Double up Cross-table Lookups

[matthiasgoergens]

The constraints for our cross-table lookups are essentially:

Z(gw) - Z(w) = filter(w)/combined(w)

Brought into purely polynomial form that becomes this quadratic equation:

(Z(gw) - Z(w)) * combined(w) = filter(w)

However, we allow cubic constraints, thus we can do combine two lookups into one constraint check:

Z(gw) - Z(w) = filter0(w)/combined0(w) + filter1(w)/combined1(w)

Or as a cubic polynomial:

(Z(gw) - Z(w)) * combined0(w) * combined1(w) = filter0(w) * combined1(w)+ filter1(w) * combined0(w)

However, now we are mixing up two potentially unrelated cross-table lookups.

To solve this problem, we can either only allow merging of related lookups. (I think that's what plonky2 now does?) Or, we can give every cross table lookup a unique 'tag' that identifies it. The tag is just a different field element constant for each CTL, and we add it as a 'virtual' CTL output columns, that we send through reduce_with_powers to get combined.