Consider a pushdown automaton $A$ with stack alphabet $\Gamma$. Let $L$ be the language on $\Gamma$ of the stack configurations encountered during accepting runs of $A$. Is $L$ a context-free language?
- $L$ is prefix-closed (any prefix of $L$ is in $L$)
- Using Bar-Hillel's lemma, pumping on the context-free language amounts to pumping on the stack language like a regular language. Hence, $L$ can be pumped as a regular language.