 In the case of the primitive recursive functions \(\mathbf{PR}\), the initial functions include the nullary zero function \(\mathbf{0}\) which returns the value 0 for all inputs (and can thus be treated as a constant symbol), \(s(x)\) denotes the unary successor function \(x \mapsto x + 1\), and \(\pi^k_i\) denotes the \(k\)-ary projection function on to the \(i\)th argument (where \(0 \leq i &lt; k\))—i.e., 