YES(?, a + 3) Initial complexity problem: 1: T: (?, 1) eval1(a, b) -> eval2(a, b) [ a >= 1 /\ b = a ] (?, 1) eval2(a, b) -> eval2(a - 1, b - 1) [ a >= 1 ] (?, 1) eval2(a, b) -> eval1(a, b) [ 0 >= a ] (1, 1) start(a, b) -> eval1(a, b) start location: start leaf cost: 0 Repeatedly removing leaves of the complexity graph in problem 1 produces the following problem: 2: T: (?, 1) eval1(a, b) -> eval2(a, b) [ a >= 1 /\ b = a ] (?, 1) eval2(a, b) -> eval2(a - 1, b - 1) [ a >= 1 ] (1, 1) start(a, b) -> eval1(a, b) start location: start leaf cost: 1 Repeatedly propagating knowledge in problem 2 produces the following problem: 3: T: (1, 1) eval1(a, b) -> eval2(a, b) [ a >= 1 /\ b = a ] (?, 1) eval2(a, b) -> eval2(a - 1, b - 1) [ a >= 1 ] (1, 1) start(a, b) -> eval1(a, b) start location: start leaf cost: 1 A polynomial rank function with Pol(eval1) = V_1 Pol(eval2) = V_1 Pol(start) = V_1 orients all transitions weakly and the transition eval2(a, b) -> eval2(a - 1, b - 1) [ a >= 1 ] strictly and produces the following problem: 4: T: (1, 1) eval1(a, b) -> eval2(a, b) [ a >= 1 /\ b = a ] (a, 1) eval2(a, b) -> eval2(a - 1, b - 1) [ a >= 1 ] (1, 1) start(a, b) -> eval1(a, b) start location: start leaf cost: 1 Complexity upper bound a + 3 Time: 0.101 sec (SMT: 0.097 sec)