WORST_CASE(?, O(n^2)) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) l0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l1(0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) l1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l2(Ar_0, Ar_1, 0, 0)) [ Ar_1 >= 1 ] (Comp: ?, Cost: 1) l2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l2(Ar_0, Ar_1, Ar_2 + 1, Ar_3 + Ar_2)) [ Ar_1 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) l2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l1(Ar_0 + Ar_3, Ar_1 - 1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l0(Ar_0, Ar_1, Ar_2, Ar_3)) [ 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, Ar_2, Ar_3) -> Com_1(l1(0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) l1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l2(Ar_0, Ar_1, 0, 0)) [ Ar_1 >= 1 ] (Comp: ?, Cost: 1) l2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l2(Ar_0, Ar_1, Ar_2 + 1, Ar_3 + Ar_2)) [ Ar_1 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) l2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l1(Ar_0 + Ar_3, Ar_1 - 1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l0(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(l0) = 3*V_2 - 2 Pol(l1) = 3*V_2 - 2 Pol(l2) = 3*V_2 - 4 Pol(koat_start) = 3*V_2 - 2 orients all transitions weakly and the transition l1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l2(Ar_0, Ar_1, 0, 0)) [ Ar_1 >= 1 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) l0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l1(0, Ar_1, Ar_2, Ar_3)) (Comp: 3*Ar_1 + 2, Cost: 1) l1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l2(Ar_0, Ar_1, 0, 0)) [ Ar_1 >= 1 ] (Comp: ?, Cost: 1) l2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l2(Ar_0, Ar_1, Ar_2 + 1, Ar_3 + Ar_2)) [ Ar_1 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) l2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l1(Ar_0 + Ar_3, Ar_1 - 1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l0(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(l2) = 1 Pol(l1) = 0 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l0(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l0(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l0(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-2) = Ar_2 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l0(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-3) = Ar_3 S("l2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l1(Ar_0 + Ar_3, Ar_1 - 1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 ]", 0-0) = ? S("l2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l1(Ar_0 + Ar_3, Ar_1 - 1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 ]", 0-1) = ? S("l2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l1(Ar_0 + Ar_3, Ar_1 - 1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 ]", 0-2) = ? S("l2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l1(Ar_0 + Ar_3, Ar_1 - 1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 ]", 0-3) = ? S("l2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l2(Ar_0, Ar_1, Ar_2 + 1, Ar_3 + Ar_2)) [ Ar_1 >= Ar_2 + 1 ]", 0-0) = ? S("l2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l2(Ar_0, Ar_1, Ar_2 + 1, Ar_3 + Ar_2)) [ Ar_1 >= Ar_2 + 1 ]", 0-1) = ? S("l2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l2(Ar_0, Ar_1, Ar_2 + 1, Ar_3 + Ar_2)) [ Ar_1 >= Ar_2 + 1 ]", 0-2) = ? S("l2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l2(Ar_0, Ar_1, Ar_2 + 1, Ar_3 + Ar_2)) [ Ar_1 >= Ar_2 + 1 ]", 0-3) = ? S("l1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l2(Ar_0, Ar_1, 0, 0)) [ Ar_1 >= 1 ]", 0-0) = ? S("l1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l2(Ar_0, Ar_1, 0, 0)) [ Ar_1 >= 1 ]", 0-1) = ? S("l1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l2(Ar_0, Ar_1, 0, 0)) [ Ar_1 >= 1 ]", 0-2) = 0 S("l1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l2(Ar_0, Ar_1, 0, 0)) [ Ar_1 >= 1 ]", 0-3) = 0 S("l0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l1(0, Ar_1, Ar_2, Ar_3))", 0-0) = 0 S("l0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l1(0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("l0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l1(0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("l0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l1(0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 orients the transitions l2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l2(Ar_0, Ar_1, Ar_2 + 1, Ar_3 + Ar_2)) [ Ar_1 >= Ar_2 + 1 ] l2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l1(Ar_0 + Ar_3, Ar_1 - 1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 ] weakly and the transition l2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l1(Ar_0 + Ar_3, Ar_1 - 1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) l0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l1(0, Ar_1, Ar_2, Ar_3)) (Comp: 3*Ar_1 + 2, Cost: 1) l1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l2(Ar_0, Ar_1, 0, 0)) [ Ar_1 >= 1 ] (Comp: ?, Cost: 1) l2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l2(Ar_0, Ar_1, Ar_2 + 1, Ar_3 + Ar_2)) [ Ar_1 >= Ar_2 + 1 ] (Comp: 3*Ar_1 + 2, Cost: 1) l2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l1(Ar_0 + Ar_3, Ar_1 - 1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l0(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(l2) = V_2 - V_3 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l0(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l0(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l0(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-2) = Ar_2 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l0(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-3) = Ar_3 S("l2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l1(Ar_0 + Ar_3, Ar_1 - 1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 ]", 0-0) = ? S("l2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l1(Ar_0 + Ar_3, Ar_1 - 1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 ]", 0-1) = 4*Ar_1 + 32 S("l2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l1(Ar_0 + Ar_3, Ar_1 - 1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 ]", 0-2) = ? S("l2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l1(Ar_0 + Ar_3, Ar_1 - 1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 ]", 0-3) = ? S("l2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l2(Ar_0, Ar_1, Ar_2 + 1, Ar_3 + Ar_2)) [ Ar_1 >= Ar_2 + 1 ]", 0-0) = ? S("l2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l2(Ar_0, Ar_1, Ar_2 + 1, Ar_3 + Ar_2)) [ Ar_1 >= Ar_2 + 1 ]", 0-1) = 4*Ar_1 + 32 S("l2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l2(Ar_0, Ar_1, Ar_2 + 1, Ar_3 + Ar_2)) [ Ar_1 >= Ar_2 + 1 ]", 0-2) = ? S("l2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l2(Ar_0, Ar_1, Ar_2 + 1, Ar_3 + Ar_2)) [ Ar_1 >= Ar_2 + 1 ]", 0-3) = ? S("l1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l2(Ar_0, Ar_1, 0, 0)) [ Ar_1 >= 1 ]", 0-0) = ? S("l1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l2(Ar_0, Ar_1, 0, 0)) [ Ar_1 >= 1 ]", 0-1) = 4*Ar_1 + 32 S("l1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l2(Ar_0, Ar_1, 0, 0)) [ Ar_1 >= 1 ]", 0-2) = 0 S("l1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l2(Ar_0, Ar_1, 0, 0)) [ Ar_1 >= 1 ]", 0-3) = 0 S("l0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l1(0, Ar_1, Ar_2, Ar_3))", 0-0) = 0 S("l0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l1(0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("l0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l1(0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("l0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l1(0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 orients the transitions l2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l2(Ar_0, Ar_1, Ar_2 + 1, Ar_3 + Ar_2)) [ Ar_1 >= Ar_2 + 1 ] weakly and the transition l2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l2(Ar_0, Ar_1, Ar_2 + 1, Ar_3 + Ar_2)) [ Ar_1 >= Ar_2 + 1 ] strictly and produces the following problem: 5: T: (Comp: 1, Cost: 1) l0(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l1(0, Ar_1, Ar_2, Ar_3)) (Comp: 3*Ar_1 + 2, Cost: 1) l1(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l2(Ar_0, Ar_1, 0, 0)) [ Ar_1 >= 1 ] (Comp: 12*Ar_1^2 + 104*Ar_1 + 64, Cost: 1) l2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l2(Ar_0, Ar_1, Ar_2 + 1, Ar_3 + Ar_2)) [ Ar_1 >= Ar_2 + 1 ] (Comp: 3*Ar_1 + 2, Cost: 1) l2(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l1(Ar_0 + Ar_3, Ar_1 - 1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(l0(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 110*Ar_1 + 12*Ar_1^2 + 69 Time: 1.137 sec (SMT: 1.106 sec)