Just to make sure I am up to date with this problem. I know (or I think I do) that it is not yet proven that there are no non-trivial cycles for the collatz sequence (please correct me if I am wrong). But have we already reduced this problem to finding a non-trivial cycle? i.e. if we suppose that there are no non-trivial cycles at all, is it already proven that with this assumption the Collatz conjecture holds? As far as I can see in the literatures, even this is not shown (i.e. we can reduce the problem to proving whether there are no non-trivial cycles). But I may be wrong, so please correct me if I am wrong.
Edit: Most of my knowledge are on par with Lagarias annotated bibliography and I believe it has been a few years since this literature review.