Pumping, With Or Without Choice

We present the first machine-checked formalization of Jaffe and Ehrenfeucht, Parikh and Rozenberg's (EPR) pumping lemmas in the Coq proof assistant. We formulate regularity in terms of finite derivatives, and prove that both Jaffe's pumping property and EPR's block pumping property precisely characterize regularity. We illuminate EPR's classical proof that the block cancellation property implies regularity, and discover that-as best we can tell-their proof relies on the Axiom of Choice. We provide a new proof which eliminates the use of Choice. We explicitly construct a function which computes block cancelable languages from well-formed short languages.