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arXiv · 2504.04942

Lemmanaid: Neuro-Symbolic Lemma Conjecturing

Yousef Alhessi·Sólrún Halla Einarsdóttir·George Granberry·Emily First·Moa Johansson·Sorin Lerner·Nicholas Smallbone
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Abstract

Mathematicians and computer scientists are increasingly leveraging proof assistants to formalize and check complex proofs, a task that demands substantial expertise. Can we lower the bar by automating the conjecturing of helpful, interesting and novel lemmas? We present the first neuro-symbolic lemma conjecturing tool, LEMMANAID, designed to discover conjectures by drawing analogies between mathematical theories. LEMMANAID uses a fine-tuned LLM to generate lemma templates that describe the shape of a lemma, and symbolic methods to fill in the details. We compare LEMMANAID against the same LLM fine-tuned to generate lemmas directly, as well as a fully symbolic conjecturing method. On test sets from Isabelle's HOL library and Archive of Formal Proofs (AFP), LEMMANAID consistently outperforms both neural and symbolic methods. Using DeepSeek-coder-6.7B as a backend, LEMMANAID discovers 50% (HOL) and 29% (AFP) of the gold standard lemmas, increasing to 55% and 35% when ensembling prompting strategies. In a case study on Octonions, LEMMANAID discovers 79% of the gold standard lemmas, compared to 62% for neural-only and 23% for the state of the art symbolic tool. Furthermore, in a targeted comparison, LEMMANAID discovers more gold standard lemmas than both Claude Opus 4.5 and GPT-5.2. Our results show that LEMMANAID can conjecture a significant number of interesting lemmas across complex formalizations in mathematics and computer science.

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Yousef Alhessi, Sólrún Halla Einarsdóttir, George Granberry, Emily First, Moa Johansson, Sorin Lerner, Nicholas Smallbone. 2025-04-07. Lemmanaid: Neuro-Symbolic Lemma Conjecturing. https://arxiv.org/abs/2504.04942

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