The definition below is from the book "Oxford Guide to English Grammar"
I can't understand why there is no auxiliary after to in the example "I haven't washed up yet, but I'm going to."
But why there is an auxiliary after to in this example "I haven't done as much work today as I'd like to have."
Is that because the first example is split by the word ", but" there?
Then what about this example "Jane was chosen for the job, although she didn't expect to be."
I want to know the grammar rule applied in the above examples.