 The foregoing definition specifies \(\mathbf{PR}\) as the minimal closure of \(I_{\mathbf{PR}}\) under the functions in \(Op_{\mathbf{PR}}\). In other words, \(\mathbf{PR}\) may be equivalently defined as the subclass of \(\bigcup_{k \in \mathbb{N}}(\mathbb{N}^k \rightarrow \mathbb{N})\) satisfying the following properties: