\begin{definition}[Definition:Primitive Recursion/Several Variables]
Let $f: \N^k \to \N$ and $g: \N^{k + 2} \to \N$ be functions.
Let $\tuple {n_1, n_2, \ldots, n_k} \in \N^k$.
Then the function $h: \N^{k + 1} \to \N$ is '''obtained from $f$ and $g$ by primitive recursion''' {{iff}}:
:$\forall n \in \N: \map h {n_1, n_2, \ldots, n_k, n} = \begin {cases}
\map f {n_1, n_2, \ldots, n_k} & : n = 0 \\
\map g {n_1, n_2, \ldots, n_k, n - 1, \map h {n_1, n_2, \ldots, n_k, n - 1} } & : n > 0 
\end {cases}$
Category:Definitions/Mathematical Logic
\end{definition}