MAYBE Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f11(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f11(Ar_0 - 1, Ar_1 - 1, Ar_2 + 1, Fresh_2, Ar_4)) [ Ar_0 >= 1 /\ Fresh_2 >= 1 ] (Comp: ?, Cost: 1) f11(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f11(Ar_0 - 1, Ar_1, Ar_2, Fresh_1, Ar_4)) [ 0 >= Fresh_1 /\ Ar_0 >= 1 /\ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f21(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f21(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) f23(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f26(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) f11(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f21(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f11(8, Fresh_0, 0, Ar_3, 8)) [ Fresh_0 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Testing for reachability in the complexity graph removes the following transition from problem 1: f23(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f26(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) We thus obtain the following problem: 2: T: (Comp: ?, Cost: 1) f21(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f21(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) f11(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f21(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) f11(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f11(Ar_0 - 1, Ar_1, Ar_2, Fresh_1, Ar_4)) [ 0 >= Fresh_1 /\ Ar_0 >= 1 /\ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f11(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f11(Ar_0 - 1, Ar_1 - 1, Ar_2 + 1, Fresh_2, Ar_4)) [ Ar_0 >= 1 /\ Fresh_2 >= 1 ] (Comp: ?, Cost: 1) f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f11(8, Fresh_0, 0, Ar_3, 8)) [ Fresh_0 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 2 produces the following problem: 3: T: (Comp: ?, Cost: 1) f21(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f21(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) f11(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f21(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) f11(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f11(Ar_0 - 1, Ar_1, Ar_2, Fresh_1, Ar_4)) [ 0 >= Fresh_1 /\ Ar_0 >= 1 /\ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f11(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f11(Ar_0 - 1, Ar_1 - 1, Ar_2 + 1, Fresh_2, Ar_4)) [ Ar_0 >= 1 /\ Fresh_2 >= 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f11(8, Fresh_0, 0, Ar_3, 8)) [ Fresh_0 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f21) = 0 Pol(f11) = 1 Pol(f0) = 1 Pol(koat_start) = 1 orients all transitions weakly and the transition f11(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f21(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_0 ] strictly and produces the following problem: 4: T: (Comp: ?, Cost: 1) f21(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f21(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) f11(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f21(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) f11(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f11(Ar_0 - 1, Ar_1, Ar_2, Fresh_1, Ar_4)) [ 0 >= Fresh_1 /\ Ar_0 >= 1 /\ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f11(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f11(Ar_0 - 1, Ar_1 - 1, Ar_2 + 1, Fresh_2, Ar_4)) [ Ar_0 >= 1 /\ Fresh_2 >= 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f11(8, Fresh_0, 0, Ar_3, 8)) [ Fresh_0 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f21) = V_1 Pol(f11) = V_1 Pol(f0) = 8 Pol(koat_start) = 8 orients all transitions weakly and the transition f11(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f11(Ar_0 - 1, Ar_1 - 1, Ar_2 + 1, Fresh_2, Ar_4)) [ Ar_0 >= 1 /\ Fresh_2 >= 1 ] strictly and produces the following problem: 5: T: (Comp: ?, Cost: 1) f21(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f21(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) f11(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f21(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) f11(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f11(Ar_0 - 1, Ar_1, Ar_2, Fresh_1, Ar_4)) [ 0 >= Fresh_1 /\ Ar_0 >= 1 /\ Ar_0 >= Ar_1 + 1 ] (Comp: 8, Cost: 1) f11(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f11(Ar_0 - 1, Ar_1 - 1, Ar_2 + 1, Fresh_2, Ar_4)) [ Ar_0 >= 1 /\ Fresh_2 >= 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f11(8, Fresh_0, 0, Ar_3, 8)) [ Fresh_0 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f21) = V_1 Pol(f11) = V_1 Pol(f0) = 8 Pol(koat_start) = 8 orients all transitions weakly and the transition f11(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f11(Ar_0 - 1, Ar_1, Ar_2, Fresh_1, Ar_4)) [ 0 >= Fresh_1 /\ Ar_0 >= 1 /\ Ar_0 >= Ar_1 + 1 ] strictly and produces the following problem: 6: T: (Comp: ?, Cost: 1) f21(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f21(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) f11(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f21(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_0 ] (Comp: 8, Cost: 1) f11(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f11(Ar_0 - 1, Ar_1, Ar_2, Fresh_1, Ar_4)) [ 0 >= Fresh_1 /\ Ar_0 >= 1 /\ Ar_0 >= Ar_1 + 1 ] (Comp: 8, Cost: 1) f11(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f11(Ar_0 - 1, Ar_1 - 1, Ar_2 + 1, Fresh_2, Ar_4)) [ Ar_0 >= 1 /\ Fresh_2 >= 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f11(8, Fresh_0, 0, Ar_3, 8)) [ Fresh_0 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Applied AI with 'oct' on problem 6 to obtain the following invariants: For symbol f11: -X_5 + 8 >= 0 /\ X_3 - X_5 + 8 >= 0 /\ -X_1 - X_5 + 16 >= 0 /\ X_5 - 8 >= 0 /\ X_3 + X_5 - 8 >= 0 /\ -X_1 + X_5 >= 0 /\ -X_1 - X_3 + 8 >= 0 /\ X_3 >= 0 /\ X_2 + X_3 - 1 >= 0 /\ -X_1 + X_3 + 8 >= 0 /\ -X_1 + X_2 + 7 >= 0 /\ -X_1 + 8 >= 0 For symbol f21: -X_5 + 8 >= 0 /\ X_3 - X_5 + 8 >= 0 /\ -X_1 - X_5 + 8 >= 0 /\ X_5 - 8 >= 0 /\ X_3 + X_5 - 8 >= 0 /\ -X_1 + X_5 - 8 >= 0 /\ -X_1 - X_3 + 8 >= 0 /\ X_3 >= 0 /\ X_2 + X_3 - 1 >= 0 /\ -X_1 + X_3 >= 0 /\ -X_1 + X_2 + 7 >= 0 /\ -X_1 >= 0 This yielded the following problem: 7: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f11(8, Fresh_0, 0, Ar_3, 8)) [ Fresh_0 >= 1 ] (Comp: 8, Cost: 1) f11(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f11(Ar_0 - 1, Ar_1 - 1, Ar_2 + 1, Fresh_2, Ar_4)) [ -Ar_4 + 8 >= 0 /\ Ar_2 - Ar_4 + 8 >= 0 /\ -Ar_0 - Ar_4 + 16 >= 0 /\ Ar_4 - 8 >= 0 /\ Ar_2 + Ar_4 - 8 >= 0 /\ -Ar_0 + Ar_4 >= 0 /\ -Ar_0 - Ar_2 + 8 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 + 8 >= 0 /\ -Ar_0 + Ar_1 + 7 >= 0 /\ -Ar_0 + 8 >= 0 /\ Ar_0 >= 1 /\ Fresh_2 >= 1 ] (Comp: 8, Cost: 1) f11(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f11(Ar_0 - 1, Ar_1, Ar_2, Fresh_1, Ar_4)) [ -Ar_4 + 8 >= 0 /\ Ar_2 - Ar_4 + 8 >= 0 /\ -Ar_0 - Ar_4 + 16 >= 0 /\ Ar_4 - 8 >= 0 /\ Ar_2 + Ar_4 - 8 >= 0 /\ -Ar_0 + Ar_4 >= 0 /\ -Ar_0 - Ar_2 + 8 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 + 8 >= 0 /\ -Ar_0 + Ar_1 + 7 >= 0 /\ -Ar_0 + 8 >= 0 /\ 0 >= Fresh_1 /\ Ar_0 >= 1 /\ Ar_0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f11(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f21(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ -Ar_4 + 8 >= 0 /\ Ar_2 - Ar_4 + 8 >= 0 /\ -Ar_0 - Ar_4 + 16 >= 0 /\ Ar_4 - 8 >= 0 /\ Ar_2 + Ar_4 - 8 >= 0 /\ -Ar_0 + Ar_4 >= 0 /\ -Ar_0 - Ar_2 + 8 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 + 8 >= 0 /\ -Ar_0 + Ar_1 + 7 >= 0 /\ -Ar_0 + 8 >= 0 /\ 0 >= Ar_0 ] (Comp: ?, Cost: 1) f21(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(f21(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ -Ar_4 + 8 >= 0 /\ Ar_2 - Ar_4 + 8 >= 0 /\ -Ar_0 - Ar_4 + 8 >= 0 /\ Ar_4 - 8 >= 0 /\ Ar_2 + Ar_4 - 8 >= 0 /\ -Ar_0 + Ar_4 - 8 >= 0 /\ -Ar_0 - Ar_2 + 8 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 >= 0 /\ -Ar_0 + Ar_1 + 7 >= 0 /\ -Ar_0 >= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound ? Time: 3.533 sec (SMT: 3.430 sec)