YES(?, 2*a + 3) Initial complexity problem: 1: T: (?, 1) f0(a, b, c) -> f3(-a, b, c) (?, 1) f3(a, b, c) -> f7(0, d, c) [ a = 0 ] (?, 1) f4(a, b, c) -> f7(0, d, c) [ a = 0 ] (1, 1) f6(a, b, c) -> f4(a, b, 1) [ a >= 1 ] (?, 1) f3(a, b, c) -> f4(-a - 1, b, 1) [ 0 >= a + 1 /\ 0 >= c ] (?, 1) f3(a, b, c) -> f4(-a - 1, b, 1) [ 0 >= a + 1 /\ c >= 2 ] (?, 1) f3(a, b, c) -> f4(-a - 1, b, 1) [ a >= 1 /\ 0 >= c ] (?, 1) f3(a, b, c) -> f4(-a - 1, b, 1) [ a >= 1 /\ c >= 2 ] (?, 1) f4(a, b, c) -> f3(-a + 1, b, 0) [ 0 >= a + 1 /\ c = 1 ] (?, 1) f4(a, b, c) -> f3(-a + 1, b, 0) [ a >= 1 /\ c = 1 ] (?, 1) f5(a, b, c) -> f3(-a + 1, b, 0) [ 0 >= a + 1 /\ c = 1 ] (?, 1) f5(a, b, c) -> f3(-a + 1, b, 0) [ a >= 1 /\ c = 1 ] start location: f6 leaf cost: 0 Repeatedly removing leaves of the complexity graph in problem 1 produces the following problem: 2: T: (?, 1) f0(a, b, c) -> f3(-a, b, c) (1, 1) f6(a, b, c) -> f4(a, b, 1) [ a >= 1 ] (?, 1) f3(a, b, c) -> f4(-a - 1, b, 1) [ 0 >= a + 1 /\ 0 >= c ] (?, 1) f3(a, b, c) -> f4(-a - 1, b, 1) [ 0 >= a + 1 /\ c >= 2 ] (?, 1) f3(a, b, c) -> f4(-a - 1, b, 1) [ a >= 1 /\ 0 >= c ] (?, 1) f3(a, b, c) -> f4(-a - 1, b, 1) [ a >= 1 /\ c >= 2 ] (?, 1) f4(a, b, c) -> f3(-a + 1, b, 0) [ 0 >= a + 1 /\ c = 1 ] (?, 1) f4(a, b, c) -> f3(-a + 1, b, 0) [ a >= 1 /\ c = 1 ] (?, 1) f5(a, b, c) -> f3(-a + 1, b, 0) [ 0 >= a + 1 /\ c = 1 ] (?, 1) f5(a, b, c) -> f3(-a + 1, b, 0) [ a >= 1 /\ c = 1 ] start location: f6 leaf cost: 2 Testing for reachability in the complexity graph removes the following transitions from problem 2: f0(a, b, c) -> f3(-a, b, c) f3(a, b, c) -> f4(-a - 1, b, 1) [ 0 >= a + 1 /\ c >= 2 ] f3(a, b, c) -> f4(-a - 1, b, 1) [ a >= 1 /\ 0 >= c ] f3(a, b, c) -> f4(-a - 1, b, 1) [ a >= 1 /\ c >= 2 ] f4(a, b, c) -> f3(-a + 1, b, 0) [ 0 >= a + 1 /\ c = 1 ] f5(a, b, c) -> f3(-a + 1, b, 0) [ 0 >= a + 1 /\ c = 1 ] f5(a, b, c) -> f3(-a + 1, b, 0) [ a >= 1 /\ c = 1 ] We thus obtain the following problem: 3: T: (?, 1) f3(a, b, c) -> f4(-a - 1, b, 1) [ 0 >= a + 1 /\ 0 >= c ] (?, 1) f4(a, b, c) -> f3(-a + 1, b, 0) [ a >= 1 /\ c = 1 ] (1, 1) f6(a, b, c) -> f4(a, b, 1) [ a >= 1 ] start location: f6 leaf cost: 2 A polynomial rank function with Pol(f3) = -V_1 Pol(f4) = V_1 Pol(f6) = V_1 orients all transitions weakly and the transitions f4(a, b, c) -> f3(-a + 1, b, 0) [ a >= 1 /\ c = 1 ] f3(a, b, c) -> f4(-a - 1, b, 1) [ 0 >= a + 1 /\ 0 >= c ] strictly and produces the following problem: 4: T: (a, 1) f3(a, b, c) -> f4(-a - 1, b, 1) [ 0 >= a + 1 /\ 0 >= c ] (a, 1) f4(a, b, c) -> f3(-a + 1, b, 0) [ a >= 1 /\ c = 1 ] (1, 1) f6(a, b, c) -> f4(a, b, 1) [ a >= 1 ] start location: f6 leaf cost: 2 Complexity upper bound 2*a + 3 Time: 0.267 sec (SMT: 0.255 sec)