Lax Theory Morphisms

Abstract

When relating formal languages, e.g., in logic or type theory, it is often important to establish representation theorems. These interpret one language in terms of another in a way that preserves semantic properties such as provability or typing. Metalanguages for stating representation theorems can be divided into two groups: First, computational languages are very expressive (usually Turing-complete), but verifying the representation theorems is very difficult (often prohibitively so); second, declarative languages are restricted to certain classes of representation theorems (often based on theory morphisms), for which correctness is decidable. Neither is satisfactory, and this article contributes to the investigation of the trade-off between these two methods. Concretely, we introduce lax theory morphisms, which combine some of the advantages of each: they are substantially more expressive than conventional theory morphisms, but they share many of the invariants that make theory morphisms easy to work with. Specifically, we introduce lax morphisms between theories of a dependently typed logical framework, but our approach and results carry over to most declarative metalanguages. We demonstrate the usefulness of lax theory morphisms by stating and verifying a type erasure translation from typed to untyped first-order logic. The translation is stated as a single lax theory morphism, and the invariants of the framework guarantee its correctness. This is the first time such a complex translation has be verified in a declarative framework.