I read the following proof so many times and I could not, for the life of me, figure out how the professor went from (1) to (2).
I know to prove
$\overline{K} \le A$ , where $\overline{K}=\{x:\phi_x(x)\uparrow\}$ and $\phi_i(x) =$ the universal function theorem $h(i, x)$
"$\overline{K}$ is strongly reducible to $A$ where the reduction is effected by $f$"
We have to find a $f$ such that $x \in \overline{K}\equiv f(x)\in A$.
$\psi(x,y)$ in (2) is not such an $f$, and neither is it a verifier for $\overline{K}$. Rather, I think it's a verifier for $K$, the halting set (complete opposite). So I don't know what it's doing here.
Can someone tell me why $\psi(x,y)$ is created and how it is created?
Edit 1: $0$ is true, $1$ is false, $\downarrow$ means defined, $\uparrow$ means undefined (loops forever), $P_*$ is the set of semi-recursive relations.
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