YES(?, 12) Initial complexity problem: 1: T: (1, 1) f0(a, b, c, d, e) -> f7(f, f, 10, 0, e) (?, 1) f7(a, b, c, d, e) -> f7(a, b, c, d + 1, f) [ c >= d + 1 ] (?, 1) f7(a, b, c, d, e) -> f19(a, b, c, d, e) [ d >= c ] 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, c, d, e) -> f7(f, f, 10, 0, e) (?, 1) f7(a, b, c, d, e) -> f7(a, b, c, d + 1, f) [ c >= d + 1 ] start location: f0 leaf cost: 1 A polynomial rank function with Pol(f0) = 10 Pol(f7) = V_3 - V_4 orients all transitions weakly and the transition f7(a, b, c, d, e) -> f7(a, b, c, d + 1, f) [ c >= d + 1 ] strictly and produces the following problem: 3: T: (1, 1) f0(a, b, c, d, e) -> f7(f, f, 10, 0, e) (10, 1) f7(a, b, c, d, e) -> f7(a, b, c, d + 1, f) [ c >= d + 1 ] start location: f0 leaf cost: 1 Complexity upper bound 12 Time: 0.080 sec (SMT: 0.076 sec)