YES(?, 86) Initial complexity problem: 1: T: (1, 1) f0(a) -> f3(0) (?, 1) f3(a) -> f3(a + 1) [ 41 >= a ] (?, 1) f3(a) -> f3(a + 1) [ 41 >= a /\ 0 >= b + 1 ] (?, 1) f3(a) -> f13(a) [ a >= 42 ] 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) -> f3(0) (?, 1) f3(a) -> f3(a + 1) [ 41 >= a ] (?, 1) f3(a) -> f3(a + 1) [ 41 >= a /\ 0 >= b + 1 ] start location: f0 leaf cost: 1 A polynomial rank function with Pol(f0) = 42 Pol(f3) = -V_1 + 42 orients all transitions weakly and the transitions f3(a) -> f3(a + 1) [ 41 >= a /\ 0 >= b + 1 ] f3(a) -> f3(a + 1) [ 41 >= a ] strictly and produces the following problem: 3: T: (1, 1) f0(a) -> f3(0) (42, 1) f3(a) -> f3(a + 1) [ 41 >= a ] (42, 1) f3(a) -> f3(a + 1) [ 41 >= a /\ 0 >= b + 1 ] start location: f0 leaf cost: 1 Complexity upper bound 86 Time: 0.124 sec (SMT: 0.119 sec)