YES(?, b + 3) Initial complexity problem: 1: T: (?, 1) f4(a, b, c) -> f5(a, b, 1) [ 0 >= a /\ 0 >= b ] (1, 1) f0(a, b, c) -> f2(a, b, 1) [ a >= 1 ] (?, 1) f4(a, b, c) -> f4(a, b - 1, c) [ b >= 1 ] (1, 1) f0(a, b, c) -> f4(a, b, 0) [ 0 >= a ] start location: f0 leaf cost: 0 Repeatedly removing leaves of the complexity graph in problem 1 produces the following problem: 2: T: (?, 1) f4(a, b, c) -> f4(a, b - 1, c) [ b >= 1 ] (1, 1) f0(a, b, c) -> f4(a, b, 0) [ 0 >= a ] start location: f0 leaf cost: 2 A polynomial rank function with Pol(f4) = V_2 Pol(f0) = V_2 orients all transitions weakly and the transition f4(a, b, c) -> f4(a, b - 1, c) [ b >= 1 ] strictly and produces the following problem: 3: T: (b, 1) f4(a, b, c) -> f4(a, b - 1, c) [ b >= 1 ] (1, 1) f0(a, b, c) -> f4(a, b, 0) [ 0 >= a ] start location: f0 leaf cost: 2 Complexity upper bound b + 3 Time: 0.090 sec (SMT: 0.086 sec)