Recursive Relation is Turing Computable/Corollary

Theorem
Let $f: S \to \set {0, 1}$, where $S \subseteq \N$ be a recursive function.

Let $x \in \N$ be encoded as:
 * $1 1 \dotsm 1$

where $1$ is repeated $x$ times.

Then there exists a Turing machine such that:
 * The input symbols of the machine are $\set 1$
 * The accepted language are the encodings of $x \in \N$ such that $\map f x = 1$
 * The machine halts on an input it is the encoding for some $x \in \N$ such that $\map f x$ is defined

Proof
By Recursive Relation is Turing Computable, there is a Turing machine $T$ that satisfies the conditions, so long as the input does not contain a $0$.

The result follows from an identical machine, except that the input symbols are restricted to be only $\set 1$.