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Nested multisets, hereditary multisets, and syntactic ordinals in Isabelle/HOL. (English) Zbl 1434.03025
Miller, Dale (ed.), 2nd international conference on formal structures for computation and deduction. FSCD 2017, Oxford, UK, September 3–9, 2017. Proceedings. Wadern: Schloss Dagstuhl – Leibniz Zentrum für Informatik. LIPIcs – Leibniz Int. Proc. Inform. 84, Article 11, 18 p. (2017).
Summary: 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 \(\epsilon_0\). 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).
For the entire collection see [Zbl 1372.68017].

MSC:
03B35 Mechanization of proofs and logical operations
03B25 Decidability of theories and sets of sentences
03F15 Recursive ordinals and ordinal notations
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