Bounded Minimization is Primitive Recursive

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

Then the function $$g: \N^{k+1} \to \N$$ defined as:
 * $$g \left({n_1, n_2, \ldots, n_k, z}\right) = \mu y \le z \left({f \left({n_1, n_2, \ldots, n_k, y}\right) = 0}\right)$$

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

is also primitive recursive.

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

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

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

is also primitive recursive.

Proof for Function
We can define $$g$$ as follows:
 * $$(1) \quad g \left({n_1, n_2, \ldots, n_k, 0}\right) = \begin{cases}

0 & : f \left({n_1, n_2, \ldots, n_k, 0}\right) = 0 \\ 1 & : \text{otherwise} \\ \end{cases}$$
 * $$(2) \quad g \left({n_1, n_2, \ldots, n_k, z + 1}\right) = \begin{cases}

g \left({n_1, n_2, \ldots, n_k, z}\right) & : g \left({n_1, n_2, \ldots, n_k, z}\right) \le z \\ z + 1 & : g \left({n_1, n_2, \ldots, n_k, z}\right) = z + 1 \text{ and } f \left({n_1, n_2, \ldots, n_k, z + 1}\right) = 0 \\ z + 2 & : \text{otherwise} \\ \end{cases}$$

The fact that $$(1)$$ is true is clear.
 * The cases in $$(1)$$ are clearly mutually exclusive and exhaustive;
 * By the corollary to Definition by Cases is Primitive Recursive, the relations defining the cases are primitive recursive;
 * The constants $0$ and $1$ are primitive recursive.

Hence by Definition by Cases is Primitive Recursive, the function defined in $$(1)$$ is primitive recursive.

Now we investigate $$(2)$$.

Note that if $$g \left({n_1, n_2, \ldots, n_k, z}\right) \le z$$ then the equation $$f \left({n_1, n_2, \ldots, n_k, y}\right) = 0$$ has a solution for some $$y \le z$$.

Then the smallest $$y \le z$$ which solves this equation is also the smallest value of $$y \le z + 1$$ which solves this equation.

So in this case, $$g \left({n_1, n_2, \ldots, n_k, z + 1}\right) = g \left({n_1, n_2, \ldots, n_k, z}\right)$$.

Otherwise there is no such $$y \le z$$ that solves the equation.

Then the value $$g \left({n_1, n_2, \ldots, n_k, z + 1}\right)$$ is $$z + 1$$ if $$f \left({n_1, n_2, \ldots, n_k, z + 1}\right) = 0$$.

But if $$f \left({n_1, n_2, \ldots, n_k, z + 1}\right) \ne 0$$ then $$g \left({n_1, n_2, \ldots, n_k, z + 1}\right) = \left({z + 1}\right) + 1 = z + 2$$.

Thus $$g$$ as defined in $$(1)$$ and $$(2)$$ are an appropriate definition of:
 * $$g \left({n_1, n_2, \ldots, n_k, z}\right) = \mu y \le z \left({f \left({n_1, n_2, \ldots, n_k, y}\right) = 0}\right)$$.

Now to show that the function defined in $$(2)$$ is primitive recursive.

By the corollary to Definition by Cases is Primitive Recursive, the relations defining the cases are primitive recursive.

Then we have that $$g \left({n_1, n_2, \ldots, n_k, z}\right)$$, $$z + 1$$ and $$z + 2$$ are expressed in terms of:
 * $$g \left({n_1, n_2, \ldots, n_k, z}\right)$$;
 * the variable $$z$$;
 * the primitive recursive function $\operatorname{add}$;
 * the constants $1$ and $2$.

So all these functions are primitive recursive.

Hence by Definition by Cases is Primitive Recursive, the function defined in $$(2)$$ is primitive recursive.

Therefore $$g$$ is defined by primitive recursion from functions which we have proved to be primitive recursive.

The result follows.

Proof for Relation
We can mimic the proof for the function.

Or we can do it this way.

From the definition of the characteristic function for $$\mathcal{R}$$, we can express $$\mu y \le z \mathcal{R} \left({n_1, n_2, \ldots, n_k, y}\right)$$ as:
 * $$\mu y \le z \left({\chi_{\mathcal{R}} \left({n_1, n_2, \ldots, n_k, y}\right) = 1}\right)$$.

Now we turn $$\chi_{\mathcal{R}} \left({n_1, n_2, \ldots, n_k, y}\right) = 1$$ into something of the form $$f \left({n_1, n_2, \ldots, n_k, y}\right) = 0$$.

We can use signum-bar function:
 * $$\chi_{\mathcal{R}} \left({n_1, n_2, \ldots, n_k, y}\right) = 1 \iff \overline{\sgn} \left({\chi_{\mathcal{R}} \left({n_1, n_2, \ldots, n_k, y}\right)}\right) = 0$$

Now we define $$f \left({n_1, n_2, \ldots, n_k, n_{k+1}}\right) = \overline{\sgn} \left({\chi_{\mathcal{R}} \left({n_1, n_2, \ldots, n_k, n_{k+1}}\right)}\right)$$.

This is primitive recursive because it is defined by substitution from:
 * the primitive recursive function $\overline{\sgn}$;
 * the primitive recursive function $$\chi_{\mathcal{R}}$$ (primitive recursive from definition, as $$\mathcal{R}$$ is so defined).

Hence from Definition by Cases is Primitive Recursive, $$g$$ is primitive recursive.