  Yamakami [2011, Theoret. Comput. Sci.] studies context-free languages with
advice functions. Here, the length of an advice is assumed to be the same as
that of an input. Let CFL and CFL/n denote the class of all context-free
languages and that with advice functions, respectively. We let CFL(2) denote
the class of intersections of two context-free languages. An interesting
direction of a research is asking how complex CFL(2) is, relative to CFL.
Yamakami raised a problem whether there is a CFL-immune set in CFL(2) - CFL/n.
The best known so far is that LSPACE - CFL/n has a CFL-immune set, where LSPACE
denotes the class of languages recognized in logarithmic-space. We present an
affirmative solution to his problem. Two key concepts of our proof are the
nested palindrome and Yamakami's swapping lemma. The swapping lemma is
applicable to the setting where the pumping lemma (Bar-Hillel's lemma) does not
work. Our proof is an example showing how useful the swapping lemma is.
