YES(?, 12) Initial complexity problem: 1: T: (1, 1) f0(a, b) -> f3(1, b) (?, 1) f3(a, b) -> f3(a + 1, -a + 10) [ 10 >= a ] (?, 1) f3(a, b) -> f10(a, b) [ a >= 11 ] 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) -> f3(1, b) (?, 1) f3(a, b) -> f3(a + 1, -a + 10) [ 10 >= a ] start location: f0 leaf cost: 1 A polynomial rank function with Pol(f0) = 10 Pol(f3) = -V_1 + 11 orients all transitions weakly and the transition f3(a, b) -> f3(a + 1, -a + 10) [ 10 >= a ] strictly and produces the following problem: 3: T: (1, 1) f0(a, b) -> f3(1, b) (10, 1) f3(a, b) -> f3(a + 1, -a + 10) [ 10 >= a ] start location: f0 leaf cost: 1 Complexity upper bound 12 Time: 0.114 sec (SMT: 0.110 sec)