Let Simp={$e:W_e$ is simple} and Creat={$e:W_e$ is creative}
I'm having troubles showing these sets are $\Sigma^0_2$-Hard, ie that any $\Sigma^0_2$ set can be many-one reduced to them.
I've already shown they are $\Pi^0_2$-Hard by many-one reducing Inf={$e:W_e$ is infinite} to both of them.
In particular for Inf$\le_m$Creat I can get a total recursive $g$ s.t.:
$\Phi_{g(e)}(y) = \begin{cases} 0 & \quad y\in K \wedge \exists z[z\in W_e\wedge z>y]\\ \uparrow & \quad \text{otherwise} \end{cases}$
This then gives that $W_e$ is infinite iff $W_{g(e)}$ is creative.
And for Inf$\le_m$Simp:
It's the same as the above except that I replaced $K$ with ran$(f)$ where $f(x)=\phi_x(\mu z(\phi_x(z)>2x))$
Any help is appreciated thanks.