Ogden's lemma

In the theory of formal languages, Ogden's lemma (named after William F. Ogden) is a generalization of the pumping lemma for context-free languages.

Ogden's lemma states that if a language L is context-free, then there exists some number p ≥ 1 (where p may or may not be a pumping length) such that for any string s of length at least p in L and every way of "marking" p or more of the positions in s, s can be written as

with strings u, v, w, x, and y, such that

In the special case where every position is marked, Ogden's lemma is equivalent to the pumping lemma for context-free languages. Ogden's lemma can be used to show that certain languages are not context-free in cases where the pumping lemma is not sufficient. An example is the language {aⁱb^jc^kd^l : i = 0 or j = k = l}. Ogden's lemma can also be used to prove the inherent ambiguity of some languages.

...
Wikipedia