YES(?, 31) Initial complexity problem: 1: T: (1, 1) f0(a, b, c, d, e, f, g, h, i) -> f7(30, 30, 1, 0, 2, f, g, h, i) (?, 1) f7(a, b, c, d, e, f, g, h, i) -> f7(a, b, c + d, c, e + 1, c, g, h, i) [ b >= e ] (?, 1) f7(a, b, c, d, e, f, g, h, i) -> f19(a, b, c, d, e, f, c, c, c) [ e >= b + 1 ] start location: f0 leaf cost: 0 Repeatedly removing leaves of the complexity graph in problem 1 produces the following problem: 2: T: (1, 1) f0(a, b, c, d, e, f, g, h, i) -> f7(30, 30, 1, 0, 2, f, g, h, i) (?, 1) f7(a, b, c, d, e, f, g, h, i) -> f7(a, b, c + d, c, e + 1, c, g, h, i) [ b >= e ] start location: f0 leaf cost: 1 A polynomial rank function with Pol(f0) = 29 Pol(f7) = V_2 - V_5 + 1 orients all transitions weakly and the transition f7(a, b, c, d, e, f, g, h, i) -> f7(a, b, c + d, c, e + 1, c, g, h, i) [ b >= e ] strictly and produces the following problem: 3: T: (1, 1) f0(a, b, c, d, e, f, g, h, i) -> f7(30, 30, 1, 0, 2, f, g, h, i) (29, 1) f7(a, b, c, d, e, f, g, h, i) -> f7(a, b, c + d, c, e + 1, c, g, h, i) [ b >= e ] start location: f0 leaf cost: 1 Complexity upper bound 31 Time: 0.176 sec (SMT: 0.167 sec)