When Agda met Vampire

2026-02-21

AI Summary

This work addresses the lack of efficient automated reasoning support in dependently typed proof assistants that is compatible with classical automated theorem provers. It proposes a lightweight integration method that automatically translates proofs generated by the classical first-order prover Vampire into type-checkable constructive proof terms in Agda, leveraging equational Horn clauses as a common fragment. The approach concisely and extensibly bridges the gap between constructive dependent type theory and classical automated reasoning while preserving correctness. A prototype implementation successfully synthesized a proof of properties of roots of unity over the complex numbers—an effort otherwise requiring two days of manual work—demonstrating substantial reduction in engineering cost and validating both the effectiveness and generalizability of the method.

Technology Category

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

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Security and Privacy: Data transparency and provenance
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

Dependently-typed proof assistants furnish expressive foundations for mechanised mathematics and verified software. However, automation for these systems has been either modest in scope or complex in implementation. We aim to improve the situation by integrating proof assistants with automated theorem provers (ATPs) in a simple way, while preserving the correctness guarantees of the former. A central difficulty arises from the fact that most ATPs operate in classical first-order logic, whereas these proof assistants are grounded in constructive dependent type theory. We identify an expressive fragment of both languages -- essentially equational Horn -- that admits sound, straightforward translations in both directions. The approach produces a prototype system for Agda forwarding proof obligations to the ATP Vampire, then transforming the resulting classical proof into a constructive proof term that Agda can type-check. The prototype automatically derives proofs concerning the properties of a complex field equipped with roots of unity, which took professional Agda developers two full days to complete. The required engineering effort is modest, and we anticipate that the methodology will extend readily to other ATPs and proof assistants.

Problem

• dependent types
• automated theorem proving
• proof assistants
• classical logic
• constructive logic

Innovation

• dependent types
• automated theorem proving
• proof translation
• constructive logic
• equational Horn fragment