# automata – Prove by pumping a lemma that the language \$ L = {a ^ ib ^ kc ^ k: i geq k geq 1 } \$ n is not without context

I am new to this forum. I have some difficulty using Lemma's Pumping to prove a non-FC language.

Let $$L = {a ^ ib ^ kc ^ k: i geq k geq 1 }$$ and the following are my attempt.

Evidence. Suppose by contradiction that $$L$$ is a language without context. Let $$p$$ the constant given by the P.L. We choose a string $$s = a ^ pb ^ pc ^ p$$. assume $$s in L$$ and it satisfies the properties of a contextless language.
that is to say. $$s$$ can be written as $$s = uvwxy$$ or $$| v | neq varepsilon$$ and $$| x | neq varepsilon$$. Further $$| vwx | leq q$$ and $$uv ^ iwx ^ iy in L, forall i geq 0$$.

• Yes $$v$$ contains at most one symbol and $$x$$ contains at most one symbol (for example, $$v = a$$, $$w = b$$, $$x = c$$.) Consider $$s = uv ^ 0wx ^ 0 = b notin L$$. And we have reached the contradiction.
• Yes $$v$$ contains more than one symbol or $$x$$ contains more than one symbol (eg. $$v = ab$$, $$w = varepsilon$$, $$x = ca$$.) Consider $$s = uv ^ 2wx ^ 2 = ababcaca notin L$$. And the contradiction is reached.

Therefore, the language $$L$$ is not without context. Q.E.D.

I am sure that I could have missed important cases, and my confusion here raises the question of what general approach to find all potentially contradictory cases. Thank you.