I am having a hard time to prove it, what i know is we cannot prove that a language is regular by using pumping lemma cause even if the "pumped string" is in the language the language could still be regular.
But since we already know that ww^r is a context free language cause we can design a pda for it, we should be able to divide a string from ww^r into 5 parts and pump it and the result should still be in the language. But i fail to do so, i have done the following:
assume w = 010 then ww^r = 010010
Then, u = 0, v = 10, x = 0, y = 1, z = 0
And then i pumped it once and i got: 010100110, which obviously isn't in the language produced by ww^r which again is contradicting because we can design a pda for it
Where am i going wrong? How do i exactly use pumping lemma for CFL?