Testing and proof are complementary. Testing, including property-based testing and fuzzing, is powerful: it catches bugs quickly, cheaply, and often in surprising ways. But testing provides confidence. Proof provides a guarantee. The difference matters, and it is hard to quantify how high the confidence from testing actually is. Software can be accompanied by proofs of its correctness, proofs that a machine checks mechanically, with no room for error. When AI makes proof cheap, it becomes the stronger path: one proof covers every possible input, every edge case, every interleaving. A verified cryptographic library is not better engineering. It is a mathematical guarantee.
it just works. no special machinery needed. the boolean operators distribute over derivatives the same way union does. derivatives are such a powerful and interesting tool that i will dedicate a separate post to them, but the main point is that they give us a simple and uniform way to handle all regular language features, including intersection and complement.
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It worked. Really well.