# Prove that Language is not regular using pumping lemma

Let's have this language:

$$L= \{ w_1 @ w_2 | w_1,w_2 \in \Sigma^*, \#_1(w_1)+(2*\#_2(w_1))=\#_1(w_2) + (2*\#_2(w_2)) \}$$

$$\Sigma = \{0,1,2\} \cup \{@\}$$

I need to prove that this language is not regular by using pumping lemma. Where $$\#_1(w_1)$$ means count of symbols "1" in string $$w_1$$. I bet there is some easy solution with that symbol "@" in the middle of string, which we could possibly omit when pumping(maybe).

Just consider the string $$111\cdots111 @ 111\cdots 111$$, where the number of $$1$$'s is the same on both sides and is large: longer than the pumping length. Then the decomposition into $$xyz$$ must be such that $$y$$ contains only $$1$$'s, and so repeating it any number of times (other than exactly once) will make $$\#_1(w_1) \neq \#_1(w_2)$$.