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I did a bit of searching on the difference between "not having" and "having not", but I could not find a convincing argument. I typed this sentence;

Congratulations on not having given up yet!

It was corrected by a friend to;

Congratulations on having not given up yet!

I am certain that my friend has a better command of English and that he is correct. Could someone explain why the first sentence is incorrect?

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Not can go before a gerund clause like having given up yet. Your friend probably didn't understand the different rule for gerunds and infinitives. Non-finite clauses like those can be preceded directly by not. Or not can occur after the first auxiliary verb (here, having) in the verb phrase of these subordinate clauses, just as it does in main clauses. – John Lawler Jul 16 '13 at 14:54
I suppose both are fine, but I like the first more. – Simon Kuang Jul 16 '13 at 19:00
If someone was saying that he was considering stopping playing rugby, I'd probably opt for 'Congratulations on not having given up yet!' However, if someone was saying that he was about to throw in the towel on a difficult crossword puzzle, I'd probably opt for 'Congratulations on having not given up yet!' – Edwin Ashworth Jul 16 '13 at 19:38

To me, having not given smells a bit like to not care, a split infinitive. Not having had much experience in this area, though, I'm interested to hear what other people think.

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Both versions are fine, but not having given up does sound a little bit more fluent. If you want to express it in the affirmative tone, I would rephrase it as:

"Congratulations on not giving up yet!"

I would say that generally when using "having + VERB" you are expected to provide more detail because it is hinting that there is something else to follow. For example: "Having to deal with this problem is giving him more things to worry about."

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