Perfect infinitive with to (to have worked)

We form the perfect infinitive with to have + the -ed form of a verb. We use the perfect infinitive after verbs such as claim, expect, hate, hope, like, love, prefer, pretend:

He pretended to have lost her number and so had been unable to contact her. (or He pretended that he had lost her number …)

The perfect infinitive often refers to things that might have happened in the past:

She claims to have met a number of famous people, but I don’t believe her. (or She claims she has met …)
I would prefer to have stayed at a small, family-run hotel than a big international chain. (I prefer to stay at small hotels, but I did not.)

The perfect infinitive can refer to something that will be completed at a point in the future:

We hope to have finished the building works by the end of March.

We can use the perfect infinitive in a clause with a verb that has no subject (a non-finite clause). It can refer to events which did happen in the past or to events that might have happened (but did not happen):

To have got the job in the face of such stiff competition was a great achievement. (The person did get the job.)
To have won the race would have been fantastic, but even coming second was a great achievement. (The person did not win the race.)