WORST_CASE(?, O(n^1)) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f2(Ar_0, Ar_1, Ar_2, Fresh_1)) [ Ar_1 >= Ar_2 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f2(1, Ar_1 + 1, Ar_2, Fresh_0)) [ 0 >= Ar_0 /\ Ar_1 + 1 = Ar_2 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(0, Ar_1 + 1, Ar_2 - 1, Ar_3)) [ Ar_2 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ Ar_2 >= Ar_1 + 2 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_3)) [ 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) f3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f2(Ar_0, Ar_1, Ar_2, Fresh_1)) [ Ar_1 >= Ar_2 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f2(1, Ar_1 + 1, Ar_2, Fresh_0)) [ 0 >= Ar_0 /\ Ar_1 + 1 = Ar_2 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(0, Ar_1 + 1, Ar_2 - 1, Ar_3)) [ Ar_2 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ Ar_2 >= Ar_1 + 2 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f3) = 1 Pol(f1) = 1 Pol(f2) = 0 Pol(koat_start) = 1 orients all transitions weakly and the transitions f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f2(1, Ar_1 + 1, Ar_2, Fresh_0)) [ 0 >= Ar_0 /\ Ar_1 + 1 = Ar_2 ] f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f2(Ar_0, Ar_1, Ar_2, Fresh_1)) [ Ar_1 >= Ar_2 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) f3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f2(Ar_0, Ar_1, Ar_2, Fresh_1)) [ Ar_1 >= Ar_2 ] (Comp: 1, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f2(1, Ar_1 + 1, Ar_2, Fresh_0)) [ 0 >= Ar_0 /\ Ar_1 + 1 = Ar_2 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(0, Ar_1 + 1, Ar_2 - 1, Ar_3)) [ Ar_2 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ Ar_2 >= Ar_1 + 2 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f3) = -V_2 + V_3 Pol(f1) = -V_2 + V_3 Pol(f2) = -V_2 + V_3 Pol(koat_start) = -V_2 + V_3 orients all transitions weakly and the transition f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(0, Ar_1 + 1, Ar_2 - 1, Ar_3)) [ Ar_2 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ Ar_2 >= Ar_1 + 2 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) f3(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f2(Ar_0, Ar_1, Ar_2, Fresh_1)) [ Ar_1 >= Ar_2 ] (Comp: 1, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f2(1, Ar_1 + 1, Ar_2, Fresh_0)) [ 0 >= Ar_0 /\ Ar_1 + 1 = Ar_2 ] (Comp: Ar_1 + Ar_2, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(0, Ar_1 + 1, Ar_2 - 1, Ar_3)) [ Ar_2 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ Ar_2 >= Ar_1 + 2 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f3(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound Ar_1 + Ar_2 + 3 Time: 0.656 sec (SMT: 0.638 sec)