Many proofs of the Kochen-Specker theorem use some form of the following argument (from Mermin's "Simple Unified Form for the major No-Hidden-Variables Theorems" )
[I]f some functional relation $$ f(A,B,C,\ldots)=0 $$ holds as an operator identity among the observables of a mutually commuting set, then since the results of the simultaneous measurements of $A,B,C,\ldots$ will be one of the sets $a,b,c,\ldots$ of simultaneous eigenvalues of $A,B,C,\ldots$, the results of those measurements must also satisfy $$ f(a,b,c,\ldots)=0 $$
Parity-type contradictions (e.g., $1=-1$ or $0=1$) are then seen to arise when $a,b,c\ldots$ are assigned values independently of the context in which they are measured. The only explicit forms of $f$ that I have seen are either (i) $A+B+C+\ldots$ or (ii) $(A)(B)(C)\ldots$ (see e.g., "Generalized Kochen-Specker Theorem" by Asher Peres, where both forms are used).
My question, then, is: are there examples of parity-type proofs where $f$ is, necessarily, not of the above forms (i) or (ii)? For example one could consider $A+(B)(C)\ldots$ etc. Ideally, I'm looking for explicit examples where $f$ is spelled out, but I would also be interested in arguments where a different kind of $f$ is implicitly used.