WORST_CASE(?, O(n^2)) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) eval1(Ar_0, Ar_1) -> Com_1(eval2(Ar_0 + 1, 1)) [ Ar_0 >= 0 ] (Comp: ?, Cost: 1) eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 + 1)) [ Ar_0 >= 0 /\ Ar_1 >= 1 /\ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) eval2(Ar_0, Ar_1) -> Com_1(eval1(Ar_0 - 2, Ar_1)) [ Ar_0 >= 0 /\ Ar_1 >= 1 /\ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) start(Ar_0, Ar_1) -> Com_1(eval1(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: ?, Cost: 1) eval1(Ar_0, Ar_1) -> Com_1(eval2(Ar_0 + 1, 1)) [ Ar_0 >= 0 ] (Comp: ?, Cost: 1) eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 + 1)) [ Ar_0 >= 0 /\ Ar_1 >= 1 /\ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) eval2(Ar_0, Ar_1) -> Com_1(eval1(Ar_0 - 2, Ar_1)) [ Ar_0 >= 0 /\ Ar_1 >= 1 /\ Ar_1 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1) -> Com_1(eval1(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(eval1) = 2*V_1 + 2 Pol(eval2) = 2*V_1 - 1 Pol(start) = 2*V_1 + 2 Pol(koat_start) = 2*V_1 + 2 orients all transitions weakly and the transition eval1(Ar_0, Ar_1) -> Com_1(eval2(Ar_0 + 1, 1)) [ Ar_0 >= 0 ] strictly and produces the following problem: 3: T: (Comp: 2*Ar_0 + 2, Cost: 1) eval1(Ar_0, Ar_1) -> Com_1(eval2(Ar_0 + 1, 1)) [ Ar_0 >= 0 ] (Comp: ?, Cost: 1) eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 + 1)) [ Ar_0 >= 0 /\ Ar_1 >= 1 /\ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) eval2(Ar_0, Ar_1) -> Com_1(eval1(Ar_0 - 2, Ar_1)) [ Ar_0 >= 0 /\ Ar_1 >= 1 /\ Ar_1 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1) -> Com_1(eval1(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(eval2) = 1 Pol(eval1) = 0 and size complexities S("koat_start(Ar_0, Ar_1) -> Com_1(start(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1) -> Com_1(start(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-1) = Ar_1 S("start(Ar_0, Ar_1) -> Com_1(eval1(Ar_0, Ar_1))", 0-0) = Ar_0 S("start(Ar_0, Ar_1) -> Com_1(eval1(Ar_0, Ar_1))", 0-1) = Ar_1 S("eval2(Ar_0, Ar_1) -> Com_1(eval1(Ar_0 - 2, Ar_1)) [ Ar_0 >= 0 /\\ Ar_1 >= 1 /\\ Ar_1 >= Ar_0 + 1 ]", 0-0) = 3*Ar_0 + 36 S("eval2(Ar_0, Ar_1) -> Com_1(eval1(Ar_0 - 2, Ar_1)) [ Ar_0 >= 0 /\\ Ar_1 >= 1 /\\ Ar_1 >= Ar_0 + 1 ]", 0-1) = ? S("eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 + 1)) [ Ar_0 >= 0 /\\ Ar_1 >= 1 /\\ Ar_0 >= Ar_1 ]", 0-0) = 3*Ar_0 + 36 S("eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 + 1)) [ Ar_0 >= 0 /\\ Ar_1 >= 1 /\\ Ar_0 >= Ar_1 ]", 0-1) = ? S("eval1(Ar_0, Ar_1) -> Com_1(eval2(Ar_0 + 1, 1)) [ Ar_0 >= 0 ]", 0-0) = 3*Ar_0 + 36 S("eval1(Ar_0, Ar_1) -> Com_1(eval2(Ar_0 + 1, 1)) [ Ar_0 >= 0 ]", 0-1) = 1 orients the transitions eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 + 1)) [ Ar_0 >= 0 /\ Ar_1 >= 1 /\ Ar_0 >= Ar_1 ] eval2(Ar_0, Ar_1) -> Com_1(eval1(Ar_0 - 2, Ar_1)) [ Ar_0 >= 0 /\ Ar_1 >= 1 /\ Ar_1 >= Ar_0 + 1 ] weakly and the transition eval2(Ar_0, Ar_1) -> Com_1(eval1(Ar_0 - 2, Ar_1)) [ Ar_0 >= 0 /\ Ar_1 >= 1 /\ Ar_1 >= Ar_0 + 1 ] strictly and produces the following problem: 4: T: (Comp: 2*Ar_0 + 2, Cost: 1) eval1(Ar_0, Ar_1) -> Com_1(eval2(Ar_0 + 1, 1)) [ Ar_0 >= 0 ] (Comp: ?, Cost: 1) eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 + 1)) [ Ar_0 >= 0 /\ Ar_1 >= 1 /\ Ar_0 >= Ar_1 ] (Comp: 2*Ar_0 + 2, Cost: 1) eval2(Ar_0, Ar_1) -> Com_1(eval1(Ar_0 - 2, Ar_1)) [ Ar_0 >= 0 /\ Ar_1 >= 1 /\ Ar_1 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1) -> Com_1(eval1(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(eval2) = V_1 - V_2 + 1 and size complexities S("koat_start(Ar_0, Ar_1) -> Com_1(start(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1) -> Com_1(start(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-1) = Ar_1 S("start(Ar_0, Ar_1) -> Com_1(eval1(Ar_0, Ar_1))", 0-0) = Ar_0 S("start(Ar_0, Ar_1) -> Com_1(eval1(Ar_0, Ar_1))", 0-1) = Ar_1 S("eval2(Ar_0, Ar_1) -> Com_1(eval1(Ar_0 - 2, Ar_1)) [ Ar_0 >= 0 /\\ Ar_1 >= 1 /\\ Ar_1 >= Ar_0 + 1 ]", 0-0) = 3*Ar_0 + 36 S("eval2(Ar_0, Ar_1) -> Com_1(eval1(Ar_0 - 2, Ar_1)) [ Ar_0 >= 0 /\\ Ar_1 >= 1 /\\ Ar_1 >= Ar_0 + 1 ]", 0-1) = ? S("eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 + 1)) [ Ar_0 >= 0 /\\ Ar_1 >= 1 /\\ Ar_0 >= Ar_1 ]", 0-0) = 3*Ar_0 + 36 S("eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 + 1)) [ Ar_0 >= 0 /\\ Ar_1 >= 1 /\\ Ar_0 >= Ar_1 ]", 0-1) = ? S("eval1(Ar_0, Ar_1) -> Com_1(eval2(Ar_0 + 1, 1)) [ Ar_0 >= 0 ]", 0-0) = 3*Ar_0 + 36 S("eval1(Ar_0, Ar_1) -> Com_1(eval2(Ar_0 + 1, 1)) [ Ar_0 >= 0 ]", 0-1) = 1 orients the transitions eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 + 1)) [ Ar_0 >= 0 /\ Ar_1 >= 1 /\ Ar_0 >= Ar_1 ] weakly and the transition eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 + 1)) [ Ar_0 >= 0 /\ Ar_1 >= 1 /\ Ar_0 >= Ar_1 ] strictly and produces the following problem: 5: T: (Comp: 2*Ar_0 + 2, Cost: 1) eval1(Ar_0, Ar_1) -> Com_1(eval2(Ar_0 + 1, 1)) [ Ar_0 >= 0 ] (Comp: 6*Ar_0^2 + 82*Ar_0 + 76, Cost: 1) eval2(Ar_0, Ar_1) -> Com_1(eval2(Ar_0, Ar_1 + 1)) [ Ar_0 >= 0 /\ Ar_1 >= 1 /\ Ar_0 >= Ar_1 ] (Comp: 2*Ar_0 + 2, Cost: 1) eval2(Ar_0, Ar_1) -> Com_1(eval1(Ar_0 - 2, Ar_1)) [ Ar_0 >= 0 /\ Ar_1 >= 1 /\ Ar_1 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) start(Ar_0, Ar_1) -> Com_1(eval1(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 86*Ar_0 + 6*Ar_0^2 + 81 Time: 0.561 sec (SMT: 0.538 sec)