Higher-order abstract syntax in classical higher-order logic

Abstract

Higher-Order Abstract Syntax, or HOAS, is a technique for using a higher-order logic as a metalanguage for an object language with binding operators. It avoids formalizing syntactic details related to variable binding. This paper gives an extension to classical higher-order logic that supports HOAS. The logic we work with is the core of the logics employed in the widely used systems HOL and Isabelle/HOL. The extension adds recursive types, and a new type constructor for parametric functions. Using these additions, we can solve, for example, the archetypal recursive type equation for a HOAS representation of the syntax of the untyped lambda-calculus: T = (T x T) + (T ↪ T), where the function type is the new parametric one. The usual HOAS induction principles can be derived. The bulk of the technical development in the paper is a semantics of the new logic, extending the usual set-theoretic semantics of classical higher-order logic.