Matěj Kripner

Jan Grebík, Pavel Hubáček, Martin Koutecký et al.

We report new results on six problems in mathematics and theoretical computer science, produced with the assistance of Bolzano, an open-source multi-agent LLM system. Bolzano orchestrates rounds of interaction between parallel prover agents and a verifier agent while maintaining a persistent knowledge base that is carried across rounds. Classified using the significance-autonomy taxonomy of Feng et al., four of the six results reach the level of publishable research, and three of the six were produced essentially autonomously by Bolzano. Our results provide evidence that LLMs can contribute meaningfully to mathematical research, complementing recent reports by Bubeck et al., Woodruff et al., and others.

Matěj Kripner, Michal Šustr, Milan Straka

Automated theorem proving (ATP) has been a classical problem in artificial intelligence since its inception, yet it remains challenging due to its vast state and action space. Large language models (LLMs) have recently emerged as a promising heuristic for ATP, but they lack correctness guarantees and thus require interaction with a proof verifier. Such interactions typically follow one of two approaches: black-box interaction, which does not utilize intermediate proof states, or white-box approaches, which allow for incremental proof construction and examination of intermediate states. While black-box approaches have directly benefited from recent LLM advances, white-box methods have comparatively lagged behind. In this paper, we address this gap by introducing LeanTree, which consists of (i) a tool built in the Lean 4 language that factorizes complex proof states into simpler, independent branches, and (ii) a dataset of these factorized intermediate states. Our white-box tooling offers several advantages over black-box approaches: it simplifies evaluation, reduces necessary context, generates richer training data, enables parallel search across multiple states, supports efficient reuse of states, and provides feedback in case of errors. Our preliminary results hint that white-box approaches outperform black-box alternatives in some settings.