>the core reasons why in the vast majority of cases either no specification/documentation exists

I feel that is much too pessimistic.

>will only cover a small case of the actual specification.

If the same applies to proofs: so be it. Don't let perfect be the enemy of good!

>For example I would bet money that not a single function in the C, C++, Java and Python standard libraries is fully specified, in the sense of nailing down the program up to observational equivalence.

I'd imagine so as well, but I think that's more indicative of how even a (superficially) simple language like C is not all that amenable to specification.

> A lot of code refactoring I've done was trivial (e.g. changing the order or arguments), but ripples through the program and proof structure.

This is not my experience. If you use sufficient proof automation, something like this should have next to no impact on proof structure. Univalence is useful, but a lot of refactoring is not just switching to isomorphic representations, so I'm convinced that larger scale proof automation is way more essential than HoTT.

> Java exception specifications

I'm not convinced that this is fundamental to specifying exceptions rather than just Java having particularly poor ergonomics for it. I've never met a person that actually liked implicit exceptions and if you ask people who dislike exceptions, that's often one key part of it.

> In contrast, expressive type-theories constantly force you to prove a lot of trivialities.

For all large dependently typed languages that actually bill themselves as programming languages (Agda, Idris, Lean) there is some sort of 'unsafe' feature that allows you to turn the termination checker off - at your own peril of course. But you only pay for what you use, if you don't need the additional correctness, you don't need to put in any more work, just like with unit tests.

(There are also ways to safely defer termination proofs, but to be fair the experience isn't the best currently.)

>I don't know what you mean by declarative (other than: leaving out some detail).

Specs tell you the what but not (necessarily) the how. Take the spec for all sorting algorithms (giving observational equivalence):

1. The output list is a permutation of the input

2. The output is sorted in regards to some predicate.

That's a couple lines at most (or a one-liner if you don't count the building blocks like the definition of a permutation), which is a good amount shorter and easier to verify than e.g. a full heapsort implementation.

>But they cannot be smaller in general: if every program P had a full specification S that was shorter [...] then you've an impossibly strong compressor

The compression is stripping away the 'how', that's why you can't 'write a spec for a spec' and compress further.

>What you see in practise is that you only specify some properties of the program you are working on.

Sure, writing a fully specified program of non-trivial size is currently really only possible if you're willing to ~waste~ invest a decade or more.

>Moreover, if you only work with a partial specification, you can ask the question: what level of partiality in my specification gives me the best software engineering results.

Why would you assume that there is a single level of partiality that gives the best results? I agree that HM style types are a great fit for 'general' programming because it has such low impedance, however I also believe that most programs have some areas where they would benefit from more specification. (I think people have a clear desire for this and that it's at least partially responsible for some of the crazy type hackery as seen in Haskell or Scala, which could often be greatly simplified with a richer type system.)

Having a richer type system doesn't mean that you always have to fully use it. It's perfectly possible to just use a HM style/System F fragment. Using dependent types just for refinement is already one of the dominant styles for verification. If dependent types ever become mainstream in any way, I imagine it will also be largely in that style.