Polymorphism Meets DHOL

2026-04-30

AI Summary

This work addresses the limitation of Dependent Higher-Order Logic (DHOL) stemming from its lack of polymorphism by systematically introducing polymorphism into DHOL for the first time. The paper develops the syntax and semantics of Polymorphic Dependent Higher-Order Logic (PDHOL) and provides a logical embedding that translates PDHOL into standard Higher-Order Logic (HOL), thereby ensuring compatibility with existing automated theorem provers. This approach enables logical modeling and automated reasoning with polymorphic dependent types. The effectiveness of the proposed method is validated through formalizations in the TPTP problem library, and a functional prototype theorem prover for PDHOL is rapidly implemented, demonstrating practical feasibility.

Abstract

DHOL is an extensional, classical logic that equips the well-known higher-order logic (HOL) with dependent types. This allows for concise encodings of important domains like size-bounded data structures, category theory, or proof theory. Automation support is obtained by translating DHOL to HOL, for which powerful modern automated theorem provers are available. However, a critically missing feature of DHOL is polymorphism. We develop the syntax and semantics of polymorphic DHOL and extend the translation accordingly. We implement the translation in the logic-embedding tool and evaluate it on a range of TPTP formalizations. The logic-embedding tool, together with an off-the-shelf HOL theorem prover easily creates a PDHOL theorem prover for experimenting.

Problem

• polymorphism
• DHOL
• dependent types
• higher-order logic
• theorem proving

Innovation

• polymorphic DHOL
• dependent types
• higher-order logic
• automated theorem proving
• logic embedding