WORST_CASE(?, O(n^1)) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f2(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0 + 1, Ar_0, Ar_2)) [ Ar_1 >= D /\ Ar_0 = Ar_1 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0, Ar_1, Fresh_1)) [ Ar_0 >= Ar_1 /\ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0, Ar_1, Fresh_0)) [ Ar_0 >= Ar_1 /\ Ar_1 >= Ar_0 + 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2)) [ 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, Ar_1, Ar_2) -> Com_1(f300(Ar_0, Ar_1, Fresh_0)) [ Ar_0 >= Ar_1 /\ Ar_1 >= Ar_0 + 1 ] We thus obtain the following problem: 2: T: (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0, Ar_1, Fresh_1)) [ Ar_0 >= Ar_1 /\ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0 + 1, Ar_0, Ar_2)) [ Ar_1 >= D /\ Ar_0 = Ar_1 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2)) [ 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, Ar_1, Ar_2) -> Com_1(f300(Ar_0, Ar_1, Fresh_1)) [ Ar_0 >= Ar_1 /\ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0 + 1, Ar_0, Ar_2)) [ Ar_1 >= D /\ Ar_0 = Ar_1 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) f2(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f1) = 1 Pol(f300) = 0 Pol(f2) = 1 Pol(koat_start) = 1 orients all transitions weakly and the transition f1(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0, Ar_1, Fresh_1)) [ Ar_0 >= Ar_1 /\ Ar_0 >= Ar_1 + 1 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0, Ar_1, Fresh_1)) [ Ar_0 >= Ar_1 /\ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0 + 1, Ar_0, Ar_2)) [ Ar_1 >= D /\ Ar_0 = Ar_1 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) f2(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f1) = -V_1 + V_2 + 1 Pol(f300) = -V_1 + V_2 + 1 Pol(f2) = -V_1 + V_2 + 1 Pol(koat_start) = -V_1 + V_2 + 1 orients all transitions weakly and the transition f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0 + 1, Ar_0, Ar_2)) [ Ar_1 >= D /\ Ar_0 = Ar_1 ] strictly and produces the following problem: 5: T: (Comp: 1, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0, Ar_1, Fresh_1)) [ Ar_0 >= Ar_1 /\ Ar_0 >= Ar_1 + 1 ] (Comp: Ar_0 + Ar_1 + 1, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0 + 1, Ar_0, Ar_2)) [ Ar_1 >= D /\ Ar_0 = Ar_1 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) f2(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f1) = -V_1 + V_2 Pol(f300) = -V_1 + V_2 Pol(f2) = -V_1 + V_2 Pol(koat_start) = -V_1 + V_2 orients all transitions weakly and the transition f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 1 ] strictly and produces the following problem: 6: T: (Comp: 1, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0, Ar_1, Fresh_1)) [ Ar_0 >= Ar_1 /\ Ar_0 >= Ar_1 + 1 ] (Comp: Ar_0 + Ar_1 + 1, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0 + 1, Ar_0, Ar_2)) [ Ar_1 >= D /\ Ar_0 = Ar_1 ] (Comp: Ar_0 + Ar_1, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) f2(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 2*Ar_0 + 2*Ar_1 + 3 Time: 0.577 sec (SMT: 0.556 sec)