[aSanchezStern]

Sounds like you've stumbled into the wonderful world of machine-learning guided proof synthesis! While I don't think the full system you're describing has been built yet, many similar systems and pieces have. In terms of the initial phase of supervised learning on existing proofs to prove new ones, there's TacticToe (https://arxiv.org/abs/1804.00596), Tactician (https://arxiv.org/pdf/2008.00120.pdf), CoqGym/ASTactic (https://arxiv.org/abs/1905.09381), Proverbot9001 (https://arxiv.org/abs/1907.07794), and Diva (https://dl.acm.org/doi/10.1145/3510003.3510138#sec-terms), among others. Most of these have some sort of language model within them, but if you're specifically looking for the LLM's that have been big recently, there's GPT-f (https://arxiv.org/abs/2009.03393), Baldur (https://arxiv.org/abs/2303.04910), and COPRA (https://arxiv.org/abs/2310.04353), though currently these models don't seem as effective as the specialized non-LLM tools. In terms of using reinforcement learning to learn beyond human written proofs, there's TacticZero (https://openreview.net/forum?id=edmYVRkYZv), this paper from OpenAI (https://arxiv.org/pdf/2202.01344.pdf), rlCoP (https://arxiv.org/abs/1805.07563), the HOList line of work (https://arxiv.org/pdf/1905.10006.pdf), and HyperTree Proof Search (https://arxiv.org/abs/2205.11491), as well as some in progress work I'm working on with a team at the University of Massachusetts.

[cubefox]

Wow, that's a great overview. While most of it goes over my head, it is interesting that quite a lot of research has been done in this direction.

The system I outlined above was inspired by this intriguing paper:

https://arxiv.org/abs/2207.14502

It is superficially about a different topic, namely generating and solving "programming puzzles", basically a type of unit test, that are then verified by the python interpreter. This system seems quite analogous to one where the programming puzzles are replaced by conjectures in a language like Lean, the proposed puzzle solutions are instead proposed proofs, and the python interpreter is replaced by the proof assistant.

I just discovered that there seems to be at least one guy working on applying a similar system to formal mathematics: https://www.cs.cmu.edu/~jlaurent/pdf/reports/thesis-proposal... He actually cites the paper above.