Researchers have developed a new method called LeanGraphSearch that helps automated theorem proving systems find the right mathematical building blocks, or lemmas, more effectively. Formal theorem proving is the process of using computers to verify mathematical statements with absolute certainty, which is essential for ensuring the correctness of software, hardware, and mathematical theories. However, locating the relevant lemmas from vast libraries of prior results can be challenging, especially when the connections between lemmas are not obvious. This new approach improves the way these systems search by leveraging the hidden web of dependencies among lemmas, potentially making automated proof discovery more reliable and efficient.
Key Takeaways
- LeanGraphSearch constructs a graph representing dependencies between mathematical lemmas, allowing the system to explore multi-step connections rather than relying on direct similarity alone.
- The method improves lemma retrieval recall by up to 12% compared to previous state-of-the-art systems, meaning it finds more relevant lemmas.
- When integrated into theorem proving workflows, LeanGraphSearch boosts the success rate of proofs by up to 40% in some benchmarks, showing it helps the automated prover find proofs more efficiently.
- This graph-based approach balances accuracy and computational cost better than prior methods, indicating practical value for real-world formal verification tasks.
Traditional lemma retrieval systems typically focus on matching the textual or semantic similarity between a target theorem and existing lemmas. While this works to some extent, it misses out on deeper relationships that exist in the structure of mathematical knowledge. For example, a lemma might not look similar to the target theorem, but if it frequently appears in proofs of related problems, it could be crucial. LeanGraphSearch addresses this by building a “dependency graph” from Mathlib, a large and widely used library of formalized mathematics in the Lean theorem prover. In this graph, each lemma is connected to the lemmas it depends on, creating a network of multi-hop links.
The system starts by identifying lemmas semantically similar to the target theorem using existing search techniques. Then, it traverses the graph to find lemmas connected through these dependency chains, effectively uncovering supporting lemmas that might otherwise be overlooked. To determine which lemmas are most relevant, LeanGraphSearch applies a graph-based ranking algorithm that propagates relevance scores through the network. Finally, it reranks the expanded set of candidate lemmas against the original query to prioritize the best matches.
To test their approach, the researchers evaluated LeanGraphSearch on two benchmarks designed to measure lemma retrieval quality and on three theorem proving benchmarks assessing end-to-end proof success. Compared to the previous leading system, LeanSearch v2, their method consistently retrieved more relevant lemmas and increased the coverage of complete proof routes. When plugged into an iterative proving loop—where the prover attempts a proof step, receives feedback, and refines its approach—LeanGraphSearch led to significantly higher proof pass rates, particularly as the number of reflection turns increased. This demonstrates that better lemma retrieval directly translates into more effective automated theorem proving.
Looking ahead, this research highlights the importance of incorporating the structural relationships between mathematical statements into search algorithms. By moving beyond simple similarity matching and embracing graph-based reasoning, automated theorem provers can become more adept at discovering proofs in complex mathematical domains. Such advancements could accelerate formal verification efforts in software and hardware development, where correctness is critical. The team’s findings open the door to further exploration of structure-aware search techniques in proof assistants, aiming to make formal theorem proving more accessible and efficient for mathematicians and engineers alike.
Based on research published on arXiv by Yihan Xu, Jia-Chen Gu, Junyi Zhang et al..