 Definition 2.5: The class of partial recursive functions \(\mathbf{PartREC}\) (also known as the \(\mu\)-recursive functions) is the smallest class of partial functions of type \(\mathbb{N}^k \rightarrow \mathbb{N}\) containing the initial functions \(I_{\mathbf{PR}} = \{\mathbf{0},s,\pi^i_k\}\) and closed under the functionals