WORST_CASE(?, O(n^1)) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) eval1(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1)) [ Ar_0 >= 1 ] (Comp: ?, Cost: 1) eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 - 1)) [ Ar_0 >= 1 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) eval2(Ar_0, Ar_1) -> Com_1(eval1(Ar_0 - 1, Ar_1)) [ Ar_0 >= 1 /\ 0 >= Ar_1 ] (Comp: ?, Cost: 1) start(Ar_0, Ar_1) -> Com_1(eval1(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(start(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: ?, Cost: 1) eval1(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1)) [ Ar_0 >= 1 ] (Comp: ?, Cost: 1) eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 - 1)) [ Ar_0 >= 1 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) eval2(Ar_0, Ar_1) -> Com_1(eval1(Ar_0 - 1, Ar_1)) [ Ar_0 >= 1 /\ 0 >= Ar_1 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1) -> Com_1(eval1(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(start(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(eval1) = 2*V_1 + 3*V_2 - 3 Pol(eval2) = 2*V_1 + 3*V_2 - 4 Pol(start) = 2*V_1 + 3*V_2 Pol(koat_start) = 2*V_1 + 3*V_2 orients all transitions weakly and the transition eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 - 1)) [ Ar_0 >= 1 /\ Ar_1 >= 1 ] strictly and produces the following problem: 3: T: (Comp: ?, Cost: 1) eval1(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1)) [ Ar_0 >= 1 ] (Comp: 2*Ar_0 + 3*Ar_1, Cost: 1) eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 - 1)) [ Ar_0 >= 1 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) eval2(Ar_0, Ar_1) -> Com_1(eval1(Ar_0 - 1, Ar_1)) [ Ar_0 >= 1 /\ 0 >= Ar_1 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1) -> Com_1(eval1(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(start(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(eval1) = 3*V_1 - 2 Pol(eval2) = 3*V_1 - 4 Pol(start) = 3*V_1 Pol(koat_start) = 3*V_1 orients all transitions weakly and the transition eval1(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1)) [ Ar_0 >= 1 ] strictly and produces the following problem: 4: T: (Comp: 3*Ar_0, Cost: 1) eval1(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1)) [ Ar_0 >= 1 ] (Comp: 2*Ar_0 + 3*Ar_1, Cost: 1) eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 - 1)) [ Ar_0 >= 1 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) eval2(Ar_0, Ar_1) -> Com_1(eval1(Ar_0 - 1, Ar_1)) [ Ar_0 >= 1 /\ 0 >= Ar_1 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1) -> Com_1(eval1(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(start(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 4 produces the following problem: 5: T: (Comp: 3*Ar_0, Cost: 1) eval1(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1)) [ Ar_0 >= 1 ] (Comp: 2*Ar_0 + 3*Ar_1, Cost: 1) eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 - 1)) [ Ar_0 >= 1 /\ Ar_1 >= 1 ] (Comp: 5*Ar_0 + 3*Ar_1, Cost: 1) eval2(Ar_0, Ar_1) -> Com_1(eval1(Ar_0 - 1, Ar_1)) [ Ar_0 >= 1 /\ 0 >= Ar_1 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1) -> Com_1(eval1(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(start(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 10*Ar_0 + 6*Ar_1 + 1 Time: 0.602 sec (SMT: 0.583 sec)