Show that the Language is irregular I was solving some problem from past test, there was this question:
Use the closure property of regular language to show the language $L$ is not regular
$$L =\{ a^3 b^n c^{n-3}  \mid  n&gt;3\} $$
I know how to solve it using Pumping Lemma, but how to solve it with closure property?
Also, I know that to prove that a language L is not regular using closure properties, the technique is to combine L with regular languages by operations that preserve regularity in order to obtain a language known to be not regular, but I can't figure out with whom to combine.
I saw some similar questions but didn't get ideas about how to do this one.
Any hints?
Thanks.
Suppose that $L$ is regular. Then so is $L' = L \circ \{ccc\} = \{a^3 b^n c^n \mid n \ge 3\}$. This implies that $L'' = (aaa)^{-1}L' = \{b^n c^n \mid n \ge 3\}$ is also regular and, in turn, that $L''' = L'' \cup \{\varepsilon, bc, bbcc\} = \{b^n c^n \mid n \in \mathbb{N}\}$ is regular. This is a contradiction since $L'''$ is known not to be regular.