I am studying Pumping Lemma for Context Free Languages, wherein, I am slightly confused in a question where one of the case doesnt obey all rules but another case does. What's the conclusion? Do we call it a contradiction and declare the language is not context free or do we say there is no contradiction?
If its the former, it will declare (a)^n(b)^n as non-context free using Pumping Lemma even though we know its a context free language, since some of the cases will not obey the 3 conditions at the same time.
By cases, I mean v and y's existence in different parts of the string. In a string
aaaaabbbbb, either v or y could fully be in
a, partially be in
a and b or fully be in
b. ( Assuming you break the string into S = uvxyz where v and y are the repeating pieces)