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