In order to answer this question, you really need to define what you mean by "provable". As Ricky pointed out, any language with a type system is a language with a built-in proof system which runs every time you compile your program. These proof systems are almost always sadly impotent — answering questions like String vs Int and avoiding questions like "is the list sorted?" — but they are proof systems none-the-less.
Unfortunately, the more sophisticated your proof goals, the harder it is to work with a system which can check your proofs. This escalates pretty quickly into undecidability when you consider the sheer size of the class of problems which are undecidable on Turing Machines. Sure, you can theoretically do basic things like proving the correctness of your sorting algorithm, but anything more than that is going to be treading on very thin ice.
Even to prove something simple like the correctness of a sorting algorithm requires a comparatively sophisticated proof system. (note: since we have already established that type systems are a proof system built into the language, I'm going to talk about things in terms of type theory, rather than waving my hands still more vigorously) I'm fairly certain that a full correctness proof on list sorting would require some form of dependent types, otherwise you have no way of referencing relative values at the type level. This immediately starts breaking into realms of type theory which have been shown to be undecidable. So, while you may be able to prove correctness on your list sorting algorithm, the only way to do it would be to use a system which will also allow you to express proofs which it cannot verify. Personally, I find this very disconcerting.
There is also the ease-of-use aspect which I alluded to earlier. The more sophisticated your type system, the less pleasant it is to use. That's not a hard and fast rule, but I think it holds true for the most part. And as with any formal system, you will often find that expressing the proof is more work (and more error prone) than creating the implementation in the first place.
With all that out of the way, it's worth noting that Scala's type system (like Haskell's) is Turing Complete, which means that you can theoretically use it to prove any decidable property about your code, provided that you have written your code in such a way that it is amenable to such proofs. Haskell is almost certainly a better language for this than Java (since type-level programming in Haskell is similar to Prolog, while type-level programming in Scala is more similar to SML). I don't advise that you use Scala's or Haskell's type systems in this way (formal proofs of algorithmic correctness), but the option is theoretically available.
Altogether, I think the reason you haven't seen formal systems in the "real world" stems from the fact that formal proof systems have yielded to the merciless tyranny of pragmatism. As I mentioned, there's so much effort involved in crafting your correctness proofs that it's almost never worthwhile. The industry decided a long time ago that it was easier to create ad hoc tests than it was to go through any sort of analytical formal reasoning process.
