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40 Chapter 2. On a conjecture about addition modulo 2 k − 12.4.4 A helpful constraint: min i (α i ) ≥ B − 1Until the end of this section we add the following constraint on t:min(α i ) ≥id∑β i − 1 = B − 1 = k − w H (t) − 1 .i=1That condition tells us that, if a is in S t,k , then a carry has to go through each subblock of 1’s,i.e. γ i′′ ≠ α i, otherwise too many carries would already be lost in the corresponding single block.Indeed, if γ i ′′ = α i , then δ i ′′ < β i and ɛ ′′i = γ′′ i + β i − δ i′′ ≥ α i + 1 ≥ B. This obviously impliesthat ∑ di=1 ɛ′′ i ≥ B and that a ∉ S t,k. Therefore, if a ∈ S t,k , then each block has to overflow. Insuch a situation, the blocks are therefore kind of independent.Recall that the definitions of the quantities γ i ′, δ′ i , γ′′ i and δ i′′in Subsection 2.4.1 trivially< ∑ di=1 δ′′ i .always impliesimply that the inequality ∑ di=1 γ′ i < ∑ di=1 δ′ i always implies the inequality ∑ di=1 γ′′ iFormulated in a different way, it means that the inequality ∑ di=1 γ′ i < ∑ di=1 δ′ ithat a ∈ S t,k . With our additional constraint the converse, which is obviously false in general,becomes true.In fact, whether the constraint is verified or not:• if ∀i δ ′′iγ ′′i= γ′ i= β i, then there are only 1’s in front of the 0’s of t, and both definitions coincide:= 0 and δ′′ i = δ′ i = β i — we always get k carries whatever values the γ i ’s take —;• if ∀i γ i′′ ≠ α i , then a carry goes out of each subblock of 1’s, and both definitions alsocoincide: γ i′′ = γ′ i and δ′′ i = δ′ i .Finally, if the constraint is verified and we are not in one of the above two situations, then thereare indices i and j such that δ i ′′ ≠ β i and γ j ′′ = α j, which implies that ∑ di=1 γ′′ i≥ B > ∑ di=1 δ′′ i .Moreover, ∑ di=1 γ′ i ≥ ∑ di=1 γ′′ i and B > ∑ di=1 δ′ i , so that ∑ di=1 γ′ i ≥ ∑ di=1 δ′ i .To summarize, we have just shown that, in our constrained case, there is an equivalencebetween a ∈ S t,k and ∑ di=1 γ′ i < ∑ di=1 δ′ i . Hence, we can formulate P t,k as follows.Proposition 2.4.9. Let k ≥ 2 and t ∈ ( Z/(2 k − 1)Z ) ∗verifying the constraintThenP t,k =min(α i ) ≥iB∑∆−1∑∆=1 Γ=0d∑β i − 1 = B − 1 = k − w H (t) − 1 .i=12 −∆−Γ−2d ∑∑d γ′ i =Γ0≤γ ′ i∑2 2#{i|δi=βi} 1 δ ′∑i =β i,γ i ′ =0 .d δ′ i =∆0≤δ ′ i ≤βi