WORST_CASE(?, O(n^2)) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f300(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0 - 99, 0, Ar_2)) [ 0 >= Ar_0 + 1 /\ Ar_1 + 1 = 0 ] (Comp: ?, Cost: 1) f300(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0 + 1, Ar_1 + 1, Ar_2)) [ 0 >= Ar_0 + 1 /\ Ar_1 >= 0 ] (Comp: ?, Cost: 1) f300(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0 + 1, Ar_1 + 1, Ar_2)) [ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + 2 ] (Comp: ?, Cost: 1) f300(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Fresh_0)) [ Ar_0 >= 0 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f2(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: ?, Cost: 1) f300(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0 - 99, 0, Ar_2)) [ 0 >= Ar_0 + 1 /\ Ar_1 + 1 = 0 ] (Comp: ?, Cost: 1) f300(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0 + 1, Ar_1 + 1, Ar_2)) [ 0 >= Ar_0 + 1 /\ Ar_1 >= 0 ] (Comp: ?, Cost: 1) f300(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0 + 1, Ar_1 + 1, Ar_2)) [ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + 2 ] (Comp: ?, Cost: 1) f300(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Fresh_0)) [ Ar_0 >= 0 ] (Comp: 1, Cost: 1) f2(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f300) = 1 Pol(f1) = 0 Pol(f2) = 1 Pol(koat_start) = 1 orients all transitions weakly and the transition f300(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Fresh_0)) [ Ar_0 >= 0 ] strictly and produces the following problem: 3: T: (Comp: ?, Cost: 1) f300(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0 - 99, 0, Ar_2)) [ 0 >= Ar_0 + 1 /\ Ar_1 + 1 = 0 ] (Comp: ?, Cost: 1) f300(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0 + 1, Ar_1 + 1, Ar_2)) [ 0 >= Ar_0 + 1 /\ Ar_1 >= 0 ] (Comp: ?, Cost: 1) f300(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0 + 1, Ar_1 + 1, Ar_2)) [ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + 2 ] (Comp: 1, Cost: 1) f300(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Fresh_0)) [ Ar_0 >= 0 ] (Comp: 1, Cost: 1) f2(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f300) = -V_1 - 198*V_2 Pol(f1) = -V_1 - 198*V_2 Pol(f2) = -V_1 - 198*V_2 Pol(koat_start) = -V_1 - 198*V_2 orients all transitions weakly and the transitions f300(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0 + 1, Ar_1 + 1, Ar_2)) [ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + 2 ] f300(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0 - 99, 0, Ar_2)) [ 0 >= Ar_0 + 1 /\ Ar_1 + 1 = 0 ] strictly and produces the following problem: 4: T: (Comp: Ar_0 + 198*Ar_1, Cost: 1) f300(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0 - 99, 0, Ar_2)) [ 0 >= Ar_0 + 1 /\ Ar_1 + 1 = 0 ] (Comp: ?, Cost: 1) f300(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0 + 1, Ar_1 + 1, Ar_2)) [ 0 >= Ar_0 + 1 /\ Ar_1 >= 0 ] (Comp: Ar_0 + 198*Ar_1, Cost: 1) f300(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0 + 1, Ar_1 + 1, Ar_2)) [ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + 2 ] (Comp: 1, Cost: 1) f300(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Fresh_0)) [ Ar_0 >= 0 ] (Comp: 1, Cost: 1) f2(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f300) = -V_1 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-2) = Ar_2 S("f2(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_0 S("f2(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("f2(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0, Ar_1, Ar_2))", 0-2) = Ar_2 S("f300(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Fresh_0)) [ Ar_0 >= 0 ]", 0-0) = Ar_0 + 99 S("f300(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Fresh_0)) [ Ar_0 >= 0 ]", 0-1) = ? S("f300(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Fresh_0)) [ Ar_0 >= 0 ]", 0-2) = ? S("f300(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0 + 1, Ar_1 + 1, Ar_2)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 2 ]", 0-0) = Ar_0 S("f300(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0 + 1, Ar_1 + 1, Ar_2)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 2 ]", 0-1) = Ar_1 S("f300(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0 + 1, Ar_1 + 1, Ar_2)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 2 ]", 0-2) = Ar_2 S("f300(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0 + 1, Ar_1 + 1, Ar_2)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 0 ]", 0-0) = Ar_0 + 99 S("f300(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0 + 1, Ar_1 + 1, Ar_2)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 0 ]", 0-1) = ? S("f300(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0 + 1, Ar_1 + 1, Ar_2)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 0 ]", 0-2) = Ar_2 S("f300(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0 - 99, 0, Ar_2)) [ 0 >= Ar_0 + 1 /\\ Ar_1 + 1 = 0 ]", 0-0) = Ar_0 + 99 S("f300(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0 - 99, 0, Ar_2)) [ 0 >= Ar_0 + 1 /\\ Ar_1 + 1 = 0 ]", 0-1) = 0 S("f300(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0 - 99, 0, Ar_2)) [ 0 >= Ar_0 + 1 /\\ Ar_1 + 1 = 0 ]", 0-2) = Ar_2 orients the transitions f300(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0 + 1, Ar_1 + 1, Ar_2)) [ 0 >= Ar_0 + 1 /\ Ar_1 >= 0 ] weakly and the transition f300(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0 + 1, Ar_1 + 1, Ar_2)) [ 0 >= Ar_0 + 1 /\ Ar_1 >= 0 ] strictly and produces the following problem: 5: T: (Comp: Ar_0 + 198*Ar_1, Cost: 1) f300(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0 - 99, 0, Ar_2)) [ 0 >= Ar_0 + 1 /\ Ar_1 + 1 = 0 ] (Comp: 100*Ar_0 + Ar_0^2 + 198*Ar_0*Ar_1 + 19602*Ar_1, Cost: 1) f300(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0 + 1, Ar_1 + 1, Ar_2)) [ 0 >= Ar_0 + 1 /\ Ar_1 >= 0 ] (Comp: Ar_0 + 198*Ar_1, Cost: 1) f300(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0 + 1, Ar_1 + 1, Ar_2)) [ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + 2 ] (Comp: 1, Cost: 1) f300(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Fresh_0)) [ Ar_0 >= 0 ] (Comp: 1, Cost: 1) f2(Ar_0, Ar_1, Ar_2) -> Com_1(f300(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f2(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 102*Ar_0 + 19998*Ar_1 + Ar_0^2 + 198*Ar_0*Ar_1 + 2 Time: 1.188 sec (SMT: 1.157 sec)