WORST_CASE(?, O(n^1)) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f3(Ar_0) -> Com_1(f1(Ar_0)) [ Ar_0 >= 0 ] (Comp: ?, Cost: 1) f1(Ar_0) -> Com_1(f1(Ar_0 - 1)) [ Ar_0 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0) -> Com_1(f3(Ar_0)) [ 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) f3(Ar_0) -> Com_1(f1(Ar_0)) [ Ar_0 >= 0 ] (Comp: ?, Cost: 1) f1(Ar_0) -> Com_1(f1(Ar_0 - 1)) [ Ar_0 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0) -> Com_1(f3(Ar_0)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f3) = V_1 Pol(f1) = V_1 Pol(koat_start) = V_1 orients all transitions weakly and the transition f1(Ar_0) -> Com_1(f1(Ar_0 - 1)) [ Ar_0 >= 1 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) f3(Ar_0) -> Com_1(f1(Ar_0)) [ Ar_0 >= 0 ] (Comp: Ar_0, Cost: 1) f1(Ar_0) -> Com_1(f1(Ar_0 - 1)) [ Ar_0 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0) -> Com_1(f3(Ar_0)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound Ar_0 + 1 Time: 0.285 sec (SMT: 0.276 sec)