WORST_CASE(?, O(n^1)) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) eval0(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= 1 ] (Comp: ?, Cost: 1) eval1(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1 + Ar_0, Ar_2)) [ Ar_0 >= Ar_1 + 1 /\ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 1 ] (Comp: ?, Cost: 1) eval1(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1, Ar_1 - Ar_0)) [ Ar_0 >= Ar_1 + 1 /\ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(eval0(Ar_0, Ar_1, Ar_2)) [ 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) eval0(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= 1 ] (Comp: ?, Cost: 1) eval1(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1 + Ar_0, Ar_2)) [ Ar_0 >= Ar_1 + 1 /\ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 1 ] (Comp: ?, Cost: 1) eval1(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1, Ar_1 - Ar_0)) [ Ar_0 >= Ar_1 + 1 /\ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(eval0(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(eval0) = -V_2 + V_3 Pol(eval1) = -V_2 + V_3 - 1 Pol(koat_start) = -V_2 + V_3 orients all transitions weakly and the transition eval1(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1, Ar_1 - Ar_0)) [ Ar_0 >= Ar_1 + 1 /\ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 1 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) eval0(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= 1 ] (Comp: ?, Cost: 1) eval1(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1 + Ar_0, Ar_2)) [ Ar_0 >= Ar_1 + 1 /\ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 1 ] (Comp: Ar_1 + Ar_2, Cost: 1) eval1(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1, Ar_1 - Ar_0)) [ Ar_0 >= Ar_1 + 1 /\ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(eval0(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(eval0) = V_1 - V_2 Pol(eval1) = V_1 - V_2 Pol(koat_start) = V_1 - V_2 orients all transitions weakly and the transition eval1(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1 + Ar_0, Ar_2)) [ Ar_0 >= Ar_1 + 1 /\ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 1 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) eval0(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= 1 ] (Comp: Ar_0 + Ar_1, Cost: 1) eval1(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1 + Ar_0, Ar_2)) [ Ar_0 >= Ar_1 + 1 /\ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 1 ] (Comp: Ar_1 + Ar_2, Cost: 1) eval1(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1, Ar_1 - Ar_0)) [ Ar_0 >= Ar_1 + 1 /\ Ar_2 >= Ar_0 + 1 /\ Ar_0 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(eval0(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound Ar_0 + 2*Ar_1 + Ar_2 + 1 Time: 0.866 sec (SMT: 0.842 sec)