requirement for pumping lemma in regular language
$|vx| &lt; p$ mean that $x$ is short and it is at the beginning of $w$. If you were to remove that constraint, all regular languages would still satisfy the lemma, but more irregular ones would too. For example $L = \{a^n \mid n ~\text {is not prime}\}$, when $n = m \cdot k $ can be pumped by taking $x = a^k$.