Signum Complement Function is Primitive Recursive
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Theorem
Let $\overline {\operatorname{sgn}}: \N \to \N$ by defined as the signum-bar function.
Then $\overline {\operatorname{sgn}}$ is primitive recursive.
Proof
From Signum Complement Function on Natural Numbers as Characteristic Function, $\map {\overline {\operatorname{sgn} } } n = \chi_{\set 0} n$.
From Set Containing Only Zero is Primitive Recursive, $\chi_{\set 0}$ is primitive recursive.
Hence the result.
$\blacksquare$