lean2/src/frontends/lean/builtin_exprs.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 "util/sstream.h"
#include "kernel/abstract.h"
#include "library/annotation.h"
#include "library/placeholder.h"
#include "library/explicit.h"
#include "library/aliases.h"
#include "library/scoped_ext.h"
#include "library/tactic/tactic.h"
#include "library/tactic/expr_to_tactic.h"
#include "library/tactic/exact_tactic.h"
#include "library/tactic/util.h"
#include "library/typed_expr.h"
#include "library/choice.h"
#include "library/let.h"
#include "library/constants.h"
#include "library/definitional/equations.h"
#include "library/tactic/assert_tactic.h"
#include "frontends/lean/builtin_exprs.h"
#include "frontends/lean/decl_cmds.h"
#include "frontends/lean/token_table.h"
#include "frontends/lean/calc.h"
#include "frontends/lean/begin_end_ext.h"
#include "frontends/lean/parser.h"
#include "frontends/lean/util.h"
#include "frontends/lean/tokens.h"
#include "frontends/lean/info_tactic.h"
#include "frontends/lean/info_annotation.h"
#include "frontends/lean/structure_cmd.h"
#include "frontends/lean/parse_rewrite_tactic.h"

namespace lean {
namespace notation {
static expr parse_Type(parser & p, unsigned, expr const *, pos_info const & pos) {
    if (p.curr_is_token(get_llevel_curly_tk())) {
        p.next();
        level l = p.parse_level();
        p.check_token_next(get_rcurly_tk(), "invalid Type expression, '}' expected");
        return p.save_pos(mk_sort(l), pos);
    } else {
        return p.save_pos(mk_sort(mk_level_placeholder()), pos);
    }
}

static expr parse_let(parser & p, pos_info const & pos);
static expr parse_let_body(parser & p, pos_info const & pos) {
    if (p.curr_is_token(get_comma_tk())) {
        p.next();
        return parse_let(p, pos);
    } else if (p.curr_is_token(get_in_tk())) {
        p.next();
        return p.parse_expr();
    } else {
        throw parser_error("invalid let declaration, 'in' or ',' expected", p.pos());
    }
}

static void parse_let_modifiers(parser & p, bool & is_visible) {
    while (true) {
        if (p.curr_is_token(get_visible_tk())) {
            is_visible = true;
            p.next();
        } else {
            break;
        }
    }
}

static expr parse_let(parser & p, pos_info const & pos) {
    parser::local_scope scope1(p);
    if (p.parse_local_notation_decl()) {
        return parse_let_body(p, pos);
    } else {
        auto id_pos     = p.pos();
        name id         = p.check_atomic_id_next("invalid let declaration, identifier expected");
        bool is_visible = false;
        optional<expr> type;
        expr value;
        parse_let_modifiers(p, is_visible);
        if (p.curr_is_token(get_assign_tk())) {
            p.next();
            value = p.parse_expr();
        } else if (p.curr_is_token(get_colon_tk())) {
            p.next();
            type = p.parse_expr();
            p.check_token_next(get_assign_tk(), "invalid declaration, ':=' expected");
            value = p.parse_expr();
        } else {
            parser::local_scope scope2(p);
            buffer<expr> ps;
            unsigned rbp = 0;
            auto lenv = p.parse_binders(ps, rbp);
            if (p.curr_is_token(get_colon_tk())) {
                p.next();
                type  = p.parse_scoped_expr(ps, lenv);
                type  = Pi(ps, *type, p);
            }
            p.check_token_next(get_assign_tk(), "invalid let declaration, ':=' expected");
            value = p.parse_scoped_expr(ps, lenv);
            value = Fun(ps, value, p);
        }
        expr v;
        if (type)
            v = p.save_pos(mk_typed_expr(*type, value), p.pos_of(value));
        else
            v = value;
        v = p.save_pos(mk_let_value(v), id_pos);
        p.add_local_expr(id, v);
        expr b = parse_let_body(p, pos);
        return p.save_pos(mk_let(id, v, b), pos);
    }
}

static expr parse_let_expr(parser & p, unsigned, expr const *, pos_info const & pos) {
    return parse_let(p, pos);
}

static expr parse_placeholder(parser & p, unsigned, expr const *, pos_info const & pos) {
    return p.save_pos(mk_explicit_expr_placeholder(), pos);
}

static expr parse_by(parser & p, unsigned, expr const *, pos_info const & pos) {
    parser::undef_id_to_local_scope scope(p);
    p.next();
    expr t = p.parse_tactic();
    return p.mk_by(t, pos);
}

static expr parse_begin_end_core(parser & p, pos_info const & pos, name const & end_token, bool nested = false) {
    if (!p.has_tactic_decls())
        throw parser_error("invalid 'begin-end' expression, tactic module has not been imported", pos);
    parser::undef_id_to_local_scope scope1(p);
    p.next();
    optional<expr> pre_tac = get_begin_end_pre_tactic(p.env());
    buffer<expr> tacs;
    bool first = true;

    auto add_tac = [&](expr tac, pos_info const & pos) {
        if (pre_tac)
            tac = p.mk_app({get_and_then_tac_fn(), *pre_tac, tac}, pos);
        tac = mk_begin_end_element_annotation(tac);
        tacs.push_back(tac);
    };

    try {
        while (!p.curr_is_token(end_token)) {
            if (first) {
                first = false;
            } else {
                auto pos = p.pos();
                p.check_token_next(get_comma_tk(), "invalid 'begin-end' expression, ',' expected");
                if (p.collecting_info()) {
                    expr info_tac = p.save_pos(mk_info_tactic_expr(), pos);
                    tacs.push_back(mk_begin_end_element_annotation(info_tac));
                }
            }
            if (p.curr_is_token(get_begin_tk())) {
                auto pos = p.pos();
                tacs.push_back(parse_begin_end_core(p, pos, get_end_tk(), true));
            } else if (p.curr_is_token(get_lcurly_tk())) {
                auto pos = p.pos();
                tacs.push_back(parse_begin_end_core(p, pos, get_rcurly_tk(), true));
            } else if (p.curr_is_token(end_token)) {
                break;
            } else if (p.curr_is_token(get_assert_tk())) {
                auto pos = p.pos();
                p.next();
                name id  = p.check_id_next("invalid 'assert' tactic, identifier expected");
                p.check_token_next(get_colon_tk(), "invalid 'assert' tactic, ':' expected");
                expr A   = p.parse_expr();
                expr assert_tac = p.save_pos(mk_assert_tactic_expr(id, A), pos);
                tacs.push_back(mk_begin_end_element_annotation(assert_tac));
            } else if (p.curr_is_token(get_have_tk())) {
                auto pos = p.pos();
                p.next();
                auto id_pos = p.pos();
                name id  = p.check_id_next("invalid 'have' tactic, identifier expected");
                p.check_token_next(get_colon_tk(), "invalid 'have' tactic, ':' expected");
                expr A   = p.parse_expr();
                expr assert_tac = p.save_pos(mk_assert_tactic_expr(id, A), pos);
                tacs.push_back(mk_begin_end_element_annotation(assert_tac));
                if (p.curr_is_token(get_bar_tk())) {
                    expr local = p.save_pos(mk_local(id, A), id_pos);
                    expr t     = parse_local_equations(p, local);
                    t      = p.mk_app(get_exact_tac_fn(), t, pos);
                    t      = p.save_pos(mk_begin_end_element_annotation(t), pos);
                    t      = p.save_pos(mk_begin_end_annotation(t), pos);
                    add_tac(t, pos);
                } else {
                    p.check_token_next(get_comma_tk(), "invalid 'have' tactic, ',' expected");
                    if (p.curr_is_token(get_from_tk())) {
                        // parse: 'from' expr
                        p.next();
                        auto pos = p.pos();
                        expr t = p.parse_expr();
                        t      = p.mk_app(get_exact_tac_fn(), t, pos);
                        t      = p.save_pos(mk_begin_end_element_annotation(t), pos);
                        t      = p.save_pos(mk_begin_end_annotation(t), pos);
                        add_tac(t, pos);
                    } else if (p.curr_is_token(get_proof_tk())) {
                        auto pos = p.pos();
                        p.next();
                        expr t = p.parse_expr();
                        p.check_token_next(get_qed_tk(), "invalid proof-qed, 'qed' expected");
                        t      = p.mk_app(get_exact_tac_fn(), t, pos);
                        t      = p.save_pos(mk_begin_end_element_annotation(t), pos);
                        t      = p.save_pos(mk_begin_end_annotation(t), pos);
                        add_tac(t, pos);
                    } else if (p.curr_is_token(get_begin_tk())) {
                        auto pos = p.pos();
                        tacs.push_back(parse_begin_end_core(p, pos, get_end_tk(), true));
                    } else if (p.curr_is_token(get_by_tk())) {
                        // parse: 'by' tactic
                        auto pos = p.pos();
                        p.next();
                        expr t = p.parse_tactic();
                        add_tac(t, pos);
                    } else {
                        throw parser_error("invalid 'have' tactic, 'by', 'begin', 'proof', or 'from' expected", p.pos());
                    }
                }
            } else if (p.curr_is_token(get_show_tk())) {
                auto pos = p.pos();
                expr t = p.parse_expr();
                t      = p.mk_app(get_exact_tac_fn(), t, pos);
                add_tac(t, pos);
            } else if (p.curr_is_token(get_match_tk()) || p.curr_is_token(get_assume_tk()) ||
                       p.curr_is_token(get_take_tk())  || p.curr_is_token(get_fun_tk())) {
                auto pos = p.pos();
                expr t = p.parse_expr();
                t      = p.mk_app(get_sexact_tac_fn(), t, pos);
                add_tac(t, pos);
            } else {
                auto pos = p.pos();
                expr t   = p.parse_tactic();
                add_tac(t, pos);
            }
        }
    } catch (exception & ex) {
        if (end_token == get_end_tk()) {
            // When the end_token is 'end', the default parser
            // sync_command does not work well because it will
            // interpret 'end' as a synchronization point because 'end' is also a command.
            while (!p.curr_is_token(get_end_tk())) {
                if (p.curr() == scanner::token_kind::Eof)
                    ex.rethrow();
                p.next();
            }
            p.next(); // consume 'end'
        }
        ex.rethrow();
    }
    auto end_pos = p.pos();
    p.next();
    if (tacs.empty()) {
        expr tac = get_id_tac_fn();
        if (pre_tac)
            tac = p.mk_app({get_and_then_tac_fn(), *pre_tac, tac}, end_pos);
        tac = mk_begin_end_element_annotation(tac);
        tacs.push_back(tac);
    }
    expr r = tacs[0];
    if (tacs.size() == 1) {
        // Hack: for having a uniform squiggle placement for unsolved goals.
        // That is, the result is always of the form and_then(...).
        r = p.mk_app({get_and_then_tac_fn(), r, mk_begin_end_element_annotation(get_id_tac_fn())}, end_pos);
    }
    for (unsigned i = 1; i < tacs.size(); i++) {
        r = p.mk_app({get_and_then_tac_fn(), r, tacs[i]}, end_pos);
    }
    r = p.save_pos(mk_begin_end_annotation(r), end_pos);
    if (nested)
        return r;
    else
        return p.mk_by(r, end_pos);
}

static expr parse_begin_end(parser & p, unsigned, expr const *, pos_info const & pos) {
    return parse_begin_end_core(p, pos, get_end_tk());
}

static expr parse_proof_qed_core(parser & p, pos_info const & pos) {
    expr r = p.parse_expr();
    p.check_token_next(get_qed_tk(), "invalid proof-qed, 'qed' expected");
    r      = p.mk_by(p.mk_app(get_exact_tac_fn(), r, pos), pos);
    return r;
}

static expr parse_proof(parser & p, expr const & prop) {
    if (p.curr_is_token(get_from_tk())) {
        // parse: 'from' expr
        p.next();
        return p.parse_expr();
    } else if (p.curr_is_token(get_proof_tk())) {
        auto pos = p.pos();
        p.next();
        return parse_proof_qed_core(p, pos);
    } else if (p.curr_is_token(get_begin_tk())) {
        auto pos = p.pos();
        return parse_begin_end_core(p, pos, get_end_tk());
    } else if (p.curr_is_token(get_by_tk())) {
        // parse: 'by' tactic
        auto pos = p.pos();
        p.next();
        expr t = p.parse_tactic();
        return p.mk_by(t, pos);
    } else if (p.curr_is_token(get_using_tk())) {
        // parse: 'using' locals* ',' proof
        auto using_pos = p.pos();
        p.next();
        parser::local_scope scope(p);
        buffer<expr> locals;
        buffer<expr> new_locals;
        while (!p.curr_is_token(get_comma_tk())) {
            auto id_pos = p.pos();
            expr l      = p.parse_id();
            if (!is_local(l))
                throw parser_error("invalid 'using' declaration for 'have', local expected", id_pos);
            expr new_l = l;
            binder_info bi = local_info(l);
            if (!bi.is_contextual())
                new_l = update_local(l, bi.update_contextual(true));
            p.add_local(new_l);
            locals.push_back(l);
            new_locals.push_back(new_l);
        }
        p.next(); // consume ','
        expr pr = parse_proof(p, prop);
        unsigned i = locals.size();
        while (i > 0) {
            --i;
            expr l     = locals[i];
            expr new_l = new_locals[i];
            pr = p.save_pos(Fun(new_l, pr), using_pos);
            pr = p.save_pos(mk_app(pr, l), using_pos);
        }
        return pr;
    } else {
        throw parser_error("invalid expression, 'by', 'begin', 'proof', 'using' or 'from' expected", p.pos());
    }
}

static expr parse_have_core(parser & p, pos_info const & pos, optional<expr> const & prev_local, bool is_visible) {
    auto id_pos       = p.pos();
    name id;
    expr prop;
    if (p.curr_is_token(get_visible_tk())) {
        p.next();
        is_visible    = true;
        id            = p.mk_fresh_name();
        prop          = p.parse_expr();
    } else if (p.curr_is_identifier()) {
        id = p.get_name_val();
        p.next();
        if (p.curr_is_token(get_visible_tk())) {
            p.next();
            p.check_token_next(get_colon_tk(), "invalid 'have/assert' declaration, ':' expected");
            is_visible = true;
            prop       = p.parse_expr();
        } else if (p.curr_is_token(get_colon_tk())) {
            p.next();
            prop      = p.parse_expr();
        } else {
            expr left = p.id_to_expr(id, id_pos);
            id        = p.mk_fresh_name();
            prop      = p.parse_led(left);
        }
    } else {
        id            = p.mk_fresh_name();
        prop          = p.parse_expr();
    }
    expr proof;
    if (p.curr_is_token(get_bar_tk()) && !prev_local) {
        expr fn = p.save_pos(mk_local(id, prop), id_pos);
        proof = parse_local_equations(p, fn);
    } else {
        p.check_token_next(get_comma_tk(), "invalid 'have/assert' declaration, ',' expected");
        if (prev_local) {
            parser::local_scope scope(p);
            p.add_local(*prev_local);
            auto proof_pos = p.pos();
            proof = parse_proof(p, prop);
            proof = p.save_pos(Fun(*prev_local, proof), proof_pos);
            proof = p.save_pos(mk_app(proof, *prev_local), proof_pos);
        } else {
            proof = parse_proof(p, prop);
        }
    }
    p.check_token_next(get_comma_tk(), "invalid 'have/assert' declaration, ',' expected");
    parser::local_scope scope(p);
    binder_info bi = mk_contextual_info(is_visible);
    expr l = p.save_pos(mk_local(id, prop, bi), pos);
    p.add_local(l);
    expr body;
    if (p.curr_is_token(get_then_tk())) {
        auto then_pos = p.pos();
        p.next();
        if (p.curr_is_token(get_assert_tk())) {
            p.next();
            is_visible = true;
        } else {
            p.check_token_next(get_have_tk(), "invalid 'then' declaration, 'have' or 'assert' expected");
            is_visible = false;
        }
        body  = parse_have_core(p, then_pos, some_expr(l), is_visible);
    } else {
        body  = p.parse_expr();
    }
    // remark: mk_contextual_info(false) informs the elaborator that prop should not occur inside metavariables.
    body = abstract(body, l);
    expr r = p.save_pos(mk_have_annotation(p.save_pos(mk_lambda(id, prop, body, bi), pos)), pos);
    return p.mk_app(r, proof, pos);
}

static expr parse_have(parser & p, unsigned, expr const *, pos_info const & pos) {
    return parse_have_core(p, pos, none_expr(), false);
}

static expr parse_assert(parser & p, unsigned, expr const *, pos_info const & pos) {
    return parse_have_core(p, pos, none_expr(), true);
}

static name * H_show = nullptr;
static expr parse_show(parser & p, unsigned, expr const *, pos_info const & pos) {
    expr prop  = p.parse_expr();
    if (p.curr_is_token(get_bar_tk())) {
        expr fn = p.save_pos(mk_local(*H_show, prop), pos);
        return parse_local_equations(p, fn);
    } else {
        p.check_token_next(get_comma_tk(), "invalid 'show' declaration, ',' expected");
        expr proof = parse_proof(p, prop);
        expr b = p.save_pos(mk_lambda(*H_show, prop, Var(0)), pos);
        expr r = p.mk_app(b, proof, pos);
        return p.save_pos(mk_show_annotation(r), pos);
    }
}

static expr parse_obtain(parser & p, unsigned, expr const *, pos_info const & pos) {
    if (!p.env().find(get_exists_elim_name()))
        throw parser_error("invalid use of 'obtain' expression, environment does not contain 'exists.elim' theorem", pos);
    // exists_elim {A : Type} {P : A → Prop} {B : Prop} (H1 : ∃ x : A, P x) (H2 : ∀ (a : A) (H : P a), B)
    buffer<expr> ps;
    auto b_pos = p.pos();
    unsigned rbp = 0;
    environment env = p.parse_binders(ps, rbp);
    unsigned num_ps = ps.size();
    if (num_ps < 2)
        throw parser_error("invalid 'obtain' expression, at least 2 binders expected", b_pos);
    p.check_token_next(get_comma_tk(), "invalid 'obtain' expression, ',' expected");
    p.check_token_next(get_from_tk(), "invalid 'obtain' expression, 'from' expected");
    expr H1 = p.parse_expr();
    p.check_token_next(get_comma_tk(), "invalid 'obtain' expression, ',' expected");
    expr b  = p.parse_scoped_expr(ps, env);
    expr H  = ps[num_ps-1];
    name H_name = local_pp_name(H);
    unsigned i = num_ps-1;
    while (i > 1) {
        --i;
        expr a      = ps[i];
        expr H_aux  = mk_local(p.mk_fresh_name(), H_name.append_after(i), mk_expr_placeholder(), mk_contextual_info(false));
        expr  H2    = Fun({a, H}, b);
        b = mk_app(mk_constant(get_exists_elim_name()), H_aux, H2);
        H = H_aux;
    }
    expr a  = ps[0];
    expr H2 = Fun({a, H}, b);
    expr r  = mk_app(mk_constant(get_exists_elim_name()), H1, H2);
    return p.rec_save_pos(r, pos);
}

static expr * g_not  = nullptr;
static unsigned g_then_else_prec = 45;

static expr parse_ite(parser & p, expr const & c, pos_info const & pos) {
    if (!p.env().find(get_ite_name()))
        throw parser_error("invalid use of 'if-then-else' expression, environment does not contain 'ite' definition", pos);
    p.check_token_next(get_then_tk(), "invalid 'if-then-else' expression, 'then' expected");
    expr t = p.parse_expr(g_then_else_prec);
    p.check_token_next(get_else_tk(), "invalid 'if-then-else' expression, 'else' expected");
    expr e = p.parse_expr(g_then_else_prec);
    return p.save_pos(mk_app(mk_constant(get_ite_name()), c, t, e), pos);
}

static expr parse_dite(parser & p, name const & H_name, expr const & c, pos_info const & pos) {
    p.check_token_next(get_then_tk(), "invalid 'if-then-else' expression, 'then' expected");
    expr t, e;
    {
        parser::local_scope scope(p);
        expr H = mk_local(H_name, c);
        p.add_local(H);
        auto pos = p.pos();
        t = p.save_pos(Fun(H, p.parse_expr(g_then_else_prec)), pos);
    }
    p.check_token_next(get_else_tk(), "invalid 'if-then-else' expression, 'else' expected");
    {
        parser::local_scope scope(p);
        expr H = mk_local(H_name, mk_app(*g_not, c));
        p.add_local(H);
        auto pos = p.pos();
        e = p.save_pos(Fun(H, p.parse_expr(g_then_else_prec)), pos);
    }
    return p.save_pos(mk_app(p.save_pos(mk_constant(get_dite_name()), pos), c, t, e), pos);
}

static expr parse_if_then_else(parser & p, unsigned, expr const *, pos_info const & pos) {
    pair<optional<name>, expr> ie = p.parse_qualified_expr();
    if (ie.first)
        return parse_dite(p, *ie.first, ie.second, pos);
    else
        return parse_ite(p, ie.second, pos);
}

static expr parse_calc_expr(parser & p, unsigned, expr const *, pos_info const &) {
    return parse_calc(p);
}

static expr parse_rewrite_tactic_expr(parser & p, unsigned, expr const *, pos_info const & pos) {
    return p.save_pos(parse_rewrite_tactic(p), pos);
}

static expr parse_esimp_tactic_expr(parser & p, unsigned, expr const *, pos_info const & pos) {
    return p.save_pos(parse_esimp_tactic(p), pos);
}

static expr parse_overwrite_notation(parser & p, unsigned, expr const *, pos_info const &) {
    name n = p.check_id_next("invalid '#' local notation, identifier expected");
    environment env = overwrite_notation(p.env(), n);
    return p.parse_scoped_expr(0, nullptr, env);
}

static expr parse_explicit_expr(parser & p, unsigned, expr const *, pos_info const & pos) {
    expr e = p.parse_expr(get_Max_prec());
    if (is_choice(e)) {
        buffer<expr> new_choices;
        for (unsigned i = 0; i < get_num_choices(e); i++)
            new_choices.push_back(p.save_pos(mk_explicit(get_choice(e, i)), pos));
        return p.save_pos(mk_choice(new_choices.size(), new_choices.data()), pos);
    } else {
        return p.save_pos(mk_explicit(e), pos);
    }
}

static expr parse_consume_args_expr(parser & p, unsigned, expr const *, pos_info const & pos) {
    expr e = p.parse_expr(get_Max_prec());
    if (is_choice(e)) {
        buffer<expr> new_choices;
        for (unsigned i = 0; i < get_num_choices(e); i++)
            new_choices.push_back(p.save_pos(mk_consume_args(get_choice(e, i)), pos));
        return p.save_pos(mk_choice(new_choices.size(), new_choices.data()), pos);
    } else {
        return p.save_pos(mk_consume_args(e), pos);
    }
}

static expr parse_sorry(parser & p, unsigned, expr const *, pos_info const & pos) {
    return p.mk_sorry(pos);
}

static expr parse_rparen(parser & p, unsigned, expr const * args, pos_info const & pos) {
    if (p.collecting_info())
        return p.save_pos(mk_extra_info(args[0], nulltag), pos);
    else
        return args[0];
}

static expr parse_proof_qed(parser & p, unsigned, expr const *, pos_info const & pos) {
    return parse_proof_qed_core(p, pos);
}

static expr parse_inaccessible(parser & p, unsigned, expr const * args, pos_info const & pos) {
    return p.save_pos(mk_inaccessible(args[0]), pos);
}

parse_table init_nud_table() {
    action Expr(mk_expr_action());
    action Skip(mk_skip_action());
    action Binders(mk_binders_action());
    expr x0 = mk_var(0);
    parse_table r;
    r = r.add({transition("_", mk_ext_action(parse_placeholder))}, x0);
    r = r.add({transition("by", mk_ext_action_core(parse_by))}, x0);
    r = r.add({transition("have", mk_ext_action(parse_have))}, x0);
    r = r.add({transition("assert", mk_ext_action(parse_assert))}, x0);
    r = r.add({transition("show", mk_ext_action(parse_show))}, x0);
    r = r.add({transition("obtain", mk_ext_action(parse_obtain))}, x0);
    r = r.add({transition("if", mk_ext_action(parse_if_then_else))}, x0);
    r = r.add({transition("(", Expr), transition(")", mk_ext_action(parse_rparen))}, x0);
    r = r.add({transition("?(", Expr), transition(")", mk_ext_action(parse_inaccessible))}, x0);
    r = r.add({transition("⌞", Expr), transition("⌟", mk_ext_action(parse_inaccessible))}, x0);
    r = r.add({transition("fun", Binders), transition(",", mk_scoped_expr_action(x0))}, x0);
    r = r.add({transition("Pi", Binders), transition(",", mk_scoped_expr_action(x0, 0, false))}, x0);
    r = r.add({transition("Type", mk_ext_action(parse_Type))}, x0);
    r = r.add({transition("let", mk_ext_action(parse_let_expr))}, x0);
    r = r.add({transition("calc", mk_ext_action(parse_calc_expr))}, x0);
    r = r.add({transition("rewrite", mk_ext_action(parse_rewrite_tactic_expr))}, x0);
    r = r.add({transition("esimp", mk_ext_action(parse_esimp_tactic_expr))}, x0);
    r = r.add({transition("#", mk_ext_action(parse_overwrite_notation))}, x0);
    r = r.add({transition("@", mk_ext_action(parse_explicit_expr))}, x0);
    r = r.add({transition("!", mk_ext_action(parse_consume_args_expr))}, x0);
    r = r.add({transition("begin", mk_ext_action_core(parse_begin_end))}, x0);
    r = r.add({transition("proof", mk_ext_action(parse_proof_qed))}, x0);
    r = r.add({transition("sorry", mk_ext_action(parse_sorry))}, x0);
    r = r.add({transition("match", mk_ext_action(parse_match))}, x0);
    init_structure_instance_parsing_rules(r);
    return r;
}

parse_table init_led_table() {
    parse_table r(false);
    r = r.add({transition("->", mk_expr_action(get_arrow_prec()-1))},    mk_arrow(Var(1), Var(1)));
    r = r.add({transition("<d", mk_expr_action(get_decreasing_prec()))}, mk_decreasing(Var(1), Var(0)));
    return r;
}
}
bool is_show_aux_name(name const & n) { return n == *notation::H_show; }

static parse_table * g_nud_table = nullptr;
static parse_table * g_led_table = nullptr;

parse_table get_builtin_nud_table() {
    return *g_nud_table;
}

parse_table get_builtin_led_table() {
    return *g_led_table;
}

void initialize_builtin_exprs() {
    notation::H_show        = new name("H_show");
    notation::g_not         = new expr(mk_constant(get_not_name()));
    g_nud_table             = new parse_table();
    *g_nud_table            = notation::init_nud_table();
    g_led_table             = new parse_table();
    *g_led_table            = notation::init_led_table();
}

void finalize_builtin_exprs() {
    delete g_led_table;
    delete g_nud_table;
    delete notation::H_show;
    delete notation::g_not;
}
}