I thank you for your response. Well, I am unsure because but I am going to have a try.
In the sentence 1: I would have done something.
I think the auxiliary is would which is in fact a modal auxiliary while done is the verb do. I would say that have is an auxiliary or a verb which is in association with done (past participle of do).
In the sentence 2: You should be eating.
Should is also a modal auxiliary while be is a an auxiliary.
In the sentencde 3: We have been going to school.
In the sentence 4: I had not been delivering the letters.
The auxiliaries are had and been and delievring is the in-form of a verb. But I taught my pupils that we don’t use two auxliaries with a verb. The reason why I asked the question above and I also taught them that have and be are auxiliaries when they are used alone such as the following below.
- I was going to school.
- She has done her homwework.
They have also been taught about the sentence structure as follow.
Affirmative form: subject + auxiliary + verb
Negative form: subject + auxiliary + not + verb
Case 1: Auxiliary + subject + verb
Case 2: Auxialiary +subject + not + verb
Now I would like to teach them about the use of a modal and an auxialiry in a sentence like that:
Affirmative form: subject + modal + auxiliary + verb
eg: We should have eaten it.
Negative form: subject +modal + not + auxiliary + verb
eg: We should not have eaten it.
Case 1: Modal + subject + not + auxiliary + verb
eg: Should we have eaten it?
Case 2: Modal subject + auxiliary + not + verb
eg: Should we not have eaten it?