On propositions-as-types in realizability models

Abstract

It is well-known that one can build models of full higher-order dependent type theory (a.k.a. calculus of constructions) using partial equivalence relations and assemblies over a partial combinatory algebra. Using the propositions as types paradigm, one can then reason about the types and kinds. But one can also use the fact that assemblies embed into a realizability topos and use the logic of the topos to reason about types and kinds, as done by Reus in his dissertation. We show that, in a suitably precise sense, reasoning using the topos logic amounts to the same as reasoning via the propositions-as-types method.