Pumping Lemma with Context Free Languages

Question

I have the language {a^i b^j c^k | i,j,k>=0 & i>j & j>k} I began by assuming some m is picked for me, such that a string

Answer 1

Note: After a bit of back-and-forth in the comments, I see that I'm wrong and William's answer is actually correct. I'll leave this answer here so I can point out where my line of reasoning failed.

I'd think about it like this:

What properties must substrings v,w,x have in order to even have a chance of remaining within the language definition after pumping? Neither v nor x can contain substrings like "ab" or "bc", or else they immediately pump out of the input language. So each of v and x must be either empty, all a's, all b's, or all c's.

Consider the string aaabbc, which is in the language.

Now what happens if we pick u="aa", v = "a", w = epsilon, x = "b", y = "bc"; and pump v and x? (Here's my mistake: I didn't consider the n=0 case, where v and x are actually removed from the string; no matter how you choose uvwxy, the proof will fail for either the n=0 or n>1 case when uvwxy is pumped to uvⁿwxⁿy).

Note: The CFL pumping lemma can be used to prove that a language is not context-free, but obeying the pumping lemma in itself is not sufficient to show that a language is context-free. There are languages that are not CF, but all the conditions of the CFL pumping lemma still hold. For such cases, you might want to have a look at Ogden's lemma, a somewhat more powerful test, and see if that can be used to show that your language is not CF.