Thus G and its string grammars must have the same language L(G) = L(String 1 (G)) = L(String 2 (G)).
by induction on the depth of the derivation trees it is also easy to see that the minimal solution of the f-description of a derivation of a terminal string s is acyclic and single-rooted, and satisfies (root b s ) = (root E) and (root b s ) = (root E'), respectively, if and only if s = s. these grammars both encode their terminal strings in their respective (root b) to (root E)/(root E') paths.