YES(?, a + 4) Initial complexity problem: 1: T: (?, 1) start(a, b, c, d, e, f) -> stop(a, b, c, d, e, f) [ 0 >= a /\ b = a /\ c = d /\ e = f ] (?, 1) start(a, b, c, d, e, f) -> lbl71(a, b - 1, c - 1, d, e + 1, f) [ a >= 1 /\ b = a /\ c = d /\ e = f ] (?, 1) lbl71(a, b, c, d, e, f) -> stop(a, b, c, d, e, f) [ d >= c + 1 /\ b = 0 /\ e + c = f + d /\ a + c = d ] (?, 1) lbl71(a, b, c, d, e, f) -> lbl71(a, b - 1, c - 1, d, e + 1, f) [ a + c >= d + 1 /\ d >= c + 1 /\ a + c >= d /\ e + c = d + f /\ b + d = a + c ] (1, 1) start0(a, b, c, d, e, f) -> start(a, a, d, d, f, f) start location: start0 leaf cost: 0 Repeatedly removing leaves of the complexity graph in problem 1 produces the following problem: 2: T: (?, 1) start(a, b, c, d, e, f) -> lbl71(a, b - 1, c - 1, d, e + 1, f) [ a >= 1 /\ b = a /\ c = d /\ e = f ] (?, 1) lbl71(a, b, c, d, e, f) -> lbl71(a, b - 1, c - 1, d, e + 1, f) [ a + c >= d + 1 /\ d >= c + 1 /\ a + c >= d /\ e + c = d + f /\ b + d = a + c ] (1, 1) start0(a, b, c, d, e, f) -> start(a, a, d, d, f, f) start location: start0 leaf cost: 2 Repeatedly propagating knowledge in problem 2 produces the following problem: 3: T: (1, 1) start(a, b, c, d, e, f) -> lbl71(a, b - 1, c - 1, d, e + 1, f) [ a >= 1 /\ b = a /\ c = d /\ e = f ] (?, 1) lbl71(a, b, c, d, e, f) -> lbl71(a, b - 1, c - 1, d, e + 1, f) [ a + c >= d + 1 /\ d >= c + 1 /\ a + c >= d /\ e + c = d + f /\ b + d = a + c ] (1, 1) start0(a, b, c, d, e, f) -> start(a, a, d, d, f, f) start location: start0 leaf cost: 2 A polynomial rank function with Pol(start) = V_1 + V_3 - V_4 Pol(lbl71) = V_1 + V_3 - V_4 + 1 Pol(start0) = V_1 orients all transitions weakly and the transition lbl71(a, b, c, d, e, f) -> lbl71(a, b - 1, c - 1, d, e + 1, f) [ a + c >= d + 1 /\ d >= c + 1 /\ a + c >= d /\ e + c = d + f /\ b + d = a + c ] strictly and produces the following problem: 4: T: (1, 1) start(a, b, c, d, e, f) -> lbl71(a, b - 1, c - 1, d, e + 1, f) [ a >= 1 /\ b = a /\ c = d /\ e = f ] (a, 1) lbl71(a, b, c, d, e, f) -> lbl71(a, b - 1, c - 1, d, e + 1, f) [ a + c >= d + 1 /\ d >= c + 1 /\ a + c >= d /\ e + c = d + f /\ b + d = a + c ] (1, 1) start0(a, b, c, d, e, f) -> start(a, a, d, d, f, f) start location: start0 leaf cost: 2 Complexity upper bound a + 4 Time: 0.280 sec (SMT: 0.268 sec)