unbounded search operator

let $\text{[math]}$ , where $\text{[math]}$ is some polynomial with integer coefficients and where $\text{[math]}$ denotes "the least $\text{[math]}$ such that $\text{[math]}$ ".
[cite:;taken from @computability_soare_1987 chapter 1 recursive functions]

if $\text{[math]}$ is a partial computable function, and
$\text{[math]}$ then $\text{[math]}$ is a partial computable function.
[cite:;taken from @computability_soare_2016 theorem 1.5.3]

let $\text{[math]}$ be a PRC class, if $\text{[math]}$ belongs to $\text{[math]}$ , then so do the functions
$\text{[math]}$ and
$\text{[math]}$ [cite:;taken from @computability_davis_1994 theorem 6.1; chapter 3.6 iterated operations and bounded quantifiers]

the sum of functions can be used to construct the $\text{[math]}$ quantifier in primitive recursive predicates, give a predicate $\text{[math]}$ in some PRC class, it follows form the theorem that $\text{[math]}$ is in PRC aswell because it simply can be represented by the function unbounded_search_operator.html. the same goes for the predicate $\text{[math]}$ which can be represented by the function unbounded_search_operator.html.

let $\text{[math]}$ belong to some given PRC class $\text{[math]}$ . then by unbounded_search_operator.html the function
$\text{[math]}$ also belongs to $\text{[math]}$ .
[cite:;taken from @computability_davis_1994 chapter 3.7 minimalization]

if $\text{[math]}$ belongs to some PRC class $\text{[math]}$ and $\text{[math]}$ , then $\text{[math]}$ also belongs to $\text{[math]}$ .
[cite:;taken from @computability_davis_1994 theorem 7.1]

the operation $\text{[math]}$ is called bounded minimalization.