Abstract: |
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 properties
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: |
@article{Complete_Non_Orders-AFP,
author = {Akihisa Yamada and Jérémy Dubut},
title = {Complete Non-Orders and Fixed Points},
journal = {Archive of Formal Proofs},
month = jun,
year = 2019,
note = {\url{http://isa-afp.org/entries/Complete_Non_Orders.html},
Formal proof development},
ISSN = {2150-914x},
}
|