$L = \{\alpha^i \beta^j \gamma^k \vert i,j,k \in \mathbb{N\textsubscript{0}}, (i=1) \Rightarrow (j=k)\}


I am asking this question here, because I am not allowed to comment on the thread that I am actually interested in, but maybe someone can still help me?

I alredy found an anwser to the Problem above (in the post linked to this question), but I still don’t understand, why I can’t just use that one case $ s = \alpha \beta^p \gamma^p$ . I can show for that case, that it doesn’t fit the pumping lemma for regular languages. Isn’t the point of condratiction, that I have to find just one case that doesn’t fit the hypothesis?

Actually I am not even supposed to use the pumping lemma, but the definitions of closure for regular languages. And that is where I started with $ (i=1) \Rightarrow (j=k) = i \neq 1 \vee j = k$ . And then I wanted to use the properties of closure, like in the first anwser in the post I linked. (I was also thinkg of using a regular expression? It seemed easier) But if I can’t just find that mentioned one word to proove the language not regular (without the PL)? I am confused. I hope it makes sense. I am genuinely interested in understanding this problem.

Irregularity of $ \{a^ib^jc^k \mid \text{if } i=1 \text{ then } j=k \}$