Definition:Minimization/Function

Definition
Let $f: \N^{k+1} \to \N$ be a total function.

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

Then the minimization operation on $f$ is written as:
 * $\mu y \left({f \left({n, y}\right) = 0}\right)$

and is specified as follows:
 * $\mu y \left({f \left({n, y}\right) = 0}\right) = \begin{cases}

\text{the smallest } y \in \N \text{ such that } f \left({n, y}\right) = 0 & : \text{if there exists such a } y \\ \text{undefined} & : \text{otherwise} \end{cases}$

Note that if $f: \N^{k+1} \to \N$ is a total function, and $g: \N^k \to \N$ is given by:
 * $g \left({n}\right) = \mu y \left({f \left({n, y}\right) = 0}\right)$

then, in general, $g$ will be a partial function.