Nested Multisets, Hereditary Multisets, and Syntactic Ordinals in Isabelle/HOL

We present a collection of formalized results about finite nested multisets, developed using the Isabelle/HOL proof assistant. The nested multiset order is a generalization of the multiset order that can be used to prove termination of processes. Hereditary multisets, a variant of nested multisets, offer a convenient representation of ordinals below 0. In Isabelle/HOL, both nested and hereditary multisets can be comfortably defined as inductive datatypes. Our formal library also provides, somewhat nonstandardly, multisets with negative multiplicities and syntactic ordinals with negative coefficients. We present applications of the library to formalizations of Goodstein’s theorem and the decidability of unary PCF (programming computable functions). 1998 ACM Subject Classification F.3.1 [Logics and Meanings of Programs]: Specifying and Verifying and Reasoning about Programs; F.4.1 [Mathematical Logic and Formal Languages]: Mathematical Logic; I.2.3 [Artificial Intelligence]: Deduction and Theorem Proving