Experiments with Choice in Dependently-Typed Higher-Order Logic

TL;DR

This paper explores how to incorporate the concept of choice into DHOL, a dependently-typed extension of higher-order logic, by extending its term structure and translation methods, and evaluates the approach on relevant problems.

Contribution

It introduces two methods for adding choice to DHOL, extending its translation to HOL, and demonstrates the effectiveness on dependent HOL problems.

Findings

Extended translation is complete and sound.

Choice can be effectively incorporated into DHOL.

Evaluation shows practical applicability on dependent HOL problems.

Abstract

Recently an extension to higher-order logic -- called DHOL -- was introduced, enriching the language with dependent types, and creating a powerful extensional type theory. In this paper we propose two ways how choice can be added to DHOL. We extend the DHOL term structure by Hilbert's indefinite choice operator $\epsilon$, define a translation of the choice terms to HOL choice that extends the existing translation from DHOL to HOL and show that the extension of the translation is complete and give an argument for soundness. We finally evaluate the extended translation on a set of dependent HOL problems that require choice.