Formalizing Fixed Point Theory in PVS

We describe an encoding of major parts of domain theory in the PVS extension of the simply typed calculus these encodings consist of Formalizations of basic structures like partial orders and complete partial orders domains Various domain constructions Notions related to monotonic functions and continuous functions Knaster Tarski xed point theorems for monotonic and continuous functions the proof of this theorem requires Zorn s lemma which has been derived from Hilbert s choice operator Scott s xed point induction for admissible predicates and various variations of xed point induction like Park s lemma Altogether these encodings form a conservative extension of the underlying PVS logic since all developments are purely de nitional Most of our proofs are straightforward transcriptions of textbook knowledge The purpose of this work however was not to merely reproduce textbook knowledge To the contrary our main motivation derived from our work on fully mechanized compiler correctness proofs which requires a full treatment of xed point induction in PVS these requirements guided our selection of which elements of domain theory were formalized 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 device of 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 Finally we exemplify the application of mechanized xed point induction in PVS by a mechanized proof in the context of relating di erent semantics of imperative programming constructs This paper appeared as the technical report UIB from the Universit at Ulm Fakult at f ur Infor matik This research has been funded in part by the Deutsche Forschungsgemeinschaft DFG under project Veri x