[Paper] Learning to Reason with Insight for Informal Theorem Proving

Source: arXiv - 2604.16278v1

Overview

The paper tackles a fundamental obstacle in using large language models (LLMs) for informal theorem proving: the models often miss the insight—the core technique or “aha” step—that guides a proof. By explicitly teaching LLMs to recognize and apply these techniques, the authors show a sizable boost in solving challenging math problems expressed in natural language.

Key Contributions

• DeepInsightTheorem dataset – a hierarchical collection of informal proofs that separates (1) the core technique, (2) a concise proof sketch, and (3) the full detailed proof.
• Progressive Multi‑Stage Supervised Fine‑Tuning (SFT) – a curriculum‑learning pipeline that first trains the model on basic proof writing, then on extracting core techniques, and finally on generating full proofs with insight‑aware guidance.
• Empirical validation – extensive experiments on established mathematical reasoning benchmarks (e.g., MATH, MiniF2F) demonstrate that the insight‑aware approach outperforms strong baselines by up to 15 % absolute accuracy.
• Analysis of insight usage – ablation studies reveal that explicitly modeling the core technique contributes the majority of the performance gain.

Methodology

1. Dataset Construction

   • Human annotators decompose each informal theorem into three layers:
     • Technique: a short phrase like “use induction on n” or “apply Cauchy‑Schwarz”.
     • Sketch: a high‑level outline of how the technique is applied.
     • Full Proof: a step‑by‑step natural‑language proof.
   • The resulting hierarchy lets a model learn what to do before how to do it.

2. Progressive Multi‑Stage SFT

   • Stage 1 – Proof Writing: fine‑tune a base LLM (e.g., LLaMA‑2‑7B) on raw proof texts to acquire basic mathematical language.
   • Stage 2 – Insight Extraction: train the same model to predict the core technique given a problem statement. This forces the model to focus on the high‑level reasoning pattern.
   • Stage 3 – Insight‑Guided Generation: combine the technique token with the problem and let the model generate the full proof, conditioning on the previously learned insight.
   • The stages are executed sequentially, mirroring how a human student first learns to write proofs, then learns to spot the key idea, and finally writes proofs that explicitly leverage that idea.

3. Evaluation

   • Accuracy is measured by exact match against reference proofs and by a semantic correctness metric using a verifier (e.g., a symbolic checker or a separate LLM).
   • Baselines include standard fine‑tuning, chain‑of‑thought prompting, and retrieval‑augmented generation.

Results & Findings

• Core technique prediction alone yields a ~7 % boost, confirming that the insight token carries substantial information.
• Models trained with the progressive curriculum converge faster (≈30 % fewer training steps) and exhibit more stable loss curves.
• Human evaluation shows that proofs generated with explicit insights are more readable and easier to follow than those from baseline models.

Practical Implications

• Developer tooling: Integrating an “insight extraction” module into code‑assistants or educational bots can make LLM‑driven math tutoring more reliable and transparent.
• Automated verification pipelines: By exposing the core technique, downstream symbolic checkers can focus on verifying a smaller, well‑defined sub‑problem, reducing computational overhead.
• Cross‑domain reasoning: The hierarchical approach can be adapted to other domains where a high‑level strategy matters (e.g., algorithm design, security proof generation, scientific hypothesis formation).
• Curriculum‑learning APIs: The progressive fine‑tuning recipe is lightweight enough to be packaged as a “training schedule” service for any LLM provider looking to boost reasoning capabilities without massive data collection.

Limitations & Future Work

• Annotation cost – Building DeepInsightTheorem required expert mathematicians to label techniques and sketches, which may not scale to all sub‑fields.
• Generalization to novel domains – The current dataset focuses on undergraduate‑level mathematics; performance on advanced research‑level proofs remains untested.
• Reliance on LLM size – Gains diminish for very small models (<2 B parameters), suggesting a lower bound on model capacity for effective insight learning.
• Future directions:
  • Semi‑automated technique extraction using weak supervision to reduce labeling effort.
  • Extending the hierarchy to include counter‑example generation for proof debugging.
  • Exploring multi‑modal insight cues (e.g., diagrams) for geometry‑heavy problems.

Bottom line: By teaching LLMs to first spot the “key idea” behind a proof, the authors unlock a new level of informal theorem‑proving performance—an approach that can be repurposed across many AI‑assisted reasoning tasks.

Authors

• Yunhe Li
• Hao Shi
• Bowen Deng
• Wei Wang
• Mengzhe Ruan
• Hanxu Hou
• Zhongxiang Dai
• Siyang Gao
• Chao Wang
• Shuang Qiu
• Linqi Song

Paper Information

• arXiv ID: 2604.16278v1
• Categories: cs.AI, cs.CL, cs.LG
• Published: April 17, 2026
• PDF: Download PDF