Automated Formal Proofs of Combinatorial Identities via Wilf–Zeilberger Guidance and LLMs

摘要

Automating formal proofs of combinatorial identities is challenging for LLM-based provers, as long-horizon proof planning is required and unconstrained search quickly explodes. Symbolic methods such as the Wilf--Zeilberger (WZ) method can achieve a mechanized proof of combinatorial identities by constructing special auxiliary functions and demonstrating that they satisfy specific recurrence relations. We propose WZ-LLM, a neuro-symbolic framework that turns WZ proof plans into executable proof sketches in Lean~4 and uses an LLM-based prover to discharge the resulting machine-checkable subgoals. We also train a dedicated WZ-Prover via a Lean-kernel-verified bootstrapping loop with expert-verified iteration, followed by DAPO-based refinement. Experiments show that WZ-LLM achieves a 34\% proof success rate on LCI-Test (100 classical combinatorial identities), outperforming strong baselines such as DeepSeek-V3 and Goedel-Prover-V2; moreover, on LCI-Test it proves 5 identities on which the symbolic-only baseline fails. WZ-LLM also improves performance on CombiBench and PutnamBench-Comb, suggesting the effectiveness of coupling symbolic proof sketches with learned formal reasoning. Experiments show that WZ-LLM achieves a 34\% proof success rate on LCI-Test (100 classic combinatorial identities), outperforming strong baselines such as DeepSeek-V3 and Goedel-Prover-V2, and delivering consistent gains on CombiBench and PutnamBench-Comb. These results indicate that our framework provides two complementary strengths: improved direct proving for identities beyond the scope of WZ, and substantially higher end-to-end success when WZ sketches guide a specialized prover.