pumping lemma, |xy| ≤ p and cycles

Question

By looking at wiki and other question, I understand $x$ is the string drive to first repeated state $q$ from starting state, and $y$ is the string that cycle back from $q$ to $q$, and $p$ is the number of state inside the automaton.

However, I still dont see why $|xy| ≤ p$ holds for general cases. If it $y$ only travel distinct states, we know all state on path of $x$ and $y$ are distinct (except $q$). Then path of $xy$ can be finished within $p$ edges.

But what if there are some "sub-cycles" inside the cycle of $y$ ? This doesn't violate the assumption of $q$ being first repeated state. But it is possible for $|xy|$ to exceed $p$, as now don't require distinct states.

I was thinking to remove those "sub-cycles", but this remove some sub-string of $y$, then cannot be used for pumping $y^n$.

Answer 1

In my opinion, if there are some "sub-cycles" in $y$, say, the first cycles is back from $q_1$ to $q_1$, we can write $y$ to $y_1y_2y_3$ so that $xy_1$ finishes at $q_1$ first time, the loop occurs within $y_2$, and $y_3$ is otherwise. We choose $w=xy_1y_2y_3z$ with $|xy_1y_2|\le p$ as the pumping lemma statement.