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Formalizing Results on Directed Sets in Isabelle/HOL (Proof Pearl)

Authors Akihisa Yamada , Jérémy Dubut

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Subject Classification

ACM Subject Classification
  • Theory of computation → Automated reasoning
  • Theory of computation → Denotational semantics
Keywords
  • Directed Sets
  • Completeness
  • Scott Continuous Functions
  • Ordinals
  • Isabelle/HOL

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Abstract

Directed sets are of fundamental interest in domain theory and topology. In this paper, we formalize some results on directed sets in Isabelle/HOL, most notably: under the axiom of choice, a poset has a supremum for every directed set if and only if it does so for every chain; and a function between such posets preserves suprema of directed sets if and only if it preserves suprema of chains. The known pen-and-paper proofs of these results crucially use uncountable transfinite sequences, which are not directly implementable in Isabelle/HOL. We show how to emulate such proofs by utilizing Isabelle/HOL’s ordinal and cardinal library. Thanks to the formalization, we relax some conditions for the above results.

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Akihisa Yamada and Jérémy Dubut. Formalizing Results on Directed Sets in Isabelle/HOL (Proof Pearl). In 14th International Conference on Interactive Theorem Proving (ITP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 268, pp. 34:1-34:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ITP.2023.34

Author Details

Akihisa Yamada
  • National Institute of Advanced Industrial Science and Technology, Tokyo, Japan
Jérémy Dubut
  • National Institute of Advanced Industrial Science and Technology, Tokyo, Japan

Funding

This work was supported by JST, CREST Grant Number JPMJCR22M1, Japan.

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