LeanGeo: Formalizing Competitional Geometry problems in Lean

Chendong Song, Zihan Wang, Frederick Pu, Haiming Wang, Xiaohan Lin, Junqi Liu, Jia LI, Zhengying Liu

Keywords: Automated Theorem Proving, Formal Geometry, Large Language Models, Reinforcement Learning, Lean 4

TL;DR: We introduce LeanGeo, a novel framework for formalizing and solving competition-level geometry problems in Lean 4, along with a benchmark of 122 problems to evaluate the geometric reasoning capabilities of large language models.

Abstract: 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. To further improve prover performance, we introduce a synthetic data generation pipeline together with a reinforcement learning training framework built on LeanGeo. We open source the theorem library and the benchmark of LeanGeo at \url{https://anonymous.4open.science/r/LeanGeo-9CE9}

Primary Area: neurosymbolic & hybrid AI systems (physics-informed, logic & formal reasoning, etc.)

Submission Number: 11867