MAYBE Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalcyclicstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicentryin(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalcyclicentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicbb3in(Ar_0, Ar_1, Ar_0 + 1)) [ Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalcyclicbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_2 = Ar_0 ] (Comp: ?, Cost: 1) evalcyclicbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalcyclicbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalcyclicbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicbbin(Ar_0, Ar_1, Ar_2)) [ 0 >= D + 1 ] (Comp: ?, Cost: 1) evalcyclicbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicbbin(Ar_0, Ar_1, Ar_2)) [ D >= 1 ] (Comp: ?, Cost: 1) evalcyclicbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicreturnin(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalcyclicbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicbb3in(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_1 >= Ar_2 ] (Comp: ?, Cost: 1) evalcyclicbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicbb3in(Ar_0, Ar_1, 0)) [ Ar_2 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalcyclicreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicstart(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: 1, Cost: 1) evalcyclicstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicentryin(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalcyclicentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicbb3in(Ar_0, Ar_1, Ar_0 + 1)) [ Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalcyclicbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_2 = Ar_0 ] (Comp: ?, Cost: 1) evalcyclicbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalcyclicbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalcyclicbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicbbin(Ar_0, Ar_1, Ar_2)) [ 0 >= D + 1 ] (Comp: ?, Cost: 1) evalcyclicbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicbbin(Ar_0, Ar_1, Ar_2)) [ D >= 1 ] (Comp: ?, Cost: 1) evalcyclicbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicreturnin(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalcyclicbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicbb3in(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_1 >= Ar_2 ] (Comp: ?, Cost: 1) evalcyclicbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicbb3in(Ar_0, Ar_1, 0)) [ Ar_2 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalcyclicreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalcyclicstart) = 2 Pol(evalcyclicentryin) = 2 Pol(evalcyclicbb3in) = 2 Pol(evalcyclicreturnin) = 1 Pol(evalcyclicbb4in) = 2 Pol(evalcyclicbbin) = 2 Pol(evalcyclicstop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalcyclicreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicstop(Ar_0, Ar_1, Ar_2)) evalcyclicbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicreturnin(Ar_0, Ar_1, Ar_2)) evalcyclicbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_2 = Ar_0 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evalcyclicstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicentryin(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalcyclicentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicbb3in(Ar_0, Ar_1, Ar_0 + 1)) [ Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalcyclicbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_2 = Ar_0 ] (Comp: ?, Cost: 1) evalcyclicbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalcyclicbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalcyclicbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicbbin(Ar_0, Ar_1, Ar_2)) [ 0 >= D + 1 ] (Comp: ?, Cost: 1) evalcyclicbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicbbin(Ar_0, Ar_1, Ar_2)) [ D >= 1 ] (Comp: 2, Cost: 1) evalcyclicbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicreturnin(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalcyclicbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicbb3in(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_1 >= Ar_2 ] (Comp: ?, Cost: 1) evalcyclicbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicbb3in(Ar_0, Ar_1, 0)) [ Ar_2 >= Ar_1 + 1 ] (Comp: 2, Cost: 1) evalcyclicreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Applied AI with 'oct' on problem 3 to obtain the following invariants: For symbol evalcyclicbb3in: X_2 - 1 >= 0 /\ X_1 + X_2 - 1 >= 0 /\ -X_1 + X_2 - 1 >= 0 /\ X_1 >= 0 For symbol evalcyclicbb4in: X_2 - 1 >= 0 /\ X_1 + X_2 - 1 >= 0 /\ -X_1 + X_2 - 1 >= 0 /\ X_1 >= 0 For symbol evalcyclicbbin: X_2 - 1 >= 0 /\ X_1 + X_2 - 1 >= 0 /\ -X_1 + X_2 - 1 >= 0 /\ X_1 >= 0 For symbol evalcyclicreturnin: X_2 - 1 >= 0 /\ X_1 + X_2 - 1 >= 0 /\ -X_1 + X_2 - 1 >= 0 /\ X_1 >= 0 This yielded the following problem: 4: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] (Comp: 2, Cost: 1) evalcyclicreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicstop(Ar_0, Ar_1, Ar_2)) [ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 ] (Comp: ?, Cost: 1) evalcyclicbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicbb3in(Ar_0, Ar_1, 0)) [ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 /\ Ar_2 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalcyclicbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicbb3in(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 /\ Ar_1 >= Ar_2 ] (Comp: 2, Cost: 1) evalcyclicbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 ] (Comp: ?, Cost: 1) evalcyclicbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicbbin(Ar_0, Ar_1, Ar_2)) [ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 /\ D >= 1 ] (Comp: ?, Cost: 1) evalcyclicbb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicbbin(Ar_0, Ar_1, Ar_2)) [ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 /\ 0 >= D + 1 ] (Comp: ?, Cost: 1) evalcyclicbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 /\ Ar_2 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalcyclicbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicbb4in(Ar_0, Ar_1, Ar_2)) [ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 /\ Ar_0 >= Ar_2 + 1 ] (Comp: 2, Cost: 1) evalcyclicbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 >= 0 /\ Ar_2 = Ar_0 ] (Comp: 1, Cost: 1) evalcyclicentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicbb3in(Ar_0, Ar_1, Ar_0 + 1)) [ Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) evalcyclicstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalcyclicentryin(Ar_0, Ar_1, Ar_2)) start location: koat_start leaf cost: 0 Complexity upper bound ? Time: 2.435 sec (SMT: 2.345 sec)