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{{Short description|Theorem in order theory and lattice theory}}
{{otheruses4|Kleene's fixed-point theorem in lattice theory|the fixed-point theorem in computability theory|Kleene's recursion theorem}}
[[File:Kleene fixpoint svg.svg|thumb|Computation of the least fixpoint of ''f''(''x'') = {{sfrac|1|10}}''x''<sup>2</sup>+[[arctangent|atan]](''x'')+1 using Kleene's theorem in the real [[interval (mathematics)|interval]] [0,7] with the usual order]]
In the [[mathematics|mathematical]] areas of [[order theory|order]] and [[lattice theory]], the '''Kleene fixed-point theorem''', named after American mathematician [[Stephen Cole Kleene]], states the following:
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Although [[Tarski's fixed point theorem]]
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
== Proof ==
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:
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