alt Hacker NewsMetamath contributor here! Each proving tool has its pros and cons, but always happy to see Metamath noted :-).
One thing that's cool about Metamath is that the axioms are not built-in. It's true that the most-used system is based on classical logic and ZFC set theory https://us.metamath.org/mpeuni/mmset.html ... but you don't have to use that system. There's a well-maintained database using intuitionistic logic: https://us.metamath.org/ileuni/mmil.html ; on the so-called "New Foundations" (a many-sorted system): https://us.metamath.org/nfeuni/mmnf.html ; on HOL https://us.metamath.org/holuni/mmhol.html ; and you can make your own if you want to.
In Metamath the proofs hide absolutely nothing. There's no hand-waving "it's obvious that". Every step in a proof must be rigorously and directly proven by some axiom or a previously-proven theorem with absolutely no exceptions. This also means that while finding proofs can be hard, verifying proofs is fast. I just ran a proof verification run of over 47,000 theorems in 6.35 seconds. In the Metamath Proof Explorer / set.mm database (the one with classical logic and ZFC), we routinely run multiple provers by different people on every proposed change. So not only is the kernel small, it's implemented by multiple different programs, making it extremely unlikely we'll accept an invalid proof.
This video I made years ago summarizes Metamath: https://www.youtube.com/watch?v=8WH4Rd4UKGE
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What semantics do you use for your HOL library? I scanned around but documentation on that page is a bit sparse. The github repo goes to a random user's page, and all I could find there was this unrelated repo: https://github.com/digama0/HOL
There are other forms of logic? is intuitionistic logic as rigorous? fascinating
edit: the link says it is a weakening. if it is weakened, how can you prove the same stuff? i am a bit confused but i can see how it is useful for smarter people than me!
Metamath is interesting for education in the principles of formal proofs, because:
1. The Metamath Book is self-contained well written, simple introduction.
2. Everything is explicit, nothing is hidden.
3. Because of extremely simple syntax and semantics it's easy write own proof verifier.
The drawbacks are:
1. Non-standard approach to free and bound variables.
2. Very weak logic in comparison to Lean. The logical syntax rules and logical semantics rules used for proving mathematical statements in Metamath, have to be explicitly specified in the Metamath database. In Lean they are part of the Lean kernel. An bug in this Metamath specification is equivalent to a bug in Lean kernel.
3. Almost no proof automation. No tactics. Long, hard to read proofs.
I would characterize Metamath as the assembly language of proof assistants, very low level and detailed way to write proofs.