WORST_CASE(?, O(1)) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f4(0, Ar_1)) (Comp: ?, Cost: 1) f4(Ar_0, Ar_1) -> Com_1(f4(Ar_0 + 1, Ar_1)) [ 1 >= Ar_0 ] (Comp: ?, Cost: 1) f10(Ar_0, Ar_1) -> Com_1(f10(Ar_0, Ar_1 + 1)) [ 1 >= Ar_1 ] (Comp: ?, Cost: 1) f10(Ar_0, Ar_1) -> Com_1(f18(Ar_0, Ar_1)) [ Ar_1 >= 2 /\ 0 >= C + 1 ] (Comp: ?, Cost: 1) f10(Ar_0, Ar_1) -> Com_1(f18(Ar_0, Ar_1)) [ Ar_1 >= 2 ] (Comp: ?, Cost: 1) f4(Ar_0, Ar_1) -> Com_1(f10(Ar_0, 0)) [ Ar_0 >= 2 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 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) f0(Ar_0, Ar_1) -> Com_1(f4(0, Ar_1)) (Comp: ?, Cost: 1) f4(Ar_0, Ar_1) -> Com_1(f4(Ar_0 + 1, Ar_1)) [ 1 >= Ar_0 ] (Comp: ?, Cost: 1) f10(Ar_0, Ar_1) -> Com_1(f10(Ar_0, Ar_1 + 1)) [ 1 >= Ar_1 ] (Comp: ?, Cost: 1) f10(Ar_0, Ar_1) -> Com_1(f18(Ar_0, Ar_1)) [ Ar_1 >= 2 /\ 0 >= C + 1 ] (Comp: ?, Cost: 1) f10(Ar_0, Ar_1) -> Com_1(f18(Ar_0, Ar_1)) [ Ar_1 >= 2 ] (Comp: ?, Cost: 1) f4(Ar_0, Ar_1) -> Com_1(f10(Ar_0, 0)) [ Ar_0 >= 2 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f0) = 2 Pol(f4) = 2 Pol(f10) = 1 Pol(f18) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions f4(Ar_0, Ar_1) -> Com_1(f10(Ar_0, 0)) [ Ar_0 >= 2 ] f10(Ar_0, Ar_1) -> Com_1(f18(Ar_0, Ar_1)) [ Ar_1 >= 2 /\ 0 >= C + 1 ] f10(Ar_0, Ar_1) -> Com_1(f18(Ar_0, Ar_1)) [ Ar_1 >= 2 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f4(0, Ar_1)) (Comp: ?, Cost: 1) f4(Ar_0, Ar_1) -> Com_1(f4(Ar_0 + 1, Ar_1)) [ 1 >= Ar_0 ] (Comp: ?, Cost: 1) f10(Ar_0, Ar_1) -> Com_1(f10(Ar_0, Ar_1 + 1)) [ 1 >= Ar_1 ] (Comp: 2, Cost: 1) f10(Ar_0, Ar_1) -> Com_1(f18(Ar_0, Ar_1)) [ Ar_1 >= 2 /\ 0 >= C + 1 ] (Comp: 2, Cost: 1) f10(Ar_0, Ar_1) -> Com_1(f18(Ar_0, Ar_1)) [ Ar_1 >= 2 ] (Comp: 2, Cost: 1) f4(Ar_0, Ar_1) -> Com_1(f10(Ar_0, 0)) [ Ar_0 >= 2 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f0) = 2 Pol(f4) = -V_1 + 2 Pol(f10) = -V_1 - V_2 - 1 Pol(f18) = -V_1 - V_2 - 1 Pol(koat_start) = 2 orients all transitions weakly and the transition f4(Ar_0, Ar_1) -> Com_1(f4(Ar_0 + 1, Ar_1)) [ 1 >= Ar_0 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f4(0, Ar_1)) (Comp: 2, Cost: 1) f4(Ar_0, Ar_1) -> Com_1(f4(Ar_0 + 1, Ar_1)) [ 1 >= Ar_0 ] (Comp: ?, Cost: 1) f10(Ar_0, Ar_1) -> Com_1(f10(Ar_0, Ar_1 + 1)) [ 1 >= Ar_1 ] (Comp: 2, Cost: 1) f10(Ar_0, Ar_1) -> Com_1(f18(Ar_0, Ar_1)) [ Ar_1 >= 2 /\ 0 >= C + 1 ] (Comp: 2, Cost: 1) f10(Ar_0, Ar_1) -> Com_1(f18(Ar_0, Ar_1)) [ Ar_1 >= 2 ] (Comp: 2, Cost: 1) f4(Ar_0, Ar_1) -> Com_1(f10(Ar_0, 0)) [ Ar_0 >= 2 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f0) = 2 Pol(f4) = 2 Pol(f10) = -V_2 + 2 Pol(f18) = -V_2 + 2 Pol(koat_start) = 2 orients all transitions weakly and the transition f10(Ar_0, Ar_1) -> Com_1(f10(Ar_0, Ar_1 + 1)) [ 1 >= Ar_1 ] strictly and produces the following problem: 5: T: (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f4(0, Ar_1)) (Comp: 2, Cost: 1) f4(Ar_0, Ar_1) -> Com_1(f4(Ar_0 + 1, Ar_1)) [ 1 >= Ar_0 ] (Comp: 2, Cost: 1) f10(Ar_0, Ar_1) -> Com_1(f10(Ar_0, Ar_1 + 1)) [ 1 >= Ar_1 ] (Comp: 2, Cost: 1) f10(Ar_0, Ar_1) -> Com_1(f18(Ar_0, Ar_1)) [ Ar_1 >= 2 /\ 0 >= C + 1 ] (Comp: 2, Cost: 1) f10(Ar_0, Ar_1) -> Com_1(f18(Ar_0, Ar_1)) [ Ar_1 >= 2 ] (Comp: 2, Cost: 1) f4(Ar_0, Ar_1) -> Com_1(f10(Ar_0, 0)) [ Ar_0 >= 2 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 11 Time: 0.750 sec (SMT: 0.725 sec)