Let S be a string of length n, and, using 1-based indexing, let r_k denote the value of r at iteration k of the Z-algorithm when run on S. Then the sequence (r_1, r_2, \dots, r_n) is always non-decreasing, with r_{i-1} \le r_i \quad \forall\, 1 < i \le n, for any S.