YES(?, a + 3) Initial complexity problem: 1: T: (?, 1) f5(a, b) -> f5(a - 1, b) [ a >= 1 ] (?, 1) f5(a, b) -> f1(a, c) [ 0 >= a ] (1, 1) f300(a, b) -> f5(a - 1, b) [ a >= 1 ] (1, 1) f300(a, b) -> f1(a, c) [ 0 >= a ] start location: f300 leaf cost: 0 Repeatedly removing leaves of the complexity graph in problem 1 produces the following problem: 2: T: (?, 1) f5(a, b) -> f5(a - 1, b) [ a >= 1 ] (1, 1) f300(a, b) -> f5(a - 1, b) [ a >= 1 ] start location: f300 leaf cost: 2 A polynomial rank function with Pol(f5) = V_1 Pol(f300) = V_1 orients all transitions weakly and the transition f5(a, b) -> f5(a - 1, b) [ a >= 1 ] strictly and produces the following problem: 3: T: (a, 1) f5(a, b) -> f5(a - 1, b) [ a >= 1 ] (1, 1) f300(a, b) -> f5(a - 1, b) [ a >= 1 ] start location: f300 leaf cost: 2 Complexity upper bound a + 3 Time: 0.119 sec (SMT: 0.115 sec)