Traversals of data structures are ubiquitous in programming. Consequently, it is important to be able to characterise those structures that are traversable and understand their algebraic properties. Traversable functors have been characterised by McBride and Paterson as those equipped with a distributive law over arbitrary applicative functors; however, laws that fully capture the intuition behind traversals are missing. This article is an attempt to remedy this situation by proposing laws for characterising traversals that capture the intuition behind them. To support our claims, we prove that finitary containers are traversable in our sense and argue that elements in a traversable structure are visited exactly once

The development of design patterns in object-oriented programming aims at capturing good software design in a re-usable generic form. However, design patterns are not expressible in conventional object-oriented programming languages. To address this shortcoming, we need to model and understand design patterns precisely. We achieve this by identifying operators characterising the most fundamental design patterns in a way that enables the construction of object-oriented programs with provable structural and behavioural properties. We use dependent-type theory to define a simplified, functional model of object-oriented programming. Design patterns are modelled in this setting as operators on object signatures and implementations. We present examples of several basic design operators and design patterns modelled in this setting and show how properties of their composition can be proven.

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