Compile to Compress: Boosting Formal Theorem Provers by Compiler Outputs

Large language models (LLMs) have demonstrated significant potential in formal theorem proving, yet state-of-the-art performance often necessitates prohibitive test-time compute via massive roll-outs or extended context windows. In this work, we address this scalability bottleneck by exploiting an informative structure in formal verification: the observation that compilers map a vast space of diverse proof attempts to a compact set of structured failure modes. We introduce a learning-to-refine framework that leverages this compression to perform efficient learning and proof exploration. We perform tree search that corrects errors locally conditioned on explicit verifier feedback, thereby circumventing the costs associated with accumulating a long history of proof attempts. Extensive evaluations show that our method consistently amplifies the reasoning capabilities of base provers across varying scales. Notably, our approach achieves state-of-the-art performance on PutnamBench among publicly reported $\sim$8B and $\sim$32B parameter models under comparable test-time budgets, offering a scalable paradigm for next-generation verifier-guided reasoning.

Large language models are becoming increasingly capable at formal mathematical reasoning, but current systems often require enormous amounts of computation by repeatedly sampling proof attempts from scratch. In this work, we show that Lean compiler error messages provide a much more efficient way to guide theorem proving. We observe that many different failed proofs produce similar error patterns, allowing the model to learn reusable correction strategies. With this notion of "compression", we train Lean provers to iteratively refine failed proofs using compiler feedback, and introduce a value-guided search strategy that prioritizes the most promising corrections. This approach substantially improves performance across multiple mathematical benchmarks while using computation more efficiently. Our results suggest that strong verifier feedback, such as Lean compiler's semantic checks, can play an important role in building more scalable AI reasoning systems.