WORST_CASE(?, O(n^1)) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f300(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f300(Ar_0 + 1, Ar_1, Ar_2, Ar_3, Ar_4)) [ 100 >= Ar_0 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f300(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, 0, 0, 0)) [ Ar_0 >= 101 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f300(1, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(0, Ar_1, 0, 0, 0)) [ 0 >= Ar_1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 1 produces the following problem: 2: T: (Comp: ?, Cost: 1) f300(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f300(Ar_0 + 1, Ar_1, Ar_2, Ar_3, Ar_4)) [ 100 >= Ar_0 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f300(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, 0, 0, 0)) [ Ar_0 >= 101 ] (Comp: 1, Cost: 1) f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f300(1, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_1 >= 1 ] (Comp: 1, Cost: 1) f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(0, Ar_1, 0, 0, 0)) [ 0 >= Ar_1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f300) = 1 Pol(f3) = 0 Pol(f2) = 1 Pol(koat_start) = 1 orients all transitions weakly and the transition f300(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, 0, 0, 0)) [ Ar_0 >= 101 ] strictly and produces the following problem: 3: T: (Comp: ?, Cost: 1) f300(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f300(Ar_0 + 1, Ar_1, Ar_2, Ar_3, Ar_4)) [ 100 >= Ar_0 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 1) f300(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, 0, 0, 0)) [ Ar_0 >= 101 ] (Comp: 1, Cost: 1) f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f300(1, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_1 >= 1 ] (Comp: 1, Cost: 1) f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(0, Ar_1, 0, 0, 0)) [ 0 >= Ar_1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f300) = -V_1 + 101*V_2 Pol(f3) = -V_1 + 101*V_2 Pol(f2) = 101*V_2 Pol(koat_start) = 101*V_2 orients all transitions weakly and the transition f300(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f300(Ar_0 + 1, Ar_1, Ar_2, Ar_3, Ar_4)) [ 100 >= Ar_0 /\ Ar_1 >= 1 ] strictly and produces the following problem: 4: T: (Comp: 101*Ar_1, Cost: 1) f300(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f300(Ar_0 + 1, Ar_1, Ar_2, Ar_3, Ar_4)) [ 100 >= Ar_0 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 1) f300(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(Ar_0, Ar_1, 0, 0, 0)) [ Ar_0 >= 101 ] (Comp: 1, Cost: 1) f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f300(1, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_1 >= 1 ] (Comp: 1, Cost: 1) f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f3(0, Ar_1, 0, 0, 0)) [ 0 >= Ar_1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 101*Ar_1 + 3 Time: 0.580 sec (SMT: 0.564 sec)