WORST_CASE(?, O(n^1)) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) eval1(Ar_0, Ar_1, Ar_2) -> Com_1(eval2(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1 + 1, Ar_2)) [ Ar_0 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0 - 1, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 ] (Comp: ?, Cost: 1) start(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(start(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: ?, Cost: 1) eval1(Ar_0, Ar_1, Ar_2) -> Com_1(eval2(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1 + 1, Ar_2)) [ Ar_0 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0 - 1, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(eval1) = V_1 - V_3 + 1 Pol(eval2) = V_1 - V_3 + 1 Pol(start) = V_1 - V_3 + 1 Pol(koat_start) = V_1 - V_3 + 1 orients all transitions weakly and the transition eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 >= Ar_2 + 1 ] strictly and produces the following problem: 3: T: (Comp: ?, Cost: 1) eval1(Ar_0, Ar_1, Ar_2) -> Com_1(eval2(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1 + 1, Ar_2)) [ Ar_0 >= Ar_2 + 1 ] (Comp: Ar_0 + Ar_2 + 1, Cost: 1) eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0 - 1, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Applied AI with 'oct' on problem 3 to obtain the following invariants: For symbol eval2: X_1 - X_2 - 1 >= 0 This yielded the following problem: 4: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0 - 1, Ar_1, Ar_2)) [ Ar_0 - Ar_1 - 1 >= 0 /\ Ar_2 >= Ar_0 ] (Comp: Ar_0 + Ar_2 + 1, Cost: 1) eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 - Ar_1 - 1 >= 0 /\ Ar_0 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1 + 1, Ar_2)) [ Ar_0 - Ar_1 - 1 >= 0 /\ Ar_0 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) eval1(Ar_0, Ar_1, Ar_2) -> Com_1(eval2(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(koat_start) = 5*V_1 - 2*V_2 - 3*V_3 Pol(start) = 5*V_1 - 2*V_2 - 3*V_3 Pol(eval1) = 5*V_1 - 2*V_2 - 3*V_3 - 3 Pol(eval2) = 5*V_1 - 2*V_2 - 3*V_3 - 4 orients all transitions weakly and the transition eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1 + 1, Ar_2)) [ Ar_0 - Ar_1 - 1 >= 0 /\ Ar_0 >= Ar_2 + 1 ] strictly and produces the following problem: 5: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0 - 1, Ar_1, Ar_2)) [ Ar_0 - Ar_1 - 1 >= 0 /\ Ar_2 >= Ar_0 ] (Comp: Ar_0 + Ar_2 + 1, Cost: 1) eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 - Ar_1 - 1 >= 0 /\ Ar_0 >= Ar_2 + 1 ] (Comp: 5*Ar_0 + 2*Ar_1 + 3*Ar_2, Cost: 1) eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1 + 1, Ar_2)) [ Ar_0 - Ar_1 - 1 >= 0 /\ Ar_0 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) eval1(Ar_0, Ar_1, Ar_2) -> Com_1(eval2(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(koat_start) = V_1 - V_2 Pol(start) = V_1 - V_2 Pol(eval1) = V_1 - V_2 Pol(eval2) = V_1 - V_2 orients all transitions weakly and the transition eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0 - 1, Ar_1, Ar_2)) [ Ar_0 - Ar_1 - 1 >= 0 /\ Ar_2 >= Ar_0 ] strictly and produces the following problem: 6: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1, Ar_2)) (Comp: Ar_0 + Ar_1, Cost: 1) eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0 - 1, Ar_1, Ar_2)) [ Ar_0 - Ar_1 - 1 >= 0 /\ Ar_2 >= Ar_0 ] (Comp: Ar_0 + Ar_2 + 1, Cost: 1) eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 - Ar_1 - 1 >= 0 /\ Ar_0 >= Ar_2 + 1 ] (Comp: 5*Ar_0 + 2*Ar_1 + 3*Ar_2, Cost: 1) eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1 + 1, Ar_2)) [ Ar_0 - Ar_1 - 1 >= 0 /\ Ar_0 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) eval1(Ar_0, Ar_1, Ar_2) -> Com_1(eval2(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 6 produces the following problem: 7: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1, Ar_2)) (Comp: Ar_0 + Ar_1, Cost: 1) eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0 - 1, Ar_1, Ar_2)) [ Ar_0 - Ar_1 - 1 >= 0 /\ Ar_2 >= Ar_0 ] (Comp: Ar_0 + Ar_2 + 1, Cost: 1) eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 - Ar_1 - 1 >= 0 /\ Ar_0 >= Ar_2 + 1 ] (Comp: 5*Ar_0 + 2*Ar_1 + 3*Ar_2, Cost: 1) eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1 + 1, Ar_2)) [ Ar_0 - Ar_1 - 1 >= 0 /\ Ar_0 >= Ar_2 + 1 ] (Comp: 7*Ar_0 + 3*Ar_1 + 4*Ar_2 + 2, Cost: 1) eval1(Ar_0, Ar_1, Ar_2) -> Com_1(eval2(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ] start location: koat_start leaf cost: 0 Complexity upper bound 14*Ar_0 + 6*Ar_1 + 8*Ar_2 + 4 Time: 1.213 sec (SMT: 1.166 sec)