Ultrametric Domain Theory and Semantics in Coq

We present a Coq formalization of ultrametric spaces and of ultrametric-enriched categories, up to and including the construction of solutions to recursive domain equations in ultrametric-enriched categories. We then show how to apply this semantic setup for giving semantics to a programming language with higher-order store. Specifically, we define a step counting operational semantics for a functional language with recursive types and higher-order store; show how to define an interpretation of the types of the programming language via a Kripke logical relation over the operational semantics, using a recursively defined metric space of worlds; and prove the fundamental theorem of logical relations.