Skill evolution improves lean theorem provers with mutation techniques
SkillEvoLean: Mutation-enhanced skill evolution for Lean provers
Machine Learning
Summary
Teaching computers to solve math problems can be tricky when they keep failing and have no examples to learn from. The authors created a way for the Lean theorem prover to improve its problem-solving skills by making smart changes when it gets stuck, rather than just copying past successes. They tested this approach on challenging math problem sets and found it worked better than earlier methods. This helps computers get better at proving math ideas without changing their core programming.
What this means in practice
- •For automated reasoning teams: Enhance automated theorem provers to generate proofs more reliably by dynamically improving solving strategies and knowledge references using mutation techniques.
- •For mathematical software developers: Integrate mutation-based skill updates in proof assistants to better handle challenging mathematical problems without retraining base models.
Authors
Kuo Zhou, ZiXion Yang, Lu Zhang
Abstract
Skill evolution offers a promising way to improve large language model agents without updating their parameters, but its use in formal theorem proving remains underexplored. Existing methods mainly target natural-language reasoning, improving skills by analyzing successful and failed trajectories and incrementally revising solving strategies. Although the Lean verifier provides reliable execution feedback, when all sampled trajectories fail, existing skill evolution methods lack successful trajectories from which to infer effective update directions. Furthermore, these methods also focus mainly on the root instruction file, thus underexploring the evolution of reference knowledge including mathematical concepts and proving techniques. To address these limitations, we propose a mutation-enhanced skill self-evolution framework for building skill-augmented Lean provers. The framework jointly evolves a high-level solving policy and its reference knowledge through progressive and mutation-based updates. Progressive evolution derives local improvements from successful and failed trajectories, while mutation is triggered when no complete proof can be generated, sampling mathematical concepts to produce and select new skill candidates under verifier feedback. We evaluate our method on MiniF2F, PutnamBench, the 2025 International Mathematical Olympiad (IMO 2025), and the 2026 USA Mathematical Olympiad (USAMO 2026). Under the same backbone model, trajectorysampling budget, and test-time compute, our method achieves proof success rates of 100.0%, 90.6%, 4/6, and 4/6, respectively, with GPT-5.5, outperforming the baseline methods. Further analysis shows that concept-guided mutation outperforms random-text-guided mutation by 6.9 and 8.2 percentage points on MiniF2F and PutnamBench, respectively, while solving one additional problem on both IMO 2025 and USAMO 2026.