Sequences whose constructor(s) are not constexpr can't always know their bounds at compile-time. What we would like is for Iterable and such typeclasses to provide defaults that also work for these runtime sequences. The difference between runtime sequences and compile-time sequences is that the is_empty method can only return a runtime bool for the former, while it returns a Bool<...> for the latter.

Attempt 1

This works, but it requires a lot of repetition:

template <typename Index, typename Iterable_>
static constexpr auto at_helper(Bool<true>, Index n, Iterable_ iterable)
{ return head(iterable); }

template <typename Index, typename Iterable_>
static constexpr auto at_helper(Bool<false>, Index n, Iterable_ iterable)
{ return at_helper(n == size_t<1>, n - size_t<1>, tail(iterable)); }

template <typename Index, typename Iterable_>
static constexpr auto at_helper(bool cond, Index n, Iterable_ iterable) {
    if (cond) return head(iterable);
    else      return at_helper(n == size_t<1>, n - size_t<1>, tail(iterable));
}

template <typename Index, typename Iterable_>
static constexpr auto at_impl(Index n, Iterable_ iterable)
{ return at_helper(n == size_t<0>, n, iterable); }

Attempt 2

The obvious improvement is to use an external if_ that factors the repetition away. Unfortunately, it does not work.

constexpr struct _lazy_if {
    template <typename Then, typename Else, typename ...Args>
    constexpr auto operator()(Bool<true>, Then t, Else, Args ...args) const
    { return t(args...); }

    template <typename Then, typename Else, typename ...Args>
    constexpr auto operator()(Bool<false>, Then, Else e, Args ...args) const
    { return e(args...); }

    template <typename Then, typename Else, typename ...Args>
    constexpr auto operator()(bool cond, Then t, Else e, Args ...args) const {
        if (cond) return t(args...);
        else      return e(args...);
    }
} lazy_if{};


template <typename Index, typename Iterable_>
static constexpr auto at_impl(Index n, Iterable_ iterable) {
    return lazy_if(n == size_t<0>,
        [](auto n, auto it) { return head(it); },
        [](auto n, auto it) { return at_impl(n - size_t<1>, tail(it)); },
        n, iterable
    );
}

Note that we pass Args... to the lazy_if because otherwise we would need to return two different lambdas from the same function in the bool case.

I think the reason for it not working is because we introduce a new intermediate lambda

[](auto n, auto it) { return at_impl(n - size_t<1>, tail(it)); },

in the then branch, which causes the compiler to hit the instantiation limit while trying to recursively instantiate at_impl when it instantiates the lambda.

Attempt 2.5

Using a different if_, but it still does not work.

constexpr struct _tri_if {
    template <typename Then, typename Else, typename Maybe, typename ...Args>
    constexpr auto operator()(Bool<true> cond, Then t, Else, Maybe, Args ...args) const
    { return t(cond, args...); }

    template <typename Then, typename Else, typename Maybe, typename ...Args>
    constexpr auto operator()(Bool<false> cond, Then, Else e, Maybe, Args ...args) const
    { return e(cond, args...); }

    template <typename Then, typename Else, typename Maybe, typename ...Args>
    constexpr auto operator()(bool cond, Then, Else, Maybe m, Args ...args) const
    { return m(cond, args...); }
} tri_if{};

template <typename Index, typename Iterable_>
static constexpr auto at_impl(Index n, Iterable_ iterable) {
    return tri_if(n == size_t<0>,
        [](auto cond, auto n, auto it) { return head(it); },
        [](auto cond, auto n, auto it) { return at_impl(n - size_t<1>, tail(it)); },
        [](bool cond, Index n, Iterable_ it) {
            return cond ? head(it) : at_impl(n - size_t<1>, tail(it));
        },
        n, iterable
    );
}

Attempt 3

Here, I'm trying to use structs to make everything more explicit. Still does not work because the problem is when instantiating the inner apply function. See the comment below.

template <typename Cond, typename = void>
struct at_helper;

template <typename Dummy>
struct at_helper<Bool<true>, Dummy> {
    template <typename Index, typename Iterable_>
    static constexpr auto apply(Bool<true>, Index n, Iterable_ iterable) {
        return head(iterable);
    }
};

template <typename Dummy>
struct at_helper<Bool<false>, Dummy> {
    template <typename Index, typename Iterable_>
    static constexpr auto apply(Bool<false>, Index n, Iterable_ iterable) {
        return at_helper<decltype(n == size_t<1>)>::apply(
            n == size_t<1>, n - size_t<1>, tail(iterable)
        );
    }
};

template <typename Dummy>
struct at_helper<bool, Dummy> {
    template <typename Index, typename Iterable_>
    static constexpr auto apply(bool cond, Index n, Iterable_ iterable) {
        if (cond)
            return head(iterable);
        else
            // This triggers infinite recursive instantiations.
            // return at_helper<Bool<false>>::apply(false_, n, iterable);

            // This works. decltype(...) is actually bool.
            return at_helper<decltype(n == size_t<1>)>::apply(
                n == size_t<1>, n - size_t<1>, tail(iterable)
            );
    }
};

template <typename Index, typename Iterable_>
static constexpr auto at_impl(Index n, Iterable_ iterable) {
    return at_helper<decltype(n == size_t<0>)>::apply(n == size_t<0>, n, iterable);
}

Attempt 4

It works, but it's not a solution. I basically use the preprocessor to generate some of the duplicate code.

#define HACK(f)                                                             \
template <TPARAMS>                                                          \
static constexpr auto f(Bool<true>, PARAMS)                                 \
{ return TRUE_BRANCH; }                                                     \
                                                                            \
template <TPARAMS>                                                          \
static constexpr auto f(Bool<false>, PARAMS)                                \
{ return FALSE_BRANCH; }                                                    \
                                                                            \
template <TPARAMS>                                                          \
static constexpr auto f(bool cond, PARAMS) {                                \
    if (cond) return TRUE_BRANCH;                                           \
    else      return FALSE_BRANCH;                                          \
}                                                                           \
/**/

#define TPARAMS typename Index, typename Iterable_
#define PARAMS Index n, Iterable_ iterable
#define TRUE_BRANCH head(iterable)
#define FALSE_BRANCH at_helper(n == size_t<1>, n - size_t<1>, tail(iterable))

HACK(at_helper);

template <typename Index, typename Iterable_>
static constexpr auto at_impl(Index n, Iterable_ iterable)
{ return at_helper(n == size_t<0>, n, iterable); }

Attempt 5

Trying to factor out the branches into lambdas. This recursively instantiates stuff too.

BOOST_HANA_CONSTEXPR_LAMBDA auto true_branch = [](auto at_impl, auto n, auto iterable)
{ return head(iterable); };

BOOST_HANA_CONSTEXPR_LAMBDA auto false_branch = [](auto at_impl, auto n, auto iterable)
{ return at_impl(n - size_t<1>, tail(iterable)); };

template <typename F, typename Index, typename Iterable_>
static constexpr auto at_helper(F at_impl, Bool<true>, Index n, Iterable_ iterable)
{ return true_branch(at_impl, n, iterable); }

template <typename F, typename Index, typename Iterable_>
static constexpr auto at_helper(F at_impl, Bool<false>, Index n, Iterable_ iterable)
{ return false_branch(at_impl, n, iterable); }

template <typename F, typename Index, typename Iterable_>
static constexpr auto at_helper(F at_impl, bool cond, Index n, Iterable_ iterable) {
    if (cond) return true_branch(at_impl, n, iterable);
    else      return false_branch(at_impl, n, iterable);
}

BOOST_HANA_CONSTEXPR_LAMBDA auto at_impl = fix(
    [](auto at_impl, auto n, auto iterable)
    { return at_helper(at_impl, n == size_t<0>, n, iterable); }
);

Attempt 6

Make everything explicit by using structs.

template <typename Condition, typename Index, typename Iterable_>
struct at_helper;

template <typename AtHelper>
struct false_branch {
    template <typename Index, typename It>
    static constexpr auto apply(Index n, It it) {
        return AtHelper::apply(n == size_t<1>, n - size_t<1>, tail(it));
    }
};

template <typename Index, typename Iterable_>
struct at_helper<Bool<true>, Index, Iterable_> {
    static constexpr auto apply(Bool<true>, Index n, Iterable_ it)
    { return head(it); }
};

template <typename Index, typename Iterable_>
struct at_helper<Bool<false>, Index, Iterable_> {
    static constexpr auto apply(Bool<false>, Index n, Iterable_ it) {
        // This works
        return false_branch<at_helper<decltype(n == size_t<1>), decltype(n - size_t<1>), decltype(tail(it))>>::apply(n, it);

        // This works too.
        // return at_helper<decltype(n == size_t<1>), decltype(n - size_t<1>), decltype(tail(it))>::
        //        apply(n == size_t<1>, n - size_t<1>, tail(it));
    }
};

template <typename Index, typename Iterable_>
struct at_helper<bool, Index, Iterable_> {
    static constexpr auto apply(bool cond, Index n, Iterable_ it) {
        if (cond) return head(it);
                  // Surprisingly, using the false_branch<...> just like above
                  // does not work here. WTF!
        else      return at_helper<decltype(n == size_t<1>), decltype(n - size_t<1>), decltype(tail(it))>::
                         apply(n == size_t<1>, n - size_t<1>, tail(it));
    }
};

template <typename Index, typename Iterable_>
static constexpr auto at_impl(Index n, Iterable_ iterable) {
    return at_helper<decltype(n == size_t<0>), decltype(n), decltype(iterable)>::
           apply(n == size_t<0>, n, iterable);
}

Rationales

Why not provide forward declaration headers as in the MPL11?

A lot of methods are in fact lambdas. Since we can't forward declare those, it makes little sense to have forward declaration headers.

Todo

Write a tutorial.
Provide a Main page for the Doxygen documentation.
Consider making function objects automatically curriable. This could allow super sexy stuff like:

template <>
struct Iterable<List> {
    static constexpr auto length_impl = foldl(some_lambda, size_t<0>);
};