Luca Anderlini Dino Gerardi Roger Lagunoff

To keep notation usage down, during the proof of this step we let mt̄−1 = m and mt̄−1 = m ′. Suppose that for some t ≥ 0 there is a pair of sequences s = (st̄, . . . , st̄+t) and γ t̄,t̄+t = (γt̄, . . . , γt̄+t) such that: (a) at̄+τ (m, st̄, . . . , st̄+τ−1, γt̄, . . . , γt̄+τ−1)> at̄+τ (m′, st̄, . . . , st̄+τ−1, γt̄, . . . , γt̄+τ−1) for every τ ≤ t, and (b) at̄+t+1(m, st̄, . . . , st̄+t, γt̄, . . . , γt̄+t) ≤ at̄+t+1(m, st̄, . . . , st̄+t, γt̄, . . . , γt̄+t). (If such a pair of sequences cannot be found then there is nothing to prove.) We need to consider two, mutually exclusive and exhaustive, cases.