formal-conjectures

>or if the initial statement in Lean is correct. I suppose that's obvious. Is it obvious? The formal statement that Astra proved was written by a human, submitted to a DeepMind repository, and reviewed by human experts in a pull request. It's completely independent of OpenAI's effort. No expert in PDEs or formal verification has disputed it as far as I'm aware. https://github.com/google-deepmind/formal-conjectures/blob/8...

This is the formalization that was proven in Lean. As of now at least, it's believed to be a correct statement of the problem. https://github.com/google-deepmind/formal-conjectures/blob/8...

Does that contradict what I said? In that quote, it says that the NL proof does not correspond to the Lean proof. However, the statement of the theorem in Lean is independent from the NL proof. It comes from a DeepMind repository, which as far as I'm aware has been accepted by the community as a valid formalization of the original Clay Institute statement. https://github.com/google-deepmind/formal-conjectures/blob/8...

They actually used Lean statements that were carefully human-written and human-reviewed, from here https://github.com/google-deepmind/formal-conjectures/blob/m... This doesn't guarantee that the statement is correct (Lean cannot do that), but makes it highly likely.

Here's your Lean proof https://github.com/google-deepmind/formal-conjectures/pull/4...

This has existed for a few months, but there aren't any reports of (unsuccessful) attempts: https://github.com/google-deepmind/formal-conjectures

It may help to look at this example concretely: The natural-language statement of the problem is (from https://www.erdosproblems.com/728): > Let C>0 and ϵ>0 be sufficiently small. Are there infinitely many integers a,b,n with a≥ϵn and b≥ϵn such that a!b!∣n!(a+b−n)! and a+b>n+Clogn? The Lean-language statement of the problem (which can be done either by hand or by AI) is (from https://github.com/plby/lean-proofs/blob/f44d8c0e433ab285541...): ∀ᶠ ε : ℝ in [>] 0, ∀ C > (0 : ℝ), ∀ C' > C, ∃ a b n : ℕ, 0 < n ∧ ε * n < a ∧ ε * n < b ∧ a ! * b ! ∣ n ! * (a + b - n)! ∧ a + b > n + C * log n ∧ a + b < n + C' * log n Yes on the one hand, one needs to know enough about Lean to be sure that this formulation matches what we intend, and isn't stating something trivial. But on the other hand, thi

Do you regularly test your AI on the https://github.com/google-deepmind/formal-conjectures collection?

Last month, before this result came out, the question "Is Every Convex Polyhedron Rupert?" was added as a formal Lean statement to Google's Formal Conjectures repository: https://github.com/google-deepmind/formal-conjectures/blob/1... I wonder how feasible it would be to formalize this new proof in Lean.

for reference, the statement has been formalized in Lean in Deepmind's open problem database: https://github.com/google-deepmind/formal-conjectures/blob/e...

They aren't - they are currently collating a repository of formalized open problems: https://github.com/google-deepmind/formal-conjectures