An elementary proof of a generalization of double Greibach normal form

Greibach normal form (GNF) introduced by Greibach, which facilitates proving several properties of context-free languages, is studied. Double Greibach normal form is a variant of GNF, where the right-hand side of each production begins and ends with a terminal symbol. Engelfriet has given an elementary proof of double GNF, while Rosenkrantz's and Hotz's proofs involve higher algebraic techniques. One can easily convert any CFG G = (V , T , P, S) in double GNF into an equivalent CFG in (m,n)-GNF for any nonnegative integers m and n such that at least one of them is not zero, by letting the new grammar consist of the rules such that α is derived from A by at most m steps of left-most derivation and at most n steps of right-most derivation. New nonterminals representing sequences of nonterminals from VLR are introduced to reduce the number of occurrences of nonterminals in production rules.