License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITP.2019.30
URN: urn:nbn:de:0030-drops-110852
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/11085/
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Yamada, Akihisa ; Dubut, Jérémy

Complete Non-Orders and Fixed Points

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Abstract

In this paper, we develop an Isabelle/HOL library of order-theoretic concepts, such as various completeness conditions and fixed-point theorems. We keep our formalization as general as possible: we reprove several well-known results about complete orders, often without any property of ordering, thus complete non-orders. In particular, we generalize the Knaster - Tarski theorem so that we ensure the existence of a quasi-fixed point of monotone maps over complete non-orders, and show that the set of quasi-fixed points is complete under a mild condition - attractivity - which is implied by either antisymmetry or transitivity. This result generalizes and strengthens a result by Stauti and Maaden. Finally, we recover Kleene's fixed-point theorem for omega-complete non-orders, again using attractivity to prove that Kleene's fixed points are least quasi-fixed points.

BibTeX - Entry

@InProceedings{yamada_et_al:LIPIcs:2019:11085,
  author =	{Akihisa Yamada and J{\'e}r{\'e}my Dubut},
  title =	{{Complete Non-Orders and Fixed Points}},
  booktitle =	{10th International Conference on Interactive Theorem Proving (ITP 2019)},
  pages =	{30:1--30:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-122-1},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{141},
  editor =	{John Harrison and John O'Leary and Andrew Tolmach},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/11085},
  URN =		{urn:nbn:de:0030-drops-110852},
  doi =		{10.4230/LIPIcs.ITP.2019.30},
  annote =	{Keywords: Order Theory, Lattice Theory, Fixed-Points, Isabelle/HOL}
}

Keywords: Order Theory, Lattice Theory, Fixed-Points, Isabelle/HOL
Collection: 10th International Conference on Interactive Theorem Proving (ITP 2019)
Issue Date: 2019
Date of publication: 05.09.2019

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