available. 2.2 Iceberg Cubes Iceberg cubes [BR99, HPDW01] take Iceberg queries a step further by proposing a variant of the CUBE operator over a relation R(c1, c2, , ck, X): SELECT c1, c2, , ck, COUNT(*) [...] threshold . A prototypical query on a re- lation R(c1, c2, , ck, rest) with threshold is SELECT c1, c2, , ck, COUNT(rest) FROM R GROUP BY c1, c2, , ck HAVING COUNT(rest) The parameter is equivalent [...] (1) We claim that # + (2) We prove Inequality (2) by induction. The base case follows from Inequality (1) directly. Assume that Inequality (2) is true for + & % . We will show …