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Interesting choice of the proof assistant though - some specific parts of the Lean's core are not completely decidable, moreover the upcoming Lean 4 version is

• ulber 5y ago
  I wasn't aware of Lean not being sound and a quick search didn't come up with anything related to that. Could you link a source?
  • martincmartin 5y ago
    See other comment. It's sound but not decidable.
    • kevinbuzzard 5y ago
      Basically it turned out that theoretical undecidability did not matter in practice, because Scholze mathematics relies so little on definitional equality. We prove theorems with `simp` not `refl`. Pierre-Marie Pédrot is quoted above as saying that various design decisions are "breaking everything around", but we don't care that our `refl` is slightly broken because it is regarded as quite a low-level tool for the tasks (eg proving theorems of Clausen and Scholze) that we are actually interested in, and I believe our interests contrast quite a lot with the things that Pédrot is interested in.
      • kmill 5y ago
        And in "Lean style", refl proofs are a bit distasteful from a software engineering point of view because they pierce through the API of a mathematical object. (In the language of OOP, it can break encapsulation.) It tends to be a good idea to give some definition, then prove a bunch of lemmas that characterize the object, and finally forget the definition.
      • bryan0 5y ago
        Can you explain to a non-mathematician how you can prove anything without refl (which I assume is the statement “x=x is true”) ?
        • digama0 5y ago
          The jargon is a bit confusing sometimes. In Lean, "refl" does a whole lot more than prove x=x. It is of course available if you want to prove x=x, but the real power of "refl" is that it also proves x=y where x and y are definitionally equal. Or at least that's the idea; it turns out that lean's definitional equality relation is not decidable so sometimes it will fail to work even when x and y are defeq, and this is the theoretically distasteful aspect that came up on the linked Coq issue. In practice, the theoretical undecidability issue never happens, however there is a related problem where depending on what is unfolded a proof by "refl" can take seconds to minutes, and if alternative external proof-checkers don't follow exactly the same unfolding heuristics it can turn a seconds long proof into a thousand-year goose chase. By comparison, methods like "simp" have much more controllable performance because they actually produce a proof term, so they tend to be preferable.
          • bryan0 5y ago
            Thanks for the explanation. Is the defeq undecidability a bug of Lean that should be fixed in the future? Or is it an intentional design decision for it to function properly for other types of proofs?
            • Jweb_Guru 5y ago
              I am quite confident that the developers of lean consider it a feature (or at least, not a bug). Though I'm also not sure why they are building on a complex metatheory like CiC if they are willing to accept undecidable typechecking since you can simplify a lot of things if you give that up.
              • foooobar 5y ago
                What simplifications do you have in mind? I think one of the reasons is that Lean's approximation of defeq still works well enough in practice. As mentioned by others, you never really run into the kind of counter examples that break theoretical properties, but instead into examples where checking defeq is inacceptably slow, which remains an issue even when defeq is decidable. From my understanding CiC is used because it allows using a unified language for proofs, specifications, programs and types, which greatly simplifies the system. For example, in Lean 4, more or less the entire system is built on the same core term language: Proofs, propositions / specifications, programs, tactics (i.e. meta programs), the output of tactics and even Lean 4 itself are all written in that same language, even if they don't always use the same method of computation. In my eyes, for the purposes of writing and checking proofs, CiC is quite complex; but if you want to capture all of the above and have them all checked by the same kernel, CiC seems like a fairly minimal metatheory to do so, even with undecidable type checking.
• deadbeef57 5y ago
  The claim that Lean's core is not completely sound is FUD. Completely bogus. You might be confused by the fact that Lean's definitional equality is not decidable, but that doesn't mean it isn't sound. Nobody has ever found a proof of `false` so far in Lean 3. The choice for Lean is actually quite natural: (i) it has a large and coherent library of mathematics to build such a project upon. And (ii), it has a substantial user base of mathematicians somewhat familiar with the subject at hand (i.e., condensed mathematics). The initial challenge by Peter Scholze was directed at all theorem proving communities. As far as I know, only the Lean community took up the challenge. (Concerning Lean 4: yes, mathlib will have to be ported. And yes, people are working hard on this. I think that none of the theorem provers so far are ready for wide-scale use in mathematics. And that's why it is important to iterate fast. The Lean community is not afraid of redesigns and large refactors when discovering better approaches.)
  • xvilka 5y ago
    > You might be confused by the fact that Lean's definitional equality is not decidable, but that doesn't mean it isn't sound. Nobody has ever found a proof of `false` so far in Lean 3. You are right, my bad. Taking my words back on that. A bit more details from the Pierre-Marie Pédrot: > Lean does not escape this limitation, as it defines the equality type inductively as in Coq. Rather, in a mastermind PR campaign, they successfully convinced non-experts of type theory that they could give them quotient types without breaking everything around. > The first thing they do is to axiomatize quotient types, which is somewhat fine and can be done the same in Coq. As any use of axioms, this breaks canonicity, meaning your computations may get stuck on some opaque term. (While slightly cumbersome, axiomatizing quotients already solves 99,9% of the problems of the working mathematician). > Now, were they do evil things that would make decent type theorists shake in terror, they add a definitional reduction rule over the quotient induction principle. We could do the same in the Coq kernel, but there is a very good reason not to do this. Namely, this definitional rule breaks subject reduction, which is a property deemed more important than loss of canonicity. > In Lean, you can write a term t : A where t reduces to u, but u doesn't have type A (or equivalently may not typecheck at all). This is BAD for a flurry of reasons, mark my words. Reinstating it while keeping quotients requires an undecidable type checking algorithm, which is another can of worms you don't want to wander into. The same kind of trouble arises from similar Lean extensions like choice.
• generationP 5y ago
  As to decidability: I suspect you mean that Lean's foundations are non-constructive. I'd rather they weren't, but with what Peter Scholze is doing, I'd be surprised if the arguments could be made constructive within just a few years from their conception. This usually takes much longer (e.g., constructive proofs of the Quillen-Suslin theorem appeared in print just a few years ago). About incompatibility: Last time I checked (some 3 or so years ago), Coq was not backwards compatible either, and libraries had to be ported manually with each update. Sadly, as this is the one greatest anti-feature that is currently putting me off proof assistants, but probably it needs some time, and the quickest way to get there is to maximize usage. From what I know about Coq and Lean, I suspect Lean will stabilize faster than Coq does, due to it being more "declarative" (Coq is based too much on complex tactics, which are hard to make stable by design).