# Church numerals and Kleene numerals

Church numerals $$\overline{0} = \lambda fx. x$$ and $$\overline{n} = \lambda f x. f^n x$$ are provisions for applying a function $$n$$ times to an argument. An alternate system of numerals, possibly attributable to Kleene, is $$\overline{0} = \lambda x. x$$ and $$\overline{n+1} = [F, \overline{n}]$$, where $$[\,,\,]$$ represents pairing and $$F$$ is the boolean value $$\textit{false}$$.

Can these numerals also be viewed as $$n$$ applications of a function? More concretely, without using the $$Y$$ combinator, is it possible to construct a lambda term $$t$$, such that $$t$$ applied to the Kleene numeral $$\overline{n}$$ and lambda terms $$f$$ and $$x$$ reduces to $$f^n x$$?