WORST_CASE(?, O(1)) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f2(Ar_0 + 1, Ar_1 + 1, Ar_2 - 1, Ar_3)) (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f2(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 /\ Ar_1 >= Ar_0 + 1 /\ Ar_2 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f300(Ar_0, Ar_1, Ar_2, Fresh_2)) [ Ar_2 >= Ar_1 + 1 /\ Ar_1 >= Ar_0 + 1 /\ Ar_2 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f300(Ar_0, Ar_1, Ar_2, Fresh_1)) [ Ar_1 >= Ar_0 + 1 /\ Ar_1 >= Ar_2 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f300(Ar_0, Ar_1, Ar_2, Fresh_0)) [ Ar_0 >= Ar_1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Testing for reachability in the complexity graph removes the following transitions from problem 1: f2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f2(Ar_0 + 1, Ar_1 + 1, Ar_2 - 1, Ar_3)) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f2(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 /\ Ar_1 >= Ar_0 + 1 /\ Ar_2 >= Ar_1 + 1 ] We thus obtain the following problem: 2: T: (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f300(Ar_0, Ar_1, Ar_2, Fresh_0)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f300(Ar_0, Ar_1, Ar_2, Fresh_1)) [ Ar_1 >= Ar_0 + 1 /\ Ar_1 >= Ar_2 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f300(Ar_0, Ar_1, Ar_2, Fresh_2)) [ Ar_2 >= Ar_1 + 1 /\ Ar_1 >= Ar_0 + 1 /\ Ar_2 >= Ar_0 + 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 2 produces the following problem: 3: T: (Comp: 1, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f300(Ar_0, Ar_1, Ar_2, Fresh_0)) [ Ar_0 >= Ar_1 ] (Comp: 1, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f300(Ar_0, Ar_1, Ar_2, Fresh_1)) [ Ar_1 >= Ar_0 + 1 /\ Ar_1 >= Ar_2 ] (Comp: 1, Cost: 1) f1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f300(Ar_0, Ar_1, Ar_2, Fresh_2)) [ Ar_2 >= Ar_1 + 1 /\ Ar_1 >= Ar_0 + 1 /\ Ar_2 >= Ar_0 + 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f1(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 3 Time: 0.300 sec (SMT: 0.295 sec)