Kleene fixpoint theorem

$f: L \to L,$

the least fixed point (lfp) of f is the least upper bound of the ascending Kleene chain of f, that is

$\textrm{bot}_L \le f(\textrm{bot}_L) \le f(f(\textrm{bot}_L)) \le ...$

obtained by iterating f on the bottom element of L. Expressed in a formula, the Kleene fixpoint theorem states that

$\textrm{lfp}(f) = \textrm{lub}\left(\left\{f^i(\textrm{bot}_L) \mid i\in\mathbb{N}\right\}\right)$

where $lfp$ denotes the least fixed point, $lub$ denotes the least upper bound, and $bot L$ is the bottom element of $L$ .

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