Removing constant symbols from language to prove irregularity
The language $L'$ is the "same" as $L$ (in fact, it's an asymmetric relation) in two steps. If $L$ were regular, then so would be $L'' = \{0^n 2^n : n &gt; 0 \}$ (take a DFA for $L$ and replace $1$ edges with $\epsilon$ edges). In its turn, $L''$ is the same as $L'$, up to change of symbols.