WORST_CASE(?, O(n^1)) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f1(Ar_0, Ar_1)) [ Ar_0 >= 1 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1) -> Com_1(f1(Ar_0, Ar_1 - Ar_0)) [ Ar_0 >= 1 /\ Ar_1 >= 0 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(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: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f1(Ar_0, Ar_1)) [ Ar_0 >= 1 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1) -> Com_1(f1(Ar_0, Ar_1 - Ar_0)) [ Ar_0 >= 1 /\ Ar_1 >= 0 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f0) = V_1 + V_2 Pol(f1) = V_2 + 1 Pol(koat_start) = V_1 + V_2 orients all transitions weakly and the transition f1(Ar_0, Ar_1) -> Com_1(f1(Ar_0, Ar_1 - Ar_0)) [ Ar_0 >= 1 /\ Ar_1 >= 0 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f1(Ar_0, Ar_1)) [ Ar_0 >= 1 ] (Comp: Ar_0 + Ar_1, Cost: 1) f1(Ar_0, Ar_1) -> Com_1(f1(Ar_0, Ar_1 - Ar_0)) [ Ar_0 >= 1 /\ Ar_1 >= 0 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound Ar_0 + Ar_1 + 1 Time: 0.361 sec (SMT: 0.351 sec)