WORST_CASE(?, O(n^1)) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Fresh_1, Ar_2, Ar_3, Ar_4)) [ 1 >= Ar_0 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f300(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= 2 /\ Ar_2 >= 2 ] (Comp: ?, Cost: 1) f300(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Fresh_0, Ar_2, Ar_3, Ar_4)) [ Ar_3 + 1 >= 0 ] (Comp: ?, Cost: 1) f300(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f300(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4 + 1)) [ Ar_4 >= 0 /\ 0 >= Ar_3 + 2 ] (Comp: ?, Cost: 1) f300(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f300(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4 + 1)) [ 0 >= Ar_4 + 2 /\ 0 >= Ar_3 + 2 ] (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: 1, Cost: 1) f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Fresh_1, Ar_2, Ar_3, Ar_4)) [ 1 >= Ar_0 ] (Comp: 1, Cost: 1) f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f300(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= 2 /\ Ar_2 >= 2 ] (Comp: ?, Cost: 1) f300(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Fresh_0, Ar_2, Ar_3, Ar_4)) [ Ar_3 + 1 >= 0 ] (Comp: ?, Cost: 1) f300(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f300(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4 + 1)) [ Ar_4 >= 0 /\ 0 >= Ar_3 + 2 ] (Comp: ?, Cost: 1) f300(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f300(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4 + 1)) [ 0 >= Ar_4 + 2 /\ 0 >= Ar_3 + 2 ] (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(f2) = 1 Pol(f1) = 0 Pol(f300) = 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(f1(Ar_0, Fresh_0, Ar_2, Ar_3, Ar_4)) [ Ar_3 + 1 >= 0 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Fresh_1, Ar_2, Ar_3, Ar_4)) [ 1 >= Ar_0 ] (Comp: 1, Cost: 1) f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f300(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= 2 /\ Ar_2 >= 2 ] (Comp: 1, Cost: 1) f300(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Fresh_0, Ar_2, Ar_3, Ar_4)) [ Ar_3 + 1 >= 0 ] (Comp: ?, Cost: 1) f300(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f300(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4 + 1)) [ Ar_4 >= 0 /\ 0 >= Ar_3 + 2 ] (Comp: ?, Cost: 1) f300(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f300(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4 + 1)) [ 0 >= Ar_4 + 2 /\ 0 >= Ar_3 + 2 ] (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(f2) = -V_4 Pol(f1) = -V_4 Pol(f300) = -V_4 Pol(koat_start) = -V_4 orients all transitions weakly and the transitions f300(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f300(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4 + 1)) [ Ar_4 >= 0 /\ 0 >= Ar_3 + 2 ] f300(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f300(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4 + 1)) [ 0 >= Ar_4 + 2 /\ 0 >= Ar_3 + 2 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Fresh_1, Ar_2, Ar_3, Ar_4)) [ 1 >= Ar_0 ] (Comp: 1, Cost: 1) f2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f300(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= 2 /\ Ar_2 >= 2 ] (Comp: 1, Cost: 1) f300(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f1(Ar_0, Fresh_0, Ar_2, Ar_3, Ar_4)) [ Ar_3 + 1 >= 0 ] (Comp: Ar_3, Cost: 1) f300(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f300(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4 + 1)) [ Ar_4 >= 0 /\ 0 >= Ar_3 + 2 ] (Comp: Ar_3, Cost: 1) f300(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f300(Ar_0, Ar_1, Ar_2, Ar_3 + 1, Ar_4 + 1)) [ 0 >= Ar_4 + 2 /\ 0 >= Ar_3 + 2 ] (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 2*Ar_3 + 3 Time: 1.049 sec (SMT: 1.022 sec)