Learning to Reason with Insight for Informal Theorem Proving
| Authors | Yunhe Li et al. |
| Year | 2026 |
| Field | AI / Agents |
| arXiv | 2604.16278 |
| Download | |
| Categories | cs.AI, cs.CL, cs.LG |
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.
Engineering Breakdown
Plain English
This paper tackles informal theorem proving—proving mathematical statements in natural language rather than formal symbolic systems—by identifying that LLMs struggle most with recognizing which core techniques to apply to complex problems. The authors propose DeepInsightTheorem, a hierarchical dataset that structures informal proofs by explicitly extracting and labeling the key techniques and proof sketches alongside final proofs, combined with a Progressive Multi-stage training framework to teach models to reason insightfully. The core insight is that informal proofs align better with how LLMs process language, but models need explicit supervision on why a technique works, not just how to execute it, to solve harder problems effectively.
Core Technical Contribution
The paper identifies 'insight'—the ability to recognize which proof techniques apply to a problem—as a bottleneck that existing automated theorem proving systems overlook. Instead of treating proof generation as a sequence-to-sequence task, the authors decompose it hierarchically: first identify the core technique(s) needed, then generate a proof sketch, then fill in details. DeepInsightTheorem dataset provides this multi-level structure explicitly, allowing models to learn that different problem classes require different reasoning strategies. This is fundamentally different from prior work that either optimizes for formal systems (where insight is implicit in symbolic manipulation) or treats informal proofs as black-box language generation tasks.
How It Works
The system operates in a progressive multi-stage pipeline during both data preparation and model training. First, the DeepInsightTheorem dataset is constructed by taking informal proofs and explicitly extracting three levels: (1) core techniques used (e.g., 'proof by contradiction', 'induction on structure'), (2) proof sketches showing the logical flow, and (3) complete formal proofs with details. During training, the model learns in stages: it first trains to identify which techniques apply given a problem statement, then learns to sketch out the high-level proof structure using those techniques, and finally learns to elaborate into full formal proofs. The model sees the same problem at each stage with different granularities of ground truth, creating a curriculum that builds reasoning skills progressively. Input is a mathematical problem statement in natural language; intermediate outputs are technique predictions and sketch structures; final output is the complete informal proof.
Production Impact
For engineering teams building mathematical reasoning systems or automated tutoring platforms, this approach offers a concrete way to improve LLM performance on hard reasoning tasks without requiring formal logic solvers. Instead of calling expensive symbolic backends or Monte Carlo tree search, you can train a single LLM end-to-end to reason through problems by learning the proof technique recognition task—this reduces latency and inference cost significantly. The hierarchical training approach is also more debuggable: you can inspect whether a model failed because it picked the wrong technique, sketched poorly, or made errors in detail-filling, enabling targeted fixes. The main trade-off is data construction cost—you must annotate proofs with technique labels and sketches, which requires expert effort, and the dataset is likely domain-specific (mathematics may not transfer to other formal reasoning domains). Integration requires a custom inference pipeline that samples techniques, then sketches, then proofs sequentially rather than one-shot generation.
Limitations and When Not to Use This
The paper's primary limitation is scalability of the dataset construction: explicitly annotating proof techniques and sketches requires mathematical expertise and does not obviously scale to new domains without human effort. The approach assumes that informal proofs decompose cleanly into recognizable techniques—it may struggle on proofs that combine many techniques in non-obvious ways or require insight that doesn't fit pre-defined categories. The abstract doesn't provide empirical results comparing against baselines, so it's unclear how much the hierarchical structure helps versus just having more data or better prompting. Additionally, informal theorem proving is easier than formal proofs but still may not reach the performance of dedicated symbolic solvers on well-defined problems; the benefit is mainly for domains where formal systems don't exist or are too brittle.
Research Context
This work builds on the observation that LLMs excel at natural language but underperform on reasoning tasks like formal mathematics (improving on work like ProofNet, Lean benchmarks, and informal proof corpora). It extends recent trends in chain-of-thought and scratchpad reasoning by making intermediate steps (techniques and sketches) explicit and supervized in the training data. The paper contributes to the broader agenda of making LLM reasoning more interpretable and controllable by structuring learning into stages rather than end-to-end generation. This opens research directions around: (1) how to automatically discover and label proof techniques from unlabeled proof corpora, (2) whether technique recognition transfers across different mathematical domains, and (3) how to combine insight-based reasoning with search-based methods for harder problems.
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