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.