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We show that the syntactically rich notion of strictly positive families can be reduced to a core type theory with a fixed number of type constructors exploiting the novel notion of indexed containers. As a result, we show indexed containers provide normal forms for strictly positive families in much the same way that containers provide normal forms for strictly positive types. Interestingly, this step from containers to indexed containers is achieved without having to extend the core type theory. Most of the construction presented here has been formalized using the Agda system.

Abstract. We define representations of continuous functions on infinite streams of discrete values, both in the case of discrete-valued functions, and in the case of stream-valued functions. We define also an operation on the representations of two continuous functions between streams that yields a representation of their composite.In the case of discrete-valued functions, the representatives are well-founded (finitepath) trees of a certain kind. The underlying idea can be traced back to Brouwer's justification of bar-induction, or to Kreisel and Troelstra's elimination of choice-sequences. In the case of stream-valued functions, the representatives are non-wellfounded trees pieced together in a coinductive fashion from well-founded trees. The definition requires an alternating fixpoint construction of some ubiquity.

This chapter considers the representation of interactive programs in dependent type theory and bases it on the monadic I/O used in Haskell. Two versions are described: in the first the interface with the real world is fixed, while in the second the potential interactions may depend on the history of previous interactions. It also looks at client and server programs, which run on opposite sides of an interface. The type of interactive program is defined as a weakly final coalgebra for a general form of polynomial functor. The chapter gives formation/introduction/elimination/equality rules for these coalgebras. The relationship of the corresponding rules with guarded induction is studied by showing that the introduction rules are a slightly restricted form of guarded induction. However, the form in which they represent guarded induction is not by means of recursive equations but by using an elimination operator in a crucial way.

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