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arXiv · 2009.13065

Fixed Points Theorems for Non-Transitive Relations

Jérémy Dubut·Akihisa Yamada
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Abstract

In this paper, we develop an Isabelle/HOL library of order-theoretic fixed-point theorems. We keep our formalization as general as possible: we reprove several well-known results about complete orders, often with only antisymmetry or attractivity, a mild condition implied by either antisymmetry or transitivity. In particular, we generalize various theorems ensuring the existence of a quasi-fixed point of monotone maps over complete relations, and show that the set of (quasi-)fixed points is itself complete. This result generalizes and strengthens theorems of Knaster-Tarski, Bourbaki-Witt, Kleene, Markowsky, Pataraia, Mashburn, Bhatta-George, and Stouti-Maaden.

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Jérémy Dubut, Akihisa Yamada. 2020-09-28. Fixed Points Theorems for Non-Transitive Relations. https://doi.org/10.46298/lmcs-18(1%3A30)2022

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