lean2/src/frontends/lean/elaborator.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 <vector>
#include "util/flet.h"
#include "util/list_fn.h"
#include "util/lazy_list_fn.h"
#include "util/sstream.h"
#include "util/name_map.h"
#include "kernel/abstract.h"
#include "kernel/instantiate.h"
#include "kernel/for_each_fn.h"
#include "kernel/find_fn.h"
#include "kernel/replace_fn.h"
#include "kernel/kernel_exception.h"
#include "kernel/error_msgs.h"
#include "kernel/free_vars.h"
#include "library/coercion.h"
#include "library/placeholder.h"
#include "library/choice.h"
#include "library/explicit.h"
#include "library/reducible.h"
#include "library/locals.h"
#include "library/let.h"
#include "library/sorry.h"
#include "library/flycheck.h"
#include "library/deep_copy.h"
#include "library/metavar_closure.h"
#include "library/typed_expr.h"
#include "library/local_context.h"
#include "library/util.h"
#include "library/tactic/expr_to_tactic.h"
#include "library/error_handling/error_handling.h"
#include "library/definitional/equations.h"
#include "frontends/lean/local_decls.h"
#include "frontends/lean/class.h"
#include "frontends/lean/tactic_hint.h"
#include "frontends/lean/info_manager.h"
#include "frontends/lean/info_annotation.h"
#include "frontends/lean/choice_iterator.h"
#include "frontends/lean/elaborator.h"
#include "frontends/lean/placeholder_elaborator.h"
#include "frontends/lean/proof_qed_elaborator.h"
#include "frontends/lean/calc_proof_elaborator.h"
#include "frontends/lean/info_tactic.h"
#include "frontends/lean/pp_options.h"
#include "frontends/lean/begin_end_ext.h"
#include "frontends/lean/elaborator_exception.h"
#include "frontends/lean/calc.h"

namespace lean {
/** \brief 'Choice' expressions <tt>(choice e_1 ... e_n)</tt> are mapped into a metavariable \c ?m
    and a choice constraints <tt>(?m in fn)</tt> where \c fn is a choice function.
    The choice function produces a stream of alternatives. In this case, it produces a stream of
    size \c n, one alternative for each \c e_i.
    This is a helper class for implementing this choice functions.
*/
struct elaborator::choice_expr_elaborator : public choice_iterator {
    elaborator &  m_elab;
    local_context m_context;
    local_context m_full_context;
    expr          m_meta;
    expr          m_choice;
    unsigned      m_idx;
    bool          m_relax_main_opaque;
    choice_expr_elaborator(elaborator & elab, local_context const & ctx, local_context const & full_ctx,
                           expr const & meta, expr const & c, bool relax):
        m_elab(elab), m_context(ctx), m_full_context(full_ctx), m_meta(meta), m_choice(c),
        m_idx(get_num_choices(m_choice)),
        m_relax_main_opaque(relax) {
    }

    virtual optional<constraints> next() {
        while (m_idx > 0) {
            --m_idx;
            expr const & c = get_choice(m_choice, m_idx);
            expr const & f = get_app_fn(c);
            m_elab.save_identifier_info(f);
            try {
                flet<local_context> set1(m_elab.m_context,      m_context);
                flet<local_context> set2(m_elab.m_full_context, m_full_context);
                pair<expr, constraint_seq> rcs = m_elab.visit(c);
                expr r = rcs.first;
                constraint_seq cs = mk_eq_cnstr(m_meta, r, justification(), m_relax_main_opaque) + rcs.second;
                return optional<constraints>(cs.to_list());
            } catch (exception &) {}
        }
        return optional<constraints>();
    }
};

elaborator::elaborator(elaborator_context & ctx, name_generator const & ngen, bool nice_mvar_names):
    m_ctx(ctx),
    m_ngen(ngen),
    m_context(),
    m_full_context(),
    m_unifier_config(ctx.m_ios.get_options(), true /* use exceptions */, true /* discard */) {
    m_has_sorry = has_sorry(m_ctx.m_env);
    m_relax_main_opaque = false;
    m_use_tactic_hints  = true;
    m_no_info           = false;
    m_in_equation_lhs   = false;
    m_tc[0]             = mk_type_checker(ctx.m_env, m_ngen.mk_child(), false);
    m_tc[1]             = mk_type_checker(ctx.m_env, m_ngen.mk_child(), true);
    m_nice_mvar_names   = nice_mvar_names;
}

expr elaborator::mk_local(name const & n, expr const & t, binder_info const & bi) {
    return ::lean::mk_local(m_ngen.next(), n, t, bi);
}

void elaborator::register_meta(expr const & meta) {
    lean_assert(is_meta(meta));
    name const & n = mlocal_name(get_app_fn(meta));
    m_mvar2meta.insert(n, meta);
    if (m_relax_main_opaque)
        m_relaxed_mvars.insert(n);
}

/** \brief Convert the metavariable to the metavariable application that captures
    the context where it was defined.
*/
optional<expr> elaborator::mvar_to_meta(expr const & mvar) {
    lean_assert(is_metavar(mvar));
    name const & m = mlocal_name(mvar);
    if (auto it = m_mvar2meta.find(m))
        return some_expr(*it);
    else
        return none_expr();
}

/** \brief Store the pair (pos(e), type(r)) in the info_data if m_info_manager is available. */
void elaborator::save_type_data(expr const & e, expr const & r) {
    if (!m_no_info && infom() && pip() &&
        (is_constant(e) || is_local(e) || is_placeholder(e) || is_as_atomic(e) ||
         is_consume_args(e) || is_notation_info(e))) {
        if (auto p = pip()->get_pos_info(e)) {
            expr t = m_tc[m_relax_main_opaque]->infer(r).first;
            m_pre_info_data.add_type_info(p->first, p->second, t);
        }
    }
}

/** \brief Store the pair (pos(e), r) in the info_data if m_info_manager is available. */
void elaborator::save_binder_type(expr const & e, expr const & r) {
    if (!m_no_info && infom() && pip()) {
        if (auto p = pip()->get_pos_info(e)) {
            m_pre_info_data.add_type_info(p->first, p->second, r);
        }
    }
}

/** \brief Store type information at pos(e) for r if \c e is marked as "extra" in the info_manager */
void elaborator::save_extra_type_data(expr const & e, expr const & r) {
    if (!m_no_info && infom() && pip()) {
        if (auto p = pip()->get_pos_info(e)) {
            expr t = m_tc[m_relax_main_opaque]->infer(r).first;
            m_pre_info_data.add_extra_type_info(p->first, p->second, r, t);
        }
    }
}

/** \brief Store proof_state information at pos(e) in the info_manager */
void elaborator::save_proof_state_info(proof_state const & ps, expr const & e) {
    if (!m_no_info && infom() && pip()) {
        if (auto p = pip()->get_pos_info(e)) {
            m_pre_info_data.add_proof_state_info(p->first, p->second, ps);
        }
    }
}

/** \brief Auxiliary function for saving information about which overloaded identifier was used by the elaborator. */
void elaborator::save_identifier_info(expr const & f) {
    if (!m_no_info && infom() && pip() && is_constant(f)) {
        if (auto p = pip()->get_pos_info(f))
            m_pre_info_data.add_identifier_info(p->first, p->second, const_name(f));
    }
}

/** \brief Store actual term that was synthesized for an explicit placeholders */
void elaborator::save_synth_data(expr const & e, expr const & r) {
    if (!m_no_info && infom() && pip() && is_placeholder(e)) {
        if (auto p = pip()->get_pos_info(e))
            m_pre_info_data.add_synth_info(p->first, p->second, r);
    }
}

void elaborator::save_placeholder_info(expr const & e, expr const & r) {
    if (is_explicit_placeholder(e)) {
        save_type_data(e, r);
        save_synth_data(e, r);
    }
}

/** \brief Auxiliary function for saving information about which coercion was used by the elaborator.
    It marks that coercion c was used on e.
*/
void elaborator::save_coercion_info(expr const & e, expr const & c) {
    if (!m_no_info && infom() && pip()) {
        if (auto p = pip()->get_pos_info(e)) {
            expr t = m_tc[m_relax_main_opaque]->infer(c).first;
            m_pre_info_data.add_coercion_info(p->first, p->second, c, t);
        }
    }
}

/** \brief Remove coercion information associated with \c e */
void elaborator::erase_coercion_info(expr const & e) {
    if (!m_no_info && infom() && pip()) {
        if (auto p = pip()->get_pos_info(e))
            m_pre_info_data.erase_coercion_info(p->first, p->second);
    }
}

void elaborator::copy_info_to_manager(substitution s) {
    if (!infom())
        return;
    m_pre_info_data.instantiate(s);
    bool overwrite = true;
    infom()->merge(m_pre_info_data, overwrite);
    m_pre_info_data.clear();
}

optional<name> elaborator::mk_mvar_suffix(expr const & b) {
    if (!infom() && !m_nice_mvar_names)
        return optional<name>();
    else
        return optional<name>(binding_name(b));
}

/** \brief Create a metavariable, and attach choice constraint for generating
    solutions using class-instances and tactic-hints.
*/
expr elaborator::mk_placeholder_meta(optional<name> const & suffix, optional<expr> const & type,
                                     tag g, bool is_strict, bool is_inst_implicit, constraint_seq & cs) {
    if (is_inst_implicit && !m_ctx.m_ignore_instances) {
        auto ec = mk_placeholder_elaborator(env(), ios(), m_context,
                                            m_ngen.next(), suffix, m_relax_main_opaque, use_local_instances(),
                                            is_strict, type, g, m_unifier_config, m_ctx.m_pos_provider);
        register_meta(ec.first);
        cs += ec.second;
        return ec.first;
    } else {
        expr m = m_context.mk_meta(m_ngen, suffix, type, g);
        register_meta(m);
        return m;
    }
}

expr elaborator::visit_expecting_type(expr const & e, constraint_seq & cs) {
    if (is_placeholder(e) && !placeholder_type(e)) {
        expr r = m_context.mk_type_meta(m_ngen, e.get_tag());
        save_placeholder_info(e, r);
        return r;
    } else {
        return visit(e, cs);
    }
}

expr elaborator::visit_expecting_type_of(expr const & e, expr const & t, constraint_seq & cs) {
    if (is_placeholder(e) && !placeholder_type(e)) {
        bool inst_imp = true;
        expr r = mk_placeholder_meta(some_expr(t), e.get_tag(), is_strict_placeholder(e), inst_imp, cs);
        save_placeholder_info(e, r);
        return r;
    } else if (is_choice(e)) {
        return visit_choice(e, some_expr(t), cs);
    } else if (is_by(e)) {
        return visit_by(e, some_expr(t), cs);
    } else if (is_calc_annotation(e)) {
        return visit_calc_proof(e, some_expr(t), cs);
    } else if (is_proof_qed_annotation(e)) {
        return visit_proof_qed(e, some_expr(t), cs);
    } else {
        return visit(e, cs);
    }
}

expr elaborator::visit_choice(expr const & e, optional<expr> const & t, constraint_seq & cs) {
    lean_assert(is_choice(e));
    // Possible optimization: try to lookahead and discard some of the alternatives.
    expr m                 = m_full_context.mk_meta(m_ngen, t, e.get_tag());
    register_meta(m);
    bool relax             = m_relax_main_opaque;
    local_context ctx      = m_context;
    local_context full_ctx = m_full_context;
    auto fn = [=](expr const & meta, expr const & /* type */, substitution const & /* s */,
                  name_generator const & /* ngen */) {
        return choose(std::make_shared<choice_expr_elaborator>(*this, ctx, full_ctx, meta, e, relax));
    };
    justification j = mk_justification("none of the overloads is applicable", some_expr(e));
    cs += mk_choice_cnstr(m, fn, to_delay_factor(cnstr_group::Basic), true, j, m_relax_main_opaque);
    return m;
}

expr elaborator::visit_by(expr const & e, optional<expr> const & t, constraint_seq & cs) {
    lean_assert(is_by(e));
    expr tac = visit(get_by_arg(e), cs);
    expr m   = m_context.mk_meta(m_ngen, t, e.get_tag());
    register_meta(m);
    m_local_tactic_hints.insert(mlocal_name(get_app_fn(m)), tac);
    return m;
}

expr elaborator::visit_calc_proof(expr const & e, optional<expr> const & t, constraint_seq & cs) {
    lean_assert(is_calc_annotation(e));
    info_manager * im = nullptr;
    if (infom())
        im = &m_pre_info_data;
    pair<expr, constraint_seq> ecs = visit(get_annotation_arg(e));
    expr m                         = m_full_context.mk_meta(m_ngen, t, e.get_tag());
    register_meta(m);
    auto fn = [=](expr const & t) { save_type_data(get_annotation_arg(e), t); };
    constraint c                   = mk_calc_proof_cnstr(env(), ios().get_options(),
                                                         m_context, m, ecs.first, ecs.second, m_unifier_config,
                                                         im, m_relax_main_opaque, fn);
    cs += c;
    return m;
}

expr elaborator::visit_proof_qed(expr const & e, optional<expr> const & t, constraint_seq & cs) {
    lean_assert(is_proof_qed_annotation(e));
    info_manager * im = nullptr;
    if (infom())
        im = &m_pre_info_data;
    pair<expr, constraint_seq> ecs = visit(get_annotation_arg(e));
    expr m                         = m_full_context.mk_meta(m_ngen, t, e.get_tag());
    register_meta(m);
    constraint c                   = mk_proof_qed_cnstr(env(), m, ecs.first, ecs.second, m_unifier_config,
                                                        im, m_relax_main_opaque);
    cs += c;
    return m;
}

static bool is_implicit_pi(expr const & e) {
    if (!is_pi(e))
        return false;
    binder_info bi = binding_info(e);
    return bi.is_strict_implicit() || bi.is_implicit() || bi.is_inst_implicit();
}

/** \brief Auxiliary function for adding implicit arguments to coercions to function-class */
expr elaborator::add_implict_args(expr e, constraint_seq & cs, bool relax) {
    type_checker & tc = *m_tc[relax];
    constraint_seq new_cs;
    expr type = tc.whnf(tc.infer(e, new_cs), new_cs);
    if (!is_implicit_pi(type))
        return e;
    cs += new_cs;
    while (true) {
        lean_assert(is_pi(type));
        tag g = e.get_tag();
        bool is_strict = true;
        bool inst_imp  = binding_info(type).is_inst_implicit();
        expr imp_arg   = mk_placeholder_meta(mk_mvar_suffix(type), some_expr(binding_domain(type)),
                                             g, is_strict, inst_imp, cs);
        e              = mk_app(e, imp_arg, g);
        type           = instantiate(binding_body(type), imp_arg);
        constraint_seq new_cs;
        type           = tc.whnf(type, new_cs);
        if (!is_implicit_pi(type))
            return e;
        cs += new_cs;
    }
}

/** \brief Make sure \c f is really a function, if it is not, try to apply coercions.
    The result is a pair <tt>new_f, f_type</tt>, where new_f is the new value for \c f,
    and \c f_type is its type (and a Pi-expression)
*/
pair<expr, expr> elaborator::ensure_fun(expr f, constraint_seq & cs) {
    expr f_type = infer_type(f, cs);
    if (!is_pi(f_type))
        f_type = whnf(f_type, cs);
    if (!is_pi(f_type) && has_metavar(f_type)) {
        constraint_seq saved_cs = cs;
        expr new_f_type = whnf(f_type, cs);
        if (!is_pi(new_f_type) && m_tc[m_relax_main_opaque]->is_stuck(new_f_type)) {
            cs = saved_cs;
            // let type checker add constraint
            f_type = m_tc[m_relax_main_opaque]->ensure_pi(f_type, f, cs);
        } else {
            f_type = new_f_type;
        }
    }
    if (!is_pi(f_type)) {
        // try coercion to function-class
        list<expr> coes  = get_coercions_to_fun(env(), f_type);
        if (is_nil(coes)) {
            throw_kernel_exception(env(), f, [=](formatter const & fmt) { return pp_function_expected(fmt, f); });
        } else if (is_nil(tail(coes))) {
            expr old_f = f;
            bool relax = m_relax_main_opaque;
            f = mk_app(head(coes), f, f.get_tag());
            f = add_implict_args(f, cs, relax);
            f_type = infer_type(f, cs);
            save_coercion_info(old_f, f);
            lean_assert(is_pi(f_type));
        } else {
            bool relax             = m_relax_main_opaque;
            local_context ctx      = m_context;
            local_context full_ctx = m_full_context;
            justification j        = mk_justification(f, [=](formatter const & fmt, substitution const & subst) {
                    return pp_function_expected(fmt, substitution(subst).instantiate(f));
                });
            auto choice_fn = [=](expr const & meta, expr const &, substitution const &, name_generator const &) {
                flet<local_context> save1(m_context,      ctx);
                flet<local_context> save2(m_full_context, full_ctx);
                list<constraints> choices = map2<constraints>(coes, [&](expr const & coe) {
                        expr new_f      = copy_tag(f, ::lean::mk_app(coe, f));
                        constraint_seq cs;
                        new_f = add_implict_args(new_f, cs, relax);
                        cs += mk_eq_cnstr(meta, new_f, j, relax);
                        return cs.to_list();
                    });
                return choose(std::make_shared<coercion_elaborator>(*this, f, choices, coes, false));
            };
            f   = m_full_context.mk_meta(m_ngen, none_expr(), f.get_tag());
            register_meta(f);
            cs += mk_choice_cnstr(f, choice_fn, to_delay_factor(cnstr_group::Basic), true, j, relax);
            lean_assert(is_meta(f));
            // let type checker add constraint
            f_type = infer_type(f, cs);
            f_type = m_tc[m_relax_main_opaque]->ensure_pi(f_type, f, cs);
            lean_assert(is_pi(f_type));
        }
    } else {
        erase_coercion_info(f);
    }
    lean_assert(is_pi(f_type));
    return mk_pair(f, f_type);
}

bool elaborator::has_coercions_from(expr const & a_type) {
    try {
        expr const & a_cls = get_app_fn(whnf(a_type).first);
        return is_constant(a_cls) && ::lean::has_coercions_from(env(), const_name(a_cls));
    } catch (exception&) {
        return false;
    }
}

bool elaborator::has_coercions_to(expr d_type) {
    try {
        d_type = whnf(d_type).first;
        expr const & fn = get_app_fn(d_type);
        if (is_constant(fn))
            return ::lean::has_coercions_to(env(), const_name(fn));
        else if (is_pi(d_type))
            return ::lean::has_coercions_to_fun(env());
        else if (is_sort(d_type))
            return ::lean::has_coercions_to_sort(env());
        else
            return false;
    } catch (exception&) {
        return false;
    }
}

expr elaborator::apply_coercion(expr const & a, expr a_type, expr d_type) {
    a_type = whnf(a_type).first;
    d_type = whnf(d_type).first;
    constraint_seq aux_cs;
    list<expr> coes = get_coercions_from_to(*m_tc[m_relax_main_opaque], a_type, d_type, aux_cs);
    if (is_nil(coes)) {
        erase_coercion_info(a);
        return a;
    } else if (is_nil(tail(coes))) {
        expr r = mk_app(head(coes), a, a.get_tag());
        save_coercion_info(a, r);
        return r;
    } else {
        for (expr const & coe : coes) {
            expr r = mk_app(coe, a, a.get_tag());
            expr r_type = infer_type(r).first;
            try {
                if (m_tc[m_relax_main_opaque]->is_def_eq(r_type, d_type).first) {
                    save_coercion_info(a, r);
                    return r;
                }
            } catch (exception&) {
            }
        }
        erase_coercion_info(a);
        return a;
    }
}

/** \brief Given a term <tt>a : a_type</tt>, and an expected type generate a metavariable with a delayed coercion. */
pair<expr, constraint_seq> elaborator::mk_delayed_coercion(
    expr const & a, expr const & a_type, expr const & expected_type,
    justification const & j) {
    bool relax = m_relax_main_opaque;
    type_checker & tc = *m_tc[relax];
    expr m       = m_full_context.mk_meta(m_ngen, some_expr(expected_type), a.get_tag());
    register_meta(m);
    constraint c = mk_coercion_cnstr(tc, *this, m, a, a_type, j, to_delay_factor(cnstr_group::Basic), relax);
    return to_ecs(m, c);
}

/** \brief Given a term <tt>a : a_type</tt>, ensure it has type \c expected_type. Apply coercions if needed

    \remark relax == true affects how opaque definitions in the main module are treated.
*/
pair<expr, constraint_seq> elaborator::ensure_has_type(
    expr const & a, expr const & a_type, expr const & expected_type,
    justification const & j, bool relax) {
    if (is_meta(expected_type) && has_coercions_from(a_type)) {
        return mk_delayed_coercion(a, a_type, expected_type, j);
    } else if (is_meta(a_type) && has_coercions_to(expected_type)) {
        return mk_delayed_coercion(a, a_type, expected_type, j);
    } else {
        pair<bool, constraint_seq> dcs;
        try {
            dcs = m_tc[relax]->is_def_eq(a_type, expected_type, j);
        } catch (exception&) {
            dcs.first = false;
        }
        if (dcs.first) {
            return to_ecs(a, dcs.second);
        } else {
            expr new_a = apply_coercion(a, a_type, expected_type);
            constraint_seq cs;
            bool coercion_worked = false;
            if (!is_eqp(a, new_a)) {
                expr new_a_type = infer_type(new_a, cs);
                try {
                    coercion_worked = m_tc[relax]->is_def_eq(new_a_type, expected_type, j, cs);
                } catch (exception&) {
                    coercion_worked = false;
                }
            }
            if (coercion_worked) {
                return to_ecs(new_a, cs);
            } else if (has_metavar(a_type) || has_metavar(expected_type)) {
                // rely on unification hints to solve this constraint
                return to_ecs(a, mk_eq_cnstr(a_type, expected_type, j, relax));
            } else {
                throw unifier_exception(j, substitution());
            }
        }
    }
}

bool elaborator::is_choice_app(expr const & e) {
    expr const & f = get_app_fn(e);
    return is_choice(f) || (is_annotation(f) && is_choice(get_nested_annotation_arg(f)));
}

/** \brief Process ((choice f_1 ... f_n) a_1 ... a_k) as
    (choice (f_1 a_1 ... a_k) ... (f_n a_1 ... a_k))
*/
expr elaborator::visit_choice_app(expr const & e, constraint_seq & cs) {
    buffer<expr> args;
    expr r = get_app_rev_args(e, args);
    expr f = get_nested_annotation_arg(r);
    lean_assert(is_choice(f));
    buffer<expr> new_choices;
    unsigned num = get_num_choices(f);
    for (unsigned i = 0; i < num; i++) {
        expr f_i = get_choice(f, i);
        f_i      = copy_annotations(r, f_i);
        new_choices.push_back(mk_rev_app(f_i, args));
    }
    return visit_choice(copy_tag(e, mk_choice(new_choices.size(), new_choices.data())), none_expr(), cs);
}

expr elaborator::visit_app(expr const & e, constraint_seq & cs) {
    if (is_choice_app(e))
        return visit_choice_app(e, cs);
    constraint_seq f_cs;
    bool expl   = is_nested_explicit(get_app_fn(e));
    expr f      = visit(app_fn(e), f_cs);
    auto f_t    = ensure_fun(f, f_cs);
    f           = f_t.first;
    expr f_type = f_t.second;
    lean_assert(is_pi(f_type));
    if (!expl) {
        bool first = true;
        while (binding_info(f_type).is_strict_implicit() ||
               (!first && binding_info(f_type).is_implicit()) ||
               (!first && binding_info(f_type).is_inst_implicit())) {
            tag g          = f.get_tag();
            bool is_strict = true;
            bool inst_imp  = binding_info(f_type).is_inst_implicit();
            expr imp_arg   = mk_placeholder_meta(mk_mvar_suffix(f_type), some_expr(binding_domain(f_type)),
                                                 g, is_strict, inst_imp, f_cs);
            f              = mk_app(f, imp_arg, g);
            auto f_t       = ensure_fun(f, f_cs);
            f              = f_t.first;
            f_type         = f_t.second;
            first          = false;
        }
        if (!first) {
            // we save the info data again for application of functions with strict implicit arguments
            save_type_data(get_app_fn(e), f);
        }
    }
    constraint_seq a_cs;
    expr d_type = binding_domain(f_type);
    if (d_type == get_tactic_expr_type()) {
        expr r = mk_app(f, mk_tactic_expr(app_arg(e)), e.get_tag());
        cs += f_cs + a_cs;
        return r;
    } else {
        expr a          = visit_expecting_type_of(app_arg(e), d_type, a_cs);
        expr a_type     = infer_type(a, a_cs);
        expr r          = mk_app(f, a, e.get_tag());
        justification j = mk_app_justification(r, a, d_type, a_type);
        auto new_a_cs   = ensure_has_type(a, a_type, d_type, j, m_relax_main_opaque);
        expr new_a      = new_a_cs.first;
        cs += f_cs + new_a_cs.second + a_cs;
        return update_app(r, app_fn(r), new_a);
    }
}

expr elaborator::visit_placeholder(expr const & e, constraint_seq & cs) {
    bool inst_implicit = true;
    expr r = mk_placeholder_meta(placeholder_type(e), e.get_tag(), is_strict_placeholder(e), inst_implicit, cs);
    save_placeholder_info(e, r);
    return r;
}

level elaborator::replace_univ_placeholder(level const & l) {
    auto fn = [&](level const & l) {
        if (is_placeholder(l))
            return some_level(mk_meta_univ(m_ngen.next()));
        else
            return none_level();
    };
    return replace(l, fn);
}

static bool contains_placeholder(level const & l) {
    bool contains = false;
    auto fn = [&](level const & l) {
        if (contains) return false;
        if (is_placeholder(l))
            contains = true;
        return true;
    };
    for_each(l, fn);
    return contains;
}

expr elaborator::visit_sort(expr const & e) {
    expr r = update_sort(e, replace_univ_placeholder(sort_level(e)));
    if (contains_placeholder(sort_level(e)))
        m_to_check_sorts.emplace_back(e, r);
    return r;
}

expr elaborator::visit_macro(expr const & e, constraint_seq & cs) {
    if (is_as_is(e)) {
        return get_as_is_arg(e);
    } else {
        buffer<expr> args;
        for (unsigned i = 0; i < macro_num_args(e); i++)
            args.push_back(visit(macro_arg(e, i), cs));
        return update_macro(e, args.size(), args.data());
    }
}

expr elaborator::visit_constant(expr const & e) {
    declaration d = env().get(const_name(e));
    buffer<level> ls;
    for (level const & l : const_levels(e))
        ls.push_back(replace_univ_placeholder(l));
    unsigned num_univ_params = length(d.get_univ_params());
    if (num_univ_params < ls.size())
        throw_kernel_exception(env(), sstream() << "incorrect number of universe levels parameters for '"
                               << const_name(e) << "', #" << num_univ_params
                               << " expected, #" << ls.size() << " provided");
    // "fill" with meta universe parameters
    for (unsigned i = ls.size(); i < num_univ_params; i++)
        ls.push_back(mk_meta_univ(m_ngen.next()));
    lean_assert(num_univ_params == ls.size());
    return update_constant(e, to_list(ls.begin(), ls.end()));
}

/** \brief Make sure \c e is a type. If it is not, then try to apply coercions. */
expr elaborator::ensure_type(expr const & e, constraint_seq & cs) {
    expr t = infer_type(e, cs);
    erase_coercion_info(e);
    if (is_sort(t))
        return e;
    t = whnf(t, cs);
    if (is_sort(t))
        return e;
    if (has_metavar(t)) {
        t = whnf(t, cs);
        if (is_sort(t))
            return e;
        if (is_meta(t)) {
            // let type checker add constraint
            m_tc[m_relax_main_opaque]->ensure_sort(t, e, cs);
            return e;
        }
    }
    list<expr> coes = get_coercions_to_sort(env(), t);
    if (is_nil(coes)) {
        throw_kernel_exception(env(), e, [=](formatter const & fmt) { return pp_type_expected(fmt, e); });
    } else {
        // Remark: we ignore other coercions to sort
        expr r = mk_app(head(coes), e, e.get_tag());
        save_coercion_info(e, r);
        return r;
    }
}

/** \brief Similar to instantiate_rev, but assumes that subst contains only local constants.
    When replacing a variable with a local, we copy the local constant and inherit the tag
    associated with the variable. This is a trick for getter better error messages */
expr elaborator::instantiate_rev_locals(expr const & a, unsigned n, expr const * subst) {
    if (closed(a))
        return a;
    auto fn = [=](expr const & m, unsigned offset) -> optional<expr> {
        if (offset >= get_free_var_range(m))
            return some_expr(m); // expression m does not contain free variables with idx >= offset
        if (is_var(m)) {
            unsigned vidx = var_idx(m);
            if (vidx >= offset) {
                unsigned h = offset + n;
                if (h < offset /* overflow, h is bigger than any vidx */ || vidx < h) {
                    expr local = subst[n - (vidx - offset) - 1];
                    lean_assert(is_local(local));
                    return some_expr(copy_tag(m, copy(local)));
                } else {
                    return some_expr(copy_tag(m, mk_var(vidx - n)));
                }
            }
        }
        return none_expr();
    };
    return replace(a, fn);
}

expr elaborator::visit_binding(expr e, expr_kind k, constraint_seq & cs) {
    flet<local_context> save1(m_context, m_context);
    flet<local_context> save2(m_full_context, m_full_context);
    buffer<expr> ds, ls, es;
    while (e.kind() == k) {
        es.push_back(e);
        expr const & d0 = binding_domain(e);
        expr d          = d0;
        d = instantiate_rev_locals(d, ls.size(), ls.data());
        d = ensure_type(visit_expecting_type(d, cs), cs);

        if (is_placeholder(d0) && !is_explicit_placeholder(d0))
            save_binder_type(d0, d);

        ds.push_back(d);
        expr l   = mk_local(binding_name(e), d, binding_info(e));
        if (binding_info(e).is_contextual())
            m_context.add_local(l);
        m_full_context.add_local(l);
        ls.push_back(l);
        e = binding_body(e);
    }
    lean_assert(ls.size() == es.size() && ls.size() == ds.size());
    e = instantiate_rev_locals(e, ls.size(), ls.data());
    e = (k == expr_kind::Pi) ? ensure_type(visit_expecting_type(e, cs), cs) : visit(e, cs);
    e = abstract_locals(e, ls.size(), ls.data());
    unsigned i = ls.size();
    while (i > 0) {
        --i;
        e = update_binding(es[i], abstract_locals(ds[i], i, ls.data()), e);
    }
    return e;
}
expr elaborator::visit_pi(expr const & e, constraint_seq & cs) {
    return visit_binding(e, expr_kind::Pi, cs);
}
expr elaborator::visit_lambda(expr const & e, constraint_seq & cs) {
    return visit_binding(e, expr_kind::Lambda, cs);
}

expr elaborator::visit_typed_expr(expr const & e, constraint_seq & cs) {
    constraint_seq t_cs;
    expr t      = visit(get_typed_expr_type(e), t_cs);
    constraint_seq v_cs;
    expr v      = visit(get_typed_expr_expr(e), v_cs);
    expr v_type = infer_type(v, v_cs);
    justification j = mk_type_mismatch_jst(v, v_type, t, e);
    auto new_vcs    = ensure_has_type(v, v_type, t, j, m_relax_main_opaque);
    v = new_vcs.first;
    cs += t_cs + new_vcs.second + v_cs;
    return v;
}

expr elaborator::visit_let_value(expr const & e, constraint_seq & cs) {
    if (auto p = m_cache.find(e)) {
        cs += p->second;
        return p->first;
    } else {
        auto ecs = visit(get_let_value_expr(e));
        expr r = copy_tag(ecs.first, mk_let_value(ecs.first));
        m_cache.insert(e, mk_pair(r, ecs.second));
        cs += ecs.second;
        return r;
    }
}

bool elaborator::is_sorry(expr const & e) const {
    return m_has_sorry && ::lean::is_sorry(e);
}

expr elaborator::visit_sorry(expr const & e) {
    level u = mk_meta_univ(m_ngen.next());
    expr t  = mk_sort(u);
    expr m  = m_full_context.mk_meta(m_ngen, some_expr(t), e.get_tag());
    return mk_app(update_constant(e, to_list(u)), m, e.get_tag());
}

expr const & elaborator::get_equation_fn(expr const & eq) const {
    expr it = eq;
    while (is_lambda(it))
        it = binding_body(it);
    if (!is_equation(it))
        throw_elaborator_exception("ill-formed equation", eq);
    expr const & fn = get_app_fn(equation_lhs(it));
    if (!is_local(fn))
        throw_elaborator_exception("ill-formed equation", eq);
    return fn;
}

/**
   \brief Given two binding expressions \c source and \c target
   s.t. they have at least \c num binders, replace the first \c num binders of \c target with \c source.
   The binder types are wrapped with \c mk_as_is to make sure the elaborator will not process
   them again.
*/
static expr copy_domain(unsigned num, expr const & source, expr const & target) {
    if (num == 0) {
        return target;
    } else {
        lean_assert(is_binding(source) && is_binding(target));
        return update_binding(source, mk_as_is(binding_domain(source)),
                              copy_domain(num-1, binding_body(source), binding_body(target)));
    }
}

/**
   \brief The left-hand-side of recursive equations may contain metavariables associated with
   implicit parameters. This procedure replaces them with fresh local constants.

   Example 1)
   Suppose we are defining
   map : Pi {n}, vec A n -> vec B n -> vec C n,
   map nil nil             := nil,
   map (a :: va) (b :: vb) := f a b :: map va vb

   After elaboration the second equation will be

   @map (succ ?M) (@cons A ?M a va) (@cons A ?M b vb) := @cons A ?M (f ab) (@map ?M va vb)

   This procedure replaces ?M with (x_1 : nat), where x_1 is a new local constant.
   The resultant eqns object is:
   [equations
     (λ (map : Π {n : ℕ}, vector A n → vector B n → vector C n), [equation (map nil nil) nil])
     (λ (map : Π {n : ℕ}, vector A n → vector B n → vector C n) (a : A) (x_1 : ℕ) (va : vector A x_1) (b : B)
        (vb : vector B x_1),
          [equation (map (a :: va) (b :: vb)) (f a b :: map va vb)])]


   Example 2)
   Suppose we are defining
   definition ideq : Π {A : Type} {a b : A}, a = b → a = b,
   ideq H := H

   After elaboration the equation is:
   @ideq ?M1 ?M2 ?M3 H := H

   This procedure replaces ?M1 ?M2 ?M3 with
   (x_1 : Type) (x_2 : x_1) (x_3 : x_1)
   The resultant eqns object is

   [equations
    (λ (ideq : ∀ {A : Type} {a b : A}, @eq A a b → @eq A a b) (x_1 : Type) (x_2 x_3 : x_1) (H : @eq x_1 x_2 x_3),
       [equation (@ideq x_1 x_2 x_3 H) H])]
*/
static expr assign_equation_lhs_metas(type_checker & tc, expr const & eqns) {
    lean_assert(is_equations(eqns));
    if (!has_metavar(eqns))
        return eqns;
    buffer<expr> eqs;
    buffer<expr> new_eqs;
    to_equations(eqns, eqs);
    unsigned num_fns = equations_num_fns(eqns);

    auto replace_meta = [](expr const & e, expr const & meta, expr const & local) {
        expr mvar = get_app_fn(meta);
        return replace(e, [&](expr const & e, unsigned) {
                if (is_meta(e) && mlocal_name(get_app_fn(e)) == mlocal_name(mvar)) {
                    return some_expr(local);
                } else if (!has_metavar(e)) {
                    return some_expr(e);
                } else {
                    return none_expr();
                }
            });
    };

    for (expr eq : eqs) {
        if (!has_metavar(eq)) {
            new_eqs.push_back(eq);
        } else {
            name x("x");
            buffer<expr> locals;
            name_generator ngen = tc.mk_ngen();
            eq = fun_to_telescope(ngen, eq, locals, optional<binder_info>());
            lean_assert(num_fns <= locals.size());
            lean_assert(is_equation(eq));
            unsigned idx = 1;
            while (true) {
                expr lhs = equation_lhs(eq);
                optional<expr> meta;
                optional<expr> meta_type;
                for_each(lhs, [&](expr const & e, unsigned offset) {
                        if (meta) return false; // already found target
                        if (offset > 0) return false;
                        if (is_meta(e)) {
                            expr type = tc.infer(e).first;
                            if (!has_expr_metavar_strict(type)) {
                                meta      = e;
                                meta_type = type;
                                return false; // stop search, found
                            }
                        }
                        return has_metavar(e);
                    });
                if (!meta)
                    break;
                expr new_local = mk_local(tc.mk_fresh_name(), x.append_after(idx), *meta_type, binder_info());
                for (expr & local : locals)
                    local = update_mlocal(local, replace_meta(mlocal_type(local), *meta, new_local));
                eq  = replace_meta(eq, *meta, new_local);
                unsigned i = num_fns;
                for (; i < locals.size(); i++) {
                    if (depends_on(mlocal_type(locals[i]), new_local)) {
                        break;
                    }
                }
                locals.insert(i, new_local);
                idx++;
            }
            new_eqs.push_back(Fun(locals, eq));
        }
    }

    return update_equations(eqns, new_eqs);
}

static constraint mk_equations_cnstr(environment const & env, io_state const & ios, expr const & m, expr const & eqns,
                                     bool relax) {
    justification j         = mk_failed_to_synthesize_jst(env, m);
    auto choice_fn = [=](expr const & meta, expr const & meta_type, substitution const & s,
                         name_generator const & ngen) {
        substitution new_s  = s;
        expr new_eqns       = substitution(s).instantiate_all(eqns);
        type_checker_ptr tc = mk_type_checker(env, ngen, relax);
        new_eqns            = assign_equation_lhs_metas(*tc, new_eqns);
        expr val            = compile_equations(*tc, ios, new_eqns, meta, meta_type, relax);
        justification j     = mk_justification("equation compilation", some_expr(eqns));
        constraint c        = mk_eq_cnstr(meta, val, j, relax);
        return lazy_list<constraints>(c);
    };
    bool owner = true;
    return mk_choice_cnstr(m, choice_fn, to_delay_factor(cnstr_group::MaxDelayed), owner, j, relax);
}

expr elaborator::visit_equations(expr const & eqns, constraint_seq & cs) {
    buffer<expr> eqs;
    buffer<expr> new_eqs;
    optional<expr> new_R;
    optional<expr> new_Hwf;

    to_equations(eqns, eqs);

    if (eqs.empty())
        throw_elaborator_exception("invalid empty set of recursive equations", eqns);

    if (is_wf_equations(eqns)) {
        new_R   = visit(equations_wf_rel(eqns), cs);
        new_Hwf = visit(equations_wf_proof(eqns), cs);
        expr Hwf_type   = infer_type(*new_Hwf, cs);
        expr wf         = visit(mk_constant("well_founded"), cs);
        wf              = ::lean::mk_app(wf, *new_R);
        justification j = mk_type_mismatch_jst(*new_Hwf, Hwf_type, wf, equations_wf_proof(eqns));
        auto new_Hwf_cs = ensure_has_type(*new_Hwf, Hwf_type, wf, j, m_relax_main_opaque);
        new_Hwf         = new_Hwf_cs.first;
        cs             += new_Hwf_cs.second;
    }

    flet<optional<expr>> set1(m_equation_R, new_R);
    unsigned num_fns = equations_num_fns(eqns);

    optional<expr> first_eq;
    for (expr const & eq : eqs) {
        expr new_eq;
        if (first_eq) {
            // Replace first num_fns domains of eq with the ones in first_eq.
            // This is a trick/hack to ensure the fns in each equation have
            // the same elaborated type.
            new_eq = visit(copy_domain(num_fns, *first_eq, eq), cs);
        } else {
            new_eq = visit(eq, cs);
            first_eq = new_eq;
        }
        new_eqs.push_back(new_eq);
    }

    expr new_eqns;
    if (new_R) {
        new_eqns = mk_equations(num_fns, new_eqs.size(), new_eqs.data(), *new_R, *new_Hwf);
    } else {
        new_eqns = mk_equations(num_fns, new_eqs.size(), new_eqs.data());
    }

    lean_assert(first_eq && is_lambda(*first_eq));
    expr type = binding_domain(*first_eq);
    expr m = m_full_context.mk_meta(m_ngen, some_expr(type), eqns.get_tag());
    register_meta(m);
    constraint c = mk_equations_cnstr(env(), ios(), m, new_eqns, m_relax_main_opaque);
    cs += c;
    return m;
}

expr elaborator::visit_equation(expr const & eq, constraint_seq & cs) {
    expr const & lhs = equation_lhs(eq);
    expr const & rhs = equation_rhs(eq);
    expr lhs_fn = get_app_fn(lhs);
    if (is_explicit(lhs_fn))
        lhs_fn = get_explicit_arg(lhs_fn);
    if (!is_local(lhs_fn))
        throw exception("ill-formed equation");
    expr new_lhs, new_rhs;
    {
        flet<bool> set(m_in_equation_lhs, true);
        new_lhs = visit(lhs, cs);
    }
    {
        optional<expr> some_new_lhs(new_lhs);
        flet<optional<expr>> set1(m_equation_lhs, some_new_lhs);
        new_rhs = visit(rhs, cs);
    }

    expr lhs_type = infer_type(new_lhs, cs);
    expr rhs_type = infer_type(new_rhs, cs);
    justification j = mk_justification(eq, [=](formatter const & fmt, substitution const & subst) {
            substitution s(subst);
            return pp_def_type_mismatch(fmt, local_pp_name(lhs_fn), s.instantiate(lhs_type), s.instantiate(rhs_type));
        });
    pair<expr, constraint_seq> new_rhs_cs = ensure_has_type(new_rhs, rhs_type, lhs_type, j, m_relax_main_opaque);
    new_rhs = new_rhs_cs.first;
    cs     += new_rhs_cs.second;
    return mk_equation(new_lhs, new_rhs);
}

expr elaborator::visit_inaccessible(expr const & e, constraint_seq & cs) {
    if (!m_in_equation_lhs)
        throw_elaborator_exception("invalid occurrence of 'inaccessible' annotation, it must only occur in the "
                                   "left-hand-side of recursive equations", e);
    return mk_inaccessible(visit(get_annotation_arg(e), cs));
}

expr elaborator::visit_decreasing(expr const & e, constraint_seq & cs) {
    if (!m_equation_lhs)
        throw_elaborator_exception("invalid occurrence of 'decreasing' annotation, it must only occur in "
                                   "the right-hand-side of recursive equations", e);
    if (!m_equation_R)
        throw_elaborator_exception("invalid occurrence of 'decreasing' annotation, it can only be used when "
                                   "recursive equations are being defined by well-founded recursion", e);
    expr const & lhs_fn = get_app_fn(*m_equation_lhs);
    if (get_app_fn(decreasing_app(e)) != lhs_fn)
        throw_elaborator_exception("invalid occurrence of 'decreasing' annotation, expression must be an "
                                   "application of the recursive function being defined", e);
    expr dec_app        = visit(decreasing_app(e), cs);
    expr dec_proof      = visit(decreasing_proof(e), cs);
    expr f_type         = mlocal_type(get_app_fn(*m_equation_lhs));
    buffer<expr> ts;
    type_checker & tc = *m_tc[m_relax_main_opaque];
    to_telescope(tc, f_type, ts, optional<binder_info>(), cs);
    buffer<expr> old_args;
    buffer<expr> new_args;
    get_app_args(*m_equation_lhs, old_args);
    get_app_args(dec_app, new_args);
    if (new_args.size() != old_args.size() || new_args.size() != ts.size())
        throw_elaborator_exception("invalid recursive application, mistmatch in the number of arguments", e);
    expr old_tuple  = mk_sigma_mk(tc, ts, old_args, cs);
    expr new_tuple  = mk_sigma_mk(tc, ts, new_args, cs);
    expr expected_dec_proof_type = mk_app(mk_app(*m_equation_R, new_tuple, e.get_tag()), old_tuple, e.get_tag());
    expr dec_proof_type = infer_type(dec_proof, cs);
    justification j = mk_type_mismatch_jst(dec_proof, dec_proof_type, expected_dec_proof_type, decreasing_proof(e));
    auto new_dec_proof_cs = ensure_has_type(dec_proof, dec_proof_type, expected_dec_proof_type, j, m_relax_main_opaque);
    dec_proof = new_dec_proof_cs.first;
    cs       += new_dec_proof_cs.second;
    return mk_decreasing(dec_app, dec_proof);
}

expr elaborator::visit_core(expr const & e, constraint_seq & cs) {
    if (is_placeholder(e)) {
        return visit_placeholder(e, cs);
    } else if (is_choice(e)) {
        return visit_choice(e, none_expr(), cs);
    } else if (is_let_value(e)) {
        return visit_let_value(e, cs);
    } else if (is_by(e)) {
        return visit_by(e, none_expr(), cs);
    } else if (is_calc_annotation(e)) {
        return visit_calc_proof(e, none_expr(), cs);
    } else if (is_proof_qed_annotation(e)) {
        return visit_proof_qed(e, none_expr(), cs);
    } else if (is_no_info(e)) {
        flet<bool> let(m_no_info, true);
        return visit(get_annotation_arg(e), cs);
    } else if (is_typed_expr(e)) {
        return visit_typed_expr(e, cs);
    } else if (is_as_atomic(e)) {
        // ignore annotation
        return visit_core(get_as_atomic_arg(e), cs);
    } else if (is_consume_args(e)) {
        // ignore annotation
        return visit_core(get_consume_args_arg(e), cs);
    } else if (is_explicit(e)) {
        // ignore annotation
        return visit_core(get_explicit_arg(e), cs);
    } else if (is_sorry(e)) {
        return visit_sorry(e);
    } else if (is_equations(e)) {
        lean_unreachable();
    } else if (is_equation(e)) {
        return visit_equation(e, cs);
    } else if (is_inaccessible(e)) {
        return visit_inaccessible(e, cs);
    } else if (is_decreasing(e)) {
        return visit_decreasing(e, cs);
    } else {
        switch (e.kind()) {
        case expr_kind::Local:      return e;
        case expr_kind::Meta:       return e;
        case expr_kind::Sort:       return visit_sort(e);
        case expr_kind::Var:        lean_unreachable();  // LCOV_EXCL_LINE
        case expr_kind::Constant:   return visit_constant(e);
        case expr_kind::Macro:      return visit_macro(e, cs);
        case expr_kind::Lambda:     return visit_lambda(e, cs);
        case expr_kind::Pi:         return visit_pi(e, cs);
        case expr_kind::App:        return visit_app(e, cs);
        }
        lean_unreachable(); // LCOV_EXCL_LINE
    }
}

pair<expr, constraint_seq> elaborator::visit(expr const & e) {
    if (is_extra_info(e)) {
        auto ecs = visit(get_annotation_arg(e));
        save_extra_type_data(e, ecs.first);
        return ecs;
    }
    if (is_notation_info(e)) {
        pair<expr, constraint_seq> ecs;
        {
            flet<bool> let(m_no_info, true);
            ecs = visit(get_annotation_arg(e));
        }
        save_type_data(e, ecs.first);
        return ecs;
    }
    expr r;
    expr b = e;
    constraint_seq cs;
    if (is_explicit(e)) {
        b    = get_explicit_arg(e);
        if (is_sorry(b)) {
            r = visit_constant(b);
        } else {
            r = visit_core(b, cs);
        }
    } else if (is_equations(e)) {
        r    = visit_equations(e, cs);
    } else if (is_explicit(get_app_fn(e))) {
        r    = visit_core(e, cs);
    } else {
        bool consume_args = false;
        if (is_as_atomic(e)) {
            flet<bool> let(m_no_info, true);
            r = get_as_atomic_arg(e);
            if (is_explicit(r)) r = get_explicit_arg(r);
            r = visit_core(r, cs);
        } else if (is_consume_args(e)) {
            consume_args = true;
            r = visit_core(get_consume_args_arg(e), cs);
        } else {
            r = visit_core(e, cs);
        }
        tag  g         = e.get_tag();
        expr r_type    = whnf(infer_type(r, cs), cs);
        expr imp_arg;
        bool is_strict = true;
        while (is_pi(r_type)) {
            binder_info bi = binding_info(r_type);
            if (!bi.is_implicit() && !bi.is_inst_implicit()) {
                if (!consume_args)
                    break;
                if (!has_free_var(binding_body(r_type), 0)) {
                    // if the rest of the type does not reference argument,
                    // then we also stop consuming arguments
                    break;
                }
            }
            bool inst_imp = bi.is_inst_implicit();
            imp_arg = mk_placeholder_meta(mk_mvar_suffix(r_type), some_expr(binding_domain(r_type)),
                                          g, is_strict, inst_imp, cs);
            r       = mk_app(r, imp_arg, g);
            r_type  = whnf(instantiate(binding_body(r_type), imp_arg), cs);
        }
    }
    save_type_data(b, r);
    return mk_pair(r, cs);
}

expr elaborator::visit(expr const & e, constraint_seq & cs) {
    auto r = visit(e);
    cs += r.second;
    return r.first;
}

unify_result_seq elaborator::solve(constraint_seq const & cs) {
    buffer<constraint> tmp;
    cs.linearize(tmp);
    return unify(env(), tmp.size(), tmp.data(), m_ngen.mk_child(), substitution(), m_unifier_config);
}

void elaborator::display_unsolved_proof_state(expr const & mvar, proof_state const & ps, char const * msg, expr const & pos) {
    lean_assert(is_metavar(mvar));
    if (!m_displayed_errors.contains(mlocal_name(mvar))) {
        m_displayed_errors.insert(mlocal_name(mvar));
        auto out = regular(env(), ios());
        flycheck_error err(out);
        display_error_pos(out, pip(), pos);
        out << " " << msg << "\n" << ps.pp(env(), ios()) << endl;
    }
}

void elaborator::display_unsolved_proof_state(expr const & mvar, proof_state const & ps, char const * msg) {
    display_unsolved_proof_state(mvar, ps, msg, mvar);
}

optional<expr> elaborator::get_pre_tactic_for(expr const & mvar) {
    if (auto it = m_local_tactic_hints.find(mlocal_name(mvar))) {
        return some_expr(*it);
    } else {
        return none_expr();
    }
}

optional<tactic> elaborator::pre_tactic_to_tactic(expr const & pre_tac) {
    try {
        bool relax = m_relax_main_opaque;
        auto fn = [=](goal const & g, name_generator const & ngen, expr const & e, bool report_unassigned) {
            elaborator aux_elaborator(m_ctx, ngen);
            // Disable tactic hints when processing expressions nested in tactics.
            // We must do it otherwise, it is easy to make the system loop.
            bool use_tactic_hints = false;
            return aux_elaborator.elaborate_nested(g.to_context(), e, relax, use_tactic_hints, report_unassigned);
        };
        return optional<tactic>(expr_to_tactic(env(), fn, pre_tac, pip()));
    } catch (expr_to_tactic_exception & ex) {
        auto out = regular(env(), ios());
        flycheck_error err(out);
        display_error_pos(out, pip(), ex.get_expr());
        out << " " << ex.what();
        out << pp_indent_expr(out.get_formatter(), pre_tac) << endl << "failed at:"
            << pp_indent_expr(out.get_formatter(), ex.get_expr()) << endl;
        return optional<tactic>();
    }
}

/** \brief Try to instantiate meta-variable \c mvar (modulo its state ps) using the given tactic.
    If it succeeds, then update subst with the solution.
    Return true iff the metavariable \c mvar has been assigned.

    If \c show_failure == true, then display reason for failure.
*/
bool elaborator::try_using(substitution & subst, expr const & mvar, proof_state const & ps, tactic const & tac,
                           bool show_failure) {
    lean_assert(length(ps.get_goals()) == 1);
    // make sure ps is a really a proof state for mvar.
    lean_assert(mlocal_name(get_app_fn(head(ps.get_goals()).get_meta())) == mlocal_name(mvar));
    try {
        proof_state_seq seq = tac(env(), ios(), ps);
        auto r = seq.pull();
        if (!r) {
            // tactic failed to produce any result
            if (show_failure)
                display_unsolved_proof_state(mvar, ps, "tactic failed");
            return false;
        } else if (!empty(r->first.get_goals())) {
            // tactic contains unsolved subgoals
            if (show_failure)
                display_unsolved_proof_state(mvar, r->first, "unsolved subgoals");
            return false;
        } else {
            subst = r->first.get_subst();
            expr v = subst.instantiate(mvar);
            subst.assign(mlocal_name(mvar), v);
            return true;
        }
    } catch (tactic_exception & ex) {
        if (show_failure) {
            auto out = regular(env(), ios());
            display_error_pos(out, pip(), ex.get_expr());
            out << " tactic failed: " << ex.what() << "\n";
        }
        return false;
    }
}

static void extract_begin_end_tactics(expr pre_tac, buffer<expr> & pre_tac_seq) {
    if (is_begin_end_element_annotation(pre_tac)) {
        pre_tac_seq.push_back(get_annotation_arg(pre_tac));
    } else {
        buffer<expr> args;
        if (get_app_args(pre_tac, args) == get_and_then_tac_fn()) {
            for (expr const & arg : args) {
                extract_begin_end_tactics(arg, pre_tac_seq);
            }
        } else {
            throw exception("internal error, invalid begin-end tactic");
        }
    }
}

void elaborator::try_using_begin_end(substitution & subst, expr const & mvar, proof_state ps, expr const & pre_tac) {
    lean_assert(is_begin_end_annotation(pre_tac));
    buffer<expr> pre_tac_seq;
    extract_begin_end_tactics(get_annotation_arg(pre_tac), pre_tac_seq);
    for (expr const & ptac : pre_tac_seq) {
        if (auto tac = pre_tactic_to_tactic(ptac)) {
            try {
                proof_state_seq seq = (*tac)(env(), ios(), ps);
                auto r = seq.pull();
                if (!r) {
                    // tactic failed to produce any result
                    display_unsolved_proof_state(mvar, ps, "tactic failed", ptac);
                    return;
                }
                if (m_ctx.m_flycheck_goals) {
                    if (auto p = pip()->get_pos_info(ptac)) {
                        auto out = regular(env(), ios());
                        flycheck_information info(out);
                        if (info.enabled()) {
                            display_information_pos(out, pip()->get_file_name(), p->first, p->second);
                            out << " proof state:\n" << ps.pp(env(), ios()) << "\n";
                        }
                    }
                }
                ps = r->first;
            } catch (tactic_exception & ex) {
                auto out = regular(env(), ios());
                display_error_pos(out, pip(), ex.get_expr());
                out << " tactic failed: " << ex.what() << "\n";
                return;
            }
        } else {
            return;
        }
    }

    if (!empty(ps.get_goals())) {
        display_unsolved_proof_state(mvar, ps, "unsolved subgoals", pre_tac);
    } else {
        subst = ps.get_subst();
        expr v = subst.instantiate(mvar);
        subst.assign(mlocal_name(mvar), v);
    }
}

void elaborator::solve_unassigned_mvar(substitution & subst, expr mvar, name_set & visited) {
    if (visited.contains(mlocal_name(mvar)))
        return;
    visited.insert(mlocal_name(mvar));
    auto meta = mvar_to_meta(mvar);
    if (!meta)
        return;
    meta = instantiate_meta(*meta, subst);
    // TODO(Leo): we are discarding constraints here
    expr type = m_tc[m_relax_main_opaque]->infer(*meta).first;
    // first solve unassigned metavariables in type
    type = solve_unassigned_mvars(subst, type, visited);
    bool relax_main_opaque = m_relaxed_mvars.contains(mlocal_name(mvar));
    proof_state ps = to_proof_state(*meta, type, subst, m_ngen.mk_child(), relax_main_opaque);
    if (auto pre_tac = get_pre_tactic_for(mvar)) {
        if (is_begin_end_annotation(*pre_tac)) {
            try_using_begin_end(subst, mvar, ps, *pre_tac);
            return;
        }

        if (auto tac = pre_tactic_to_tactic(*pre_tac)) {
            bool show_failure = true;
            try_using(subst, mvar, ps, *tac, show_failure);
            return;
        }
    }

    if (m_use_tactic_hints) {
        // using tactic_hints
        for (expr const & pre_tac : get_tactic_hints(env())) {
            if (auto tac = pre_tactic_to_tactic(pre_tac)) {
                bool show_failure = false;
                if (try_using(subst, mvar, ps, *tac, show_failure))
                    return;
            }
        }
    }
}

expr elaborator::solve_unassigned_mvars(substitution & subst, expr e, name_set & visited) {
    e = subst.instantiate(e);
    metavar_closure mvars(e);
    mvars.for_each_expr_mvar([&](expr const & mvar) {
            check_interrupted();
            solve_unassigned_mvar(subst, mvar, visited);
        });
    return subst.instantiate(e);
}

expr elaborator::solve_unassigned_mvars(substitution & subst, expr const & e) {
    name_set visited;
    return solve_unassigned_mvars(subst, e, visited);
}

void elaborator::display_unassigned_mvars(expr const & e, substitution const & s) {
    if (check_unassigned() && has_metavar(e)) {
        substitution tmp_s(s);
        for_each(e, [&](expr const & e, unsigned) {
                if (!is_metavar(e))
                    return has_metavar(e);
                if (auto it = mvar_to_meta(e)) {
                    expr meta      = tmp_s.instantiate(*it);
                    expr meta_type = tmp_s.instantiate(type_checker(env()).infer(meta).first);
                    goal g(meta, meta_type);
                    bool relax     = true;
                    proof_state ps(goals(g), s, m_ngen, constraints(), relax);
                    display_unsolved_proof_state(e, ps, "don't know how to synthesize placeholder");
                }
                return false;
            });
    }
}

/** \brief Check whether the solution found by the elaborator is producing too specific
    universes.

    \remark For now, we only check if a term Type.{?u} was solved by assigning ?u to 0.
    In this case, the user should write Prop instead of Type.
*/
void elaborator::check_sort_assignments(substitution const & s) {
    for (auto const & p : m_to_check_sorts) {
        expr pre  = p.first;
        expr post = p.second;
        lean_assert(is_sort(post));
        for_each(sort_level(post), [&](level const & u) {
                if (is_meta(u) && s.is_assigned(u)) {
                    level r = *s.get_level(u);
                    if (is_explicit(r)) {
                        substitution saved_s(s);
                        throw_kernel_exception(env(), pre, [=](formatter const & fmt) {
                                options o = fmt.get_options();
                                o  = o.update(get_pp_universes_option_name(), true);
                                format r("solution computed by the elaborator forces a universe placeholder"
                                         " to be a fixed value, computed sort is");
                                r += pp_indent_expr(fmt.update_options(o), substitution(saved_s).instantiate(post));
                                return r;
                            });
                    }
                }
                return true;
            });
    }
}

/** \brief Apply substitution and solve remaining metavariables using tactics. */
expr elaborator::apply(substitution & s, expr const & e, name_set & univ_params, buffer<name> & new_params) {
    expr r = s.instantiate(e);
    if (has_univ_metavar(r))
        r = univ_metavars_to_params(env(), lls(), s, univ_params, new_params, r);
    r = solve_unassigned_mvars(s, r);
    display_unassigned_mvars(r, s);
    return r;
}

std::tuple<expr, level_param_names> elaborator::apply(substitution & s, expr const & e) {
    auto ps = collect_univ_params(e);
    buffer<name> new_ps;
    expr r = apply(s, e, ps, new_ps);
    return std::make_tuple(r, to_list(new_ps.begin(), new_ps.end()));
}

auto elaborator::operator()(list<expr> const & ctx, expr const & e, bool _ensure_type, bool relax_main_opaque)
-> std::tuple<expr, level_param_names> {
    m_context.set_ctx(ctx);
    m_full_context.set_ctx(ctx);
    flet<bool> set_relax(m_relax_main_opaque, relax_main_opaque);
    constraint_seq cs;
    expr r = visit(e, cs);
    if (_ensure_type)
        r = ensure_type(r, cs);
    auto p  = solve(cs).pull();
    lean_assert(p);
    substitution s = p->first.first;
    auto result = apply(s, r);
    check_sort_assignments(s);
    copy_info_to_manager(s);
    return result;
}

std::tuple<expr, expr, level_param_names> elaborator::operator()(
    expr const & t, expr const & v, name const & n, bool is_opaque) {
    constraint_seq t_cs;
    expr r_t      = ensure_type(visit(t, t_cs), t_cs);
    // Opaque definitions in the main module may treat other opaque definitions (in the main module) as transparent.
    flet<bool> set_relax(m_relax_main_opaque, is_opaque);
    constraint_seq v_cs;
    expr r_v      = visit(v, v_cs);
    expr r_v_type = infer_type(r_v, v_cs);
    justification j = mk_justification(r_v, [=](formatter const & fmt, substitution const & subst) {
            substitution s(subst);
            return pp_def_type_mismatch(fmt, n, s.instantiate(r_t), s.instantiate(r_v_type));
        });
    pair<expr, constraint_seq> r_v_cs = ensure_has_type(r_v, r_v_type, r_t, j, is_opaque);
    r_v = r_v_cs.first;
    constraint_seq cs = t_cs + r_v_cs.second + v_cs;
    auto p  = solve(cs).pull();
    lean_assert(p);
    substitution s = p->first.first;
    name_set univ_params = collect_univ_params(r_v, collect_univ_params(r_t));
    buffer<name> new_params;
    expr new_r_t = apply(s, r_t, univ_params, new_params);
    expr new_r_v = apply(s, r_v, univ_params, new_params);
    check_sort_assignments(s);
    copy_info_to_manager(s);
    return std::make_tuple(new_r_t, new_r_v, to_list(new_params.begin(), new_params.end()));
}

// Auxiliary procedure for #translate
static expr translate_local_name(list<expr> const & ctx, name const & local_name,
                                 expr const & src) {
    for (expr const & local : ctx) {
        if (local_pp_name(local) == local_name)
            return copy(local);
    }
    throw_elaborator_exception(sstream() << "unknown identifier '" << local_name << "'", src);
}

/** \brief Translated local constants (and undefined constants) occurring in \c e into
    local constants provided by \c ctx.
    Throw exception is \c ctx does not contain the local constant.
*/
static expr translate(environment const & env, list<expr> const & ctx, expr const & e) {
    auto fn = [&](expr const & e) {
        if (is_placeholder(e) || is_by(e)) {
            return some_expr(e); // ignore placeholders
        } else if (is_constant(e)) {
            if (!env.find(const_name(e))) {
                expr new_e = copy_tag(e, translate_local_name(ctx, const_name(e), e));
                return some_expr(new_e);
            } else {
                return none_expr();
            }
        } else if (is_local(e)) {
            expr new_e = copy_tag(e, translate_local_name(ctx, local_pp_name(e), e));
            return some_expr(new_e);
        } else {
            return none_expr();
        }
    };
    return replace(e, fn);
}

/** \brief Elaborate expression \c e in context \c ctx. */
pair<expr, constraints> elaborator::elaborate_nested(list<expr> const & ctx, expr const & n,
                                                     bool relax, bool use_tactic_hints, bool report_unassigned) {
    if (infom()) {
        if (auto ps = get_info_tactic_proof_state()) {
            save_proof_state_info(*ps, n);
        }
    }
    expr e = translate(env(), ctx, n);
    m_context.set_ctx(ctx);
    m_full_context.set_ctx(ctx);
    flet<bool> set_relax(m_relax_main_opaque, relax);
    flet<bool> set_discard(m_unifier_config.m_discard, false);
    flet<bool> set_use_hints(m_use_tactic_hints, use_tactic_hints);
    constraint_seq cs;
    expr r  = visit(e, cs);
    auto p  = solve(cs).pull();
    lean_assert(p);
    substitution s  = p->first.first;
    constraints rcs = p->first.second;
    r = s.instantiate_all(r);
    r = solve_unassigned_mvars(s, r);
    copy_info_to_manager(s);
    if (report_unassigned)
        display_unassigned_mvars(r, s);
    return mk_pair(r, rcs);
}

static name * g_tmp_prefix = nullptr;

std::tuple<expr, level_param_names> elaborate(elaborator_context & env, list<expr> const & ctx, expr const & e,
                                              bool relax_main_opaque, bool ensure_type, bool nice_mvar_names) {
    return elaborator(env, name_generator(*g_tmp_prefix), nice_mvar_names)(ctx, e, ensure_type, relax_main_opaque);
}

std::tuple<expr, expr, level_param_names> elaborate(elaborator_context & env, name const & n, expr const & t, expr const & v,
                                                    bool is_opaque) {
    return elaborator(env, name_generator(*g_tmp_prefix))(t, v, n, is_opaque);
}

void initialize_elaborator() {
    g_tmp_prefix = new name(name::mk_internal_unique_name());
}

void finalize_elaborator() {
    delete g_tmp_prefix;
}
}