# Correct Proof Of ZBC Theorem From Odifreddi? Also Extension Question

So I’m looking at the proof of the ZBC lemma in Odifreddi’s Classical Recursion Theory volume 2 page 808 and I don’t see why $$0′ \oplus C$$ produced computes $$B’$$ as claimed. The positive requirements try and code $$P^C_e: \; x \in C^{[e]} \iff (\exists z > x) B^{[e]}(x) \not= B^{[e]}(z)$$

Now if these requirements always succeeded we would be great. We could read off from $$C^{[e]}$$ the point at which $$B^{[e]}$$ achieved its limit and from that compute $$W$$ which in turn computes $$B’$$. However, higher priority negative requirements might restrain $$P^C_e$$ from acting. These requirements have the form. $$N^C_e : \; (\exists_\infty s)(\phi^{B\oplus C_s}_{e,s}(e)\downarrow) \implies \phi^{B\oplus C}_{e}(e)\downarrow$$

However, since $$B$$ isn’t low we can’t guarantee that $$0’$$ can tells us whether or not such a higher priority negative requirement might be falsely resulting in $$C^{[e]}(x)=0$$ so how do we actually conclude that $$B’ \leq_T 0′ \oplus C$$?

While I’m at it am I correct in presuming that the only reason that one can assume we meet the negative requirement for $$B$$ is that we assumed $$C$$ meets its own negative requirements above. This almost makes me think the right proof would build them simultaneously so they satisfied a requirement more like

$$N^C_e : \; (\exists_\infty s)(\phi^{B^{e}_s\oplus C_s}_{e,s}(e)\downarrow) \implies \phi^{B\oplus C}_{e}(e)\downarrow$$

where $$B^{e}_s(x) = \begin{cases} B(x) & \text{ if } x = \land i \leq e \ B_s(x) & \text{ otherwise } \end{cases}$$

That way one could get some kind of induction off the ground where $$0’$$ would be able to determine if the next negative requirement ever engaged using the knowledge of $$B’$$ restricted to $$e$$. Or is there some easier trick I’m missing?

Note that the reason I’m interested is that I wanted to see if I could extend the ZBC theorem so that given r.e. set $$A$$ and set $$W$$ r.e. in $$A$$ it produced $$B, C$$ with $$B \oplus C \leq_T A \oplus W$$ and $$(B \oplus C)’ \equiv_T 0’\oplus B \oplus C \equiv_T 0′ \oplus W$$. So thoughts about that are appreciated as well.