YES(?, 17) Initial complexity problem: 1: T: (1, 1) f0(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s) -> f12(3, t, 3, 1, 0, f, g, h, i, j, k, l, m, n, o, p, q, r, s) (?, 1) f12(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s) -> f12(a, b, c, t, e + 1, f, g, h, i, j, k, l, m, n, o, p, q, r, s) [ c >= e + 1 ] (?, 1) f24(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s) -> f24(a, b, c, d, e, f, g + 1, t, i, j, k, l, m, n, o, p, q, r, s) [ f >= g + 1 ] (?, 1) f36(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s) -> f36(a, b, c, d, e, f, g, h, i, j + 1, t, l, m, n, o, p, q, r, s) [ i >= j + 1 ] (?, 1) f36(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s) -> f46(a, b, c, d, e, f, g, h, i, j, k, k, k, n, o, p, q, r, s) [ j >= i ] (?, 1) f24(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s) -> f36(a, b, c, d, e, f, g, h, a, 0, 1, l, m, h, h, t, q, r, s) [ g >= f ] (?, 1) f12(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s) -> f24(a, b, c, d, e, a, 0, 1, i, j, k, l, m, n, o, p, d, d, t) [ e >= c ] start location: f0 leaf cost: 0 Slicing away variables that do not contribute to conditions from problem 1 leaves variables [a, c, e, f, g, i, j]. We thus obtain the following problem: 2: T: (?, 1) f12(a, c, e, f, g, i, j) -> f24(a, c, e, a, 0, i, j) [ e >= c ] (?, 1) f24(a, c, e, f, g, i, j) -> f36(a, c, e, f, g, a, 0) [ g >= f ] (?, 1) f36(a, c, e, f, g, i, j) -> f46(a, c, e, f, g, i, j) [ j >= i ] (?, 1) f36(a, c, e, f, g, i, j) -> f36(a, c, e, f, g, i, j + 1) [ i >= j + 1 ] (?, 1) f24(a, c, e, f, g, i, j) -> f24(a, c, e, f, g + 1, i, j) [ f >= g + 1 ] (?, 1) f12(a, c, e, f, g, i, j) -> f12(a, c, e + 1, f, g, i, j) [ c >= e + 1 ] (1, 1) f0(a, c, e, f, g, i, j) -> f12(3, 3, 0, f, g, i, j) start location: f0 leaf cost: 0 Repeatedly removing leaves of the complexity graph in problem 2 produces the following problem: 3: T: (?, 1) f12(a, c, e, f, g, i, j) -> f24(a, c, e, a, 0, i, j) [ e >= c ] (?, 1) f24(a, c, e, f, g, i, j) -> f36(a, c, e, f, g, a, 0) [ g >= f ] (?, 1) f36(a, c, e, f, g, i, j) -> f36(a, c, e, f, g, i, j + 1) [ i >= j + 1 ] (?, 1) f24(a, c, e, f, g, i, j) -> f24(a, c, e, f, g + 1, i, j) [ f >= g + 1 ] (?, 1) f12(a, c, e, f, g, i, j) -> f12(a, c, e + 1, f, g, i, j) [ c >= e + 1 ] (1, 1) f0(a, c, e, f, g, i, j) -> f12(3, 3, 0, f, g, i, j) start location: f0 leaf cost: 1 A polynomial rank function with Pol(f12) = 2 Pol(f24) = 1 Pol(f36) = -1 Pol(f0) = 2 orients all transitions weakly and the transitions f24(a, c, e, f, g, i, j) -> f36(a, c, e, f, g, a, 0) [ g >= f ] f12(a, c, e, f, g, i, j) -> f24(a, c, e, a, 0, i, j) [ e >= c ] strictly and produces the following problem: 4: T: (2, 1) f12(a, c, e, f, g, i, j) -> f24(a, c, e, a, 0, i, j) [ e >= c ] (2, 1) f24(a, c, e, f, g, i, j) -> f36(a, c, e, f, g, a, 0) [ g >= f ] (?, 1) f36(a, c, e, f, g, i, j) -> f36(a, c, e, f, g, i, j + 1) [ i >= j + 1 ] (?, 1) f24(a, c, e, f, g, i, j) -> f24(a, c, e, f, g + 1, i, j) [ f >= g + 1 ] (?, 1) f12(a, c, e, f, g, i, j) -> f12(a, c, e + 1, f, g, i, j) [ c >= e + 1 ] (1, 1) f0(a, c, e, f, g, i, j) -> f12(3, 3, 0, f, g, i, j) start location: f0 leaf cost: 1 A polynomial rank function with Pol(f12) = V_2 - V_3 Pol(f24) = 4*V_1 + V_2 - V_3 - 4*V_4 - V_5 - 2 Pol(f36) = V_1 + V_2 - V_3 + V_4 - 6*V_5 + 3*V_6 - V_7 - 2 Pol(f0) = 3 orients all transitions weakly and the transition f12(a, c, e, f, g, i, j) -> f12(a, c, e + 1, f, g, i, j) [ c >= e + 1 ] strictly and produces the following problem: 5: T: (2, 1) f12(a, c, e, f, g, i, j) -> f24(a, c, e, a, 0, i, j) [ e >= c ] (2, 1) f24(a, c, e, f, g, i, j) -> f36(a, c, e, f, g, a, 0) [ g >= f ] (?, 1) f36(a, c, e, f, g, i, j) -> f36(a, c, e, f, g, i, j + 1) [ i >= j + 1 ] (?, 1) f24(a, c, e, f, g, i, j) -> f24(a, c, e, f, g + 1, i, j) [ f >= g + 1 ] (3, 1) f12(a, c, e, f, g, i, j) -> f12(a, c, e + 1, f, g, i, j) [ c >= e + 1 ] (1, 1) f0(a, c, e, f, g, i, j) -> f12(3, 3, 0, f, g, i, j) start location: f0 leaf cost: 1 A polynomial rank function with Pol(f12) = V_1 + 2 Pol(f24) = V_4 - V_5 + 2 Pol(f36) = V_1 + V_4 - V_5 - V_6 - V_7 + 2 Pol(f0) = 5 orients all transitions weakly and the transition f24(a, c, e, f, g, i, j) -> f24(a, c, e, f, g + 1, i, j) [ f >= g + 1 ] strictly and produces the following problem: 6: T: (2, 1) f12(a, c, e, f, g, i, j) -> f24(a, c, e, a, 0, i, j) [ e >= c ] (2, 1) f24(a, c, e, f, g, i, j) -> f36(a, c, e, f, g, a, 0) [ g >= f ] (?, 1) f36(a, c, e, f, g, i, j) -> f36(a, c, e, f, g, i, j + 1) [ i >= j + 1 ] (5, 1) f24(a, c, e, f, g, i, j) -> f24(a, c, e, f, g + 1, i, j) [ f >= g + 1 ] (3, 1) f12(a, c, e, f, g, i, j) -> f12(a, c, e + 1, f, g, i, j) [ c >= e + 1 ] (1, 1) f0(a, c, e, f, g, i, j) -> f12(3, 3, 0, f, g, i, j) start location: f0 leaf cost: 1 A polynomial rank function with Pol(f12) = V_1 Pol(f24) = V_1 Pol(f36) = V_6 - V_7 Pol(f0) = 3 orients all transitions weakly and the transition f36(a, c, e, f, g, i, j) -> f36(a, c, e, f, g, i, j + 1) [ i >= j + 1 ] strictly and produces the following problem: 7: T: (2, 1) f12(a, c, e, f, g, i, j) -> f24(a, c, e, a, 0, i, j) [ e >= c ] (2, 1) f24(a, c, e, f, g, i, j) -> f36(a, c, e, f, g, a, 0) [ g >= f ] (3, 1) f36(a, c, e, f, g, i, j) -> f36(a, c, e, f, g, i, j + 1) [ i >= j + 1 ] (5, 1) f24(a, c, e, f, g, i, j) -> f24(a, c, e, f, g + 1, i, j) [ f >= g + 1 ] (3, 1) f12(a, c, e, f, g, i, j) -> f12(a, c, e + 1, f, g, i, j) [ c >= e + 1 ] (1, 1) f0(a, c, e, f, g, i, j) -> f12(3, 3, 0, f, g, i, j) start location: f0 leaf cost: 1 Complexity upper bound 17 Time: 0.282 sec (SMT: 0.262 sec)