I'm not sure whether I should use 'meet' or 'meet with':
- if a Python implementation meets our performance requirements, then...
- if a Python implementation meets with our performance requirements, then...
Which one is grammatically correct?
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The definition of meet in this context is fulfil or satisfy, so to append with is redundant, and it should be omitted from formal or semi-formal writing.
In this context, either. It doesn't really matter for the sake of a software analysis, and I think they might mean the same anyway.
The difference between "meets" and "meets with", generally, is that "meets" usually means "meets for the first time", while "meets with" simply means "has a meeting with", "joins", or "encounters". See sense #27 here http://www.dictionary.com/browse/meet--with
Both "meets" and "meets with" have also come to be used with requirements, to mean "satisfies", and that's the sense you're asking about. So, either.