We review retrenchment as a liberalisation of refinement, for the description of applications too rich (e.g. using continuous and infinite types) for refinement. A specialisation of the notion, evolving retrenchment is introduced, motivated by the need for an approximate, evolving notion of simulation. The focus of the paper is the case study, a substantial second-order linear control system. The design step from continuous to zero-order hold discrete system is expressible as an evolving retrenchment. Thus we demonstrate that the retrenchment approach can formalise the development of useful applications, which are outside the scope of refinement. The work is presented in a data type-enriched language containing the B language of J.-R. Abrial. 1

This paper gives a formal description of -- at least a part of -- the type system of Aldor, the extension language of the computer algebra system AXIOM. In the process of doing this a critique of the design of the system emerges.

We demonstrate the use of formal methods tools to provide a semantics for the type hierarchy of the AXIOM computer algebra system, and a methodology for Aldor program analysis and verification. We give a case study of abstract specifications of AXIOM primitives, and provide an interface between these abstractions and Aldor code.

The hierarchic structures of abstract algebra pose challenges to the module systems of both programming and specification languages. We relate two existing module systems that are designed for this purpose: the type system of the computer algebra system Axiom, and the module system of the theorem prover Isabelle.