This work proposes a dependent type theory that combines functions and session-typed processes (with value dependencies) through a contextual monad, internalising typed processes in a dependently-typed λ -calculus. The proposed framework, by allowing session processes to depend on functions and vice-versa, enables us to specify and statically verify protocols where the choice of the next communication action can depend on specific values of received data. Moreover, the type theoretic nature of the framework endows us with the ability to internally describe and prove predicates on process behaviours. Our main results are type soundness of the framework, and a faithful embedding of the functional layer of the calculus within the session-typed layer, showcasing the expressiveness of dependent session types.
