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 "util/interrupt.h"
#include "util/luaref.h"
#include "util/lazy_list_fn.h"
#include "util/sstream.h"
#include "util/lbool.h"
#include "kernel/for_each_fn.h"
#include "kernel/abstract.h"
#include "kernel/instantiate.h"
#include "kernel/type_checker.h"
#include "library/occurs.h"
#include "library/unifier.h"
#include "library/opaque_hints.h"
#include "library/unifier_plugin.h"
#include "library/kernel_bindings.h"

namespace lean {
static name g_unifier_max_steps      {"unifier", "max_steps"};
RegisterUnsignedOption(g_unifier_max_steps, LEAN_DEFAULT_UNIFIER_MAX_STEPS, "(unifier) maximum number of steps");
unsigned get_unifier_max_steps(options const & opts) { return opts.get_unsigned(g_unifier_max_steps, LEAN_DEFAULT_UNIFIER_MAX_STEPS); }

// 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) || std::find(args.begin(), it, *it) != 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) && std::find(locals.begin(), locals.end(), e) == locals.end()) {
                failed = true;
                return false;
            }
            return true;
        });
    return !failed;
}

// 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.
lbool occurs_context_check(expr const & e, expr const & m, buffer<expr> const & locals) {
    expr root = e;
    lbool r = l_true;
    for_each(e, [&](expr const & e, unsigned) {
            if (r == l_false) {
                return false;
            } else if (is_local(e) && std::find(locals.begin(), locals.end(), e) == locals.end()) {
                // right-hand-side contains variable that is not in the scope
                // of metavariable.
                r = l_false;
                return false;
            } else if (is_meta(e)) {
                if (!context_check(e, locals) || occurs(m, e))
                    r = l_undef;
                if (get_app_fn(e) == m)
                    r = l_false;
                return false; // do not visit children
            } else {
                // we only need to continue exploring e if it contains
                // metavariables and/or local constants.
                return has_metavar(e) || has_local(e);
            }
        });
    return r;
}

// Create a lambda abstraction by abstracting the local constants \c locals in \c e
expr lambda_abstract_locals(expr const & e, buffer<expr> const & locals) {
    expr v = abstract_locals(e, locals.size(), locals.data());
    unsigned i = locals.size();
    while (i > 0) {
        --i;
        expr t = abstract_locals(mlocal_type(locals[i]), i, locals.data());
        v = mk_lambda(local_pp_name(locals[i]), t, v);
    }
    return v;
}

static std::pair<unify_status, substitution> unify_simple_core(substitution const & 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 mk_pair(unify_status::Unsupported, s);
    } else {
        switch (occurs_context_check(rhs, *m, args)) {
        case l_false: return mk_pair(unify_status::Failed, s);
        case l_undef: mk_pair(unify_status::Unsupported, s);
        case l_true: {
            expr v = lambda_abstract_locals(rhs, args);
            return mk_pair(unify_status::Solved, s.assign(mlocal_name(*m), v, j));
        }}
    }
    lean_unreachable(); // LCOV_EXCL_LINE
}

std::pair<unify_status, substitution> unify_simple(substitution const & s, expr const & lhs, expr const & rhs, justification const & j) {
    if (lhs == rhs)
        return mk_pair(unify_status::Solved, s);
    else if (!has_metavar(lhs) && !has_metavar(rhs))
        return mk_pair(unify_status::Failed, s);
    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 mk_pair(unify_status::Unsupported, s);
}

// 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;
}

std::pair<unify_status, substitution> unify_simple_core(substitution const & 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 mk_pair(unify_status::Failed, s);
        else
            return mk_pair(unify_status::Unsupported, s);
    }
    return mk_pair(unify_status::Solved, s.assign(meta_id(lhs), rhs, j));
}

std::pair<unify_status, substitution> unify_simple(substitution const & s, level const & lhs, level const & rhs, justification const & j) {
    if (lhs == rhs)
        return mk_pair(unify_status::Solved, s);
    else if (!has_meta(lhs) && !has_meta(rhs))
        return mk_pair(unify_status::Failed, s);
    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 mk_pair(unify_status::Unsupported, s);
}

std::pair<unify_status, substitution> unify_simple(substitution const & 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 mk_pair(unify_status::Unsupported, s);
}


/** \brief Given \c type of the form <tt>(Pi ctx, r)</tt>, return <tt>(Pi ctx, new_range)</tt> */
static expr replace_range(expr const & type, expr const & new_range) {
    if (is_pi(type))
        return update_binding(type, binding_domain(type), replace_range(binding_body(type), new_range));
    else
        return new_range;
}

/** \brief Return the "arity" of the given type. The arity is the number of nested pi-expressions. */
static unsigned get_arity(expr type) {
    unsigned r = 0;
    while (is_pi(type)) {
        type = binding_body(type);
        r++;
    }
    return r;
}

/**
   \brief Given a type \c t of the form
   <tt>Pi (x_1 : A_1) ... (x_n : A_n[x_1, ..., x_{n-1}]), B[x_1, ..., x_n]</tt>
   return a new metavariable \c m1 with type
   <tt>Pi (x_1 : A_1) ... (x_n : A_n[x_1, ..., x_{n-1}]), Type.{u}</tt>
   where \c u is a new universe metavariable.
*/
expr mk_aux_type_metavar_for(name_generator & ngen, expr const & t) {
    expr new_type = replace_range(t, mk_sort(mk_meta_univ(ngen.next())));
    name n        = ngen.next();
    return mk_metavar(n, new_type);
}

/**
   \brief Given a type \c t of the form
   <tt>Pi (x_1 : A_1) ... (x_n : A_n[x_1, ..., x_{n-1}]), B[x_1, ..., x_n]</tt>
   return a new metavariable \c m1 with type
   <tt>Pi (x_1 : A_1) ... (x_n : A_n[x_1, ..., x_{n-1}]), (m2 x_1 ... x_n)</tt>
   where \c m2 is a new metavariable with type
   <tt>Pi (x_1 : A_1) ... (x_n : A_n[x_1, ..., x_{n-1}]), Type.{u}</tt>
   where \c u is a new universe metavariable.
*/
expr mk_aux_metavar_for(name_generator & ngen, expr const & t) {
    unsigned num  = get_arity(t);
    expr r        = mk_app_vars(mk_aux_type_metavar_for(ngen, t), num);
    expr new_type = replace_range(t, r);
    name n        = ngen.next();
    return mk_metavar(n, new_type);
}

static constraint g_dont_care_cnstr = mk_eq_cnstr(expr(), expr(), justification());

/**
    The unifier divides the constraints in 6 groups: Simple, Basic, FlexRigid, Reduction, FlexFlex, DelayedChoice

    1) Simple: constraints that never create case-splits. Example: pattern matching constraints (?M l_1 ... l_n) =?= t.
       The are not even inserted in the constraint priority queue.

    2) Basic: contains user choice constraints used to model coercions and overloaded constraints, and constraints
       that cannot be solved, and the unification plugin must be invoked.

    3) FlexRigid constraints (?M t_1 ... t_n) =?= t, where t_n is not an introduction application

    4) PluginDelayed: contraints delayed by the unifier_plugin. Examples: (elim ... (?m ...)) and (?m ... (intro ...)),
       where elim is an eliminator/recursor and intro is an introduction/constructor.
       This constraints are delayed because after ?m is assigned we may be able to reduce them.

    5) FlexFlex:  (?m1 ...) =?= (?m2 ...) we don't try to solve this constraint, we delay them and hope the other
       ones instantiate ?m1 or ?m2. If this kind of constraint is the next to be processed in the queue, then
       we simply discard it.

    6) DelayedChoice: delayed choice constraints, they are used to model features such as class-instance
       They are really term synthesizers. We use the unifier just to exploit the non-chronological backtracking
       infrastructure
*/
enum class cnstr_group { Basic = 0, FlexRigid, PluginDelayed, FlexFlex, DelayedChoice };
static unsigned g_group_size = 1u << 28;
static unsigned g_cnstr_group_first_index[5] = { 0, g_group_size, 2*g_group_size, 3*g_group_size, 4*g_group_size};
static unsigned get_group_first_index(cnstr_group g) {
    return g_cnstr_group_first_index[static_cast<unsigned>(g)];
}

/** \brief Auxiliary functional object for implementing simultaneous higher-order unification */
struct unifier_fn {
    typedef std::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 rb_map<name, cnstr_idx_set, name_quick_cmp> name_to_cnstrs;
    typedef std::unique_ptr<type_checker> type_checker_ptr;
    environment      m_env;
    name_generator   m_ngen;
    substitution     m_subst;
    unifier_plugin   m_plugin;
    type_checker_ptr m_tc;
    bool             m_use_exception; //!< True if we should throw an exception when there are no more solutions.
    unsigned         m_max_steps;
    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 two maps are indices. The map a metavariable name \c m to the se of all constraint indices that contain \c m.
        We use these indices whenever a metavariable \c m is assigned. In this case, we used these indices to
        remove any constraint that contains \c m from \c m_cnstrs, instantiate \c m, and reprocess them.

        \remark \c m_mvar_occs is for regular metavariables, and \c m_mlvl_occs is for universe metavariables.
    */
    name_to_cnstrs   m_mvar_occs;
    name_to_cnstrs   m_mlvl_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_failed_justifications; // justifications for failed branches
        // snapshot of unifier's state
        substitution     m_subst;
        cnstr_set        m_cnstrs;
        name_to_cnstrs   m_mvar_occs;
        name_to_cnstrs   m_mlvl_occs;

        /** \brief Save unifier's state */
        case_split(unifier_fn & u):
            m_assumption_idx(u.m_next_assumption_idx), m_subst(u.m_subst), m_cnstrs(u.m_cnstrs),
            m_mvar_occs(u.m_mvar_occs), m_mlvl_occs(u.m_mlvl_occs) {
            u.m_next_assumption_idx++;
            u.m_tc->push();
        }

        /** \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_tc->pop();   // restore type checker state
            u.m_tc->push();
            u.m_subst     = m_subst;
            u.m_cnstrs    = m_cnstrs;
            u.m_mvar_occs = m_mvar_occs;
            u.m_mlvl_occs = m_mlvl_occs;
            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>();
        }

        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, lazy_list<constraints> const & tail):case_split(u), 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, list<constraints> const & tail):case_split(u), m_tail(tail) {}
        virtual bool next(unifier_fn & u) { return u.next_simple_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,
               bool use_exception, unsigned max_steps):
        m_env(env), m_ngen(ngen), m_subst(s), m_plugin(get_unifier_plugin(env)),
        m_tc(mk_type_checker_with_hints(env, m_ngen.mk_child())),
        m_use_exception(use_exception), m_max_steps(max_steps), m_num_steps(0) {
        m_next_assumption_idx = 0;
        m_next_cidx = 0;
        m_first     = true;
        for (unsigned i = 0; i < num_cs; i++) {
            process_constraint(cs[i]);
        }
    }

    void check_system() {
        check_interrupted();
        if (m_num_steps > m_max_steps)
            throw exception(sstream() << "unifier maximum number of steps (" << 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()); }

    /**
       \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</tt>
    */
    expr mk_lambda_for(expr const & t, expr const & v) {
        if (is_pi(t)) {
            return mk_lambda(binding_name(t), binding_domain(t), mk_lambda_for(binding_body(t), v), binding_info(t));
        } else {
            return v;
        }
    }

    /**
        \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.

        \remark \c MVar is true if \c m is a regular metavariable, and false if it is a universe metavariable.
    */
    template<bool MVar>
    void add_occ(name const & m, unsigned cidx) {
        cnstr_idx_set s;
        name_to_cnstrs & map = MVar ? m_mvar_occs : m_mlvl_occs;
        auto it = map.find(m);
        if (it)
            s = *it;
        if (!s.contains(cidx)) {
            s.insert(cidx);
            map.insert(m, s);
        }
    }

    /** \see add_occ */
    void add_mvar_occ(name const & m, unsigned cidx) { add_occ<true>(m, cidx); }
    /** \see add_occ */
    void add_mlvl_occ(name const & m, unsigned cidx) { add_occ<false>(m, cidx); }

    /**
        \brief Update the indices \c m_mvar_occs and \c m_mlvl_occs.
        For every metavariable name \c m in \c mlvl_occs and \c mvar_occs, add an entry to \c cidx.

        \remark \c cidx is the index of some constraint in \c m_cnstrs.
    */
    void add_occs(unsigned cidx, name_set const * mlvl_occs, name_set const * mvar_occs) {
        if (mlvl_occs) {
            mlvl_occs->for_each([=](name const & m) {
                    add_mlvl_occ(m, cidx);
                });
        }
        if (mvar_occs) {
            mvar_occs->for_each([=](name const & m) {
                    add_mvar_occ(m, cidx);
                });
        }
    }

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

    bool is_def_eq(expr const & t1, expr const & t2, justification const & j) {
        if (m_tc->is_def_eq(t1, t2, j)) {
            return true;
        } else {
            set_conflict(j);
            return false;
        }
    }

    /**
       \brief Assign \c v to metavariable \c m with justification \c j.
       The type of v and m are inferred, and is_def_eq is invoked.
       Any constraint that contains \c m is revisited.
    */
    bool assign(expr const & m, expr const & v, justification const & j) {
        lean_assert(is_metavar(m));
        m_subst.d_assign(m, v, j);
        expr m_type = mlocal_type(m);
        expr v_type = m_tc->infer(v);
        if (in_conflict())
            return false;
        if (!is_def_eq(m_type, v_type, 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.d_assign(m, v, j);
        auto it = m_mlvl_occs.find(meta_id(m));
        if (it) {
            cnstr_idx_set s = *it;
            m_mlvl_occs.erase(meta_id(m));
            s.for_each([&](unsigned cidx) {
                    process_constraint_cidx(cidx);
                });
            return !in_conflict();
        } else {
            return true;
        }
    }

    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;
        switch (occurs_context_check(rhs, *m, locals)) {
        case l_false:
            set_conflict(j);
            return Failed;
        case l_undef:
            return Continue;
        case l_true:
            lean_assert(!m_subst.is_assigned(*m));
            if (assign(*m, lambda_abstract_locals(rhs, locals), j)) {
                return Solved;
            } else {
                return Failed;
            }
        }
        lean_unreachable(); // LCOV_EXCL_LINE
    }

    optional<declaration> is_delta(expr const & e) { return ::lean::is_delta(m_env, 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));
    }

    /** \brief Process an equality constraints. */
    bool process_eq_constraint(constraint const & c) {
        lean_assert(is_eq_cnstr(c));
        // instantiate assigned metavariables
        name_set unassigned_lvls, unassigned_exprs;
        auto lhs_jst = m_subst.instantiate_metavars(cnstr_lhs_expr(c), &unassigned_lvls, &unassigned_exprs);
        auto rhs_jst = m_subst.instantiate_metavars(cnstr_rhs_expr(c), &unassigned_lvls, &unassigned_exprs);
        expr lhs = lhs_jst.first;
        expr rhs = rhs_jst.first;

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

        // Update justification using the justification of the instantiated metavariables
        justification new_jst = mk_composite1(mk_composite1(c.get_justification(), lhs_jst.second), rhs_jst.second);
        if (!has_metavar(lhs) && !has_metavar(rhs)) {
            return is_def_eq(lhs, rhs, new_jst);
        }

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

        // If lhs or rhs were updated, then invoke is_def_eq again.
        if (lhs != cnstr_lhs_expr(c) || rhs != cnstr_rhs_expr(c)) {
            // some metavariables were instantiated, try is_def_eq again
            return is_def_eq(lhs, rhs, new_jst);
        }

        if (is_eq_deltas(lhs, rhs)) {
            // we need to create a backtracking point for this one
            add_cnstr(c, &unassigned_lvls, &unassigned_exprs, cnstr_group::Basic);
        } else if (m_plugin->delay_constraint(*m_tc, c)) {
            add_cnstr(c, &unassigned_lvls, &unassigned_exprs, cnstr_group::PluginDelayed);
        } else if (is_meta(lhs) && is_meta(rhs)) {
            // flex-flex constraints are delayed the most.
            add_cnstr(c, &unassigned_lvls, &unassigned_exprs, cnstr_group::FlexFlex);
        } else if (is_meta(lhs) || is_meta(rhs)) {
            // flex-rigid constraints are delayed.
            add_cnstr(c, &unassigned_lvls, &unassigned_exprs, cnstr_group::FlexRigid);
        } else {
            // this constraints require the unifier plugin to be solved
            add_cnstr(c, &unassigned_lvls, &unassigned_exprs, 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))
                return Failed;
            else
                return Continue;
        }
        lean_assert(!m_subst.is_assigned(lhs));
        if (assign(lhs, rhs, j)) {
            return Solved;
        } else {
            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
        name_set unassigned_lvls;
        auto lhs_jst = m_subst.instantiate_metavars(cnstr_lhs_level(c), &unassigned_lvls);
        auto rhs_jst = m_subst.instantiate_metavars(cnstr_rhs_level(c), &unassigned_lvls);
        level lhs = lhs_jst.first;
        level rhs = rhs_jst.first;

        // 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

        justification new_jst = mk_composite1(mk_composite1(c.get_justification(), lhs_jst.second), rhs_jst.second);
        if (!has_meta(lhs) && !has_meta(rhs)) {
            set_conflict(new_jst);
            return false; // trivial failure
        }

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

        if (lhs != cnstr_lhs_level(c) || rhs != cnstr_rhs_level(c)) {
            constraint new_c = mk_level_eq_cnstr(lhs, rhs, new_jst);
            add_cnstr(new_c, &unassigned_lvls, nullptr, cnstr_group::FlexRigid);
        } else {
            add_cnstr(c, &unassigned_lvls, nullptr, cnstr_group::FlexRigid);
        }

        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_system();
        switch (c.kind()) {
        case constraint_kind::Choice:
            // Choice constraints are never considered easy.
            if (cnstr_delayed(c))
                add_cnstr(c, nullptr, nullptr, cnstr_group::DelayedChoice);
            else
                add_cnstr(c, nullptr, nullptr, cnstr_group::Basic);
            return true;
        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_simple_formatter(), options(), nullptr, m_subst) << "\n";
        if (j.is_composite()) {
            display(out, composite_child1(j), indent+2);
            display(out, composite_child2(j), indent+2);
        }
    }

    bool resolve_conflict() {
        lean_assert(in_conflict());
        // Remark: we must save the value of m_conflict because d->next(*this) may change it.
        justification conflict = *m_conflict;
        while (!m_case_splits.empty()) {
            std::unique_ptr<case_split> & d = m_case_splits.back();
            if (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;
                }
            }
            m_tc->pop();
            m_case_splits.pop_back();
        }
        return false;
    }

    optional<substitution> failure() {
        lean_assert(in_conflict());
        if (m_use_exception)
            throw unifier_exception(*m_conflict, m_subst);
        else
            return optional<substitution>();
    }

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

    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_assumption_justification(cs.m_assumption_idx));
        } else {
            // update conflict
            update_conflict(mk_composite1(*m_conflict, cs.m_failed_justifications));
            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, r->second)));
                return process_constraints(r->first, mk_composite1(j, a));
            }
        } else {
            set_conflict(j);
            return false;
        }
    }

    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());
        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);
        auto m_type_jst             = m_subst.instantiate_metavars(m_tc->infer(m), nullptr, nullptr);
        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_assumption_justification(cs.m_assumption_idx));
        } else {
            // update conflict
            update_conflict(mk_composite1(*m_conflict, cs.m_failed_justifications));
            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(lhs_fn);
        if (lhs_args.size() != rhs_args.size() ||
            length(lhs_lvls) != length(rhs_lvls) ||
            length(d.get_univ_params()) != length(lhs_lvls)) {
            // the constraint is not well-formed, this can happen when users are abusing the API
            return false;
        }
        buffer<constraint> cs_buffer;
        while (!is_nil(lhs_lvls)) {
            cs_buffer.push_back(mk_level_eq_cnstr(head(lhs_lvls), head(rhs_lvls), j));
            lhs_lvls = tail(lhs_lvls);
            rhs_lvls = tail(rhs_lvls);
        }
        unsigned i = lhs_args.size();
        while (i > 0) {
            --i;
            cs_buffer.push_back(mk_eq_cnstr(lhs_args[i], rhs_args[i], j));
        }
        constraints cs1  = to_list(cs_buffer.begin(), cs_buffer.end());

        expr lhs_fn_val = instantiate_univ_params(d.get_value(), d.get_univ_params(), const_levels(lhs_fn));
        expr rhs_fn_val = instantiate_univ_params(d.get_value(), d.get_univ_params(), 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());
        constraints cs2(mk_eq_cnstr(t, s, j));

        justification a = mk_assumption_justification(m_next_assumption_idx);
        add_case_split(std::unique_ptr<case_split>(new simple_case_split(*this, list<constraints>(cs2))));
        return process_constraints(cs1, a);
    }

    /** \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 Given
               m      := a metavariable ?m
               margs  := [a_1 ... a_k]
               rhs    := (g 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), f (?m_1 x_1 ... x_k) ... (?m_n x_1 ... x_k)

       Remark: The term f is:
                  - g (if g is a constant), OR
                  - variable (if g is a local constant equal to a_i)
    */
    constraints mk_flex_rigid_app_cnstrs(expr const & m, buffer<expr> const & margs,
                                         expr const & f, expr const & rhs, justification const & j) {
        lean_assert(is_metavar(m));
        lean_assert(is_app(rhs));
        lean_assert(is_constant(f) || is_var(f));
        buffer<constraint> cs;
        expr const & mtype = mlocal_type(m);
        buffer<expr> rargs;
        get_app_args(rhs, rargs);
        buffer<expr> sargs;
        for (expr const & rarg : rargs) {
            expr maux = mk_aux_metavar_for(m_ngen, mtype);
            cs.push_back(mk_eq_cnstr(mk_app(maux, margs), rarg, j));
            sargs.push_back(mk_app_vars(maux, margs.size()));
        }
        expr v = mk_app(f, sargs);
        v = mk_lambda_for(mtype, v);
        cs.push_back(mk_eq_cnstr(m, v, j));
        return to_list(cs.begin(), cs.end());
    }

    /**
       \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.
    */
    constraints mk_bindings_imitation(expr const & m, buffer<expr> const & margs, expr const & rhs, justification const & j) {
        lean_assert(is_metavar(m));
        lean_assert(is_binding(rhs));
        expr const & mtype = mlocal_type(m);
        expr maux1  = mk_aux_metavar_for(m_ngen, mtype);
        buffer<constraint> cs;
        cs.push_back(mk_eq_cnstr(mk_app(maux1, margs), binding_domain(rhs), j));
        expr dontcare;
        expr tmp_pi = mk_pi(binding_name(rhs), mk_app_vars(maux1, margs.size()), dontcare); // trick for "extending" the context
        expr mtype2 = replace_range(mtype, tmp_pi); // trick for "extending" the context
        expr maux2  = mk_aux_metavar_for(m_ngen, mtype2);
        expr new_local = mk_local(m_ngen.next(), binding_name(rhs), binding_domain(rhs), binding_info(rhs));
        cs.push_back(mk_eq_cnstr(mk_app(mk_app(maux2, margs), new_local), instantiate(binding_body(rhs), new_local), j));
        expr v = update_binding(rhs, mk_app_vars(maux1, margs.size()), mk_app_vars(maux2, margs.size() + 1));
        v = mk_lambda_for(mtype, v);
        cs.push_back(mk_eq_cnstr(m, v, j));
        return to_list(cs.begin(), cs.end());
    }

    /**
       \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
    */
    constraints mk_simple_imitation(expr const & m, expr const & rhs, justification const & j) {
        lean_assert(is_metavar(m));
        lean_assert(is_sort(rhs) || is_constant(rhs));
        expr const & mtype = mlocal_type(m);
        expr v = mk_lambda_for(mtype, rhs);
        return constraints(mk_eq_cnstr(m, v, j));
    }

    /**
       \brief Given
               m      := a metavariable ?m
               margs  := [a_1 ... a_k]
               rhs    := M(b_1 ... b_n) where M is a macro with arguments 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), M((?m_1 x_1 ... x_k) ... (?m_n x_1 ... x_k))
    */
    constraints mk_macro_imitation(expr const & m, buffer<expr> const & margs, expr const & rhs, justification const & j) {
        lean_assert(is_metavar(m));
        lean_assert(is_macro(rhs));
        expr const & mtype = mlocal_type(m);
        buffer<constraint> cs;
        // create an auxiliary metavariable for each macro argument
        buffer<expr> sargs;
        for (unsigned i = 0; i < macro_num_args(rhs); i++) {
            expr maux = mk_aux_metavar_for(m_ngen, mtype);
            cs.push_back(mk_eq_cnstr(mk_app(maux, margs), macro_arg(rhs, i), j));
            sargs.push_back(mk_app_vars(maux, margs.size()));
        }
        expr v = mk_macro(macro_def(rhs), sargs.size(), sargs.data());
        v = mk_lambda_for(mtype, v);
        cs.push_back(mk_eq_cnstr(m, v, j));
        return to_list(cs.begin(), cs.end());
    }

    /**
       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 add_simple_projections(expr const & m, buffer<expr> const & margs, expr const & rhs, justification const & j,
                                buffer<constraints> & alts) {
        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
                constraint c1 = mk_eq_cnstr(marg, rhs, j);
                constraint c2 = mk_eq_cnstr(m, mk_lambda_for(mtype, mk_var(vidx)), j);
                alts.push_back(constraints({c1, c2}));
            } 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 c1 = mk_eq_cnstr(m, mk_lambda_for(mtype, mk_var(vidx)), j);
                alts.push_back(constraints(c1));
            }
        }
    }

    /** \brief Process a flex rigid constraint */
    bool process_flex_rigid(expr const & lhs, expr const & rhs, justification const & j) {
        lean_assert(is_meta(lhs));
        lean_assert(!is_meta(rhs));
        buffer<expr> margs;
        expr m     = get_app_args(lhs, margs);
        for (expr & marg : margs)
            marg = m_tc->whnf(marg);
        buffer<constraints> alts;
        switch (rhs.kind()) {
        case expr_kind::Var: case expr_kind::Meta:
            lean_unreachable(); // LCOV_EXCL_LINE
        case expr_kind::Local:
            add_simple_projections(m, margs, rhs, j, alts);
            break;
        case expr_kind::Sort: case expr_kind::Constant:
            add_simple_projections(m, margs, rhs, j, alts);
            alts.push_back(mk_simple_imitation(m, rhs, j));
            break;
        case expr_kind::Pi: case expr_kind::Lambda:
            add_simple_projections(m, margs, rhs, j, alts);
            alts.push_back(mk_bindings_imitation(m, margs, rhs, j));
            break;
        case expr_kind::Macro:
            add_simple_projections(m, margs, rhs, j, alts);
            alts.push_back(mk_macro_imitation(m, margs, rhs, j));
            break;
        case expr_kind::App: {
            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) {
                    unsigned vidx = margs.size() - i;
                    --i;
                    expr const & marg = margs[i];
                    if (is_local(marg) && mlocal_name(marg) == mlocal_name(f))
                        alts.push_back(mk_flex_rigid_app_cnstrs(m, margs, mk_var(vidx), rhs, j));
                }
            } else if (is_constant(f)) {
                add_simple_projections(m, margs, rhs, j, alts);
                alts.push_back(mk_flex_rigid_app_cnstrs(m, margs, f, rhs, j));
            } else {
                lean_unreachable(); // LCOV_EXCL_LINE
            }
            break;
        }}

        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, 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));
        if (is_meta(cnstr_lhs_expr(c)))
            return process_flex_rigid(cnstr_lhs_expr(c), cnstr_rhs_expr(c), c.get_justification());
        else
            return process_flex_rigid(cnstr_rhs_expr(c), cnstr_lhs_expr(c), c.get_justification());
    }

    bool process_flex_flex(constraint const &) {
        // We just ignore flex-flex constraints.
        // This kind of constraint does not occur very often.
        return true;
    }

    void consume_tc_cnstrs() {
        while (true) {
            if (in_conflict())
                return;
            if (auto c = m_tc->next_cnstr()) {
                process_constraint(*c);
            } else {
                break;
            }
        }
    }

    /** \brief Process the next constraint in the constraint queue m_cnstrs */
    bool process_next() {
        lean_assert(!m_cnstrs.empty());
        constraint c = m_cnstrs.min()->first;
        m_cnstrs.erase_min();
        if (is_choice_cnstr(c))
            return process_choice_constraint(c);
        else if (is_delta_cnstr(c))
            return process_delta(c);
        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();
    }

    /** \brief Produce the next solution */
    optional<substitution> 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 optional<substitution>();
        }
        while (true) {
            consume_tc_cnstrs();
            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());
        return optional<substitution>(m_subst.forget_justifications());
    }
};

lazy_list<substitution> 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 lazy_list<substitution>();
    } else {
        return mk_lazy_list<substitution>([=]() {
                auto s = u->next();
                if (s)
                    return some(mk_pair(*s, unify(u)));
                else
                    return lazy_list<substitution>::maybe_pair();
            });
    }
}

lazy_list<substitution> unify(environment const & env,  unsigned num_cs, constraint const * cs, name_generator const & ngen,
                              bool use_exception, unsigned max_steps) {
    return unify(std::make_shared<unifier_fn>(env, num_cs, cs, ngen, substitution(), use_exception, max_steps));
}

lazy_list<substitution> unify(environment const & env,  unsigned num_cs, constraint const * cs, name_generator const & ngen,
                              bool use_exception, options const & o) {
    return unify(env, num_cs, cs, ngen, use_exception, get_unifier_max_steps(o));
}

lazy_list<substitution> unify(environment const & env, expr const & lhs, expr const & rhs, name_generator const & ngen, substitution const & s,
                              unsigned max_steps) {
    auto u = std::make_shared<unifier_fn>(env, 0, nullptr, ngen, s, false, max_steps);
    expr _lhs = s.instantiate(lhs);
    expr _rhs = s.instantiate(rhs);
    if (!u->m_tc->is_def_eq(_lhs, _rhs))
        return lazy_list<substitution>();
    else
        return unify(u);
}

lazy_list<substitution> unify(environment const & env, expr const & lhs, expr const & rhs, name_generator const & ngen,
                              substitution const & s, options const & o) {
    return unify(env, lhs, rhs, ngen, s, get_unifier_max_steps(o));
}

static int unify_simple(lua_State * L) {
    int nargs = lua_gettop(L);
    std::pair<unify_status, substitution> 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));
    push_integer(L, static_cast<unsigned>(r.first));
    push_substitution(L, r.second);
    return 2;
}

typedef lazy_list<substitution> substitution_seq;
DECL_UDATA(substitution_seq)

static const struct luaL_Reg substitution_seq_m[] = {
    {"__gc", substitution_seq_gc},
    {0, 0}
};

static int substitution_seq_next(lua_State * L) {
    substitution_seq seq = to_substitution_seq(L, lua_upvalueindex(1));
    substitution_seq::maybe_pair p;
    p = seq.pull();
    if (p) {
        push_substitution_seq(L, p->second);
        lua_replace(L, lua_upvalueindex(1));
        push_substitution(L, p->first);
    } else {
        lua_pushnil(L);
    }
    return 1;
}

static int push_substitution_seq_it(lua_State * L, substitution_seq const & seq) {
    push_substitution_seq(L, seq);
    lua_pushcclosure(L, &safe_function<substitution_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 = name::mk_internal_unique_name();

static int unify(lua_State * L) {
    int nargs = lua_gettop(L);
    lazy_list<substitution> 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), to_options(L, 6));
        else
            r = unify(env, to_expr(L, 2), to_expr(L, 3), to_name_generator(L, 4), to_substitution(L, 5), options());
    } else {
        buffer<constraint> cs;
        to_constraint_buffer(L, 2, cs);
        if (nargs == 4)
            r = unify(env, cs.size(), cs.data(), to_name_generator(L, 3), false, to_options(L, 4));
        else
            r = unify(env, cs.size(), cs.data(), to_name_generator(L, 3), false, options());
    }
    return push_substitution_seq_it(L, r);
}

void open_unifier(lua_State * L) {
    luaL_newmetatable(L, substitution_seq_mt);
    lua_pushvalue(L, -1);
    lua_setfield(L, -2, "__index");
    setfuncs(L, substitution_seq_m, 0);
    SET_GLOBAL_FUN(substitution_seq_pred, "is_substitution_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");
}
}