MAYBE Initial complexity problem: 1: T: (1, 1) f3(a, b, c) -> f2(a, b, c) (?, 1) f2(a, b, c) -> f2(a - 1, d, c) [ d >= 1 /\ a >= 0 ] (?, 1) f2(a, b, c) -> f2(a - 1, d, c) [ 0 >= d + 1 /\ a >= 0 ] (?, 1) f2(a, b, c) -> f2(a, 0, c) [ a >= 0 ] (?, 1) f2(a, b, c) -> f300(a, b, d) [ 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, c) -> f2(a, b, c) (?, 1) f2(a, b, c) -> f2(a - 1, d, c) [ d >= 1 /\ a >= 0 ] (?, 1) f2(a, b, c) -> f2(a - 1, d, c) [ 0 >= d + 1 /\ a >= 0 ] (?, 1) f2(a, b, c) -> f2(a, 0, c) [ a >= 0 ] start location: f3 leaf cost: 1 A polynomial rank function with Pol(f3) = V_1 + 1 Pol(f2) = V_1 + 1 orients all transitions weakly and the transitions f2(a, b, c) -> f2(a - 1, d, c) [ d >= 1 /\ a >= 0 ] f2(a, b, c) -> f2(a - 1, d, c) [ 0 >= d + 1 /\ a >= 0 ] strictly and produces the following problem: 3: T: (1, 1) f3(a, b, c) -> f2(a, b, c) (a + 1, 1) f2(a, b, c) -> f2(a - 1, d, c) [ d >= 1 /\ a >= 0 ] (a + 1, 1) f2(a, b, c) -> f2(a - 1, d, c) [ 0 >= d + 1 /\ a >= 0 ] (?, 1) f2(a, b, c) -> f2(a, 0, c) [ a >= 0 ] start location: f3 leaf cost: 1 Complexity upper bound ? Time: 0.266 sec (SMT: 0.249 sec)