这是Ardens Lemma的作品吗？ | 有限自动机

Question

I need your help. When I get this question like this

and when I see the answer it look like this

This must be by Ardens Lemma right? I just want to why teacher eliminated first E1 then E2, why not E2 then E1?

Second Question: I want to know do I have to first convert NFA to DFA then do by Ardens Lemma?

Answer 1

Question 1: Probably because $E_1$ does not appear on the right hand side of its own equation. Otherwise you could not eliminate it so easily, becuase replacing it by the right hand side would just lead to new occurences.

Question 2: Arden's rule applies to equations of sets. It does not matter which kind of automaton these equations are derived from. But from a deterministic one you get easier equations.