We achieved a new state-of-the-art (41.2% vs 29.3%) on the miniF2F⁠(opens in a new window) benchmark, a challenging collection of high-school olympiad problems. Our approach, which we call statement curriculum learning, consists of manually collecting a set of statements of varying difficulty levels (without proof) where the hardest statements are similar to the benchmark we target. Initially our neural prover is weak and can only prove a few of them. We iteratively search for new proofs and re-train our neural network on the newly discovered proofs, and after 8 iterations, our prover ends up being vastly superior when tested on miniF2F.

Formal mathematics is an exciting domain to study because of (i) its richness, letting you prove arbitrary theorems which require reasoning, creativity and insight and (ii) its similarity to games—where AI has been spectacularly successful—in that it has an automated way of determining whether a proof is successful (i.e., verified by the formal system). As demonstrated in the trivial example below, proving a formal statement requires generating a sequence of proof steps, each proof step consisting in a call to a tactic.B

These tactics take mathematical terms as arguments and each tactic call will transform the current statement to prove, into statements that are easier to prove, until nothing is left to prove.

We observe that the capability to generate original mathematical terms required as arguments of tactics, which cannot be done without a neural language model, emerges from our training procedure. The proof below is an example of it: the proof step use n + 1 (entirely generated by our models) proposes to use n + 1 as a solution, the rest of the formal proof relying on the [ring_exp](https://leanprover-community.github.io/mathlib_docs/tactic/ring_exp.html) tactic to verify that it is indeed valid.