2026-07-22T16:00:23Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:11088 2024-03-06T10:47:20Z ddc:004 open_access

The DPRM Theorem in Isabelle (Short Paper) Bayer, Jonas David, Marco Pal, Abhik Stock, Benedikt Schleicher, Dierk DPRM theorem Hilbert’s tenth problem Diophantine predicates Register machines Recursively enumerable sets Isabelle Formal verification Hilbert’s 10th problem asks for an algorithm to tell whether or not a given diophantine equation has a solution over the integers. The non-existence of such an algorithm was shown in 1970 by Yuri Matiyasevich. The key step is known as the DPRM theorem: every recursively enumerable set of natural numbers is Diophantine. We present the formalization of Matiyasevich’s proof of the DPRM theorem in Isabelle. To represent recursively enumerable sets in equations, we implement and arithmetize register machines. Using several number-theoretic lemmas, we prove that exponentiation has a diophantine representation. Further, we contribute a small library of number-theoretic implementations of binary digit-wise relations. Finally, we discuss and contribute an is_diophantine predicate. We expect the complete formalization of the DPRM theorem in the near future; at present it is complete except for a minor gap in the arithmetization proofs of register machines and extending the is_diophantine predicate by two binary digit-wise relations. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Jonas Bayer and Marco David and Abhik Pal and Benedikt Stock and Dierk Schleicher 2019 Is Part Of LIPIcs, Volume 141, 10th International Conference on Interactive Theorem Proving (ITP 2019) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ITP.2019.33 urn:nbn:de:0030-drops-110883 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2019.33 eng https://creativecommons.org/licenses/by/3.0/legalcode