WORST_CASE(?, O(n^1)) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0 + 1, Ar_1)) [ Ar_0 >= Ar_1 + 1 /\ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0 + 1, Ar_1)) [ Ar_1 >= Ar_0 + 1 /\ Ar_1 >= Ar_0 ] (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 Testing for reachability in the complexity graph removes the following transitions from problem 1: eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= Ar_1 + 1 ] eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0 + 1, Ar_1)) [ Ar_0 >= Ar_1 + 1 /\ Ar_1 >= Ar_0 ] We thus obtain the following problem: 2: T: (Comp: ?, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0 + 1, Ar_1)) [ Ar_1 >= Ar_0 + 1 /\ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 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 2 produces the following problem: 3: T: (Comp: ?, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0 + 1, Ar_1)) [ Ar_1 >= Ar_0 + 1 /\ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 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_1 - V_2 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(eval(Ar_0, Ar_1))", 0-0) = Ar_0 S("start(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_1))", 0-1) = Ar_1 S("eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ]", 0-1) = ? S("eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0 + 1, Ar_1)) [ Ar_1 >= Ar_0 + 1 /\\ Ar_1 >= Ar_0 ]", 0-0) = ? S("eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0 + 1, Ar_1)) [ Ar_1 >= Ar_0 + 1 /\\ Ar_1 >= Ar_0 ]", 0-1) = Ar_1 orients the transitions eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ] weakly and the transition eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ] strictly and produces the following problem: 4: T: (Comp: ?, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0 + 1, Ar_1)) [ Ar_1 >= Ar_0 + 1 /\ Ar_1 >= Ar_0 ] (Comp: Ar_0 + Ar_1, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 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_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(eval(Ar_0, Ar_1))", 0-0) = Ar_0 S("start(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_1))", 0-1) = Ar_1 S("eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 1 ]", 0-1) = 2*Ar_0 + 2*Ar_1 S("eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0 + 1, Ar_1)) [ Ar_1 >= Ar_0 + 1 /\\ Ar_1 >= Ar_0 ]", 0-0) = ? S("eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0 + 1, Ar_1)) [ Ar_1 >= Ar_0 + 1 /\\ Ar_1 >= Ar_0 ]", 0-1) = Ar_1 orients the transitions eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0 + 1, Ar_1)) [ Ar_1 >= Ar_0 + 1 /\ Ar_1 >= Ar_0 ] weakly and the transition eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0 + 1, Ar_1)) [ Ar_1 >= Ar_0 + 1 /\ Ar_1 >= Ar_0 ] strictly and produces the following problem: 5: T: (Comp: Ar_0 + Ar_1 + 1, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0 + 1, Ar_1)) [ Ar_1 >= Ar_0 + 1 /\ Ar_1 >= Ar_0 ] (Comp: Ar_0 + Ar_1, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_1 + 1)) [ Ar_0 >= Ar_1 + 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_0 + 2*Ar_1 + 2 Time: 0.561 sec (SMT: 0.538 sec)