeff > 0);
            unsigned slack = p.max_sum() - coeff;
            
            // we need antecedents to be deeper than alit.
            for (; j < p.size(); ++j) {
                literal lit = p[j].second;
                unsigned w = p[j].first;
                if (l_false != value(lit)) {
                    // skip
                }
                else if (lvl(lit) > lvl(l)) {
                    // skip
                }
                else if (lvl(lit) == lvl(l) && assigned_above(~lit, l)) {
                    // skip
                }
                else if (slack + w < k) {
                    slack += w;
                }
                else {
                    r.push_back(~lit); 
                }
            }
        }
        SASSERT(validate_unit_propagation(p, r, l));
    }

    bool ba_solver::is_extended_binary(ext_justification_idx idx, literal_vector & r) {
        constraint const& c = index2constraint(idx);
        switch (c.tag()) {
        case card_t: {
            card const& ca = c.to_card();
            if (ca.size() == ca.k() + 1 && ca.lit() == null_literal) {
                r.reset();
                for (literal l : ca) r.push_back(l);
                return true;
            }
            else {
                return false;
            }
        }
        default:
            return false;
        }
    }


    void ba_solver::get_antecedents(literal l, card const& c, literal_vector& r) {
        if (l == ~c.lit()) {
            for (unsigned i = c.k() - 1; i < c.size(); ++i) {
                VERIFY(value(c[i]) == l_false);
                r.push_back(~c[i]);
            }   
            return;
        }
        DEBUG_CODE(
            bool found = false;
            for (unsigned i = 0; !found && i < c.k(); ++i) {
                found = c[i] == l;
            }
            CTRACE("ba",!found, s().display(tout << l << ":" << c << "\n"););
            SASSERT(found););
        
        VERIFY(c.lit() == null_literal || value(c.lit()) != l_false);
        if (c.lit() != null_literal) r.push_back(value(c.lit()) == l_true ? c.lit() : ~c.lit());
        for (unsigned i = c.k(); i < c.size(); ++i) {
            SASSERT(value(c[i]) == l_false);
            r.push_back(~c[i]);
        }
    }


    // ----------------------------
    // constraint generic methods

    void ba_solver::get_antecedents(literal l, ext_justification_idx idx, literal_vector & r) {
        get_antecedents(l, index2constraint(idx), r);
    }

    bool ba_solver::is_watched(literal lit, constraint const& c) const {
        return get_wlist(~lit).contains(watched(c.cindex()));
    }
    
    void ba_solver::unwatch_literal(literal lit, constraint& c) {
        watched w(c.cindex());
        get_wlist(~lit).erase(w);
        SASSERT(!is_watched(lit, c));
    }

    void ba_solver::watch_literal(literal lit, constraint& c) {
        if (c.is_pure() && lit == ~c.lit()) return;
        SASSERT(!is_watched(lit, c));
        watched w(c.cindex());
        get_wlist(~lit).push_back(w);
    }

    void ba_solver::get_antecedents(literal l, constraint const& c, literal_vector& r) {
        switch (c.tag()) {
        case card_t: get_antecedents(l, c.to_card(), r); break;
        case pb_t: get_antecedents(l, c.to_pb(), r); break;
        case xr_t: get_antecedents(l, c.to_xr(), r); break;
        default: UNREACHABLE(); break;
        }
    }

    void ba_solver::nullify_tracking_literal(constraint& c) {
        if (c.lit() != null_literal) {
            unwatch_literal(c.lit(), c);
            unwatch_literal(~c.lit(), c);
            c.nullify_literal();
        }
    }

    void ba_solver::clear_watch(constraint& c) {
        switch (c.tag()) {
        case card_t:
            clear_watch(c.to_card());
            break;
        case pb_t:
            clear_watch(c.to_pb());
            break;
        case xr_t:
            clear_watch(c.to_xr());
            break;
        default:
            UNREACHABLE();
        }
    }

    void ba_solver::remove_constraint(constraint& c, char const* reason) {
        TRACE("ba", display(tout << "remove ", c, true) << " " << reason << "\n";);
        IF_VERBOSE(21, display(verbose_stream() << "remove " << reason << " ", c, true););
        nullify_tracking_literal(c);
        clear_watch(c);
        c.set_removed();
        m_constraint_removed = true;
    }

    // --------------------------------
    // validation

    bool ba_solver::validate_unit_propagation(constraint const& c, literal l) const {
        return true;
        switch (c.tag()) {
        case card_t:  return validate_unit_propagation(c.to_card(), l); 
        case pb_t: return validate_unit_propagation(c.to_pb(), l);
        case xr_t: return true;
        default: UNREACHABLE(); break;
        }
        return false;
    }

    bool ba_solver::validate_conflict(constraint const& c) const {
        return eval(c) == l_false;
    }

    lbool ba_solver::eval(constraint const& c) const {
        lbool v1 = c.lit() == null_literal ? l_true : value(c.lit());
        switch (c.tag()) {
        case card_t: return eval(v1, eval(c.to_card())); 
        case pb_t:   return eval(v1, eval(c.to_pb()));
        case xr_t:  return eval(v1, eval(c.to_xr()));
        default: UNREACHABLE(); break;
        }
        return l_undef;        
    }

    lbool ba_solver::eval(model const& m, constraint const& c) const {
        lbool v1 = c.lit() == null_literal ? l_true : value(m, c.lit());
        switch (c.tag()) {
        case card_t: return eval(v1, eval(m, c.to_card())); 
        case pb_t:   return eval(v1, eval(m, c.to_pb()));
        case xr_t:   return eval(v1, eval(m, c.to_xr()));
        default: UNREACHABLE(); break;
        }
        return l_undef;        
    }

    lbool ba_solver::eval(lbool a, lbool b) const {
        if (a == l_undef || b == l_undef) return l_undef;
        return (a == b) ? l_true : l_false;
    }

    lbool ba_solver::eval(card const& c) const {
        unsigned trues = 0, undefs = 0;
        for (literal l : c) {
            switch (value(l)) {
            case l_true: trues++; break;
            case l_undef: undefs++; break;
            default: break;
            }
        }
        if (trues + undefs < c.k()) return l_false;
        if (trues >= c.k()) return l_true;
        return l_undef;        
    }

    lbool ba_solver::eval(model const& m, card const& c) const {
        unsigned trues = 0, undefs = 0;
        for (literal l : c) {
            switch (value(m, l)) {
            case l_true: trues++; break;
            case l_undef: undefs++; break;
            default: break;
            }
        }
        if (trues + undefs < c.k()) return l_false;
        if (trues >= c.k()) return l_true;
        return l_undef;        
    }

    lbool ba_solver::eval(model const& m, pb const& p) const {
        unsigned trues = 0, undefs = 0;
        for (wliteral wl : p) {
            switch (value(m, wl.second)) {
            case l_true: trues += wl.first; break;
            case l_undef: undefs += wl.first; break;
            default: break;
            }
        }
        if (trues + undefs < p.k()) return l_false;
        if (trues >= p.k()) return l_true;
        return l_undef;        
    }

    lbool ba_solver::eval(pb const& p) const {
        unsigned trues = 0, undefs = 0;
        for (wliteral wl : p) {
            switch (value(wl.second)) {
            case l_true: trues += wl.first; break;
            case l_undef: undefs += wl.first; break;
            default: break;
            }
        }
        if (trues + undefs < p.k()) return l_false;
        if (trues >= p.k()) return l_true;
        return l_undef;        
    }


    bool ba_solver::validate() {
        if (!validate_watch_literals()) {
            return false;
        }
        for (constraint* c : m_constraints) {
            if (!validate_watched_constraint(*c)) 
                return false;
        }
        for (constraint* c : m_learned) {
            if (!validate_watched_constraint(*c)) 
                return false;
        }
        return true;
    }

    bool ba_solver::validate_watch_literals() const {
        for (unsigned v = 0; v < s().num_vars(); ++v) {
            literal lit(v, false);
            if (lvl(lit) == 0) continue;
            if (!validate_watch_literal(lit)) return false;
            if (!validate_watch_literal(~lit)) return false;
        }        
        return true;
    }

    bool ba_solver::validate_watch_literal(literal lit) const {
        if (lvl(lit) == 0) return true;
        for (auto const & w : get_wlist(lit)) {
            if (w.get_kind() == watched::EXT_CONSTRAINT) {
                constraint const& c = index2constraint(w.get_ext_constraint_idx());
                if (!c.is_watching(~lit) && lit.var() != c.lit().var()) {
                    IF_VERBOSE(0, display(verbose_stream() << lit << " " << lvl(lit) << " is not watched in " << c << "\n", c, true););
                    UNREACHABLE();
                    return false;
                }
            }
        }
        return true;
    }

    bool ba_solver::validate_watched_constraint(constraint const& c) const {
        if (c.is_pb() && !validate_watch(c.to_pb(), null_literal)) {
            return false;
        }
        if (c.lit() != null_literal && value(c.lit()) != l_true) return true;
        SASSERT(c.lit() == null_literal || lvl(c.lit()) == 0 || (is_watched(c.lit(), c) && is_watched(~c.lit(), c)));
        if (eval(c) == l_true) {
            return true;
        }
        literal_vector lits(c.literals());
        for (literal l : lits) {
            if (lvl(l) == 0) continue;
            bool found = is_watched(l, c);
            if (found != c.is_watching(l)) {

                IF_VERBOSE(0, 
                           verbose_stream() << "Discrepancy of watched literal: " << l << " id: " << c.id() 
                           << " clause: " << c << (found?" is watched, but shouldn't be":" not watched, but should be") << "\n";
                           s().display_watch_list(verbose_stream() <<  l << ": ", get_wlist(l)) << "\n";
                           s().display_watch_list(verbose_stream() << ~l << ": ", get_wlist(~l)) << "\n";
                           verbose_stream() << "value: " << value(l) << " level: " << lvl(l) << "\n";
                           display(verbose_stream(), c, true);
                           if (c.lit() != null_literal) verbose_stream() << value(c.lit()) << "\n";);

                IF_VERBOSE(0, s().display_watches(verbose_stream()));

                UNREACHABLE();
                return false;
            }
        }
        return true;
    }

    bool ba_solver::validate_watch(pb const& p, literal alit) const {
        for (unsigned i = 0; i < p.size(); ++i) {
            literal l = p[i].second;
            if (l != alit && lvl(l) != 0 && is_watched(l, p) != (i < p.num_watch())) {
                IF_VERBOSE(0, display(verbose_stream(), p, true);
                           verbose_stream() << "literal " << l << " at position " << i << " " << is_watched(l, p) << "\n";);
                UNREACHABLE();
                return false;
            }
        }
        unsigned slack = 0;
        for (unsigned i = 0; i < p.num_watch(); ++i) {
            slack += p[i].first;
        }
        if (slack != p.slack()) {
            IF_VERBOSE(0, display(verbose_stream(), p, true););
            UNREACHABLE();
            return false;
        }
        return true;
    }

    
    /**
       \brief Lex on (glue, size)
    */
    struct constraint_glue_psm_lt {
        bool operator()(ba_solver::constraint const * c1, ba_solver::constraint const * c2) const {
            return 
                (c1->glue()  < c2->glue()) ||
                (c1->glue() == c2->glue() && 
                 (c1->psm() < c2->psm() || 
                  (c1->psm() == c2->psm() && c1->size() < c2->size())));
        }
    };

    void ba_solver::update_psm(constraint& c) const {
        unsigned r = 0;
        switch (c.tag()) {            
        case card_t: 
            for (literal l : c.to_card()) {                
                if (s().m_phase[l.var()] == !l.sign()) ++r;
            }
            break;
        case pb_t:
            for (wliteral l : c.to_pb()) {                
                if (s().m_phase[l.second.var()] == !l.second.sign()) ++r;
            }
            break;
        default:
            break;
        }
        c.set_psm(r);
    }
    
    unsigned ba_solver::max_var(unsigned w) const {
        for (constraint* cp : m_constraints) {
            w = cp->fold_max_var(w);
        }
        for (constraint* cp : m_learned) {
            w = cp->fold_max_var(w);
        }
        return w;
    }

    void ba_solver::gc() {
        if (m_learned.size() >= 2 * m_constraints.size() && 
            (s().at_search_lvl() || s().at_base_lvl())) {
            for (auto & c : m_learned) update_psm(*c);
            std::stable_sort(m_learned.begin(), m_learned.end(), constraint_glue_psm_lt());
            gc_half("glue-psm");
            cleanup_constraints(m_learned, true);
        }
    }

    void ba_solver::gc_half(char const* st_name) {
        TRACE("ba", tout << "gc\n";);
        unsigned sz     = m_learned.size();
        unsigned new_sz = sz/2;
        unsigned removed = 0;
        for (unsigned i = new_sz; i < sz; i++) {
            constraint* c = m_learned[i];
            if (!m_constraint_to_reinit.contains(c)) {
                remove_constraint(*c, "gc");
                c->deallocate(m_allocator);
                ++removed;
            }
            else {
                m_learned[new_sz++] = c;
            }
        }
        m_stats.m_num_gc += removed;
        m_learned.shrink(new_sz);
        IF_VERBOSE(2, verbose_stream() << "(sat-gc :strategy " << st_name << " :deleted " << removed << ")\n";);

    }

    lbool ba_solver::add_assign(card& c, literal alit) {
        // literal is assigned to false.        
        unsigned sz = c.size();
        unsigned bound = c.k();
        TRACE("ba", tout << "assign: " << c.lit() << ": " << ~alit << "@" << lvl(~alit) << " " << c << "\n";);

        SASSERT(0 < bound && bound <= sz);
        if (bound == sz) {
            if (c.lit() != null_literal && value(c.lit()) == l_undef) {
                assign(c, ~c.lit());
                return inconsistent() ? l_false : l_true;
            }
            set_conflict(c, alit);
            return l_false;
        }
        SASSERT(value(alit) == l_false);
        VERIFY(c.lit() == null_literal || value(c.lit()) != l_false);
        unsigned index = 0;
        for (index = 0; index <= bound; ++index) {
            if (c[index] == alit) {
                break;
            }
        }
        if (index == bound + 1) {
            // literal is no longer watched.
            return l_undef;
        }
        VERIFY(index <= bound);
        VERIFY(c[index] == alit);
        
        // find a literal to swap with:
        for (unsigned i = bound + 1; i < sz; ++i) {
            literal lit2 = c[i];
            if (value(lit2) != l_false) {
                c.swap(index, i);
                watch_literal(lit2, c);
                return l_undef;
            }
        }

        // conflict
        if (bound != index && value(c[bound]) == l_false) {
            TRACE("ba", tout << "conflict " << c[bound] << " " << alit << "\n";);
            if (c.lit() != null_literal && value(c.lit()) == l_undef) {
                if (index + 1 < bound) c.swap(index, bound - 1);
                assign(c, ~c.lit());
                return inconsistent() ? l_false : l_true;
            }
            set_conflict(c, alit);
            return l_false;
        }

        if (index != bound) {
            c.swap(index, bound);
        }        

        // TRACE("ba", tout << "no swap " << index << " " << alit << "\n";);
        // there are no literals to swap with,
        // prepare for unit propagation by swapping the false literal into 
        // position bound. Then literals in positions 0..bound-1 have to be
        // assigned l_true.

        if (c.lit() != null_literal && value(c.lit()) == l_undef) {
            return l_true;
        }

        for (unsigned i = 0; i < bound; ++i) {
            assign(c, c[i]);
        }

        if (c.learned() && c.glue() > 2) {
            unsigned glue;
            if (s().num_diff_false_levels_below(c.size(), c.begin(), c.glue()-1, glue)) {
                c.set_glue(glue);
            }
        }

        return inconsistent() ? l_false : l_true;
    }

    void ba_solver::asserted(literal l) {
    }


    check_result ba_solver::check() { return CR_DONE; }

    void ba_solver::push() {
        m_constraint_to_reinit_lim.push_back(m_constraint_to_reinit.size());
    }

    void ba_solver::pop(unsigned n) {        
        TRACE("sat_verbose", tout << "pop:" << n << "\n";);
        unsigned new_lim = m_constraint_to_reinit_lim.size() - n;
        m_constraint_to_reinit_last_sz = m_constraint_to_reinit_lim[new_lim];
        m_constraint_to_reinit_lim.shrink(new_lim);
        m_num_propagations_since_pop = 0;
    }

    void ba_solver::pop_reinit() {
        unsigned sz = m_constraint_to_reinit_last_sz;
        for (unsigned i = sz; i < m_constraint_to_reinit.size(); ++i) {
            constraint* c = m_constraint_to_reinit[i];
            if (!init_watch(*c) && !s().at_base_lvl()) {
                m_constraint_to_reinit[sz++] = c;
            }
        }
        m_constraint_to_reinit.shrink(sz);        
    }

    
    void ba_solver::simplify(constraint& c) {
        SASSERT(s().at_base_lvl());
        switch (c.tag()) {
        case card_t:
            simplify(c.to_card());
            break;
        case pb_t:
            simplify(c.to_pb());
            break;
        case xr_t:
            simplify(c.to_xr());
            break;
        default:
            UNREACHABLE();
        }                
    }

    void ba_solver::simplify() {        
        if (!s().at_base_lvl()) 
            s().pop_to_base_level();
        if (s().inconsistent())
            return;
        unsigned trail_sz, count = 0;
        do {
            trail_sz = s().init_trail_size();
            m_simplify_change = false;
            m_clause_removed = false;
            m_constraint_removed = false;
            for (unsigned sz = m_constraints.size(), i = 0; i < sz; ++i) simplify(*m_constraints[i]);
            for (unsigned sz = m_learned.size(), i = 0; i < sz; ++i) simplify(*m_learned[i]);
            init_use_lists();
            remove_unused_defs();
            set_non_external();
            elim_pure();
            for (unsigned sz = m_constraints.size(), i = 0; i < sz; ++i) subsumption(*m_constraints[i]);
            for (unsigned sz = m_learned.size(), i = 0; i < sz; ++i) subsumption(*m_learned[i]);    
            unit_strengthen();
            extract_xor();
            merge_xor();
            cleanup_clauses();
            cleanup_constraints();
            update_pure();
            count++;
        }        
        while (count < 10 && (m_simplify_change || trail_sz < s().init_trail_size()));

        // validate_eliminated();

        IF_VERBOSE(1, 
                   unsigned subs = m_stats.m_num_bin_subsumes + m_stats.m_num_clause_subsumes + m_stats.m_num_pb_subsumes;
                   verbose_stream() << "(ba.simplify" << " :constraints " << m_constraints.size();
                   if (!m_learned.empty()) verbose_stream() << " :lemmas " << m_learned.size();
                   if (subs > 0) verbose_stream() << " :subsumes " << subs;
                   if (m_stats.m_num_gc > 0) verbose_stream() << " :gc " << m_stats.m_num_gc;
                   verbose_stream() << ")\n";);

        // IF_VERBOSE(0, s().display(verbose_stream()));
        // mutex_reduction();
        // if (s().m_clauses.size() < 80000) lp_lookahead_reduction();
    }

    /*
     * ~lit does not occur in clauses
     * ~lit is only in one constraint use list
     * lit == C 
     * -> ignore assignments to ~lit for C
     * 
     * ~lit does not occur in clauses
     * lit is only in one constraint use list
     * lit == C
     * -> negate: ~lit == ~C
     */
    void ba_solver::update_pure() {
        //return;
        for (constraint* cp : m_constraints) {
            literal lit = cp->lit();
            if (lit != null_literal && 
                !cp->is_pure() &&
                value(lit) == l_undef && 
                get_wlist(~lit).size() == 1 &&              
                m_clause_use_list.get(lit).empty()) {
                clear_watch(*cp);
                cp->negate();
                lit.neg();
            }
            if (lit != null_literal && 
                !cp->is_pure() &&
                m_cnstr_use_list[(~lit).index()].size() == 1 &&
                get_wlist(lit).size() == 1 && 
                m_clause_use_list.get(~lit).empty()) { 
                cp->set_pure();
                get_wlist(~lit).erase(watched(cp->cindex())); // just ignore assignments to false
            }
        }
    }

    void ba_solver::mutex_reduction() {
        literal_vector lits;
        for (unsigned v = 0; v < s().num_vars(); ++v) {
            lits.push_back(literal(v, false));
            lits.push_back(literal(v, true));
        }
        vector<literal_vector> mutexes;
        s().find_mutexes(lits, mutexes);
        for (literal_vector& mux : mutexes) {
            if (mux.size() > 2) {
                IF_VERBOSE(1, verbose_stream() << "mux: " << mux << "\n";);
                for (unsigned i = 0; i < mux.size(); ++i) mux[i].neg(); 
                add_at_least(null_literal, mux, mux.size() - 1, false);
            }
        }
    }

    // -------------------------
    // sorting networks
    literal ba_solver::ba_sort::mk_false() {
        return ~mk_true();
    }

    literal ba_solver::ba_sort::mk_true() {
        if (m_true == null_literal) {
            bool_var v = s.s().mk_var(false, false);
            m_true = literal(v, false);
            s.s().mk_clause(1,&m_true);
        }
        VERIFY(m_true != null_literal);
        return m_true;
    }

    literal ba_solver::ba_sort::mk_not(literal l) { 
        return ~l; 
    }

    literal ba_solver::ba_sort::fresh(char const*) {
        bool_var v = s.s().mk_var(false, true);
        return literal(v, false);
    }


    literal ba_solver::ba_sort::mk_max(unsigned n, literal const* lits) {
        m_lits.reset();        
        for (unsigned i = 0; i < n; ++i) {
            if (lits[i] == m_true) return m_true;
            if (lits[i] == ~m_true) continue;
            m_lits.push_back(lits[i]);
        }
        switch (m_lits.size()) {
        case 0: 
            return ~m_true;
        case 1: 
            return m_lits[0];
        default: {
            literal max = fresh("max");
            for (unsigned i = 0; i < n; ++i) {
                s.s().mk_clause(~m_lits[i], max);
            }
            m_lits.push_back(~max);
            s.s().mk_clause(m_lits.size(), m_lits.c_ptr());
            return max;
        }
        }
    }

    literal ba_solver::ba_sort::mk_min(unsigned n, literal const* lits) {
        m_lits.reset();        
        for (unsigned i = 0; i < n; ++i) {
            if (lits[i] == ~m_true) return ~m_true;
            if (lits[i] == m_true) continue;
            m_lits.push_back(lits[i]);
        }
        switch (m_lits.size()) {
        case 0: 
            return m_true;
        case 1: 
            return m_lits[0];
        default: {
            literal min = fresh("min");
            for (unsigned i = 0; i < n; ++i) {
                s.s().mk_clause(~min, m_lits[i]);
                m_lits[i] = ~m_lits[i];
            }
            m_lits.push_back(min);
            s.s().mk_clause(m_lits.size(), m_lits.c_ptr());
            return min;
        }
        }
    }

    void ba_solver::ba_sort::mk_clause(unsigned n, literal const* lits) {
        m_lits.reset();
        m_lits.append(n, lits);
        s.s().mk_clause(n, m_lits.c_ptr());
    }

    std::ostream& ba_solver::ba_sort::pp(std::ostream& out, literal l) const {
        return out << l;
    }

    
    // -------------------------------
    // set literals equivalent

    void ba_solver::reserve_roots() {
        m_root_vars.reserve(s().num_vars(), false);
        for (unsigned i = m_roots.size(); i < 2 * s().num_vars(); ++i) {
            m_roots.push_back(to_literal(i));
        }
    }

    bool ba_solver::set_root(literal l, literal r) { 
        if (s().is_assumption(l.var())) {
            return false;
        }
        reserve_roots();
        m_roots[l.index()] = r;
        m_roots[(~l).index()] = ~r;
        m_root_vars[l.var()] = true;
        return true;
    }

    void ba_solver::flush_roots() {
        if (m_roots.empty()) return;
        reserve_roots();
        // validate();
        m_constraint_removed = false;
        for (unsigned sz = m_constraints.size(), i = 0; i < sz; ++i) 
            flush_roots(*m_constraints[i]);
        for (unsigned sz = m_learned.size(), i = 0; i < sz; ++i) 
            flush_roots(*m_learned[i]);
        cleanup_constraints();
        // validate();

        // validate_eliminated();
    }

    void ba_solver::validate_eliminated() {
        validate_eliminated(m_constraints);
        validate_eliminated(m_learned);
    }

    void ba_solver::validate_eliminated(ptr_vector<constraint> const& cs) {
        for (constraint const* c : cs) {
            if (c->learned()) continue;
            switch (c->tag()) {
            case tag_t::card_t:
                for (literal l : c->to_card()) {
                    VERIFY(!s().was_eliminated(l.var()));
                }
                break;
            case tag_t::pb_t:
                for (wliteral wl : c->to_pb()) {
                    VERIFY(!s().was_eliminated(wl.second.var()));
                }
                break;
            case tag_t::xr_t:
                for (literal l : c->to_xr()) {
                    VERIFY(!s().was_eliminated(l.var()));
                }
                break;                
            }
        }
    }

    void ba_solver::recompile(constraint& c) {
        if (c.id() == _bad_id) {
            IF_VERBOSE(0, display(verbose_stream() << "recompile\n", c, true););
        }
        switch (c.tag()) {
        case card_t:
            recompile(c.to_card());
            break;
        case pb_t:
            recompile(c.to_pb());
            break;
        case xr_t:
            add_xr(c.to_xr().literals(), c.learned());
            remove_constraint(c, "recompile xor");
            break;
        default:
            UNREACHABLE();
        }                
    }

    void ba_solver::recompile(card& c) {
        SASSERT(c.lit() == null_literal || is_watched(c.lit(), c));

        // pre-condition is that the literals, except c.lit(), in c are unwatched.
        if (c.id() == _bad_id) std::cout << "recompile: " << c << "\n";
        m_weights.resize(2*s().num_vars(), 0);
        for (literal l : c) {
            ++m_weights[l.index()];
        }
        unsigned k = c.k();
        bool all_units = true;
        unsigned sz = c.size();
        unsigned_vector coeffs;
        literal_vector lits;
        unsigned j = 0;
        for (unsigned i = 0; i < sz && 0 < k; ++i) {
            literal l = c[i];
            unsigned w = m_weights[l.index()];
            unsigned w2 = m_weights[(~l).index()];
            if (w == 0 || w < w2) {
                continue;
            }
            else if (k <= w2) {
                k = 0;
                break;
            }
            else {
                SASSERT(w2 <= w && w2 < k);
                k -= w2;
                w -= w2;
                m_weights[(~l).index()] = 0;        
                m_weights[l.index()] = 0;        
                if (w == 0) {
                    continue;
                }    
                else {
                    all_units &= (w == 1);
                    coeffs.push_back(w);
                    c[j++] = l;
                }
            }
        }
        sz = j;

        // clear weights
        for (literal l : c) {
            m_weights[l.index()] = 0;
            m_weights[(~l).index()] = 0;
        }       

        if (k == 0 && c.lit() == null_literal) {
            remove_constraint(c, "recompiled to true");
            return;
        }

        if (k == 1 && c.lit() == null_literal) {
            literal_vector lits(sz, c.literals().c_ptr());
            s().mk_clause(sz, lits.c_ptr(), c.learned());
            remove_constraint(c, "recompiled to clause");
            return;
        }

        if (sz == 0) {
            if (c.lit() == null_literal) {
                if (k > 0) {
                    s().mk_clause(0, nullptr, true);
                }
            }
            else if (k > 0) {
                literal lit = ~c.lit();
                s().mk_clause(1, &lit, c.learned());
            }
            else {
                literal lit = c.lit();
                s().mk_clause(1, &lit, c.learned());
            }
            remove_constraint(c, "recompiled to clause");
            return;
        }
        if (all_units && sz < k) {
            if (c.lit() == null_literal) {
                s().mk_clause(0, nullptr, true);            
            }
            else {
                literal lit = ~c.lit();
                s().mk_clause(1, &lit, c.learned());
            }
            remove_constraint(c, "recompiled to clause");
            return;            
        }
        VERIFY(!all_units || c.size() - c.k() >= sz - k);
        c.set_size(sz);
        c.set_k(k);    

        if (all_units && clausify(c)) {
            return;
        }

        if (!all_units) {            
            TRACE("ba", tout << "replacing by pb: " << c << "\n";);
            m_wlits.reset();
            for (unsigned i = 0; i < sz; ++i) {
                m_wlits.push_back(wliteral(coeffs[i], c[i]));
            }
            literal root = c.lit();
            remove_constraint(c, "recompiled to pb");
            add_pb_ge(root, m_wlits, k, c.learned());
        }
        else {
            if (c.lit() == null_literal || value(c.lit()) == l_true) {
                init_watch(c);
            }
            SASSERT(c.lit() == null_literal || is_watched(c.lit(), c));
            SASSERT(c.well_formed());
        }        
    }

    bool ba_solver::clausify(literal lit, unsigned n, literal const* lits, unsigned k) {
        return false;
        bool is_def = lit != null_literal;
        if ((!is_def || !s().was_eliminated(lit)) && 
            !std::any_of(lits, lits + n, [&](literal l) { return s().was_eliminated(l); })) {
            literal def_lit = m_sort.ge(is_def, k, n, lits);
            if (is_def) {
                s().mk_clause(~lit,  def_lit);
                s().mk_clause( lit, ~def_lit);
            }
            return true;
        }
        return false;
    }


    bool ba_solver::clausify(card& c) {
        return false;
        if (get_config().m_card_solver)
            return false;

        //
        // TBD: conditions for when to clausify are TBD and
        // handling of conditional cardinality as well.
        // 
        if (!c.learned() && clausify(c.lit(), c.size(), c.begin(), c.k())) {
            IF_VERBOSE(1, verbose_stream() << "clausify " << c << "\n";);                       
            // compiled
        }        
        remove_constraint(c, "recompiled to clauses");
        return true;        
    }

    bool ba_solver::clausify(pb& p) {
        return false;
        if (get_config().m_card_solver) 
            return false;
        
        bool ok = !p.learned();
        bool is_def = p.lit() != null_literal;
        for (wliteral wl : p) {
            ok &= !s().was_eliminated(wl.second);
        }
        ok &= !is_def || !s().was_eliminated(p.lit());
        if (!ok) {
            remove_constraint(p, "recompiled to clauses");
            return true;
        }

        if (is_cardinality(p, m_lemma)) {
            literal lit = m_sort.ge(is_def, p.k(), m_lemma.size(), m_lemma.c_ptr());
            if (is_def) {
                s().mk_clause(p.lit(), ~lit);
                s().mk_clause(~p.lit(), lit);
            }
            remove_constraint(p, "recompiled to clauses");
            return true;
        }
        return false;
    }

    bool ba_solver::is_cardinality(pb const& p, literal_vector& lits) {
        lits.reset();
        p.size();
        for (wliteral wl : p) {
            if (lits.size() > 2*p.size() + wl.first) {
                return false;
            }
            for (unsigned i = 0; i < wl.first; ++i) {
                lits.push_back(wl.second);
            }
        }
        return true;
    }

    void ba_solver::split_root(constraint& c) {
        switch (c.tag()) {
        case card_t: split_root(c.to_card()); break;
        case pb_t: split_root(c.to_pb()); break;
        case xr_t: NOT_IMPLEMENTED_YET(); break;
        }
    }

    void ba_solver::flush_roots(constraint& c) {
        if (c.lit() != null_literal && !is_watched(c.lit(), c)) {
            watch_literal(c.lit(), c);
            watch_literal(~c.lit(), c);
        }
        SASSERT(c.lit() == null_literal || is_watched(c.lit(), c));
        bool found = c.lit() != null_literal && m_root_vars[c.lit().var()];
        for (unsigned i = 0; !found && i < c.size(); ++i) {
            found = m_root_vars[c.get_lit(i).var()];
        }
        if (!found) return;
        clear_watch(c);
        
        // this could create duplicate literals
        for (unsigned i = 0; i < c.size(); ++i) {
            literal lit = m_roots[c.get_lit(i).index()];
            c.set_lit(i, lit);
        }

        literal root = c.lit();
        if (root != null_literal && m_roots[root.index()] != root) {
            root = m_roots[root.index()];
            nullify_tracking_literal(c);
            c.update_literal(root);
            watch_literal(root, c);
            watch_literal(~root, c);
        }

        bool found_dup = false;
        bool found_root = false;
        init_visited();
        for (unsigned i = 0; i < c.size(); ++i) {
            literal l = c.get_lit(i);
            if (is_visited(l)) {
                found_dup = true;
                break;
            }
            else {
                mark_visited(l);
                mark_visited(~l);
            }
        }
        for (unsigned i = 0; i < c.size(); ++i) {
            found_root |= c.get_lit(i).var() == root.var();
        }

        if (found_root) {
            split_root(c);
            c.negate();
            split_root(c);
            remove_constraint(c, "flush roots");
        }
        else if (found_dup) {
            recompile(c);
        }
        else {
            if (c.lit() == null_literal || value(c.lit()) != l_undef) init_watch(c);            
            SASSERT(c.well_formed());
        }
    }

    unsigned ba_solver::get_num_unblocked_bin(literal l) {
        return s().m_simplifier.num_nonlearned_bin(l);
    }

    /*
      \brief garbage collection.
      This entails 
      - finding pure literals, 
      - setting literals that are not used in the extension to non-external.
      - subsumption
      - resolution 
      - blocked literals
     */
    void ba_solver::init_use_lists() {
        m_clause_use_list.init(s().num_vars());
        m_cnstr_use_list.reset();
        m_cnstr_use_list.resize(2*s().num_vars());
        for (clause* c : s().m_clauses) {
            if (!c->frozen()) 
                m_clause_use_list.insert(*c);
        }
        for (constraint* cp : m_constraints) {
            literal lit = cp->lit();
            if (lit != null_literal) {
                m_cnstr_use_list[lit.index()].push_back(cp);
                m_cnstr_use_list[(~lit).index()].push_back(cp);
            }
            switch (cp->tag()) {
            case card_t: {
                card& c = cp->to_card();
                for (literal l : c) {
                    m_cnstr_use_list[l.index()].push_back(&c);
                    if (lit != null_literal) m_cnstr_use_list[(~l).index()].push_back(&c);
                }  
                break;
            }
            case pb_t: {
                pb& p = cp->to_pb();
                for (wliteral wl : p) {
                    literal l = wl.second;
                    m_cnstr_use_list[l.index()].push_back(&p);
                    if (lit != null_literal) m_cnstr_use_list[(~l).index()].push_back(&p);
                }
                break;
            }
            case xr_t: {
                xr& x = cp->to_xr();
                for (literal l : x) {
                    m_cnstr_use_list[l.index()].push_back(&x);
                    m_cnstr_use_list[(~l).index()].push_back(&x);
                }             
                break;
            }
            }
        }
    }

    void ba_solver::remove_unused_defs() {
        if (incremental_mode()) return;
        // remove constraints where indicator literal isn't used.
        for (constraint* cp : m_constraints) {
            constraint& c = *cp;
            literal lit = c.lit();
            switch (c.tag()) {
            case card_t: 
            case pb_t: {
                if (lit != null_literal &&
                    value(lit) == l_undef && 
                    use_count(lit) == 1 &&
                    use_count(~lit) == 1 &&
                    get_num_unblocked_bin(lit) == 0 && 
                    get_num_unblocked_bin(~lit) == 0) {
                    remove_constraint(c, "unused def");
                }
                break;
            }
            default:
                break;
            }
        }
    }

    bool ba_solver::incremental_mode() const {
        sat_simplifier_params p(s().m_params);
        bool incremental_mode = s().get_config().m_incremental && !p.override_incremental();
        incremental_mode |=  s().tracking_assumptions();
        return incremental_mode;
    }
    
    unsigned ba_solver::set_non_external() {
        // set variables to be non-external if they are not used in theory constraints.
        unsigned ext = 0;
        if (!incremental_mode()) {
            for (unsigned v = 0; v < s().num_vars(); ++v) {
                literal lit(v, false);
                if (s().is_external(v) && 
                    m_cnstr_use_list[lit.index()].empty() && 
                    m_cnstr_use_list[(~lit).index()].empty()) {
                    s().set_non_external(v);
                    ++ext;
                }            
            }
        }
        // ensure that lemmas use only non-eliminated variables
        for (constraint* cp : m_learned) {
            constraint& c = *cp;
            if (c.was_removed()) continue;
            SASSERT(c.lit() == null_literal);
            for (unsigned i = 0; i < c.size(); ++i) {
                bool_var v = c.get_lit(i).var();
                if (s().was_eliminated(v)) {
                    remove_constraint(c, "contains eliminated var");                    
                    break;
                }
            }
        }
        return ext;
    }

    bool ba_solver::elim_pure(literal lit) {
        if (value(lit) == l_undef && !m_cnstr_use_list[lit.index()].empty() && 
            use_count(~lit) == 0 && get_num_unblocked_bin(~lit) == 0) {
            IF_VERBOSE(100, verbose_stream() << "pure literal: " << lit << "\n";);
            s().assign_scoped(lit);
            return true;
        }
        return false;
    }

    unsigned ba_solver::elim_pure() {
        if (!get_config().m_elim_vars || incremental_mode()) {
            return 0;
        }
        // eliminate pure literals
        unsigned pure_literals = 0;
        for (unsigned v = 0; v < s().num_vars(); ++v) {
            literal lit(v, false);
            if (value(v) != l_undef) continue;
            if (m_cnstr_use_list[lit.index()].empty() &&
                m_cnstr_use_list[(~lit).index()].empty()) continue;

            if (elim_pure(lit) || elim_pure(~lit)) {
                ++pure_literals;
            }
        }
        return pure_literals;
    }

    /**
     * Strengthen inequalities using binary implication information.
     *
     * x -> ~y, x -> ~z,   y + z + u >= 2
     * ----------------------------------
     *       y + z + u + ~x >= 3
     * 
     * for c : constraints
     *   for l : c:
     *    slack <- of c under root(~l)
     *    if slack < 0:
     *       add ~root(~l) to c, k <- k + 1
     */ 
    void ba_solver::unit_strengthen() {
        big big(s().m_rand);
        big.init(s(), true);
        for (unsigned sz = m_constraints.size(), i = 0; i < sz; ++i) 
            unit_strengthen(big, *m_constraints[i]);
        for (unsigned sz = m_learned.size(), i = 0; i < sz; ++i) 
            unit_strengthen(big, *m_learned[i]);          
    }

    void ba_solver::unit_strengthen(big& big, constraint& c) {
        if (c.was_removed()) return;
        switch (c.tag()) {
        case card_t: 
            unit_strengthen(big, c.to_card());
            break;
        case pb_t: 
            unit_strengthen(big, c.to_pb());
            break;
        default:
            break;                
        }
    }

    void ba_solver::unit_strengthen(big& big, pb_base& p) {
        if (p.lit() != null_literal) return;
        unsigned sz = p.size();
        for (unsigned i = 0; i < sz; ++i) {
            literal u = p.get_lit(i);
            literal r = big.get_root(u);
            if (r == u) continue;
            unsigned k = p.k(), b = 0;
            for (unsigned j = 0; j < sz; ++j) {
                literal v = p.get_lit(j);
                if (r == big.get_root(v)) {
                    b += p.get_coeff(j);
                }
            }            
            if (b > k) {
                r.neg();
                unsigned coeff = b - k;
                
                svector<wliteral> wlits;
                // add coeff * r to p
                wlits.push_back(wliteral(coeff, r));
                for (unsigned j = 0; j < sz; ++j) {
                    u = p.get_lit(j);
                    unsigned c = p.get_coeff(j);
                    if (r == u) {
                        wlits[0].first += c;
                    }
                    else if (~r == u) {
                        if (coeff == c) {
                            wlits[0] = wlits.back();
                            wlits.pop_back();
                            b -= c;
                        }
                        else if (coeff < c) {
                            wlits[0].first = c - coeff;
                            wlits[0].second.neg();
                            b -= coeff;
                        }
                        else {
                            // coeff > c
                            wlits[0].first = coeff - c;
                            b -= c;
                        }
                    }
                    else {
                        wlits.push_back(wliteral(c, u));
                    }
                }
                ++m_stats.m_num_big_strengthenings;
                p.set_removed();
                add_pb_ge(null_literal, wlits, b, p.learned());
                return;
            }
        }
    }

    void ba_solver::subsumption(constraint& cnstr) {
        if (cnstr.was_removed()) return;
        switch (cnstr.tag()) {
        case card_t: {
            card& c = cnstr.to_card();
            if (c.k() > 1) subsumption(c);
            break;
        }
        case pb_t: {
            pb& p = cnstr.to_pb();
            if (p.k() > 1) subsumption(p);
            break;
        }
        default:
            break;                
        }
    }

    void ba_solver::init_visited() { s().init_visited(); }
    void ba_solver::mark_visited(literal l) { s().mark_visited(l); }
    void ba_solver::mark_visited(bool_var v) { s().mark_visited(v); }
    bool ba_solver::is_visited(bool_var v) const { return s().is_visited(v); }    
    bool ba_solver::is_visited(literal l) const { return s().is_visited(l); }

    void ba_solver::cleanup_clauses() {
        if (m_clause_removed) {
            cleanup_clauses(s().m_clauses);
            cleanup_clauses(s().m_learned);
        }
    }

    void ba_solver::cleanup_clauses(clause_vector& clauses) {
        // version in simplify first clears 
        // all watch literals, then reinserts them.
        // this ensures linear time cleanup.
        clause_vector::iterator it = clauses.begin();
        clause_vector::iterator end = clauses.end();
        clause_vector::iterator it2 = it;
        for (; it != end; ++it) {
            clause* c = *it;
            if (c->was_removed() && s().can_delete(*c)) {
                s().detach_clause(*c);
                s().del_clause(*c);
            }
            else {
                if (it2 != it) {
                    *it2 = *it;
                }
                ++it2;
            }
        }
        clauses.set_end(it2);        
    }
    
    void ba_solver::cleanup_constraints() {
        if (m_constraint_removed) {
            cleanup_constraints(m_constraints, false);
            cleanup_constraints(m_learned, true);
            m_constraint_removed = false;
        }
    }

    void ba_solver::cleanup_constraints(ptr_vector<constraint>& cs, bool learned) {
        ptr_vector<constraint>::iterator it = cs.begin();
        ptr_vector<constraint>::iterator it2 = it;
        ptr_vector<constraint>::iterator end = cs.end();
        for (; it != end; ++it) {
            constraint& c = *(*it);
            if (c.was_removed()) {
                clear_watch(c);
                nullify_tracking_literal(c);
                c.deallocate(m_allocator);
            }
            else if (learned && !c.learned()) {
                m_constraints.push_back(&c);
            }
            else {
                if (it != it2) {
                    *it2 = *it;
                }
                ++it2;
            }
        }
        cs.set_end(it2);
    }


    /*
      \brief subsumption between two cardinality constraints
      - A >= k       subsumes A + B >= k' for k' <= k
      - A + A' >= k  subsumes A + B >= k' for k' + |A'| <= k
      - A + lit >= k self subsumes A + ~lit + B >= k' into A + B >= k' for k' <= k
      - version that generalizes self-subsumption to more than one literal
        A + ~L + B >= k'   =>    A + B >= k'    if A + A' + L >= k and k' + |L| + |A'| <= k
     */
    bool ba_solver::subsumes(card& c1, card& c2, literal_vector & comp) {
        if (c2.lit() != null_literal) return false; 

        unsigned c2_exclusive = 0;
        unsigned common = 0;
        comp.reset();
        for (literal l : c2) {
            if (is_visited(l)) {
                ++common;
            }
            else if (is_visited(~l)) {
                comp.push_back(l);
            }
            else {
                ++c2_exclusive;
            }
        }

        unsigned c1_exclusive = c1.size() - common - comp.size();
        return c1_exclusive + c2.k() + comp.size() <= c1.k();
    }

    /*
      \brief L + A >= k subsumes L + C if |A| < k
             A + L + B >= k self-subsumes A + ~L + C >= 1
             if k + 1 - |B| - |C| - |L| > 0
     */
    bool ba_solver::subsumes(card& c1, clause& c2, bool& self) {
        unsigned common = 0, complement = 0, c2_exclusive = 0;
        self = false;
        
        for (literal l : c2) {
            if (is_visited(l)) {
                ++common;
            }
            else if (is_visited(~l)) {
                ++complement;
            }
            else {
                ++c2_exclusive;
            }
        }
        unsigned c1_exclusive = c1.size() - common - complement;
        if (complement > 0 && c1.k() + 1 > c1_exclusive + c2_exclusive + common) {
            self = true;
            return true; 
        }
        return c1.size() - common < c1.k();
    }

    /*
      \brief Ax >= k subsumes By >= k' if
      all coefficients in A are <= B and k >= k'
     */
    bool ba_solver::subsumes(pb const& p1, pb_base const& p2) {
        if (p1.k() < p2.k() || p1.size() > p2.size()) return false;
        unsigned num_sub = 0;
        for (unsigned i = 0; i < p2.size(); ++i) {
            literal l = p2.get_lit(i);
            if (is_visited(l) && m_weights[l.index()] <= p2.get_coeff(i)) {
                ++num_sub;
            }
            if (p1.size() + i > p2.size() + num_sub) return false;
        }
        return num_sub == p1.size();
    }

    void ba_solver::subsumes(pb& p1, literal lit) {
        for (constraint* c : m_cnstr_use_list[lit.index()]) {
            if (c == &p1 || c->was_removed()) continue;
            bool s = false;
            switch (c->tag()) {
            case card_t: 
                s = subsumes(p1, c->to_card()); 
                break;
            case pb_t: 
                s = subsumes(p1, c->to_pb()); 
                break;
            default: 
                break;
            }
            if (s) {
                ++m_stats.m_num_pb_subsumes;                
                set_non_learned(p1);
                remove_constraint(*c, "subsumed");
            }
        }
    }

    literal ba_solver::get_min_occurrence_literal(card const& c) {
        unsigned occ_count = UINT_MAX;
        literal lit = null_literal;
        for (literal l : c) {
            unsigned occ_count1 = m_cnstr_use_list[l.index()].size();
            if (occ_count1 < occ_count) {
                lit = l;
                occ_count = occ_count1;
            }
        }
        return lit;
    }

    void ba_solver::card_subsumption(card& c1, literal lit) {
        literal_vector slit;
        for (constraint* c : m_cnstr_use_list[lit.index()]) {
            if (!c->is_card() || c == &c1 || c->was_removed()) {
                continue;
            }
            card& c2 = c->to_card();

            SASSERT(&c1 != &c2);
            if (subsumes(c1, c2, slit)) {
                if (slit.empty()) {
                    TRACE("ba", tout << "subsume cardinality\n" << c1 << "\n" << c2 << "\n";);
                    remove_constraint(c2, "subsumed");
                    ++m_stats.m_num_pb_subsumes;
                    set_non_learned(c1);
                }
                else {
                    TRACE("ba", tout << "self subsume cardinality\n";);
                    IF_VERBOSE(11, 
                               verbose_stream() << "self-subsume cardinality\n"; 
                               verbose_stream() << c1 << "\n";
                               verbose_stream() << c2 << "\n";);
                    clear_watch(c2);                    
                    unsigned j = 0;
                    for (unsigned i = 0; i < c2.size(); ++i) {
                        if (!is_visited(~c2[i])) {
                            c2[j++] = c2[i];
                        }
                    }
                    c2.set_size(j);
                    init_watch(c2);
                    m_simplify_change = true;
                }
            }
        }
    }

    void ba_solver::clause_subsumption(card& c1, literal lit, clause_vector& removed_clauses) {
        SASSERT(!c1.was_removed());
        clause_use_list& occurs = m_clause_use_list.get(lit);
        clause_use_list::iterator it = occurs.mk_iterator();
        while (!it.at_end()) {
            clause& c2 = it.curr();
            bool self;
            if (!c2.was_removed() && subsumes(c1, c2, self)) {
                if (self) {
                    // self-subsumption is TBD
                }
                else {
                    TRACE("ba", tout << "remove\n" << c1 << "\n" << c2 << "\n";);
                    removed_clauses.push_back(&c2);
                    ++m_stats.m_num_clause_subsumes;
                    set_non_learned(c1);
                }            
            }
            it.next();
        }
    }

    void ba_solver::set_non_learned(constraint& c) {
        literal lit = c.lit();
        if (lit != null_literal) {
            s().set_external(lit.var());
        }
        switch (c.tag()) {
        case card_t:
            for (literal lit : c.to_card()) {
                s().set_external(lit.var());             
                SASSERT(!s().was_eliminated(lit.var()));
            }
            break;
        case pb_t:
            for (wliteral wl : c.to_pb()) {
                s().set_external(wl.second.var());
                SASSERT(!s().was_eliminated(wl.second.var()));
            }
            break;
        default:
            for (literal lit : c.to_xr()) {
                s().set_external(lit.var());
                SASSERT(!s().was_eliminated(lit.var()));
            }
            break;
        }
        c.set_learned(false);
    }

    void ba_solver::binary_subsumption(card& c1, literal lit) {
        if (c1.k() + 1 != c1.size()) return;
        SASSERT(is_visited(lit));
        SASSERT(!c1.was_removed());
        watch_list & wlist = get_wlist(~lit);
        watch_list::iterator it = wlist.begin();
        watch_list::iterator it2 = it;
        watch_list::iterator end = wlist.end();
        for (; it != end; ++it) {
            watched w = *it;
            if (w.is_binary_clause() && is_visited(w.get_literal())) {
                ++m_stats.m_num_bin_subsumes;
                IF_VERBOSE(20, verbose_stream() << c1 << " subsumes (" << lit << " " << w.get_literal() << ")\n";);
                if (!w.is_learned()) {
                    set_non_learned(c1);
                }
            }
            else {
                if (it != it2) {
                    *it2 = *it;
                }
                ++it2;
            }
        }
        wlist.set_end(it2);        
    }

    void ba_solver::subsumption(card& c1) {
        if (c1.was_removed() || c1.lit() != null_literal) {
            return;
        }
        clause_vector removed_clauses;
        init_visited();
        for (literal l : c1) mark_visited(l);  
        for (unsigned i = 0; i < std::min(c1.size(), c1.k() + 1); ++i) {
            literal lit = c1[i];            
            card_subsumption(c1, lit);
            clause_subsumption(c1, lit, removed_clauses);
            binary_subsumption(c1, lit);                   
        }
        m_clause_removed |= !removed_clauses.empty();
        for (clause *c : removed_clauses) {
            c->set_removed(true);
            m_clause_use_list.erase(*c);
        }
    }

    void ba_solver::subsumption(pb& p1) {
        if (p1.was_removed() || p1.lit() != null_literal) {
            return;
        }
        init_visited();
        for (wliteral l : p1) {
            SASSERT(m_weights.size() <= l.second.index() || m_weights[l.second.index()] == 0);
            m_weights.setx(l.second.index(), l.first, 0);
            mark_visited(l.second);  
        }
        for (unsigned i = 0; i < std::min(10u, p1.num_watch()); ++i) {
            unsigned j = s().m_rand() % p1.num_watch();
            subsumes(p1, p1[j].second);
        }
        for (wliteral l : p1) {
            m_weights[l.second.index()] = 0;
        }        
    }

    void ba_solver::clauses_modifed() {}

    lbool ba_solver::get_phase(bool_var v) { return l_undef; }

    /*
      \brief lit <=> conjunction of unconstrained lits
     */
    void ba_solver::assert_unconstrained(literal lit, literal_vector const& lits) {
        if (lit == null_literal) {
            for (literal l : lits) {
                if (value(l) == l_undef) {
                    s().assign_scoped(l);
                }
            }
        }
        else {
            // add clauses for: lit <=> conjunction of undef literals
            SASSERT(value(lit) == l_undef);
            literal_vector cl;
            cl.push_back(lit);
            for (literal l : lits) {
                if (value(l) == l_undef) {
                    s().mk_clause(~lit, l);
                    cl.push_back(~l);
                }
            }    
            s().mk_clause(cl);
        }
    }

    extension* ba_solver::copy(solver* s) {
        return fresh(s, m, si);
    }

    th_solver* ba_solver::fresh(solver* s, ast_manager& m, sat_internalizer& si) {
        ba_solver* result = alloc(ba_solver, m, si);
        result->set_solver(s);
        copy_constraints(result, m_constraints);
        return result;
    }

    void ba_solver::copy_constraints(ba_solver* result, ptr_vector<constraint> const& constraints) {
        literal_vector lits;
        svector<wliteral> wlits;
        for (constraint* cp : constraints) {
            switch (cp->tag()) {
            case card_t: {
                card const& c = cp->to_card();
                lits.reset();
                for (literal l : c) lits.push_back(l);
                result->add_at_least(c.lit(), lits, c.k(), c.learned());        
                break;
            }
            case pb_t: {
                pb const& p = cp->to_pb();
                wlits.reset();
                for (wliteral w : p) {
                    wlits.push_back(w);
                }
                result->add_pb_ge(p.lit(), wlits, p.k(), p.learned());
                break;
            }
            case xr_t: {
                xr const& x = cp->to_xr();
                lits.reset();
                for (literal l : x) lits.push_back(l);
                result->add_xr(lits, x.learned());        
                break;
            }
            default:
                UNREACHABLE();
            }                
        }
    }

    void ba_solver::init_use_list(ext_use_list& ul) {
        ul.init(s().num_vars());
        for (constraint const* cp : m_constraints) {
            ext_constraint_idx idx = cp->cindex();
            if (cp->lit() != null_literal) {
                ul.insert(cp->lit(), idx);
                ul.insert(~cp->lit(), idx);                
            }
            switch (cp->tag()) {
            case card_t: {
                card const& c = cp->to_card();
                for (literal l : c) {
                    ul.insert(l, idx);
                }
                break;
            }
            case pb_t: {
                pb const& p = cp->to_pb();
                for (wliteral w : p) {
                    ul.insert(w.second, idx);
                }
                break;
            }
            case xr_t: {
                xr const& x = cp->to_xr();
                for (literal l : x) {
                    ul.insert(l, idx);
                    ul.insert(~l, idx);
                }
                break;
            }
            default:
                UNREACHABLE();
            }                            
        }
    }

    //
    // literal is used in a clause (C or l), it
    // it occurs negatively in constraint c.
    // all literals in C are marked
    // 
    bool ba_solver::is_blocked(literal l, ext_constraint_idx idx) {
        constraint const& c = index2constraint(idx);
        simplifier& sim = s().m_simplifier;
        if (c.lit() != null_literal) return false;
        switch (c.tag()) {
        case card_t: {
            card const& ca = c.to_card();
            unsigned weight = 0;
            for (literal l2 : ca) {
                if (sim.is_marked(~l2)) ++weight;
            }
            return weight >= ca.k();
        }
        case pb_t: {
            pb const& p = c.to_pb();
            unsigned weight = 0, offset = 0;
            for (wliteral l2 : p) {
                if (~l2.second == l) {
                    offset = l2.first;
                    break;
                }
            }
            SASSERT(offset != 0);
            for (wliteral l2 : p) {
                if (sim.is_marked(~l2.second)) {
                    weight += std::min(offset, l2.first);
                }
            }
            return weight >= p.k();
        }
        default:
            break;
        }
        return false;
    }


    void ba_solver::find_mutexes(literal_vector& lits, vector<literal_vector> & mutexes) {
        literal_set slits(lits);
        bool change = false;
        for (constraint* cp : m_constraints) {
            if (!cp->is_card()) continue;
            card const& c = cp->to_card();
            if (c.size() == c.k() + 1) {
                literal_vector mux;
                for (literal lit : c) {
                    if (slits.contains(~lit)) {
                        mux.push_back(~lit);
                    }
                }
                if (mux.size() <= 1) {
                    continue;
                }

                for (literal m : mux) {
                    slits.remove(m);
                }
                change = true;
                mutexes.push_back(mux);
            }
        }        
        if (!change) return;
        lits.reset();
        for (literal l : slits) {
            lits.push_back(l);
        }
    }

    void ba_solver::display(std::ostream& out, ineq const& ineq, bool values) const {
        for (unsigned i = 0; i < ineq.size(); ++i) {
            if (ineq.coeff(i) != 1) out << ineq.coeff(i) << "*";
            out << ineq.lit(i) << " ";
            if (values) out << value(ineq.lit(i)) << " ";
        }
        out << ">= " << ineq.m_k << "\n";
    }


    void ba_solver::display_lit(std::ostream& out, literal lit, unsigned sz, bool values) const {
        if (lit != null_literal) {
            if (values) {
                out << lit << "[" << sz << "]";
                out << "@(" << value(lit);
                if (value(lit) != l_undef) {
                    out << ":" << lvl(lit);
                }
                out << "): ";
            }
            else {
                out << lit << " == ";
            }
        }
    }

    void ba_solver::display(std::ostream& out, card const& c, bool values) const {
        display_lit(out, c.lit(), c.size(), values);
        for (unsigned i = 0; i < c.size(); ++i) {
            literal l = c[i];
            out << l;
            if (values) {
                out << "@(" << value(l);
                if (value(l) != l_undef) {
                    out << ":" << lvl(l);
                }
                out << ") ";
            }
            else {
                out << " ";
            }
        }
        out << ">= " << c.k()  << "\n";
    }

    std::ostream& ba_solver::display(std::ostream& out) const {
        for (constraint const* c : m_constraints) {
            out << (*c) << "\n";
        }
        if (!m_learned.empty()) {
            out << "learned:\n";
        }
        for (constraint const* c : m_learned) {
            out << (*c) << "\n";
        }
        return out;
    }

    std::ostream& ba_solver::display_justification(std::ostream& out, ext_justification_idx idx) const {
        return out << index2constraint(idx);
    }

    std::ostream& ba_solver::display_constraint(std::ostream& out, ext_constraint_idx idx) const {
        return out << index2constraint(idx);
    }

    std::ostream& ba_solver::display(std::ostream& out, constraint const& c, bool values) const {
        switch (c.tag()) {
        case card_t: display(out, c.to_card(), values); break;
        case pb_t: display(out, c.to_pb(), values); break;
        case xr_t: display(out, c.to_xr(), values); break;
        default: UNREACHABLE(); break;
        }
        return out;
    }

    void ba_solver::collect_statistics(statistics& st) const {
        st.update("ba propagations", m_stats.m_num_propagations);
        st.update("ba conflicts", m_stats.m_num_conflicts);
        st.update("ba resolves", m_stats.m_num_resolves);
        st.update("ba cuts", m_stats.m_num_cut);
        st.update("ba gc", m_stats.m_num_gc);
        st.update("ba overflow", m_stats.m_num_overflow);
        st.update("ba big strengthenings", m_stats.m_num_big_strengthenings);
        st.update("ba lemmas", m_stats.m_num_lemmas);
        st.update("ba subsumes", m_stats.m_num_bin_subsumes + m_stats.m_num_clause_subsumes + m_stats.m_num_pb_subsumes);
    }

    bool ba_solver::validate_unit_propagation(card const& c, literal alit) const { 
        (void) alit;
        if (c.lit() != null_literal && value(c.lit()) != l_true) return false;
        for (unsigned i = c.k(); i < c.size(); ++i) {
            if (value(c[i]) != l_false) return false;
        }
        return true;
    }

    bool ba_solver::validate_unit_propagation(pb const& p, literal alit) const { 
        if (p.lit() != null_literal && value(p.lit()) != l_true) {            
            return false;
        }

        unsigned sum = 0;
        TRACE("ba", display(tout << "validate: " << alit << "\n", p, true););
        for (wliteral wl : p) {
            literal lit = wl.second;
            lbool val = value(lit);
            if (val != l_false && lit != alit) {
                sum += wl.first;
            }
        }
        return sum < p.k();
    }

    bool ba_solver::validate_unit_propagation(pb const& p, literal_vector const& r, literal alit) const {
        // all elements of r are true, 
        for (literal l : r) {
            if (value(l) != l_true) {
                IF_VERBOSE(0, verbose_stream() << "value of " << l << " is " << value(l) << "\n";
                           display(verbose_stream(), p, true););
                return false;
            }
            if (value(alit) == l_true && lvl(l) > lvl(alit)) {
                IF_VERBOSE(0, 
                           verbose_stream() << "level of premise " << l << " is " << lvl(l) << "\n";
                           verbose_stream() << "level of asserting literal " << alit << " is " << lvl(alit) << "\n";
                           display(verbose_stream(), p, true););
                return false;                
            }
            // if (value(alit) == l_true && lvl(l) == lvl(alit)) {
            // std::cout << "same level " << alit << " " << l << "\n";
            // }
        }
        // the sum of elements not in r or alit add up to less than k.
        unsigned sum = 0;
        // 
        // a*x + b*alit + c*r >= k
        // sum a < k
        // val(r) = false
        // hence alit has to be true.
        for (wliteral wl : p) {
            literal lit = wl.second;
            if (lit != alit && !r.contains(~lit)) {
                sum += wl.first;
            }
        }
        if (sum >= p.k()) {
            IF_VERBOSE(0,             
                       verbose_stream() << "sum is " << sum << " >= " << p.k() << "\n";
                       display(verbose_stream(), p, true);
                       verbose_stream() << "id: " << p.id() << "\n";
                       sum = 0;
                       for (wliteral wl : p) sum += wl.first;
                       verbose_stream() << "overall sum " << sum << "\n";
                       verbose_stream() << "asserting literal: " << alit << "\n";
                       verbose_stream() << "reason: " << r << "\n";);
            return false;
        }
        for (wliteral wl : p) {
            if (alit == wl.second) {
                return true;
            }
        }
        IF_VERBOSE(0, verbose_stream() << alit << " not found among literals\n";);
        return false;
    }


    bool ba_solver::validate_lemma() { 
        int64_t bound64 = m_bound;
        int64_t val = -bound64;
        reset_active_var_set();
        for (bool_var v : m_active_vars) {
            if (!test_and_set_active(v)) continue;
            wliteral wl = get_wliteral(v);
            if (wl.first == 0) continue;
            if (!is_false(wl.second)) {
                val += wl.first;
            }
        }
        CTRACE("ba", val >= 0, active2pb(m_A); display(tout, m_A, true););
        return val < 0;
    }

    /**
     * the slack of inequalities on the stack should be non-positive.
     */
    bool ba_solver::validate_ineq(ineq const& ineq) const {
        int64_t k = -static_cast<int64_t>(ineq.m_k);
        for (wliteral wl : ineq.m_wlits) {
            if (!is_false(wl.second))
                k += wl.first;
        }
        CTRACE("ba", k > 0, display(tout, ineq, true););
        return k <= 0;
    }

    void ba_solver::reset_active_var_set() {
        while (!m_active_var_set.empty()) m_active_var_set.erase();
    }

    bool ba_solver::test_and_set_active(bool_var v) {
        if (m_active_var_set.contains(v)) {
            return false;
        }
        else {
            m_active_var_set.insert(v);
            return true;
        }
    }

    void ba_solver::active2pb(ineq& p) {
        p.reset(m_bound);
        active2wlits(p.m_wlits);
    }

    void ba_solver::active2wlits() {
        m_wlits.reset();
        active2wlits(m_wlits);
    }

    void ba_solver::active2wlits(svector<wliteral>& wlits) {
        uint64_t sum = 0;        
        reset_active_var_set();
        for (bool_var v : m_active_vars) {
            if (!test_and_set_active(v)) continue;
            wliteral wl = get_wliteral(v);
            if (wl.first == 0) continue;
            wlits.push_back(wl);
            sum += wl.first;
        }
        m_overflow |= sum >= UINT_MAX/2;
    }

    ba_solver::constraint* ba_solver::active2lemma() {
        switch (s().m_config.m_pb_lemma_format) {
        case PB_LEMMA_CARDINALITY:
            return active2card();
        case PB_LEMMA_PB:
            return active2constraint();
        default:
            UNREACHABLE();
            return nullptr;
        }
    }

    ba_solver::constraint* ba_solver::active2constraint() {
        active2wlits();
        if (m_overflow) {
            return nullptr;
        }
        constraint* c = add_pb_ge(null_literal, m_wlits, m_bound, true);                
        TRACE("ba", if (c) display(tout, *c, true););
        ++m_stats.m_num_lemmas;
        return c;
    }
    
    /*
      Chai Kuhlmann:
      
      a1*l1 + ... + a_n*l_n >= k
      s.t.
      a1 >= a2 >= .. >= a_n

      let m be such that

         sum_{i = 1}^{m-1} a_i < k <= sum_{i = 1}^{m}

      then 

      l1 + ... + l_n >= m

      furthermore, for the largest n' <= n, such that

         sum_{i = n'+1}^n a_i + sum_{i = 1}^{m-1} a_i < k

      then 

         l1 + ... + l_n' >= m
      
     */
    struct compare_wlit {
        bool operator()(ba_solver::wliteral l1, ba_solver::wliteral l2) const {
            return l1.first > l2.first;
        }
    };


    ba_solver::constraint* ba_solver::active2card() {
        active2wlits();
        if (m_overflow) {
            return nullptr;
        }
        std::sort(m_wlits.begin(), m_wlits.end(), compare_wlit());
        unsigned k = 0;
        uint64_t sum = 0, sum0 = 0;
        for (wliteral wl : m_wlits) {
            if (sum >= m_bound) break;
            sum0 = sum;
            sum += wl.first;
            ++k;
        }
        if (k == 1) {
            return nullptr;
        }
        while (!m_wlits.empty()) {
            wliteral wl = m_wlits.back();
            if (wl.first + sum0 >= m_bound) break;
            m_wlits.pop_back();
            sum0 += wl.first;
        }

        unsigned slack = 0;
        unsigned max_level = 0;
        unsigned num_max_level = 0;
        for (wliteral wl : m_wlits) {
            if (value(wl.second) != l_false) ++slack;
            unsigned level = lvl(wl.second);
            if (level > max_level) {
                max_level = level;
                num_max_level = 1;
            }
            else if (max_level == level) {
                ++num_max_level;
            }
        }
        if (m_overflow) {
            return nullptr;
        }

        if (slack >= k) {
#if 0
            return active2constraint();
            active2pb(m_A);
            std::cout << "not asserting\n";
            display(std::cout, m_A, true);
#endif
            return nullptr;
        }

        // produce asserting cardinality constraint
        literal_vector lits;
        for (wliteral wl : m_wlits) { 
            lits.push_back(wl.second);
        }       
        constraint* c = add_at_least(null_literal, lits, k, true);      

        ++m_stats.m_num_lemmas;

        if (c) {
            lits.reset();
            for (wliteral wl : m_wlits) {
                if (value(wl.second) == l_false) lits.push_back(wl.second);        
            }
            unsigned glue = s().num_diff_levels(lits.size(), lits.c_ptr());
            c->set_glue(glue);
        }
        return c;
    }


    void ba_solver::justification2pb(justification const& js, literal lit, unsigned offset, ineq& ineq) {
        switch (js.get_kind()) {
        case justification::NONE:
            SASSERT(lit != null_literal);
            ineq.reset(offset);
            ineq.push(lit, offset);
            break;
        case justification::BINARY:
            SASSERT(lit != null_literal);
            ineq.reset(offset);
            ineq.push(lit, offset);
            ineq.push(js.get_literal(), offset);
            break;
        case justification::TERNARY:
            SASSERT(lit != null_literal);
            ineq.reset(offset);
            ineq.push(lit, offset);
            ineq.push(js.get_literal1(), offset);
            ineq.push(js.get_literal2(), offset);
            break;
        case justification::CLAUSE: {
            ineq.reset(offset);
            clause & c = s().get_clause(js);
            for (literal l : c) ineq.push(l, offset);
            break;
        }
        case justification::EXT_JUSTIFICATION: {
            ext_justification_idx index = js.get_ext_justification_idx();
            constraint& cnstr = index2constraint(index);
            constraint2pb(cnstr, lit, offset, ineq);
            break;
        }
        default:
            UNREACHABLE();
            break;
        }
    }

    void ba_solver::constraint2pb(constraint& cnstr, literal lit, unsigned offset, ineq& ineq) {
        switch (cnstr.tag()) {
        case card_t: {
            card& c = cnstr.to_card();
            ineq.reset(offset*c.k());
            for (literal l : c) ineq.push(l, offset);
            if (c.lit() != null_literal) ineq.push(~c.lit(), offset*c.k());                
            break;
        }
        case pb_t: {
            pb& p = cnstr.to_pb();
            ineq.reset(offset * p.k());
            for (wliteral wl : p) ineq.push(wl.second, offset * wl.first);
            if (p.lit() != null_literal) ineq.push(~p.lit(), offset * p.k());
            break;
        }
        case xr_t: {
            xr& x = cnstr.to_xr();
            literal_vector ls;
            SASSERT(lit != null_literal);
            get_antecedents(lit, x, ls);                
            ineq.reset(offset);
            for (literal l : ls) ineq.push(~l, offset);
            literal lxr = x.lit();                
            if (lxr != null_literal) ineq.push(~lxr, offset);
            break;
        }
        default:
            UNREACHABLE();
            break;
        }
    }

    // validate that m_A & m_B implies m_C

    bool ba_solver::validate_resolvent() {
        return true;
        u_map<uint64_t> coeffs;
        uint64_t k = m_A.m_k + m_B.m_k;
        for (unsigned i = 0; i < m_A.size(); ++i) {
            uint64_t coeff = m_A.coeff(i);
            SASSERT(!coeffs.contains(m_A.lit(i).index()));
            coeffs.insert(m_A.lit(i).index(), coeff);
        }
        for (unsigned i = 0; i < m_B.size(); ++i) {
            uint64_t coeff1 = m_B.coeff(i), coeff2;
            literal lit = m_B.lit(i);
            if (coeffs.find((~lit).index(), coeff2)) {
                if (coeff1 == coeff2) {
                    coeffs.remove((~lit).index());
                    k += coeff1;
                }
                else if (coeff1 < coeff2) {
                    coeffs.insert((~lit).index(), coeff2 - coeff1);
                    k += coeff1;
                }
                else {
                    SASSERT(coeff2 < coeff1);
                    coeffs.remove((~lit).index());
                    coeffs.insert(lit.index(), coeff1 - coeff2);
                    k += coeff2;
                }
            }
            else if (coeffs.find(lit.index(), coeff2)) {
                coeffs.insert(lit.index(), coeff1 + coeff2);
            }
            else {
                coeffs.insert(lit.index(), coeff1);
            }
        }
        // C is above the sum of A and B
        for (unsigned i = 0; i < m_C.size(); ++i) {
            literal lit = m_C.lit(i);
            uint64_t coeff;
            if (coeffs.find(lit.index(), coeff)) {
                if (coeff > m_C.coeff(i) && m_C.coeff(i) < m_C.m_k) {
                    goto violated;
                }
                coeffs.remove(lit.index());
            }
        }
        if (!coeffs.empty()) goto violated;
        if (m_C.m_k > k) goto violated;
        SASSERT(coeffs.empty());
        SASSERT(m_C.m_k <= k);
        return true;

    violated:
        // last ditch effort by translating to SAT.
        solver s0(s().m_params, s().rlimit());
        u_map<bool_var> translation;
        literal l1 = translate_to_sat(s0, translation, m_A);
        if (l1 == null_literal) return true;
        literal l2 = translate_to_sat(s0, translation, m_B);
        if (l2 == null_literal) return true;
        ineq notC = negate(m_B);
        literal l3 = translate_to_sat(s0, translation, notC);
        if (l3 == null_literal) return true;
        s0.assign_scoped(l1);
        s0.assign_scoped(l2);
        s0.assign_scoped(l3);
        lbool is_sat = s0.check();
        TRACE("ba", s0.display(tout << "trying sat encoding"););
        if (is_sat == l_false) return true;

        IF_VERBOSE(0, 
                   display(verbose_stream(), m_A);
                   display(verbose_stream(), m_B);
                   display(verbose_stream(), m_C);
                   for (auto& e : coeffs) {
                       verbose_stream() << to_literal(e.m_key) << ": " << e.m_value << "\n";
                   });
        
        UNREACHABLE();
        return false;
    }

    /**
       \brief translate PB inequality to SAT formula.
     */
    literal ba_solver::translate_to_sat(solver& s, u_map<bool_var>& translation, ineq const& pb) {
        SASSERT(pb.m_k > 0);
        if (pb.size() > 1) {
            ineq a, b;
            a.reset(pb.m_k);
            b.reset(pb.m_k);
            for (unsigned i = 0; i < pb.size()/2; ++i) {
                a.push(pb.lit(i), pb.coeff(i));
            }
            for (unsigned i = pb.size()/2; i < pb.size(); ++i) {
                b.push(pb.lit(i), pb.coeff(i));
            }
            bool_var v = s.mk_var();
            literal lit(v, false);
            literal_vector lits;
            lits.push_back(~lit);            
            push_lit(lits, translate_to_sat(s, translation, a));
            push_lit(lits, translate_to_sat(s, translation, b));
            push_lit(lits, translate_to_sat(s, translation, a, b));
            s.mk_clause(lits);
            return lit;
        }
        if (pb.coeff(0) >= pb.m_k) {
            return translate_to_sat(s, translation, pb.lit(0));
        }
        else {
            return null_literal;
        }
    }

    /*
      \brief encode the case where Sum(a) >= k-1 & Sum(b) >= 1 \/ ... \/ Sum(a) >= 1 & Sum(b) >= k-1
     */
    literal ba_solver::translate_to_sat(solver& s, u_map<bool_var>& translation, ineq& a, ineq& b) {
        uint64_t k0 = a.m_k;
        literal_vector lits;
        for (unsigned k = 1; k < a.m_k - 1; ++k) {
            a.m_k = k; b.m_k = k0 - k;
            literal lit1 = translate_to_sat(s, translation, a);
            literal lit2 = translate_to_sat(s, translation, b);
            if (lit1 != null_literal && lit2 != null_literal) {
                bool_var v = s.mk_var();
                literal lit(v, false);
                s.mk_clause(~lit, lit1);
                s.mk_clause(~lit, lit2);
                lits.push_back(lit);
            }
        }
        a.m_k = k0;
        b.m_k = k0;
        switch (lits.size()) {
        case 0: return null_literal;
        case 1: return lits[0];
        default: {
            bool_var v = s.mk_var();
            literal lit(v, false);
            lits.push_back(~lit);
            s.mk_clause(lits);
            return lit;
        }
        }
    }
    
    literal ba_solver::translate_to_sat(solver& s, u_map<bool_var>& translation, literal lit) {
        bool_var v;
        if (!translation.find(lit.var(), v)) {            
            v = s.mk_var();
            translation.insert(lit.var(), v);
        }
        return literal(v, lit.sign());
    }

    ba_solver::ineq ba_solver::negate(ineq const& a) const {
        ineq result;
        uint64_t sum = 0;
        for (unsigned i = 0; i < a.size(); ++i) { 
            result.push(~a.lit(i), a.coeff(i));
            sum += a.coeff(i);
        }
        SASSERT(sum >= a.m_k + 1);
        result.m_k = sum + 1 - a.m_k;
        return result;
    }

    void ba_solver::push_lit(literal_vector& lits, literal lit) {
        if (lit != null_literal) {
            lits.push_back(lit);
        }
    }

    bool ba_solver::validate_conflict(literal_vector const& lits, ineq& p) { 
        for (literal l : lits) {
            if (value(l) != l_false) {
                TRACE("ba", tout << "literal " << l << " is not false\n";);
                return false;
            }
            if (!p.contains(l)) {
                TRACE("ba", tout << "lemma contains literal " << l << " not in inequality\n";);
                return false;
            }
        }
        uint64_t value = 0;        
        for (unsigned i = 0; i < p.size(); ++i) {
            uint64_t coeff = p.coeff(i);
            if (!lits.contains(p.lit(i))) {
                value += coeff;
            }
        }
        CTRACE("ba", value >= p.m_k, tout << "slack: " << value << " bound " << p.m_k << "\n";
               display(tout, p);
               tout << lits << "\n";);
        return value < p.m_k;
    }

    bool ba_solver::check_model(model const& m) const {
        bool ok = true;
        for (constraint const* c : m_constraints) {
            if (c->was_removed()) {
                continue;
            }
            if (c->is_pure() && c->lit() != null_literal && m[c->lit().var()] == (c->lit().sign() ? l_true : l_false)) {
                continue;
            }
            switch (eval(m, *c)) {
            case l_false: 
                IF_VERBOSE(0, verbose_stream() << "failed checking " << c->id() << ": " << *c << "\n";);
                ok = false;
                break;
            case l_true:
                break;
            case l_undef:
                IF_VERBOSE(0, verbose_stream() << "undef " << c->id() << ": " << *c << "\n";);
                break;
            }
        }
        return ok;
    }

    bool ba_solver::extract_pb(std::function<void(unsigned sz, literal const* c, unsigned k)>& add_cardinality,
                               std::function<void(unsigned sz, literal const* c, unsigned const* coeffs, unsigned k)>& add_pb) {
        
        unsigned_vector coeffs;
        literal_vector lits;
        for (constraint* cp : m_constraints) {
            switch (cp->tag()) {
            case card_t: {
                card const& c = cp->to_card();
                unsigned n = c.size();
                unsigned k = c.k();
                
                if (c.lit() == null_literal) {
                    //    c.lits() >= k 
                    // <=> 
                    //    ~c.lits() <= n - k
                    lits.reset();
                    for (unsigned j = 0; j < n; ++j) lits.push_back(c[j]);
                    add_cardinality(lits.size(), lits.c_ptr(), n - k);
                }
                else {
                    //
                    // c.lit() <=> c.lits() >= k
                    // 
                    //    (c.lits() < k) or c.lit()
                    // =  (c.lits() + (n - k + 1)*~c.lit()) <= n    
                    //
                    //    ~c.lit() or (c.lits() >= k)
                    // =  ~c.lit() or (~c.lits() <= n - k)
                    // =  k*c.lit() + ~c.lits() <= n 
                    // 
                    lits.reset();
                    coeffs.reset();
                    for (literal l : c) lits.push_back(l), coeffs.push_back(1);
                    lits.push_back(~c.lit()); coeffs.push_back(n - k + 1);
                    add_pb(lits.size(), lits.c_ptr(), coeffs.c_ptr(), n);
                    
                    lits.reset();
                    coeffs.reset();
                    for (literal l : c) lits.push_back(~l), coeffs.push_back(1);
                    lits.push_back(c.lit()); coeffs.push_back(k);
                    add_pb(lits.size(), lits.c_ptr(), coeffs.c_ptr(), n);
                }
                break;
            }
            case ba_solver::pb_t: {
                ba_solver::pb const& p = cp->to_pb();
                lits.reset();
                coeffs.reset();
                unsigned sum = 0;
                for (ba_solver::wliteral wl : p) sum += wl.first;
                
                if (p.lit() == null_literal) {
                    //   w1 + .. + w_n >= k
                    // <=> 
                    //  ~wl + ... + ~w_n <= sum_of_weights - k
                    for (ba_solver::wliteral wl : p) lits.push_back(~(wl.second)), coeffs.push_back(wl.first);
                    add_pb(lits.size(), lits.c_ptr(), coeffs.c_ptr(), sum - p.k());
                }
                else {
                    //    lit <=> w1 + .. + w_n >= k
                    // <=>
                    //     lit or w1 + .. + w_n <= k - 1
                    //    ~lit or w1 + .. + w_n >= k
                    // <=> 
                    //     (sum - k + 1)*~lit + w1 + .. + w_n <= sum
                    //     k*lit + ~wl + ... + ~w_n <= sum
                    lits.push_back(p.lit()), coeffs.push_back(p.k());
                    for (ba_solver::wliteral wl : p) lits.push_back(~(wl.second)), coeffs.push_back(wl.first);
                    add_pb(lits.size(), lits.c_ptr(), coeffs.c_ptr(), sum);
                    
                    lits.reset();
                    coeffs.reset();
                    lits.push_back(~p.lit()), coeffs.push_back(sum + 1 - p.k());
                    for (ba_solver::wliteral wl : p) lits.push_back(wl.second), coeffs.push_back(wl.first);
                    add_pb(lits.size(), lits.c_ptr(), coeffs.c_ptr(), sum);
                }
                break;
            }
            case ba_solver::xr_t:
                return false;
            }
        }
        return true;
    }
    
};