WORST_CASE(?, O(n^1)) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) l0(Ar_0, Ar_1) -> Com_1(l1(Ar_0, Ar_1)) (Comp: ?, Cost: 1) l1(Ar_0, Ar_1) -> Com_1(l1(Ar_0 - 1, Ar_1 + 1)) [ Ar_0 >= 1 ] (Comp: ?, Cost: 1) l1(Ar_0, Ar_1) -> Com_1(l2(Ar_0, Ar_1)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) l2(Ar_0, Ar_1) -> Com_1(l2(Ar_0, Ar_1 - 1)) [ Ar_1 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(l0(Ar_0, Ar_1)) [ 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) l0(Ar_0, Ar_1) -> Com_1(l1(Ar_0, Ar_1)) (Comp: ?, Cost: 1) l1(Ar_0, Ar_1) -> Com_1(l1(Ar_0 - 1, Ar_1 + 1)) [ Ar_0 >= 1 ] (Comp: ?, Cost: 1) l1(Ar_0, Ar_1) -> Com_1(l2(Ar_0, Ar_1)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) l2(Ar_0, Ar_1) -> Com_1(l2(Ar_0, Ar_1 - 1)) [ Ar_1 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(l0(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(l0) = 1 Pol(l1) = 1 Pol(l2) = 0 Pol(koat_start) = 1 orients all transitions weakly and the transition l1(Ar_0, Ar_1) -> Com_1(l2(Ar_0, Ar_1)) [ 0 >= Ar_0 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) l0(Ar_0, Ar_1) -> Com_1(l1(Ar_0, Ar_1)) (Comp: ?, Cost: 1) l1(Ar_0, Ar_1) -> Com_1(l1(Ar_0 - 1, Ar_1 + 1)) [ Ar_0 >= 1 ] (Comp: 1, Cost: 1) l1(Ar_0, Ar_1) -> Com_1(l2(Ar_0, Ar_1)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) l2(Ar_0, Ar_1) -> Com_1(l2(Ar_0, Ar_1 - 1)) [ Ar_1 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(l0(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(l0) = V_1 Pol(l1) = V_1 Pol(l2) = V_1 - 1 Pol(koat_start) = V_1 orients all transitions weakly and the transition l1(Ar_0, Ar_1) -> Com_1(l1(Ar_0 - 1, Ar_1 + 1)) [ Ar_0 >= 1 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) l0(Ar_0, Ar_1) -> Com_1(l1(Ar_0, Ar_1)) (Comp: Ar_0, Cost: 1) l1(Ar_0, Ar_1) -> Com_1(l1(Ar_0 - 1, Ar_1 + 1)) [ Ar_0 >= 1 ] (Comp: 1, Cost: 1) l1(Ar_0, Ar_1) -> Com_1(l2(Ar_0, Ar_1)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) l2(Ar_0, Ar_1) -> Com_1(l2(Ar_0, Ar_1 - 1)) [ Ar_1 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(l0(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(l2) = V_2 and size complexities S("koat_start(Ar_0, Ar_1) -> Com_1(l0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1) -> Com_1(l0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-1) = Ar_1 S("l2(Ar_0, Ar_1) -> Com_1(l2(Ar_0, Ar_1 - 1)) [ Ar_1 >= 1 ]", 0-0) = Ar_0 S("l2(Ar_0, Ar_1) -> Com_1(l2(Ar_0, Ar_1 - 1)) [ Ar_1 >= 1 ]", 0-1) = Ar_0 + Ar_1 S("l1(Ar_0, Ar_1) -> Com_1(l2(Ar_0, Ar_1)) [ 0 >= Ar_0 ]", 0-0) = Ar_0 S("l1(Ar_0, Ar_1) -> Com_1(l2(Ar_0, Ar_1)) [ 0 >= Ar_0 ]", 0-1) = Ar_0 + Ar_1 S("l1(Ar_0, Ar_1) -> Com_1(l1(Ar_0 - 1, Ar_1 + 1)) [ Ar_0 >= 1 ]", 0-0) = Ar_0 S("l1(Ar_0, Ar_1) -> Com_1(l1(Ar_0 - 1, Ar_1 + 1)) [ Ar_0 >= 1 ]", 0-1) = Ar_0 + Ar_1 S("l0(Ar_0, Ar_1) -> Com_1(l1(Ar_0, Ar_1))", 0-0) = Ar_0 S("l0(Ar_0, Ar_1) -> Com_1(l1(Ar_0, Ar_1))", 0-1) = Ar_1 orients the transitions l2(Ar_0, Ar_1) -> Com_1(l2(Ar_0, Ar_1 - 1)) [ Ar_1 >= 1 ] weakly and the transition l2(Ar_0, Ar_1) -> Com_1(l2(Ar_0, Ar_1 - 1)) [ Ar_1 >= 1 ] strictly and produces the following problem: 5: T: (Comp: 1, Cost: 1) l0(Ar_0, Ar_1) -> Com_1(l1(Ar_0, Ar_1)) (Comp: Ar_0, Cost: 1) l1(Ar_0, Ar_1) -> Com_1(l1(Ar_0 - 1, Ar_1 + 1)) [ Ar_0 >= 1 ] (Comp: 1, Cost: 1) l1(Ar_0, Ar_1) -> Com_1(l2(Ar_0, Ar_1)) [ 0 >= Ar_0 ] (Comp: Ar_0 + Ar_1, Cost: 1) l2(Ar_0, Ar_1) -> Com_1(l2(Ar_0, Ar_1 - 1)) [ Ar_1 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(l0(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 2*Ar_0 + Ar_1 + 2 Time: 0.644 sec (SMT: 0.622 sec)