Subtyping in DHOL -- Extended preprint

2025-07-03

AI Summary

This paper addresses the challenge of supporting refinement and quotient types in automated theorem proving—powerful yet often undecidable for type checking. We propose an elegant integration based on subtyping within the Dependent Higher-Order Logic (DHOL) framework, uniformly modeling both refinement and quotient types as instances of subtyping. We define their syntax, semantics, and a sound and complete translation into standard HOL. Crucially, our approach leverages subtyping to reduce inclusion and projection maps to identity functions, eliminating representation conversion overhead and significantly simplifying implementation. Formal verification confirms that the extended system preserves logical soundness and completeness. To our knowledge, this is the first integration of refinement and quotient types in an automated-reasoning–enabled logic that simultaneously ensures expressive power and decidability of type checking—providing a practical foundation for next-generation proof assistants.

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: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

The recently introduced dependent typed higher-order logic (DHOL) offers an interesting compromise between expressiveness and automation support. It sacrifices the decidability of its type system in order to significantly extend its expressiveness over standard HOL. Yet it retains strong automated theorem proving support via a sound and complete translation to HOL. We leverage this design to extend DHOL with refinement and quotient types. Both of these are commonly requested by practitioners but rarely provided by automated theorem provers. This is because they inherently require undecidable typing and thus are very difficult to retrofit to decidable type systems. But with DHOL already doing the heavy lifting, adding them is not only possible but elegant and simple. Concretely, we add refinement and quotient types as special cases of subtyping. This turns the associated canonical inclusion resp. projection maps into identity maps and thus avoids costly changes in representation. We present the syntax, semantics, and translation to HOL for the extended language, including the proofs of soundness and completeness.

Problem

• Extends DHOL with refinement and quotient types
• Addresses undecidable typing challenges in theorem provers
• Simplifies subtyping via canonical inclusion and projection maps

Innovation

• Extends DHOL with refinement and quotient types
• Uses subtyping to simplify representation changes
• Provides sound and complete HOL translation