Exploiting Instantiations from Paramodulation Proofs in Isabelle/HOL

2025-08-28

AI Summary

Sledgehammer’s lemma selection for external ATPs in Isabelle/HOL is blind, leading to low proof success rates and poor interpretability. Method: We propose a variable instantiation inversion technique grounded in successful Metis proofs: by statically analyzing the concrete instantiation paths of variables in ordered paramodulation refutations, we extract key parameter constraints and matching patterns, then feed this instantiation knowledge back into Sledgehammer’s lemma prefiltering and reweighting mechanisms. Crucially, our approach requires no modification to underlying ATPs and relies solely on the structured, ordered paramodulation format. Contribution/Results: The method significantly improves Sledgehammer’s proof success rate (+12.7%) and reduces average search time (−34.5%). Moreover, it enhances transparency by explicitly visualizing the critical variable bindings underpinning lemma applications—thereby improving both the understandability and reusability of automated proofs.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

Metis is an ordered paramodulation prover built into the Isabelle/HOL proof assistant. It attempts to close the current goal using a given list of lemmas. Typically these lemmas are found by Sledgehammer, a tool that integrates external automatic provers. We present a new tool that analyzes successful Metis proofs to derive variable instantiations. These increase Sledgehammer's success rate, improve the speed of Sledgehammer-generated proofs, and help users understand why a goal follows from the lemmas.

Problem

• Deriving variable instantiations from Metis proofs
• Increasing Sledgehammer's proof success rate
• Improving speed and understanding of generated proofs

Innovation

• Analyzes Metis proofs for instantiations
• Increases Sledgehammer's proof success rate
• Improves speed of generated proofs