Typal Heterogeneous Equality Types

Abstract

The usual homogeneous form of equality type in Martin-Löf Type Theory contains identifications between elements of the same type. By contrast, the heterogeneous form of equality contains identifications between elements of possibly different types. This short note introduces a simple set of axioms for such types. The axioms are shown to be equivalent to the combination of systematic elimination rules for both forms of equality, albeit with typal (also known as “propositional”) computation properties, together with Streicher’s Axiom K, or equivalently, the principle of uniqueness of identity proofs.