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Lean allows for third party type checkers. There are relatively small alternative type checkers for Lean, e.g. [1].
Lean's power lies in its elaborator that breaks down complex tactic-based proofs to a core proof language. This elaboration process can be extended with custom tactics, making it way more powerful than metamath.
[1] https://github.com/gebner/trepplein/tree/master/src/main/sca...
> Both tactics and proof terms are already quite old (for CS concepts, that is) and there hasn't been any real competition, so I imagine in the medium-term we'll just see refinements of them.
So it's really more of an issue of presentation? The techniques are fine? (I'm a professional programmer but an amateur logician, I really don't know what the big kids do.)
> I can't imagine anyone wanting to read latex source code over tactics/proof term code. Unless you're talking about rendered latex?
Yeah, you would generally only be looking at LaTeX source to debug your tools.
> But that's not something people can realistically work with.
I don't understand. I rarely work with it, but I was under the impression that it's pretty standard for writing math and science papers? Are there no WYSIWYG tools for working with rendered LaTeX? How do people work with it now, I guess is what I'm asking.
> Graphical proof assistants exist, but nobody uses them.
I just did a quick search and found two but they seem obscure:
https://en.wikipedia.org/wiki/Jape_(software)
https://github.com/liamoc/holbert
I guess the question I have is why does no one use them? Is it just inertia? I mean this is a thread about promoting the use of Lean et. al., so even the non-graphical, well-known tools are still kind of a niche, no?
Are graphical proof assistants only good for students and teaching, not "heavy lifting"?
In any event, I still feel that we can do better on the presentation side of things. (That's not controversial is it? The Lean folks are working on it?) I want to understand what kinds of software would help mathematicians.
> In a sense, tactics are a very weak form of this. Instead of just describing a proof as its structured in the system, they allow a proof author to also describe some of their intent or intuition. It's definitely why some people prefer tactics-based proofs.
That's pretty cool. :)
> I can't even imagine would formalizing something so subjective would even look beyond this. I'm not sure if it's even possible.
The Turing Machine is itself a formalization of a subjective process, eh?
If we get to the point where the machines can "read our minds" then it will be really easy. :) Heck, mathematicians can just watch videos of each other's mental imagery!
In the meantime, externalizing and formalizing these subjective intuitive processes with the machinery we've got seems like a fun and useful challenge, eh?