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I found this statement listed as a theorem in a textbook:

If L ⊆ Σ∗ is any language, then L is regular iff it has finitely many right derivatives. Furthermore, if L is regular, then all its right derivatives are regular and their number is equal to the number of states of the minimal DFA’s for L.*

Unfortunately, no proof was provided. Why is this statement true?

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See Janusz Brzozowski's 1964 paper Derivatives of Regular Expressions. This is theorem 5.2 in the paper, and the proof is in Appendix 2.

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