In the answer to this question, I'm not understanding how the string is derived for a given $l$.
For example,
Case 1: $vx = a^i$ where $i > 0$. Choose $l = 2$ to get $a^{n+i} b^{n+1} c^{n+1} d^n \notin L$.
Why is $l = 2$ chosen and how is $a^{n+i}b^{n+1}c^{n+1}d^n$ derived from $l = 2$?
Also, how can $vx$ be chosen instead of $vwx$ as the OP chose? What do we do about $w$? Is it the empty string?