Fermat's Last Theorem in Lean 4
github.comI wonder if any piece of the lean code is in a shape which means it could be contributed to one of the existing Lean libraries.
My experience is that it takes a lot of human input to make Fable write code nice enough for a formalisation library others can work on. But since this is certainly a lot of prerequisites formalised as well, it would be nice if not all of the effort was wasted on one capstone proof! (Repost of a earlier comment, but I feel it fits better here)
Serious question: how do you prove that the Lean interpreter itself (not to mention the toolchain built around it) is error-free? Isn't this turtles all the way down to some degree?
You can’t, so you keep the kernel small. The Lean tactics language is rich, so users can autogenerate proofs for the truly trivial bits, but the core language is checkable in dependent type theory.
Kernel bugs, like compiler bugs, exist. As of now, a prover is considered good if it has no known bugs that would thwart a mathematician working in good faith. It’s not considered responsible yet for being impervious to adverse users, but that may change in the age of Ai.
So is there a chance that some of these AI-discovered proofs are actually Lean exploits?
This has happened: https://github.com/leanprover/lean4/issues/14576
would you assess Metamath systems more robust in adversarial settings, because the verifier is so short?
Metamath is short, which does make it easier to verify. In addition, because it's simple, there are many implementions. The set.mm Metamath database, the most popular, is checked by 5 independently implemented proof verifiers.
In addition to what others added below you might be interested in this postmortem https://leodemoura.github.io/blog/2026-8-24-postmortem-for-t...
You can only do so in another framework that might itself have bugs.
Lean is called that because the hope is the part that has to be correct by inspection ("the kernel") is small or "lean".
The kernel does have bugs sometimes.
I consider Metamath a lot more lean than "Lean", calling something such and so doesn't make it so in comparison to its peers.
You haven't thought that through. The regress obviously isn't infinite, and it bottoms out in things that are immediately true by inspection. And seriously, how likely is it that you have stumbled upon a fundamental problem with the whole notion of automated proof that no one in the field has thought of?
https://www.youtube.com/watch?v=RxV4PQcJ1fw ("The Proof in the Code: How Lean Is Quietly Rewriting Trust in Math")
My anecdotal experience is that while LLMs are quite good at closing theorems given an LSP to inspect the proof-tree, they suffer from similar kind of problems with proofs as they do with bigger codebases in any language -- finding reusable parts that can be built into libraries (that's lemmas in Lean 4 sense). However, Buzzard has many times said that he wouldn't care how big the proof is and how ugly it would be, as long as there would be a proof.
Kevin might not care, but I care more about building the foundation for future proofs and human understanding than I do about this particular result.
Is any piece you've seen in good enough shape to be in a Lean library?
I believe Lean supports a signature search mechanism. E.g. Haskell has Hoogle, Lean has Loogle. So in many ways it's actually easier to search for "library" code than in most languages, because the type tells you everything you need to know and you don't need to care about the implementation.
What a time to be alive stuff.
https://github.com/anthropics/fermats-last-theorem/blob/main...
I think an interesting problem, perhaps even more interesting problem, would be the shortest / most concise / easiest to understand (formally verifiably) proof.
This is a very impressive result. Bravo to that team.
Front-page discussion: https://news.ycombinator.com/item?id=49568506
I love that “grind” is a keyword.
It’s a tactic.
So is “simp”
Now we have what Fermat tried to write in the margin: aa2d8b34692b16c70f699536de0d8e75b9a3e9ef
Mine is much shorter though...
There is a shorter proof but since thinking ossified in the 20th century we won't be sociologicaly ready to accept it at this time. Much of math is playing according to arbitrary culturally enforced rules that are not natural in the sense of being minimum logical requirements. Take the axiom of infinity or the axiom of choice for example. Fundamental math need not be based on zfc but that is what we have chosen as our foundation because we elevated continuity, infinity to ontological higher status than distinguishability. In the past similar cultural barriers were present in math for example imaginary numbers are so called because the name originated as derision. It seems unlikely to suggest that math today is not similarly culturally constrained in certain areas and some things we find confounding are more so due to our choice of foundation than their intrinsic nature.