LeanGeo: Formalizing Competitional Geometry problems in Lean
This work addresses the problem of rigorous verification and integration of geometry problems for researchers and AI developers, though it is incremental as it builds on existing theorem proving frameworks.
The authors tackled the challenge of formalizing and verifying competition-level geometry problems by introducing LeanGeo, a unified formal system within the Lean 4 theorem prover, which includes a comprehensive theorem library and a benchmark (LeanGeo-Bench) based on problems from sources like the International Mathematical Olympiad, and they evaluated state-of-the-art Large Language Models on this benchmark to highlight limitations in automated geometric reasoning.
Geometry problems are a crucial testbed for AI reasoning capabilities. Most existing geometry solving systems cannot express problems within a unified framework, thus are difficult to integrate with other mathematical fields. Besides, since most geometric proofs rely on intuitive diagrams, verifying geometry problems is particularly challenging. To address these gaps, we introduce LeanGeo, a unified formal system for formalizing and solving competition-level geometry problems within the Lean 4 theorem prover. LeanGeo features a comprehensive library of high-level geometric theorems with Lean's foundational logic, enabling rigorous proof verification and seamless integration with Mathlib. We also present LeanGeo-Bench, a formal geometry benchmark in LeanGeo, comprising problems from the International Mathematical Olympiad (IMO) and other advanced sources. Our evaluation demonstrates the capabilities and limitations of state-of-the-art Large Language Models on this benchmark, highlighting the need for further advancements in automated geometric reasoning. We open source the theorem library and the benchmark of LeanGeo at https://github.com/project-numina/LeanGeo/tree/master.