Learning to Reason with Insight for Informal Theorem Proving

2026-04-17

AI Summary

Current large language models lack the ability to identify and apply core proof techniques in informal theorem proving. To address this limitation, this work proposes DeepInsightTheorem, the first framework that explicitly models “insight” as a learnable reasoning capability. The approach constructs a hierarchical dataset to extract proof sketches and key techniques, and introduces a progressive multi-stage supervised fine-tuning strategy that mimics the human learning trajectory from shallow to deep understanding. Experimental results demonstrate that the proposed method significantly outperforms existing baselines across multiple challenging mathematical benchmarks, thereby validating the effectiveness of an insight-driven generation strategy in enhancing the mathematical reasoning capabilities of language models.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Natural Language Processing: (Large) Language Models
• Machine Learning: Large Multimodal Models (LMMs)

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Search and Retrieval-Augmented AI: Search Tool Learning with LLM: Teaching LLMs to invoke search and make use of retrieved information
• Web Mining and Content Analysis: Large pretrained models with web data

Abstract

Although most of the automated theorem-proving approaches depend on formal proof systems, informal theorem proving can align better with large language models' (LLMs) strength in natural language processing. In this work, we identify a primary bottleneck in informal theorem proving as a lack of insight, namely the difficulty of recognizing the core techniques required to solve complex problems. To address this, we propose a novel framework designed to cultivate this essential reasoning skill and enable LLMs to perform insightful reasoning. We propose $\mathtt{DeepInsightTheorem}$, a hierarchical dataset that structures informal proofs by explicitly extracting core techniques and proof sketches alongside the final proof. To fully exploit this dataset, we design a Progressive Multi-Stage SFT strategy that mimics the human learning process, guiding the model from basic proof writing to insightful thinking. Our experiments on challenging mathematical benchmarks demonstrate that this insight-aware generation strategy significantly outperforms baselines. These results demonstrate that teaching models to identify and apply core techniques can substantially improve their mathematical reasoning.

Problem

• informal theorem proving
• insight
• mathematical reasoning
• core techniques
• large language models

Innovation

• insightful reasoning
• informal theorem proving
• hierarchical dataset
• progressive multi-stage SFT
• core technique extraction