Helper function involving std::move - c++

Question

codereview would not allow hypothetical code, so I am posting here instead. Given this code:
#include <string>
struct foo {
std::string a, b;
foo (const std::string& a, const std::string& b) : a(a), b(b) { }
};
template <typename T> void bar (const T&) { }
template <typename T> void bar (T&&) { }
void execute (const foo& f) {
// etc...
bar(f.a);
// etc...
bar(f.b);
// etc...
bar(f);
// etc...
}
void execute (foo&& f) {
// etc...
bar(std::move(f.a));
// etc...
bar(std::move(f.b));
// etc...
bar(std::move(f));
// etc...
}
int main() {}
I want to write execute_helper to handle both execute overloads, since they are identical except for the std::move calls. Here is what I have, and I want to know if I did it correctly. I'm unsure about using T& and T in my check_move overloads, and whether passing T by value worsens the time complexity or not.
#include <string>
#include <iostream>
struct foo {
std::string a, b;
foo (const std::string& a, const std::string& b) : a(a), b(b) { }
};
template <typename T> void bar (const T&) { }
template <typename T> void bar (T&&) { }
template <typename T>
T check_move (const foo&, T& t) { std::cout << "std::move not called.\n"; return t; }
template <typename T>
T&& check_move (foo&&, T t) { std::cout << "std::move called.\n"; return std::move(t); }
template <typename F>
void execute_helper (F&& f) {
// etc...
bar(check_move(std::forward<F>(f), f.a));
// etc...
bar(check_move(std::forward<F>(f), f.b));
// etc...
bar(check_move(std::forward<F>(f), std::forward<F>(f)));
// etc...
}
void execute (const foo& f) {
execute_helper(f);
}
void execute (foo&& f) {
execute_helper(std::forward<foo>(f));
}
// Test
foo make_foo() {
return foo("hello", "bye");
}
int main() {
foo f("hello", "bye");
execute(f);
execute(make_foo());
}
Output:
std::move not called.
std::move not called.
std::move not called.
std::move called.
std::move called.
std::move called.