For a language $L$ with pumping length $p$, and a string $s\in L$, the pumping lemmas are as follows:Regular version:If $|s| \geq p$, then $s$ can be written as $xyz$, satisfying the following conditions:$|y|\geq 1$$|xy|\leq p$$ \forall i\geq 0: xy^iz\in L$Context-free version:If $|s| \geq p$, then $s$ can be written as $uvxyz$, satisfying the following conditions:$|vy|\geq 1$$|vxy|\leq p$$ \forall i\geq 0: uv^ixy^iz\in L$My question is this: Can someone give a concise and clear explanation of how regularity (context-freeness) imply the first and second conditions above? The pumping length is determined by (finite) properties (finite number of states or finite properties of production rules, respectively), the third properties guarantee that a state (production rule) can be skipped or repeated arbitrarily many times, but where do the first and second conditions originate? How are they justified?
Where do the length restrictions of the pumping lemma come from?
formal languages;regular languages;context free;pumping lemma