WORST_CASE(?, O(1)) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f0(Ar_0) -> Com_1(f1(300)) (Comp: ?, Cost: 1) f1(Ar_0) -> Com_1(f1(Ar_0 - 1)) [ Ar_0 >= 102 ] (Comp: ?, Cost: 1) f1(Ar_0) -> Com_1(f1(Ar_0 - 1)) [ 100 >= Ar_0 ] (Comp: 1, Cost: 0) koat_start(Ar_0) -> Com_1(f0(Ar_0)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Testing for reachability in the complexity graph removes the following transition from problem 1: f1(Ar_0) -> Com_1(f1(Ar_0 - 1)) [ 100 >= Ar_0 ] We thus obtain the following problem: 2: T: (Comp: ?, Cost: 1) f1(Ar_0) -> Com_1(f1(Ar_0 - 1)) [ Ar_0 >= 102 ] (Comp: ?, Cost: 1) f0(Ar_0) -> Com_1(f1(300)) (Comp: 1, Cost: 0) koat_start(Ar_0) -> Com_1(f0(Ar_0)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 2 produces the following problem: 3: T: (Comp: ?, Cost: 1) f1(Ar_0) -> Com_1(f1(Ar_0 - 1)) [ Ar_0 >= 102 ] (Comp: 1, Cost: 1) f0(Ar_0) -> Com_1(f1(300)) (Comp: 1, Cost: 0) koat_start(Ar_0) -> Com_1(f0(Ar_0)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f1) = V_1 - 101 Pol(f0) = 199 Pol(koat_start) = 199 orients all transitions weakly and the transition f1(Ar_0) -> Com_1(f1(Ar_0 - 1)) [ Ar_0 >= 102 ] strictly and produces the following problem: 4: T: (Comp: 199, Cost: 1) f1(Ar_0) -> Com_1(f1(Ar_0 - 1)) [ Ar_0 >= 102 ] (Comp: 1, Cost: 1) f0(Ar_0) -> Com_1(f1(300)) (Comp: 1, Cost: 0) koat_start(Ar_0) -> Com_1(f0(Ar_0)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 200 Time: 0.331 sec (SMT: 0.323 sec)