Understanding and Improving Automated Proof Synthesis for Interactive Theorem Provers

2026-04-27

AI Summary

Existing automated proof synthesis methods struggle with complex theorems in interactive theorem provers and rely heavily on expert knowledge. This work presents the first systematic analysis of failed proof attempts, uncovering critical correlations between human expert proof patterns and successful proofs. Building on these insights, we propose Pattern-Guided Tactic Search (PGTS), a novel approach that integrates deep learning–driven proof synthesis, empirical analysis of proof scripts, and heuristic tactic search guided by expert-derived patterns. Experimental results demonstrate that PGTS improves upon existing tools by proving 8.05% more theorems on standard benchmarks on average and achieves a 20% higher success rate on previously unproven theorems, while also generating more concise proof scripts.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Search and Retrieval-Augmented AI: Agentic search
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

Formal verification using interactive theorem provers ensures high-quality software. However, writing proof scripts for interactive theorem provers is labor-intensive and requires deep expertise. Recent studies have leveraged deep learning to automate theorem proving by learning from manually written proof corpora. Nevertheless, these techniques still achieve limited success rates in proof synthesis. This paper investigates the theorems that current proof synthesis techniques are unable to prove and analyzes their characteristics. Through an in-depth analysis of the proven theorems, proof scripts, and the proof search process, we identify the limitations of existing tools and summarize the key factors influencing proof success rates. Our research provides valuable insights for the future optimization of automated proof tools. Based on our empirical study, we discover that tactic selections conforming to human expert proof patterns are more likely to lead to successful proofs. Inspired by this finding, we propose a pattern-guided tactic search (PGTS) method to heuristically enhance the search process of existing proof synthesis tools. Our evaluation experiments demonstrate that PGTS improves the number of theorems proved by existing proof synthesis tools by an average of 8.05\%, while also achieving an average 20\% increase in previously unproven theorems. Furthermore, PGTS enhances the capability of proof synthesis tools to prove complex theorems and generates more concise proof scripts.

Problem

• automated proof synthesis
• interactive theorem provers
• proof success rate
• tactic selection
• formal verification

Innovation

• pattern-guided tactic search
• automated theorem proving
• proof synthesis
• interactive theorem provers
• deep learning for formal verification