Are both of these Raising’s?

[i] Equi
Billᵢ wants [[(for) Billᵢ] to leave at noon]
[Billᵢ wants [ __ ᵢ to leave at noon]]
[ii] Raising
[[(For) the casserole] to be done] seems
The casserole seems [to be done]
(Prof.John Lawler's article)

[case 1] She was last to arrive.(OALD)
[i] Equi: She was last [[for her] to arrive].
[ii] Raising: [[For her] to arrive] was last.

[case 2] They were the last to arrive.(OALD)
[i] Equi: They were the last [[for them] to arrive]].
[ii] Raising: [[For them] to arrive] was the last.

Are both, case 1 and case 2, Raising’s?
I tilt my head once one way, then the other. May I get the idea clear?

Solution 1:

This is really not about Raising or Equi; let's look at at the sentences:

1. She was last to arrive.
2. They were the last to arrive.

There are a couple of irrelevant differences between them.

• Sentence (2) uses the last, while (1) uses no article (both versions are grammatical here)
• Sentence (1) has a singular subject and verb, while (2) has plurals.
  Aside from these differences, however, the sentences are the same in structure.
  I'll concentrate on (1) here.

(1) has a lot of deletions -- i.e, lots of words are missing.
(This is grammatical; it's all done by rules)
(1) is equivalent to

• She was the last to arrive.
• She was the last one to arrive.
• She was the last person to arrive.
• She was the last one who arrived.
• She was the last one of the people who arrived.

The infinitive to arrive in (1) and (2) is a Relative Infinitive, not a Complement Infinitive.
Since it's not a subject or object complement clause of last, Raising is not applicable.
More on relative infinitives here.

In this case the predicate is a definite noun phrase containing a relative infinitive

• [vp (was) [np (the) last (one) (of the people) to arrive np] vp]

and there is no complement clause.