WORST_CASE(?, O(n^2)) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f3(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_1 >= Ar_2 + 2 /\ Ar_1 >= Ar_0 ] (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: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f3(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_1 >= Ar_2 + 2 /\ Ar_1 >= Ar_0 ] (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(f1) = 5*V_1 - 5*V_2 - 4 Pol(f3) = 5*V_1 - 5*V_2 Pol(koat_start) = 5*V_1 - 5*V_2 orients all transitions weakly and the transition f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ] strictly and produces the following problem: 3: T: (Comp: 5*Ar_0 + 5*Ar_1, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f3(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_1 >= Ar_2 + 2 /\ Ar_1 >= Ar_0 ] (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(f1) = V_2 - V_3 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(f3(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-2) = Ar_2 S("f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_1 >= Ar_2 + 2 /\\ Ar_1 >= Ar_0 ]", 0-0) = Ar_0 S("f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_1 >= Ar_2 + 2 /\\ Ar_1 >= Ar_0 ]", 0-1) = 6*Ar_0 + 6*Ar_1 S("f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_1 >= Ar_2 + 2 /\\ Ar_1 >= Ar_0 ]", 0-2) = ? S("f3(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ]", 0-0) = Ar_0 S("f3(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ]", 0-1) = Ar_1 S("f3(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ]", 0-2) = Ar_2 S("f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ]", 0-1) = 6*Ar_0 + 6*Ar_1 S("f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ]", 0-2) = Ar_2 orients the transitions f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_1 >= Ar_2 + 2 /\ Ar_1 >= Ar_0 ] weakly and the transition f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_1 >= Ar_2 + 2 /\ Ar_1 >= Ar_0 ] strictly and produces the following problem: 4: T: (Comp: 5*Ar_0 + 5*Ar_1, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1 + 1, Ar_2)) [ Ar_0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f3(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Ar_2)) [ Ar_1 >= Ar_2 + 1 ] (Comp: Ar_1 + Ar_2 + 30*Ar_0^2 + 60*Ar_0*Ar_1 + 30*Ar_1^2 + 5*Ar_0*Ar_2 + 5*Ar_1*Ar_2, Cost: 1) f1(Ar_0, Ar_1, Ar_2) -> Com_1(f1(Ar_0, Ar_1, Ar_2 + 1)) [ Ar_1 >= Ar_2 + 2 /\ Ar_1 >= Ar_0 ] (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 5*Ar_0 + 6*Ar_1 + Ar_2 + 30*Ar_0^2 + 60*Ar_0*Ar_1 + 30*Ar_1^2 + 5*Ar_0*Ar_2 + 5*Ar_1*Ar_2 + 1 Time: 0.462 sec (SMT: 0.444 sec)