Abstract Free theorems establish interesting properties of parametrically polymorphic functions, solely from their types, and serve as a nice proof tool. For pure and lazy functional programming languages, they can be used with very few preconditions. Unfortunately, in the presence of selective strictness, as provided in languages like Haskell, their original strength is reduced. In this paper we present an approach for overcoming this weakness in specific situations. Employing a refined type system which tracks the use of enforced strict evaluation, we rule out unnecessary restrictions that otherwise emerge. Additionally, we provide (and implement) an algorithm determining all refined types for a given term. 1

We begin with a functional reactive programming (FRP) model in which every program is viewed as a signal function that converts a stream of input values into a stream of output values. We observe that objects in the real world – such as a keyboard or sound card – can be thought of as signal functions as well. This leads us to a radically different approach to I/O – instead of treating real-world objects as being external to the program, we expand the sphere of influence of program execution to include them within the program. We call this virtualizing real-world objects. We explore how even virtual objects, such as GUI widgets, and non-local effects, such as are needed for debugging (using something that we call a “wormhole”) and random number generation, can be handled in the same way. Our methodology may at first seem naïve – one may ask how we prevent a virtualized device from being copied, thus potentially introducing non-determinism as one part of a program competes for the same resource as another. To solve this problem, we introduce the notion of a resource type that assures that a virtualized object is not duplicated and that I/Oandnon-localeffectsaresafe. Resourcetypesalsoprovideadeeperleveloftransparency: