Non-constructive Proofs and Productivity

Languages in the Dependent ML family allow users to ensure termination by using a termination metric, a number that is strictly decreasing on recursive calls in the definition of the function.

There is a counterpart to termination for co-recursive structures, where we need to show that a function produces values indefinitely. So, for the Collatz conjecture, we want to show termination for:

collatz n = collatz' n []
collatz' 1 r = 1:r
collatz' n r =
    if n `mod` 2 == 0
    then collatz' (n `div` 2) (n:r)
    else collatz' (3*n + 1) (n:r)

But for the twin prime conjecture we want to show the productivity of:

data Stream a = Stream a (Stream a)
twins = twins' 3
twins' n =
    if (primep n) && (primep (n+2))
    then Stream (n,n+2) (twins' (n+2))
    else twins (n+2)

I don't know if any of the members of the DML family can check productivity. Imagine they could, and imagine we had a non-constructive proof of the twin primes conjecture. Could we use that proof to allow a definition of twins to be marked productive?

Most of the time, we talk about the Curry-Howard correspondence in reference to intuitionist logic and functional programming, but non-constructive proofs of productivity seem like a good way to be able to use more powerful logic in programming, since they don't have to change the implementation of the function they are attached to.