YES(?, a + 3) Initial complexity problem: 1: T: (?, 1) f2(a, b, c) -> f2(-b + a, b + 1, c) [ a >= 1 ] (1, 1) f3(a, b, c) -> f2(a, b, c) [ b >= 1 ] (?, 1) f2(a, b, c) -> f4(a, b, d) [ 0 >= a ] (1, 1) f3(a, b, c) -> f4(a, b, d) [ 0 >= b ] start location: f3 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(-b + a, b + 1, c) [ a >= 1 ] (1, 1) f3(a, b, c) -> f2(a, b, c) [ b >= 1 ] start location: f3 leaf cost: 2 Applied AI with 'oct' on problem 2 to obtain the following invariants: For symbol f2: X_2 - 1 >= 0 This yielded the following problem: 3: T: (1, 1) f3(a, b, c) -> f2(a, b, c) [ b >= 1 ] (?, 1) f2(a, b, c) -> f2(-b + a, b + 1, c) [ b - 1 >= 0 /\ a >= 1 ] start location: f3 leaf cost: 2 A polynomial rank function with Pol(f3) = V_1 Pol(f2) = V_1 orients all transitions weakly and the transition f2(a, b, c) -> f2(-b + a, b + 1, c) [ b - 1 >= 0 /\ a >= 1 ] strictly and produces the following problem: 4: T: (1, 1) f3(a, b, c) -> f2(a, b, c) [ b >= 1 ] (a, 1) f2(a, b, c) -> f2(-b + a, b + 1, c) [ b - 1 >= 0 /\ a >= 1 ] start location: f3 leaf cost: 2 Complexity upper bound a + 3 Time: 0.240 sec (SMT: 0.228 sec)