Naoto Onda

Kazumi Kasaura, Naoto Onda, Yuta Oriike et al.

Large Language Models have demonstrated significant promise in formal theorem proving. However, previous works mainly focus on solving existing problems. In this paper, we focus on the ability of LLMs to find novel theorems. We propose Conjecturing-Proving Loop pipeline for automatically generating mathematical conjectures and proving them in Lean 4 format. A feature of our approach is that we generate and prove further conjectures with context including previously generated theorems and their proofs, which enables the generation of more difficult proofs by in-context learning of proof strategies without changing parameters of LLMs. We demonstrated that our framework rediscovered theorems with verification, which were published in past mathematical papers and have not yet formalized. Moreover, at least one of these theorems could not be proved by the LLM without in-context learning, even in natural language, which means that in-context learning was effective for neural theorem proving. The source code is available at https://github.com/auto-res/ConjecturingProvingLoop.

Sho Sonoda, Kazumi Kasaura, Yuma Mizuno et al.

We formalize the generalization error bound using the Rademacher complexity for the Lean 4 theorem prover based on the probability theory in the Mathlib 4 library. Generalization error quantifies the gap between a learning machine's performance on given training data versus unseen test data, and the Rademacher complexity is a powerful tool to upper-bound the generalization error of a variety of modern learning problems. Previous studies have only formalized extremely simple cases such as bounds by parameter counts and analyses for very simple models (decision stumps). Formalizing the Rademacher complexity bound, also known as the uniform law of large numbers, requires substantial development and is achieved for the first time in this study. In the course of development, we formalize the Rademacher complexity and its unique arguments such as symmetrization, and clarify the topological assumptions on hypothesis classes under which the bound holds. As an application, we also present the formalization of generalization error bound for $L^2$-regularization models.

Naoto Onda, Kazumi Kasaura, Yuta Oriike et al.

We introduce LeanConjecturer, a pipeline for automatically generating university-level mathematical conjectures in Lean 4 using Large Language Models (LLMs). Our hybrid approach combines rule-based context extraction with LLM-based theorem statement generation, addressing the data scarcity challenge in formal theorem proving. Through iterative generation and evaluation, LeanConjecturer produced 12,289 conjectures from 40 Mathlib seed files, with 3,776 identified as syntactically valid and non-trivial, that is, cannot be proven by \texttt{aesop} tactic. We demonstrate the utility of these generated conjectures for reinforcement learning through Group Relative Policy Optimization (GRPO), showing that targeted training on domain-specific conjectures can enhance theorem proving capabilities. Our approach generates 103.25 novel conjectures per seed file on average, providing a scalable solution for creating training data for theorem proving systems. Our system successfully verified several non-trivial theorems in topology, including properties of semi-open, alpha-open, and pre-open sets, demonstrating its potential for mathematical discovery beyond simple variations of existing results.