Kleene fixed-point theorem: Difference between revisions

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does not consider how fixed points can be computed by iterating ''f'' from some seed (also, it pertains to [[monotone function]]s on [[complete lattices]]), this result is often attributed to [[Alfred Tarski]] who proves it for additive functions.<ref>{{cite journal | author=Alfred Tarski | url=http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.pjm/1103044538 | title=A lattice-theoretical fixpoint theorem and its applications | journal = Pacific Journal of Mathematics | volume=5:2 | year=1955 | pages=285&ndash;309}}, page 305.</ref> Moreover, Kleene Fixed-Point Theorem can be extended to [[monotone function]]s using transfinite iterations.<ref>{{cite journal | author=Patrick Cousot and Radhia Cousot | url=https://projecteuclid.org/euclid.pjm/1102785059 | title=Constructive versions of Tarski's fixed point theorems | journal = Pacific Journal of Mathematics | volume=82:1 | year=1979 | pages=43&ndash;57}}</ref>

== Proof ==

== ProofSource:<ref>{{Cite book|title=Mathematical Theory of Domains by V. Stoltenberg-Hansen|last=Stoltenberg-Hansen|first=V.|last2=Lindstrom|first2=I.|last3=Griffor|first3=E. R.|publisher=Cambridge University Press|year=1994|isbn=0521383447|pages=[https://archive.org/details/mathematicaltheo0000stol/page/24 24]|language=en|doi=10.1017/cbo9781139166386|url=https://archive.org/details/mathematicaltheo0000stol/page/24}}</ref> ==

We first have to show that the ascending Kleene chain of <math>f</math> exists in <math>L</math>. To show that, we prove the following: