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I am not a native English speaker. Can we use "must have" when we want to express a regret about something in the past that did not happen?

For example:

If I had come earlier, I must have met them.

A friend of mine said we cannot use "must have" for the third conditional sentences. We must use "would have," "could have," or "might have" instead.

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Your friend is right. We use 'must have' for something that we are certain, or very sure, did happen. Example: My mother must have visited yesterday while I was out, because my kitchen is all clean, and there is a bunch of flowers on the table.

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With any unreal past conditional ("third conditional") sentence, the main clause needs [ past modal + "have" + past participle ].

"Must" is a non-past modal and doesn't have an equivalent past modal, so it cannot be used at all in unreal past conditionals. The only past modals are "would", "could", "should" and "might".

To get the meaning of ["must" + past] + "have" + past participle, you need to use [ "would have had to" + base form ]. So to express your example sentence with correct grammar, instead of a "must" structure, we can use "would" with "have to", which has the past form "had to":

If I had come earlier, I would have had to meet them.

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