Definition:Primitive Recursion/Several Variables

Definition
Let $f: \N^k \to \N$ and $g: \N^{k+2} \to \N$ be functions.

Let $\left({n_1, n_2, \ldots, n_k}\right) \in \N^k$.

Then the function $h: \N^{k+1} \to \N$ is obtained from $f$ and $g$ by primitive recursion if:
 * $\forall n \in \N: h \left({n_1, n_2, \ldots, n_k, n}\right) = \begin{cases}

f \left({n_1, n_2, \ldots, n_k}\right) & : n = 0 \\ g \left({n_1, n_2, \ldots, n_k, n-1, h \left({n_1, n_2, \ldots, n_k, n-1}\right)}\right) & : n > 0 \end{cases}$