Failure of the Pumping Lemma On the Wikipedia page for the Pumping Lemma for Context-Free Languages, a language, 
$$
\{b^j c^k d^l | j, k, l \in N\} \cup \{a^i b^j c^k d^l | i &gt; 0, j = k = l\}
$$
is introduced. The pumping lemma proof for this supposedly fails because, given a string without a's, you can pump the b's and, given a string with a's, you can pump the a's.  In either case, all of the possible cuttings are pumpable according to the constraints of the language.

For example, given the string
$$
a^k b^k c^k d^k
$$
you can split vwx over |a| and pump, leaving |b| = |c| = |d|.

Why can the string
$$
a b^k c^k d^k
$$
not be used?

This seems to meet all the constraints of the pumping lemma, and you cannot split vwx across |a|.  If v = a then x must contain at least one b.  The string cannot be pumped for any i > 2 as the |b| will exceed the length of |c| and |b| will exceed the length of |d|.  

Is there some reason that |a| > 1 or |a| = k?
The condition of the Pumping lemma is fulfilled for words of the form $ab^kc^kd^k$.

Pick $v=a$, $u=x = \varepsilon$, and $w$, $y$ arbitrary.