WORST_CASE(?, O(n^1)) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) eval1(Ar_0, Ar_1, Ar_2) -> Com_1(eval2(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval2(Ar_0, Ar_1, Ar_2 - 1)) [ Ar_0 >= Ar_1 + 1 /\ Ar_2 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0 - 1, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 /\ Ar_1 >= Ar_2 ] (Comp: ?, Cost: 1) start(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 1 produces the following problem: 2: T: (Comp: ?, Cost: 1) eval1(Ar_0, Ar_1, Ar_2) -> Com_1(eval2(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval2(Ar_0, Ar_1, Ar_2 - 1)) [ Ar_0 >= Ar_1 + 1 /\ Ar_2 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0 - 1, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 /\ Ar_1 >= Ar_2 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(eval1) = 3*V_1 - 3*V_2 - 2 Pol(eval2) = 3*V_1 - 3*V_2 - 4 Pol(start) = 3*V_1 - 3*V_2 Pol(koat_start) = 3*V_1 - 3*V_2 orients all transitions weakly and the transition eval1(Ar_0, Ar_1, Ar_2) -> Com_1(eval2(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ] strictly and produces the following problem: 3: T: (Comp: 3*Ar_0 + 3*Ar_1, Cost: 1) eval1(Ar_0, Ar_1, Ar_2) -> Com_1(eval2(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval2(Ar_0, Ar_1, Ar_2 - 1)) [ Ar_0 >= Ar_1 + 1 /\ Ar_2 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0 - 1, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 /\ Ar_1 >= Ar_2 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(eval2) = 1 Pol(eval1) = 0 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-2) = Ar_2 S("start(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_0 S("start(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("start(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1, Ar_2))", 0-2) = Ar_2 S("eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0 - 1, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 /\\ Ar_1 >= Ar_2 ]", 0-0) = ? S("eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0 - 1, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 /\\ Ar_1 >= Ar_2 ]", 0-1) = Ar_1 S("eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0 - 1, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 /\\ Ar_1 >= Ar_2 ]", 0-2) = ? S("eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval2(Ar_0, Ar_1, Ar_2 - 1)) [ Ar_0 >= Ar_1 + 1 /\\ Ar_2 >= Ar_1 + 1 ]", 0-0) = ? S("eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval2(Ar_0, Ar_1, Ar_2 - 1)) [ Ar_0 >= Ar_1 + 1 /\\ Ar_2 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval2(Ar_0, Ar_1, Ar_2 - 1)) [ Ar_0 >= Ar_1 + 1 /\\ Ar_2 >= Ar_1 + 1 ]", 0-2) = ? S("eval1(Ar_0, Ar_1, Ar_2) -> Com_1(eval2(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ]", 0-0) = ? S("eval1(Ar_0, Ar_1, Ar_2) -> Com_1(eval2(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("eval1(Ar_0, Ar_1, Ar_2) -> Com_1(eval2(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ]", 0-2) = ? orients the transitions eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval2(Ar_0, Ar_1, Ar_2 - 1)) [ Ar_0 >= Ar_1 + 1 /\ Ar_2 >= Ar_1 + 1 ] eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0 - 1, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 /\ Ar_1 >= Ar_2 ] weakly and the transition eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0 - 1, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 /\ Ar_1 >= Ar_2 ] strictly and produces the following problem: 4: T: (Comp: 3*Ar_0 + 3*Ar_1, Cost: 1) eval1(Ar_0, Ar_1, Ar_2) -> Com_1(eval2(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval2(Ar_0, Ar_1, Ar_2 - 1)) [ Ar_0 >= Ar_1 + 1 /\ Ar_2 >= Ar_1 + 1 ] (Comp: 3*Ar_0 + 3*Ar_1, Cost: 1) eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0 - 1, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 /\ Ar_1 >= Ar_2 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(eval1) = -V_2 + V_3 Pol(eval2) = -V_2 + V_3 Pol(start) = -V_2 + V_3 Pol(koat_start) = -V_2 + V_3 orients all transitions weakly and the transition eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval2(Ar_0, Ar_1, Ar_2 - 1)) [ Ar_0 >= Ar_1 + 1 /\ Ar_2 >= Ar_1 + 1 ] strictly and produces the following problem: 5: T: (Comp: 3*Ar_0 + 3*Ar_1, Cost: 1) eval1(Ar_0, Ar_1, Ar_2) -> Com_1(eval2(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ] (Comp: Ar_1 + Ar_2, Cost: 1) eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval2(Ar_0, Ar_1, Ar_2 - 1)) [ Ar_0 >= Ar_1 + 1 /\ Ar_2 >= Ar_1 + 1 ] (Comp: 3*Ar_0 + 3*Ar_1, Cost: 1) eval2(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0 - 1, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 + 1 /\ Ar_1 >= Ar_2 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1, Ar_2) -> Com_1(eval1(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(start(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 6*Ar_0 + 7*Ar_1 + Ar_2 + 1 Time: 0.524 sec (SMT: 0.502 sec)