Supplementary proofs from Section 2

We begin with a simple lemma showing that the values of the levels are monotone: Lemma A.1. First, we note that the second part of the lemma holds by lines 15-16. Let zil and zih be the value of zland zhin Algorithm 2 on line 9 on window i. There are two cases, depending on whether an element e? was added to the solutions or not. Suppose no element e? was added to the solution. Then all the levels remain the same.