WORST_CASE(?, O(n^1)) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_0)) [ 0 >= Ar_0 /\ Ar_1 = 1 ] (Comp: ?, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_0)) [ Ar_1 >= 1 /\ Ar_1 + 1 >= 0 /\ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, 0)) [ Ar_0 >= 1 /\ Ar_1 = 1 ] (Comp: ?, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_1 - 1)) [ Ar_1 >= 1 /\ Ar_1 + 1 >= 0 /\ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) start(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(start(Ar_0, Ar_1)) [ 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) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_0)) [ 0 >= Ar_0 /\ Ar_1 = 1 ] (Comp: 1, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_0)) [ Ar_1 >= 1 /\ Ar_1 + 1 >= 0 /\ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, 0)) [ Ar_0 >= 1 /\ Ar_1 = 1 ] (Comp: ?, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_1 - 1)) [ Ar_1 >= 1 /\ Ar_1 + 1 >= 0 /\ Ar_0 >= Ar_1 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(start(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(eval) = V_2 + 2 Pol(start) = V_2 + 2 Pol(koat_start) = V_2 + 2 orients all transitions weakly and the transition eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_1 - 1)) [ Ar_1 >= 1 /\ Ar_1 + 1 >= 0 /\ Ar_0 >= Ar_1 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_0)) [ 0 >= Ar_0 /\ Ar_1 = 1 ] (Comp: 1, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_0)) [ Ar_1 >= 1 /\ Ar_1 + 1 >= 0 /\ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, 0)) [ Ar_0 >= 1 /\ Ar_1 = 1 ] (Comp: Ar_1 + 2, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_1 - 1)) [ Ar_1 >= 1 /\ Ar_1 + 1 >= 0 /\ Ar_0 >= Ar_1 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(start(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 3 produces the following problem: 4: T: (Comp: 1, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_0)) [ 0 >= Ar_0 /\ Ar_1 = 1 ] (Comp: 1, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_0)) [ Ar_1 >= 1 /\ Ar_1 + 1 >= 0 /\ Ar_1 >= Ar_0 + 1 ] (Comp: Ar_1 + 4, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, 0)) [ Ar_0 >= 1 /\ Ar_1 = 1 ] (Comp: Ar_1 + 2, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_1 - 1)) [ Ar_1 >= 1 /\ Ar_1 + 1 >= 0 /\ Ar_0 >= Ar_1 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(start(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 2*Ar_1 + 9 Time: 0.553 sec (SMT: 0.538 sec)