On the Proof Theory of Program Transformations

We provide an intensional semantics for certain elementary program transformations by describing a translation from these transformations to the derivations of a simple theory of operations and types and we show that this semantics is intensionally faithful. Our objective is to understand precisely the `folk-lore' view that program transformations are induction proofs in disguise and thus to understand more clearly the intensional (i.e. proof-theoretic) structure of a class of semi-formal program derivations. Keywords: program transformation, constructive set theory, intensional semantics 1 Introduction This paper makes precise a view which has had, for many years, a folk-lore status: program transformations are induction proofs in disguise. There have been, perhaps, two main reasons why this view has remained a conjecture rather than a theorem. Firstly, the obvious induction proof to which a transformation from a program f to a program f 0 corresponds is the proof that f and f 0..

On the Proof Theory of Program Transformations

We provide an intensional semantics for certain elementary program transformations by describing a translation from these transformations to the derivations of a simple theory of operations and types and we show that this semantics is intensionally faithful. Our objective is to understand more precisely the intensional structure of a class of semi-formal program derivations. Keywords: constructive set theory, program transformation, intensional semantics 2 Introduction This paper continues our study of the proof theory of certain elementary program derivations: those obtained by the techniques of transformational programming (e.g. [BuD77] [Bir84] [Hen88]) from functional programs. In our earlier work [Hen93] we concentrated exclusively on transformations over the natural numbers. In this paper we wish to extend this work towards algebraic types in general. These data-types are, in their full generality, significantly more problematic than the almost pathologically simple special cas..