Concatenating tuples as types

I'm trying to practice some template programming. Maybe there's a standard way to do this, and I would be thankful for such answers, but my main goal is to practice the template programming techniques, so I tried to implement it myself:

I need to concatenate multiple tuples, but as types, not like std::cat_tuple does it. So I need something like cat<std::tuple<int, float>, std::tuple<char, bool>, ...> to get std::tuple<int, float, char, bool, ...> as a type.

My current attempt failed with a is not a template error:

/* Concat tuples as types: */
template <typename first_t, typename... rest_t> struct cat {
    using type = typename _cat<first_t, typename cat<rest_t...>::type>::type;
                          ^^^^ cat is not a template
};
template <typename first_t, typename second_t>
struct cat<first_t, second_t> {
    using type = typename _cat<first_t, second_t>::type;
                          ^^^^ cat is not a template
};
// Concat two tuples:
template <typename, typename> struct _cat;
template <typename tuple_t, typename first_t, typename... rest_t>
struct _cat<tuple_t, std::tuple<first_t, rest_t...>> {
    using type = typename _cat<typename append<first_t, tuple_t>::type, std::tuple<rest_t...>>::type;
};
template <typename tuple_t, typename first_t>
struct _cat<tuple_t, std::tuple<first_t>> {
    using type = typename append<first_t, tuple_t>::type;
};
// Prepend element to tuple:
template <typename, typename> struct prepend;
template <typename elem_t, typename... tuple_elem_t>
struct prepend<elem_t, std::tuple<tuple_elem_t...>> {
    using type = std::tuple<elem_t, tuple_elem_t...>;
};
// Apppend element to tuple:
template <typename, typename> struct append;
template <typename elem_t, typename... tuple_elem_t>
struct append<elem_t, std::tuple<tuple_elem_t...>> {
    using type = std::tuple<tuple_elem_t..., elem_t>;
};

What may be causing the error?

Is this a good approach? It might be solved in a simpler way, but I wanted it to be multi-purpose (with the append/prepend operations etc.).