We proved that the language $ L = { Omega Omega | omega in {0,1 } ^ {*} } $ It's not a CFL, and we did it using the pumping lemma. And the proof is clear to me. But I thought about the following CFG:

$ G = ( {S, S_ {1} }, {0,1 }, R, S) $ where R has the following rules:

$ S rightarrow S_ {1} S_ {1} | epsilon $

$ S_ {1} rightarrow 0S_ {1} | 1S_ {1} | epsilon $

It seems that the language of this CFG should be the language $ L $ that I've defined above since each substitution adds the same letter on both sides. But it can not be so since we can use the pumping lemma on the word $ 0 ^ {l} 1 ^ {l} 0 ^ {l} 1 ^ {l} $ (or $ l $ is the pumping length). So, either I do not do the substitution incorrectly, or the CFG language contains $ L $ and has more words than I currently do not see …

Can any one help me and point out where is my mistake?