= Multiplicative block-count language
{title2=$\{a^m b^{mn}c^n:m,n\geq0\}$}
This <formal language> is not a <context-free language>. For <pumping length> $p$, use $a^p b^{p^2}c^p$. A pumpable window of length at most $p$ cannot meet both outer blocks. If pumping adds $\alpha,\beta,\gamma$ copies of $a,b,c$, then $\alpha\gamma=0$ and $1\leq\alpha+\beta+\gamma\leq p$. Preserving the product constraint would require $\beta=p(\alpha+\gamma)$, impossible both when $\alpha+\gamma=0$ and when it is positive. This contradicts the <pumping lemma for context-free languages>.
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