WORST_CASE(?, O(1)) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6) -> Com_1(f5(0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6)) (Comp: ?, Cost: 1) f5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6) -> Com_1(f5(Ar_0 + 1, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6)) [ 99 >= Ar_0 ] (Comp: ?, Cost: 1) f17(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6) -> Com_1(f17(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6)) (Comp: ?, Cost: 1) f17(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6) -> Com_1(f17(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6)) [ 0 >= H + 1 ] (Comp: ?, Cost: 1) f17(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6) -> Com_1(f17(Ar_0, Ar_1 + 1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6)) (Comp: ?, Cost: 1) f32(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6) -> Com_1(f32(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6)) (Comp: ?, Cost: 1) f32(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6) -> Com_1(f32(Ar_0, Ar_1, Ar_2 + 1, Ar_3, Ar_4, Ar_5, Ar_6)) [ 0 >= H + 1 ] (Comp: ?, Cost: 1) f32(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6) -> Com_1(f32(Ar_0, Ar_1, Ar_2 + 1, Ar_3, Ar_4, Ar_5, Ar_6)) (Comp: ?, Cost: 1) f32(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6) -> Com_1(f13(Ar_0, Ar_1, Ar_2, Ar_2, Ar_2, Ar_5, Ar_6)) (Comp: ?, Cost: 1) f17(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6) -> Com_1(f32(Ar_0, Ar_1, Ar_1, Ar_1, Ar_4, Ar_1, Fresh_2)) [ 0 >= I + 1 ] (Comp: ?, Cost: 1) f17(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6) -> Com_1(f32(Ar_0, Ar_1, Ar_1, Ar_1, Ar_4, Ar_1, Fresh_1)) (Comp: ?, Cost: 1) f17(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6) -> Com_1(f13(Ar_0, Ar_1, Ar_2, Ar_1, Ar_4, Ar_1, Fresh_0)) (Comp: ?, Cost: 1) f5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6) -> Com_1(f13(Ar_0, Ar_1, Ar_2, Ar_0 - 2, Ar_4, Ar_5, Ar_6)) [ Ar_0 >= 100 ] (Comp: ?, Cost: 1) f5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6) -> Com_1(f17(Ar_0, Ar_0 - 2, Ar_2, Ar_0 - 2, Ar_4, Ar_5, Ar_6)) [ 0 >= Ar_0 + 1 /\ Ar_0 >= 100 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6) -> Com_1(f0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4, Ar_5, Ar_6)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Slicing away variables that do not contribute to conditions from problem 1 leaves variables [Ar_0]. We thus obtain the following problem: 2: T: (Comp: 1, Cost: 0) koat_start(Ar_0) -> Com_1(f0(Ar_0)) [ 0 <= 0 ] (Comp: ?, Cost: 1) f5(Ar_0) -> Com_1(f17(Ar_0)) [ 0 >= Ar_0 + 1 /\ Ar_0 >= 100 ] (Comp: ?, Cost: 1) f5(Ar_0) -> Com_1(f13(Ar_0)) [ Ar_0 >= 100 ] (Comp: ?, Cost: 1) f17(Ar_0) -> Com_1(f13(Ar_0)) (Comp: ?, Cost: 1) f17(Ar_0) -> Com_1(f32(Ar_0)) (Comp: ?, Cost: 1) f17(Ar_0) -> Com_1(f32(Ar_0)) [ 0 >= I + 1 ] (Comp: ?, Cost: 1) f32(Ar_0) -> Com_1(f13(Ar_0)) (Comp: ?, Cost: 1) f32(Ar_0) -> Com_1(f32(Ar_0)) (Comp: ?, Cost: 1) f32(Ar_0) -> Com_1(f32(Ar_0)) [ 0 >= H + 1 ] (Comp: ?, Cost: 1) f32(Ar_0) -> Com_1(f32(Ar_0)) (Comp: ?, Cost: 1) f17(Ar_0) -> Com_1(f17(Ar_0)) (Comp: ?, Cost: 1) f17(Ar_0) -> Com_1(f17(Ar_0)) [ 0 >= H + 1 ] (Comp: ?, Cost: 1) f17(Ar_0) -> Com_1(f17(Ar_0)) (Comp: ?, Cost: 1) f5(Ar_0) -> Com_1(f5(Ar_0 + 1)) [ 99 >= Ar_0 ] (Comp: ?, Cost: 1) f0(Ar_0) -> Com_1(f5(0)) start location: koat_start leaf cost: 0 Testing for reachability in the complexity graph removes the following transitions from problem 2: f5(Ar_0) -> Com_1(f17(Ar_0)) [ 0 >= Ar_0 + 1 /\ Ar_0 >= 100 ] f17(Ar_0) -> Com_1(f13(Ar_0)) f17(Ar_0) -> Com_1(f32(Ar_0)) f17(Ar_0) -> Com_1(f32(Ar_0)) [ 0 >= I + 1 ] f32(Ar_0) -> Com_1(f13(Ar_0)) f32(Ar_0) -> Com_1(f32(Ar_0)) f32(Ar_0) -> Com_1(f32(Ar_0)) [ 0 >= H + 1 ] f32(Ar_0) -> Com_1(f32(Ar_0)) f17(Ar_0) -> Com_1(f17(Ar_0)) f17(Ar_0) -> Com_1(f17(Ar_0)) [ 0 >= H + 1 ] f17(Ar_0) -> Com_1(f17(Ar_0)) We thus obtain the following problem: 3: T: (Comp: ?, Cost: 1) f5(Ar_0) -> Com_1(f13(Ar_0)) [ Ar_0 >= 100 ] (Comp: ?, Cost: 1) f5(Ar_0) -> Com_1(f5(Ar_0 + 1)) [ 99 >= Ar_0 ] (Comp: ?, Cost: 1) f0(Ar_0) -> Com_1(f5(0)) (Comp: 1, Cost: 0) koat_start(Ar_0) -> Com_1(f0(Ar_0)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 3 produces the following problem: 4: T: (Comp: ?, Cost: 1) f5(Ar_0) -> Com_1(f13(Ar_0)) [ Ar_0 >= 100 ] (Comp: ?, Cost: 1) f5(Ar_0) -> Com_1(f5(Ar_0 + 1)) [ 99 >= Ar_0 ] (Comp: 1, Cost: 1) f0(Ar_0) -> Com_1(f5(0)) (Comp: 1, Cost: 0) koat_start(Ar_0) -> Com_1(f0(Ar_0)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f5) = 1 Pol(f13) = 0 Pol(f0) = 1 Pol(koat_start) = 1 orients all transitions weakly and the transition f5(Ar_0) -> Com_1(f13(Ar_0)) [ Ar_0 >= 100 ] strictly and produces the following problem: 5: T: (Comp: 1, Cost: 1) f5(Ar_0) -> Com_1(f13(Ar_0)) [ Ar_0 >= 100 ] (Comp: ?, Cost: 1) f5(Ar_0) -> Com_1(f5(Ar_0 + 1)) [ 99 >= Ar_0 ] (Comp: 1, Cost: 1) f0(Ar_0) -> Com_1(f5(0)) (Comp: 1, Cost: 0) koat_start(Ar_0) -> Com_1(f0(Ar_0)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f5) = -V_1 + 100 Pol(f13) = -V_1 + 100 Pol(f0) = 100 Pol(koat_start) = 100 orients all transitions weakly and the transition f5(Ar_0) -> Com_1(f5(Ar_0 + 1)) [ 99 >= Ar_0 ] strictly and produces the following problem: 6: T: (Comp: 1, Cost: 1) f5(Ar_0) -> Com_1(f13(Ar_0)) [ Ar_0 >= 100 ] (Comp: 100, Cost: 1) f5(Ar_0) -> Com_1(f5(Ar_0 + 1)) [ 99 >= Ar_0 ] (Comp: 1, Cost: 1) f0(Ar_0) -> Com_1(f5(0)) (Comp: 1, Cost: 0) koat_start(Ar_0) -> Com_1(f0(Ar_0)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 102 Time: 0.838 sec (SMT: 0.818 sec)