# Non existence of certain hyperarithmetical functions

In what follows, $$\phi_n$$ is the $$n$$th partial recursive function, and $$\phi_n^g$$ is the $$n$$th partial recursive function with oracle $$g$$.

We say $$x\in\mathbb{N}$$ is pre-total if the following two conditions hold:

• For all distinct $$y,z<_\mathcal{O} x$$, we either have $$y<_{\mathcal{O}} z$$ or $$z<_\mathcal{O} y$$, where $$<_\mathcal{O}$$ is the ordering on Kleene's $$\mathcal{O}$$.
• If $$x=3\cdot 5^e$$ then $$\phi_e$$ is total.

We say $$x$$ is total if it is pre-total and all $$y<_\mathcal{O} x$$ are pre-total.

Consider the following conditions on a binary function $$f$$:

1. $$f(x,y)=0$$ or $$f(x,y)=1$$
2. $$f(1,y)=0$$
3. $$f(2^x,y)=1\iff \phi_y^{f(x,\cdot)}(y)\downarrow$$
4. $$f(3\cdot 5^x,\langle u,v\rangle)=1\iff$$ there exists an $$n$$ such that $$u<_\mathcal{O} \phi_x(n)$$ and $$f(u,v)=1$$.

I'd like to show that no hyperarithmetical (that is, $$\Delta_1^1$$ and total) function $$f$$ can satisfy conditions (1)-(5) for all total $$x$$ and $$y$$.

I'm having trouble getting started, in part because I don't really see what these properties are encoding. Any help would be appreciated.