Embedding First‐order Logic in a Pure Type System with Parameters

A standard way to code first‐order predicate logic in a propositions‐as‐types style uses a type system that is a variant of the pure type system λP. Types in this system are not necessarily in β‐normal form. Therefore, checking whether two types are β‐equal is decidable, albeit sometimes costly in terms of time and/or memory. In this paper we present an alternative to Berardi's system, based on pure type systems with parameters. We show that all types in our system are in β‐normal form, which is an advantage for implementations. Moreover, the syntactical structure of the system with parameters is similar to the syntax of first‐order predicate logic. This is not the case for the system without parameters.