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, Ar_1)) [ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 1 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0 - Ar_1, Ar_1)) [ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 1 /\ Ar_1 >= 1 /\ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_1 - Ar_0)) [ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 1 /\ Ar_1 >= 1 /\ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_1 - Ar_0)) [ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 1 /\ Ar_1 >= 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, Ar_1)) [ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 1 /\ Ar_1 >= 1 /\ Ar_0 >= Ar_1 + 1 ] eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_1 - Ar_0)) [ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 1 /\ 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, Ar_1 - Ar_0)) [ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 1 /\ Ar_1 >= 1 /\ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0 - Ar_1, Ar_1)) [ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 1 /\ 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, Ar_1 - Ar_0)) [ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 1 /\ Ar_1 >= 1 /\ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0 - Ar_1, Ar_1)) [ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 1 /\ 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 Pol(start) = V_1 + V_2 Pol(koat_start) = V_1 + V_2 orients all transitions weakly and the transition eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_1 - Ar_0)) [ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 1 /\ Ar_1 >= 1 /\ Ar_1 >= Ar_0 ] strictly and produces the following problem: 4: T: (Comp: Ar_0 + Ar_1, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_1 - Ar_0)) [ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 1 /\ Ar_1 >= 1 /\ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0 - Ar_1, Ar_1)) [ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 1 /\ 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 Pol(start) = V_1 Pol(koat_start) = V_1 orients all transitions weakly and the transition eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0 - Ar_1, Ar_1)) [ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 1 /\ Ar_1 >= 1 ] strictly and produces the following problem: 5: T: (Comp: Ar_0 + Ar_1, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0, Ar_1 - Ar_0)) [ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 1 /\ Ar_1 >= 1 /\ Ar_1 >= Ar_0 ] (Comp: Ar_0, Cost: 1) eval(Ar_0, Ar_1) -> Com_1(eval(Ar_0 - Ar_1, Ar_1)) [ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 1 /\ 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 + Ar_1 + 1 Time: 0.715 sec (SMT: 0.694 sec)