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?


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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