Definition of a simple function in fine structure theory I have a few questions on the definition of a simple function and I hope someone can help me with them.
First let me give some context. Let $E$ be a set or a proper class. We say a function or relation is $\text{rud}_E,$ if it is rudimentary in $E$. And for any transitive set $U$, $\text{rud}_E(U)$ denotes the closure of $U$ under $\text{rud}_E$ functions. We say $U$ is $\text{rud}_E$ closed iff $\text{rud}_E(U) \subseteq U$. Also the structures we deal with here are of the form $\langle M, \in, E\cap M\rangle$.
This is the definition:

Call a function $f:V^k \rightarrow V$, where $k \lt \omega$, simple iff the following holds true: if $\varphi(v_0, v_1, \dots, v_k)$ is $\Sigma_0$ in the $\mathcal{L}_{\in,E}$, then $\varphi(f(v_1', \dots, v_k'), v_1, \dots, v_k)$ is equivalent over transitive $\text{rud}_E$ closed structures to a $\Sigma_0$ formula in the same language.

So here are my questions:


*

*Since we say simple with no mention of $E$(like simple$_E$ for example), does it mean that we quantify over all $E$, in the definition? Or we don't mention it, because we have already fixed $E$?

*The equivalent formula that we get in the definition, is it uniform? Meaning that: Do we expect to have one formula that is equivalent over all structures mentioned above to the original formula? Or do we get one equivalent formula for each structure?(The latter seems extreme to me, as it would require lots of coding of syntax and such.)

*And lastly, it seems to me that the restriction to $\text{rud}_E$ closed structures is redundant. Since if this holds for all transitive structures, then it certainly does for $\text{rud}_E$ closed structures. And if it holds for all $\text{rud}_E$ closed structures, given an arbitrary structure $\langle M, \in, E\cap M\rangle$, we can look at $\langle \text{rud}_E(M), \in, E\cap \text{rud}_E(M)\rangle$ and then by absoluteness we can come back down, because the formula in question is $\Sigma_0$. So is the restriction to $\text{rud}_E$ closed structures necessary?

EDIT I:
The definition here can be found in Ralf Schindler's book "Set Theory: Exploring Independence and Truth", the edition for Feb. $28$ $2014$, Page $70$, in the middle of the proof of lemma $5.11$.

EDIT II:
I also would really appreciate if someone could put the definition above in terms of symbols and mathematical language, since I think that could resolve both my first two questions and some other minor questions that I have.
 A: Let me try to give some input. So first of all, this definition appears in a proof, so it should be understood in the context of the proof. The $E$ is fixed in the statement of the lemma and thus it is not quantified over in the definition, so what simple means in the proof maybe should be called $E$-simple and being $E$-simple can be different from being $F$-simple for $E\neq F$. 
Regarding your second question, in the definition it is not required that the equivalent $\Sigma_0$-formula is uniform in the structures. To finish the proof, one only needs that all $\operatorname{rud}_E$-functions are simple in this sense (as this is quite tedious to do, this statement was packaged as an exercise). The uniformity is not relevant. Anyhow, doing this exercise reveals that for $\operatorname{rud}_E$-functions these formulas can be chosen uniformly. Indeed even more is true: In the same way as one can associate natural numbers to first order formulas by looking at how they are build from the atomic formulas and the connectives, one can do this with rudimentary functions. There is then a recursive map $\eta:\operatorname{Fml}_{\in, E}\times\omega\rightarrow\operatorname{Fml}_{\in, E}$ so that whenever $f$ is $\operatorname{rud}_E$ and $\varphi$ is a $\Sigma_0$ $\{\in, E\}$-formula then $\varphi(f(v_0, \dots, v_n), w_0, \dots , w_m)$ is equivalent to $\eta(\varphi, k)(v_0, \dots, v_n, w_0,\dots, w_m)$ over any transitive $\operatorname{rud}_E$-closed structure, where $k$ is the natural number associated to $f$. [Again, E is fixed here]
Lastly, the reason why one quantifies over not just all transitive but moreover $\operatorname{rud}_E$-closed strucures is simply that the question whether $\varphi(f(v_0, \dots, v_n), w_0, \dots , w_m)$ is equivalent to $\psi(v_0, \dots, v_n, w_0, \dots , w_m)$ over a structure $\mathcal M=(M, \in, E)$ only makes sense if $M$ is closed under $f$. It means 
$$\text{for all }x_0,\dots, x_n, y_0,\dots y_m\in M\ \mathcal M\models \varphi(f(x_0, \dots, x_n), y_0, \dots , y_m)\Leftrightarrow\psi(x_0, \dots, x_n, y_0, \dots , y_m)$$
after all.
