Definition:Bounded Minimization

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

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

Then the bounded minimization operation on $$f$$ is written as:
 * $$\mu y \le z \left({f \left({n_1, n_2, \ldots, n_k, y}\right) = 0}\right)$$

and is specified as follows:
 * $$\mu y \le z \left({f \left({n_1, n_2, \ldots, n_k, y}\right) = 0}\right) = \begin{cases}

\text{the smallest } y \in \N \text{ such that } f \left({n_1, n_2, \ldots, n_k, y}\right) = 0 & : \exists y \in \N: y \le z \\ z + 1 & : \text{otherwise} \end{cases}$$

Relation
Let $$\mathcal{R} \left({n_1, n_2, \ldots, n_k, y}\right) $$ be a $k+1$-ary relation on $$\N^{k+1}$$.

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

Then the bounded minimization operation on $$\mathcal{R}$$ is written as:
 * $$\mu y \le z \mathcal{R}\left({n_1, n_2, \ldots, n_k, y}\right)$$

and is specified as follows:
 * $$\mu y \le z \mathcal{R}\left({n_1, n_2, \ldots, n_k, y}\right) = \begin{cases}

\text{the smallest } y \in \N \text{ for which } \mathcal{R}\left({n_1, n_2, \ldots, n_k, y}\right) \text{ holds} & : \exists y \in \N: y \le z \\ z + 1 & : \text{otherwise} \end{cases}$$

We can consider the definition for a function to be a special case of this.

The no-solution case
The choice of $$z + 1$$ for the value when there is no solution $$y$$ less than or equal to $$z$$ is arbitrary, but convenient.

It ensures a well-defined solution for every $$z$$.