I understand that this topic has been discussed, and I have reviewed numerous posts about it. However, my question remains unresolved. Specifically, I am seeking clarification on the use of the Pigeonhole Principle in determining the regularity of a language.

Consider the string a^{12}.

Suppose we have a DFA with 5 states and we process the string a^{12}. By the Pigeonhole Principle, the DFA will revisit a state more than once. If we remove the segment between the revisited states, this could result in an odd number of a’s, for instance, a^7

The string a^7 would end in the same accepting state as a^{12}, yet a^7 does not belong to the language L={a^2n∣n≥0}.

My lecturer mentioned that the Pigeonhole Principle is used to prove that languages are non-regular. Based on this principle, I demonstrated how L fails the test, leading to my confusion. So how can L still be classified as a regular language?

Like I said I reviewed numerous posts on stack overflow about it and was still confused.