MAYBE Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f7(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) f6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f7(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 ] (Comp: ?, Cost: 1) f0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_1 + 1)) [ Ar_1 >= 0 /\ Ar_2 >= Ar_1 ] (Comp: ?, Cost: 1) f4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f6(Fresh_1, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) f4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f6(Fresh_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f7(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_2 >= Ar_3 ] (Comp: ?, Cost: 1) f7(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f4(Ar_0, Ar_1, Ar_2, 0)) [ Ar_3 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) f6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f14(0, Ar_1, Ar_2, Ar_3)) [ Ar_0 = 0 ] (Comp: ?, Cost: 1) f4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f14(Ar_0, Ar_1, Ar_2, Ar_1)) [ Ar_1 = Ar_3 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 1 produces the following problem: 2: T: (Comp: ?, Cost: 1) f6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f7(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) f6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f7(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_1 + 1)) [ Ar_1 >= 0 /\ Ar_2 >= Ar_1 ] (Comp: ?, Cost: 1) f4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f6(Fresh_1, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) f4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f6(Fresh_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f7(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_2 >= Ar_3 ] (Comp: ?, Cost: 1) f7(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f4(Ar_0, Ar_1, Ar_2, 0)) [ Ar_3 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) f6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f14(0, Ar_1, Ar_2, Ar_3)) [ Ar_0 = 0 ] (Comp: ?, Cost: 1) f4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f14(Ar_0, Ar_1, Ar_2, Ar_1)) [ Ar_1 = Ar_3 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f6) = 1 Pol(f7) = 1 Pol(f0) = 1 Pol(f4) = 1 Pol(f14) = 0 Pol(koat_start) = 1 orients all transitions weakly and the transitions f6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f14(0, Ar_1, Ar_2, Ar_3)) [ Ar_0 = 0 ] f4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f14(Ar_0, Ar_1, Ar_2, Ar_1)) [ Ar_1 = Ar_3 ] strictly and produces the following problem: 3: T: (Comp: ?, Cost: 1) f6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f7(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) f6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f7(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_1 + 1)) [ Ar_1 >= 0 /\ Ar_2 >= Ar_1 ] (Comp: ?, Cost: 1) f4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f6(Fresh_1, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) f4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f6(Fresh_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f7(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_2 >= Ar_3 ] (Comp: ?, Cost: 1) f7(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f4(Ar_0, Ar_1, Ar_2, 0)) [ Ar_3 >= Ar_2 + 1 ] (Comp: 1, Cost: 1) f6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f14(0, Ar_1, Ar_2, Ar_3)) [ Ar_0 = 0 ] (Comp: 1, Cost: 1) f4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f14(Ar_0, Ar_1, Ar_2, Ar_1)) [ Ar_1 = Ar_3 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Applied AI with 'oct' on problem 3 to obtain the following invariants: For symbol f4: X_3 - X_4 + 1 >= 0 /\ X_3 >= 0 /\ X_2 + X_3 >= 0 /\ -X_2 + X_3 >= 0 /\ X_2 >= 0 For symbol f6: X_3 - X_4 + 1 >= 0 /\ X_3 >= 0 /\ X_2 + X_3 >= 0 /\ -X_2 + X_3 >= 0 /\ X_2 >= 0 For symbol f7: X_3 - X_4 + 1 >= 0 /\ X_3 >= 0 /\ X_2 + X_3 >= 0 /\ -X_2 + X_3 >= 0 /\ X_2 >= 0 This yielded the following problem: 4: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] (Comp: 1, Cost: 1) f4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f14(Ar_0, Ar_1, Ar_2, Ar_1)) [ Ar_2 - Ar_3 + 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 >= 0 /\ -Ar_1 + Ar_2 >= 0 /\ Ar_1 >= 0 /\ Ar_1 = Ar_3 ] (Comp: 1, Cost: 1) f6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f14(0, Ar_1, Ar_2, Ar_3)) [ Ar_2 - Ar_3 + 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 >= 0 /\ -Ar_1 + Ar_2 >= 0 /\ Ar_1 >= 0 /\ Ar_0 = 0 ] (Comp: ?, Cost: 1) f7(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f4(Ar_0, Ar_1, Ar_2, 0)) [ Ar_2 - Ar_3 + 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 >= 0 /\ -Ar_1 + Ar_2 >= 0 /\ Ar_1 >= 0 /\ Ar_3 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) f7(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_2 - Ar_3 + 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 >= 0 /\ -Ar_1 + Ar_2 >= 0 /\ Ar_1 >= 0 /\ Ar_2 >= Ar_3 ] (Comp: ?, Cost: 1) f4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f6(Fresh_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 - Ar_3 + 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 >= 0 /\ -Ar_1 + Ar_2 >= 0 /\ Ar_1 >= 0 /\ Ar_3 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f6(Fresh_1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - Ar_3 + 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 >= 0 /\ -Ar_1 + Ar_2 >= 0 /\ Ar_1 >= 0 /\ Ar_1 >= Ar_3 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_1 + 1)) [ Ar_1 >= 0 /\ Ar_2 >= Ar_1 ] (Comp: ?, Cost: 1) f6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f7(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 - Ar_3 + 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 >= 0 /\ -Ar_1 + Ar_2 >= 0 /\ Ar_1 >= 0 /\ Ar_0 >= 1 ] (Comp: ?, Cost: 1) f6(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f7(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 - Ar_3 + 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 >= 0 /\ -Ar_1 + Ar_2 >= 0 /\ Ar_1 >= 0 /\ 0 >= Ar_0 + 1 ] start location: koat_start leaf cost: 0 Complexity upper bound ? Time: 2.037 sec (SMT: 1.961 sec)