Since in our assumptions, (X_{ij}) and (Y_{ij}) are mutually independent, the assumption of independence between (delta_{i,i-1,j}^{(1)}) and (delta _{i,i-1,j}^{(2)}) cannot cause a contradiction.
Furthermore, as any S ∈ T X [ 1 : i ] for i < i q, h i t(H′)[ S] = t (otherwise, it would cause a contradiction that H ℓ is a solution as no element in { u i q, …, u i ℓ } will cover S).
Note that an incorrect classification into the Hgs might cause a contradiction to the assumption of independent occurrences of mutations, but this is not a real concern, since Hg classification is done with very high confidence according to SNPs (Hammer et al., 2009; Karafet et al., 2008; The Y Chromosome Consortium, 2002).
Thus we have a contradiction.
So we have a contradiction.
The causes leading to such a contradiction might be that, on the one hand, different measures were used to test reading and writing abilities, on the other, summarization task served different purposes in these studies.
If weight (H ) < weight (H j ℓ ), then by Lemma 0.1, H ∪ H′ will be a t-cover hitting set with a smaller weight than the weight of H ℓ, which causes contradiction.
This late integration causes some contradictions between productivity and safety.
Similarly, it will cause the contradiction as before.
