Separating Timing and Calculation in Real-Time Refinement

We consider the specification and refinement of sequential real-time programs. Our realtime specifications describe the allowable behaviours of an implementation in terms of the values of variables over time. Hence within a specification the values of the variables and the times at which they have those values are intertwined. However, in a real-time program some commands are concerned with calculating the right outputs, while other commands, such as delays and deadlines, are concerned with making sure the outputs appear at the right time. During the refinement process we would like to decompose the overall problem into those aspects dealing with time and those that are purely calculation. We need refinement rules that allow us to separate these concerns. Further, given a component that is only concerned with calculation, the complexities of the real-time calculus that deal with timing behaviour are an unnecessary burden. Such calculational components can be developed more straightforwardly in the standard refinement calculus. We would like to allow the use of the untimed calculus for the development of such components. To do that we need to embed the untimed calculus within the real-time calculus.