z3-src 503.0.0

/**++
Copyright (c) Arie Gurfinkel

Module Name:

    qe_term_graph.cpp

Abstract:

    Equivalence graph of terms

Author:

    Arie Gurfinkel
    Hari Govind V K (hgvk94)
    Isabel Garcia (igcontreras)

Revision History:

    Added implementation of qe_lite using term graph

Notes:

--*/

#include "qe/mbp/mbp_term_graph.h"
#include "ast/array_peq.h"
#include "ast/ast.h"
#include "ast/ast_pp.h"
#include "ast/ast_util.h"
#include "ast/for_each_expr.h"
#include "ast/occurs.h"
#include "ast/rewriter/th_rewriter.h"
#include "model/model_evaluator.h"
#include "util/bit_vector.h"
#include "util/obj_pair_hashtable.h"
#include "util/uint_set.h"
#include "util/util.h"

namespace mbp {

static expr_ref mk_neq(ast_manager &m, expr *e1, expr *e2) {
    expr *t = nullptr;
    // x != !x  == true
    if ((m.is_not(e1, t) && t == e2) || (m.is_not(e2, t) && t == e1))
        return expr_ref(m.mk_true(), m);
    else if (m.are_distinct(e1, e2))
        return expr_ref(m.mk_true(), m);
    return expr_ref(m.mk_not(m.mk_eq(e1, e2)), m);
}

namespace {
struct sort_lt_proc { // for representatives in model_complete
    bool operator()(const expr *a, const expr *b) const {
        return a->get_sort()->get_id() < b->get_sort()->get_id();
    }
};
struct mark_all_sub_expr {
    expr_sparse_mark &m_mark;
    mark_all_sub_expr(expr_sparse_mark &mark) : m_mark(mark) {}
    void operator()(var *n) const {}
    void operator()(app *n) const { m_mark.mark(n); }
    void operator()(quantifier *n) const {}
};
} // namespace

namespace is_pure_ns {
struct found {};
struct proc {
    is_variable_proc &m_is_var;
    proc(is_variable_proc &is_var) : m_is_var(is_var) {}
    void operator()(var *n) const {
        if (m_is_var(n)) throw found();
    }
    void operator()(app const *n) const {
        if (m_is_var(n)) throw found();
    }
    void operator()(quantifier *n) const {}
};
} // namespace is_pure_ns

bool is_pure(is_variable_proc &is_var, expr *e) {
    try {
        is_pure_ns::proc v(is_var);
        quick_for_each_expr(v, e);
    } catch (const is_pure_ns::found &) { return false; }
    return true;
}

bool term_graph::is_ground(expr *e) {
    try {
        is_ground_ns::proc v(m_is_var);
        quick_for_each_expr(v, e);
    } catch (const is_ground_ns::found &) { return false; }
    return true;
}

class term {
    // -- an app represented by this term
    expr_ref m_expr; // NSB: to make usable with exprs
    // -- root of the equivalence class
    term *m_root;
    // -- representative of the equivalence class
    term *m_repr;
    // -- next element in the equivalence class (cyclic linked list)
    term *m_next;
    // -- general purpose mark
    unsigned m_mark : 1;
    // -- general purpose second mark
    unsigned m_mark2 : 1;
    // -- is an interpreted constant
    unsigned m_interpreted : 1;
    // caches whether m_expr is an equality
    unsigned m_is_eq : 1;
    // caches whether m_expr is an inequality
    unsigned m_is_neq : 1;
    // caches whether m_expr is a distinct
    unsigned m_is_distinct : 1;
    // caches whether m_expr is a partial equality
    unsigned m_is_peq : 1;
    // caches whether m_expr is the child of not
    unsigned m_is_neq_child : 1;
    // caches whether m_expr is peq and the child of not
    unsigned m_is_npeq_child : 1;

    // -- the term is a compound term can be rewritten to be ground or it is a
    // ground constant
    unsigned m_cgr : 1;
    // -- the term is ground
    unsigned m_gr : 1;

    // -- terms that contain this term as a child (only maintained for root
    // nodes)
    ptr_vector<term> m_parents;

    // arguments of term.
    ptr_vector<term> m_children;

    struct class_props {
        // TODO: parents should be here
        // -- the class has a ground representative
        unsigned m_gr_class : 1;
        // -- eq class size
        unsigned m_class_size;
        // -- disequality sets that the class belongs to
        term_graph::deqs m_deqs;

        class_props() : m_gr_class(0), m_class_size(1) {}
        void merge(class_props &b) {
            m_class_size += b.m_class_size;
            m_gr_class |= b.m_gr_class;
            m_deqs |= b.m_deqs; // merge disequalities
            // -- reset (useful for debugging)
            b.m_class_size = 0;
            b.m_gr_class = false;
            b.m_deqs.reset();
        }
        void transfer(class_props &b) {
            // TODO replace by std::swap of the whole struct?
            m_class_size = b.m_class_size;
            b.m_class_size = 0;
            std::swap(m_deqs, b.m_deqs);
            m_gr_class = b.m_gr_class;
            b.m_gr_class = false;
        }
    };
    class_props m_class_props;

public:
    term(expr_ref const &v, u_map<term *> &app2term)
        : m_expr(v), m_root(this), m_repr(nullptr), m_next(this), m_mark(false),
          m_mark2(false), m_interpreted(false),
          m_is_eq(m_expr.get_manager().is_eq(m_expr)), m_is_peq(false),
          m_is_neq_child(false), m_is_npeq_child(false),
          m_cgr(0), m_gr(0) {
        m_is_neq = m_expr.get_manager().is_not(m_expr) &&
                   m_expr.get_manager().is_eq(to_app(m_expr)->get_arg(0));
        m_is_distinct = m_expr.get_manager().is_distinct(m_expr);
        m_children.reset();
        if (!is_app(m_expr)) return;
        for (expr *e : *to_app(m_expr)) {
            term *t = app2term[e->get_id()];
            t->get_root().m_parents.push_back(this);
            m_children.push_back(t);
        }
        m_is_peq = is_partial_eq(m_expr);
    }

    class parents {
        term const &t;

    public:
        parents(term const &_t) : t(_t) {}
        parents(term const *_t) : t(*_t) {}
        ptr_vector<term>::const_iterator begin() const {
            return t.m_parents.begin();
        }
        ptr_vector<term>::const_iterator end() const {
            return t.m_parents.end();
        }
    };

    class children {
        term const &t;

    public:
        children(term const &_t) : t(_t) {}
        children(term const *_t) : t(*_t) {}
        ptr_vector<term>::const_iterator begin() const {
            return t.m_children.begin();
        }
        ptr_vector<term>::const_iterator end() const {
            return t.m_children.end();
        }
    };

    // Congruence table hash function is based on
    // roots of children and function declaration.

    unsigned get_hash() const {
        unsigned a, b, c;
        b = c = get_decl_id();
        for (term *ch : children(this)) {
            a = ch->get_root().get_id();
            mix(a, b, c);
        }
        return c;
    }

    static bool cg_eq(term const *t1, term const *t2) {
        if (t1->get_decl_id() != t2->get_decl_id()) return false;
        if (t1->m_children.size() != t2->m_children.size()) return false;
        for (unsigned i = 0, sz = t1->m_children.size(); i < sz; ++i) {
            if (t1->m_children[i]->get_root().get_id() !=
                t2->m_children[i]->get_root().get_id())
                return false;
        }
        return true;
    }

    unsigned deg() const { return m_children.size(); }
    unsigned get_id() const { return m_expr->get_id(); }
    bool is_eq_or_neq() const { return m_is_eq || m_is_neq || m_is_distinct; }
    bool is_eq_or_peq() const { return m_is_eq || m_is_peq; }
    bool is_neq() const { return m_is_neq; }
    void set_neq_child() { m_is_neq_child = true; }
    void set_npeq_child() { m_is_npeq_child = true; }
    bool is_neq_child() const { return m_is_neq_child; }
    bool is_npeq_child() const { return m_is_npeq_child; }
    unsigned get_decl_id() const {
        return is_app(m_expr) ? to_app(m_expr)->get_decl()->get_id()
                              : m_expr->get_id();
    }

    bool is_marked() const { return m_mark; }
    void set_mark(bool v) { m_mark = v; }
    bool is_marked2() const { return m_mark2; } // NSB: where is this used?
    void set_mark2(bool v) { m_mark2 = v; }     // NSB: where is this used?

    bool is_cgr() const { return m_cgr; }
    void set_cgr(bool v) { m_cgr = v; }

    bool is_gr() const { return m_gr; }
    void set_gr(bool v) { m_gr = v; }

    bool is_class_gr_root() const {
        SASSERT(is_root());
        return m_class_props.m_gr_class;
    }
    void set_class_gr_root(bool v) {
        SASSERT(is_root());
        m_class_props.m_gr_class = v;
    }
    bool is_class_gr() const { return m_root->is_class_gr_root(); }
    void set_class_gr(bool v) { m_root->set_class_gr_root(v); }

    static bool are_deq(const term &t1, const term &t2) {
        term_graph::deqs const &ds1 = t1.get_root().get_deqs();
        term_graph::deqs const &ds2 = t2.get_root().get_deqs();

        term_graph::deqs tmp(ds1); // copy

        tmp &= ds2;
        return tmp != 0;
    }

    static void set_deq(term_graph::deqs &ds, unsigned idx) {
        ds.resize(idx + 1);
        ds.set(idx);
    }

    bool all_children_ground() {
        SASSERT(deg() != 0);
        return all_of(m_children,
                      [&](const term *t) { return t->is_class_gr(); });
    }

    void set_mark2_terms_class(bool v) { // TODO: remove
        if (is_marked2()) return;
        term *curr = this;
        do {
            curr->set_mark2(v);
            curr = &curr->get_next();
        }
        while (curr != this);
    }

    bool is_interpreted() const { return m_interpreted; }
    bool is_theory() const {
        return !is_app(m_expr) ||
               to_app(m_expr)->get_family_id() != null_family_id;
    }
    void mark_as_interpreted() { m_interpreted = true; }
    expr *get_expr() const { return m_expr; }
    unsigned get_num_args() const {
        return is_app(m_expr) ? to_app(m_expr)->get_num_args() : 0;
    }

    term &get_root() const { return *m_root; }
    bool is_root() const { return m_root == this; }
    void set_root(term &r) { m_root = &r; }
    term *get_repr() const { return m_repr; }
    bool is_repr() const { return m_repr == this; }
    void set_repr(term *t) {
        SASSERT(get_root().get_id() == t->get_root().get_id());
        m_repr = t;
    }
    void reset_repr() { m_repr = nullptr; }
    term &get_next() const { return *m_next; }
    void add_parent(term *p) { m_parents.push_back(p); }

    unsigned get_class_size() const { return m_class_props.m_class_size; }

    void merge_eq_class(term &b) {
        std::swap(this->m_next, b.m_next);
        m_class_props.merge(b.m_class_props);
    }

    // -- make this term the repr of its equivalence class
    void mk_repr() {
        term *curr = this;
        do {
            curr->set_repr(this);
            curr = &curr->get_next();
        }
        while (curr != this);
    }

    std::ostream &display(std::ostream &out) const {
        out << get_id() << ": " << m_expr << (is_repr() ? " R" : "")
            << (is_gr() ? " G" : "") << (is_class_gr() ? " clsG" : "")
            << (is_cgr() ? " CG" : "") << " deg:" << deg() << " - ";
        term const *r = &this->get_next();
        while (r != this) {
            out << r->get_id() << " " << (r->is_cgr() ? " CG" : "") << " ";
            r = &r->get_next();
        }
        out << "\n";
        return out;
    }

    term_graph::deqs &get_deqs() { return m_class_props.m_deqs; }
};

static std::ostream &operator<<(std::ostream &out, term const &t) {
    return t.display(out);
}


// t1 != t2
void term_graph::add_deq_proc::operator()(term *t1, term *t2) {
    term::set_deq(t1->get_root().get_deqs(), m_deq_cnt); 
    term::set_deq(t2->get_root().get_deqs(), m_deq_cnt);
    inc_count();
}

// distinct(ts)
void term_graph::add_deq_proc::operator()(ptr_vector<term> &ts) {
    for (auto t : ts)
        term::set_deq(t->get_root().get_deqs(), m_deq_cnt);
    inc_count();
}

void term_graph::add_deq_proc::inc_count() {
    m_deq_cnt++;
    if (m_deq_cnt == 0)
        throw default_exception("unexpected wrap-around on m_deq_cnt");
}

bool term_graph::is_variable_proc::operator()(const expr *e) const {
    if (!is_app(e)) return false;
    const app *a = ::to_app(e);
    TRACE(qe_verbose, tout << a->get_family_id() << " "
                             << m_solved.contains(a->get_decl()) << " "
                             << m_decls.contains(a->get_decl()) << "\n";);
    return a->get_family_id() == null_family_id &&
           !m_solved.contains(a->get_decl()) &&
           m_exclude == m_decls.contains(a->get_decl());
}

bool term_graph::is_variable_proc::operator()(const term &t) const {
    return (*this)(t.get_expr());
}

void term_graph::is_variable_proc::set_decls(const func_decl_ref_vector &decls,
                                             bool exclude) {
    reset();
    m_exclude = exclude;
    for (auto *d : decls) m_decls.insert(d);
}

void term_graph::is_variable_proc::add_decls(const app_ref_vector &decls) {
    for (auto *d : decls) m_decls.insert(d->get_decl());
}

void term_graph::is_variable_proc::add_decl(app *d) {
    m_decls.insert(d->get_decl());
}

void term_graph::is_variable_proc::set_decls(const app_ref_vector &vars,
                                             bool exclude) {
    reset();
    m_exclude = exclude;
    for (auto *v : vars) m_decls.insert(v->get_decl());
}

void term_graph::is_variable_proc::mark_solved(const expr *e) {
    if ((*this)(e) && is_app(e)) m_solved.insert(::to_app(e)->get_decl());
}

unsigned term_graph::term_hash::operator()(term const *t) const {
    return t->get_hash();
}

bool term_graph::term_eq::operator()(term const *a, term const *b) const {
    return term::cg_eq(a, b);
}

term_graph::term_graph(ast_manager &man)
    : m(man), m_lits(m), m_pinned(m) {
    m_is_var.reset();
    m_plugins.register_plugin(mbp::mk_basic_solve_plugin(m, m_is_var));
    m_plugins.register_plugin(mbp::mk_arith_solve_plugin(m, m_is_var));
}

term_graph::~term_graph() {
    dealloc(m_projector);
    reset();
}

bool term_graph::is_pure_def(expr *atom, expr *&v) {
    expr *e = nullptr;
    return m.is_eq(atom, v, e) && m_is_var(v) && is_pure(m_is_var, e);
}

static family_id get_family_id(ast_manager &m, expr *lit) {
    if (m.is_not(lit, lit))
        return get_family_id(m, lit);

    expr *a = nullptr, *b = nullptr;
    if (m.is_eq(lit, a, b))   // deal with equality using sort of range
        return a->get_sort()->get_family_id(); 
    else if (is_app(lit))     // extract family_id of top level app
        return to_app(lit)->get_decl()->get_family_id(); 
    else
        return null_family_id; 
}
    
void term_graph::add_lit(expr *l) {
    expr_ref lit(m);
    expr_ref_vector lits(m);
    lits.push_back(l);
    for (unsigned i = 0; i < lits.size(); ++i) {
        l = lits.get(i);
        family_id fid = get_family_id(m, l);
        mbp::solve_plugin *pin = m_plugins.get_plugin(fid);
        lit = pin ? (*pin)(l) : l;
        if (m.is_and(lit)) {
            lits.append(::to_app(lit)->get_num_args(),
                        ::to_app(lit)->get_args());
        }
        else {
            m_lits.push_back(lit);
            internalize_lit(lit);
        }
    }
}

// collect expressions of all terms in the term graph
// optionally, exclude constructively ground nodes that are not equalities
// overwrites res
void term_graph::get_terms(expr_ref_vector &res, bool exclude_cground) {
    std::function<bool(term *)> fil = nullptr;
    if (exclude_cground) {
        fil = [](term *t) {
            return !t->is_neq_child() && !t->is_npeq_child() && (t->is_eq_or_peq() || !t->is_cgr());
        };
    }
    else {
        fil = [](term *t) { return !t->is_neq_child() && !t->is_npeq_child(); };
    }
    auto terms = m_terms.filter_pure(fil);
    res.resize(terms.size());
    unsigned i = 0;
    for (term *t : terms)
        res[i++] = t->get_expr();
}

bool term_graph::is_cgr(expr *e) {
    if (!is_internalized(e)) return false;
    term *t = get_term(e);
    return (!t->is_eq_or_peq() && t->is_cgr());
}

bool term_graph::is_internalized(expr *a) {
    return m_app2term.contains(a->get_id());
}

term *term_graph::get_term(expr *a) {
    term *res;
    return m_app2term.find(a->get_id(), res) ? res : nullptr;
}

term *term_graph::mk_term(expr *a) {
    expr_ref e(a, m);
    term *t = alloc(term, e, m_app2term);
    if (is_ground(a)) {
        t->set_gr(true);
        t->set_cgr(true);
        t->set_class_gr(true);
    }
    else if (t->deg() > 0 && t->all_children_ground()) {
        t->set_cgr(true);
        t->set_class_gr(true);
    }
    if (t->get_num_args() == 0 && m.is_unique_value(a))
        t->mark_as_interpreted();

    m_terms.push_back(t);
    m_app2term.insert(a->get_id(), t);
    return t;
}

term *term_graph::internalize_term(expr *t) {
    term *res = get_term(t);
    if (res) return res;
    ptr_buffer<expr> todo;
    todo.push_back(t);
    while (!todo.empty()) {
        t = todo.back();
        res = get_term(t);
        if (res) {
            todo.pop_back();
            continue;
        }
        unsigned sz = todo.size();
        if (is_app(t)) 
            for (expr *arg : *::to_app(t)) 
                if (!get_term(arg))
                    todo.push_back(arg);

        if (sz < todo.size())
            continue;
        todo.pop_back();
        res = mk_term(t);

        // the term was not internalized in this syntactic form, but it
        // could be congruent with some other term, if that is the case, we
        // need to merge them.
        term *res_old = m_cg_table.insert_if_not_there(res);
        if (res->is_cgr())
            res_old->set_cgr(true);
        SASSERT(res_old->is_cgr() == res->is_cgr());
        if (res_old->get_root().get_id() != res->get_root().get_id()) 
            m_merge.push_back({res, res_old});            
    }
    merge_flush();
    SASSERT(res);
    expr* arg = nullptr;
    if (m.is_not(t, arg) && is_partial_eq(arg)) {
        term* p = get_term(arg);
        SASSERT(p);
        p->set_npeq_child();
    }
    return res;
}

void term_graph::internalize_eq(expr *a1, expr *a2) {
    SASSERT(m_merge.empty());
    merge(*internalize_term(a1), *internalize_term(a2));
    merge_flush();
    SASSERT(m_merge.empty());
    if (!m_explicit_eq)
        return;
    expr_ref eq(m.mk_eq(a1, a2), m);
    term *res = get_term(eq);
    if (!res)
        mk_term(eq);
}

void term_graph::internalize_distinct(expr *d) {
    app *a = to_app(d);
    ptr_vector<term> ts(a->get_decl()->get_arity());
    auto tsit = ts.begin();
    for (auto arg : *a) {
        *tsit = internalize_term(arg);
        tsit++;
    }
    m_add_deq(ts);
    m_deq_distinct.push_back(ts);
    if (!m_explicit_eq) return;
    term *t = get_term(d);
    if (!t) mk_term(d);
}

// Assumes that a1 != a2 is satisfiable
void term_graph::internalize_deq(expr *a1, expr *a2) {
    term *t1 = internalize_term(a1);
    term *t2 = internalize_term(a2);
    m_add_deq(t1, t2);
    m_deq_pairs.push_back({t1, t2});
    if (!m_explicit_eq)
        return;
    expr_ref eq(m.mk_eq(a1, a2), m);
    term *eq_term = mk_term(eq);
    eq_term->set_neq_child();
    expr_ref deq(m.mk_not(eq), m);
    term *res = get_term(deq);
    if (!res)
        mk_term(deq);
}

void term_graph::internalize_lit(expr *lit) {
    expr *e1 = nullptr, *e2 = nullptr, *ne = nullptr, *v = nullptr;
    if (m.is_eq(lit, e1, e2))  // internalize equality
        internalize_eq(e1, e2);
    else if (m.is_distinct(lit))
        internalize_distinct(lit); 
    else if (m.is_not(lit, ne) && m.is_eq(ne, e1, e2)) 
        internalize_deq(e1, e2);
    else
        internalize_term(lit); 
    if (is_pure_def(lit, v))
        m_is_var.mark_solved(v); 
}

void term_graph::merge_flush() {
    while (!m_merge.empty()) {
        term *t1 = m_merge.back().first;
        term *t2 = m_merge.back().second;
        m_merge.pop_back();
        merge(*t1, *t2);
    }
}

void term_graph::merge(term &t1, term &t2) {
    term *a = &t1.get_root();
    term *b = &t2.get_root();

    if (a == b)
        return;

    // -- merge might invalidate term2app cache
    m_term2app.reset();
    m_pinned.reset();
    m_repick_repr = true;

    if (a->get_class_size() > b->get_class_size())
        std::swap(a, b); 

    // Remove parents of b from the cg table
    for (term *p : term::parents(b)) {
        if (!p->is_marked()) {
            p->set_mark(true);
            m_cg_table.erase(p);
        }
    }

    bool prop_cgroundness = (b->is_class_gr() != a->is_class_gr());
    // make 'a' be the root of the equivalence class of 'b'
    b->set_root(*a);
    for (term *it = &b->get_next(); it != b; it = &it->get_next()) 
        it->set_root(*a);

    // merge equivalence classes
    a->merge_eq_class(*b);

    // Insert parents of b's old equivalence class into the cg table
    // bottom-up merge of parents
    for (term *p : term::parents(b)) {
        if (p->is_marked()) {
            term *p_old = m_cg_table.insert_if_not_there(p);
            p->set_mark(false);
            a->add_parent(p);
            // propagate new equalities.
            if (p->get_root().get_id() != p_old->get_root().get_id()) 
                m_merge.push_back({p, p_old});            
        }
    }
    if (prop_cgroundness)
        cground_percolate_up(a);

    SASSERT(marks_are_clear());
}

expr *term_graph::mk_app_core(expr *e) {
    if (!is_app(e))
        return e;
    expr_ref_buffer kids(m);
    app *a = ::to_app(e);
    for (expr *arg : *a) {
        // Keep literal values (e.g. the coefficient 2 in (* 2 x)) as-is.
        // Replacing a value with its non-value class representative (e.g. x1
        // when (= x1 2) was added to the graph) can turn a linear coefficient
        // into a free variable, producing a nonlinear term such as (* x1 x).
        // Values are canonical and never need to be expressed through a
        // symbolic representative.
        if (m.is_value(arg))
            kids.push_back(arg);
        else
            kids.push_back(mk_app(arg));
    }
    app *res = m.mk_app(a->get_decl(), a->get_num_args(), kids.data());
    m_pinned.push_back(res);
    return res;
}

expr_ref term_graph::mk_app(term &r) {
    SASSERT(r.is_repr());

    if (r.get_num_args() == 0)
        return expr_ref(r.get_expr(), m); 

    expr *res = nullptr;
    if (m_term2app.find(r.get_id(), res))
        return expr_ref(res, m); 

    res = mk_app_core(r.get_expr());
    m_term2app.insert(r.get_id(), res);
    return expr_ref(res, m);
}

expr_ref term_graph::mk_app(expr *a) {
    term *t = get_term(a);
    SASSERT(!t || t->get_repr());
    if (!t)
        return expr_ref(a, m);
    else 
        return mk_app(*t->get_repr());
}

void term_graph::mk_equalities(term &t, expr_ref_vector &out) {
    SASSERT(t.is_repr());
    if (t.get_class_size() == 1)
        return;
    expr_ref rep(mk_app(t), m);
    for (term *it = &t.get_next(); it != &t; it = &it->get_next()) {
        expr *mem = mk_app_core(it->get_expr());
        out.push_back(m.mk_eq(rep, mem));
    }
}

void term_graph::mk_all_equalities(term &t, expr_ref_vector &out) {
    if (t.get_class_size() == 1)
        return;

    mk_equalities(t, out);

    for (term *it = &t.get_next(); it != &t; it = &it->get_next()) {
        expr *a1 = mk_app_core(it->get_expr());
        for (term *it2 = &it->get_next(); it2 != &t; it2 = &it2->get_next()) {
            expr *a2 = mk_app_core(it2->get_expr());
            out.push_back(m.mk_eq(a1, a2));
        }
    }
}

void term_graph::mk_qe_lite_equalities(term &t, expr_ref_vector &out,
                                       check_pred &contains_nc) {
    SASSERT(t.is_repr());
    if (t.get_class_size() == 1) return;
    expr_ref rep(m);
    rep = mk_app(t);
    if (contains_nc(rep)) {
        TRACE(
            qe_debug, tout << "repr not in core " << t;
            for (term *it = &t.get_next(); it != &t;
                 it = &it->get_next()) { tout << *it << "\n"; };);
        DEBUG_CODE(
            for (term *it = &t.get_next(); it != &t; it = &it->get_next())
                SASSERT(!it->is_cgr() || it->is_eq_or_neq() ||
                        contains_nc(mk_app_core(it->get_expr()))););
        return;
    }
    for (term *it = &t.get_next(); it != &t; it = &it->get_next()) {
        expr *e = it->get_expr();
        SASSERT(is_app(e));
        app *a = to_app(e);
        // don't add equalities for vars to eliminate
        if (m_is_var.contains(a->get_decl())) continue;
        expr *mem = mk_app_core(e);
        if (rep != mem && !contains_nc(mem))
            out.push_back(m.mk_eq(rep, mem));
    }
}

void term_graph::reset_marks() {
    for (term *t : m_terms) t->set_mark(false); 
}

void term_graph::reset_marks2() {
    for (term *t : m_terms) t->set_mark2(false); 
}

bool term_graph::marks_are_clear() {
    return all_of(m_terms, [](term* t) { return !t->is_marked(); });
}

/// Order of preference for roots of equivalence classes
/// XXX This should be factored out to let clients control the preference
bool term_graph::term_lt(term const &t1, term const &t2) {
    // prefer constants over applications (ground)
    // prefer applications over variables (for non-ground)
    // prefer uninterpreted constants over values
    // prefer smaller expressions over larger ones
    //
    // Keeping uninterpreted constants as representatives is required by
    // model-based projection clients such as Spacer: substituting a
    // non-eliminated state constant with its concrete model value over-grounds
    // the projected lemma (see Z3Prover/bench discussion #3420, benchmark
    // iss-5561/bug-2.smt2).
    //
    // The complementary concern — that a literal value used as a coefficient
    // (e.g. the 2 in (* 2 x)) could be displaced by a free variable x1 when
    // (= x1 2) is in scope, yielding a nonlinear term (* x1 x) — is addressed
    // separately in mk_app_core, which keeps literal values as-is and never
    // replaces them with a non-value representative.

    if (t1.get_num_args() == 0 || t2.get_num_args() == 0) {
        if (t1.get_num_args() == t2.get_num_args()) {
            if (m.is_value(t1.get_expr()) == m.is_value(t2.get_expr()))
                return t1.get_id() < t2.get_id();
            return m.is_value(t2.get_expr());
        }
        return t1.get_num_args() < t2.get_num_args();
    }

    // XXX this is the internalized size, not the size with the new
    // representatives
    unsigned sz1 = get_num_exprs(t1.get_expr());
    unsigned sz2 = get_num_exprs(t2.get_expr());
    return sz1 < sz2;
}

bool all_children_picked(term *t) {
    if (t->deg() == 0)
        return true;
    for (term *c : term::children(t)) 
        if (!c->get_repr())
            return false;
    return true;
}

// pick representatives for all terms in todo. Then, pick representatives for
// all terms whose children have representatives
void term_graph::pick_repr_percolate_up(ptr_vector<term> &todo) {
    term *t;
    while (!todo.empty()) {
        t = todo.back();
        todo.pop_back();
        if (t->get_repr())
            continue;
        pick_repr_class(t);
        for (auto it : term::parents(t->get_root()))
            if (all_children_picked(it))
                todo.push_back(it);
    }
}

// iterate through all terms in a class and pick a representative that:
//  1. is cgr and 2. least according to term_lt
void term_graph::pick_repr_class(term *t) {
    SASSERT(all_children_picked(t));
    term *r = t;
    for (term *it = &t->get_next(); it != t; it = &it->get_next()) {
        if (!all_children_picked(it))
            continue;
        if ((it->is_cgr() && !r->is_cgr()) ||
            (it->is_cgr() == r->is_cgr() && term_lt(*it, *r)))
            r = it;
    }
    r->mk_repr();
}

// Choose repr for equivalence classes
// repr has the following properties:
// 1. acyclicity (mk_app terminates)
// 2. maximal wrt cgr
// 3. each class has exactly one repr
// assumes that cgroundness has been computed
void term_graph::pick_repr() {
    // invalidates cache
    m_term2app.reset();
    DEBUG_CODE(for (term *t : m_terms)
                   SASSERT(t->deg() == 0 || !t->all_children_ground() ||
                           t->is_cgr()););
    for (term *t : m_terms) t->reset_repr();
    ptr_vector<term> todo;
    for (term *t : m_terms) 
        if (t->deg() == 0 && t->is_cgr())
            todo.push_back(t);
    pick_repr_percolate_up(todo);
    DEBUG_CODE(for (term *t : m_terms) SASSERT(!t->is_cgr() || t->get_repr()););

    for (term *t : m_terms) {
        if (t->get_repr())
            continue;
        if (t->deg() == 0)
            todo.push_back(t);
    }
    pick_repr_percolate_up(todo);
    DEBUG_CODE(for (term *t : m_terms) SASSERT(t->get_repr()););
    DEBUG_CODE(for (auto t
                    : m_terms)
                   SASSERT(!t->is_cgr() || t->get_repr()->is_cgr()););
}

// if t is a variable, attempt to pick non-var
void term_graph::refine_repr_class(term *t) {
    SASSERT(t->is_repr());
    auto is_var = [&](term *p) {
        SASSERT(is_app(p->get_expr()));
        return m_is_var.contains(to_app(p->get_expr())->get_decl());
    };
    if (!is_var(t))
        return;
    term *r = t;
    for (term *it = &t->get_next(); it != t; it = &it->get_next()) 
        if (!makes_cycle(it) && is_var(r) && !is_var(it))
            r = it;
    r->mk_repr();
}

// check if t makes a cycle if chosen as repr. This function assumes that the
// current repr doesn't have cycles. If there is a cycle in a child class the
// function doesn't terminate.
bool term_graph::makes_cycle(term *t) {
    term &r = t->get_root();
    ptr_vector<term> todo;
    for (auto *it : term::children(t))
        todo.push_back(it->get_repr()); 
    term *it;
    while (!todo.empty()) {
        it = todo.back();
        todo.pop_back();
        if (it->get_root().get_id() == r.get_id())
            return true;
        for (auto *ch : term::children(it))
            todo.push_back(ch->get_repr()); 
    }
    return false;
}

void term_graph::refine_repr() {
    // invalidates cache
    m_term2app.reset();
    for (term *t : m_terms) 
        if (!t->get_repr()->is_cgr())
            refine_repr_class(t->get_repr());
}

// returns true if tg ==> e = v where v is a value
bool term_graph::has_val_in_class(expr *e) {
    term *r = get_term(e);
    if (!r) return false;
    auto is_val = [&](term *t) { return m.is_value(t->get_expr()); };
    if (is_val(r))
        return true;
    for (term *it = &r->get_next(); it != r; it = &it->get_next())
        if (is_val(it))
            return true;
    return false;
}

// if there exists an uninterpreted const c s.t. tg ==> e = c, return c
// else return nullptr
app *term_graph::get_const_in_class(expr *e) {
    term *r = get_term(e);
    if (!r)
        return nullptr;
    auto is_const = [](term *t) { return is_uninterp_const(t->get_expr()); };
    if (is_const(r))
        return ::to_app(r->get_expr());
    for (term *it = &r->get_next(); it != r; it = &it->get_next())
        if (is_const(it))
            return ::to_app(it->get_expr());
    return nullptr;
}

void term_graph::display(std::ostream &out) {
    for (term *t : m_terms) { out << *t; }
}

void term_graph::to_lits(expr_ref_vector &lits, bool all_equalities,
                         bool repick_repr) {
    if (m_repick_repr || repick_repr) pick_repr();

    for (expr *a : m_lits) {
        if (is_internalized(a)) {
            if (m_explicit_eq && get_term(a)->is_eq_or_neq()) continue;
            lits.push_back(::to_app(mk_app(a)));
        }
    }

    for (term *t : m_terms) {
        if (t->is_eq_or_neq()) continue;
        if (!t->is_repr())
            continue;
        else if (all_equalities)
            mk_all_equalities(*t, lits);
        else
            mk_equalities(*t, lits);
    }

    // TODO: use seen to prevent duplicate disequalities
    for (auto p : m_deq_pairs) {
        expr_ref lhs = mk_app(p.first->get_expr());
        expr_ref rhs = mk_app(p.second->get_expr());
        lits.push_back(mk_neq(m, lhs, rhs));
    }

    for (auto t : m_deq_distinct) {
        ptr_vector<expr> args(t.size());
        for (auto c : t) args.push_back(mk_app(c->get_expr()));
        lits.push_back(m.mk_distinct(args.size(), args.data()));
    }
}

// assumes that representatives have already been picked
void term_graph::to_lits_qe_lite(expr_ref_vector &lits,
                                 std::function<bool(expr *)> *non_core) {
    SASSERT(all_of(m_terms, [](term* t) { return !!t->get_repr(); }));
    SASSERT(all_of(m_terms, [](term* t) { return !t->is_cgr() || t->get_repr()->is_cgr(); }));
    is_non_core not_in_core(non_core);
    check_pred contains_nc(not_in_core, m, false);
    // literals other than eq, neq, distinct
    for (expr *a : m_lits) {
        if (!is_internalized(a)) 
            continue;
        if (m_explicit_eq && get_term(a)->is_eq_or_neq()) 
            continue;
        expr_ref r(mk_app(a), m);        
        if (non_core == nullptr || !contains_nc(r)) 
            lits.push_back(r);
    }

    // equalities
    for (term *t : m_terms) {
        if (t->is_eq_or_neq()) continue;
        if (!t->is_repr()) continue;
        mk_qe_lite_equalities(*t, lits, contains_nc);
    }
    // disequalities and distinct
    // TODO: use seen to prevent duplicate disequalities
    expr_ref e1(m), e2(m), d(m), distinct(m);
    expr_ref_vector args(m);
    for (auto p : m_deq_pairs) {
        e1 = mk_app(*(p.first->get_repr()));
        e2 = mk_app(*(p.second->get_repr()));
        if (non_core == nullptr || (!contains_nc(e1) && !contains_nc(e2)))
            lits.push_back(mk_neq(m, e1, e2));
    }

    for (auto t : m_deq_distinct) {
        args.reset();
        for (auto c : t) {
            d = mk_app(*(c->get_repr()));
            if (non_core == nullptr || !contains_nc(d)) args.push_back(d);
        }
        if (args.size() < 2) 
            continue;
        if (args.size() == 2)
            distinct = mk_neq(m, args.get(0), args.get(1));
        else
            distinct = m.mk_distinct(args.size(), args.data());
        lits.push_back(distinct);
    }
}

expr_ref term_graph::to_expr(bool repick_repr) {
    expr_ref_vector lits(m);
    to_lits(lits, false, repick_repr);
    return mk_and(lits);
}

void term_graph::reset() {
    m_term2app.reset();
    m_pinned.reset();
    m_app2term.reset();
    std::for_each(m_terms.begin(), m_terms.end(), delete_proc<term>());
    m_terms.reset();
    m_lits.reset();
    m_cg_table.reset();
}

class term_graph::projector {
    term_graph &m_tg;
    ast_manager &m;
    u_map<expr *> m_term2app;
    u_map<expr *> m_root2rep;
    th_rewriter m_rewriter;

    model* m_model = nullptr;
    expr_ref_vector m_pinned; // tracks expr in the maps

    expr *mk_pure(term const &t) {
        TRACE(qe, t.display(tout));
        expr *e = nullptr;
        if (find_term2app(t, e))
            return e;
        e = t.get_expr();
        if (!is_app(e))
            return nullptr;
        app *a = ::to_app(e);
        expr_ref_buffer kids(m);
        for (term *ch : term::children(t)) {
            // prefer a node that resembles current child,
            // otherwise, pick a root representative, if present.
            if (find_term2app(*ch, e))
                kids.push_back(e); 
            else if (m_root2rep.find(ch->get_root().get_id(), e)) 
                kids.push_back(e);
            else
                return nullptr; 
            TRACE(qe_verbose, tout << *ch << " -> " << mk_pp(e, m) << "\n";);
        }
        expr_ref pure = m_rewriter.mk_app(a->get_decl(), kids.size(), kids.data());
        m_pinned.push_back(pure);
        add_term2app(t, pure);
        return pure;
    }

    bool is_better_rep(expr *t1, expr *t2) {
        if (!t2) return t1 != nullptr;
        return m.is_unique_value(t1) && !m.is_unique_value(t2);
    }

    struct term_depth {
        bool operator()(term const *t1, term const *t2) const {
            unsigned d1 = get_depth(t1->get_expr());
            unsigned d2 = get_depth(t2->get_expr());
            return d1 != d2 ? d1 < d2 : t1->get_expr()->get_id() < t2->get_expr()->get_id();
        }
    };

    void solve_core() {
        ptr_vector<term> worklist;
        for (term *t : m_tg.m_terms) {
            // skip pure terms
            if (!in_term2app(*t) && !t->is_eq_or_neq()) {
                worklist.push_back(t);
                t->set_mark(true);
            }
        }
        term_depth td;
        std::sort(worklist.begin(), worklist.end(), td);

        for (unsigned i = 0; i < worklist.size(); ++i) {
            term *t = worklist[i];
            t->set_mark(false);
            if (in_term2app(*t)) continue;

            expr *pure = mk_pure(*t);
            if (!pure) continue;

            add_term2app(*t, pure);
            expr *rep = nullptr;
            // ensure that the root has a representative
            m_root2rep.find(t->get_root().get_id(), rep);

            if (!rep) {
                m_root2rep.insert(t->get_root().get_id(), pure);
                for (term *p : term::parents(t->get_root())) {
                    SASSERT(!in_term2app(*p));
                    if (!p->is_marked()) {
                        p->set_mark(true);
                        worklist.push_back(p);
                    }
                }
            }
        }
        m_tg.reset_marks();
    }

    bool find_app(term const &t, expr *&res) {
        return find_term2app(t, res) ||
               m_root2rep.find(t.get_root().get_id(), res);
    }

    bool find_app(expr *lit, expr *&res) {
        term const *t = m_tg.get_term(lit);
        return find_term2app(*t, res) ||
               m_root2rep.find(t->get_root().get_id(), res);
    }

    void mk_lits(expr_ref_vector &res) {
        expr *e = nullptr;
        for (auto *lit : m_tg.m_lits) 
            if (!m.is_eq(lit) && find_app(lit, e))
                res.push_back(e);
        TRACE(qe, tout << "literals: " << res << "\n";);
    }

    void lits2pure(expr_ref_vector &res) {
        expr *e1 = nullptr, *e2 = nullptr, *e = nullptr, *p1 = nullptr, *p2 = nullptr;
        for (auto *lit : m_tg.m_lits) {
            if (m.is_eq(lit, e1, e2)) {
                if (find_app(e1, p1) && find_app(e2, p2)) {
                    if (p1 != p2) res.push_back(m.mk_eq(p1, p2));
                }
                else
                    TRACE(qe, tout << "skipping " << mk_pp(lit, m) << "\n";);
            }
            else if (m.is_not(lit, e) && m.is_eq(e, e1, e2)) {
                if (find_app(e1, p1) && find_app(e2, p2)) {
                    res.push_back(mk_neq(m, p1, p2));
                }
                else
                    TRACE(qe, tout << "skipping " << mk_pp(lit, m) << "\n";);
            }
            else if (m.is_distinct(lit)) {
                ptr_buffer<expr> diff;
                for (expr *arg : *to_app(lit))
                    if (find_app(arg, p1)) diff.push_back(p1);
                if (diff.size() > 1)
                    res.push_back(m.mk_distinct(diff.size(), diff.data()));
                else
                    TRACE(qe, tout << "skipping " << mk_pp(lit, m) << "\n";);
            }
            else if (find_app(lit, p1))
                res.push_back(p1);
            else
                TRACE(qe, tout << "skipping " << mk_pp(lit, m) << "\n";);
        }
        remove_duplicates(res);
        TRACE(qe, tout << "literals: " << res << "\n";);
    }

    void remove_duplicates(expr_ref_vector &v) {
        expr_fast_mark1 seen;
        unsigned j = 0;
        for (expr *e : v) {
            if (!seen.is_marked(e)) {
                v[j++] = e;
                seen.mark(e);
            }
        }
        v.shrink(j);
    }

    vector<ptr_vector<term>> m_decl2terms; // terms that use function f
    ptr_vector<func_decl> m_decls;

    void collect_decl2terms() {
        // Collect the projected function symbols.
        m_decl2terms.reset();
        m_decls.reset();
        for (term *t : m_tg.m_terms) {
            if (t->is_eq_or_neq()) continue;
            expr *e = t->get_expr();
            if (!is_app(e)) continue;
            if (!is_projected(*t)) continue;
            app *a = to_app(e);
            func_decl *d = a->get_decl();
            if (d->get_arity() == 0) continue;
            unsigned id = d->get_small_id();
            m_decl2terms.reserve(id + 1);
            if (m_decl2terms[id].empty()) m_decls.push_back(d);
            m_decl2terms[id].push_back(t);
        }
    }

    void args_are_distinct(expr_ref_vector &res) {
        //
        // for each projected function that occurs
        // (may occur) in multiple congruence classes,
        // produce assertions that non-congruent arguments
        // are distinct.
        //
        for (func_decl *d : m_decls) {
            unsigned id = d->get_small_id();
            ptr_vector<term> const &terms = m_decl2terms[id];
            if (terms.size() <= 1) continue;
            unsigned arity = d->get_arity();
            for (unsigned i = 0; i < arity; ++i) {
                obj_hashtable<expr> roots, root_vals;
                expr_ref_vector pinned(m);
                for (term *t : terms) {
                    expr *arg = to_app(t->get_expr())->get_arg(i);
                    term const &root = m_tg.get_term(arg)->get_root();
                    expr *r = root.get_expr();
                    // if a model is given, then use the equivalence class
                    // induced by the model. Otherwise, use the congruence
                    // class.
                    if (m_model) {
                        expr_ref tmp(m);
                        tmp = (*m_model)(r);
                        if (!root_vals.contains(tmp)) {
                            root_vals.insert(tmp);
                            roots.insert(r);
                            pinned.push_back(tmp);
                        }
                    }
                    else { roots.insert(r); }
                }
                if (roots.size() > 1) {
                    ptr_buffer<expr> args;
                    for (expr *r : roots) { args.push_back(r); }
                    TRACE(qe, tout << "function: " << d->get_name() << "\n";);
                    res.push_back(m.mk_distinct(args.size(), args.data()));
                }
            }
        }
    }

    void mk_distinct(expr_ref_vector &res) {
        collect_decl2terms();
        args_are_distinct(res);
        TRACE(qe, tout << res << "\n";);
    }

    void mk_pure_equalities(const term &t, expr_ref_vector &res) {
        SASSERT(t.is_root());
        expr *rep = nullptr;
        if (!m_root2rep.find(t.get_id(), rep)) return;
        obj_hashtable<expr> members;
        members.insert(rep);
        term const *r = &t;
        do {
            expr *member = nullptr;
            if (find_term2app(*r, member) && !members.contains(member)) {
                res.push_back(m.mk_eq(rep, member));
                members.insert(member);
            }
            r = &r->get_next();
        }
        while (r != &t);
    }

    bool is_projected(const term &t) { return m_tg.m_is_var(t); }

    void mk_unpure_equalities(const term &t, expr_ref_vector &res) {
        expr *rep = nullptr;
        if (!m_root2rep.find(t.get_id(), rep)) return;
        obj_hashtable<expr> members;
        members.insert(rep);
        term const *r = &t;
        do {
            expr *member = mk_pure(*r);
            SASSERT(member);
            if (!members.contains(member) &&
                (!is_projected(*r) || !is_solved_eq(rep, member))) {
                res.push_back(m.mk_eq(rep, member));
                members.insert(member);
            }
            r = &r->get_next();
        }
        while (r != &t);
    }

    template <bool pure> void mk_equalities(expr_ref_vector &res) {
        for (term *t : m_tg.m_terms) {
            if (t->is_eq_or_neq()) continue;
            if (!t->is_root()) continue;
            if (!m_root2rep.contains(t->get_id())) continue;
            if (pure)
                mk_pure_equalities(*t, res);
            else
                mk_unpure_equalities(*t, res);
        }
        TRACE(qe, tout << "literals: " << res << "\n";);
    }

    void mk_pure_equalities(expr_ref_vector &res) { mk_equalities<true>(res); }

    void mk_unpure_equalities(expr_ref_vector &res) {
        mk_equalities<false>(res);
    }

    // TBD: generalize for also the case of a (:var n)
    bool is_solved_eq(expr *lhs, expr *rhs) {
        return is_uninterp_const(rhs) && !occurs(rhs, lhs);
    }

    /// Add equalities and disequalities for all pure representatives
    /// based on their equivalence in the model
    void model_complete(expr_ref_vector &res) {
        if (!m_model) return;
        obj_map<expr, expr *> val2rep;
        model_evaluator mev(*m_model);
        for (auto &kv : m_root2rep) {
            expr *rep = kv.m_value;
            expr_ref val(m);
            expr *u = nullptr;
            if (!mev.eval(rep, val)) continue;
            if (val2rep.find(val, u)) { res.push_back(m.mk_eq(u, rep)); }
            else { val2rep.insert(val, rep); }
        }

        // TBD: optimize further based on implied values (e.g.,
        // some literals are forced to be true/false) and based on
        // unique_values (e.g., (x=1 & y=1) does not require
        // (x!=y) to be added
        ptr_buffer<expr> reps;
        for (auto &kv : val2rep) {
            expr *rep = kv.m_value;
            if (!m.is_unique_value(rep)) reps.push_back(kv.m_value);
        }

        if (reps.size() <= 1) return;

        // -- sort representatives, call mk_distinct on any range
        // -- of the same sort longer than 1
        std::sort(reps.data(), reps.data() + reps.size(), sort_lt_proc());
        unsigned i = 0;
        unsigned sz = reps.size();
        while (i < sz) {
            sort *last_sort = res.get(i)->get_sort();
            unsigned j = i + 1;
            while (j < sz && last_sort == reps.get(j)->get_sort()) { ++j; }
            if (j - i == 2) {
                expr_ref d(m);
                d = mk_neq(m, reps.get(i), reps.get(i + 1));
                if (!m.is_true(d)) res.push_back(d);
            }
            else if (j - i > 2)
                res.push_back(m.mk_distinct(j - i, reps.data() + i));
            i = j;
        }
        TRACE(qe, tout << "after distinct: " << res << "\n";);
    }

    std::ostream &display(std::ostream &out) const {
        m_tg.display(out);
        out << "term2app:\n";
        for (auto const &kv : m_term2app) {
            out << kv.m_key << " |-> " << mk_pp(kv.m_value, m) << "\n";
        }
        out << "root2rep:\n";
        for (auto const &kv : m_root2rep) {
            out << kv.m_key << " |-> " << mk_pp(kv.m_value, m) << "\n";
        }
        return out;
    }

public:
    projector(term_graph &tg)
        : m_tg(tg), m(m_tg.m), m_rewriter(m), m_pinned(m) {}

    void add_term2app(term const &t, expr *a) {
        m_term2app.insert(t.get_id(), a);
    }

    void del_term2app(term const &t) { m_term2app.remove(t.get_id()); }

    bool find_term2app(term const &t, expr *&r) {
        return m_term2app.find(t.get_id(), r);
    }

    expr* rep_of(term const &t) {
        expr *r = nullptr;
        find_app(t, r);
        return r;
    }

    bool in_term2app(term const &t) { return m_term2app.contains(t.get_id()); }

    void set_model(model &mdl) { m_model = &mdl; }

    void reset() {
        m_tg.reset_marks();
        m_term2app.reset();
        m_root2rep.reset();
        m_pinned.reset();
        m_model = nullptr;
    }

    expr_ref_vector project() {
        expr_ref_vector res(m);
        purify();
        lits2pure(res);
        mk_distinct(res);
        reset();
        return res;
    }

    expr_ref_vector get_ackerman_disequalities() {
        expr_ref_vector res(m);
        purify();
        lits2pure(res);
        unsigned sz = res.size();
        mk_distinct(res);
        reset();
        unsigned j = 0;
        for (unsigned i = sz; i < res.size(); ++i) { res[j++] = res.get(i); }
        res.shrink(j);
        return res;
    }

    expr_ref_vector solve() {
        expr_ref_vector res(m);
        purify();
        solve_core();
        mk_lits(res);
        mk_unpure_equalities(res);
        reset();
        return res;
    }

    vector<expr_ref_vector> get_partition(model &mdl, bool include_bool) {
        vector<expr_ref_vector> result;
        expr_ref_vector pinned(m);
        obj_map<expr, unsigned> pid;
        auto insert_val = [&](expr *a, expr *val) {
            unsigned p = 0;
            // NB. works for simple domains Integers, Rationals,
            // but not for algebraic numerals.
            if (!pid.find(val, p)) {
                p = pid.size();
                pid.insert(val, p);
                pinned.push_back(val);
                result.push_back(expr_ref_vector(m));
            }
            result[p].push_back(a);
        };
        model::scoped_model_completion _smc(mdl, true);
        for (term *t : m_tg.m_terms) {
            if (t->is_eq_or_neq()) continue;
            expr *a = t->get_expr();
            if (!is_app(a)) continue;
            if (m.is_bool(a) && !include_bool) continue;
            expr_ref val = mdl(a);
            insert_val(a, val);
        }
        return result;
    }

    expr_ref_vector shared_occurrences(family_id fid) {
        expr_ref_vector result(m);
        for (term *t : m_tg.m_terms) {
            if (t->is_eq_or_neq()) continue;
            expr *e = t->get_expr();
            if (e->get_sort()->get_family_id() != fid) continue;
            for (term *p : term::parents(t->get_root())) {
                expr *pe = p->get_expr();
                if (!is_app(pe)) continue;
                if (to_app(pe)->get_family_id() == fid) continue;
                if (to_app(pe)->get_family_id() == m.get_basic_family_id())
                    continue;
                result.push_back(e);
                break;
            }
        }
        return result;
    }

    void purify() {
        // - propagate representatives up over parents.
        //   use work-list + marking to propagate.
        // - produce equalities over represented classes.
        // - produce other literals over represented classes
        //   (walk disequalities in m_lits and represent
        //   lhs/rhs over decls or excluding decls)

        ptr_vector<term> worklist;
        for (term *t : m_tg.m_terms) {
            if (t->is_eq_or_neq()) continue;
            worklist.push_back(t);
            t->set_mark(true);
        }
        // traverse worklist in order of depth.
        term_depth td;
        std::sort(worklist.begin(), worklist.end(), td);

        for (unsigned i = 0; i < worklist.size(); ++i) {
            term *t = worklist[i];
            t->set_mark(false);
            if (in_term2app(*t)) continue;
            if (!t->is_theory() && is_projected(*t)) continue;

            expr *pure = mk_pure(*t);
            if (!pure) continue;

            add_term2app(*t, pure);
            TRACE(qe_verbose,
                  tout << "purified " << *t << " " << mk_pp(pure, m) << "\n";);
            expr *rep = nullptr; // ensure that the root has a representative
            m_root2rep.find(t->get_root().get_id(), rep);

            // update rep with pure if it is better
            if (pure != rep && is_better_rep(pure, rep)) {
                m_root2rep.insert(t->get_root().get_id(), pure);
                for (term *p : term::parents(t->get_root())) {
                    del_term2app(*p);
                    if (!p->is_marked()) {
                        p->set_mark(true);
                        worklist.push_back(p);
                    }
                }
            }
        }

        // Here we could also walk equivalence classes that
        // contain interpreted values by sort and extract
        // disequalities between non-unique value
        // representatives.  these disequalities are implied
        // and can be mined using other means, such as theory
        // aware core minimization
        m_tg.reset_marks();
        TRACE(qe, display(tout << "after purify\n"););
    }
};

// produce a quantifier reduction of the formula stored in the term graph
//  removes from `vars` the variables that have a ground representative
//  modifies `vars` to keep the variables that could not be eliminated
void term_graph::qel(app_ref_vector &vars, expr_ref &fml,
                     std::function<bool(expr *)> *non_core) {

    filter(vars, [this](auto v) { return is_internalized(v); });
    pick_repr();
    refine_repr();

    expr_ref_vector lits(m);
    to_lits_qe_lite(lits, non_core);
    fml = mk_and(lits);

    // Remove all variables that are do not appear in the formula
    expr_sparse_mark mark;
    mark_all_sub_expr marker(mark);
    quick_for_each_expr(marker, fml);
    filter(vars, [&mark](auto v) { return mark.is_marked(v); });
}

void term_graph::set_vars(func_decl_ref_vector const &decls, bool exclude) {
    m_is_var.set_decls(decls, exclude);
}

void term_graph::set_vars(app_ref_vector const &vars, bool exclude) {
    m_is_var.set_decls(vars, exclude);
}

void term_graph::add_vars(app_ref_vector const &vars) {
    m_is_var.add_decls(vars);
}

void term_graph::add_var(app *var) { m_is_var.add_decl(var); }

expr_ref_vector term_graph::project() {
    // reset solved vars so that they are not considered pure by projector
    m_is_var.reset_solved();
    term_graph::projector p(*this);
    return p.project();
}

expr_ref_vector term_graph::project(model &mdl) {
    m_is_var.reset_solved();
    term_graph::projector p(*this);
    p.set_model(mdl);
    return p.project();
}

expr_ref_vector term_graph::solve() {
    // reset solved vars so that they are not considered pure by projector
    m_is_var.reset_solved();
    term_graph::projector p(*this);
    return p.solve();
}

expr_ref_vector term_graph::get_ackerman_disequalities() {
    m_is_var.reset_solved();
    dealloc(m_projector);
    m_projector = alloc(term_graph::projector, *this);
    return m_projector->get_ackerman_disequalities();
}

vector<expr_ref_vector> term_graph::get_partition(model &mdl) {
    dealloc(m_projector);
    m_projector = alloc(term_graph::projector, *this);
    return m_projector->get_partition(mdl, false);
}

expr_ref_vector term_graph::shared_occurrences(family_id fid) {
    term_graph::projector p(*this);
    return p.shared_occurrences(fid);
}

void term_graph::add_model_based_terms(model &mdl,
                                       expr_ref_vector const &terms) {
    for (expr *t : terms) { internalize_term(t); }
    m_is_var.reset_solved();

    SASSERT(!m_projector);
    m_projector = alloc(term_graph::projector, *this);

    // retrieve partition of terms
    vector<expr_ref_vector> equivs = m_projector->get_partition(mdl, true);

    // merge term graph on equal terms.
    for (auto const &cs : equivs) {
        term *t0 = get_term(cs[0]);
        for (unsigned i = 1; i < cs.size(); ++i) {
            merge(*t0, *get_term(cs[i]));
        }
    }
    TRACE(
        qe, for (auto &es
                   : equivs) {
            tout << "equiv: ";
            for (expr *t : es) tout << expr_ref(t, m) << " ";
            tout << "\n";
        } display(tout););
    // create representatives for shared/projected variables.
    m_projector->set_model(mdl);
    m_projector->purify();
}

expr* term_graph::rep_of(expr *e) {
    SASSERT(m_projector);
    term *t = get_term(e);
    SASSERT(t && "only get representatives");
    return m_projector->rep_of(*t);    
}

expr_ref_vector term_graph::dcert(model &mdl, expr_ref_vector const &lits) {
    TRACE(qe, tout << "dcert " << lits << "\n";);
    struct pair_t {
        expr *a, *b;
        pair_t() : a(nullptr), b(nullptr) {}
        pair_t(expr *_a, expr *_b) : a(_a), b(_b) {
            if (a->get_id() > b->get_id()) std::swap(a, b);
        }
        struct hash {
            unsigned operator()(pair_t const &p) const {
                return mk_mix(p.a ? p.a->hash() : 0, p.b ? p.b->hash() : 0, 1);
            }
        };
        struct eq {
            bool operator()(pair_t const &a, pair_t const &b) const {
                return a.a == b.a && a.b == b.b;
            }
        };
    };
    hashtable<pair_t, pair_t::hash, pair_t::eq> diseqs;
    expr_ref_vector result(m);
    add_lits(lits);
    svector<pair_t> todo;

    for (expr *e : lits) {
        expr *ne, *a, *b;
        if (m.is_not(e, ne) && m.is_eq(ne, a, b) &&
            (is_uninterp(a) || is_uninterp(b))) {
            diseqs.insert(pair_t(a, b));
        }
        else if (is_uninterp(e)) { diseqs.insert(pair_t(e, m.mk_false())); }
        else if (m.is_not(e, ne) && is_uninterp(ne)) {
            diseqs.insert(pair_t(ne, m.mk_true()));
        }
    }
    for (auto &p : diseqs) todo.push_back(p);

    auto const partitions = get_partition(mdl);
    obj_map<expr, unsigned> term2pid;
    unsigned id = 0;
    for (auto const &vec : partitions) {
        for (expr *e : vec) term2pid.insert(e, id);
        ++id;
    }
    expr_ref_vector empty(m);
    auto partition_of = [&](expr *e) {
        unsigned pid;
        if (!term2pid.find(e, pid)) return empty;
        return partitions[pid];
    };
    auto in_table = [&](expr *a, expr *b) {
        return diseqs.contains(pair_t(a, b));
    };
    auto same_function = [](expr *a, expr *b) {
        return is_app(a) && is_app(b) &&
               to_app(a)->get_decl() == to_app(b)->get_decl() &&
               to_app(a)->get_family_id() == null_family_id;
    };

    // make sure that diseqs is closed under function applications
    // of uninterpreted functions.
    for (unsigned idx = 0; idx < todo.size(); ++idx) {
        auto p = todo[idx];
        for (expr *t1 : partition_of(p.a)) {
            for (expr *t2 : partition_of(p.b)) {
                if (same_function(t1, t2)) {
                    unsigned sz = to_app(t1)->get_num_args();
                    bool found = false;
                    pair_t q(t1, t2);
                    for (unsigned i = 0; i < sz; ++i) {
                        expr *arg1 = to_app(t1)->get_arg(i);
                        expr *arg2 = to_app(t2)->get_arg(i);
                        if (mdl(arg1) == mdl(t2)) { continue; }
                        if (in_table(arg1, arg2)) {
                            found = true;
                            break;
                        }
                        q = pair_t(arg1, arg2);
                    }
                    if (!found) {
                        diseqs.insert(q);
                        todo.push_back(q);
                        result.push_back(m.mk_not(m.mk_eq(q.a, q.b)));
                    }
                }
            }
        }
    }
    for (auto const &terms : partitions) {
        expr *a = nullptr;
        for (expr *b : terms) {
            if (is_uninterp(b)) {
                if (a)
                    result.push_back(m.mk_eq(a, b));
                else
                    a = b;
            }
        }
    }
    TRACE(qe, tout << result << "\n";);
    return result;
}

void term_graph::cground_percolate_up(term *t) {
    SASSERT(t->is_class_gr());
    term *it = t;
    // there is a cgr term in all ground classes
    while (!it->is_cgr()) {
        it = &it->get_next();
        SASSERT(it != t);
    }

    ptr_vector<term> todo;
    todo.push_back(it);
    cground_percolate_up(todo);
}

void term_graph::cground_percolate_up(ptr_vector<term> &todo) {
    while (!todo.empty()) {
        term* t = todo.back();
        todo.pop_back();
        t->set_cgr(true);
        t->set_class_gr(true);
        for (auto p : term::parents(t->get_root()))
            if (!p->is_cgr() && p->all_children_ground())
                todo.push_back(p);
    }
}

void term_graph::compute_cground() {
    for (auto t : m_terms) {
        t->set_cgr(false);
        t->set_class_gr(false);
    }
    ptr_vector<term> todo;
    for (auto t : m_terms) 
        if (t->is_gr())
            todo.push_back(t); 
    cground_percolate_up(todo);
    DEBUG_CODE(for (auto t : m_terms) {
        bool isclsg = true;
        for (auto c : term::children(t)) isclsg &= c->is_class_gr();
        SASSERT(t->deg() == 0 || !isclsg || t->is_cgr());
        SASSERT(t->deg() == 0 || isclsg || !t->is_cgr());
    });
}
} // namespace mbp