State Dependent IO-Monads in Type Theory

We introduce the notion of state dependent interactive programs for Martin-Löf Type Theory. These programs are elements of coalgebras of an endofunctor on the presheaf category S → Set. We prove the existence of final coalgebras for these functors. This shows as well the consistency of type theory plus rules expressing the existence of weakly final coalgebras for these functors, which represents the type of interactive programs. We define in this type theory the bisimulation relation, and give some simple examples for interactive programs. A generalised monad operation is defined by corecursion on interactive programs with return value, and a generalised version of the monad laws for this operation is proved. All results have been verified in the theorem prover Agda which is based on intensional type theory.