 Many-one completeness of the halting problem

Every recursively enumerable set many-one reduces to the diagonal halting set. If a program halts exactly on $X$, the S-m-n theorem produces from $x$ an index $g(x)$ for a unary program that ignores its input and performs that computation on $x$. ThenConversely, a computable preimage of a recursively enumerable set is recursively enumerable.