MAYBE Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0 - 1, Fresh_2, Ar_2)) [ Fresh_2 >= 1 /\ Ar_0 >= 0 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0 - 1, Fresh_1, Ar_2)) [ 0 >= Fresh_1 + 1 /\ Ar_0 >= 0 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, 0, Ar_2)) [ Ar_0 >= 0 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0, Ar_1, Fresh_0)) [ 0 >= Ar_0 + 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f3(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) f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0 - 1, Fresh_2, Ar_2)) [ Fresh_2 >= 1 /\ Ar_0 >= 0 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0 - 1, Fresh_1, Ar_2)) [ 0 >= Fresh_1 + 1 /\ Ar_0 >= 0 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, 0, Ar_2)) [ Ar_0 >= 0 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0, Ar_1, Fresh_0)) [ 0 >= Ar_0 + 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f3) = 1 Pol(f2) = 1 Pol(f300) = 0 Pol(koat_start) = 1 orients all transitions weakly and the transition f2(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0, Ar_1, Fresh_0)) [ 0 >= Ar_0 + 1 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0 - 1, Fresh_2, Ar_2)) [ Fresh_2 >= 1 /\ Ar_0 >= 0 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0 - 1, Fresh_1, Ar_2)) [ 0 >= Fresh_1 + 1 /\ Ar_0 >= 0 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, 0, Ar_2)) [ Ar_0 >= 0 ] (Comp: 1, Cost: 1) f2(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0, Ar_1, Fresh_0)) [ 0 >= Ar_0 + 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f3) = V_1 + 1 Pol(f2) = V_1 + 1 Pol(f300) = V_1 + 1 Pol(koat_start) = V_1 + 1 orients all transitions weakly and the transitions f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0 - 1, Fresh_2, Ar_2)) [ Fresh_2 >= 1 /\ Ar_0 >= 0 ] f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0 - 1, Fresh_1, Ar_2)) [ 0 >= Fresh_1 + 1 /\ Ar_0 >= 0 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) f3(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2)) (Comp: Ar_0 + 1, Cost: 1) f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0 - 1, Fresh_2, Ar_2)) [ Fresh_2 >= 1 /\ Ar_0 >= 0 ] (Comp: Ar_0 + 1, Cost: 1) f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0 - 1, Fresh_1, Ar_2)) [ 0 >= Fresh_1 + 1 /\ Ar_0 >= 0 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, 0, Ar_2)) [ Ar_0 >= 0 ] (Comp: 1, Cost: 1) f2(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0, Ar_1, Fresh_0)) [ 0 >= Ar_0 + 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound ? Time: 1.275 sec (SMT: 1.228 sec)