According to this ThoughtCo. article titled "NICE Properties of Auxiliary Verbs":
NICE is an acronym for the four syntactic characteristics that distinguish auxiliary verbs from lexical verbs in English grammar: negation, inversion, code, emphasis.
The NICE properties were identified as such by linguist Rodney Huddleston in the article "Some Theoretical Issues in the Description of the English Verb" (Lingua, 1976).
I won't be explaining in detail what the NICE properties are, because they're fairly straightforward. If you're not familiar with them, please visit the link above for reference.
Here are some NICE property examples shown in The Cambridge Grammar of the English Language (page 93):
In the above examples, only finite auxiliary verbs are shown to have the NICE properties. Do non-finite auxiliary verbs too have the NICE properties? If the NICE properties are "the four syntactic characteristics that distinguish auxiliary verbs from lexical verbs in English grammar", I think non-finite auxiliary verbs (as well as finite auxiliary verbs) should have the NICE properties. But they don't seem to.
Here are my failed attempts at producing NICE property examples of non-finite auxiliary verbs:
(1) Not having seen it, he had no clue. [Negation]
(2) N/A [Inversion]
(3) He may have seen it and I may have. [Code]
(4) ??They don't think he has seen it but he must HAVE seen it. [Emphasis]
In (1), not typically comes before having rather than after. It's not clear whether the not coming before having should be considered as negation of having. But let's assume for the moment that it should. Also, some speakers might put not after having, in which case not does negate having. It's possible, therefore, to argue that the non-finite auxiliary having does have the Negation property.
I can't think of any examples for (2) because there's no subject-auxiliary inversion in non-finite clauses.
In (3), although some speakers may choose to omit the second have, at least for those speakers who allow the second have, it can be said that the second have, a non-finite auxiliary, has the Code property.
In (4), it's unlikely that you could pronounce HAVE emphatically.
All in all, however hard I try, non-finite auxiliaries cannot seem to allow the Inversion and Emphasis constructions, which means that non-finite auxiliaries don't seem to have the NICE properties.
If non-finite auxiliary verbs don't have the NICE properties, which they don't seem to, is it really correct to say that the NICE properties are "the four syntactic characteristics that distinguish auxiliary verbs from lexical verbs in English grammar"?