User blog comment:Tomtom2357/Complete Proper Combinators/@comment-1605058-20130713065502/@comment-10217029-20130713083128

I have figured out a way of testing combinators for completeness. All I do is I pick a combinator, and figure out whether it can express I. For example: If I check the combinator Axyz=x(yx), I will check two possibilities: Aa=I, and Aab=I. All other combinators will reduce to those two forms. If Aa=I for some a, then AaIc=IIc=c, so a(Ia)=aa=c for all c, which is impossible, so Aa is not I for any combinator a. If Aab=I for some a and b, then Aabc=Ic=c, so a(ba)=c for all c, which is impossible. Therefore I is not expressable using A, so A is not universal.