Is that a result of how painful the tool is to use? (And by “tool” I mean any such tool - coq or whatever.)
What would need to be improved in the state-of-the-art of automated proof validation for the process to be less painful?
I think there are two parts:
1) Improve the theorem proving languages syntax so it's more intuitive, and so there is a wealth of "libraries" to build upon so you don't have to go from the basic axioms (of arithmetic or the reals for example) any time you wanted to prove something.
2) Provide some sort of computational intelligence to fill in natural gaps of proofs. I believe humans tend to leave a large number of more or less trivial gaps even in rigorous mathematical proofs that couldn't be avoided by language syntax. For this probably some kind of AI system would be ideal that tries to derive successive results and complete the proof automatically.
It's used for clustering statements and proofs, the idea being that if you're stuck trying to prove something, you might be able to find something "similar" in an existing library to use as inspiration. It's mostly to help with manual searches, rather than automatically solving anything (although there is some related work on doing that for Coq called SEPIA https://arxiv.org/abs/1505.07987 and ACL2 http://www.macs.hw.ac.uk/~ek19/ACL2ml/index.html ).
You can help its heuristics by providing it a set of potential proof facts explicitly.
The libraries available for it are already quite huge... sometimes in a few encodings as well. (valued, typed, functional etc.)
And none of its dependencies actually exist either. Like someone just said... relational database? Yeah we could write one of those...
To better understand current challenges with provers I can recommend this episode of the Type Theory podcast with Dan Licata as a guest: http://typetheorypodcast.com/2015/01/episode-3-dan-licata-on...
It's quite dense, but there are some bits that I, as a programmer, could understand.