Mechanizing Domain Theory

We describe an encoding of major parts of domain theory and xed-point theory in the Pvs extension of the simply-typed-calculus; these formaliza-tions comprise the encoding of mathematical structures like complete partial orders (domains), domain constructions, the Knaster-Tarski xed-point theorem for mono-tonic functions, and variations of xed-point induction. Altogether, these encodings form a conservative extension of the underlying Pvs logic. A major problem of embedding mathematical theories like domain theory lies in the fact that developing and working with those theories usually generates myriads of applicability and type-correctness conditions. Our approach to exploiting the Pvs devices of predicate subtypes and judgements to establish many applicability conditions behind the scenes leads to a considerable reduction in the number of the conditions that actually need to be proved. We illustrate the applicability of our encodings by means of simple examples including a mechanized xed-point induction proof in the context of relating diierent semantics of imperative programming constructs.