YES(?, 25) Initial complexity problem: 1: T: (?, 1) f4(a) -> f5(a) [ 0 >= b + 1 ] (?, 1) f4(a) -> f5(a) (1, 1) f0(a) -> f4(0) (?, 1) f5(a) -> f11(a) [ a >= 3 ] (?, 1) f4(a) -> f11(a) (?, 1) f5(a) -> f4(a + 1) [ 2 >= a ] (?, 1) f11(a) -> f14(a) [ 1 >= a ] (?, 1) f11(a) -> f14(a) [ a >= 2 ] 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) -> f5(a) [ 0 >= b + 1 ] (?, 1) f4(a) -> f5(a) (1, 1) f0(a) -> f4(0) (?, 1) f5(a) -> f4(a + 1) [ 2 >= a ] start location: f0 leaf cost: 4 A polynomial rank function with Pol(f4) = -2*V_1 + 6 Pol(f5) = -2*V_1 + 5 Pol(f0) = 6 orients all transitions weakly and the transition f5(a) -> f4(a + 1) [ 2 >= a ] strictly and produces the following problem: 3: T: (?, 1) f4(a) -> f5(a) [ 0 >= b + 1 ] (?, 1) f4(a) -> f5(a) (1, 1) f0(a) -> f4(0) (6, 1) f5(a) -> f4(a + 1) [ 2 >= a ] start location: f0 leaf cost: 4 Repeatedly propagating knowledge in problem 3 produces the following problem: 4: T: (7, 1) f4(a) -> f5(a) [ 0 >= b + 1 ] (7, 1) f4(a) -> f5(a) (1, 1) f0(a) -> f4(0) (6, 1) f5(a) -> f4(a + 1) [ 2 >= a ] start location: f0 leaf cost: 4 Complexity upper bound 25 Time: 0.118 sec (SMT: 0.114 sec)