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Proof of pumping lemma

Example: $L = \{1^{n^2}:n\geq 1\}$

(All strings with only 1s and the number of 1s is a perfect square)

Suppose that $L$ is regular. Then, there is some number $p$ such that the pumping lemma holds.

Let $x$ be a string with 1s $p^2$ times.

By the PL, we can write $x=abc$ such that $abbc\in L$.

$|abbc| = p^2 = |b| \leq p^2 + p$

There is a DFA $D$ that computes some $L$.

Let $p =$ number of states in the DFA $D$.

Let $x$ have length $\geq p$.

We have $S_0, S_1,…,S_p$ is a sequence of $p+1$ state indices.

However, since the DFA has $p$ states, there are only $p$ state indices.

By the pigeonhole principle, we must have repeated a state.

Let’s say these repeated states are $S_i$ to $S_j$.

$x=\underset{\mathrm{a}}{S_0…,}\;\underset{\mathrm{b}}{S_i…,\,S_j...,}\; \underset{\mathrm{c}}{S_p,\,S_{p+1}}$

Let $S_0 = 0$ as the starting state.