Composition

Given $f : \mathbb{N}^k \rightarrow \mathbb{N}$ and $g_1,g_2,\ldots,g_k : \mathbb{N}^h \to \mathbb{N}$, the composition of $f$ and $g_1,g_2,\ldots,g_k$, denoted $f \circ (g_1,g_2,\ldots,g_k)$, is the function

$$\begin{array}{l} f \circ (g_1,g_2,\ldots,g_k) (n_1,n_2,\ld... ...,n_2,\ldots,n_h),\ldots, g_k(n_1,n_2,\ldots,n_h)) \end{array}$$

For example, the function $f(n) = n{+2}$ can be defined as the composition $f = S \circ S$ of the successor function with itself.