Is the language $\{a^n b^n c^i | i \leq n\}$ context free?

Question

I'm trying to apply the CFL pumping lemma. And, I've already tried words $a^pb^p$ and $a^pb^pc^p$. Not sure where to go from here.

Answer 1

Classical pumping works. According to the pumping lemma we can find a segmentation $uvwxy$ such that $uv^iwx^iy$ is in the language for all $i$.

Start with your own suggestion $a^pb^pc^p $.

Three easy steps.

• Each of the "pumping segments" $v$, $x$ can only be within the boundaries of one the "letter segments" $a^p$, $b^p$, $c^p$.

• It is not possible to have one of $v$ or $x$ to be in the $a^p$ segment without the other being inside the $b^p$ segment. And the other way around. In that case pump down ($i=0$).

• Final possibility both $v$ and $x$ are in $c^p$. Pump up.