Polytope Enumeration System hits new high

Running for quite some time now we have got > 400K (d=5) polytopes and > 150K (d=7)
polytopes, alongwith their Face lattice.

Face lattice construction from VERTICES_IN_FACETS uses the co-atomic property of polytopes, that faces are proper intersection of facets;

Consider
VERTICES_IN_FACETS
{ 0 1 2 3 5 6 7 8 9 11 12 13 14 15}
{ 0 1 2 4 5 6 7 8 10 11 12 13 14 16}
{ 0 1 3 4 5 6 7 9 10 11 12 13 15 16}
{ 0 2 3 4 5 6 8 9 10 11 12 14 15 16}
{ 1 2 3 4 5 7 8 9 10 11 13 14 15 16}
{ 0 1 2 3 4 6 7 8 9 10}
{ 0 1 2 3 4 12 13 14 15 16}
{ 5 6 7 8 9 10}
{ 11 12 13 14 15 16}

Using "polymake" for reference gives us:
SIMPLE
1

DIAMETER
3

F_VECTOR
17 51 75 65 33 9

CD_INDEX
c^6 + 7c^4d + 24c^3dc + 39c^2dc^2 + 51c^2d^2 + 34cdc^3 + 85cdcd + 102cd^2c + 15dc^4 + 54dc^2d + 105dcdc + 75d^2c^2 + 102d^3

Also we get:
F2_VECTOR
17 102 255 340 255 102
102 51 255 510 510 255
255 255 75 300 450 300
340 510 300 65 195 195
255 510 450 195 33 66
102 255 300 195 66 9

SIMPLE
1

FLAG_VECTOR
1 17 51 75 255 65 340 510 33 255 510 450 1530

CD_INDEX_COEFFICIENTS
1 7 24 39 51 34 85 102 15 54 105 75 102

The constructed graph is:
GRAPH
{1 2 3 4 6 12}
{0 2 3 4 7 13}
{0 1 3 4 8 14}
{0 1 2 4 9 15}
{0 1 2 3 10 16}
{6 7 8 9 10 11}
{0 5 7 8 9 10}
{1 5 6 8 9 10}
{2 5 6 7 9 10}
{3 5 6 7 8 10}
{4 5 6 7 8 9}
{5 12 13 14 15 16}
{0 11 13 14 15 16}
{1 11 12 14 15 16}
{2 11 12 13 15 16}
{3 11 12 13 14 16}
{4 11 12 13 14 15}

Yohoo! another step towards full classification of simple polytopes using Cohen-Macaulay ring algebra.