Automated theorem proving (ATP) has become an appealing domain for exploring the reasoning ability of the recent successful generative language models. However, current ATP benchmarks mainly focus on symbolic inference, but rarely involve the understanding of complex number combination reasoning. In this work, we propose TRIGO, an ATP benchmark that not only requires a model to reduce a trigonometric expression with step-by-step proofs but also evaluates a generative LM's reasoning ability on formulas and its capability to manipulate, group, and factor number terms. We gather trigonometric expressions and their reduced forms from the web, annotate the simplification process manually, and translate it into the ``Lean'' formal language system. We then automatically generate additional examples from the annotated samples to expand the dataset. Furthermore, we develop an automatic generator based on Lean-Gym to create dataset splits of varying difficulties and distributions in order to thoroughly analyze the model's generalization ability. Our extensive experiments show our proposed TRIGO poses a new challenge for advanced generative LM's including GPT-4 which is pre-trained on a considerable amount of open-source formal theorem-proving language data, and provide a new tool to study the generative LM's ability on both formal and mathematical reasoning.

翻译：自动定理证明（ATP）已成为探索近期生成式语言模型推理能力的热门领域。然而，现有ATP基准主要关注符号推理，鲜有涉及复杂数值组合推理的理解。本研究提出TRIGO基准，该基准不仅要求模型通过逐步证明约简三角表达式，还评估生成式语言模型在公式推理、数值项操作、分组及因子分解方面的能力。我们从网络收集三角表达式及其约简形式，手动标注简化过程，并将其转化为"Lean"形式语言系统。基于标注样本自动生成更多实例以扩展数据集，并开发基于Lean-Gym的自动生成器，构建不同难度与分布的数据集划分以全面分析模型泛化能力。大量实验表明，本工作提出的TRIGO为包括预训练于大量开源形式定理证明语言数据的GPT-4在内的先进生成式语言模型带来了新挑战，并为研究生成式语言模型在形式推理与数学推理方面的能力提供了新工具。