WORST_CASE(?, O(n^1)) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f(Ar_0, Ar_1, Ar_2) -> Com_1(f(Ar_0, Ar_1 + 1, Ar_2)) [ Ar_0 >= Ar_1 + Ar_2 + 1 ] (Comp: ?, Cost: 1) f(Ar_0, Ar_1, Ar_2) -> Com_1(f(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 >= Ar_1 + Ar_2 + 1 ] (Comp: ?, Cost: 1) start(Ar_0, Ar_1, Ar_2) -> Com_1(f(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) f(Ar_0, Ar_1, Ar_2) -> Com_1(f(Ar_0, Ar_1 + 1, Ar_2)) [ Ar_0 >= Ar_1 + Ar_2 + 1 ] (Comp: ?, Cost: 1) f(Ar_0, Ar_1, Ar_2) -> Com_1(f(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 >= Ar_1 + Ar_2 + 1 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2) -> Com_1(f(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(f) = V_1 - V_2 - V_3 + 1 Pol(start) = V_1 - V_2 - V_3 + 1 Pol(koat_start) = V_1 - V_2 - V_3 + 1 orients all transitions weakly and the transition f(Ar_0, Ar_1, Ar_2) -> Com_1(f(Ar_0, Ar_1 + 1, Ar_2)) [ Ar_0 >= Ar_1 + Ar_2 + 1 ] strictly and produces the following problem: 3: T: (Comp: Ar_0 + Ar_1 + Ar_2 + 1, Cost: 1) f(Ar_0, Ar_1, Ar_2) -> Com_1(f(Ar_0, Ar_1 + 1, Ar_2)) [ Ar_0 >= Ar_1 + Ar_2 + 1 ] (Comp: ?, Cost: 1) f(Ar_0, Ar_1, Ar_2) -> Com_1(f(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 >= Ar_1 + Ar_2 + 1 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2) -> Com_1(f(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(f) = V_1 - V_2 - V_3 Pol(start) = V_1 - V_2 - V_3 Pol(koat_start) = V_1 - V_2 - V_3 orients all transitions weakly and the transition f(Ar_0, Ar_1, Ar_2) -> Com_1(f(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 >= Ar_1 + Ar_2 + 1 ] strictly and produces the following problem: 4: T: (Comp: Ar_0 + Ar_1 + Ar_2 + 1, Cost: 1) f(Ar_0, Ar_1, Ar_2) -> Com_1(f(Ar_0, Ar_1 + 1, Ar_2)) [ Ar_0 >= Ar_1 + Ar_2 + 1 ] (Comp: Ar_0 + Ar_1 + Ar_2, Cost: 1) f(Ar_0, Ar_1, Ar_2) -> Com_1(f(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_0 >= Ar_1 + Ar_2 + 1 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2) -> Com_1(f(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 Complexity upper bound 2*Ar_0 + 2*Ar_1 + 2*Ar_2 + 2 Time: 0.433 sec (SMT: 0.417 sec)