Language of all even-length words with no 1's in left half
Two points:

the reasoning is incomplete: you need to show that pumping $y$ in the two first cases also leads to elements outside $L$
the reasoning does not use $ |xy|&lt;d$ which entails that only the first case occurs.

But it is not wrong. If you want to ignore $|xy|&lt;d $ then you need to argue the three cases. The first case holds because deleting $y$ creates a string of odd length or one in which a $1$ appears in the first half. The second case holds because if we pump $y$ $ n &gt;1$ times a $1$ will flow into the first part.