MAYBE Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f0(Ar_0, Ar_1, Ar_2) -> Com_1(f1(0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1 - 1, Fresh_4)) [ Ar_1 >= 1 /\ Fresh_4 >= 1 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1 - 2, Fresh_3)) [ Ar_1 >= 1 /\ 0 >= Fresh_3 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f4(Ar_0, Ar_1, Fresh_2)) [ 0 >= Ar_1 ] (Comp: ?, Cost: 1) f4(Ar_0, Ar_1, Ar_2) -> Com_1(f4(1, Ar_1, Fresh_1)) [ Ar_2 >= 1 ] (Comp: ?, Cost: 1) f4(Ar_0, Ar_1, Ar_2) -> Com_1(f4(2, Ar_1, Fresh_0)) [ 0 >= Ar_2 ] (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 1 produces the following problem: 2: T: (Comp: 1, Cost: 1) f0(Ar_0, Ar_1, Ar_2) -> Com_1(f1(0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1 - 1, Fresh_4)) [ Ar_1 >= 1 /\ Fresh_4 >= 1 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1 - 2, Fresh_3)) [ Ar_1 >= 1 /\ 0 >= Fresh_3 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f4(Ar_0, Ar_1, Fresh_2)) [ 0 >= Ar_1 ] (Comp: ?, Cost: 1) f4(Ar_0, Ar_1, Ar_2) -> Com_1(f4(1, Ar_1, Fresh_1)) [ Ar_2 >= 1 ] (Comp: ?, Cost: 1) f4(Ar_0, Ar_1, Ar_2) -> Com_1(f4(2, Ar_1, Fresh_0)) [ 0 >= Ar_2 ] (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) = 1 Pol(f1) = 1 Pol(f4) = 0 Pol(koat_start) = 1 orients all transitions weakly and the transition f1(Ar_0, Ar_1, Ar_2) -> Com_1(f4(Ar_0, Ar_1, Fresh_2)) [ 0 >= Ar_1 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) f0(Ar_0, Ar_1, Ar_2) -> Com_1(f1(0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1 - 1, Fresh_4)) [ Ar_1 >= 1 /\ Fresh_4 >= 1 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1 - 2, Fresh_3)) [ Ar_1 >= 1 /\ 0 >= Fresh_3 ] (Comp: 1, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f4(Ar_0, Ar_1, Fresh_2)) [ 0 >= Ar_1 ] (Comp: ?, Cost: 1) f4(Ar_0, Ar_1, Ar_2) -> Com_1(f4(1, Ar_1, Fresh_1)) [ Ar_2 >= 1 ] (Comp: ?, Cost: 1) f4(Ar_0, Ar_1, Ar_2) -> Com_1(f4(2, Ar_1, Fresh_0)) [ 0 >= Ar_2 ] (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) = V_2 Pol(f1) = V_2 Pol(f4) = V_2 Pol(koat_start) = V_2 orients all transitions weakly and the transitions f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1 - 1, Fresh_4)) [ Ar_1 >= 1 /\ Fresh_4 >= 1 ] f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1 - 2, Fresh_3)) [ Ar_1 >= 1 /\ 0 >= Fresh_3 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) f0(Ar_0, Ar_1, Ar_2) -> Com_1(f1(0, Ar_1, Ar_2)) (Comp: Ar_1, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1 - 1, Fresh_4)) [ Ar_1 >= 1 /\ Fresh_4 >= 1 ] (Comp: Ar_1, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1 - 2, Fresh_3)) [ Ar_1 >= 1 /\ 0 >= Fresh_3 ] (Comp: 1, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f4(Ar_0, Ar_1, Fresh_2)) [ 0 >= Ar_1 ] (Comp: ?, Cost: 1) f4(Ar_0, Ar_1, Ar_2) -> Com_1(f4(1, Ar_1, Fresh_1)) [ Ar_2 >= 1 ] (Comp: ?, Cost: 1) f4(Ar_0, Ar_1, Ar_2) -> Com_1(f4(2, Ar_1, Fresh_0)) [ 0 >= Ar_2 ] (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 Applied AI with 'oct' on problem 4 to obtain the following invariants: For symbol f1: -X_1 >= 0 /\ X_1 >= 0 For symbol f4: -X_2 >= 0 /\ X_1 - X_2 >= 0 /\ X_1 >= 0 This yielded 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: ?, Cost: 1) f4(Ar_0, Ar_1, Ar_2) -> Com_1(f4(2, Ar_1, Fresh_0)) [ -Ar_1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_0 >= 0 /\ 0 >= Ar_2 ] (Comp: ?, Cost: 1) f4(Ar_0, Ar_1, Ar_2) -> Com_1(f4(1, Ar_1, Fresh_1)) [ -Ar_1 >= 0 /\ Ar_0 - Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_2 >= 1 ] (Comp: 1, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f4(Ar_0, Ar_1, Fresh_2)) [ -Ar_0 >= 0 /\ Ar_0 >= 0 /\ 0 >= Ar_1 ] (Comp: Ar_1, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1 - 2, Fresh_3)) [ -Ar_0 >= 0 /\ Ar_0 >= 0 /\ Ar_1 >= 1 /\ 0 >= Fresh_3 ] (Comp: Ar_1, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1 - 1, Fresh_4)) [ -Ar_0 >= 0 /\ Ar_0 >= 0 /\ Ar_1 >= 1 /\ Fresh_4 >= 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1, Ar_2) -> Com_1(f1(0, Ar_1, Ar_2)) start location: koat_start leaf cost: 0 Complexity upper bound ? Time: 2.532 sec (SMT: 2.446 sec)