z3-src 501.0.0

/*++
Copyright (c) 2025 Microsoft Corporation

Module Name:

    ho_matcher

Abstract:

    second and higher-order matcher
    
Author:

    Nikolaj Bjorner (nbjorner) 2025-07-07

   Ho-matcher performs a tree search over match_goals.
   Let Q be the current set of match_goals
   Let B be the current state of a backtracking stack.
   Each element in Q is a work item. 
   The workflow is as follows:
   elements w of Q are removed and added to B, 
   if processing of w results in a unitary match, it is removed from B and the match_goals are added to Q.

   match_goals in Q can be processed independently on when they are generated.
   If a subgoal of Q fails to match, the algorithm backtracks.
   Backtracking can be tuned by keeping track of dependencies.

   Elements in Q are simplified relative to a current substitution. 
   The level where the current substitution affects simplification determines 
   the persistence level of simplification.
  
   A v1 ignores dependencies and assumes always that side-effects are relative to the current backtracking scope.
   A v2 should address and measure the overhead.

   Elements in Q need to persist relative to changes made within a backtracking scope.
   Every operation on Q should be replayed. 
   Thus, elements removed from Q are re-inserted.

   (pat, offset) (t, offset)
   - a variable is considered bound if it's index is below offset.
   - meta variables are adjusted relative to offset

--*/

#include "ast/euf/ho_matcher.h"
#include "ast/well_sorted.h"



namespace euf {

    namespace {
        struct scoped_trail_level {
            trail_stack &m_trail;
            unsigned m_base;
            scoped_trail_level(trail_stack &t) : m_trail(t), m_base(t.get_num_scopes()) {}
            ~scoped_trail_level() {
                m_trail.pop_scope(m_trail.get_num_scopes() - m_base);
            }
        };
    }

    expr_ref_vector const &ho_subst::get_binding(quantifier *q) {
        ast_manager &m = m_subst.get_manager();
        m_binding.reset();
        m_binding.append(m_subst);

        // Shrink binding to original quantifier's num_decls
        // The HO quantifier has extra vars at higher indices; drop them.
        // Binding is indexed by var index: binding[i] = value for var i.
        // First substitute any remaining vars, then keep only original vars.
        TRACE(
            ho_matching, tout << "num bound variables " << q->get_num_decls() << " for " << mk_bounded_pp(q, m) << "\n"
                              << m_binding << "\n";
            for (unsigned i = 0; i < m_binding.size(); ++i) {
                tout << i << " - " << mk_pp(m_binding.get(i)->get_sort(), m) << ": " << mk_ll_pp(m_binding.get(i), m)
                     << "\n";
            });
        if (m_binding.size() > q->get_num_decls()) {
            // binding is indexed directly (binding[k] = value for var k),
            // so the substitution must use direct (non-standard) order to
            // resolve chained HO variable references; the sort guard below
            // is checked with the matching order.
            var_subst sub(m, false);
            bool change = true;
            while (change) {
                change = false;
                for (unsigned i = 0; i < m_binding.size(); ++i) {
                    if (!m_binding.get(i))
                        continue;
                    // A misaligned higher-order binding would build an
                    // ill-sorted term. Abandon this refinement (no instance)
                    // rather than aborting the whole solve.
                    SASSERT(is_well_sorted(m, m_binding.get(i)));
                    auto r = sub(m_binding.get(i), m_binding);
                    change |= r != m_binding.get(i);
                    m_binding[i] = r;
                    SASSERT(is_well_sorted(m, m_binding.get(i)));
                    TRACE(ho_matching, tout << "setting v" << i << " <- " << r << "\n");
                }
            }
            m_binding.shrink(q->get_num_decls());
        }
        SASSERT(m_binding.size() == q->get_num_decls());

        m_binding.reverse();
        return m_binding;
    }

    void ho_matcher::operator()(expr *pat, expr *t, unsigned num_vars) {
        (*this)(pat, t, 0, num_vars);
    }

    void ho_matcher::operator()(expr *pat, expr *t, unsigned num_bound, unsigned num_vars) {
        scoped_trail_level _restore(m_trail);
        m_trail.push_scope();
        m_subst.resize(0);
        m_subst.resize(num_vars);
        m_goals.reset();
        m_goals.push(nullptr, 0, num_bound, pat, t);
        search();
    }

    void ho_matcher::search() {
        IF_VERBOSE(10, display(verbose_stream()));

        struct drain_backtrack {
            ho_matcher &m;
            drain_backtrack(ho_matcher &m) : m(m) {}
            ~drain_backtrack() {
                while (!m.m_backtrack.empty()) {
                    m.backtrack();
                }
            }
        };

        drain_backtrack _drain(*this);

        unsigned budget = m_max_iterations;
        while (m.inc()) {
            if (budget-- == 0) {
                IF_VERBOSE(2, verbose_stream() << "ho_matcher: search budget exhausted\n");
                break;
            }
            // Q, B -> Q', B'. Push work on the backtrack stack and new work items
            // e, Bw -> Q', B'. Consume backtrack stack
            if (!m_goals.empty())
                push_backtrack();
            else if (!m_backtrack.empty())
                resume();
            else
                break;
        }

        IF_VERBOSE(10, display(verbose_stream() << "ho_matcher: done\n"));
    }

    void ho_matcher::backtrack() {
        SASSERT(!m_backtrack.empty());
        auto wi = m_backtrack.back();
        if (wi->in_scope())
            m_trail.pop_scope(1);
        m_backtrack.pop_back();
    }

    void ho_matcher::resume() {
        while (!m_backtrack.empty()) {
            auto &wi = *m_backtrack.back();
            bool st = consume_work(wi);
            TRACE(ho_matching, display(tout << "ho_matcher::consume_work: " << mk_bounded_pp(wi.pat, m) << " =?= "
                                            << mk_bounded_pp(wi.t, m) << " -> " << (st ? "true" : "false") << "\n"););
            if (st) {
                if (m_goals.empty())
                    m_on_match(m_subst);
                break;
            }
            else
                backtrack();
        }
    }

    void ho_matcher::push_backtrack() {
        SASSERT(!m_goals.empty());
        m_backtrack.push_back(m_goals.pop());
        resume();
    }

    lbool ho_matcher::are_equal(unsigned o1, expr *p, unsigned o2, expr *t) const {
        if (p->get_sort() != t->get_sort()) {
            TRACE(ho_matching, tout << "sort mismatch: " << mk_pp(p, m) << " : " << mk_pp(p->get_sort(), m) << " vs "
                                    << mk_pp(t, m) << " : " << mk_pp(t->get_sort(), m) << "\n";);
            return l_false;
        }
        if (o1 == o2 && p == t)
            return l_true;

        if (is_ground(p) && is_ground(t)) {
            if (use_cgr())
                return to_lbool(m_are_equal(p, t));
            else
                return to_lbool(p == t);
        }

        if (is_lambda(p) && is_lambda(t)) {
            auto q1 = to_quantifier(p);
            auto q2 = to_quantifier(t);
            SASSERT(q1->get_num_decls() == q2->get_num_decls());
            return are_equal(o1 + q1->get_num_decls(), q1->get_expr(), o2 + q2->get_num_decls(), q2->get_expr());
        }

        if (is_meta_var(p, o1)) {
            auto p1 = m_subst.get(to_var(p)->get_idx() - o1);
            if (p1)
                return are_equal(0, p1, o2, t);
            return l_undef;
        }

        if (is_meta_var(t, o2)) {
            auto t1 = m_subst.get(to_var(t)->get_idx() - o2);
            if (t1)
                return are_equal(o1, p, 0, t1);
            return l_undef;
        }

        if (m_unitary.is_flex(o1, p)) {
            expr_ref h = whnf(p, o1);
            if (h != p)
                return are_equal(o1, h, o2, t);
            return l_undef;
        }
        if (m_unitary.is_flex(o2, t)) {
            expr_ref h = whnf(t, o2);
            if (h != t)
                return are_equal(o1, p, o2, h);
            return l_undef;
        }
        if (m_array.is_select(p))
            return l_undef;  // TODO: interleave check with whnf expansion

        if (is_app(p) && is_app(t)) {
            auto a1 = to_app(p);
            auto a2 = to_app(t);
            if (a1->get_decl() != a2->get_decl())
                return l_false;
            if (a1->get_num_args() != a2->get_num_args())
                return l_false;
            for (unsigned i = 0; i < a1->get_num_args(); ++i) {
                auto r = are_equal(o1, a1->get_arg(i), o2, a2->get_arg(i));
                if (r != l_true)
                    return r;
            }
            return l_true;
        }
        if (is_bound_var(p, o1) || is_bound_var(t, o2))
            return to_lbool(p == t);

        return l_undef;
    }

    //
    // reduces (M N)
    // and recurisvely ((M N) K)
    // such that M is not a lambda
    //
    expr_ref ho_matcher::whnf(expr *e, unsigned offset) const {
        expr_ref r(e, m), t(m);
        //        verbose_stream() << "ho_matcher::whnf: " << r << "\n";
        if (is_meta_var(r, offset)) {
            auto v = to_var(e);
            auto idx = v->get_idx();
            if (idx >= offset && m_subst.get(idx - offset)) {
                auto e = m_subst.get(idx - offset);
                r = e;
                if (offset > 0) {
                    var_shifter sh(m);
                    sh(e, offset, r);
                }
                return r;
            }
        }
        while (m_array.is_select(r)) {
            app *a = to_app(r);
            auto arg0 = a->get_arg(0);
            //  apply substitution:
            if (is_meta_var(arg0, offset)) {
                auto v = to_var(arg0);
                auto idx = v->get_idx();
                if (idx >= offset && m_subst.get(idx - offset)) {
                    auto e = m_subst.get(idx - offset);
                    if (offset > 0) {
                        var_shifter sh(m);
                        sh(e, offset, r);
                        e = r;
                    }
                    expr_ref_vector args(m);
                    args.push_back(e);
                    for (unsigned i = 1; i < a->get_num_args(); ++i)
                        args.push_back(a->get_arg(i));

                    r = m.mk_app(a->get_decl(), args.size(), args.data());
                    //                  verbose_stream() << "ho_matcher::whnf: " << r << "\n";
                    continue;
                }
            }
            if (m_array.is_select(arg0)) {
                t = whnf(arg0, offset);
                if (t != arg0) {
                    ptr_vector<expr> args(a->get_num_args(), a->get_args());
                    args[0] = t;
                    r = m_array.mk_select(args.size(), args.data());
                    //                 verbose_stream() << "ho_matcher::whnf-rec: " << r << "\n";
                    continue;
                }
            }
            // outer beta reduction
            auto st = m_rewriter.mk_app_core(a->get_decl(), a->get_num_args(), a->get_args(), t);
            if (st != BR_FAILED)
                r = t;
            else
                break;
        }
        return r;
    }

    expr_ref ho_matcher::whnf_star(expr *e, unsigned offset) const {
        expr_ref r(e, m);
        while (true) {
            auto q = whnf(r, offset);
            if (q == r)
                return r;
            r = q;
        }
    }

    void ho_matcher::reduce(match_goal &wi) {
        wi.pat = whnf_star(wi.pat, wi.pat_offset());
        wi.t = whnf_star(wi.t, wi.term_offset());
    }

    bool ho_matcher::consume_work(match_goal &wi) {
        //        IF_VERBOSE(1, display(verbose_stream() << "ho_matcher::consume_work: " << wi.pat << " =?= " << wi.t <<
        //        "\n"););

        if (wi.in_scope())
            m_trail.pop_scope(1);

        wi.set_in_scope();
        m_trail.push_scope();

        if (wi.is_done())
            return false;

        if (wi.level > m_max_depth) {
            wi.set_done();
            return false;
        }

        reduce(wi);

        auto t = wi.t;
        auto p = wi.pat;

        switch (are_equal(wi.pat_offset(), p, wi.term_offset(), t)) {
        case l_false: wi.set_done(); return false;
        case l_true: wi.set_done(); return true;
        case l_undef: break;
        }

        // v >= offset
        // v - offset |-> t
        if (is_meta_var(p, wi.pat_offset()) && is_closed(t, 0, wi.term_offset())) {
            auto v = to_var(p);
            SASSERT(!m_subst.get(v->get_idx() -
                                 wi.pat_offset()));  // reduce ensures meta variables are not in substitutions
            add_binding(v, wi.pat_offset(), t);
            wi.set_done();
            return true;
        }

        // N = \ x. T => ((shift1 N) x) = T
        if (is_lambda(t) && !is_lambda(p)) {
            auto q = to_quantifier(t);
            auto t_body = q->get_expr();
            auto nd = q->get_num_decls();
            var_shifter vs(m);
            expr_ref r(m);
            vs(p, nd, r);
            expr_ref_vector args(m);
            args.push_back(r);
            for (unsigned i = 0; i < nd; ++i)
                args.push_back(m.mk_var(nd - 1 - i, q->get_decl_sort(i)));
            r = m_array.mk_select(args);
            m_goals.push(&wi, wi.level, wi.term_offset() + nd, r, t_body);
            wi.set_done();
            return true;
        }

        // \x . N = T => N = ((shift1 T) x)
        if (is_lambda(p) && !is_lambda(t)) {
            auto q = to_quantifier(p);
            auto p_body = q->get_expr();
            auto nd = q->get_num_decls();
            var_shifter vs(m);
            expr_ref r(m);
            vs(t, nd, r);
            expr_ref_vector args(m);
            args.push_back(r);
            for (unsigned i = 0; i < nd; ++i)
                args.push_back(m.mk_var(nd - 1 - i, q->get_decl_sort(i)));
            r = m_array.mk_select(args);
            m_goals.push(&wi, wi.level, wi.term_offset() + nd, p_body, r);
            wi.set_done();
            return true;
        }

        //
        // lambda x . p == lambda x . t
        //
        if (is_quantifier(p) && is_quantifier(t)) {
            auto qp = to_quantifier(p);
            auto qt = to_quantifier(t);
            unsigned pd = qp->get_num_decls();
            unsigned td = qt->get_num_decls();
            if (qp->get_kind() != qt->get_kind())
                return false;
            if (pd != td)
                return false;
            for (unsigned i = 0; i < pd; ++i)
                if (qp->get_decl_sort(i) != qt->get_decl_sort(i))
                    return false;
            m_goals.push(&wi, wi.level, wi.term_offset() + td, qp->get_expr(), qt->get_expr());
            wi.set_done();
            return true;
        }

        if (m.is_not(p) && m.is_true(t)) {
            m_goals.push(&wi, wi.level, wi.term_offset(), to_app(p)->get_arg(0), m.mk_false());
            wi.set_done();
            return true;
        }

        if (m.is_not(p) && m.is_false(t)) {
            m_goals.push(&wi, wi.level, wi.term_offset(), to_app(p)->get_arg(0), m.mk_true());
            wi.set_done();
            return true;
        }

        expr *cond, *th, *el;
        if (m.is_ite(p, cond, th, el) && use_cgr()) {
            if (wi.is_init())
                wi.set_project();
            SASSERT(wi.is_project());
            auto lvl = wi.level;
            auto off = wi.term_offset();
            if (wi.index() == 0) {
                m_goals.push(&wi, lvl, off, cond, m.mk_true());
                m_goals.push(&wi, lvl, off, th, t);
                wi.set_index(1);
            }
            else {
                m_goals.push(&wi, lvl, off, cond, m.mk_false());
                m_goals.push(&wi, lvl, off, el, t);
                wi.set_done();
            }
            return true;
        }

        if (m.is_ite(t, cond, th, el) && use_cgr()) {
            if (m_are_equal(cond, m.mk_true())) {
                m_goals.push(&wi, wi.level, wi.term_offset(), p, th);
                wi.set_done();
                return true;
            }
            if (m_are_equal(cond, m.mk_false())) {
                m_goals.push(&wi, wi.level, wi.term_offset(), p, el);
                wi.set_done();
                return true;
            }
        }

        expr *p1, *p2;
        if (m.is_eq(p, p1, p2) && use_cgr() && m.is_true(t)) {
            if (is_ground(p1)) {
                m_goals.push(&wi, wi.level, wi.term_offset(), p2, p1);
                wi.set_done();
                return true;
            }
            if (is_ground(p2)) {
                m_goals.push(&wi, wi.level, wi.term_offset(), p1, p2);
                wi.set_done();
                return true;
            }
        }

        // Flex head unitary
        // H(pat) = t

        if (m_array.is_select(p) && m_unitary.is_flex(wi.pat_offset(), p) && is_pattern(p, wi.pat_offset(), t)) {
            auto lam = abstract_pattern(p, wi.pat_offset(), t);
            while (m_array.is_select(p))
                p = to_app(p)->get_arg(0);
            SASSERT(is_meta_var(p, wi.pat_offset()));
            add_binding(to_var(p), wi.pat_offset(), lam);
            wi.set_done();
            return true;
        }

        // Flex head general case
        // X p = t = f(ts)
        // - project : X -> \lambda x . x_i | p_i = t
        // - imitate : X -> \lambda x . f (X1 p) (X2 p) | (Xi p) = ts_i
        // - eq for t = T: X -> \lambda x . x = t
        // - eq for t = F: X -> \lambda x . x = t' for t in diff(t)
        // - ite: X -> \lambda x . if (X1 p) then (X2 p) else (X3 p)

        if (m_array.is_select(p) && m_unitary.is_flex(wi.pat_offset(), p)) {
            // innermost select is a meta variable,
            // order patterns from inner-most application to outer-most.
            ptr_vector<app> pats;
            auto p1 = p;
            while (m_array.is_select(p1)) {
                pats.push_back(to_app(p1));
                p1 = to_app(p1)->get_arg(0);
            }
            pats.reverse();
            auto v = to_var(p1);

            if (wi.is_init())
                wi.set_project();

            if (wi.is_project() && process_project(wi, v, pats, t))
                return true;

            SASSERT(!is_lambda(t));

            if (wi.is_project())
                wi.set_imitate();

            if (wi.is_imitate() && !block_inference(wi)) {
                auto f = [&wi, v, &pats, this](expr *s) { return process_imitation(wi, v, pats, s); };
                if (process_equiv_class(wi, t, f))
                    return true;
            }

            if (wi.is_imitate())
                wi.set_eq();

            if (process_eq(wi, v, pats, t))
                return true;

            if (wi.is_eq())
                wi.set_ite();

            if (process_ite(wi, v, pats, t))
                return true;

            return false;
        }

        // f(p) ~ f(ts)
        auto f = [&wi, &p, this](expr *s) { return process_app(wi, p, s); };
        if (is_app(p) && process_equiv_class(wi, t, f))
            return true;

        return false;
    }

    // v -> lambda xs . x_i = p
    bool ho_matcher::process_eq(match_goal &wi, var *v, ptr_vector<app> const &pats, expr *t) {
        if (!wi.is_eq())
            return false;

        if (m.is_true(t)) {
            unsigned start = wi.index();
            unsigned i = 0;
            unsigned num_bound = 0;
            for (auto pa : pats)
                for (auto pi : array_select_indices(pa)) {
                    (void)pi;
                    ++num_bound;
                }
            for (auto pa : pats) {
                for (auto pi : array_select_indices(pa)) {
                    if (start == i) {
                        // create binding into pi:
                        auto mk_eq = [&](expr *x) {
                            var_shifter vs(m);
                            expr_ref shifted_pi(m);
                            vs(pi, num_bound, shifted_pi);
                            return m.mk_eq(shifted_pi, x);
                        };
                        auto e = mk_project(pats.size(), i, v->get_sort(), mk_eq);
                        add_binding(v, wi.pat_offset(), e);
                        wi.set_index(i + 1);
                        return true;
                    }
                    ++i;
                }
            }
            return false;
        }
        if (m.is_false(t)) {
        }
        return false;
    }

    // v -> lambda xs . ite(C xs, Tr xs, El xs)
    // create subgoal ite(C ps, Tr ps, El ps) = t
    bool ho_matcher::process_ite(match_goal &wi, var *v, ptr_vector<app> const &pats, expr *t) {
        if (!wi.is_ite())
            return false;
        if (block_inference(wi))
            return false;

        // Introduce three fresh meta-variables C, Tr, El and bind
        //   v -> \xs . ite(C xs, Tr xs, El xs)
        // with the single residual subgoal
        //   ite(C ps, Tr ps, El ps) ~ t
        // The condition/branch split is performed later when the ite subgoal
        // is consumed (see the is_ite(p, ...) case in consume_work).

        m_trail.push(undo_resize(m_subst));

        flex_frame fr(m);
        init_flex_frame(pats, fr);

        // ranges of the three fresh meta-variables: condition is boolean,
        // both branches have the sort of the matched term t.
        sort *ranges[3] = { m.mk_bool_sort(), t->get_sort(), t->get_sort() };
        expr_ref_vector sub_args(m), body_args(m);
        for (auto range : ranges) {
            expr_ref subgoal_app(m), body_app(m);
            mk_flex_app(fr, wi.pat_offset(), range, subgoal_app, body_app);
            sub_args.push_back(subgoal_app);
            body_args.push_back(body_app);
        }

        expr_ref sub(m);
        sub = m.mk_ite(sub_args.get(0), sub_args.get(1), sub_args.get(2));
        m_goals.push(&wi, wi.level + 1, wi.term_offset(), sub, t);

        expr_ref body(m);
        body = m.mk_ite(body_args.get(0), body_args.get(1), body_args.get(2));
        add_binding(v, wi.pat_offset(), mk_flex_lambda(fr, pats, body));
        return true;
    }

    bool ho_matcher::block_inference(match_goal const &wi) {
        // block imitation on the same term
        if (wi.is_imitate() && use_cgr()) {
            auto p = wi.parent();
            while (p) {
                if (m_are_equal(p->t, wi.t))
                    return true;
                p = p->parent();
            }
            return false;
        }

        // disallow nested ite
        if (wi.is_ite()) {
            auto p = wi.parent();
            while (p) {
                if (p->is_ite())
                    return true;
                p = p->parent();
            }
            return false;
        }

        return false;
    }

    bool ho_matcher::process_equiv_class(match_goal &wi, expr *t, std::function<bool(expr *)> const &f) {
        expr *s = wi.get_term();
        if (s == t) 
            return false;
        if (!use_cgr()) {
            wi.set_term(t);
            return f(t);
        }        
        if (!s)  // we are just starting.
            s = t;

        do {
            if (is_app(s) && m_is_cgr_root(s) && f(s)) {
                s = m_next(s);
                wi.set_term(s);
                return true;
            }
            s = m_next(s);
        } while (s != t);
        return false;
    }

    bool ho_matcher::process_app(match_goal &wi, expr *p, expr *t) {
        if (!is_app(p) || !is_app(t))
            return false;
        auto ta = to_app(t);
        auto tp = to_app(p);
        if (ta->get_decl() != tp->get_decl())
            return false;
        if (ta->get_num_args() != tp->get_num_args())
            return false;
        for (unsigned i = 0; i < ta->get_num_args(); ++i)
            m_goals.push(&wi, wi.level, wi.term_offset(), tp->get_arg(i), ta->get_arg(i));
        return true;
    }


    void ho_matcher::init_flex_frame(ptr_vector<app> const &pats, flex_frame &fr) {
        // The select chain `pats` is ordered from the inner-most application
        // (which binds the outer-most lambda) to the outer-most. Distinct
        // pattern indices are collected once into pat_domain/pat_args, while
        // domain/names track every binder in application order.
        svector<unsigned> pat_pos;  // forward binder position (in domain) of each distinct index
        fr.pat_args.push_back(nullptr);
        fr.pat_vars.push_back(nullptr);
        pat_pos.push_back(0);  // placeholder for the flex-head slot 0
        expr_mark seen;
        for (auto pat : pats) {
            for (auto pi : array_select_indices(pat)) {
                if (!seen.is_marked(pi)) {
                    fr.pat_domain.push_back(pi->get_sort());
                    fr.pat_args.push_back(pi);
                    pat_pos.push_back(fr.num_bound);
                    seen.mark(pi);
                }
                ++fr.num_bound;
                fr.domain.push_back(pi->get_sort());
                fr.names.push_back(symbol(fr.num_bound));
            }
        }
        for (unsigned k = 1; k < fr.pat_args.size(); ++k) {
            unsigned db = fr.num_bound - 1 - pat_pos[k];
            fr.pat_vars.push_back(m.mk_var(db, fr.pat_args.get(k)->get_sort()));
        }
    }

    void ho_matcher::mk_flex_app(flex_frame &fr, unsigned pat_offset, sort *range, expr_ref &subgoal_app,
                                 expr_ref &body_app) {
        sort *v_sort = m_array.mk_array_sort(fr.pat_domain.size(), fr.pat_domain.data(), range);
        auto mv = m.mk_var(m_subst.size() + pat_offset, v_sort);
        auto mw = m.mk_var(m_subst.size() + pat_offset + fr.num_bound, v_sort);  // shifted by number of bound
        m_subst.resize(m_subst.size() + 1);
        fr.pat_args[0] = mv;
        subgoal_app = m_array.mk_select(fr.pat_args.size(), fr.pat_args.data());
        fr.pat_vars[0] = mw;
        body_app = m_array.mk_select(fr.pat_vars.size(), fr.pat_vars.data());
    }

    expr_ref ho_matcher::mk_flex_lambda(flex_frame const &fr, ptr_vector<app> const &pats, expr *body) {
        expr_ref lam(body, m);
        unsigned num_bound = fr.num_bound;
        for (unsigned i = pats.size(); i-- > 0;) {
            auto sz = pats[i]->get_num_args() - 1;
            num_bound -= sz;
            lam = m.mk_lambda(sz, fr.domain.data() + num_bound, fr.names.data() + num_bound, lam);
        }
        return lam;
    }

    bool ho_matcher::process_imitation(match_goal& wi, var* v, ptr_vector<app> const& pats, expr* t) {
        if (!is_app(t))
            return false;
        if (!wi.is_imitate())
            return false;
        app *ta = to_app(t);       
        unsigned sz = ta->get_num_args();
        if (sz > 0) {
            wi.inc_index();
            m_trail.push(undo_resize(m_subst));
        }

        // H (p1) (p2) = f(t1, .., tn)
        // H -> \x1 \x2 f(H1(x1, x2), .., Hn(x1, x2))
        // H1(p1, p2) = t1, .., Hn(p1, p2) = tn

        flex_frame fr(m);
        init_flex_frame(pats, fr);

        expr_ref_vector bound_args(m);
        for (auto ti : *ta) {
            expr_ref subgoal_app(m), body_app(m);
            mk_flex_app(fr, wi.pat_offset(), ti->get_sort(), subgoal_app, body_app);
            m_goals.push(&wi, wi.level + 1, wi.term_offset(), subgoal_app, ti);
            bound_args.push_back(body_app);
        }

        expr_ref body(m);
        body = m.mk_app(ta->get_decl(), bound_args.size(), bound_args.data());
        add_binding(v, wi.pat_offset(), mk_flex_lambda(fr, pats, body));
        return true;
    }

    bool ho_matcher::process_project(match_goal &wi, var* v, ptr_vector<app> const& pats, expr* t) {
        if (!wi.is_project())
            return false;
        // v -> \x\y . x_i
        unsigned start = wi.index();
        unsigned i = 0;
        for (auto pa : pats) {
            for (auto pi : array_select_indices(pa)) {
                if (start <= i && pi->get_sort() == t->get_sort()) {
                    auto eq = are_equal(wi.pat_offset(), pi, wi.term_offset(), t);
                    if (eq == l_false) {
                        ++i;
                        continue;
                    }
                    auto e = mk_project(pats.size(), i, v->get_sort(), [](expr * e) { return e; });
                    add_binding(v, wi.pat_offset(), e);
                    if (eq == l_undef)
                        m_goals.push(&wi, wi.level + 1, wi.pat_offset(), pi, t);
                    wi.set_index(i + 1);
                    return true;
                }
                // pi has sort T1 -> T2 -> T, and t has sort T.
                // we can project \vars . x_i (H1 vars) (H2 vars) to get a term of sort T.
                if (start <= i && maps_to_sort(pi->get_sort(), t->get_sort())) {
                    IF_VERBOSE(3, verbose_stream() << "maps to " << mk_pp(pi->get_sort(), m) << " "
                                                   << mk_pp(t->get_sort(), m) << "\n");
                    // TODO: implement this case
                    // v->get_sort() determines vars
                    // x := bound variable from "project" function.
                    // add_meta_var_apps(sort *s, sort *t, expr_ref& x, expr_ref_vector const& vars, unsigned
                    // offset)
                }
                ++i;
            }
        }
        return false;
    }

    // M p1 p2 ... pk
    // where M is a meta-variable and p1, .., pk are distinct bound variables
    // and t does not contain a bound variable not mentioned in p1,..,pk
    // or terms that don't occur in t.
    bool ho_matcher::is_pattern(expr* p, unsigned offset, expr* t) {
        uint_set vars;
        while (m_array.is_select(p)) {
            auto a = to_app(p);
            for (auto arg : *a) {
                if (!is_bound_var(arg, offset))
                    return false;
                auto idx = to_var(arg)->get_idx();
                if (vars.contains(idx))
                    return false;
                vars.insert(idx);
            }
            p = a->get_arg(0);
        }
        if (!is_meta_var(p, offset))
            return false;
        // check that every free variable in t is in vars.
        vector<ptr_vector<expr>> btodo;
        btodo.push_back(ptr_vector<expr>());
        btodo[0].push_back(t);
        for (unsigned i = 0; i < btodo.size(); ++i) {
            expr_fast_mark1 visited;
            for (unsigned j = 0; j < btodo[i].size(); ++j) {
                auto t = btodo[i][j];
                if (visited.is_marked(t))
                    continue;
                visited.mark(t);
                
                if (is_app(t)) {
                    auto a = to_app(t);
                    btodo[i].append(a->get_num_args(), a->get_args());
                }
                else if (is_var(t)) {
                    auto idx = to_var(t)->get_idx();
                    if (idx >= i && !vars.contains(idx - i))
                        return false;
                }
                else {
                    quantifier* q = to_quantifier(t);
                    btodo.reserve(i + q->get_num_decls() + 1);
                    btodo[i + q->get_num_decls()].push_back(q->get_expr());
                }
            }
        }

        return true;
    }

    // s is of the form T1 -> T2 -> .. -> Tn -> t
    bool ho_matcher::maps_to_sort(sort* s, sort* t) const {
        SASSERT(s != t);
        while (m_array.is_array(s)) {
            s = get_array_range(s); 
            if (s == t)
                return true;
        }
        return false;
    }

    // s := (T1*T1' -> T2 -> t)
    // x is of type s
    // x := (select (select x (H1 vars) (H2 vars)) (H3 vars)) of type t
    void ho_matcher::add_meta_var_apps(sort *s, sort *t, expr_ref& x, expr_ref_vector const& vars, unsigned offset) {
        
        expr_ref_vector args(m), hargs(m);
        ptr_buffer<sort> domain;
        for (auto v : vars) 
            domain.push_back(v->get_sort());        

        SASSERT(s == x->get_sort());
        while (s != t) {

            SASSERT(m_array.is_array(s));
            unsigned arity = get_array_arity(s);
            args.reset();
            args.push_back(x);
            for (unsigned i = 0; i < arity; ++i) {
                sort *d = get_array_domain(s, i);
                auto r = m_array.mk_array_sort(domain.size(), domain.data(), d);
                hargs.reset();
                hargs.push_back(m.mk_var(++offset, r));
                hargs.append(vars);
                auto h = m_array.mk_select(hargs);
                args.push_back(h);                       
            }
            x = m_array.mk_select(args);
            s = get_array_range(s); 
            SASSERT(s == x->get_sort());
        }     
    }

    // create a lambda abstraction for the meta variable such that
    // when applied to patterns, the result is t.
    // pre-condition: is_pattern(p, offset, t);
    // H (v1, v2) (v3, v4, v5)
    // lambda x1 x2 . lambda x3 x4 x5. t[v5 -> 0, v4 -> 1, v3 -> 2, v2 -> 3, v1 -> 4] 
    expr_ref ho_matcher::abstract_pattern(expr* p, unsigned offset, expr* t) {
        unsigned num_bound = 0;
        expr* p1 = p;
        ptr_vector<app> pats;
        while (m_array.is_select(p1)) {
            pats.push_back(to_app(p1));
            num_bound += to_app(p1)->get_num_args() - 1;
            p1 = to_app(p1)->get_arg(0);            
        }
        expr_ref_vector pat2bound(m);        
        for (auto a : pats) {
            for (auto arg : *a) {
                SASSERT(is_bound_var(arg, offset));
                auto idx = to_var(arg)->get_idx();
                pat2bound.reserve(idx + 1);
                pat2bound[idx] = m.mk_var(--num_bound, arg->get_sort());
            }
        }
        var_subst sub(m, false);
        expr_ref lam = sub(t, pat2bound);
        for (unsigned i = pats.size(), j = 0; i-- > 0; ) {
            vector<symbol> names;
            ptr_vector<sort> sorts;
            for (unsigned k = 1; k < pats[i]->get_num_args(); ++k) {
                names.push_back(symbol(j++));
                sorts.push_back(pats[i]->get_arg(k)->get_sort());
            }
            lam = m.mk_lambda(names.size(), sorts.data(), names.data(), lam);
        }
        SASSERT(is_well_sorted(m, lam));
        return lam;
    }

    //
    // keep track of number of internal scopes and offset to non-capture variables.
    // a variable is captured if its index is in the interval [scopes, offset[.
    //
    bool ho_matcher::is_closed(expr* v, unsigned scopes, unsigned offset) const {
        if (is_ground(v))
            return true;
        if (is_app(v)) 
            return all_of(*to_app(v), [&](expr* arg) { return is_closed(arg, scopes, offset); });
        if (is_var(v)) 
            return to_var(v)->get_idx() < scopes || to_var(v)->get_idx() >= offset;
        if (is_quantifier(v)) {
            auto q = to_quantifier(v);
            return is_closed(q->get_expr(), scopes + q->get_num_decls(), offset + q->get_num_decls());
        }
        return false;
    }

    // (T1, T2,.., Tn) -> (Tn+1,.., Ti,.., Tm) -> Ti => lambda x1...xn . lambda x_n+1,..x_m . x_i
    expr_ref ho_matcher::mk_project(unsigned num_lambdas, unsigned i, sort* s, std::function<expr*(expr*)> const& mk_body) {
        SASSERT(num_lambdas > 0);
        SASSERT(m_array.is_array(s));
        unsigned num_binders = 0;
        expr_ref body(m);
        sort* a = s, * var_sort = nullptr;
        unsigned j = i;
        for (unsigned k = 0; k < num_lambdas; ++k) {
            auto arity = get_array_arity(a);
            if (j >= arity)
                j -= arity;
            else if (j < arity && !var_sort)
                var_sort = get_array_domain(a, j);
            num_binders += arity;
            a = get_array_range(a);
        }
        SASSERT(var_sort);
        body = m.mk_var(num_binders - i - 1, var_sort);
        bind_lambdas(num_lambdas, s, body);
        SASSERT(body->get_sort() == s);
        SASSERT(is_well_sorted(m, body));
        return body;
    }

    void ho_matcher::bind_lambdas(unsigned j, sort* s, expr_ref& body) {
        if (j > 1) 
            bind_lambdas(j - 1, get_array_range(s), body);        
        unsigned sz = get_array_arity(s);
        ptr_vector<sort> decl_sorts;
        vector<symbol> decl_names;
        for (unsigned i = 0; i < sz; ++i) {
            decl_sorts.push_back(get_array_domain(s, i));
            decl_names.push_back(symbol(i));
        }
        body = m.mk_lambda(sz, decl_sorts.data(), decl_names.data(), body);
        SASSERT(s == body->get_sort());
        SASSERT(is_well_sorted(m, body));
    }

    void ho_matcher::add_binding(var* v, unsigned offset, expr* _t) {
        SASSERT(v->get_idx() >= offset);
        expr_ref t(_t, m);
        inv_var_shifter vs(m);
        vs(_t, offset, t);
        m_subst.set(v->get_idx() - offset, t);
        SASSERT(is_well_sorted(m, t));
        SASSERT(v->get_sort() == t->get_sort());
        TRACE(ho_matching, tout << "ho_matcher::add_binding: " << offset << " v" << v->get_idx() - offset << " -> " << mk_pp(t, m) << "\n";);
        m_trail.push(undo_set(m_subst, v->get_idx() - offset));
    }


    std::pair<quantifier*, app*> ho_matcher::compile_ho_pattern(quantifier* q, app* p) {
        app* p1 = nullptr;
        quantifier *q1 = nullptr;
        if (m_pat2hopat.find(p, p1) && m_q2hoq.find(q, q1)) {
            return { q1, p1 };
        }
        auto is_ho = any_of(subterms::all(expr_ref(p, m)), [&](expr* t) { 
            return m_unitary.is_flex(0, t) || 
                   is_lambda(t); 
        });
        if (!is_ho)
            return { q, p };
        vector<std::pair<expr*, unsigned>> todo;
        ptr_buffer<var> bound;
        expr_ref_vector cache(m);
        unsigned nb = q->get_num_decls();
        bool contains_pat2abs = m_pat2abs.contains(p);
        SASSERT(m.is_pattern(p));
        todo.push_back({p, 0});
        while (!todo.empty()) {
            auto [t, lvl] = todo.back();
            if (is_var(t)) {
                cache.setx(t->get_id(), t);
                todo.pop_back();
                continue;
            }
            if ((m_unitary.is_flex(0, t) && lvl > 1) || is_lambda(t)) {
                if (!contains_pat2abs)
                    m_pat2abs.insert_if_not_there(p, svector<std::pair<unsigned, expr*>>()).push_back({ nb, t });
                auto v = m.mk_var(nb++, t->get_sort());
                bound.push_back(v);
                cache.setx(t->get_id(), v);
                todo.pop_back();
                continue;
            }            
            if (is_app(t)) {
                auto a = to_app(t);

                unsigned sz = a->get_num_args();
                ptr_buffer<expr> args;
                for (auto arg : *a) {
                    cache.reserve(arg->get_id() + 1);
                    expr* arg1 = cache.get(arg->get_id());
                    if (!arg1)
                        todo.push_back({arg, lvl + 1});
                    else
                        args.push_back(arg1);
                }
                if (sz != args.size())
                    continue;
                todo.pop_back();
                cache.setx(t->get_id(), m.mk_app(a->get_decl(), args.size(), args.data()));
            }
            if (is_quantifier(t)) {
                if (!contains_pat2abs)
                    m_pat2abs.remove(p);
                return { q, p };
            }
        }
        p1 = to_app(cache.get(p->get_id()));

        if (p1 == p)
            return {q, p};
        expr_free_vars free_vars;
        free_vars(p1);
        app_ref_vector new_ground(m);
        app_ref_vector new_patterns(m);

        ptr_buffer<sort> sorts;
        vector<symbol> names;
        for (unsigned i = bound.size(); i-- > 0; ) {
            sorts.push_back(bound[i]->get_sort());
            names.push_back(symbol(bound[i]->get_idx()));
        }
        unsigned sz = q->get_num_decls();
        for (unsigned i = 0; i < sz; ++i) {
            unsigned idx = sz - i - 1;
            auto s = q->get_decl_sort(i);
            sorts.push_back(s);
            names.push_back(q->get_decl_name(i));
            if (!free_vars.contains(idx)) {
                auto p = m.mk_fresh_func_decl("p", 1, &s, m.mk_bool_sort());
                new_patterns.push_back(m.mk_app(p, m.mk_var(idx, s)));
                new_ground.push_back(m.mk_app(p, m.mk_fresh_const(symbol("c"), s)));
            }
        }
        auto body = q->get_expr();
        if (!new_patterns.empty()) {
            ptr_vector<app> pats;
            CTRACE(ho_matching, !m.is_pattern(p1), 
                tout << mk_pp(p, m) << "\n" << mk_pp(p1, m) << "\n";);
            VERIFY(m.is_pattern(p1, pats));
            for (auto p : new_patterns) // patterns for variables that are not free in new pattern
                pats.push_back(p);
            for (auto g : new_ground) // ensure ground terms are in pattern so they have enodes
                pats.push_back(g);
            p1 = m.mk_pattern(pats.size(), pats.data());
        }

        q1 = m.mk_forall(sorts.size(), sorts.data(), names.data(), body);

        m_ho_patterns.push_back(p1);
        m_ho_qs.push_back(q1);
        trail().push(push_back_vector(m_ho_patterns));
        trail().push(push_back_vector(m_ho_qs));

        if (!m_pat2hopat.contains(p)) {
            m_pat2hopat.insert(p, p1);
            trail().push(insert_map(m_pat2hopat, p));
        }
        if (!m_hopat2pat.contains(p1)) {
            m_hopat2pat.insert(p1, p);
            trail().push(insert_map(m_hopat2pat, p1));
        }
        if (!m_q2hoq.contains(q)) {
            m_q2hoq.insert(q, q1);
            trail().push(insert_map(m_q2hoq, q));
        }
        if (!m_hoq2q.contains(q1)) {
            m_hoq2q.insert(q1, q);
            trail().push(insert_map(m_hoq2q, q1));
        }
        if (!m_hopat2free_vars.contains(p1)) {
            m_hopat2free_vars.insert(p1, std::move(free_vars));
            trail().push(insert_map(m_hopat2free_vars, p1));
        }
        if (!contains_pat2abs)
            trail().push(insert_map(m_pat2abs, p));

        TRACE(ho_matching, tout << mk_pp(q, m) << "\n"
                                << mk_pp(p, m) << "\n->\n"
                                << mk_pp(q1, m) << "\n"
                                << mk_pp(p1, m) << "\n");
        return { q1, p1 };
    }

    bool ho_matcher::is_ho_pattern(app* p) {
        return m_hopat2pat.contains(p);
    }

    void ho_matcher::register_ho_pattern(app* alias_p, app* full_p) {
        if (alias_p == full_p) return;
        auto orig_p = m_hopat2pat[full_p];
        m_hopat2pat.insert(alias_p, orig_p);
        m_hopat2free_vars.insert(alias_p, m_hopat2free_vars[full_p]);
        m_ho_patterns.push_back(alias_p);
        trail().push(push_back_vector(m_ho_patterns));
        trail().push(insert_map(m_hopat2pat, alias_p));
        trail().push(insert_map(m_hopat2free_vars, alias_p));
    }

    void ho_matcher::refine_ho_match(app* p, expr_ref_vector& s) {
        auto fo_pat = m_hopat2pat[p];
        IF_VERBOSE(10, verbose_stream() << "refine_ho_match: p=" << mk_pp(p, m) << "\n  fo_pat=" << mk_pp(fo_pat, m) << "\n";
                   verbose_stream() << "  m_pat2abs has fo_pat: " << m_pat2abs.contains(fo_pat) << "\n";
                   auto& abs = m_pat2abs[fo_pat];
                   verbose_stream() << "  m_pat2abs size: " << abs.size() << "\n";
                   for (auto [v, pat] : abs) verbose_stream() << "    v=" << v << " pat=" << mk_pp(pat, m) << "\n";);
        scoped_trail_level _restore(m_trail);
        m_trail.push_scope();
        m_subst.resize(0);
        m_subst.resize(s.size());
        m_goals.reset();
        // MAM bindings are reversed: s[i] = binding for var idx = s.size()-1-i
        // m_subst is indexed by var index directly
        for (unsigned i = 0; i < s.size(); ++i) {
            auto idx = s.size() - i - 1;
            if (!m_hopat2free_vars[p].contains(idx))
                s[i] = m.mk_var(idx, s[i]->get_sort());
            else if (s.get(i))
                m_subst.set(idx, s.get(i));
        }

        TRACE(ho_matching, tout << "refine " << mk_pp(p, m) << "\n" << s << "\n");

        unsigned num_bound = 0, level = 0;
        for (auto [v, pat] : m_pat2abs[fo_pat]) {
            // If a binding's sort disagrees with the pattern variable it would
            // fill, substituting it would build an ill-sorted term. This can
            // arise for deeply nested multi-select patterns whose de Bruijn
            // remapping does not line up, or when E-matching delivers a
            // candidate binding whose sort is incompatible with the abstracted
            // higher-order pattern. Discard this refinement candidate (produce
            // no instance) instead of aborting the whole solve.
            if (!subst_sorts_match(m, pat, s, true)) {
                IF_VERBOSE(0, verbose_stream() << "refine_ho_match: sorts do not match for " << mk_pp(pat, m) << " and "
                                               << s << "\n";);
                UNREACHABLE();
                return;
            }
        }
        for (auto [v, pat] : m_pat2abs[fo_pat]) {
            var_subst sub(m, true);
            auto pat_refined = sub(pat, s);
            TRACE(ho_matching, tout << mk_pp(pat, m) << " -> " << pat_refined << "\n");
            m_goals.push(nullptr, level, num_bound, pat_refined, m_subst.get(v));
        }
        search();
    }

    bool ho_matcher::subst_sorts_match(ast_manager& m, expr* t, expr_ref_vector const& s, bool std_order) {
        unsigned sz = s.size();
        ptr_buffer<expr> es;
        svector<unsigned> offs;
        es.push_back(t);
        offs.push_back(0);
        while (!es.empty()) {
            expr* e = es.back(); es.pop_back();
            unsigned off = offs.back(); offs.pop_back();
            if (is_var(e)) {
                unsigned idx = to_var(e)->get_idx();
                if (idx < off)
                    continue;
                unsigned k = idx - off;
                if (k >= sz)
                    continue;
                expr* r = std_order ? s.get(sz - k - 1) : s.get(k);
                if (r && r->get_sort() != e->get_sort())
                    return false;
            }
            else if (is_app(e)) {
                for (expr* arg : *to_app(e)) {
                    es.push_back(arg);
                    offs.push_back(off);
                }
            }
            else if (is_quantifier(e)) {
                quantifier* q = to_quantifier(e);
                es.push_back(q->get_expr());
                offs.push_back(off + q->get_num_decls());
            }
        }
        return true;
    }

    std::ostream& ho_matcher::display(std::ostream& out) const {
        m_subst.display(out << "subst\n");
        m_goals.display(out << "goals\n");
        out << "stack\n";
        for (auto const* mi : m_backtrack)
            mi->display(out);       
        return out;
    }

    struct retire_match_goal : public trail {
        match_goal& mg;        
        retire_match_goal(match_goal& mg) : mg(mg) {}
        void undo() override {
            mg.reset();
        }
    };

    void match_goals::push(match_goal* parent, unsigned level, unsigned offset, expr_ref const& pat, expr_ref const& t) {
        SASSERT(pat->get_sort() == t->get_sort());
        match_goal* wi = new (ho.trail().get_region()) match_goal(parent, level, offset, pat, t);
        ho.trail().push(retire_match_goal(*wi)); // reset on undo
        wi->init(wi);
        if (ho.is_cheap(*wi)) {
            match_goal::push_to_front(m_cheap, wi);
            ho.trail().push(remove_dll(m_cheap, wi));    // remove from cheap
        }
        else {
            ho.reduce(*wi);
            if (ho.is_cheap(*wi)) {
                match_goal::push_to_front(m_cheap, wi);
                ho.trail().push(remove_dll(m_cheap, wi));
            }
            else {
                match_goal::push_to_front(m_expensive, wi);
                ho.trail().push(remove_dll(m_expensive, wi)); // remove from expensive
            }
        }
    }

    match_goal* match_goals::pop() {
        SASSERT(!empty());
        if (m_cheap)
            return pop(m_cheap);
        auto* wi = m_expensive;
        do {
            expr_ref pat(wi->pat), t(wi->t);
            ho.reduce(*wi);
            if (pat == wi->pat && t == wi->t)
                continue;
            if (pat != wi->pat)
                ho.trail().push(update_value(wi->pat, pat));
            if (t != wi->t)
                ho.trail().push(update_value(wi->t, t));
            if (ho.is_cheap(*wi)) {
                dll_base<match_goal>::remove_from(m_expensive, wi);
                match_goal::push_to_front(m_cheap, wi);

                ho.trail().push(insert_dll(m_expensive, wi)); // remove from expensive
                ho.trail().push(remove_dll(m_cheap, wi));
                return pop(m_cheap);
            }
            wi = wi->next();
        } 
        while (wi != m_expensive);
                
        return pop(m_expensive);
    }    

    match_goal* match_goals::pop(match_goal* &ws) {
        SASSERT(ws);
        auto* wi = ws->pop(ws);
        ho.trail().push(insert_dll(ws, wi)); // insert wi into ws   
        wi->set_init();
        return wi;
    }

    std::ostream& match_goals::display(std::ostream& out) const {
        for (auto const& wi : dll_elements(m_cheap))
            wi.display(out << "c ");
        
        for (auto const& wi : dll_elements(m_expensive))
            wi.display(out << "e ");
        
        return out;
    }

}