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

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

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