Learning to Reason with Insight for Informal Theorem Proving

Forum topic · 小凯 · 2026-04-21

Summary

This paper addresses a key bottleneck in informal theorem proving with large language models: a lack of insight, i.e., the ability to recognize the core techniques needed to solve complex mathematical problems. The authors introduce DeepInsightTheorem, a hierarchical dataset that structures informal proofs by explicitly extracting core techniques and proof sketches alongside the final proof. To exploit this dataset, they propose a Progressive Multi-Stage SFT (supervised fine-tuning) strategy that mimics human learning, guiding models from basic proof writing to insightful reasoning. Experiments on challenging mathematical benchmarks show that this insight-aware generation approach significantly outperforms baselines, indicating that teaching models to identify and apply core techniques can substantially improve their mathematical reasoning capabilities. The work aligns theorem proving with LLMs' natural language strengths rather than relying on formal proof systems.

Overview

Field: NLP / Mathematical Reasoning
Authors: Yunhe Li, Hao Shi, Bowen Deng, Wei Wang, Mengzhe Ruan, Hanxu Hou, Zhongxiang Dai, Siyang Gao, Chao Wang, Shuang Qiu, Linqi Song
Published: 2026-04-17
arXiv: 2604.16278

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 imitates the human learning process, guiding models from basic proof writing to insightful thinking. Experiments on challenging mathematical benchmarks show that this insight-aware generation strategy significantly outperforms baselines. These results suggest that teaching models to recognize and apply core techniques can substantially improve their mathematical reasoning capabilities.

Key Contributions

1. Identifies insight as the bottleneck: informal theorem proving fails mainly because models cannot recognize the core techniques needed for complex problems. 2. DeepInsightTheorem dataset: a hierarchical dataset structuring informal proofs with explicit core techniques, proof sketches, and final proofs. 3. Progressive Multi-Stage SFT: a training strategy mimicking human learning, from basic proof writing to insightful reasoning. 4. Empirical gains: significant improvements over baselines on challenging mathematical benchmarks.

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