The argument below is made by a student who tries to prove a context-free grammar is unambiguous.

"For each variable and each symbol, there is only one production rule by which the variable can produce the symbol at the start of the remainder of the string. Thus, in a leftmost derivation, there is only one possible rule to apply at any time. Hence a leftmost derivation is unique and the grammar is unambiguous."

What is wrong with this argument? Please list all the errors it makes, and be as specific as possible. Thank you.