YES(?, 4*a + 5*b + c + 1) Initial complexity problem: 1: T: (?, 1) eval1(a, b, c) -> eval2(a, b, c) [ a >= b + 1 ] (?, 1) eval2(a, b, c) -> eval2(a, b, c - 1) [ a >= b + 1 /\ c >= b + 1 ] (?, 1) eval2(a, b, c) -> eval1(a - 1, b, c) [ a >= b + 1 /\ b >= c ] (1, 1) start(a, b, c) -> eval1(a, b, c) start location: start leaf cost: 0 A polynomial rank function with Pol(eval1) = 2*V_1 - 2*V_2 Pol(eval2) = 2*V_1 - 2*V_2 - 1 Pol(start) = 2*V_1 - 2*V_2 orients all transitions weakly and the transitions eval2(a, b, c) -> eval1(a - 1, b, c) [ a >= b + 1 /\ b >= c ] eval1(a, b, c) -> eval2(a, b, c) [ a >= b + 1 ] strictly and produces the following problem: 2: T: (2*a + 2*b, 1) eval1(a, b, c) -> eval2(a, b, c) [ a >= b + 1 ] (?, 1) eval2(a, b, c) -> eval2(a, b, c - 1) [ a >= b + 1 /\ c >= b + 1 ] (2*a + 2*b, 1) eval2(a, b, c) -> eval1(a - 1, b, c) [ a >= b + 1 /\ b >= c ] (1, 1) start(a, b, c) -> eval1(a, b, c) start location: start leaf cost: 0 A polynomial rank function with Pol(eval1) = -V_2 + V_3 Pol(eval2) = -V_2 + V_3 Pol(start) = -V_2 + V_3 orients all transitions weakly and the transition eval2(a, b, c) -> eval2(a, b, c - 1) [ a >= b + 1 /\ c >= b + 1 ] strictly and produces the following problem: 3: T: (2*a + 2*b, 1) eval1(a, b, c) -> eval2(a, b, c) [ a >= b + 1 ] (b + c, 1) eval2(a, b, c) -> eval2(a, b, c - 1) [ a >= b + 1 /\ c >= b + 1 ] (2*a + 2*b, 1) eval2(a, b, c) -> eval1(a - 1, b, c) [ a >= b + 1 /\ b >= c ] (1, 1) start(a, b, c) -> eval1(a, b, c) start location: start leaf cost: 0 Complexity upper bound 4*a + 5*b + c + 1 Time: 0.566 sec (SMT: 0.531 sec)