e Theorem 1.3 (Coiteration). For all f : C ! TC there exists a unique g : C ! X:TX such that C g ## f ## X:TX out ## TC Tg ## TX:TX commutes. Example 1.4 (Coiteration on Streams). Let TX = L X for some xed set L and denote the set of (innite) sequences over L by L N = ff j f : N ! Lg. Then hhd; tli : L N ! L L N is nal, where hd(f) = f(0) and tl(f<F10.9

this paper express that the above principles work under di#erent additional assumptions which are needed to show that the large system can actually be constructed inside the category. The basic Theorem requires the existence of countable coproducts. Later we also present a variant where the functor T comes a as a monad, the functor F is taken from a copointed functor, and the distributive law # is assumed to interact nicely with this additional structure (i.e. # should be a distributive law of the monad over the copointed functor, see again (Lenisa et al., 2000))

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In this article we give an accessible introduction to stream differential equations, ie., equations that take the shape of differential equations from analysis and that are used to define infinite streams. Furthermore we discuss a syntactic format for stream differential equations that ensures that any system of equations that fits into the format has a unique solution. It turns out that the stream functions that can be defined using our format are precisely the causal stream functions. Finally, we are going to discuss non-standard stream calculus that uses basic (co-)operations different from the usual head and tail operations in order to define and to reason about streams and stream functions. 1

Abstract. We present a coinductive proof system for bisimilarity in transition systems specifiable in the de Simone SOS format. Our coinduction is incremental, in that it allows building incrementally an a priori unknown bisimulation, and pattern-based, in that it works on equalities of process patterns (i.e., universally quantified equations of process terms containing process variables), thus taking advantage of equational reasoning in a “circular ” manner, inside coinductive proof loops. The proof system has been formalized and proved sound in Isabelle/HOL. 1