YES(?, a + 3) Initial complexity problem: 1: T: (1, 1) f3(a, b) -> f1(a, b) (?, 1) f1(a, b) -> f1(a - 1, b) [ a >= 0 ] (?, 1) f1(a, b) -> f300(a, c) [ 0 >= a + 1 ] start location: f3 leaf cost: 0 Repeatedly removing leaves of the complexity graph in problem 1 produces the following problem: 2: T: (1, 1) f3(a, b) -> f1(a, b) (?, 1) f1(a, b) -> f1(a - 1, b) [ a >= 0 ] start location: f3 leaf cost: 1 A polynomial rank function with Pol(f3) = V_1 + 1 Pol(f1) = V_1 + 1 orients all transitions weakly and the transition f1(a, b) -> f1(a - 1, b) [ a >= 0 ] strictly and produces the following problem: 3: T: (1, 1) f3(a, b) -> f1(a, b) (a + 1, 1) f1(a, b) -> f1(a - 1, b) [ a >= 0 ] start location: f3 leaf cost: 1 Complexity upper bound a + 3 Time: 0.080 sec (SMT: 0.076 sec)