 The techniques used in this proof can also be used to show that \(\alpha(x,y)\), the universal \(k\)-ary primitive recursive evaluation function \(u_k(i,\vec{x})\), and the modified diagonal function \(\delta(x)\) are all recursive (despite the fact that we have seen above that they are not primitive recursive). For instance note that the coding of definitions of \(k\)-ary partial recursive functions described above also allows us to uniformly enumerate all primitive recursive functions \(g_0(\vec{x}),g_1(\vec{x}),\ldots\) by considering the codes of terms not containing \(\mathcal{Min}_k\). We can define in this manner a primitive recursive function \(r(i)\) enumerating the indices for these functions such that we can obtain the universal function for \(k\)-ary primitive recursive function as \(u_k(i,\vec{x}) = u(\mu s T_1(r(i),\vec{x},s)) = g_i(\vec{x})\). But note that since \(g_i(\vec{x})\) is always defined, \(u_1(i,\vec{x})\) is not only partial recursive but also total, and hence recursive.