MAYBE Initial complexity problem: 1: T: (?, 1) f2(a, b, c) -> f2(a - 1, d, c) [ d >= 1 /\ a >= 2 ] (?, 1) f2(a, b, c) -> f2(a - 1, d, c) [ 0 >= d + 1 /\ a >= 2 ] (?, 1) f2(a, b, c) -> f1(a - 1, d, e) [ d >= 1 /\ 1 >= a ] (?, 1) f2(a, b, c) -> f1(a - 1, d, e) [ 0 >= d + 1 /\ 1 >= a ] (?, 1) f2(a, b, c) -> f2(a, 0, c) [ a >= 1 ] (?, 1) f2(a, b, c) -> f1(a, 0, d) [ 0 >= a ] (1, 1) f300(a, b, c) -> f2(a, b, c) start location: f300 leaf cost: 0 Repeatedly removing leaves of the complexity graph in problem 1 produces the following problem: 2: T: (?, 1) f2(a, b, c) -> f2(a - 1, d, c) [ d >= 1 /\ a >= 2 ] (?, 1) f2(a, b, c) -> f2(a - 1, d, c) [ 0 >= d + 1 /\ a >= 2 ] (?, 1) f2(a, b, c) -> f2(a, 0, c) [ a >= 1 ] (1, 1) f300(a, b, c) -> f2(a, b, c) start location: f300 leaf cost: 3 A polynomial rank function with Pol(f2) = V_1 - 1 Pol(f300) = V_1 - 1 orients all transitions weakly and the transitions f2(a, b, c) -> f2(a - 1, d, c) [ d >= 1 /\ a >= 2 ] f2(a, b, c) -> f2(a - 1, d, c) [ 0 >= d + 1 /\ a >= 2 ] strictly and produces the following problem: 3: T: (a + 1, 1) f2(a, b, c) -> f2(a - 1, d, c) [ d >= 1 /\ a >= 2 ] (a + 1, 1) f2(a, b, c) -> f2(a - 1, d, c) [ 0 >= d + 1 /\ a >= 2 ] (?, 1) f2(a, b, c) -> f2(a, 0, c) [ a >= 1 ] (1, 1) f300(a, b, c) -> f2(a, b, c) start location: f300 leaf cost: 3 Complexity upper bound ? Time: 0.269 sec (SMT: 0.253 sec)