lean2/src/library/unifier.cpp

/*
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.

Author: Leonardo de Moura
*/
#include <utility>
#include <memory>
#include <vector>
#include <limits>
#include <algorithm>
#include "util/interrupt.h"
#include "util/luaref.h"
#include "util/lazy_list_fn.h"
#include "util/sstream.h"
#include "util/lbool.h"
#include "util/flet.h"
#include "util/sexpr/option_declarations.h"
#include "kernel/for_each_fn.h"
#include "kernel/abstract.h"
#include "kernel/instantiate.h"
#include "kernel/type_checker.h"
#include "kernel/kernel_exception.h"
#include "kernel/error_msgs.h"
#include "library/normalize.h"
#include "library/occurs.h"
#include "library/locals.h"
#include "library/module.h"
#include "library/unifier.h"
#include "library/reducible.h"
#include "library/unifier_plugin.h"
#include "library/kernel_bindings.h"
#include "library/print.h"
#include "library/expr_lt.h"

#ifndef LEAN_DEFAULT_UNIFIER_MAX_STEPS
#define LEAN_DEFAULT_UNIFIER_MAX_STEPS 20000
#endif

#ifndef LEAN_DEFAULT_UNIFIER_COMPUTATION
#define LEAN_DEFAULT_UNIFIER_COMPUTATION false
#endif

#ifndef LEAN_DEFAULT_UNIFIER_EXPENSIVE_CLASSES
#define LEAN_DEFAULT_UNIFIER_EXPENSIVE_CLASSES false
#endif

#ifndef LEAN_DEFAULT_UNIFIER_CONSERVATIVE
#define LEAN_DEFAULT_UNIFIER_CONSERVATIVE false
#endif

#ifndef LEAN_DEFAULT_UNIFIER_NONCHRONOLOGICAL
#define LEAN_DEFAULT_UNIFIER_NONCHRONOLOGICAL true
#endif

namespace lean {
static name * g_unifier_max_steps               = nullptr;
static name * g_unifier_computation             = nullptr;
static name * g_unifier_expensive_classes       = nullptr;
static name * g_unifier_conservative            = nullptr;
static name * g_unifier_nonchronological        = nullptr;

unsigned get_unifier_max_steps(options const & opts) {
    return opts.get_unsigned(*g_unifier_max_steps, LEAN_DEFAULT_UNIFIER_MAX_STEPS);
}

bool get_unifier_computation(options const & opts) {
    return opts.get_bool(*g_unifier_computation, LEAN_DEFAULT_UNIFIER_COMPUTATION);
}

bool get_unifier_expensive_classes(options const & opts) {
    return opts.get_bool(*g_unifier_expensive_classes, LEAN_DEFAULT_UNIFIER_EXPENSIVE_CLASSES);
}

bool get_unifier_conservative(options const & opts) {
    return opts.get_bool(*g_unifier_conservative, LEAN_DEFAULT_UNIFIER_CONSERVATIVE);
}

bool get_unifier_nonchronological(options const & opts) {
    return opts.get_bool(*g_unifier_nonchronological, LEAN_DEFAULT_UNIFIER_NONCHRONOLOGICAL);
}

unifier_config::unifier_config(bool use_exceptions, bool discard):
    m_use_exceptions(use_exceptions),
    m_max_steps(LEAN_DEFAULT_UNIFIER_MAX_STEPS),
    m_computation(LEAN_DEFAULT_UNIFIER_COMPUTATION),
    m_expensive_classes(LEAN_DEFAULT_UNIFIER_EXPENSIVE_CLASSES),
    m_discard(discard),
    m_nonchronological(LEAN_DEFAULT_UNIFIER_NONCHRONOLOGICAL) {
    m_kind    = unifier_kind::Liberal;
    m_pattern = false;
    m_ignore_context_check = false;
}

unifier_config::unifier_config(options const & o, bool use_exceptions, bool discard):
    m_use_exceptions(use_exceptions),
    m_max_steps(get_unifier_max_steps(o)),
    m_computation(get_unifier_computation(o)),
    m_expensive_classes(get_unifier_expensive_classes(o)),
    m_discard(discard),
    m_nonchronological(get_unifier_nonchronological(o)) {
    if (get_unifier_conservative(o))
        m_kind = unifier_kind::Conservative;
    else
        m_kind = unifier_kind::Liberal;
    m_pattern = false;
    m_ignore_context_check = false;
}

// If \c e is a metavariable ?m or a term of the form (?m l_1 ... l_n) where
// l_1 ... l_n are distinct local variables, then return ?m, and store l_1 ... l_n in args.
// Otherwise return none.
optional<expr> is_simple_meta(expr const & e, buffer<expr> & args) {
    expr const & m = get_app_args(e, args);
    if (!is_metavar(m))
        return none_expr();
    for (auto it = args.begin(); it != args.end(); it++) {
        if (!is_local(*it) || contains_local(*it, args.begin(), it))
            return none_expr();
    }
    return some_expr(m);
}

bool is_simple_meta(expr const & e) {
    buffer<expr> args;
    return (bool)is_simple_meta(e, args); // NOLINT
}

// Return true if all local constants in \c e are in locals
bool context_check(expr const & e, buffer<expr> const & locals) {
    bool failed = false;
    for_each(e, [&](expr const & e, unsigned) {
            if (failed)
                return false;
            if (is_local(e)) {
                if (!contains_local(e, locals))
                    failed = true;
                return false; // do not visit type
            }
            if (is_metavar(e))
                return false; // do not visit type
            return has_local(e);
        });
    return !failed;
}

enum class occurs_check_status { Ok, Maybe, FailCircular, FailLocal };

// Return
//   - l_undef if \c e contains a metavariable application that contains
//     a local constant not in locals
//   - l_true if \c e does not contain the metavariable \c m, and all local
//     constants are in \c e are in \c locals.
//   - l_false if \c e contains \c m or it contains a local constant \c l
//     not in locals that is not in a metavariable application.
occurs_check_status occurs_context_check(substitution & s, expr const & e, expr const & m, buffer<expr> const & locals, expr & bad_local) {
    expr root = e;
    occurs_check_status r = occurs_check_status::Ok;
    for_each(e, [&](expr const & e, unsigned) {
            if (r == occurs_check_status::FailLocal || r == occurs_check_status::FailCircular) {
                return false;
            } else if (is_local(e)) {
                if (!contains_local(e, locals)) {
                    // right-hand-side contains variable that is not in the scope
                    // of metavariable.
                    bad_local = e;
                    r = occurs_check_status::FailLocal;
                }
                return false; // do not visit type
            } else if (is_meta(e)) {
                if (r == occurs_check_status::Ok) {
                    if (!context_check(e, locals))
                        r = occurs_check_status::Maybe;
                    if (s.occurs(m, e))
                        r = occurs_check_status::Maybe;
                }
                if (mlocal_name(get_app_fn(e)) == mlocal_name(m))
                    r = occurs_check_status::FailCircular;
                return false; // do not visit children
            } else {
                // we only need to continue exploring e if it contains
                // metavariables and/or local constants.
                return has_expr_metavar(e) || has_local(e);
            }
        });
    if (r != occurs_check_status::Ok)
        return r;
    for (expr const & local : locals) {
        if (s.occurs(m, mlocal_type(local)))
            return occurs_check_status::Maybe;
    }
    return r;
}

occurs_check_status occurs_context_check(substitution & s, expr const & e, expr const & m, buffer<expr> const & locals) {
    expr bad_local;
    return occurs_context_check(s, e, m, locals, bad_local);
}

unify_status unify_simple_core(substitution & s, expr const & lhs, expr const & rhs, justification const & j) {
    lean_assert(is_meta(lhs));
    buffer<expr> args;
    auto m = is_simple_meta(lhs, args);
    if (!m || is_meta(rhs)) {
        return unify_status::Unsupported;
    } else {
        switch (occurs_context_check(s, rhs, *m, args)) {
        case occurs_check_status::FailLocal:
        case occurs_check_status::FailCircular:
            return unify_status::Failed;
        case occurs_check_status::Maybe:
            return unify_status::Unsupported;
        case occurs_check_status::Ok: {
            s.assign(*m, args, rhs, j);
            return unify_status::Solved;
        }}
    }
    lean_unreachable(); // LCOV_EXCL_LINE
}

unify_status unify_simple(substitution & s, expr const & lhs, expr const & rhs, justification const & j) {
    if (lhs == rhs)
        return unify_status::Solved;
    else if (!has_metavar(lhs) && !has_metavar(rhs))
        return unify_status::Failed;
    else if (is_meta(lhs))
        return unify_simple_core(s, lhs, rhs, j);
    else if (is_meta(rhs))
        return unify_simple_core(s, rhs, lhs, j);
    else
        return unify_status::Unsupported;
}

// Return true if m occurs in e
bool occurs_meta(level const & m, level const & e) {
    lean_assert(is_meta(m));
    bool contains = false;
    for_each(e, [&](level const & l) {
            if (contains)
                return false;
            if (l == m) {
                contains = true;
                return false;
            }
            return has_meta(l);
        });
    return contains;
}

unify_status unify_simple_core(substitution & s, level const & lhs, level const & rhs, justification const & j) {
    lean_assert(is_meta(lhs));
    bool contains = occurs_meta(lhs, rhs);
    if (contains) {
        if (is_succ(rhs))
            return unify_status::Failed;
        else
            return unify_status::Unsupported;
    }
    s.assign(meta_id(lhs), rhs, j);
    return unify_status::Solved;
}

unify_status unify_simple(substitution & s, level const & lhs, level const & rhs, justification const & j) {
    if (lhs == rhs)
        return unify_status::Solved;
    else if (!has_meta(lhs) && !has_meta(rhs))
        return unify_status::Failed;
    else if (is_meta(lhs))
        return unify_simple_core(s, lhs, rhs, j);
    else if (is_meta(rhs))
        return unify_simple_core(s, rhs, lhs, j);
    else if (is_succ(lhs) && is_succ(rhs))
        return unify_simple(s, succ_of(lhs), succ_of(rhs), j);
    else
        return unify_status::Unsupported;
}

unify_status unify_simple(substitution & s, constraint const & c) {
    if (is_eq_cnstr(c))
        return unify_simple(s, cnstr_lhs_expr(c), cnstr_rhs_expr(c), c.get_justification());
    else if (is_level_eq_cnstr(c))
        return unify_simple(s, cnstr_lhs_level(c), cnstr_rhs_level(c), c.get_justification());
    else
        return unify_status::Unsupported;
}

static constraint * g_dont_care_cnstr = nullptr;
static unsigned g_group_size = 1u << 28;
constexpr unsigned g_num_groups = 8;
static unsigned g_cnstr_group_first_index[g_num_groups] = { 0, g_group_size, 2*g_group_size, 3*g_group_size, 4*g_group_size, 5*g_group_size, 6*g_group_size, 7*g_group_size};
static unsigned get_group_first_index(cnstr_group g) {
    return g_cnstr_group_first_index[static_cast<unsigned>(g)];
}
static cnstr_group to_cnstr_group(unsigned d) {
    if (d >= g_num_groups)
        d = g_num_groups-1;
    return static_cast<cnstr_group>(d);
}

/** \brief Convert choice constraint delay factor to cnstr_group */
cnstr_group get_choice_cnstr_group(constraint const & c) {
    lean_assert(is_choice_cnstr(c));
    unsigned f = cnstr_delay_factor(c);
    if (f > static_cast<unsigned>(cnstr_group::MaxDelayed)) {
        return cnstr_group::MaxDelayed;
    } else {
        return static_cast<cnstr_group>(f);
    }
}

/** \brief Auxiliary functional object for implementing simultaneous higher-order unification */
struct unifier_fn {
    typedef pair<constraint, unsigned> cnstr; // constraint + idx
    struct cnstr_cmp {
        int operator()(cnstr const & c1, cnstr const & c2) const {
            return c1.second < c2.second ? -1 : (c1.second == c2.second ? 0 : 1);
        }
    };
    typedef rb_tree<cnstr, cnstr_cmp> cnstr_set;
    typedef rb_tree<unsigned, unsigned_cmp> cnstr_idx_set;
    typedef name_map<cnstr_idx_set> name_to_cnstrs;
    typedef name_map<unsigned> owned_map;
    typedef rb_map<expr, pair<expr, justification>, expr_quick_cmp> expr_map;
    typedef std::shared_ptr<type_checker> type_checker_ptr;
    environment      m_env;
    name_generator   m_ngen;
    substitution     m_subst;
    constraints      m_postponed; // constraints that will not be solved
    owned_map        m_owned_map; // mapping from metavariable name m to delay factor of the choice constraint that owns m
    expr_map         m_type_map;  // auxiliary map for storing the type of the expr in choice constraints
    unifier_plugin   m_plugin;
    type_checker_ptr m_tc;
    type_checker_ptr m_flex_rigid_tc; // type checker used when solving flex rigid constraints. By default,
                                      // only the definitions from the main module are treated as transparent.
    unifier_config   m_config;
    unsigned         m_num_steps;
    bool             m_first; //!< True if we still have to generate the first solution.
    unsigned         m_next_assumption_idx; //!< Next assumption index.
    unsigned         m_next_cidx; //!< Next constraint index.
    /**
       \brief "Queue" of constraints to be solved.

       We implement it using a red-black-tree because:
       1- Our red-black-trees support a O(1) copy operation. So, it is cheap to create a snapshot
       whenever we create a backtracking point.

       2- We can easily remove any constraint from the queue in O(n log n). We do that when
       a metavariable \c m is assigned, and we want to instantiate it in all constraints that
       contains it.
    */
    cnstr_set        m_cnstrs;
    /**
        \brief The following map is an index. The map a metavariable name \c m to the set of constraint indices that contain \c m.
        We use these indices whenever a metavariable \c m is assigned.
        When the metavariable is assigned, we used this index to remove constraints that contains \c m from \c m_cnstrs,
        instantiate \c m, and reprocess them.

        \remark \c m_mvar_occs is for regular metavariables.
    */
    name_to_cnstrs   m_mvar_occs;

    /**
        \brief Base class for the case-splits created by the unifier.

        We have three different kinds of case splits:
        1- unifier plugin
        2- choice constraints
        3- higher-order unification
    */
    struct case_split {
        unsigned         m_assumption_idx; // idx of the current assumption
        justification    m_jst;
        justification    m_failed_justifications; // justifications for failed branches
        // snapshot of unifier's state
        substitution     m_subst;
        constraints      m_postponed;
        cnstr_set        m_cnstrs;
        expr_map         m_type_map;
        name_to_cnstrs   m_mvar_occs;
        owned_map        m_owned_map;

        /** \brief Save unifier's state */
        case_split(unifier_fn & u, justification const & j):
            m_assumption_idx(u.m_next_assumption_idx), m_jst(j), m_subst(u.m_subst),
            m_postponed(u.m_postponed), m_cnstrs(u.m_cnstrs), m_type_map(u.m_type_map),
            m_mvar_occs(u.m_mvar_occs), m_owned_map(u.m_owned_map) {
            u.m_next_assumption_idx++;
        }

        /** \brief Restore unifier's state with saved values, and update m_assumption_idx and m_failed_justifications. */
        void restore_state(unifier_fn & u) {
            lean_assert(u.in_conflict());
            u.m_subst     = m_subst;
            u.m_postponed = m_postponed;
            u.m_cnstrs    = m_cnstrs;
            u.m_mvar_occs = m_mvar_occs;
            u.m_owned_map = m_owned_map;
            u.m_type_map  = m_type_map;
            m_assumption_idx = u.m_next_assumption_idx;
            m_failed_justifications = mk_composite1(m_failed_justifications, *u.m_conflict);
            u.m_next_assumption_idx++;
            u.m_conflict  = optional<justification>();
        }
        justification get_jst() const { return m_jst; }
        virtual ~case_split() {}
        virtual bool next(unifier_fn & u) = 0;
    };
    typedef std::vector<std::unique_ptr<case_split>> case_split_stack;

    struct lazy_constraints_case_split : public case_split {
        lazy_list<constraints> m_tail;
        lazy_constraints_case_split(unifier_fn & u, justification const & j, lazy_list<constraints> const & tail):
            case_split(u, j), m_tail(tail) {}
        virtual bool next(unifier_fn & u) { return u.next_lazy_constraints_case_split(*this); }
    };

    struct simple_case_split : public case_split {
        list<constraints> m_tail;
        simple_case_split(unifier_fn & u, justification const & j, list<constraints> const & tail):case_split(u, j), m_tail(tail) {}
        virtual bool next(unifier_fn & u) { return u.next_simple_case_split(*this); }
    };

    struct delta_unfold_case_split : public case_split {
        bool       m_done;
        constraint m_cnstr;
        delta_unfold_case_split(unifier_fn & u, justification const & j, constraint const & c):
            case_split(u, j), m_done(false), m_cnstr(c) {}
        virtual bool next(unifier_fn & u) { return u.next_delta_unfold_case_split(*this); }
    };

    case_split_stack        m_case_splits;
    optional<justification> m_conflict; //!< if different from none, then there is a conflict.

    unifier_fn(environment const & env, unsigned num_cs, constraint const * cs,
               name_generator const & ngen, substitution const & s,
               unifier_config const & cfg):
        m_env(env), m_ngen(ngen), m_subst(s), m_plugin(get_unifier_plugin(env)),
        m_config(cfg), m_num_steps(0) {
        switch (m_config.m_kind) {
        case unifier_kind::Cheap:
            m_tc = mk_opaque_type_checker(env, m_ngen.mk_child());
            m_flex_rigid_tc = m_tc;
            m_config.m_computation = false;
            break;
        case unifier_kind::VeryConservative:
            m_tc = mk_type_checker(env, m_ngen.mk_child(), UnfoldReducible);
            m_flex_rigid_tc = m_tc;
            m_config.m_computation = false;
            break;
        case unifier_kind::Conservative:
            m_tc = mk_type_checker(env, m_ngen.mk_child(), UnfoldQuasireducible);
            m_flex_rigid_tc = m_tc;
            m_config.m_computation = false;
            break;
        case unifier_kind::Liberal:
            m_tc = mk_type_checker(env, m_ngen.mk_child());
            if (!cfg.m_computation)
                m_flex_rigid_tc = mk_type_checker(env, m_ngen.mk_child(), UnfoldQuasireducible);
            break;
        default:
            lean_unreachable();
        }
        m_next_assumption_idx = 0;
        m_next_cidx = 0;
        m_first     = true;
        process_input_constraints(num_cs, cs);
    }

    void process_input_constraints(unsigned num_cs, constraint const * cs) {
        // Input choice constraints may have ownership over a metavariable.
        // So, we must first process them, to make sure the ownership table is initialized before
        // we solve the remaining constraints
        for (unsigned i = 0; i < num_cs; i++) {
            if (cs[i].kind() == constraint_kind::Choice)
                preprocess_choice_constraint(cs[i]);
        }
        for (unsigned i = 0; i < num_cs; i++) {
            if (cs[i].kind() != constraint_kind::Choice)
                process_constraint(cs[i]);
        }
    }

    void check_system() {
        ::lean::check_system("unifier");
    }

    void check_full() {
        check_system();
        if (m_num_steps > m_config.m_max_steps)
            throw exception(sstream() << "unifier maximum number of steps (" << m_config.m_max_steps << ") exceeded, " <<
                            "the maximum number of steps can be increased by setting the option unifier.max_steps " <<
                            "(remark: the unifier uses higher order unification and unification-hints, " <<
                            "which may trigger non-termination");
        m_num_steps++;
    }

    bool in_conflict() const { return (bool)m_conflict; } // NOLINT
    void set_conflict(justification const & j) { m_conflict = j; }
    void update_conflict(justification const & j) { m_conflict = j; }
    void reset_conflict() { m_conflict = optional<justification>(); lean_assert(!in_conflict()); }

    expr mk_local_for(expr const & b) {
        return mk_local(m_ngen.next(), binding_name(b), binding_domain(b), binding_info(b));
    }

    /**
        \brief Update occurrence index with entry <tt>m -> cidx</tt>, where \c m is the name of a metavariable,
        and \c cidx is the index of a constraint that contains \c m.
    */
    void add_mvar_occ(name const & m, unsigned cidx) {
        cnstr_idx_set s;
        auto it = m_mvar_occs.find(m);
        if (it)
            s = *it;
        if (!s.contains(cidx)) {
            s.insert(cidx);
            m_mvar_occs.insert(m, s);
        }
    }

    void add_meta_occ(expr const & m, unsigned cidx) {
        lean_assert(is_meta(m));
        add_mvar_occ(mlocal_name(get_app_fn(m)), cidx);
    }

    /** \brief For each metavariable m in e add an entry m -> cidx at m_mvar_occs.
        Return true if at least one entry was added.
    */
    bool add_meta_occs(expr const & e, unsigned cidx) {
        bool added = false;
        if (has_expr_metavar(e)) {
            for_each(e, [&](expr const & e, unsigned) {
                    if (is_meta(e)) {
                        add_meta_occ(e, cidx);
                        added = true;
                        return false;
                    }
                    if (is_local(e))
                        return false;
                    return has_expr_metavar(e);
                });
        }
        return added;
    }

    /** \brief Add constraint to the constraint queue */
    unsigned add_cnstr(constraint const & c, cnstr_group g) {
        unsigned cidx = m_next_cidx + get_group_first_index(g);
        m_cnstrs.insert(cnstr(c, cidx));
        m_next_cidx++;
        return cidx;
    }

    /** \brief Check if \c t1 and \c t2 are definitionally equal, if they are not, set a conflict with justification \c j
    */
    bool is_def_eq(expr const & t1, expr const & t2, justification const & j) {
        try {
            auto dcs = m_tc->is_def_eq(t1, t2, j);
            if (!dcs.first) {
                // std::cout << "conflict: " << t1 << " =?= " << t2 << "\n";
                set_conflict(j);
                return false;
            } else {
                return process_constraints(dcs.second);
            }
        } catch (exception&) {
            set_conflict(j);
            return false;
        }
    }

    /** \brief Process the given constraints. Return true iff no conflict was detected. */
    bool process_constraints(constraint_seq const & cs) {
        return cs.all_of([&](constraint const & c) { return process_constraint(c); });
    }

    bool process_constraints(buffer<constraint> const & cs) {
        for (auto const & c : cs) {
            if (!process_constraint(c))
                return false;
        }
        return true;
    }


    /** \brief Process constraints in \c cs, and append justification \c j to them. */
    bool process_constraints(constraint_seq const & cs, justification const & j) {
        return cs.all_of([&](constraint const & c) {
                return process_constraint(update_justification(c, mk_composite1(c.get_justification(), j)));
            });
    }

    template<typename Constraints>
    bool process_constraints(Constraints const & cs, justification const & j) {
        for (auto const & c : cs) {
            if (!process_constraint(update_justification(c, mk_composite1(c.get_justification(), j))))
                return false;
        }
        return true;
    }

    /** \brief Put \c e in weak head normal form.

        \remark Constraints generated in the process are stored in \c cs.
    */
    expr whnf(expr const & e, constraint_seq & cs) {
        return m_tc->whnf(e, cs);
    }

    /** \brief Infer \c e type.

        \remark Return none if an exception was throw when inferring the type.
        \remark Constraints generated in the process are stored in \c cs.
    */
    optional<expr> infer(expr const & e, constraint_seq & cs) {
        try {
            return some_expr(m_tc->infer(e, cs));
        } catch (exception &) {
            return none_expr();
        }
    }

    expr whnf(expr const & e, justification const & j, buffer<constraint> & cs) {
        constraint_seq _cs;
        expr r = whnf(e, _cs);
        to_buffer(_cs, j, cs);
        return r;
    }

    expr flex_rigid_whnf(expr const & e, justification const & j, buffer<constraint> & cs) {
        if (m_config.m_computation) {
            return whnf(e, j, cs);
        } else {
            constraint_seq _cs;
            expr r = m_flex_rigid_tc->whnf(e, _cs);
            to_buffer(_cs, j, cs);
            return r;
        }
    }

    justification mk_assign_justification(expr const & m, expr const & m_type, expr const & v_type, justification const & j) {
        auto r = j.get_main_expr();
        if (!r) r = m;
        justification new_j = mk_justification(r, [=](formatter const & fmt, substitution const & subst) {
                substitution s(subst);
                format r;
                expr _m = s.instantiate(m);
                if (is_meta(_m)) {
                    r = format("type error in placeholder assignment");
                } else {
                    r  = format("type error in placeholder assigned to");
                    r += pp_indent_expr(fmt, _m);
                }
                format expected_fmt, given_fmt;
                std::tie(expected_fmt, given_fmt) = pp_until_different(fmt, m_type, v_type);
                r += compose(line(), format("placeholder has type"));
                r += given_fmt;
                r += compose(line(), format("but is expected to have type"));
                r += expected_fmt;
                r += compose(line(), format("the assignment was attempted when trying to solve"));
                r += nest(2*get_pp_indent(fmt.get_options()), compose(line(), j.pp(fmt, nullptr, subst)));
                return r;
            });
        return mk_composite1(new_j, j);
    }

    /**
       \brief Given lhs of the form (m args), assign (m args) := rhs with justification j.
       The type of lhs and rhs are inferred, and is_def_eq is invoked.

       Any other constraint that contains \c m is revisited
    */
    bool assign(expr const & lhs, expr const & m, buffer<expr> const & args, expr const & rhs, justification const & j) {
        lean_assert(is_metavar(m));
        lean_assert(!in_conflict());
        m_subst.assign(m, args, rhs, j);
        constraint_seq cs;
        auto lhs_type = infer(lhs, cs);
        auto rhs_type = infer(rhs, cs);
        if (lhs_type && rhs_type) {
            if (!process_constraints(cs, j))
                return false;
            justification new_j = mk_assign_justification(m, *lhs_type, *rhs_type, j);
            if (!is_def_eq(*lhs_type, *rhs_type, new_j))
                return false;
        } else {
            set_conflict(j);
            return false;
        }
        auto it = m_mvar_occs.find(mlocal_name(m));
        if (it) {
            cnstr_idx_set s = *it;
            m_mvar_occs.erase(mlocal_name(m));
            s.for_each([&](unsigned cidx) {
                    process_constraint_cidx(cidx);
                });
            return !in_conflict();
        } else {
            return true;
        }
    }

    /**
       \brief Assign \c v to universe metavariable \c m with justification \c j.
       Any constraint that contains \c m is revisted.
    */
    bool assign(level const & m, level const & v, justification const & j) {
        lean_assert(is_meta(m));
        m_subst.assign(m, v, j);
        return true;
    }

    justification mk_invalid_local_ctx_justification(expr const & lhs, expr const & rhs, justification const & j,
                                                     expr const & bad_local) {
        justification new_j = mk_justification(get_app_fn(lhs), [=](formatter const & fmt, substitution const & subst) {
                format r = format("invalid local context when tried to assign");
                r += pp_indent_expr(fmt, rhs);
                buffer<expr> locals;
                auto m = get_app_args(lhs, locals);
                r += line() + format("containing '") + fmt(bad_local) + format("', to placeholder '") + fmt(m) + format("'");
                if (locals.empty()) {
                    r += format(", in the empty local context");
                } else {
                    r += format(", in the local context");
                    format aux;
                    bool first = true;
                    for (expr const l : locals) {
                        if (first) first = false; else aux += space();
                        aux += fmt(l);
                    }
                    r += nest(get_pp_indent(fmt.get_options()), compose(line(), aux));
                }
                r += compose(line(), format("the assignment was attempted when trying to solve"));
                r += nest(2*get_pp_indent(fmt.get_options()), compose(line(), j.pp(fmt, nullptr, subst)));
                return r;
            });
        return mk_composite1(new_j, j);
    }

    enum status { Solved, Failed, Continue };
    /**
       \brief Process constraints of the form <tt>lhs =?= rhs</tt> where lhs is of the form <tt>?m</tt> or <tt>(?m l_1 .... l_n)</tt>,
       where all \c l_i are distinct local variables. In this case, the method returns Solved, if the method assign succeeds.
       The method returns \c Failed if \c rhs contains <tt>?m</tt>, or it contains a local constant not in <tt>{l_1, ..., l_n}</tt>.
       Otherwise, it returns \c Continue.
    */
    status process_metavar_eq(expr const & lhs, expr const & rhs, justification const & j) {
        if (!is_meta(lhs))
            return Continue;
        buffer<expr> locals;
        auto m = is_simple_meta(lhs, locals);
        if (!m || is_meta(rhs))
            return Continue;
        expr bad_local;
        occurs_check_status status;
        if (m_config.m_ignore_context_check)
            status = occurs_check_status::Ok;
        else
            status = occurs_context_check(m_subst, rhs, *m, locals, bad_local);
        if (status == occurs_check_status::FailLocal || status == occurs_check_status::FailCircular) {
            // Try to normalize rhs
            // Example:  ?M := f (pr1 (pair 0 ?M))
            constraint_seq cs;
            auto is_target_fn = [&](expr const & e) {
                if (status == occurs_check_status::FailLocal && occurs(bad_local, e))
                    return true;
                else if (status == occurs_check_status::FailCircular && occurs(*m, e))
                    return true;
                return false;
            };
            expr rhs_n = normalize(*m_tc, rhs, is_target_fn, cs);
            if (rhs != rhs_n && process_constraints(cs))
                return process_metavar_eq(lhs, rhs_n, j);
        }
        switch (status) {
        case occurs_check_status::FailLocal:
            set_conflict(mk_invalid_local_ctx_justification(lhs, rhs, j, bad_local));
            return Failed;
        case occurs_check_status::FailCircular:
            set_conflict(j);
            return Failed;
        case occurs_check_status::Maybe:
            return Continue;
        case occurs_check_status::Ok:
            lean_assert(!m_subst.is_assigned(*m));
            if (assign(lhs, *m, locals, rhs, j)) {
                return Solved;
            } else {
                return Failed;
            }
        }
        lean_unreachable(); // LCOV_EXCL_LINE
    }

    optional<declaration> is_delta(expr const & e) {
        return m_tc->is_delta(e);
    }

    /** \brief Return true if lhs and rhs are of the form (f ...) where f can be expanded */
    bool is_eq_deltas(expr const & lhs, expr const & rhs) {
        auto lhs_d = is_delta(lhs);
        auto rhs_d = is_delta(rhs);
        return lhs_d && rhs_d && is_eqp(*lhs_d, *rhs_d);
    }

    /** \brief Return true if the constraint is of the form (f ...) =?= (f ...), where f can be expanded. */
    bool is_delta_cnstr(constraint const & c) {
        return is_eq_cnstr(c) && is_eq_deltas(cnstr_lhs_expr(c), cnstr_rhs_expr(c));
    }

    pair<constraint, bool> instantiate_metavars(constraint const & c) {
        if (is_eq_cnstr(c)) {
            auto lhs_jst = m_subst.instantiate_metavars(cnstr_lhs_expr(c));
            auto rhs_jst = m_subst.instantiate_metavars(cnstr_rhs_expr(c));
            expr lhs = lhs_jst.first;
            expr rhs = rhs_jst.first;
            if (lhs != cnstr_lhs_expr(c) || rhs != cnstr_rhs_expr(c)) {
                return mk_pair(mk_eq_cnstr(lhs, rhs,
                                           mk_composite1(mk_composite1(c.get_justification(), lhs_jst.second), rhs_jst.second)),
                               true);
            }
        } else if (is_level_eq_cnstr(c)) {
            auto lhs_jst = m_subst.instantiate_metavars(cnstr_lhs_level(c));
            auto rhs_jst = m_subst.instantiate_metavars(cnstr_rhs_level(c));
            level lhs = lhs_jst.first;
            level rhs = rhs_jst.first;
            if (lhs != cnstr_lhs_level(c) || rhs != cnstr_rhs_level(c)) {
                return mk_pair(mk_level_eq_cnstr(lhs, rhs,
                                                 mk_composite1(mk_composite1(c.get_justification(), lhs_jst.second), rhs_jst.second)),
                               true);
            }
        }
        return mk_pair(c, false);
    }

    status process_eq_constraint_core(constraint const & c) {
        expr const & lhs = cnstr_lhs_expr(c);
        expr const & rhs = cnstr_rhs_expr(c);
        justification const & jst = c.get_justification();

        if (lhs == rhs)
            return Solved; // trivial constraint

        // Update justification using the justification of the instantiated metavariables
        if (!has_metavar(lhs) && !has_metavar(rhs)) {
            return is_def_eq(lhs, rhs, jst) ? Solved : Failed;
        }

        // Handle higher-order pattern matching.
        status st = process_metavar_eq(lhs, rhs, jst);
        if (st != Continue) return st;
        st = process_metavar_eq(rhs, lhs, jst);
        if (st != Continue) return st;

        return Continue;
    }

    expr instantiate_meta(expr e, justification & j) {
        while (true) {
            if (auto p = m_subst.expand_metavar_app(e)) {
                // The following check_system is defensive programming.
                // If the unifier is correct, and no loops are introduced in the substituion,
                // then this loop should always terminate.
                // Anyway, we may have bugs, and we should interrupt the loop if all resources are being consumed.
                check_system();
                e = p->first;
                j = mk_composite1(j, p->second);
            } else {
                return e;
            }
        }
    }

    expr instantiate_meta_args(expr const & e, justification & j) {
        if (!is_app(e))
            return e;
        buffer<expr> args;
        bool modified = false;
        expr const & f = get_app_rev_args(e, args);
        unsigned i = args.size();
        while (i > 0) {
            --i;
            expr new_arg = instantiate_meta(args[i], j);
            if (new_arg != args[i]) {
                modified = true;
                args[i]  = new_arg;
            }
        }
        if (!modified)
            return e;
        return mk_rev_app(f, args.size(), args.data());
    }

    status instantiate_eq_cnstr(constraint const & c) {
        justification j = c.get_justification();
        expr lhs = instantiate_meta(cnstr_lhs_expr(c), j);
        expr rhs = instantiate_meta(cnstr_rhs_expr(c), j);
        if (lhs != cnstr_lhs_expr(c) || rhs != cnstr_rhs_expr(c))
            return is_def_eq(lhs, rhs, j) ? Solved : Failed;
        lhs = instantiate_meta_args(lhs, j);
        rhs = instantiate_meta_args(rhs, j);
        if (lhs != cnstr_lhs_expr(c) || rhs != cnstr_rhs_expr(c))
            return is_def_eq(lhs, rhs, j) ? Solved : Failed;
        return Continue;
    }

    /** \brief Return a delay factor if e is of the form (?m ...) and ?m is a metavariable owned by
        a choice constraint. The delay factor is the delay of the choice constraint.
        Return none otherwise. */
    optional<unsigned> is_owned(expr const & e) {
        expr const & m = get_app_fn(e);
        if (!is_metavar(m))
            return optional<unsigned>();
        if (auto it = m_owned_map.find(mlocal_name(m)))
            return optional<unsigned>(*it);
        else
            return optional<unsigned>();
    }

    /** \brief Applies previous method to the left and right hand sides of the equality constraint */
    optional<unsigned> is_owned(constraint const & c) {
        if (auto d = is_owned(cnstr_lhs_expr(c)))
            return d;
        else
            return is_owned(cnstr_rhs_expr(c));
    }

    /** \brief Process an equality constraints. */
    bool process_eq_constraint(constraint const & c) {
        lean_assert(is_eq_cnstr(c));
        // instantiate assigned metavariables
        status st = instantiate_eq_cnstr(c);
        if (st != Continue) return st == Solved;

        if (auto d = is_owned(c)) {
            // Metavariable in the constraint is owned by choice constraint.
            // So, we postpone this constraint.
            add_cnstr(c, to_cnstr_group(*d+1));
            return true;
        }

        st = process_eq_constraint_core(c);
        if (st != Continue) return st == Solved;

        expr const & lhs = cnstr_lhs_expr(c);
        expr const & rhs = cnstr_rhs_expr(c);

        if (is_eq_deltas(lhs, rhs)) {
            // we need to create a backtracking point for this one
            add_cnstr(c, cnstr_group::Basic);
        } else if (is_meta(lhs) && is_meta(rhs)) {
            // flex-flex constraints are delayed the most.
            unsigned cidx = add_cnstr(c, cnstr_group::FlexFlex);
            add_meta_occ(lhs, cidx);
            add_meta_occ(rhs, cidx);
        } else if (m_tc->may_reduce_later(lhs) ||
                   m_tc->may_reduce_later(rhs) ||
                   m_plugin->delay_constraint(*m_tc, c)) {
            unsigned cidx = add_cnstr(c, cnstr_group::PluginDelayed);
            add_meta_occs(lhs, cidx);
            add_meta_occs(rhs, cidx);
        } else if (is_meta(lhs)) {
            // flex-rigid constraints are delayed.
            unsigned cidx = add_cnstr(c, cnstr_group::FlexRigid);
            add_meta_occ(lhs, cidx);
        } else if (is_meta(rhs)) {
            // flex-rigid constraints are delayed.
            unsigned cidx = add_cnstr(c, cnstr_group::FlexRigid);
            add_meta_occ(rhs, cidx);
        } else {
            // this constraints require the unifier plugin to be solved
            add_cnstr(c, cnstr_group::Basic);
        }
        return true;
    }

    /**
       \brief Process a universe level constraints of the form <tt>?m =?= rhs</tt>. It fails if rhs contains \c ?m and
       is definitely bigger than \c ?m.

       TODO(Leo): we should improve this method in the future. It is doing only very basic things.
    */
    status process_metavar_eq(level const & lhs, level const & rhs, justification const & j) {
        if (!is_meta(lhs))
            return Continue;
        bool contains = occurs_meta(lhs, rhs);
        if (contains) {
            if (is_succ(rhs)) {
                set_conflict(j);
                return Failed;
            } else {
                return Continue;
            }
        }
        lean_assert(!m_subst.is_assigned(lhs));
        if (assign(lhs, rhs, j)) {
            return Solved;
        } else {
            set_conflict(j);
            return Failed;
        }
    }

    /** \brief Process a universe level contraints. */
    bool process_level_eq_constraint(constraint const & c) {
        lean_assert(is_level_eq_cnstr(c));
        // instantiate assigned metavariables
        constraint new_c = instantiate_metavars(c).first;
        level lhs = cnstr_lhs_level(new_c);
        level rhs = cnstr_rhs_level(new_c);
        justification jst = new_c.get_justification();

        // normalize lhs and rhs
        lhs = normalize(lhs);
        rhs = normalize(rhs);
        // eliminate outermost succs
        while (is_succ(lhs) && is_succ(rhs)) {
            lhs = succ_of(lhs);
            rhs = succ_of(rhs);
        }

        if (lhs == rhs)
            return true; // trivial constraint

        if (!has_meta(lhs) && !has_meta(rhs)) {
            set_conflict(jst);
            return false; // trivial failure
        }

        status st = process_metavar_eq(lhs, rhs, jst);
        if (st != Continue) return st == Solved;
        st = process_metavar_eq(rhs, lhs, jst);
        if (st != Continue) return st == Solved;

        if (lhs != cnstr_lhs_level(new_c) || rhs != cnstr_rhs_level(new_c))
            new_c = mk_level_eq_cnstr(lhs, rhs, new_c.get_justification());

        add_cnstr(new_c, cnstr_group::FlexRigid);
        return true;
    }

    bool preprocess_choice_constraint(constraint c) {
        if (!cnstr_on_demand(c)) {
            if (cnstr_is_owner(c)) {
                expr m = get_app_fn(cnstr_expr(c));
                lean_assert(is_metavar(m));
                m_owned_map.insert(mlocal_name(m), cnstr_delay_factor(c));
            }
            add_cnstr(c, get_choice_cnstr_group(c));
            return true;
        } else {
            expr m = cnstr_expr(c);
            // choice constraints that are marked as "on demand"
            // are only processed when all metavariables in the
            // type of m have been instantiated.
            expr type;
            justification jst;
            if (auto it = m_type_map.find(m)) {
                // Type of m is already cached in m_type_map
                type = it->first;
                jst  = it->second;
            } else {
                // Type of m is not cached yet, we
                // should infer it, process generated
                // constraints and save the result in
                // m_type_map.
                constraint_seq cs;
                optional<expr> t = infer(m, cs);
                if (!t) {
                    set_conflict(c.get_justification());
                    return false;
                }
                if (!process_constraints(cs, c.get_justification()))
                    return false;
                type = *t;
                m_type_map.insert(m, mk_pair(type, justification()));
            }
            // Try to instantiate metavariables in type
            pair<expr, justification> type_jst = m_subst.instantiate_metavars(type);
            if (type_jst.first != type) {
                // Type was modified by instantiation,
                // we update the constraint justification,
                // and store the new type in m_type_map
                jst  = mk_composite1(jst, type_jst.second);
                type = type_jst.first;
                c    = update_justification(c, mk_composite1(c.get_justification(), jst));
                m_type_map.insert(m, mk_pair(type, jst));
            }
            unsigned cidx = add_cnstr(c, cnstr_group::ClassInstance);
            if (!add_meta_occs(type, cidx)) {
                // type does not contain metavariables...
                // so this "on demand" constraint is ready to be solved
                m_cnstrs.erase(cnstr(c, cidx));
                add_cnstr(c, cnstr_group::Basic);
                m_type_map.erase(m);
            }
            return true;
        }
    }

    /**
        \brief Process the given constraint \c c. "Easy" constraints are solved, and the remaining ones
        are added to the constraint queue m_cnstrs. By "easy", see the methods
        #process_eq_constraint and #process_level_eq_constraint.
    */
    bool process_constraint(constraint const & c) {
        if (in_conflict())
            return false;
        check_full();
        // std::cout << "process: " << c << "\n";
        switch (c.kind()) {
        case constraint_kind::Choice:
            return preprocess_choice_constraint(c);
        case constraint_kind::Eq:
            return process_eq_constraint(c);
        case constraint_kind::LevelEq:
            return process_level_eq_constraint(c);
        }
        lean_unreachable(); // LCOV_EXCL_LINE
    }

    /**
       \brief Process constraint with index \c cidx. The constraint is removed
       from the constraint queue, and the method #process_constraint is invoked.
    */
    bool process_constraint_cidx(unsigned cidx) {
        if (in_conflict())
            return false;
        cnstr c(*g_dont_care_cnstr, cidx);
        if (auto it = m_cnstrs.find(c)) {
            constraint c2 = it->first;
            m_cnstrs.erase(c);
            return process_constraint(c2);
        }
        return true;
    }

    void add_case_split(std::unique_ptr<case_split> && cs) {
        m_case_splits.push_back(std::move(cs));
    }

    // This method is used only for debugging purposes.
    void display(std::ostream & out, justification const & j, unsigned indent = 0) {
        for (unsigned i = 0; i < indent; i++)
            out << " ";
        out << j.pp(mk_print_formatter_factory()(m_env, options()), nullptr, m_subst) << "\n";
        if (j.is_composite()) {
            display(out, composite_child1(j), indent+2);
            display(out, composite_child2(j), indent+2);
        }
    }

    void pop_case_split() {
        m_case_splits.pop_back();
    }

    bool resolve_conflict() {
        lean_assert(in_conflict());
        while (!m_case_splits.empty()) {
            check_system();
            justification conflict = *m_conflict;
            std::unique_ptr<case_split> & d = m_case_splits.back();
            if (!m_config.m_nonchronological || depends_on(conflict, d->m_assumption_idx)) {
                d->m_failed_justifications = mk_composite1(d->m_failed_justifications, conflict);
                if (d->next(*this)) {
                    reset_conflict();
                    return true;
                }
            } else {
                pop_case_split();
            }
        }
        return false;
    }

    bool next_lazy_constraints_case_split(lazy_constraints_case_split & cs) {
        auto r = cs.m_tail.pull();
        if (r) {
            cs.restore_state(*this);
            lean_assert(!in_conflict());
            cs.m_tail = r->second;
            return process_constraints(r->first, mk_composite1(cs.get_jst(), mk_assumption_justification(cs.m_assumption_idx)));
        } else {
            // update conflict
            update_conflict(mk_composite1(*m_conflict, cs.m_failed_justifications));
            pop_case_split();
            return false;
        }
    }

    bool process_lazy_constraints(lazy_list<constraints> const & l, justification const & j) {
        auto r = l.pull();
        if (r) {
            if (r->second.is_nil()) {
                // there is only one alternative
                return process_constraints(r->first, j);
            } else {
                justification a = mk_assumption_justification(m_next_assumption_idx);
                add_case_split(std::unique_ptr<case_split>(new lazy_constraints_case_split(*this, j, r->second)));
                return process_constraints(r->first, mk_composite1(j, a));
            }
        } else {
            set_conflict(j);
            return false;
        }
    }

    /** \brief Given a constraint of the form
            f a_1 ... a_n =?= f b_1 ... b_n
        Return singleton stream with the possible solution
             a_i =?= b_i
        If c is not of the expected form, then return the empty stream.
    */
    lazy_list<constraints> process_const_const_cnstr(constraint const & c) {
        if (!is_eq_cnstr(c))
            return lazy_list<constraints>();
        expr const & lhs = cnstr_lhs_expr(c);
        expr const & rhs = cnstr_rhs_expr(c);
        expr const & f_lhs = get_app_fn(lhs);
        expr const & f_rhs = get_app_fn(rhs);
        if (!is_constant(f_lhs) || !is_constant(f_rhs) || const_name(f_lhs) != const_name(f_rhs))
            return lazy_list<constraints>();
        justification const & j = c.get_justification();
        constraint_seq cs;
        auto fcs = m_tc->is_def_eq(f_lhs, f_rhs, j);
        if (!fcs.first)
            return lazy_list<constraints>();
        cs = fcs.second;
        buffer<expr> args_lhs, args_rhs;
        get_app_args(lhs, args_lhs);
        get_app_args(rhs, args_rhs);
        if (args_lhs.size() != args_rhs.size())
            return lazy_list<constraints>();
        for (unsigned i = 0; i < args_lhs.size(); i++) {
            auto acs = m_tc->is_def_eq(args_lhs[i], args_rhs[i], j);
            if (!acs.first)
                return lazy_list<constraints>();
            cs = acs.second + cs;
        }
        return lazy_list<constraints>(cs.to_list());
    }

    bool process_plugin_constraint(constraint const & c) {
        lean_assert(!is_choice_cnstr(c));
        lazy_list<constraints> alts = m_plugin->solve(*m_tc, c, m_ngen.mk_child());
        alts = append(alts, process_const_const_cnstr(c));
        return process_lazy_constraints(alts, c.get_justification());
    }

    bool process_choice_constraint(constraint const & c) {
        lean_assert(is_choice_cnstr(c));
        expr const &   m            = cnstr_expr(c);
        choice_fn const & fn        = cnstr_choice_fn(c);
        if (cnstr_is_owner(c)) {
            // choice will have a chance to assign m, so
            // we remove the "barrier" that was preventing m from being assigned.
            m_owned_map.erase(mlocal_name(get_app_fn(m)));
        }
        expr m_type;

        constraint_seq cs;
        if (auto type = infer(m, cs)) {
            m_type = *type;
            if (!process_constraints(cs))
                return false;
        } else {
            set_conflict(c.get_justification());
            return false;
        }
        auto m_type_jst             = m_subst.instantiate_metavars(m_type);
        lazy_list<constraints> alts = fn(m, m_type_jst.first, m_subst, m_ngen.mk_child());
        return process_lazy_constraints(alts, mk_composite1(c.get_justification(), m_type_jst.second));
    }

    bool next_simple_case_split(simple_case_split & cs) {
        if (!is_nil(cs.m_tail)) {
            cs.restore_state(*this);
            lean_assert(!in_conflict());
            constraints c = head(cs.m_tail);
            cs.m_tail     = tail(cs.m_tail);
            return process_constraints(c, mk_composite1(cs.get_jst(), mk_assumption_justification(cs.m_assumption_idx)));
        } else {
            // update conflict
            update_conflict(mk_composite1(*m_conflict, cs.m_failed_justifications));
            pop_case_split();
            return false;
        }
    }

    bool unfold_delta(constraint const & c, justification const & extra_j) {
        expr const & lhs = cnstr_lhs_expr(c);
        expr const & rhs = cnstr_rhs_expr(c);
        buffer<expr> lhs_args, rhs_args;
        justification j = c.get_justification();
        expr lhs_fn     = get_app_rev_args(lhs, lhs_args);
        expr rhs_fn     = get_app_rev_args(rhs, rhs_args);
        declaration d   = *m_env.find(const_name(lhs_fn));
        expr lhs_fn_val = instantiate_value_univ_params(d, const_levels(lhs_fn));
        expr rhs_fn_val = instantiate_value_univ_params(d, const_levels(rhs_fn));
        expr t = apply_beta(lhs_fn_val, lhs_args.size(), lhs_args.data());
        expr s = apply_beta(rhs_fn_val, rhs_args.size(), rhs_args.data());
        auto dcs = m_tc->is_def_eq(t, s, j);
        if (dcs.first) {
            constraints cnstrs = dcs.second.to_list();
            return process_constraints(cnstrs, extra_j);
        } else {
            set_conflict(j);
            return false;
        }
    }

    bool next_delta_unfold_case_split(delta_unfold_case_split & cs) {
        if (!cs.m_done) {
            cs.restore_state(*this);
            cs.m_done = true;
            constraint const & c = cs.m_cnstr;
            justification j      = mk_composite1(cs.get_jst(), mk_assumption_justification(cs.m_assumption_idx));
            return unfold_delta(c, j);
        } else {
            // update conflict
            update_conflict(mk_composite1(*m_conflict, cs.m_failed_justifications));
            pop_case_split();
            return false;
        }
    }

    /**
        \brief Solve constraints of the form (f a_1 ... a_n) =?= (f b_1 ... b_n) where f can be expanded.
        We consider two possible solutions:
        1)   a_1 =?= b_1, ..., a_n =?= b_n
        2)   t =?= s, where t and s are the terms we get after expanding f
    */
    bool process_delta(constraint const & c) {
        lean_assert(is_delta_cnstr(c));
        expr const & lhs = cnstr_lhs_expr(c);
        expr const & rhs = cnstr_rhs_expr(c);
        buffer<expr> lhs_args, rhs_args;
        justification j = c.get_justification();
        expr lhs_fn     = get_app_rev_args(lhs, lhs_args);
        expr rhs_fn     = get_app_rev_args(rhs, rhs_args);
        declaration d   = *m_env.find(const_name(lhs_fn));
        levels lhs_lvls = const_levels(lhs_fn);
        levels rhs_lvls = const_levels(rhs_fn);
        if (length(lhs_lvls) != length(rhs_lvls) ||
            d.get_num_univ_params() != length(lhs_lvls)) {
            // the constraint is not well-formed, this can happen when users are abusing the API
            set_conflict(j);
            return false;
        }

        if (lhs_args.size() != rhs_args.size())
            return unfold_delta(c, justification());

        justification a;
        if (m_config.m_kind == unifier_kind::Liberal &&
            (m_config.m_computation || module::is_definition(m_env, d.get_name()) || is_at_least_quasireducible(m_env, d.get_name()))) {
            // add case_split for t =?= s
            a = mk_assumption_justification(m_next_assumption_idx);
            add_case_split(std::unique_ptr<case_split>(new delta_unfold_case_split(*this, j, c)));
        }

        // process first case
        justification new_j = mk_composite1(j, a);
        while (!is_nil(lhs_lvls)) {
            level lhs = head(lhs_lvls);
            level rhs = head(rhs_lvls);
            if (!process_constraint(mk_level_eq_cnstr(lhs, rhs, new_j)))
                return false;
            lhs_lvls = tail(lhs_lvls);
            rhs_lvls = tail(rhs_lvls);
        }

        unsigned i = lhs_args.size();
        while (i > 0) {
            --i;
            if (!is_def_eq(lhs_args[i], rhs_args[i], new_j))
                return false;
        }
        return true;
    }

    /** \brief Return true iff \c c is a flex-rigid constraint. */
    static bool is_flex_rigid(constraint const & c) {
        if (!is_eq_cnstr(c))
            return false;
        bool is_lhs_meta = is_meta(cnstr_lhs_expr(c));
        bool is_rhs_meta = is_meta(cnstr_rhs_expr(c));
        return is_lhs_meta != is_rhs_meta;
    }

    /** \brief Return true iff \c c is a flex-flex constraint */
    static bool is_flex_flex(constraint const & c) {
        return is_eq_cnstr(c) && is_meta(cnstr_lhs_expr(c)) && is_meta(cnstr_rhs_expr(c));
    }

    /** \brief Append the auxiliary constraints \c aux to each alternative in \c alts */
    void append_auxiliary_constraints(buffer<constraints> & alts, list<constraint> const & aux) {
        if (is_nil(aux))
            return;
        for (constraints & cs : alts)
            cs = append(aux, cs);
    }

    /** \brief Auxiliary functional object for implementing process_flex_rigid_core */
    class flex_rigid_core_fn {
        unifier_fn &          u;
        expr const &          lhs;
        buffer<expr>          margs;
        expr const &          m;
        expr const &          rhs;
        justification         j;
        buffer<constraints> & alts; // result: alternatives
        bool                  imitation_only; // if true, then only imitation step is used

        optional<bool>        _has_meta_args;

        bool cheap() const { return u.m_config.m_kind == unifier_kind::Cheap; }
        bool pattern() const { return u.m_config.m_pattern; }

        type_checker & tc() {
            return *u.m_tc;
        }

        type_checker & restricted_tc() {
            if (u.m_config.m_computation)
                return *u.m_tc;
            else
                return *u.m_flex_rigid_tc;
        }

        /** \brief Return true if margs contains an expression \c e s.t. is_meta(e) */
        bool has_meta_args() {
            if (!_has_meta_args) {
                _has_meta_args = std::any_of(margs.begin(), margs.end(),
                                              [](expr const & e) { return is_meta(e); });
            }
            return *_has_meta_args;
        }

        /**
           \brief Given t
           <tt>Pi (x_1 : A_1) ... (x_n : A_n[x_1, ..., x_{n-1}]), B[x_1, ..., x_n]</tt>
           return
           <tt>fun (x_1 : A_1) ... (x_n : A_n[x_1, ..., x_{n-1}]), v[x_1, ... x_n]</tt>

           \remark v has free variables.
        */
        expr mk_lambda_for(unsigned i, expr const & t, expr const & v) {
            if (i < margs.size()) {
                return mk_lambda(binding_name(t), binding_domain(t), mk_lambda_for(i+1, binding_body(t), v), binding_info(t));
            } else {
                return v;
            }
        }

        expr mk_lambda_for(expr const & t, expr const & v) {
            return mk_lambda_for(0, t, v);
        }

        /** \brief Return true if \c local occurs once in the buffer \c es. */
        bool local_occurs_once(expr const & local, buffer<expr> const & es) {
            bool found = false;
            for (expr const & e : es) {
                if (is_local(e) && mlocal_name(e) == mlocal_name(local)) {
                    if (found)
                        return false;
                    found = true;
                }
            }
            return true;
        }

        /** \see ensure_sufficient_args */
        expr ensure_sufficient_args_core(expr mtype, unsigned i, constraint_seq & cs) {
            if (i == margs.size())
                return mtype;
            mtype = tc().ensure_pi(mtype, cs);
            expr local = u.mk_local_for(mtype);
            expr body  = instantiate(binding_body(mtype), local);
            return Pi(local, ensure_sufficient_args_core(body, i+1, cs));
        }

        /** \brief Make sure mtype is a Pi of size at least margs.size().
            If it is not, we use ensure_pi and (potentially) add new constaints to enforce it.
        */
        expr ensure_sufficient_args(expr const & mtype, constraint_seq & cs) {
            expr t = mtype;
            unsigned num = 0;
            while (is_pi(t)) {
                num++;
                t = binding_body(t);
            }
            if (num >= margs.size())
                return mtype;
            return ensure_sufficient_args_core(mtype, 0, cs);
        }

        /**
           \brief Given
                 m      := a metavariable ?m
                 rhs    := sort, constant
           Then solve (?m a_1 ... a_k) =?= rhs, by returning the constraint
                 ?m  =?= fun (x1 ... x_k), rhs
        */
        void mk_simple_imitation() {
            lean_assert(is_metavar(m));
            lean_assert(is_sort(rhs) || is_constant(rhs));
            expr const & mtype = mlocal_type(m);
            constraint_seq cs;
            expr new_mtype = ensure_sufficient_args(mtype, cs);
            cs = cs + mk_eq_cnstr(m, mk_lambda_for(new_mtype, rhs), j);
            alts.push_back(cs.to_list());
        }

        bool restricted_is_def_eq(expr const & marg, expr const & rhs, justification const & j, constraint_seq & cs) {
            try {
                if (restricted_tc().is_def_eq(marg, rhs, j, cs)) {
                    return true;
                } else {
                    return false;
                }
            } catch (exception & ex) {
                return false;
            }
        }

        /**
           Given,
               m      := a metavariable ?m
               margs  := [a_1 ... a_k]
           We say a unification problem (?m a_1 ... a_k) =?= rhs uses "simple nonlocal i-th projection" when

           1) rhs is not a local constant
           2) is_def_eq(a_i, rhs) does not fail

           In this case, we add
                a_i =?= rhs
                ?m  =?= fun x_1 ... x_k, x_i
           to alts as a possible solution.
        */
        void mk_simple_nonlocal_projection(unsigned i) {
            expr const & mtype = mlocal_type(m);
            unsigned vidx = margs.size() - i - 1;
            expr const & marg = margs[i];
            constraint_seq cs;
            auto new_mtype = ensure_sufficient_args(mtype, cs);
            // Remark: we should not use mk_eq_cnstr(marg, rhs, j) since is_def_eq may be able to reduce them.
            // The unifier assumes the eq constraints are reduced.
            if (tc().is_def_eq_types(marg, rhs, j, cs) &&
                restricted_is_def_eq(marg, rhs, j, cs)) {
                expr v = mk_lambda_for(new_mtype, mk_var(vidx));
                cs = cs + mk_eq_cnstr(m, v, j);
                alts.push_back(cs.to_list());
            }
        }

        /**
           Given,
              m      := a metavariable ?m
              margs  := [a_1 ... a_k]
           We say a unification problem (?m a_1 ... a_k) =?= rhs uses "simple projections" when

           If (rhs and a_i are *not* local constants) OR (rhs is a local constant and a_i is a metavariable application),
           then we add the constraints
                a_i =?= rhs
                ?m  =?= fun x_1 ... x_k, x_i
                to alts as a possible solution.

           If rhs is a local constant and a_i == rhs, then we add the constraint
                ?m =?= fun x_1 ... x_k, x_i
           to alts as a possible solution when a_i is the same local constant or a metavariable application
        */
        void mk_simple_projections() {
            lean_assert(is_metavar(m));
            lean_assert(!is_meta(rhs));
            expr const & mtype = mlocal_type(m);
            unsigned i = margs.size();
            while (i > 0) {
                unsigned vidx = margs.size() - i;
                --i;
                expr const & marg = margs[i];
                if ((!is_local(marg) && !is_local(rhs)) || (is_meta(marg) && is_local(rhs))) {
                    // if rhs is not local, then we only add projections for the nonlocal arguments of lhs
                    mk_simple_nonlocal_projection(i);
                    if (cheap())
                        return;
                } else if (is_local(marg) && is_local(rhs) && mlocal_name(marg) == mlocal_name(rhs)) {
                    // if the argument is local, and rhs is equal to it, then we also add a projection
                    constraint_seq cs;
                    auto new_mtype = ensure_sufficient_args(mtype, cs);
                    expr v = mk_lambda_for(new_mtype, mk_var(vidx));
                    cs = cs + mk_eq_cnstr(m, v, j);
                    alts.push_back(cs.to_list());
                    if (cheap())
                        return;
                }
            }
        }

        void mk_app_projections() {
            lean_assert(is_metavar(m));
            lean_assert(is_app(rhs));
            if (!pattern() && !cheap()) {
                expr const & f = get_app_fn(rhs);
                lean_assert(is_constant(f) || is_local(f));
                if (is_local(f)) {
                    unsigned i = margs.size();
                    while (i > 0) {
                        --i;
                        if (!(is_local(margs[i]) && mlocal_name(margs[i]) == mlocal_name(f)))
                            mk_simple_nonlocal_projection(i);
                    }
                } else {
                    mk_simple_projections();
                }
            }
        }

        /** \brief Create the local context \c locals for the imitation step.
         */
        void mk_local_context(buffer<expr> & locals, constraint_seq & cs) {
            expr mtype     = mlocal_type(m);
            unsigned nargs = margs.size();
            mtype = ensure_sufficient_args(mtype, cs);
            expr it = mtype;
            for (unsigned i = 0; i < nargs; i++) {
                expr d     = instantiate_rev(binding_domain(it), locals.size(), locals.data());
                auto d_jst = u.m_subst.instantiate_metavars(d);
                d = d_jst.first;
                j = mk_composite1(j, d_jst.second);
                name n;
                if (is_local(margs[i]) && local_occurs_once(margs[i], margs)) {
                    n = mlocal_name(margs[i]);
                } else {
                    n = u.m_ngen.next();
                }
                expr local = mk_local(n, binding_name(it), d, binding_info(it));
                locals.push_back(local);
                it = binding_body(it);
            }
        }

        expr mk_imitation_arg(expr const & arg, expr const & type, buffer<expr> const & locals,
                               constraint_seq & cs) {
            // The following optimization is broken. It does not really work when we have dependent
            // types. The problem is that the type of arg may depend on other arguments,
            // and constraints are not generated to enforce it.
            //
            //  Here is a minimal counterexample
            //     ?M A B a b H B b  =?=  heq.type_eq A B a b H
            //  with this optimization the imitation is
            //
            //    ?M := fun (A B a b H B' b'), heq.type_eq A (?M1 A B a b H B' b') a (?M2 A B a b H B' b') H
            //
            //  This imitation is only correct if
            //     typeof(H) is (heq A a (?M1 A B a b H B' b') (?M2 A B a b H B' b'))
            //
            //  Adding an extra constraint is problematic since typeof(H) may contain local constants,
            //  and these local constants may have been "renamed" by mk_local_context above
            //
            //  For now, we simply comment the optimization.
            //

            // Broken optimization
            // if (!has_meta_args() && is_local(arg) && contains_local(arg, locals)) {
            //    return arg;
            // }

            // The following code is not affected by the problem above because we
            // attach the type \c type to the new metavariables being created.

            // std::cout << "type: " << type << "\n";
            if (context_check(type, locals)) {
                expr maux    = mk_metavar(u.m_ngen.next(), Pi(locals, type));
                // std::cout << "  >> " << maux << " : " << mlocal_type(maux) << "\n";
                cs = mk_eq_cnstr(mk_app(maux, margs), arg, j) + cs;
                return mk_app(maux, locals);
            } else {
                expr maux_type   = mk_metavar(u.m_ngen.next(), Pi(locals, mk_sort(mk_meta_univ(u.m_ngen.next()))));
                expr maux        = mk_metavar(u.m_ngen.next(), Pi(locals, mk_app(maux_type, locals)));
                cs = mk_eq_cnstr(mk_app(maux, margs), arg, j) + cs;
                return mk_app(maux, locals);
            }
        }

        void mk_app_imitation_core(expr const & f, buffer<expr> const & locals, constraint_seq & cs) {
            buffer<expr> rargs;
            get_app_args(rhs, rargs);
            buffer<expr> sargs;
            try {
                expr f_type = tc().infer(f, cs);
                for (expr const & rarg : rargs) {
                    f_type      = tc().ensure_pi(f_type, cs);
                    expr d_type = binding_domain(f_type);
                    expr sarg   = mk_imitation_arg(rarg, d_type, locals, cs);
                    sargs.push_back(sarg);
                    f_type      = instantiate(binding_body(f_type), sarg);
                }
            } catch (exception&) {}
            expr v = Fun(locals, mk_app(f, sargs));
            cs += mk_eq_cnstr(m, v, j);
            alts.push_back(cs.to_list());
        }

        /**
           \brief Given
               m      := a metavariable ?m
               margs  := [a_1 ... a_k]
               rhs    := (f b_1 ... b_n)
           Then create the constraints
               (?m_1 a_1 ... a_k) =?= b_1
               ...
               (?m_n a_1 ... a_k) =?= b_n
               ?m  =?= fun (x_1 ... x_k), g (?m_1 x_1 ... x_k) ... (?m_n x_1 ... x_k)

           If f is a constant (or a macro), then g is f.
           If f is a local constant, then we consider each a_i that is equal to f.

           Remark: we try to minimize the number of constraints (?m_i a_1 ... a_k) =?= b_i by detecting "easy" cases
           that can be solved immediately. See \c mk_imitation_arg
        */
        void mk_app_imitation() {
            lean_assert(is_metavar(m));
            lean_assert(is_app(rhs));
            buffer<expr> locals;
            constraint_seq cs;
            flet<justification> let(j, j); // save j value
            mk_local_context(locals, cs);
            lean_assert(margs.size() == locals.size());
            expr const & f = get_app_fn(rhs);
            lean_assert(is_constant(f) || is_macro(f) || is_local(f));
            if (is_local(f)) {
                unsigned i = margs.size();
                while (i > 0) {
                    --i;
                    if (is_local(margs[i]) && mlocal_name(margs[i]) == mlocal_name(f)) {
                        constraint_seq new_cs = cs;
                        mk_app_imitation_core(locals[i], locals, new_cs);
                    }
                }
            } else {
                lean_assert(is_constant(f) || is_macro(f));
                mk_app_imitation_core(f, locals, cs);
            }
        }

        /**
           \brief Given
               m      := a metavariable ?m
               margs  := [a_1 ... a_k]
               rhs    := (fun/Pi (y : A), B y)
           Then create the constraints
              (?m_1 a_1 ... a_k)   =?= A
              (?m_2 a_1 ... a_k l) =?= B l
              ?m   =?= fun (x_1 ... x_k), fun/Pi (y : ?m_1 x_1 ... x_k), ?m_2 x_1 ... x_k y
           where l is a fresh local constant.
        */
        void mk_bindings_imitation() {
            lean_assert(is_metavar(m));
            lean_assert(is_binding(rhs));
            constraint_seq cs;
            buffer<expr> locals;
            flet<justification> let(j, j); // save j value
            mk_local_context(locals, cs);
            lean_assert(margs.size() == locals.size());
            try {
                // create a scope to make sure no constraints "leak" into the current state
                expr rhs_A     = binding_domain(rhs);
                expr A_type    = tc().infer(rhs_A, cs);
                expr A         = mk_imitation_arg(rhs_A, A_type, locals, cs);
                expr local     = mk_local(u.m_ngen.next(), binding_name(rhs), A, binding_info(rhs));
                locals.push_back(local);
                margs.push_back(local);
                expr rhs_B     = instantiate(binding_body(rhs), local);
                expr B_type    = tc().infer(rhs_B, cs);
                expr B         = mk_imitation_arg(rhs_B, B_type, locals, cs);
                expr binding   = is_pi(rhs) ? Pi(local, B) : Fun(local, B);
                locals.pop_back();
                expr v         = Fun(locals, binding);
                cs += mk_eq_cnstr(m, v, j);
                alts.push_back(cs.to_list());
            } catch (exception&) {}
            margs.pop_back();
        }

    public:
        flex_rigid_core_fn(unifier_fn & _u, expr const & _lhs, expr const & _rhs,
                           justification const & _j, buffer<constraints> & _alts,
                           bool _imitation_only):
            u(_u), lhs(_lhs), m(get_app_args(lhs, margs)), rhs(_rhs), j(_j), alts(_alts),
            imitation_only(_imitation_only) {}

        void operator()() {
            switch (rhs.kind()) {
            case expr_kind::Var: case expr_kind::Meta:
                lean_unreachable(); // LCOV_EXCL_LINE
            case expr_kind::Local:
                mk_simple_projections();
                break;
            case expr_kind::Sort: case expr_kind::Constant:
                if (!pattern() && !cheap() && !imitation_only)
                    mk_simple_projections();
                mk_simple_imitation();
                break;
            case expr_kind::Pi: case expr_kind::Lambda:
                if (!pattern() && !cheap() && !imitation_only)
                    mk_simple_projections();
                mk_bindings_imitation();
                break;
            case expr_kind::Macro:
                lean_unreachable(); // LCOV_EXCL_LINE
            case expr_kind::App:
                if (!imitation_only)
                    mk_app_projections();
                mk_app_imitation();
                break;
            }
        }
    };

    void process_flex_rigid_core(expr const & lhs, expr const & rhs, justification const & j,
                                 buffer<constraints> & alts, bool imitation_only) {
        flex_rigid_core_fn(*this, lhs, rhs, j, alts, imitation_only)();
    }

    /** \brief When lhs is an application (f ...), make sure that if any argument that is reducible to a
        local constant is replaced with a local constant.

        \remark We store auxiliary constraints created in the reductions in \c aux. We return the new
        "reduce" application.
    */
    expr expose_local_args(expr const & lhs, justification const & j, buffer<constraint> & aux) {
        buffer<expr> margs;
        expr m        = get_app_args(lhs, margs);
        bool modified = false;
        for (expr & marg : margs) {
            // Make sure that if marg is reducible to a local constant, then it is replaced with it.
            if (!is_local(marg)) {
                expr new_marg = whnf(marg, j, aux);
                if (is_local(new_marg)) {
                    marg     = new_marg;
                    modified = true;
                }
            }
        }
        return modified ? mk_app(m, margs) : lhs;
    }

    optional<expr> expand_rhs(expr const & rhs) {
        buffer<expr> args;
        expr const & f = get_app_rev_args(rhs, args);
        lean_assert(!is_local(f) && !is_constant(f) && !is_var(f) && !is_metavar(f));
        if (is_lambda(f)) {
            return some_expr(apply_beta(f, args.size(), args.data()));
        } else if (is_macro(f)) {
            if (optional<expr> new_f = m_tc->expand_macro(f))
                return some_expr(mk_rev_app(*new_f, args.size(), args.data()));
        }
        return none_expr();
    }

    /** \brief When solving flex-rigid constraints lhs =?= rhs (lhs is of the form ?M a_1 ... a_n),
        we consider an additional case-split where rhs is put in weak-head-normal-form when

        1- Option unifier.computation is true
        2- At least one a_i is not a local constant
        3- rhs contains a local constant that is not equal to any a_i.
    */
    bool use_flex_rigid_whnf_split(expr const & lhs, expr const & rhs) {
        lean_assert(is_meta(lhs));
        if (m_config.m_kind != unifier_kind::Liberal)
            return false;
        if (m_config.m_computation)
            return true; // if unifier.computation is true, we always consider the additional whnf split
        buffer<expr> locals;
        expr const * it = &lhs;
        while (is_app(*it)) {
            expr const & arg = app_arg(*it);
            if (!is_local(arg))
                return true; // lhs contains non-local constant
            locals.push_back(arg);
            it = &(app_fn(*it));
        }
        if (!context_check(rhs, locals))
            return true; // rhs contains local constant that is not in locals
        return false;
    }

    /** \brief Process a flex rigid constraint */
    bool process_flex_rigid(expr lhs, expr const & rhs, justification const & j) {
        lean_assert(is_meta(lhs));
        lean_assert(!is_meta(rhs));
        if (is_app(rhs)) {
            expr const & f = get_app_fn(rhs);
            if (!is_local(f) && !is_constant(f)) {
                if (auto new_rhs = expand_rhs(rhs)) {
                    lean_assert(*new_rhs != rhs);
                    return is_def_eq(lhs, *new_rhs, j);
                } else {
                    return false;
                }
            }
        } else if (is_macro(rhs)) {
            if (auto new_rhs = expand_rhs(rhs)) {
                lean_assert(*new_rhs != rhs);
                return is_def_eq(lhs, *new_rhs, j);
            } else {
                return false;
            }
        }

        buffer<constraint> aux;
        lhs = expose_local_args(lhs, j, aux);
        buffer<constraints> alts;
        process_flex_rigid_core(lhs, rhs, j, alts, false);
        append_auxiliary_constraints(alts, to_list(aux.begin(), aux.end()));
        if (use_flex_rigid_whnf_split(lhs, rhs)) {
            expr rhs_whnf = flex_rigid_whnf(rhs, j, aux);
            if (rhs_whnf != rhs) {
                if (is_meta(rhs_whnf)) {
                    // it become a flex-flex constraint
                    alts.push_back(constraints(mk_eq_cnstr(lhs, rhs_whnf, j)));
                } else {
                    buffer<constraints> alts2;
                    process_flex_rigid_core(lhs, rhs_whnf, j, alts2, true);
                    append_auxiliary_constraints(alts2, to_list(aux.begin(), aux.end()));
                    alts.append(alts2);
                }
            }
        }

        // std::cout << "FlexRigid\n";
        // for (auto cs : alts) {
        //     std::cout << " alternative\n";
        //     for (auto c : cs) {
        //         std::cout << "   >> " << c << "\n";
        //     }
        // }

        if (alts.empty()) {
            set_conflict(j);
            return false;
        } else if (alts.size() == 1) {
            // we don't need to create a backtracking point
            return process_constraints(alts[0], justification());
        } else {
            justification a = mk_assumption_justification(m_next_assumption_idx);
            add_case_split(std::unique_ptr<case_split>(new simple_case_split(*this, j, to_list(alts.begin() + 1, alts.end()))));
            return process_constraints(alts[0], a);
        }
    }

    /** \brief Process a flex rigid constraint */
    bool process_flex_rigid(constraint const & c) {
        lean_assert(is_flex_rigid(c));
        expr lhs   = cnstr_lhs_expr(c);
        expr rhs   = cnstr_rhs_expr(c);
        if (is_meta(lhs))
            return process_flex_rigid(lhs, rhs, c.get_justification());
        else
            return process_flex_rigid(rhs, lhs, c.get_justification());
    }

    void postpone(constraint const & c) {
        m_postponed = cons(c, m_postponed);
    }

    void discard(constraint const & c) {
        if (!m_config.m_discard)
            postpone(c);
    }

    bool process_flex_flex(constraint const & c) {
        expr const & lhs = cnstr_lhs_expr(c);
        expr const & rhs = cnstr_rhs_expr(c);
        // We ignore almost all flex-flex constraints.
        // We just handle flex_flex "first-order" case
        //   ?M_1 l_1 ... l_k =?= ?M_2 l_1 ... l_k
        //   ?M_1 l_1 ... l_k =?= ?M_2 l_1 ... l_k ... l_n
        //   ?M_1 l_1 ... l_k ... l_n =?= ?M_2 l_1 ... l_k
        if (!is_simple_meta(lhs) || !is_simple_meta(rhs)) {
            discard(c);
            return true;
        }
        buffer<expr> lhs_args, rhs_args;
        expr ml = get_app_args(lhs, lhs_args);
        expr mr = get_app_args(rhs, rhs_args);
        if (mlocal_name(ml) == mlocal_name(mr)) {
            discard(c);
            return true;
        }
        unsigned min_sz = std::min(lhs_args.size(), rhs_args.size());
        lean_assert(!m_subst.is_assigned(ml));
        lean_assert(!m_subst.is_assigned(mr));
        unsigned i = 0;
        for (; i < min_sz; i++)
            if (mlocal_name(lhs_args[i]) != mlocal_name(rhs_args[i]))
                break;
        if (i == min_sz) {
            if (lhs_args.size() >= rhs_args.size()) {
                return assign(lhs, ml, lhs_args, rhs, c.get_justification());
            } else {
                return assign(rhs, mr, rhs_args, lhs, c.get_justification());
            }
        } else {
            discard(c);
            return true;
        }
    }

    /** \brief Return true iff \c rhs is of the form <tt> max(l_1 ... lhs ... l_k) </tt>,
        such that l_i's do not contain lhs.

        If the result is true, then all l_i's are stored in rest.
    */
    static bool generalized_check_meta(level const & m, level const & rhs, bool & found_m, buffer<level> & rest) {
        lean_assert(is_meta(m));
        if (is_max(rhs)) {
            return
                generalized_check_meta(m, max_lhs(rhs), found_m, rest) &&
                generalized_check_meta(m, max_rhs(rhs), found_m, rest);
        } else if (m == rhs) {
            found_m = true;
            return true;
        } else if (occurs_meta(m, rhs)) {
            return false;
        } else {
            rest.push_back(rhs);
            return true;
        }
    }

    status process_l_eq_max_core(level const & lhs, level const & rhs, justification const & jst) {
        lean_assert(is_meta(lhs));
        buffer<level> rest;
        bool found_lhs = false;
        if (generalized_check_meta(lhs, rhs, found_lhs, rest)) {
            level r;
            if (found_lhs) {
                // rhs is of the form max(rest, lhs)
                // Solution is lhs := max(rest, ?u) where ?u is fresh metavariable
                r = mk_meta_univ(m_ngen.next());
                rest.push_back(r);
                unsigned i = rest.size();
                while (i > 0) {
                    --i;
                    r = mk_max(rest[i], r);
                }
                r = normalize(r);
            } else {
                // lhs does not occur in rhs
                r = rhs;
            }

            if (assign(lhs, r, jst)) {
                return Solved;
            } else {
                set_conflict(jst);
                return Failed;
            }
        } else {
            return Continue;
        }
    }

    /** \brief Return solved iff \c c is a constraint of the form
                   lhs =?= max(rest, lhs)
               and is successfully solved.
    */
    status process_l_eq_max(constraint const & c) {
        lean_assert(is_level_eq_cnstr(c));
        level lhs = cnstr_lhs_level(c);
        level rhs = cnstr_rhs_level(c);
        justification jst = c.get_justification();
        if (is_meta(lhs))
            return process_l_eq_max_core(lhs, rhs, jst);
        else if (is_meta(rhs))
            return process_l_eq_max_core(rhs, lhs, jst);
        else
            return Continue;
    }

    /** Auxiliary method for process_succ_eq_max */
    status process_succ_eq_max_core(level const & lhs, level const & rhs, justification const & jst) {
        if (!is_succ(lhs) || !is_max(rhs))
            return Continue;
        level m = rhs;
        while (is_max(m)) {
            level m1 = max_lhs(m);
            level m2 = max_rhs(m);
            if (is_geq(lhs, m1)) {
                m = m2;
            } else if (is_geq(lhs, m2)) {
                m = m1;
            } else {
                return Continue;
            }
        }
        if (!is_meta(m))
            return Continue;
        if (assign(m, lhs, jst)) {
            return Solved;
        } else {
            set_conflict(jst);
            return Failed;
        }
    }

    /** \brief Return Solved iff \c c is a constraint of the form
                  succ^k_1 a =?= max(succ^k_2 b, ?m)
               where k_1 >= k_2 and a == b or b == zero

              and is successfully solved by assigning ?m := succ^k_1 a
    */
    status process_succ_eq_max(constraint const & c) {
        lean_assert(is_level_eq_cnstr(c));
        level lhs = cnstr_lhs_level(c);
        level rhs = cnstr_rhs_level(c);
        justification jst = c.get_justification();
        status st = process_succ_eq_max_core(lhs, rhs, jst);
        if (st != Continue) return st;
        return process_succ_eq_max_core(rhs, lhs, jst);
    }

    /**
       \brief Process the following constraints
          1. (max l1 l2) =?= 0    OR
                 solution:  l1 =?= 0, l2 =?= 0
          2. (imax l1 l2) =?= 0
                 solution:  l2 =?= 0
    */
    status try_level_eq_zero(level const & lhs, level const & rhs, justification const & j) {
        if (!is_zero(rhs))
            return Continue;
        if (is_max(lhs)) {
            if (process_constraint(mk_level_eq_cnstr(max_lhs(lhs), rhs, j)) &&
                process_constraint(mk_level_eq_cnstr(max_rhs(lhs), rhs, j)))
                return Solved;
            else
                return Failed;
        } else if (is_imax(lhs)) {
            return process_constraint(mk_level_eq_cnstr(imax_rhs(lhs), rhs, j)) ? Solved : Failed;
        } else {
            return Continue;
        }
    }

    status try_level_eq_zero(constraint const & c) {
        lean_assert(is_level_eq_cnstr(c));
        level const & lhs = cnstr_lhs_level(c);
        level const & rhs = cnstr_rhs_level(c);
        justification const & j = c.get_justification();
        status st = try_level_eq_zero(lhs, rhs, j);
        if (st != Continue) return st;
        return try_level_eq_zero(rhs, lhs, j);
    }

    /** \brief Try to solve constraints of the form
              (?m1 =?= max ?m2 ?m3)
              (?m1 =?= max ?m2 ?m3)
        by assigning ?m1 =?= ?m2 and ?m1 =?= ?m3

        \remark we may miss solutions.
    */
    status try_merge_max_core(level const & lhs, level const & rhs, justification const & j) {
        level m1 = lhs;
        level m2, m3;
        if (is_max(rhs)) {
            m2 = max_lhs(rhs);
            m3 = max_rhs(rhs);
        } else if (is_imax(rhs)) {
            m2 = imax_lhs(rhs);
            m3 = imax_rhs(rhs);
        } else {
            return Continue;
        }
        if (process_constraint(mk_level_eq_cnstr(m1, m2, j)) &&
            process_constraint(mk_level_eq_cnstr(m1, m3, j)))
            return Solved;
        else
            return Failed;
    }

    /** \see try_merge_max_core */
    status try_merge_max(constraint const & c) {
        lean_assert(is_level_eq_cnstr(c));
        level const & lhs = cnstr_lhs_level(c);
        level const & rhs = cnstr_rhs_level(c);
        justification const & j = c.get_justification();
        status st = try_merge_max_core(lhs, rhs, j);
        if (st != Continue) return st;
        return try_merge_max_core(rhs, lhs, j);
    }

    /** \brief Process the next constraint in the constraint queue m_cnstrs */
    bool process_next() {
        lean_assert(!m_cnstrs.empty());
        auto const * p = m_cnstrs.min();
        constraint c   = p->first;
        unsigned cidx  = p->second;
        if (cidx >= get_group_first_index(cnstr_group::ClassInstance) &&
            is_choice_cnstr(c) && cnstr_on_demand(c)) {
            // we postpone class-instance constraints whose type still contains metavariables
            m_cnstrs.erase_min();
            postpone(c);
            return true;
        }
        // std::cout << "process_next: " << c << "\n";
        m_cnstrs.erase_min();
        if (is_choice_cnstr(c)) {
            return process_choice_constraint(c);
        } else {
            auto r = instantiate_metavars(c);
            c = r.first;
            bool modified = r.second;
            if (is_level_eq_cnstr(c)) {
                if (modified) {
                    return process_constraint(c);
                }
                status st = process_l_eq_max(c);
                if (st != Continue) return st == Solved;
                st = process_succ_eq_max(c);
                if (st != Continue) return st == Solved;
                if (m_config.m_discard) {
                    // we only try try_level_eq_zero and try_merge_max when we are discarding
                    // constraints that canno be solved.
                    st = try_level_eq_zero(c);
                    if (st != Continue) return st == Solved;
                    if (cidx < get_group_first_index(cnstr_group::FlexFlex)) {
                        add_cnstr(c, cnstr_group::FlexFlex);
                        return true;
                    }
                    st = try_merge_max(c);
                    if (st != Continue) return st == Solved;
                    return process_plugin_constraint(c);
                } else {
                    discard(c);
                    return true;
                }
            } else {
                lean_assert(is_eq_cnstr(c));
                if (is_delta_cnstr(c)) {
                    return process_delta(c);
                } else if (modified) {
                    return is_def_eq(cnstr_lhs_expr(c), cnstr_rhs_expr(c), c.get_justification());
                } else if (auto d = is_owned(c)) {
                    // Metavariable in the constraint is owned by choice constraint.
                    // choice constraint was postponed... since c was not modifed
                    // So, we should postpone this one too.
                    add_cnstr(c, to_cnstr_group(*d+1));
                    return true;
                } else if (is_flex_rigid(c)) {
                    return process_flex_rigid(c);
                } else if (is_flex_flex(c)) {
                    return process_flex_flex(c);
                } else {
                    return process_plugin_constraint(c);
                }
            }
        }
    }

    /** \brief Return true if unifier may be able to produce more solutions */
    bool more_solutions() const {
        return !in_conflict() || !m_case_splits.empty();
    }

    typedef optional<pair<substitution, constraints>> next_result;

    next_result failure() {
        lean_assert(in_conflict());
        if (m_config.m_use_exceptions)
            throw unifier_exception(*m_conflict, m_subst);
        else
            return next_result();
    }

    /** \brief Produce the next solution */
    next_result next() {
        if (!more_solutions())
            return failure();
        if (!m_first && !m_case_splits.empty()) {
            justification all_assumptions;
            for (auto const & cs : m_case_splits)
                all_assumptions = mk_composite1(all_assumptions, mk_assumption_justification(cs->m_assumption_idx));
            set_conflict(all_assumptions);
            if (!resolve_conflict())
                return failure();
        } else if (m_first) {
            m_first = false;
        } else {
            // This is not the first run, and there are no case-splits.
            // We don't throw an exception since there are no more solutions.
            return next_result();
        }

        while (true) {
            if (!in_conflict()) {
                if (m_cnstrs.empty())
                    break;
                process_next();
            }
            if (in_conflict() && !resolve_conflict())
                return failure();
        }
        lean_assert(!in_conflict());
        lean_assert(m_cnstrs.empty());
        substitution s = m_subst;
        s.forget_justifications();
        return next_result(mk_pair(s, m_postponed));
    }
};

unify_result_seq unify(std::shared_ptr<unifier_fn> u) {
    if (!u->more_solutions()) {
        u->failure(); // make sure exception is thrown if u->m_use_exception is true
        return unify_result_seq();
    } else {
        return mk_lazy_list<pair<substitution, constraints>>([=]() {
                auto s = u->next();
                if (s)
                    return some(mk_pair(*s, unify(u)));
                else
                    return unify_result_seq::maybe_pair();
            });
    }
}

unify_result_seq unify(environment const & env,  unsigned num_cs, constraint const * cs, name_generator const & ngen,
                       substitution const & s, unifier_config const & cfg) {
    return unify(std::make_shared<unifier_fn>(env, num_cs, cs, ngen, s, cfg));
}

unify_result_seq unify(environment const & env, expr const & lhs, expr const & rhs, name_generator const & ngen,
                       substitution const & s, unifier_config const & cfg) {
    substitution new_s = s;
    expr _lhs = new_s.instantiate(lhs);
    expr _rhs = new_s.instantiate(rhs);
    auto u = std::make_shared<unifier_fn>(env, 0, nullptr, ngen, new_s, cfg);
    constraint_seq cs;
    if (!u->m_tc->is_def_eq(_lhs, _rhs, justification(), cs) || !u->process_constraints(cs)) {
        return unify_result_seq();
    } else {
        return unify(u);
    }
}

static int unify_simple(lua_State * L) {
    int nargs = lua_gettop(L);
    unify_status r;
    if (nargs == 2)
        r = unify_simple(to_substitution(L, 1), to_constraint(L, 2));
    else if (nargs == 3 && is_expr(L, 2))
        r = unify_simple(to_substitution(L, 1), to_expr(L, 2), to_expr(L, 3), justification());
    else if (nargs == 3 && is_level(L, 2))
        r = unify_simple(to_substitution(L, 1), to_level(L, 2), to_level(L, 3), justification());
    else if (is_expr(L, 2))
        r = unify_simple(to_substitution(L, 1), to_expr(L, 2), to_expr(L, 3), to_justification(L, 4));
    else
        r = unify_simple(to_substitution(L, 1), to_level(L, 2), to_level(L, 3), to_justification(L, 4));
    return push_integer(L, static_cast<unsigned>(r));
}

DECL_UDATA(unify_result_seq)

static const struct luaL_Reg unify_result_seq_m[] = {
    {"__gc", unify_result_seq_gc},
    {0, 0}
};

static int unify_result_seq_next(lua_State * L) {
    unify_result_seq seq = to_unify_result_seq(L, lua_upvalueindex(1));
    unify_result_seq::maybe_pair p;
    p = seq.pull();
    if (p) {
        push_unify_result_seq(L, p->second);
        lua_replace(L, lua_upvalueindex(1));
        push_substitution(L, p->first.first);
        // TODO(Leo): return postponed constraints
    } else {
        lua_pushnil(L);
    }
    return 1;
}

static int push_unify_result_seq_it(lua_State * L, unify_result_seq const & seq) {
    push_unify_result_seq(L, seq);
    lua_pushcclosure(L, &safe_function<unify_result_seq_next>, 1); // create closure with 1 upvalue
    return 1;
}

static void to_constraint_buffer(lua_State * L, int idx, buffer<constraint> & cs) {
    luaL_checktype(L, idx, LUA_TTABLE);
    lua_pushvalue(L, idx); // put table on top of the stack
    int n = objlen(L, idx);
    for (int i = 1; i <= n; i++) {
        lua_rawgeti(L, -1, i);
        cs.push_back(to_constraint(L, -1));
        lua_pop(L, 1);
    }
    lua_pop(L, 1);
}

#if 0
static constraints to_constraints(lua_State * L, int idx) {
    buffer<constraint> cs;
    to_constraint_buffer(L, idx, cs);
    return to_list(cs.begin(), cs.end());
}

static unifier_plugin to_unifier_plugin(lua_State * L, int idx) {
    luaL_checktype(L, idx, LUA_TFUNCTION); // user-fun
    luaref f(L, idx);
    return unifier_plugin([=](constraint const & c, name_generator const & ngen) {
            lua_State * L = f.get_state();
            f.push();
            push_constraint(L, c);
            push_name_generator(L, ngen);
            pcall(L, 2, 1, 0);
            lazy_list<constraints> r;
            if (is_constraint(L, -1)) {
                // single constraint
                r = lazy_list<constraints>(constraints(to_constraint(L, -1)));
            } else if (lua_istable(L, -1)) {
                int num = objlen(L, -1);
                if (num == 0) {
                    // empty table
                    r = lazy_list<constraints>();
                } else {
                    lua_rawgeti(L, -1, 1);
                    if (is_constraint(L, -1)) {
                        // array of constraints case
                        lua_pop(L, 1);
                        r = lazy_list<constraints>(to_constraints(L, -1));
                    } else {
                        lua_pop(L, 1);
                        buffer<constraints> css;
                        // array of array of constraints
                        for (int i = 1; i <= num; i++) {
                            lua_rawgeti(L, -1, i);
                            css.push_back(to_constraints(L, -1));
                            lua_pop(L, 1);
                        }
                        r = to_lazy(to_list(css.begin(), css.end()));
                    }
                }
            } else if (lua_isnil(L, -1)) {
                // nil case
                r = lazy_list<constraints>();
            } else {
                throw exception("invalid unifier plugin, the result value must be a constrant, "
                                "nil, an array of constraints, or an array of arrays of constraints");
            }
            lua_pop(L, 1);
            return r;
        });
}
#endif

static name * g_tmp_prefix = nullptr;

static int unify(lua_State * L) {
    int nargs = lua_gettop(L);
    unify_result_seq r;
    environment const & env = to_environment(L, 1);
    if (is_expr(L, 2)) {
        if (nargs == 6)
            r = unify(env, to_expr(L, 2), to_expr(L, 3), to_name_generator(L, 4), to_substitution(L, 5),
                      unifier_config(to_options(L, 6)));
        else if (nargs == 5)
            r = unify(env, to_expr(L, 2), to_expr(L, 3), to_name_generator(L, 4), to_substitution(L, 5));
        else
            r = unify(env, to_expr(L, 2), to_expr(L, 3), to_name_generator(L, 4));
    } else {
        buffer<constraint> cs;
        to_constraint_buffer(L, 2, cs);
        if (nargs == 5)
            r = unify(env, cs.size(), cs.data(), to_name_generator(L, 3), to_substitution(L, 4), unifier_config(to_options(L, 5)));
        else if (nargs == 4)
            r = unify(env, cs.size(), cs.data(), to_name_generator(L, 3), to_substitution(L, 4));
        else
            r = unify(env, cs.size(), cs.data(), to_name_generator(L, 3));
    }
    return push_unify_result_seq_it(L, r);
}

void open_unifier(lua_State * L) {
    luaL_newmetatable(L, unify_result_seq_mt);
    lua_pushvalue(L, -1);
    lua_setfield(L, -2, "__index");
    setfuncs(L, unify_result_seq_m, 0);
    SET_GLOBAL_FUN(unify_result_seq_pred, "is_unify_result_seq");

    SET_GLOBAL_FUN(unify_simple,  "unify_simple");
    SET_GLOBAL_FUN(unify,         "unify");

    lua_newtable(L);
    SET_ENUM("Solved",       unify_status::Solved);
    SET_ENUM("Failed",       unify_status::Failed);
    SET_ENUM("Unsupported",  unify_status::Unsupported);
    lua_setglobal(L, "unify_status");
}

void initialize_unifier() {
    g_unifier_max_steps         = new name{"unifier", "max_steps"};
    g_unifier_computation       = new name{"unifier", "computation"};
    g_unifier_expensive_classes = new name{"unifier", "expensive_classes"};
    g_unifier_conservative      = new name{"unifier", "conservative"};
    g_unifier_nonchronological  = new name{"unifier", "nonchronological"};

    register_unsigned_option(*g_unifier_max_steps, LEAN_DEFAULT_UNIFIER_MAX_STEPS, "(unifier) maximum number of steps");
    register_bool_option(*g_unifier_computation, LEAN_DEFAULT_UNIFIER_COMPUTATION,
                         "(unifier) always case-split on reduction/computational steps when solving flex-rigid and delta-delta constraints");
    register_bool_option(*g_unifier_expensive_classes, LEAN_DEFAULT_UNIFIER_EXPENSIVE_CLASSES,
                         "(unifier) use \"full\" higher-order unification when solving class instances");
    register_bool_option(*g_unifier_conservative, LEAN_DEFAULT_UNIFIER_CONSERVATIVE,
                         "(unifier) unfolds only constants marked as reducible, avoid expensive case-splits (it is faster but less complete)");
    register_bool_option(*g_unifier_nonchronological, LEAN_DEFAULT_UNIFIER_NONCHRONOLOGICAL,
                         "(unifier) enable/disable nonchronological backtracking in the unifier (this option is only available for debugging and benchmarking purposes, and running experiments)");

    g_dont_care_cnstr = new constraint(mk_eq_cnstr(expr(), expr(), justification()));
    g_tmp_prefix      = new name(name::mk_internal_unique_name());
}

void finalize_unifier() {
    delete g_tmp_prefix;
    delete g_dont_care_cnstr;
    delete g_unifier_max_steps;
    delete g_unifier_computation;
    delete g_unifier_expensive_classes;
    delete g_unifier_conservative;
    delete g_unifier_nonchronological;
}
}