 By performing an unbounded search over codes of computation sequences in this manner, we achieve the dual purposes of both determining if the computation described by \(\tau\) on input \(\vec{n}\) halts after a finite number of steps and, if so, also finding a code \(s\) of a computation sequence which witnesses this fact. The function \(u(s)\) can then be defined by formalizing the operation which extracts the output of the computation from the last step \((s)_{\textit{len}(s)-1}\) of the sequence encoded by \(s\). In the case that \(T_k(e,\vec{n},s)\) holds, \(u(s)\) will thus correspond to the value \(f(\vec{n})\). Since the foregoing steps require only performing bounded search and checking the local combinatorial properties of finite sequences, it can additionally be shown that \(T_k(e,\vec{n},s)\) and \(u(x)\) are primitive recursive.