Enhancing Theorem Proving Performance Through Simple Structural Guidance

Researchers are increasingly exploring how to enhance the performance of artificial intelligence in formal reasoning, and a new study investigates whether even highly-trained neural theorem provers can benefit from basic structural guidance. Zachary Burton from Massachusetts Institute of Technology, alongside colleagues, demonstrate that a lightweight intervention , a fixed prompt schedule utilising common tactic skeletons , significantly improves performance on the miniF2F benchmark. Their work reveals a 43% relative improvement in pass@16 rates, achieving 21.7% compared to 15.2% with standard sampling, using the same computational resources. This finding is significant because it suggests that even state-of-the-art reinforcement learning-trained provers don’t fully exploit existing structural knowledge within theorem proving, and that simple, cost-effective guidance at inference time can provide a substantial boost to their capabilities.

Structural Hints Boost Neural Theorem Proving performance significantly

The team achieved these results by introducing a Lean-aware Intermediate Representation (IR) with a fixed prompt schedule, effectively guiding the model with structural skeletons. This work probes the behavioural limits of current RL-trained provers, suggesting they underutilise structural priors inherent in the tactic language. Experiments show that enforcing these priors at inference time provides a cheap, complementary boost to performance, even for models already proficient in formal proof techniques. The research establishes that this lightweight guidance doesn’t merely trade off between problem types; a paired analysis revealed a strong asymmetry with 19 successful completions versus only 0.3 losses, indicating a robust and consistent improvement.

This breakthrough reveals a surprising disconnect between mathematical insight and structural correctness in current neural provers. Even with sophisticated RL training, models frequently stumble on low-level structural errors, invalid syntax, identifier hallucinations, or getting lost in the proof state space, rather than lacking the necessary mathematical understanding. The study addresses this by focusing on syntactic skeletons, enforcing valid tactic structures like induction or cases at the start of generation, serving as a hint for the Language Model’s approach. By conditioning the model on a structural tuple separating proof planning from tactic execution, the researchers constrained the search space and improved the efficiency of the theorem proving process.

Furthermore, the work opens possibilities for resource-constrained settings where full lemma synthesis is too costly. The researchers rigorously controlled the inference budget, limiting generation to 1024 tokens and restricting sampling to k=16 attempts per theorem, to evaluate structural efficiency under strict conditions. This approach provides a low-latency performance floor, demonstrating that even with limited compute, structural guidance can significantly enhance performance. The0.6 to generate the proof completion, balancing structure with creative exploration. Experiments employed DeepSeek-Prover-V1.5-RL2 in completion-style prompting on the miniF2F-test set, comprising 244 theorems.

Structural Guidance Boosts Theorem Proving Success significantly

Experiments revealed that the implemented Intermediate Representation (IR) with a fixed prompt schedule significantly outperforms standard sampling under identical inference budgets. Specifically, the research team measured a performance increase from 15.16% to 21.72% using the k=16 and 1024 token parameters, highlighting the effectiveness of the structural guidance. A paired analysis conducted by the scientists showed a strong asymmetry, with 19 wins versus only 0.3 losses, indicating the tactic skeletons consistently provide a robust performance boost. Measurements confirm that the gains aren’t simply a trade-off between problem types, but a genuine improvement in solving capability.

The study meticulously analysed error distributions, discovering they remained similar between the methods tested. Data shows that the observed performance gains stem from a higher rate of successful completions, rather than systematic avoidance of specific error types. Scientists defined a Structured Query as a tuple (x, s), where ‘x’ represents the theorem statement and ‘s’ is a tactic skeleton representing the high-level proof strategy. The model, constrained to complete the proof given the enforced structural start, effectively reduces the search space for valid next tokens, improving efficiency.

Researchers established that the generation process can be modelled as Pθ(y | x, s), where θ is the model constrained to complete the proof ‘y’ given the enforced structural start ‘s’. Tests prove that this approach, conditioning the model on a structural tuple separating proof planning from tactic execution, offers a valuable enhancement to existing neural theorem proving systems. The breakthrough delivers a cheap, complementary boost to performance, particularly in resource-constrained settings where full lemma synthesis is impractical.

Tactic skeletons boost theorem proving performance significantly

By enforcing a valid tactic skeleton at the start of generation, the model appears to avoid early errors that can lead to overall failure, effectively providing a “warm start” towards successful proof completion. Although the distribution of failure modes remained similar between methods, the skeleton-guided approach consistently achieved a higher rate of successful proofs, particularly under strict resource constraints. The authors acknowledge a primary limitation of the study lies in the restricted computational budget, specifically the 1024 token limit, which may have truncated some valid proofs. Furthermore, results were based on a single run with a fixed seed and decoding configuration, leaving run-to-run variability unmeasured. Future research will focus on scaling this structural guidance to larger computational budgets and exploring the potential of a learned “skeleton retriever” to dynamically predict the most effective structure for a given theorem. This work highlights that improvements in theorem proving do not solely rely on increasing model scale, but can also be achieved through the strategic incorporation of structural priors.