I have the following context-free grammar: $$ S\to aAb|aaBb|ab\\ A\to S|B\\ B\to C|S\\ C\to aC $$ and I need to convert the grammar to Chomsky normal form (CNF).
After removing unit rules I get: $$ S\to aAb|aaBb|ab\\ A\to aAb|aaBb|ab|aC\\ B\to aAb|aaBb|ab|aC\\ C\to aC $$
Before I convert the grammar to CNF, I think the production rule $C\to aC$ is redundant because $A$ or $B$ already lead to $aC$ so can I delete the rule $C\to aC$?
Also the rule $A$ is identical to the rule $B$. Can either of them be removed as well?
This would the result of the $2$ improvements above (not yet CNF): $$ S\to aAb|aaBb|ab\\ A\to aAb|aaBb|ab|aC\\ $$