Proving a language is not a CFL although it can be pumped

Question

Let $L=\{a^mb^nc^k\mid k\le \min(m,n)\}$

$L$ can be pumped with the pumping lemma for Context Free Languages which makes it very difficult to prove it is not a Context Free Language.

Any idea how to prove such a tricky case?

This is homework of course.

Answer 1

Hint: Suppose L is regular and the pumping length is $n$, then consider the string $w = a^{n}b^{n}c^{n} \in L$. By the pumping lemma, there exists a decomposition $w = uvxyz$ such that $\left|vxy\right| \leq n$ (and $\left|vy\right| \gt 0$) and $uv^ixy^iz \in L$ for all $i \in \{0, 1, 2, \ldots\}$. What are the possibilities for $vxy$?