U8Air Extended

[hecmas]

A new air has been added:

airtemplate U8AirExtended(const int N = 2**16) {
    // The same as U8Air, but with the cartisian product of U8 with itself to allow for lookups from two to two

    if (N != 2**16) {
        error(`The number of rows N should be 2**16 to use the predefined range U8xU8, got N=${N} instead`);
    }

    col witness mul;

    col fixed U8_A = [0..BYTE]...;
    col fixed U8_B = [0:P2_8..BYTE:P2_8];
    lookup_proves(DEFAULT_OPIDS[2], [U8_A, U8_B], mul, PIOP_NAME_RANGE_CHECK);
}

that allows to optimize multiple range checks to the range $[0,2^8-1]$ by a factor of two.

• Used for the case where the the range is of the form $[a,b]$, with $a \geq 0$ and $b \leq 2^8-1$, except for the range $[0,2^8-1]$.
• Used to divide the number of assumes to the exact range $[0,2^8-1]$ by a factor of two, since two range checks to the range $[0,2^8-1]$ can be compressed to a single range check to the (cartesian product) range $[0,2^8-1] \times [0,2^8-1]$.
• Do the second one for the generic case!!!! $[0,2^N-1]$ as long as the cartesian product is not too large. We can also do it for the cartesian product of the same range three times or more as long as it is not too large.
• We whould do it with a new api like range_check_double(), range_check_triple...
• Fix witness computation issue where the std needs to group columns into pairs and keep this grouping along all the std_range_check wintess computation.

Type	#RangeChecks	#Cols	#Const	#Evals
Old	1	11	3	8
Old	2	16	4	11
Old	4	23	4	16
Old	8	37	6	26
Old	16	65	10	46
New	1	11	3	8
New	2	13	3	10
New	4	20	3	15
New	8	31	4	24
New	16	53	6	42