Guarded Induction and Weakly Final Coalgebras in Dependent Type Theory

We introduce concepts for representing interactive programs in dependent type theory. The representation uses a monad, as in Haskell. We consider two versions, one, in which the interface with the real world is fixed, and another one, in which the interface varies depending on previous interactions. We then generalise the monadic construction to polynomial functors. Then we look at rules needed in order to introduce weakly final coalgebras in dependent type theory. We arrive at the notion of coiteration, and investigate its relationship to guarded induction. Finally we explore the relationship between state dependent coalgebras and bisimulation.