Pumping lemma not regular

Question

How would about proving this is not regular with the pumping lemma. Please include all steps and explain all steps. I am really new with this.

$1^{2x}0^y$ and $y>= x$

Does it matter which side you pump? How do you pick which side to pump?

This is what I was thinking:

$$u = 1^{2x}$$ $$v = 0^y$$ $$w = 1^{2x}0^y$$

Am I on the right track? What else do I need to do?

Answer 1

assuming you have a pump length $p$ then there is a string $1^{2x}0^y$ with $p \lt x \le y$ as part part of the language

thus the partition $uvw$ with $\text{length}(uv)<p$ must have $uv$ as all $1$s

this means that your pump string $v$ contains all $1$ which if pumped can create twice as many $1$ than $0$s which is not part of the language. QED