Experiments with Choice in Dependently-Typed Higher-Order Logic
This work addresses a theoretical gap in type theory for logicians and formal verification researchers, but it is incremental as it builds on existing DHOL and HOL frameworks.
The paper tackles the problem of adding choice to DHOL, a dependently-typed higher-order logic, by proposing two methods involving Hilbert's ε operator and a translation to HOL, and shows the translation is complete and sound, with evaluation on dependent HOL problems.
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 $ε$, 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.