 The proof of  Proposition 3.2  is largely a matter of unpacking definitions. For instance, to see that iv implies i, suppose that \(A = \textrm{dom}(\phi_e)\)—i.e., \(A = \{n \in \mathbb{N} : \phi_e(n) \darrow\}\). If \(A = \emptyset\) it is automatically c.e. Otherwise, there is an element \(a \in A\). We may now define 