OMCC 2007 Problem 4

In a remote island, a language in which every word can be written using only the letters $a$ , $b$ , $c$ , $d$ , $e$ , $f$ , $g$ is spoken. Let's say two words are synonymous if we can transform one into the other according to the following rules:

i) Change a letter by another two in the following way: $$a \rightarrow bc,\ b \rightarrow cd,\ c \rightarrow de,\ d \rightarrow ef,\ e \rightarrow fg,\ f\rightarrow ga,\ g\rightarrow ab$$ ii) If a letter is between other two equal letters, these can be removed. For example, $dfd \rightarrow f$ .

Show that all words in this language are synonymous.