Adjective complements: what are they?

[Soup]

[1] He says he has all the proof. (verb complement)
=> Wouldn't this be the verb's object?

[2] He runs the fastest in the team. (adverb complement)
=> I'm not sure about this one. Complements are different from adjuncts.

[donnach]

Thanks, Soup. I still think it's a rather diffuse notion. Since so many different structures fall into the adjective complement type, what's the idea of using it? By analogy, we can introduce noun complements, verb complements, even prepositional complements.

Yes. In the example studying for TOEFL is exhausting, why can't "studying for TOEFL" simply be a gerund phrase (composed of a gerund + prep. phrase modifying studying, but when we try to break it down further it gets so confusing to me, i.e., whether "for TOEFL" acts as an adj. modifying the noun "studying" or as an adverb modifying the verb "studying" (verb within the phrase itself), but isn't my main question)....sorry, resuming the main question....why can't it simply be a gerund phrase in form and the subject of the sentence in function?

Also, "exhausting" is termed both predicate adjective and adjective complement, so, we then have an adjective complement "Studying for TOEFL" complementing an adjective complement "exhausting".

I find adjective complements just add complication but don't really give me additional understanding of the English language. But maybe I just need to keep making the effort to understand and recognize these kind of slippery (to me) things called adjective complements.