YES(?, 499) Initial complexity problem: 1: T: (1, 1) f0(a, b, c, d, e) -> f4(0, b, c, d, e) (?, 1) f20(a, b, c, d, e) -> f20(a, b + 1, b, d, e) [ 199 >= b ] (?, 1) f20(a, b, c, d, e) -> f31(a, b, c, d, e) [ b >= 200 ] (?, 1) f4(a, b, c, d, e) -> f4(a + 1, b, c, a, a) [ 99 >= a ] (?, 1) f4(a, b, c, d, e) -> f20(a, 100, c, d, e) [ a >= 100 ] 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) -> f4(0, b, c, d, e) (?, 1) f20(a, b, c, d, e) -> f20(a, b + 1, b, d, e) [ 199 >= b ] (?, 1) f4(a, b, c, d, e) -> f4(a + 1, b, c, a, a) [ 99 >= a ] (?, 1) f4(a, b, c, d, e) -> f20(a, 100, c, d, e) [ a >= 100 ] start location: f0 leaf cost: 1 A polynomial rank function with Pol(f0) = 99 Pol(f4) = 99 Pol(f20) = 98 orients all transitions weakly and the transition f4(a, b, c, d, e) -> f20(a, 100, c, d, e) [ a >= 100 ] strictly and produces the following problem: 3: T: (1, 1) f0(a, b, c, d, e) -> f4(0, b, c, d, e) (?, 1) f20(a, b, c, d, e) -> f20(a, b + 1, b, d, e) [ 199 >= b ] (?, 1) f4(a, b, c, d, e) -> f4(a + 1, b, c, a, a) [ 99 >= a ] (99, 1) f4(a, b, c, d, e) -> f20(a, 100, c, d, e) [ a >= 100 ] start location: f0 leaf cost: 1 A polynomial rank function with Pol(f0) = 298 Pol(f4) = -V_1 + 298 Pol(f20) = -V_1 - V_2 + 398 orients all transitions weakly and the transition f4(a, b, c, d, e) -> f4(a + 1, b, c, a, a) [ 99 >= a ] strictly and produces the following problem: 4: T: (1, 1) f0(a, b, c, d, e) -> f4(0, b, c, d, e) (?, 1) f20(a, b, c, d, e) -> f20(a, b + 1, b, d, e) [ 199 >= b ] (298, 1) f4(a, b, c, d, e) -> f4(a + 1, b, c, a, a) [ 99 >= a ] (99, 1) f4(a, b, c, d, e) -> f20(a, 100, c, d, e) [ a >= 100 ] start location: f0 leaf cost: 1 A polynomial rank function with Pol(f0) = 100 Pol(f4) = 100 Pol(f20) = -V_2 + 200 orients all transitions weakly and the transition f20(a, b, c, d, e) -> f20(a, b + 1, b, d, e) [ 199 >= b ] strictly and produces the following problem: 5: T: (1, 1) f0(a, b, c, d, e) -> f4(0, b, c, d, e) (100, 1) f20(a, b, c, d, e) -> f20(a, b + 1, b, d, e) [ 199 >= b ] (298, 1) f4(a, b, c, d, e) -> f4(a + 1, b, c, a, a) [ 99 >= a ] (99, 1) f4(a, b, c, d, e) -> f20(a, 100, c, d, e) [ a >= 100 ] start location: f0 leaf cost: 1 Complexity upper bound 499 Time: 0.195 sec (SMT: 0.184 sec)