WORST_CASE(?, O(n^2)) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f1(Ar_0, Ar_1) -> Com_1(f3(Ar_0, Ar_0)) [ Ar_0 >= 1 ] (Comp: ?, Cost: 1) f3(Ar_0, Ar_1) -> Com_1(f3(Ar_0, Ar_1 - 1)) [ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f3(Ar_0, Ar_1) -> Com_1(f3(Ar_0 - 1, Ar_0 - 1)) [ 0 >= Ar_1 /\ Ar_0 >= 2 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f1(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) f1(Ar_0, Ar_1) -> Com_1(f3(Ar_0, Ar_0)) [ Ar_0 >= 1 ] (Comp: ?, Cost: 1) f3(Ar_0, Ar_1) -> Com_1(f3(Ar_0, Ar_1 - 1)) [ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f3(Ar_0, Ar_1) -> Com_1(f3(Ar_0 - 1, Ar_0 - 1)) [ 0 >= Ar_1 /\ Ar_0 >= 2 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f1(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f1) = V_1 Pol(f3) = V_1 Pol(koat_start) = V_1 orients all transitions weakly and the transition f3(Ar_0, Ar_1) -> Com_1(f3(Ar_0 - 1, Ar_0 - 1)) [ 0 >= Ar_1 /\ Ar_0 >= 2 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) f1(Ar_0, Ar_1) -> Com_1(f3(Ar_0, Ar_0)) [ Ar_0 >= 1 ] (Comp: ?, Cost: 1) f3(Ar_0, Ar_1) -> Com_1(f3(Ar_0, Ar_1 - 1)) [ Ar_1 >= 1 ] (Comp: Ar_0, Cost: 1) f3(Ar_0, Ar_1) -> Com_1(f3(Ar_0 - 1, Ar_0 - 1)) [ 0 >= Ar_1 /\ Ar_0 >= 2 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f1(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f3) = V_2 and size complexities S("koat_start(Ar_0, Ar_1) -> Com_1(f1(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1) -> Com_1(f1(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-1) = Ar_1 S("f3(Ar_0, Ar_1) -> Com_1(f3(Ar_0 - 1, Ar_0 - 1)) [ 0 >= Ar_1 /\\ Ar_0 >= 2 ]", 0-0) = Ar_0 S("f3(Ar_0, Ar_1) -> Com_1(f3(Ar_0 - 1, Ar_0 - 1)) [ 0 >= Ar_1 /\\ Ar_0 >= 2 ]", 0-1) = Ar_0 S("f3(Ar_0, Ar_1) -> Com_1(f3(Ar_0, Ar_1 - 1)) [ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f3(Ar_0, Ar_1) -> Com_1(f3(Ar_0, Ar_1 - 1)) [ Ar_1 >= 1 ]", 0-1) = Ar_0 S("f1(Ar_0, Ar_1) -> Com_1(f3(Ar_0, Ar_0)) [ Ar_0 >= 1 ]", 0-0) = Ar_0 S("f1(Ar_0, Ar_1) -> Com_1(f3(Ar_0, Ar_0)) [ Ar_0 >= 1 ]", 0-1) = Ar_0 orients the transitions f3(Ar_0, Ar_1) -> Com_1(f3(Ar_0, Ar_1 - 1)) [ Ar_1 >= 1 ] weakly and the transition f3(Ar_0, Ar_1) -> Com_1(f3(Ar_0, Ar_1 - 1)) [ Ar_1 >= 1 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) f1(Ar_0, Ar_1) -> Com_1(f3(Ar_0, Ar_0)) [ Ar_0 >= 1 ] (Comp: Ar_0^2 + Ar_0, Cost: 1) f3(Ar_0, Ar_1) -> Com_1(f3(Ar_0, Ar_1 - 1)) [ Ar_1 >= 1 ] (Comp: Ar_0, Cost: 1) f3(Ar_0, Ar_1) -> Com_1(f3(Ar_0 - 1, Ar_0 - 1)) [ 0 >= Ar_1 /\ Ar_0 >= 2 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f1(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound Ar_0^2 + 2*Ar_0 + 1 Time: 0.847 sec (SMT: 0.824 sec)