arXiv:2607.19374cs.AI2026-07被引 1

将几何题自动形式化为Lean语言,构建大规模可验证数据集。

Euclean: Automated Geometry Problem Formalization with Unified Verification in Lean

论文配图:Euclean: Automated Geometry Problem Formalization with Unified Verification in Lean
图 1 · 摘自论文原文
  • 四阶段流程自动处理几何题中的隐含假设并映射到Mathlib
  • 建成768道竞赛题的OMNI-Geometry和17.7万道的Numina-Geometry数据集
  • 支持统一神经定理证明,适合形式化推理与AI数学研究者

近期形式化推理系统已达到国际数学奥林匹克水平,但领域分裂严重:代数与数论在Lean中处理,几何仍依赖特定领域语言且缺乏形式化保证。现有几何形式化工作(LeanEuclid、LeanGeo)使用不兼容Mathlib的自定义公理体系,规模有限(<1,100题),阻碍统一模型发展。原生Mathlib几何形式化面临新挑战:需显式表达隐含图示假设(如拓扑配置、非退化性),且模型须适应小而快速演化的几何基础设施。我们提出Euclean,一种四阶段框架——约束显化、配置锚定、形式化映射与迭代修复,实现原生Mathlib下的几何自动形式化。构建了包含768道竞赛题的OMNI-Geometry和177,597道题的Numina-Geometry,是当前最大几何形式化数据集。人工评估显示TOP1准确率48.89%,TOP5准确率73.33%。在该数据集上训练Goedel v2后,证明成功率从13.6%提升至15.1%,验证了数据集对统一神经定理证明的有效性。代码与数据集:https://github.com/tlb-22/Euclean。

原文摘要 · Abstract (English)

Recent formal reasoning systems have reached IMO-level performance, yet they leave a fragmented landscape: algebra and number theory are handled in Lean, while geometry still relies on domain-specific languages with limited formal guarantees. This split increases the trusted computing base and hinders unified model development. Existing geometry-in-Lean efforts (LeanEuclid, LeanGeo) introduce custom axiom systems incompatible with standard Mathlib, and their small scale ($<$ 1,100 problems) limits large-scale training. Native Mathlib autoformalization of geometry, however, poses distinct challenges: implicit diagrammatic assumptions (e.g., topological configuration and non-degeneracy) must be made explicit rather than deferred to external solvers, and models must adapt to Mathlib's small, rapidly evolving geometry infrastructure. We present Euclean, a four-stage framework - constraint explication, configuration anchoring, formalization mapping, and iterative repair - for automatically formalizing geometry in native Mathlib. We construct OMNI-Geometry (768 competition problems) and Numina-Geometry (177,597 problems), the largest geometry formalization dataset in Lean. Human evaluation shows 48.89% TOP1 and 73.33% TOP5 accuracy. Training Goedel v2 on our formalizations improves proof success from 13.6% to 15.1%, validating dataset quality for unified neural theorem proving. Code and datasets: https://github.com/tlb-22/Euclean.

形式化推理几何证明Lean数据集

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