WORST_CASE(?, O(n^1)) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f1(Ar_0 - 1, Fresh_0)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ Ar_0 >= 1 ] (Comp: ?, Cost: 1) f300(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f300(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) f2(Ar_0, Ar_1) -> Com_1(f1(Ar_0 - 1, Fresh_0)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ Ar_0 >= 1 ] (Comp: 1, Cost: 1) f300(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f300(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f2) = 1 Pol(f1) = 0 Pol(f300) = 1 Pol(koat_start) = 1 orients all transitions weakly and the transition f2(Ar_0, Ar_1) -> Com_1(f1(Ar_0 - 1, Fresh_0)) [ 0 >= Ar_0 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f1(Ar_0 - 1, Fresh_0)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ Ar_0 >= 1 ] (Comp: 1, Cost: 1) f300(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f300(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f2) = V_1 Pol(f1) = V_1 + 1 Pol(f300) = V_1 Pol(koat_start) = V_1 orients all transitions weakly and the transition f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ Ar_0 >= 1 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f1(Ar_0 - 1, Fresh_0)) [ 0 >= Ar_0 ] (Comp: Ar_0, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ Ar_0 >= 1 ] (Comp: 1, Cost: 1) f300(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f300(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound Ar_0 + 2 Time: 0.448 sec (SMT: 0.435 sec)