WORST_CASE(?, O(n^1)) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f4(0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) f4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f8(Ar_0 + 1, Ar_1, 0, Ar_3)) [ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) f8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f16(Ar_0, Ar_1, Ar_2, 0)) [ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) f8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f16(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) f8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f8(Ar_0 + 1, Ar_1, Ar_2 + 1, Fresh_1)) [ Ar_1 >= Ar_0 + 1 /\ 0 >= Fresh_1 + 1 ] (Comp: ?, Cost: 1) f8(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f8(Ar_0 + 1, Ar_1, Ar_2 + 1, Fresh_0)) [ Ar_1 >= Ar_0 + 1 /\ Fresh_0 >= 1 ] (Comp: ?, Cost: 1) f16(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f4(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_2 ] (Comp: ?, Cost: 1) f16(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f4(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= 1 ] (Comp: ?, Cost: 1) f4(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(f20(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ] (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 Slicing away variables that do not contribute to conditions from problem 1 leaves variables [Ar_0, Ar_1, Ar_2]. We thus obtain the following problem: 2: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f0(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] (Comp: ?, Cost: 1) f4(Ar_0, Ar_1, Ar_2) -> Com_1(f20(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) f16(Ar_0, Ar_1, Ar_2) -> Com_1(f4(Ar_0 - 1, Ar_1, Ar_2)) [ Ar_2 >= 1 ] (Comp: ?, Cost: 1) f16(Ar_0, Ar_1, Ar_2) -> Com_1(f4(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_2 ] (Comp: ?, Cost: 1) f8(Ar_0, Ar_1, Ar_2) -> Com_1(f8(Ar_0 + 1, Ar_1, Ar_2 + 1)) [ Ar_1 >= Ar_0 + 1 /\ Fresh_0 >= 1 ] (Comp: ?, Cost: 1) f8(Ar_0, Ar_1, Ar_2) -> Com_1(f8(Ar_0 + 1, Ar_1, Ar_2 + 1)) [ Ar_1 >= Ar_0 + 1 /\ 0 >= Fresh_1 + 1 ] (Comp: ?, Cost: 1) f8(Ar_0, Ar_1, Ar_2) -> Com_1(f16(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) f8(Ar_0, Ar_1, Ar_2) -> Com_1(f16(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) f4(Ar_0, Ar_1, Ar_2) -> Com_1(f8(Ar_0 + 1, Ar_1, 0)) [ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) f0(Ar_0, Ar_1, Ar_2) -> Com_1(f4(0, Ar_1, Ar_2)) start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 2 produces the following problem: 3: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f0(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] (Comp: ?, Cost: 1) f4(Ar_0, Ar_1, Ar_2) -> Com_1(f20(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) f16(Ar_0, Ar_1, Ar_2) -> Com_1(f4(Ar_0 - 1, Ar_1, Ar_2)) [ Ar_2 >= 1 ] (Comp: ?, Cost: 1) f16(Ar_0, Ar_1, Ar_2) -> Com_1(f4(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_2 ] (Comp: ?, Cost: 1) f8(Ar_0, Ar_1, Ar_2) -> Com_1(f8(Ar_0 + 1, Ar_1, Ar_2 + 1)) [ Ar_1 >= Ar_0 + 1 /\ Fresh_0 >= 1 ] (Comp: ?, Cost: 1) f8(Ar_0, Ar_1, Ar_2) -> Com_1(f8(Ar_0 + 1, Ar_1, Ar_2 + 1)) [ Ar_1 >= Ar_0 + 1 /\ 0 >= Fresh_1 + 1 ] (Comp: ?, Cost: 1) f8(Ar_0, Ar_1, Ar_2) -> Com_1(f16(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) f8(Ar_0, Ar_1, Ar_2) -> Com_1(f16(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) f4(Ar_0, Ar_1, Ar_2) -> Com_1(f8(Ar_0 + 1, Ar_1, 0)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1, Ar_2) -> Com_1(f4(0, Ar_1, Ar_2)) start location: koat_start leaf cost: 0 A polynomial rank function with Pol(koat_start) = 1 Pol(f0) = 1 Pol(f4) = 1 Pol(f20) = 0 Pol(f16) = 1 Pol(f8) = 1 orients all transitions weakly and the transition f4(Ar_0, Ar_1, Ar_2) -> Com_1(f20(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f0(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] (Comp: 1, Cost: 1) f4(Ar_0, Ar_1, Ar_2) -> Com_1(f20(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) f16(Ar_0, Ar_1, Ar_2) -> Com_1(f4(Ar_0 - 1, Ar_1, Ar_2)) [ Ar_2 >= 1 ] (Comp: ?, Cost: 1) f16(Ar_0, Ar_1, Ar_2) -> Com_1(f4(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_2 ] (Comp: ?, Cost: 1) f8(Ar_0, Ar_1, Ar_2) -> Com_1(f8(Ar_0 + 1, Ar_1, Ar_2 + 1)) [ Ar_1 >= Ar_0 + 1 /\ Fresh_0 >= 1 ] (Comp: ?, Cost: 1) f8(Ar_0, Ar_1, Ar_2) -> Com_1(f8(Ar_0 + 1, Ar_1, Ar_2 + 1)) [ Ar_1 >= Ar_0 + 1 /\ 0 >= Fresh_1 + 1 ] (Comp: ?, Cost: 1) f8(Ar_0, Ar_1, Ar_2) -> Com_1(f16(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) f8(Ar_0, Ar_1, Ar_2) -> Com_1(f16(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) f4(Ar_0, Ar_1, Ar_2) -> Com_1(f8(Ar_0 + 1, Ar_1, 0)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1, Ar_2) -> Com_1(f4(0, Ar_1, Ar_2)) start location: koat_start leaf cost: 0 A polynomial rank function with Pol(koat_start) = 2*V_2 Pol(f0) = 2*V_2 Pol(f4) = -2*V_1 + 2*V_2 - 1 Pol(f20) = -2*V_1 + 2*V_2 - 1 Pol(f16) = -2*V_1 + 2*V_2 + 1 Pol(f8) = -2*V_1 + 2*V_2 + 1 orients all transitions weakly and the transitions f8(Ar_0, Ar_1, Ar_2) -> Com_1(f8(Ar_0 + 1, Ar_1, Ar_2 + 1)) [ Ar_1 >= Ar_0 + 1 /\ Fresh_0 >= 1 ] f8(Ar_0, Ar_1, Ar_2) -> Com_1(f8(Ar_0 + 1, Ar_1, Ar_2 + 1)) [ Ar_1 >= Ar_0 + 1 /\ 0 >= Fresh_1 + 1 ] strictly and produces the following problem: 5: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f0(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] (Comp: 1, Cost: 1) f4(Ar_0, Ar_1, Ar_2) -> Com_1(f20(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) f16(Ar_0, Ar_1, Ar_2) -> Com_1(f4(Ar_0 - 1, Ar_1, Ar_2)) [ Ar_2 >= 1 ] (Comp: ?, Cost: 1) f16(Ar_0, Ar_1, Ar_2) -> Com_1(f4(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_2 ] (Comp: 2*Ar_1, Cost: 1) f8(Ar_0, Ar_1, Ar_2) -> Com_1(f8(Ar_0 + 1, Ar_1, Ar_2 + 1)) [ Ar_1 >= Ar_0 + 1 /\ Fresh_0 >= 1 ] (Comp: 2*Ar_1, Cost: 1) f8(Ar_0, Ar_1, Ar_2) -> Com_1(f8(Ar_0 + 1, Ar_1, Ar_2 + 1)) [ Ar_1 >= Ar_0 + 1 /\ 0 >= Fresh_1 + 1 ] (Comp: ?, Cost: 1) f8(Ar_0, Ar_1, Ar_2) -> Com_1(f16(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) f8(Ar_0, Ar_1, Ar_2) -> Com_1(f16(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) f4(Ar_0, Ar_1, Ar_2) -> Com_1(f8(Ar_0 + 1, Ar_1, 0)) [ Ar_1 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1, Ar_2) -> Com_1(f4(0, Ar_1, Ar_2)) start location: koat_start leaf cost: 0 Applied AI with 'oct' on problem 5 to obtain the following invariants: For symbol f16: X_2 - X_3 - 1 >= 0 /\ X_1 - X_3 - 1 >= 0 /\ X_3 >= 0 /\ X_2 + X_3 - 1 >= 0 /\ X_1 + X_3 - 1 >= 0 /\ X_2 - 1 >= 0 /\ X_1 + X_2 - 2 >= 0 /\ -X_1 + X_2 >= 0 /\ X_1 - 1 >= 0 For symbol f4: X_1 >= 0 For symbol f8: X_2 - X_3 - 1 >= 0 /\ X_1 - X_3 - 1 >= 0 /\ X_3 >= 0 /\ X_2 + X_3 - 1 >= 0 /\ X_1 + X_3 - 1 >= 0 /\ X_2 - 1 >= 0 /\ X_1 + X_2 - 2 >= 0 /\ -X_1 + X_2 >= 0 /\ X_1 - 1 >= 0 This yielded the following problem: 6: T: (Comp: 1, Cost: 1) f0(Ar_0, Ar_1, Ar_2) -> Com_1(f4(0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) f4(Ar_0, Ar_1, Ar_2) -> Com_1(f8(Ar_0 + 1, Ar_1, 0)) [ Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) f8(Ar_0, Ar_1, Ar_2) -> Com_1(f16(Ar_0, Ar_1, Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) f8(Ar_0, Ar_1, Ar_2) -> Com_1(f16(Ar_0, Ar_1, Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= Ar_1 ] (Comp: 2*Ar_1, Cost: 1) f8(Ar_0, Ar_1, Ar_2) -> Com_1(f8(Ar_0 + 1, Ar_1, Ar_2 + 1)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Fresh_1 + 1 ] (Comp: 2*Ar_1, Cost: 1) f8(Ar_0, Ar_1, Ar_2) -> Com_1(f8(Ar_0 + 1, Ar_1, Ar_2 + 1)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Fresh_0 >= 1 ] (Comp: ?, Cost: 1) f16(Ar_0, Ar_1, Ar_2) -> Com_1(f4(Ar_0, Ar_1, Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_2 ] (Comp: ?, Cost: 1) f16(Ar_0, Ar_1, Ar_2) -> Com_1(f4(Ar_0 - 1, Ar_1, Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_2 >= 1 ] (Comp: 1, Cost: 1) f4(Ar_0, Ar_1, Ar_2) -> Com_1(f20(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= 0 /\ Ar_0 >= Ar_1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f0(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f0) = 4*V_2 Pol(f4) = -3*V_1 + 4*V_2 - 1 Pol(f8) = -3*V_1 + 4*V_2 + 3*V_3 + 1 Pol(f16) = -3*V_1 + 4*V_2 + 3*V_3 Pol(f20) = -3*V_1 + 4*V_2 - 1 Pol(koat_start) = 4*V_2 orients all transitions weakly and the transitions f8(Ar_0, Ar_1, Ar_2) -> Com_1(f16(Ar_0, Ar_1, Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= Ar_1 ] f16(Ar_0, Ar_1, Ar_2) -> Com_1(f4(Ar_0, Ar_1, Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_2 ] strictly and produces the following problem: 7: T: (Comp: 1, Cost: 1) f0(Ar_0, Ar_1, Ar_2) -> Com_1(f4(0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) f4(Ar_0, Ar_1, Ar_2) -> Com_1(f8(Ar_0 + 1, Ar_1, 0)) [ Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) f8(Ar_0, Ar_1, Ar_2) -> Com_1(f16(Ar_0, Ar_1, Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_0 + 1 ] (Comp: 4*Ar_1, Cost: 1) f8(Ar_0, Ar_1, Ar_2) -> Com_1(f16(Ar_0, Ar_1, Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= Ar_1 ] (Comp: 2*Ar_1, Cost: 1) f8(Ar_0, Ar_1, Ar_2) -> Com_1(f8(Ar_0 + 1, Ar_1, Ar_2 + 1)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Fresh_1 + 1 ] (Comp: 2*Ar_1, Cost: 1) f8(Ar_0, Ar_1, Ar_2) -> Com_1(f8(Ar_0 + 1, Ar_1, Ar_2 + 1)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Fresh_0 >= 1 ] (Comp: 4*Ar_1, Cost: 1) f16(Ar_0, Ar_1, Ar_2) -> Com_1(f4(Ar_0, Ar_1, Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_2 ] (Comp: ?, Cost: 1) f16(Ar_0, Ar_1, Ar_2) -> Com_1(f4(Ar_0 - 1, Ar_1, Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_2 >= 1 ] (Comp: 1, Cost: 1) f4(Ar_0, Ar_1, Ar_2) -> Com_1(f20(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= 0 /\ Ar_0 >= Ar_1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f0(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f0) = 5*V_2 - 5 Pol(f4) = -3*V_1 + 5*V_2 - 6 Pol(f8) = -3*V_1 + 5*V_2 + 3*V_3 - 4 Pol(f16) = -3*V_1 + 5*V_2 + 3*V_3 - 5 Pol(f20) = -3*V_1 + 5*V_2 - 6 Pol(koat_start) = 5*V_2 - 5 orients all transitions weakly and the transition f16(Ar_0, Ar_1, Ar_2) -> Com_1(f4(Ar_0 - 1, Ar_1, Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_2 >= 1 ] strictly and produces the following problem: 8: T: (Comp: 1, Cost: 1) f0(Ar_0, Ar_1, Ar_2) -> Com_1(f4(0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) f4(Ar_0, Ar_1, Ar_2) -> Com_1(f8(Ar_0 + 1, Ar_1, 0)) [ Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) f8(Ar_0, Ar_1, Ar_2) -> Com_1(f16(Ar_0, Ar_1, Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_0 + 1 ] (Comp: 4*Ar_1, Cost: 1) f8(Ar_0, Ar_1, Ar_2) -> Com_1(f16(Ar_0, Ar_1, Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= Ar_1 ] (Comp: 2*Ar_1, Cost: 1) f8(Ar_0, Ar_1, Ar_2) -> Com_1(f8(Ar_0 + 1, Ar_1, Ar_2 + 1)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Fresh_1 + 1 ] (Comp: 2*Ar_1, Cost: 1) f8(Ar_0, Ar_1, Ar_2) -> Com_1(f8(Ar_0 + 1, Ar_1, Ar_2 + 1)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Fresh_0 >= 1 ] (Comp: 4*Ar_1, Cost: 1) f16(Ar_0, Ar_1, Ar_2) -> Com_1(f4(Ar_0, Ar_1, Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_2 ] (Comp: 5*Ar_1 + 5, Cost: 1) f16(Ar_0, Ar_1, Ar_2) -> Com_1(f4(Ar_0 - 1, Ar_1, Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_2 >= 1 ] (Comp: 1, Cost: 1) f4(Ar_0, Ar_1, Ar_2) -> Com_1(f20(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= 0 /\ Ar_0 >= Ar_1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f0(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 8 produces the following problem: 9: T: (Comp: 1, Cost: 1) f0(Ar_0, Ar_1, Ar_2) -> Com_1(f4(0, Ar_1, Ar_2)) (Comp: 9*Ar_1 + 6, Cost: 1) f4(Ar_0, Ar_1, Ar_2) -> Com_1(f8(Ar_0 + 1, Ar_1, 0)) [ Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 ] (Comp: 13*Ar_1 + 6, Cost: 1) f8(Ar_0, Ar_1, Ar_2) -> Com_1(f16(Ar_0, Ar_1, Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_0 + 1 ] (Comp: 4*Ar_1, Cost: 1) f8(Ar_0, Ar_1, Ar_2) -> Com_1(f16(Ar_0, Ar_1, Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= Ar_1 ] (Comp: 2*Ar_1, Cost: 1) f8(Ar_0, Ar_1, Ar_2) -> Com_1(f8(Ar_0 + 1, Ar_1, Ar_2 + 1)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Fresh_1 + 1 ] (Comp: 2*Ar_1, Cost: 1) f8(Ar_0, Ar_1, Ar_2) -> Com_1(f8(Ar_0 + 1, Ar_1, Ar_2 + 1)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Fresh_0 >= 1 ] (Comp: 4*Ar_1, Cost: 1) f16(Ar_0, Ar_1, Ar_2) -> Com_1(f4(Ar_0, Ar_1, Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_2 ] (Comp: 5*Ar_1 + 5, Cost: 1) f16(Ar_0, Ar_1, Ar_2) -> Com_1(f4(Ar_0 - 1, Ar_1, Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 2 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_2 >= 1 ] (Comp: 1, Cost: 1) f4(Ar_0, Ar_1, Ar_2) -> Com_1(f20(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= 0 /\ Ar_0 >= Ar_1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f0(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 39*Ar_1 + 19 Time: 2.958 sec (SMT: 2.841 sec)