YES(?, 103) Initial complexity problem: 1: T: (1, 1) f0(a, b, c) -> f9(0, d, 0) (?, 1) f9(a, b, c) -> f9(a, b, c + 1) [ 49 >= c ] (?, 1) f17(a, b, c) -> f17(a + 1, b, c) [ 49 >= a ] (?, 1) f17(a, b, c) -> f24(a, b, c) [ a >= 50 ] (?, 1) f9(a, b, c) -> f17(0, b, c) [ c >= 50 ] 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) -> f9(0, d, 0) (?, 1) f9(a, b, c) -> f9(a, b, c + 1) [ 49 >= c ] (?, 1) f17(a, b, c) -> f17(a + 1, b, c) [ 49 >= a ] (?, 1) f9(a, b, c) -> f17(0, b, c) [ c >= 50 ] start location: f0 leaf cost: 1 A polynomial rank function with Pol(f0) = 1 Pol(f9) = 1 Pol(f17) = 0 orients all transitions weakly and the transition f9(a, b, c) -> f17(0, b, c) [ c >= 50 ] strictly and produces the following problem: 3: T: (1, 1) f0(a, b, c) -> f9(0, d, 0) (?, 1) f9(a, b, c) -> f9(a, b, c + 1) [ 49 >= c ] (?, 1) f17(a, b, c) -> f17(a + 1, b, c) [ 49 >= a ] (1, 1) f9(a, b, c) -> f17(0, b, c) [ c >= 50 ] start location: f0 leaf cost: 1 A polynomial rank function with Pol(f0) = 50 Pol(f9) = 50 Pol(f17) = -V_1 + 50 orients all transitions weakly and the transition f17(a, b, c) -> f17(a + 1, b, c) [ 49 >= a ] strictly and produces the following problem: 4: T: (1, 1) f0(a, b, c) -> f9(0, d, 0) (?, 1) f9(a, b, c) -> f9(a, b, c + 1) [ 49 >= c ] (50, 1) f17(a, b, c) -> f17(a + 1, b, c) [ 49 >= a ] (1, 1) f9(a, b, c) -> f17(0, b, c) [ c >= 50 ] start location: f0 leaf cost: 1 A polynomial rank function with Pol(f0) = 50 Pol(f9) = -V_3 + 50 Pol(f17) = -V_3 orients all transitions weakly and the transition f9(a, b, c) -> f9(a, b, c + 1) [ 49 >= c ] strictly and produces the following problem: 5: T: (1, 1) f0(a, b, c) -> f9(0, d, 0) (50, 1) f9(a, b, c) -> f9(a, b, c + 1) [ 49 >= c ] (50, 1) f17(a, b, c) -> f17(a + 1, b, c) [ 49 >= a ] (1, 1) f9(a, b, c) -> f17(0, b, c) [ c >= 50 ] start location: f0 leaf cost: 1 Complexity upper bound 103 Time: 0.171 sec (SMT: 0.161 sec)