ion. A common form of transformation, which is easily justified by appealing to reversibility, is abstraction. The abstraction transformation lifts some instances of subexpressions from the right-hand sides of a set of definitions and replaces them with function calls for some new functions. The abstraction process can be used in conjunction with a call-by-need implementation to avoid repeated evaluation of subexpressions. A well-known example is Hughes' supercombinator abstraction [Hughes 1982]. Another form of abstraction which is common in program transformation is syntactic generalization in which an expression e is replaced by a function call g e 1 : : : e n , where g is a new function defined by g x 1 : : : xn \Delta = e 0 , such that e j e 0 f e 1 : : : e n= x 1 : : : xn g. General statements about abstractions and their correctness are notationally rather complex. In practice we have found it is easier to appeal to a reversibility argument on a case-by-case basis than...

Co-induction is an important tool for reasoning about unbounded structures. This tutorial explains the foundations of co-induction, and shows how it justifies intuitive arguments about lazy streams, of central importance to lazy functional programmers. We explain from first principles a theory based on a new formulation of bisimilarity for functional programs, which coincides exactly with Morris-style contextual equivalence. We show how to prove properties of lazy streams by co-induction and derive Bird and Wadler's Take Lemma, a well-known proof technique for lazy streams.

We present a natural semantics for the untyped lazy -calculus plus McCarthy's amb, a nondeterministic choice operator. The natural semantics includes rules for both convergent behaviour (dened inductively) and divergent behaviour (dened co-inductively). This semantics is equivalent to a small step reduction semantics that corresponds closely to our operational intuitions about McCarthy's amb. We present equivalences for convergent and divergent behaviour based on the natural semantics and prove a Context Lemma for the convergence equivalence. We then give a -theory l 8 , based on the equivalences for convergent and divergent behaviour. Since it is able to distinguish between programs that dier only in their divergent behaviour, the -theory is more discriminating than equational theories based on current domain-theoretic models. It is therefore more suitable for reasoning about functional programs containing McCarthy's amb. Contents 1 Introduction 2 2 Related Work 3 3 ...