When I take a look at the proofs for pumping lemma, I have a feeling that I am often missing the intuition behind the condition |xy| ≤ p.
What exactly is the reason behind this condition? All the literature I have taken a look at are either silent (no proof, no discussion, only a statement) about this point, or state that we want the first repetition to occur not too far from the start.
But when there is an inequality involved, with very precise symbols and operators, don't we expect there to be a clear proof where we ultimately reach this condition?
What I am looking for are,
- Mathematical proof for the condition, |xy| ≤ p.
- Intuition behind the same condition.
To make the strings sufficiently long we have the condition string size at least p. My specific issue is why it is needed that |xy| ≤ p? And what does it break if |xy| ≤ p is not true?
(What's the reason for the second condition of the pumping lemma(s)?, does not exacly answer my question. The question is perhaps alright, but the answers only give some examples without much deep intuition.)