I want to show that $L = \{a^n b^m a^{n+m} \mid n, m \geq 0\}$ is not regular.Can I say that the complement of $L$ intersected with $a^*b^*$ equals $\{a^n b^n \mid n \geq 0\}$ and since I know that $\{a^n b^n \mid n \geq 0\}$ is not regular, then $L$ is not regular?Or would I have to use the pumping lemma?
Using closure properties to show that $a^n b^m a^{n+m}$ is not regular
formal languages;pumping lemma
That line of reasoning can work in principle; you are trying to use closure properties of REG for a proof of contradiction. Make sure you understand that that's what you're doing.Unfortunately, you do not get $\{a^nb^n\}$. For instance, $a^3 b^4 \in \overline{L} \cap a^*b^*$. Check out our reference question for more techniques you can try.