Automated Discovery of Tactic Libraries for Interactive Theorem Proving

2025-03-31

AI Summary

To address the challenges of low reusability of high-level tactics and high proof redundancy in interactive theorem provers (ITPs), this paper proposes an automated tactic discovery and refactoring method based on Tactic Dependency Graphs (TDGs). TDGs introduce a semantic-dependency abstraction—replacing syntactic matching—to decouple logical structure from implementation details, thereby enabling cross-proof identification of reusable high-level tactics and modular refactoring of existing proofs. Integrating program synthesis with proof-structure analysis, our custom tool TacMiner implements this approach. Experiments show that TacMiner discovers three times as many reusable tactics as baseline methods; reduces average proof size by 26%; and improves the success rate of downstream automation tools by 172%. This work establishes a new paradigm for tactic-level knowledge capture and reuse in ITPs.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Data Mining & Knowledge Management: Representing, Reasoning, and Using Provenance, Trust

Application Category

• Web Mining and Content Analysis: Topic discovery and tracking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Enabling more concise and modular proofs is essential for advancing formal reasoning using interactive theorem provers (ITPs). Since many ITPs, such as Rocq and Lean, use tactic-style proofs, learning higher-level custom tactics is crucial for proof modularity and automation. This paper presents a novel approach to tactic discovery, which leverages Tactic Dependence Graphs (TDGs) to identify reusable proof strategies across multiple proofs. TDGs capture logical dependencies between tactic applications while abstracting away irrelevant syntactic details, allowing for both the discovery of new tactics and the refactoring of existing proofs into more modular forms. We have implemented this technique in a tool called TacMiner and compare it against an anti-unification-based approach Peano to tactic discovery. Our evaluation demonstrates that TacMiner can learn 3x as many tactics as Peano and reduces the size of proofs by 26% across all benchmarks. Furthermore, our evaluation demonstrates the benefits of learning custom tactics for proof automation, allowing a state-of-the-art proof automation tool to achieve a relative increase of 172% in terms of success rate.

Problem

• Automating discovery of reusable tactics for interactive theorem provers
• Improving proof modularity and conciseness via tactic dependence graphs
• Enhancing proof automation success rates with custom tactic learning

Innovation

• Uses Tactic Dependence Graphs for tactic discovery
• Implements TacMiner for reusable proof strategies
• Reduces proof size and boosts automation success