Prove that the language

L$1=\{0^i{1}^j{0}^k|i$k$\}$

is not regular by using the pumping lemma

Template for using the pumping lemma

We usually use the Pumping Lemma to prove that a language $C$ is not

regular.

Pumping Lemma $($Textbook Theorm $1\mathrm{.}70)$

If $A$ is a regular language, then there is a number $p($the pumping length$)$

where if $s$ is any string in $A$ of length at least $p,$ then $s$ may be divided into

three pieces, $s=xyz,$ satisfying the following conditions:

for each $i0,x{y}^i$zinA,

$|y|>0,$ and

$|xy|p.$

Theorem.

Language $C$ is not regular.

Proof using the pumping lemma.

Leading to a contradiction, assume $C$ is regular. Then, the pumping lemma

applies to $C.$ Let $p$ be the pumping length that applies to $C$ from the pumping

lemma. Let $s=$dots $($Insert a string $sinC$ that depends on $p$ and will lead

to a contradiction$).$ Since $s$ has length at least $p$ and is in $C,$ the pumping

lemma applies to $s,$ so $s$ can be divided into three strings $s=xyz$ where

the conditions $1,2$ and $3$ of the pumping lemma hold for $x,y$ and $z.$ Since

conditions $2$ and $3$ hold, we know that $...($explain the appropriate property

of $x,y$ and $z)$ Then for $i=$dots $($find an i that will lead to a contradiction$)x{y}^{i}z$

is not in $C$ since $...($explain why $x{y}^{i}z$ is not in $C),$ contradicting condition

$1$ of the pumping lemma.