Why does it take 2n-1 rules to make a string in Chomsky Normal Form?

Here we give a full proof that for a grammar in Chomsky Normal Form, generating a string w will take 2|w|-1 rule applications, except if the string is empty.

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I am a professor of Computer Science, and am passionate about CS theory. I have taught many courses at several different universities, including several sections of undergraduate and graduate theory-level classes.