 \(\mathcal{Iter}\) takes as arguments a function \(\phi:\mathbb{N} \rightarrow \mathbb{N}\) as well as a number \(x \in \mathbb{N}\) and is defined so that \(\mathcal{Iter}(\phi,n) = \phi^{n+1}(x)\)—i.e., the function which is the \(n\)th iterate of \(\phi\). We may now define a function \(\beta\) of type \(\mathbb{N} \rightarrow (\mathbb{N} \rightarrow \mathbb{N})\) as follows: 