WORST_CASE(?, O(n^1)) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) sqrt(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f(0, 1, 1, Ar_3)) (Comp: ?, Cost: 1) f(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f(Ar_0 + 1, Ar_1 + 2, Ar_2 + Ar_1 + 2, Ar_3)) [ Ar_3 >= Ar_2 /\ Ar_1 >= 0 ] (Comp: ?, Cost: 1) f(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(end(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(sqrt(Ar_0, Ar_1, Ar_2, Ar_3)) [ 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) sqrt(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f(0, 1, 1, Ar_3)) (Comp: ?, Cost: 1) f(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f(Ar_0 + 1, Ar_1 + 2, Ar_2 + Ar_1 + 2, Ar_3)) [ Ar_3 >= Ar_2 /\ Ar_1 >= 0 ] (Comp: ?, Cost: 1) f(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(end(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(sqrt(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(sqrt) = 1 Pol(f) = 1 Pol(end) = 0 Pol(koat_start) = 1 orients all transitions weakly and the transition f(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(end(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) sqrt(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f(0, 1, 1, Ar_3)) (Comp: ?, Cost: 1) f(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f(Ar_0 + 1, Ar_1 + 2, Ar_2 + Ar_1 + 2, Ar_3)) [ Ar_3 >= Ar_2 /\ Ar_1 >= 0 ] (Comp: 1, Cost: 1) f(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(end(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(sqrt(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(sqrt) = V_4 Pol(f) = -V_3 + V_4 + 1 Pol(end) = -V_3 + V_4 + 1 Pol(koat_start) = V_4 orients all transitions weakly and the transition f(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f(Ar_0 + 1, Ar_1 + 2, Ar_2 + Ar_1 + 2, Ar_3)) [ Ar_3 >= Ar_2 /\ Ar_1 >= 0 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) sqrt(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f(0, 1, 1, Ar_3)) (Comp: Ar_3, Cost: 1) f(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f(Ar_0 + 1, Ar_1 + 2, Ar_2 + Ar_1 + 2, Ar_3)) [ Ar_3 >= Ar_2 /\ Ar_1 >= 0 ] (Comp: 1, Cost: 1) f(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(end(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(sqrt(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound Ar_3 + 2 Time: 0.657 sec (SMT: 0.639 sec)