a context$-$free grammar G in Chomsky Normal Form, and you treasure it dearly. Unfortunately, context$-$sensitive goblins recently broke into your house and tampered with your grammar. They added various rules of the form AB $->$ c$,$ where A$,$ B are variables and c is a terminal. Of course, these rules mean that your new grammar H is no longer context$-$free. Since you love context$-$free grammars, you want to produce a grammar without these rules, while preserving the language of H$.$

Given a grammar H$,$ whose rules are of the form

A $->$ BC

A $->$ a

AB $->$ c

where capital letters are variables and lowercase letters are terminals $($in particular, S $->$ is not a rule$),$ show how to produce an equivalent context$-$free grammar.

partial for the case where there is only a single rule of the form AB $->$ c$.$