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

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 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.

CombiBench: Benchmarking LLM Capability for Combinatorial Mathematics

Neurosymbolic approaches integrating large language models with formal reasoning have recently achieved human-level performance on mathematics competition problems in algebra, geometry and number theory. In comparison, combinatorics remains a challenging domain, characterized by a lack of appropriate benchmarks and theorem libraries. To address this gap, we introduce CombiBench, a comprehensive benchmark comprising 100 combinatorial problems, each formalized in Lean~4 and paired with its corresponding informal statement. The problem set covers a wide spectrum of difficulty levels, ranging from middle school to IMO and university level, and span over ten combinatorial topics. CombiBench is suitable for testing IMO solving capabilities since it includes all IMO combinatorial problems since 2000 (except IMO 2004 P3 as its statement contain an images). Furthermore, we provide a comprehensive and standardized evaluation framework, dubbed Fine-Eval (for $\textbf{F}$ill-in-the-blank $\textbf{in}$ L$\textbf{e}$an Evaluation), for formal mathematics. It accommodates not only proof-based problems but also, for the first time, the evaluation of fill-in-the-blank questions. Using Fine-Eval as the evaluation method and Kimina Lean Server as the backend, we benchmark several LLMs on CombiBench and observe that their capabilities for formally solving combinatorial problems remain limited. Among all models tested (none of which has been trained for this particular task), Kimina-Prover attains the best results, solving 7 problems (out of 100) under both ``with solution'' and ``without solution'' scenarios. We open source the benchmark dataset alongside with the code of the proposed evaluation method at https://github.com/MoonshotAI/CombiBench/.

A Combinatorial Identities Benchmark for Theorem Proving via Automated Theorem Generation

Large language models (LLMs) have significantly advanced formal theorem proving, yet the scarcity of high-quality training data constrains their capabilities in complex mathematical domains. Combinatorics, a cornerstone of mathematics, provides essential tools for analyzing discrete structures and solving optimization problems. However, its inherent complexity makes it particularly challenging for automated theorem proving (ATP) for combinatorial identities. To address this, we manually construct LeanComb, combinatorial identities benchmark in Lean, which is, to our knowledge, the first formalized theorem proving benchmark built for combinatorial identities. We develop an Automated Theorem Generator for Combinatorial Identities, ATG4CI, which combines candidate tactics suggested by a self-improving large language model with a Reinforcement Learning Tree Search approach for tactic prediction. By utilizing ATG4CI, we generate a LeanComb-Enhanced dataset comprising 260K combinatorial identities theorems, each with a complete formal proof in Lean, and experimental evaluations demonstrate that models trained on this dataset can generate more effective tactics, thereby improving success rates in automated theorem proving for combinatorial identities.