I have the following function:
template<class T>
T Check(int index);
How can I write a function, CheckTuple, which, given a tuple type, populates a tuple with calls to Check?
For example:
CheckTuple< std::tuple<int, float, std::string> >()
would return the following tuple:
std::make_tuple( Check<int>(1), Check<float>(2), Check<std::string>(3) )
The other questions I see involve unpacking a given tuple, not building one up this way.
Implementing what you're looking for becomes pretty simple using C++14's integer_sequence. If you don't have that available, here's a C++11 implementation written by Jonathan Wakely.
template<typename Tuple, int... I>
Tuple CallCheck(std::integer_sequence<int, I...>)
{
    return std::make_tuple(Check<typename std::tuple_element<I, Tuple>::type>(I)...);
}
template<typename Tuple>
Tuple CheckTuple()
{
    return CallCheck<Tuple>(std::make_integer_sequence<int, std::tuple_size<Tuple>::value>());
}
// Use it as 
auto tup = CheckTuple<std::tuple<int, float, std::string>>();
Live demo
Here is my working test implementation. (Perhaps someone has an idea of how to improve upon it in terms of conciseness. Can I get rid of TupleInfo somehow?)
#include <typeinfo>
#include <tuple>
#include <iostream>
template<class T>
T Check(int i) {
    std::cout << "getting a " << typeid(T).name() << " at position " << i << std::endl;
    return T();
}
template<typename Signature>
struct TupleInfo;
template<class T, class... Args>
struct TupleInfo< std::tuple<T, Args...> > {
    using Head = T;
    using Tail = std::tuple<Args...>;
};
template<int N, class Tuple>
struct TupleChecker {
    static Tuple CheckTuple() {
        auto t = std::make_tuple(Check<typename TupleInfo<Tuple>::Head>(N));
        return std::tuple_cat(t, TupleChecker<N+1, typename TupleInfo<Tuple>::Tail >::CheckTuple());
    }
};
template<int N>
struct TupleChecker<N, std::tuple<> > {
    static std::tuple<> CheckTuple() {
        return std::tuple<>();
    }
};
template<class Tuple>
Tuple CheckTuple() {
    return TupleChecker<1, Tuple>::CheckTuple();
}
int main() {
    std::tuple<> t0 = CheckTuple<std::tuple<> >();
    std::tuple<int> t1 = CheckTuple<std::tuple<int> >();
    std::tuple<int, float, std::string> t2 = CheckTuple<std::tuple<int, float, std::string> >();
    return 0;
}
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