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16/03/2014, 19:20
... must have |uv|a ≤ and |uv|b ≤ Case |uv|c ≥ Since |u|c ≤ and |v|c ≤ 2, we must have |u|c ≥ and |v|c ≥ Also, ψ(uv) ∈ [−2, 2] × [−2, 2] implies that |uv|a ≥ and |uv|b ≥ Since u and v are short, it ... vm−1,2 ) Lemma If u and v are short strings and ψ(uv) ∈ [−2, 2] × [−2, 2], then |uv|d ≤ for each d ∈ {a, b, c} Proof Since u and v are short, we have |u|a ≤ 4, |u|b ≤ 4, |u|c ≤ and |v|a ≤ 4, |v|b ... string, and (iii) both ψ(u1 u2 ) and ψ(v1 v2 ) are in [−2, 2] × [−2, 2] Then either (u1 , u2 ) or (v1 , v2 ) is short Proof Suppose (u1 , u2 ) and (v1 , v2 ) are both long Since (u1 , u2 ) and (v1...