I hate the Pumping Lemma for regular languages. It’s a complicated way to express an idea that is fundamentally very simple, and it isn’t even a very good way to prove that a language is not regular.
It’s easy enough to see that any derivative of a regular language is again regular: taking a derivative just corresponds to changing the start state in a deterministic automaton. By the same argument, any regular language has only a finite number of different derivatives.
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