static transition parse_transition(parser & p, optional<parse_table> const & pt, name const & tk,
                                   buffer<expr> & locals, buffer<token_entry> & new_tokens, notation_entry_group grp,
                                   name const & pp_tk) {
    if (p.curr_is_token_or_id(get_binder_tk())) {
        p.next();
        unsigned rbp = parse_binders_rbp(p);
        return transition(tk, mk_binder_action(rbp), pp_tk);
    } else if (p.curr_is_token_or_id(get_binders_tk())) {
        p.next();
        unsigned rbp = parse_binders_rbp(p);
        return transition(tk, mk_binders_action(rbp), pp_tk);
    } else if (p.curr_is_identifier()) {
        unsigned default_prec = get_default_prec(pt, tk);
        name n   = p.get_name_val();
        p.next();
        action a = parse_action(p, tk, default_prec, locals, new_tokens, grp);
        expr local_type = mk_Prop(); // type used in notation local declarations, it is irrelevant
        expr l = mk_local(n, local_type);
        p.add_local(l);
        locals.push_back(l);
        return transition(tk, a, pp_tk);
    } else if (p.curr_is_quoted_symbol() || p.curr_is_keyword() ||
               p.curr_is_token(get_assign_tk()) || p.curr_is_command() || p.curr_is_eof()) {
        return transition(tk, mk_skip_action(), pp_tk);
    } else {
        throw parser_error("invalid notation declaration, quoted-symbol, identifier, "
                           "'binder', 'binders' expected", p.pos());
    }
}
static name parse_quoted_symbol_or_token(parser & p, buffer<token_entry> & new_tokens, bool & used_default, notation_entry_group grp) {
    used_default = false;
    if (p.curr_is_quoted_symbol()) {
        environment const & env = p.env();
        auto pp_tk = p.get_name_val();
        auto tks   = utf8_trim(pp_tk.to_string());
        auto tkcs  = tks.c_str();
        check_not_forbidden(tkcs);
        p.next();
        if (p.curr_is_token(get_colon_tk())) {
            p.next();
            unsigned prec = parse_precedence(p);
            new_tokens.push_back(mk_token_entry(tkcs, prec, grp));
        } else if (!get_precedence(env, tkcs, grp)) {
            new_tokens.push_back(mk_token_entry(tkcs, LEAN_DEFAULT_PRECEDENCE, grp));
            used_default = true;
        }
        return pp_tk;
    } else if (p.curr_is_keyword()) {
        auto tk = p.get_token_info().token();
        check_not_forbidden(tk.to_string().c_str());
        p.next();
        return tk;
    } else {
        throw parser_error("invalid notation declaration, symbol expected", p.pos());
    }
}
static notation_entry parse_notation_core(parser & p, bool overload, notation_entry_group grp, buffer<token_entry> & new_tokens, bool parse_only,
                                          unsigned priority) {
    buffer<expr>       locals;
    buffer<transition> ts;
    parser::local_scope scope(p);
    bool is_nud = true;
    optional<parse_table> pt;
    optional<parse_table> reserved_pt;
    if (p.curr_is_numeral()) {
        lean_assert(p.curr_is_numeral());
        mpz num = p.get_num_val().get_numerator();
        p.next();
        p.check_token_next(get_assign_tk(), "invalid numeral notation, `:=` expected");
        auto e_pos = p.pos();
        expr e     = p.parse_expr();
        check_notation_expr(e, e_pos);
        return notation_entry(num, e, overload, parse_only);
    } else if (p.curr_is_identifier()) {
        parse_notation_local(p, locals);
        is_nud = false;
        pt = get_led_table(p.env());
        if (grp != notation_entry_group::Reserve)
            reserved_pt = get_reserved_led_table(p.env());
    } else {
        pt = get_nud_table(p.env());
        if (grp != notation_entry_group::Reserve)
            reserved_pt = get_reserved_nud_table(p.env());
    }
    bool used_default = false;
    while ((grp != notation_entry_group::Reserve && !p.curr_is_token(get_assign_tk())) ||
           (grp == notation_entry_group::Reserve && !p.curr_is_command() && !p.curr_is_eof())) {
        name pp_tk = parse_quoted_symbol_or_token(p, new_tokens, used_default, grp).to_string();
        name tk = utf8_trim(pp_tk.to_string());
        if (auto at = find_next(reserved_pt, tk)) {
            // Remark: we are ignoring multiple actions in the reserved notation table
            transition const & trans = head(at).first;
            action const & a = trans.get_action();
            reserved_pt      = head(at).second;
            if (!p.curr_is_quoted_symbol())
                pp_tk = trans.get_pp_token();
            switch (a.kind()) {
            case notation::action_kind::Skip:
                if (!p.curr_is_quoted_symbol() && !p.curr_is_keyword() && !p.curr_is_token(get_assign_tk())) {
                    if (g_allow_local && !p.curr_is_token_or_id(get_binders_tk())) {
                        ts.push_back(parse_transition(p, pt, tk, locals, new_tokens, grp, pp_tk));
                        break;
                    }
                    p.check_token_or_id_next(get_binders_tk(),
                                             "invalid notation declaration, quoted-symbol, keyword or `:=` expected "
                                             "(declaration prefix matches reserved notation)");
                }
                ts.push_back(transition(tk, a, pp_tk));
                break;
            case notation::action_kind::Binder:
                if (g_allow_local && !p.curr_is_token_or_id(get_binder_tk())) {
                    ts.push_back(parse_transition(p, pt, tk, locals, new_tokens, grp, pp_tk));
                    break;
                }
                p.check_token_or_id_next(get_binder_tk(),
                                         "invalid notation declaration, 'binder' expected "
                                         "(declaration prefix matches reserved notation)");
                ts.push_back(transition(tk, a, pp_tk));
                break;
            case notation::action_kind::Binders:
                if (g_allow_local && !p.curr_is_token_or_id(get_binders_tk())) {
                    ts.push_back(parse_transition(p, pt, tk, locals, new_tokens, grp, pp_tk));
                    break;
                }
                p.check_token_or_id_next(get_binders_tk(),
                                         "invalid notation declaration, 'binders' expected "
                                         "(declaration prefix matches reserved notation)");
                ts.push_back(transition(tk, a, pp_tk));
                break;
            case notation::action_kind::Expr: case notation::action_kind::Exprs: case notation::action_kind::ScopedExpr:
            case notation::action_kind::Ext:  {
                if (g_allow_local && !p.curr_is_identifier()) {
                    ts.push_back(parse_transition(p, pt, tk, locals, new_tokens, grp, pp_tk));
                    break;
                }
                name n = p.check_id_next("invalid notation declaration, identifier expected "
                                         "(declaration prefix matches reserved notation)");
                if (p.curr_is_token(get_colon_tk())) {
                    if (g_allow_local) {
                        unsigned default_prec = get_default_prec(pt, tk);
                        action a = parse_action(p, tk, default_prec, locals, new_tokens, grp);
                        expr local_type = mk_Prop(); // type used in notation local declarations, it is irrelevant
                        expr l = mk_local(n, local_type);
                        p.add_local(l);
                        locals.push_back(l);
                        ts.push_back(transition(tk, a, pp_tk));
                        break;
                    } else {
                        throw parser_error("invalid notation declaration, invalid ':' occurrence "
                                           "(declaration prefix matches reserved notation)", p.pos());
                    }
                } else {
                    expr local_type = mk_Prop(); // type used in notation local declarations, it is irrelevant
                    expr l = mk_local(n, local_type);
                    p.add_local(l);
                    locals.push_back(l);
                    ts.push_back(transition(tk, a, pp_tk));
                    break;
                }
            }}
        } else {
            reserved_pt = optional<parse_table>();
            ts.push_back(parse_transition(p, pt, tk, locals, new_tokens, grp, pp_tk));
        }
        pt = find_match(pt, ts.back());
    }
    // for atomic notation where binding power was not set, we set it to max
    if (used_default && ts.size() == 1 && ts.back().get_action().kind() == notation::action_kind::Skip) {
        lean_assert(!new_tokens.empty());
        new_tokens.back().m_prec = get_max_prec();
    }
    expr n;
    if (grp == notation_entry_group::Reserve) {
        // reserve notation commands do not have a denotation
        lean_assert(p.curr_is_command() || p.curr_is_eof());
        expr dummy = mk_Prop(); // any expression without free variables will do
        n = dummy;
    } else {
        lean_assert(p.curr_is_token(get_assign_tk()));
        p.next();
        if (ts.empty())
            throw parser_error("invalid notation declaration, empty notation is not allowed", p.pos());
        n = parse_notation_expr(p, locals);
    }
    return notation_entry(is_nud, to_list(ts.begin(), ts.end()), n, overload, priority, grp, parse_only);
}
static action parse_action(parser & p, name const & prev_token, unsigned default_prec,
                           buffer<expr> & locals, buffer<token_entry> & new_tokens, notation_entry_group grp) {
    if (p.curr_is_token(get_colon_tk())) {
        p.next();
        if (p.curr_is_numeral() || p.curr_is_token_or_id(get_max_tk())) {
            unsigned prec = parse_precedence(p);
            return mk_expr_action(prec);
        } else if (p.curr_is_token_or_id(get_prev_tk())) {
            p.next();
            return mk_expr_action(get_precedence(p.env(), new_tokens, prev_token));
        } else if (p.curr_is_token_or_id(get_scoped_tk())) {
            p.next();
            return mk_scoped_expr_action(mk_var(0));
        } else {
            p.check_token_next(get_lparen_tk(), "invalid notation declaration, '(', numeral or 'scoped' expected");
            if (p.curr_is_token_or_id(get_foldl_tk()) || p.curr_is_token_or_id(get_foldr_tk())) {
                bool is_fold_right = p.curr_is_token_or_id(get_foldr_tk());
                p.next();
                auto prec = parse_optional_precedence(p);
                name sep  = parse_quoted_symbol_or_token(p, new_tokens, grp);
                expr rec;
                {
                    parser::local_scope scope(p);
                    p.check_token_next(get_lparen_tk(), "invalid fold notation argument, '(' expected");
                    parse_notation_local(p, locals);
                    parse_notation_local(p, locals);
                    p.check_token_next(get_comma_tk(),  "invalid fold notation argument, ',' expected");
                    rec  = parse_notation_expr(p, locals);
                    p.check_token_next(get_rparen_tk(), "invalid fold notation argument, ')' expected");
                    locals.pop_back();
                    locals.pop_back();
                }
                optional<expr> ini;
                if (!p.curr_is_token(get_rparen_tk()) && !p.curr_is_quoted_symbol())
                    ini = parse_notation_expr(p, locals);
                optional<name> terminator;
                if (!p.curr_is_token(get_rparen_tk()))
                    terminator = parse_quoted_symbol_or_token(p, new_tokens, grp);
                p.check_token_next(get_rparen_tk(), "invalid fold notation argument, ')' expected");
                return mk_exprs_action(sep, rec, ini, terminator, is_fold_right, prec ? *prec : 0);
            } else if (p.curr_is_token_or_id(get_scoped_tk())) {
                p.next();
                auto prec = parse_optional_precedence(p);
                expr rec;
                {
                    parser::local_scope scope(p);
                    parse_notation_local(p, locals);
                    p.check_token_next(get_comma_tk(),  "invalid scoped notation argument, ',' expected");
                    rec  = parse_notation_expr(p, locals);
                    locals.pop_back();
                }
                p.check_token_next(get_rparen_tk(), "invalid scoped notation argument, ')' expected");
                return mk_scoped_expr_action(rec, prec ? *prec : 0);
            } else {
                throw parser_error("invalid notation declaration, 'foldl', 'foldr' or 'scoped' expected", p.pos());
            }
        }
    } else {
        return mk_expr_action(default_prec);
    }
}