If $L_1$ is a regular language and $L_1 \cap L_2$ is a context-free language, does it mean that $L_2$ is a context-free language too?I attempted to prove that $L_2$ was not required to be context-free by the following: If $L_1 = \{ a^n \}$ and $L_2 = \{ a^nb^nc^n \} $ then $L_1 \cap L_2 = \{ a^n \}$ which is context-free. Is this a counterexample to show that $L_2$ is not required to be context-free? If not, then do you have any suggestions?
If $L_1$ is regular and $L_1 \cap L_2$ context-free, is $L_2$ always context-free?
formal languages;regular languages;context free;closure properties