Function Obtained by Minimization from URM Computable Relations

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

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

Let $$g: \N^k \to \N$$ be the function obtained by minimization from $$f$$ thus:
 * $$g \left({n_1, n_2, \ldots, n_k}\right) \approx \mu y \mathcal{R} \left({n_1, n_2, \ldots, n_k, y}\right)$$.

Then $$g$$ is also URM computable.

Proof
From Minimization on Relation Equivalent to Minimization on Function, minimization on $$\mathcal{R}$$ is equivalent to minimization on $$\overline{\sgn} \circ \chi_{\mathcal{R}}$$.

We have that a Primitive Recursive Function is URM Computable.

By definition, if $$\mathcal{R}$$ is URM computable then so is its characteristic function $$\chi_{\mathcal{R}}$$.

We have that $\overline{\sgn}$ is primitive recursive and thus URM computable.

Thus, from Function Obtained by Substitution from URM Computable Functions, $$\overline{\sgn} \circ \chi_{\mathcal{R}}$$ is URM computable.