ImProver 2: Iteratively Self-Improving LMs for Neurosymbolic Proof Optimization

2026-05-20

AI Summary

This work addresses the challenges of proof repair in formal mathematical libraries, including scarce training data and high inference costs of large language models, by introducing ImProver 2, a neurosymbolic framework. The approach integrates lightweight informal abstractions with formal structures to construct a neurosymbolic scaffold, designs multidimensional structured metrics for proof evaluation, and leverages efficient expert iteration to automate proof optimization in Lean 4. Experimental results demonstrate that the resulting 7B-parameter model outperforms larger models within the same family across multiple metrics and approaches the performance of mid-tier state-of-the-art systems, thereby validating the efficacy and scalable learning capacity of smaller models in proof optimization tasks.

Abstract

Formal mathematics libraries are rapidly expanding, creating a growing need to refactor verified proofs for maintainability and to improve training data quality for neural provers. However, scalable proof optimization is hindered by heterogeneous and heuristically specified objectives, scarce data, and high training and inference costs. To overcome these challenges, we introduce ImProver 2, a neurosymbolic framework for automated proof optimization in Lean 4. ImProver 2 combines a data-efficient expert-iteration pipeline with a scaffold that exposes formal structure alongside lightweight informal abstractions. We further introduce a suite of metrics capturing structural proof properties. Using ImProver 2, we train a 7B-parameter model that outperforms orders-of-magnitude larger models within the same model family, and is competitive with mid-tier frontier models across metrics. We additionally demonstrate that our neurosymbolic scaffold significantly improves performance across both small and frontier models. We show that with proper scaffolding and training, small models can effectively restructure research-level proofs over complex and varied metrics, matching substantially larger systems and establishing proof optimization as a scalable, learnable task.

Problem

• proof optimization
• formal mathematics
• neural provers
• maintainability
• training data quality

Innovation

• neurosymbolic
• proof optimization
• expert iteration
• formal mathematics
• Lean 4