I am going to present a brand-new (about a week old) proof of the classic theorem of I. Simon showing the equivalence of piecewise testability of regular languages and J-triviality of the syntactic monoid. Mikolaj Bojanczyk and I found this proof while contemplating recent results, discovered independently by Czerwinski-Martens-Masopust and by van Rooijen-Zeitoun, which show that one can determine in polynomial time in the size of NFAs recognizing L and M, whether or not L and M can be separated by a piecewise testable language. (In particular, if M is the complement of L, this gives an efficient algorithm for determining if a given language is piecewise testable.) After giving the proof of Simon’s Theorem, I will explain this more general result.