Note finally that if \(f(\vec{x})\) is recursive it may be defined via some finite number of applications of composition, primitive recursion, and unbounded minimization in a manner which preserves the totality of intermediate functions in its definition. Thus although the specification of \(f(\vec{x})\) may involve one or more applications of unbounded search, each search required to compute its value is guaranteed to terminate in a finite number of steps. It thus follows that all of functions