Given a string w, a string u is a prefix of w if and only if v, w = uv. Therefore, un is a prefix of the word unhappy. Given a language L, Lpre = {u | w L, and u is a prefix of w}. Show that if L is a context free language, then Lpre is also context free.

Hint: The most elegant way to do this is to use recursive constructions. Given a context free language L, think of proving Lrev, the reverse of strings in the language to be context free. You can apply similar ideas for constructing Lpre.