. A new treatment of data refinement in typed lambda calculus is proposed, based on pre-logical relations [HS99] rather than logical relations as in [Ten94], and incorporating a constructive element. Constructive data refinement is shown to have desirable properties, and a substantial example of refinement is presented. 1 Introduction Various treatments of data refinement in the context of typed lambda calculus, beginning with Tennent's in [Ten94], have used logical relations to formalize the intuitive notion of refinement. This work has its roots in [Hoa72], which proposes that the correctness of a concrete version of an abstract program be verified using an invariant on the domain of concrete values together with a function mapping concrete values (that satisfy the invariant) to abstract values. In algebraic terms, what is required is a homomorphism from a subalgebra of the concrete algebra to the abstract algebra. A strictly more general method is to take a homomorphic relatio...

: The usual approach to real arithmetic on computers consists of using oating point approximations. Unfortunately, oating point arithmetic can sometimes produce wildly erroneous results. One alternative approach is to use exact real arithmetic. Exact real arithmetic allows exact real number computation to be performed without the roundo errors characteristic of other methods. Conventional representations such as decimal and binary notation are inadequate for this purpose. We consider an alternative representation of reals, using the golden ratio. Firstly we look at the golden ratio and its relation to the Fibonacci series, nding some interesting identities. Then we implement algorithms for basic arithmetic operations, trigonometric and logarithmic functions, conversion and integration. These include new algorithms for addition, multiplication, multiplication by 2, division by 2 and manipulating nite and innite streams. Acknowledgements I would especially like to than my supe...