We develop a modular method to build algebraic structures. Our approach is categorical: we describe the layers of our construct as monads, and combine them using distributive laws. Finding such laws is known to be difficult and our method identifies precise sufficient conditions for two monads to distribute. We either (i) concretely build a distributive law which then provides a monad structure to the composition of layers, or (ii) pinpoint the algebraic obstacles to the existence of a distributive law and suggest a weakening of one layer that ensures distributivity. This method can be applied to a step-by-step construction of a programming language. Our running example will involve three layers: a basic imperative language enriched first by adding non-determinism and then probabilistic choice. The first extension works seamlessly, but the second encounters an obstacle, resulting in an 'approximate' language very similar to the probabilistic network specification language ProbNetKAT.

Dedicated to Frank de Boer on the occasion of his 60th birthday.Abstract. Moessner's Theorem describes a construction of the sequence of powers (1 n , 2 n , 3 n , . . . ), by repeatedly dropping and summing elements from the sequence of positive natural numbers. The theorem was presented by Moessner in 1951 without a proof and later proved and generalized in several directions. More recently, a coinductive proof of the original theorem was given by Niqui and Rutten. We present a formalization of their proof in the Coq proof assistant. This formalization serves as a non-trivial illustration of the use of coinduction in Coq. During the formalization, we discovered that Long and Salié's generalizations could also be proved using (almost) the same bisimulation.

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