Minimization on Relation Equivalent to Minimization on Function

Theorem
Let $\mathcal R$ be a $k+1$-ary relation on $\N^{k+1}$.

Then the function $g: \N^{k+1} \to \N$ defined as:
 * $g \left({n_1, n_2, \ldots, n_k, z}\right) = \mu y \ \mathcal R \left({n_1, n_2, \ldots, n_k, y}\right)$

where $\mu y \ \mathcal R \left({n_1, n_2, \ldots, n_k, y}\right)$ is the minimization operation on $\mathcal R$

is equivalent to minimization on a total function.

Proof
We have that $\mathcal R \left({n_1, n_2, \ldots, n_k, y}\right)$ holds iff $\chi_\mathcal R \left({n_1, n_2, \ldots, n_k, y}\right) = 1$, from the definition of the characteristic function of a relation.

This in turn holds iff $\overline{\operatorname{sgn}} \left({\chi_\mathcal R \left({n_1, n_2, \ldots, n_k, y}\right)}\right) = 0$, where $\overline{\operatorname{sgn}}$ is the signum-bar function.

Hence we have:
 * $\mu y \ \mathcal R \left({n_1, n_2, \ldots, n_k, y}\right) \iff \mu y \left({\overline{\operatorname{sgn}} \left({\chi_\mathcal R \left({n_1, n_2, \ldots, n_k, y}\right)}\right) = 0}\right)$.

Since $\overline{\operatorname{sgn}}$ and $\chi_\mathcal R$ are total functions, then so is $\overline{\operatorname{sgn}} \circ \chi_\mathcal R$.