AI RESEARCH ANALYSIS
Structured Hints for Sample-Efficient Lean Theorem Proving
New research demonstrates that even highly-trained neural theorem provers benefit significantly from simple structural guidance at inference time. A lightweight intervention using fixed prompt schedules and tactic skeletons yields a 43% relative improvement in proof-solving performance on the miniF2F benchmark, challenging assumptions about LLM self-sufficiency in complex formal reasoning tasks.
Executive Summary: Boosting AI Prover Efficiency
This paper addresses the surprising brittleness of state-of-the-art neural theorem provers, which often fail due to structural errors rather than a lack of mathematical insight. By introducing a Lean-aware Intermediate Representation (IR) and a fixed prompt schedule that provides structural guidance through tactic skeletons, the authors achieve substantial gains. The method significantly outperforms standard sampling under strict inference budget constraints, demonstrating that explicit structural priors can be a cost-effective complement to advanced reinforcement learning.
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Relative Improvement
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Structured IR Solved
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Unique Wins for IR
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Structured Intermediate Representation
The core of the proposed method is a Structured Query q = (x, s), where x is the formal theorem statement (the goal) and s is a tactic skeleton (a high-level proof strategy like simp, intro, constructor). This explicitly separates proof planning from tactic execution, constraining the model's generation process to complete P(y | x, s).
Fixed Prompt Schedule for Diversity
Instead of relying solely on stochastic sampling, a fixed prompt schedule is used. For each theorem, k=16 distinct queries are generated by cycling through 15 pre-defined tactic skeletons (including an empty one). This ensures a diverse range of proof strategies are attempted, increasing the chance of alignment with a viable proof path, even under a low inference budget.
Enterprise Process Flow
Formal Theorem Statement (x)
→
Select Tactic Skeleton (s)
→
Generate Structured Query q=(x,s)
→
LLM Generates Proof Completion (y)
→
Lean Verification
→
Proof Solved / Failed
43.2% Relative Performance Gain
On the miniF2F-test benchmark, the Structured IR approach solved 53 theorems (21.72%) compared to the baseline's 37 theorems (15.16%). This represents an absolute gain of 6.56 percentage points and a remarkable 43.2% relative improvement under identical, strict inference constraints (k=16 samples, 1024 tokens max).
Dominant Performance: 19 Wins vs. 3 Losses
A paired analysis revealed strong asymmetry: our method uniquely solved 19 theorems that the baseline failed, while the baseline only uniquely solved 3 that our method missed. Both methods solved 34 theorems in common. This statistically significant result (p < 0.001) demonstrates that structural guidance provides a robust boost rather than merely trading off problem types.
| Feature | Structured IR (Ours) | Baseline (Unguided) |
|---|---|---|
| Pass@16 (%) | 21.72% | 15.16% |
| Relative Improvement | +43.2% | N/A |
| Theorems Solved (N) | 53 | 37 |
| Unique Wins | 19 | 3 |
| Shared Solves | 34 | 34 |
| Computational Budget | k=16, 1024 tokens | k=16, 1024 tokens |
Similar Error Distributions
Intriguingly, the failure mode analysis showed that the distribution of error types (e.g., unsolved_goals, syntax_error, unknown_identifier) was qualitatively similar between the Structured IR and baseline methods. This suggests that the performance gains are not due to systematically avoiding specific error types. Instead, the structural hints guide the model towards successful proof completions more often when a skeleton aligns with the correct strategy.
By providing a 'warm start' with a valid tactic skeleton, the model avoids early missteps that can cascade into failures, effectively increasing the hit rate on successful proofs rather than fundamentally altering the types of errors encountered.
Challenging the 'More Scale' Assumption
This work challenges the prevailing assumption that simply scaling LLMs or applying more compute is the sole path to better theorem proving. By demonstrating that lightweight, explicit structural priors can yield significant gains (43% relative improvement) under strict resource constraints, the research highlights the complementary value of inference-time guidance.
Efficiency for Resource-Constrained Environments
For applications where massive tree searches or extensive sampling (e.g., k > 1000) are infeasible, structural guidance via a fixed prompt schedule offers a more efficient lever than pure stochastic sampling. This makes advanced theorem proving more accessible and practical in real-world, low-latency scenarios.
Future Research Directions
Future work includes investigating the scalability of this structural guidance to larger budgets (e.g., k=128) and stronger base models. Additionally, developing 'dynamic skeletons'—a learned system that predicts the optimal tactic structure for a given theorem—could further enhance the query generation process and lead to even greater efficiency.
43.2%
Relative Performance Improvement
This significant gain was achieved under strict computational budgets (k=16 samples, 1024 max tokens), proving the efficiency of structural hints.
Case Study: Enhancing Lean Theorem Proving with Structural Hints
In the domain of formal verification with Lean, state-of-the-art neural provers often struggle with basic structural correctness, leading to failures despite possessing advanced mathematical reasoning. Our intervention involved guiding the DeepSeek-Prover-V1.5-RL model with pre-defined 'tactic skeletons' at inference time. This simple yet effective approach demonstrated that by providing a high-level proof strategy upfront, the model's success rate on the miniF2F benchmark increased by 43.2% relative to unguided sampling. This outcome highlights the critical role of explicit structural priors in making AI-driven formal reasoning more robust and efficient for enterprise applications.
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