YES(?, 23) Initial complexity problem: 1: T: (1, 1) f0(a, b, c) -> f8(d, 0, c) (?, 1) f8(a, b, c) -> f8(a, b + 1, c) [ 9 >= b ] (?, 1) f19(a, b, c) -> f19(a, b, c + 1) [ 9 >= c ] (?, 1) f19(a, b, c) -> f29(a, b, c) [ c >= 10 ] (?, 1) f8(a, b, c) -> f19(a, b, 0) [ b >= 10 ] 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) -> f8(d, 0, c) (?, 1) f8(a, b, c) -> f8(a, b + 1, c) [ 9 >= b ] (?, 1) f19(a, b, c) -> f19(a, b, c + 1) [ 9 >= c ] (?, 1) f8(a, b, c) -> f19(a, b, 0) [ b >= 10 ] start location: f0 leaf cost: 1 A polynomial rank function with Pol(f0) = 1 Pol(f8) = 1 Pol(f19) = 0 orients all transitions weakly and the transition f8(a, b, c) -> f19(a, b, 0) [ b >= 10 ] strictly and produces the following problem: 3: T: (1, 1) f0(a, b, c) -> f8(d, 0, c) (?, 1) f8(a, b, c) -> f8(a, b + 1, c) [ 9 >= b ] (?, 1) f19(a, b, c) -> f19(a, b, c + 1) [ 9 >= c ] (1, 1) f8(a, b, c) -> f19(a, b, 0) [ b >= 10 ] start location: f0 leaf cost: 1 A polynomial rank function with Pol(f0) = 10 Pol(f8) = 10 Pol(f19) = -V_3 + 10 orients all transitions weakly and the transition f19(a, b, c) -> f19(a, b, c + 1) [ 9 >= c ] strictly and produces the following problem: 4: T: (1, 1) f0(a, b, c) -> f8(d, 0, c) (?, 1) f8(a, b, c) -> f8(a, b + 1, c) [ 9 >= b ] (10, 1) f19(a, b, c) -> f19(a, b, c + 1) [ 9 >= c ] (1, 1) f8(a, b, c) -> f19(a, b, 0) [ b >= 10 ] start location: f0 leaf cost: 1 A polynomial rank function with Pol(f0) = 10 Pol(f8) = -V_2 + 10 Pol(f19) = -V_2 orients all transitions weakly and the transition f8(a, b, c) -> f8(a, b + 1, c) [ 9 >= b ] strictly and produces the following problem: 5: T: (1, 1) f0(a, b, c) -> f8(d, 0, c) (10, 1) f8(a, b, c) -> f8(a, b + 1, c) [ 9 >= b ] (10, 1) f19(a, b, c) -> f19(a, b, c + 1) [ 9 >= c ] (1, 1) f8(a, b, c) -> f19(a, b, 0) [ b >= 10 ] start location: f0 leaf cost: 1 Complexity upper bound 23 Time: 0.158 sec (SMT: 0.149 sec)