Is it part of the present perfect tense?
What is the grammatical form and function of "after" here?
'after' is a preposition. If we place a verb form after a preposition, it has to be in the -ing form. This can be just the -ing form, or the perfect -ing form, which has 'having' and the past participle.
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'after' is a preposition. If we place a verb form after a preposition, it has to be in the -ing form. This can be just the -ing form, or the perfect -ing form, which has 'having' and the past participle.
The second form is not the present perfect tense because the auxiliary 'have' is in the -ing form. The present perfect tense never has the auxiliary 'have' in the -ing form.
So