Expressive logics, such as the modal $\mu$-calculus, can be used to specify and verify functional requirements of program models. While such verification is useful, a key challenge is to guarantee that the model being verified actually corresponds to the (typically effectful) program it is supposed to. We explore an approach that bridges this gap between effectful programming and functional requirement verification. Using dependently-typed programming in Agda, we develop an embedding of the modal $\mu$-calculus for defining and verifying functional properties of possibly-non-terminating effectful programs which we represent in Agda using the coinductive free monad.