Page 296 - Practical English Usage 3ed - Michael Swan, Oxford

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infinitives (10): I'm glad to have left perfect or past meaning
Perfect infinitives (to have gone, to have left etc) can have the same kind of meaning as perfect or past tenses.
I'm glad to have left school. (= I'm glad that I have left ...)
She was sorry to have missed BilL (= •.• that she had missed Bill.)
We hope to have finished the job by next Saturday. (= .•• that we will have
finished ...)
You seem to have annoyed Anne yesterday. (= It seems that you annoyed
Anne yesterday.)
perfect infinitive for 'unreal' past
After some verbs (e.g. mean, be, would like), perfect infinitives can refer to 'unreal' past situations that are the opposite of what really happened.
I meant to have telephoned, but I forgot. (OR I meant to telephone . ..) He was to have been the new ambassador, but he fell ill.
I wish I'd been there - I would like to have seen Harry's face when Nan
walked in.
With would like, would prefer and one or two other verbs, a double perfect infinitive is sometimes used in informal speech: the extra perfect infinitive does not change the meaning.
I would have liked to have seen Harry's face.
modals: He could have killed himself
After the modal verbs could, might, ought, should, would and needn't, we often use perfect infinitives to refer to unreal situations.
Did you see him fall? He could have killed himself.
(He did not kill himself.)
You should have written - I was getting worried.
(The person did not write.)
I would have gone to university if my parents had had more money. (The speaker did not go to university.)
She needn't have sent me flowers.
(She did send flowers.)
Modal verbs with perfect infinitives can also refer to situations that are not unreal, but uncertain.
She could/should/ought tolmaylwilllmust have arrived by now.
For more details. see the entries for the different modal verbs.
infinitives (11): purpose I sat down to rest
We often use an infinitive to talk about a person's purpose - why he or she does something.
I sat down to rest. (NOT I M t ti8ftffl for r-esting I fer t6 Fest.) He went abroad to forget.
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