YES(?, 31) Initial complexity problem: 1: T: (?, 1) f12(a, b, c, d, e) -> f12(a + 1, b, c, d, e) [ 9 >= a ] (?, 1) f25(a, b, c, d, e) -> f25(a, b + 1, c, d, e) [ 9 >= b ] (?, 1) f25(a, b, c, d, e) -> f36(a, b, c, d, e) [ b >= 10 ] (?, 1) f12(a, b, c, d, e) -> f25(a, 0, f, d, e) [ a >= 10 ] (1, 1) f0(a, b, c, d, e) -> f12(0, b, c, f, g) start location: f0 leaf cost: 0 Repeatedly removing leaves of the complexity graph in problem 1 produces the following problem: 2: T: (?, 1) f12(a, b, c, d, e) -> f12(a + 1, b, c, d, e) [ 9 >= a ] (?, 1) f25(a, b, c, d, e) -> f25(a, b + 1, c, d, e) [ 9 >= b ] (?, 1) f12(a, b, c, d, e) -> f25(a, 0, f, d, e) [ a >= 10 ] (1, 1) f0(a, b, c, d, e) -> f12(0, b, c, f, g) start location: f0 leaf cost: 1 A polynomial rank function with Pol(f12) = 1 Pol(f25) = 0 Pol(f0) = 1 orients all transitions weakly and the transition f12(a, b, c, d, e) -> f25(a, 0, f, d, e) [ a >= 10 ] strictly and produces the following problem: 3: T: (?, 1) f12(a, b, c, d, e) -> f12(a + 1, b, c, d, e) [ 9 >= a ] (?, 1) f25(a, b, c, d, e) -> f25(a, b + 1, c, d, e) [ 9 >= b ] (1, 1) f12(a, b, c, d, e) -> f25(a, 0, f, d, e) [ a >= 10 ] (1, 1) f0(a, b, c, d, e) -> f12(0, b, c, f, g) start location: f0 leaf cost: 1 A polynomial rank function with Pol(f12) = 18 Pol(f25) = -V_2 + 18 Pol(f0) = 18 orients all transitions weakly and the transition f25(a, b, c, d, e) -> f25(a, b + 1, c, d, e) [ 9 >= b ] strictly and produces the following problem: 4: T: (?, 1) f12(a, b, c, d, e) -> f12(a + 1, b, c, d, e) [ 9 >= a ] (18, 1) f25(a, b, c, d, e) -> f25(a, b + 1, c, d, e) [ 9 >= b ] (1, 1) f12(a, b, c, d, e) -> f25(a, 0, f, d, e) [ a >= 10 ] (1, 1) f0(a, b, c, d, e) -> f12(0, b, c, f, g) start location: f0 leaf cost: 1 A polynomial rank function with Pol(f12) = -V_1 + 10 Pol(f25) = -V_1 Pol(f0) = 10 orients all transitions weakly and the transition f12(a, b, c, d, e) -> f12(a + 1, b, c, d, e) [ 9 >= a ] strictly and produces the following problem: 5: T: (10, 1) f12(a, b, c, d, e) -> f12(a + 1, b, c, d, e) [ 9 >= a ] (18, 1) f25(a, b, c, d, e) -> f25(a, b + 1, c, d, e) [ 9 >= b ] (1, 1) f12(a, b, c, d, e) -> f25(a, 0, f, d, e) [ a >= 10 ] (1, 1) f0(a, b, c, d, e) -> f12(0, b, c, f, g) start location: f0 leaf cost: 1 Complexity upper bound 31 Time: 0.199 sec (SMT: 0.189 sec)