/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.koat /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(1)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1). (0) CpxIntTrs (1) Koat Proof [FINISHED, 13 ms] (2) BOUNDS(1, 1) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_easy1_start(v_0, v_x_0) -> Com_1(eval_easy1_bb0_in(v_0, v_x_0)) :|: TRUE eval_easy1_bb0_in(v_0, v_x_0) -> Com_1(eval_easy1_1(nondef_0, v_x_0)) :|: TRUE eval_easy1_1(v_0, v_x_0) -> Com_1(eval_easy1_2(v_0, v_x_0)) :|: TRUE eval_easy1_2(v_0, v_x_0) -> Com_1(eval_easy1_3(v_0, v_x_0)) :|: TRUE eval_easy1_3(v_0, v_x_0) -> Com_1(eval_easy1_4(v_0, v_x_0)) :|: TRUE eval_easy1_4(v_0, v_x_0) -> Com_1(eval_easy1_5(v_0, v_x_0)) :|: TRUE eval_easy1_5(v_0, v_x_0) -> Com_1(eval_easy1_6(v_0, v_x_0)) :|: TRUE eval_easy1_6(v_0, v_x_0) -> Com_1(eval_easy1_bb1_in(v_0, 0)) :|: TRUE eval_easy1_bb1_in(v_0, v_x_0) -> Com_1(eval_easy1_bb2_in(v_0, v_x_0)) :|: v_x_0 < 40 eval_easy1_bb1_in(v_0, v_x_0) -> Com_1(eval_easy1_bb3_in(v_0, v_x_0)) :|: v_x_0 >= 40 eval_easy1_bb2_in(v_0, v_x_0) -> Com_1(eval_easy1_bb1_in(v_0, v_x_0 + 1)) :|: v_0 >= 0 && v_0 <= 0 eval_easy1_bb2_in(v_0, v_x_0) -> Com_1(eval_easy1_bb1_in(v_0, v_x_0 + 2)) :|: v_0 < 0 eval_easy1_bb2_in(v_0, v_x_0) -> Com_1(eval_easy1_bb1_in(v_0, v_x_0 + 2)) :|: v_0 > 0 eval_easy1_bb3_in(v_0, v_x_0) -> Com_1(eval_easy1_stop(v_0, v_x_0)) :|: TRUE The start-symbols are:[eval_easy1_start_2] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 172) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evaleasy1start(Ar_0, Ar_1) -> Com_1(evaleasy1bb0in(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evaleasy1bb0in(Ar_0, Ar_1) -> Com_1(evaleasy11(Fresh_0, Ar_1)) (Comp: ?, Cost: 1) evaleasy11(Ar_0, Ar_1) -> Com_1(evaleasy12(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evaleasy12(Ar_0, Ar_1) -> Com_1(evaleasy13(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evaleasy13(Ar_0, Ar_1) -> Com_1(evaleasy14(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evaleasy14(Ar_0, Ar_1) -> Com_1(evaleasy15(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evaleasy15(Ar_0, Ar_1) -> Com_1(evaleasy16(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evaleasy16(Ar_0, Ar_1) -> Com_1(evaleasy1bb1in(Ar_0, 0)) (Comp: ?, Cost: 1) evaleasy1bb1in(Ar_0, Ar_1) -> Com_1(evaleasy1bb2in(Ar_0, Ar_1)) [ 39 >= Ar_1 ] (Comp: ?, Cost: 1) evaleasy1bb1in(Ar_0, Ar_1) -> Com_1(evaleasy1bb3in(Ar_0, Ar_1)) [ Ar_1 >= 40 ] (Comp: ?, Cost: 1) evaleasy1bb2in(Ar_0, Ar_1) -> Com_1(evaleasy1bb1in(Ar_0, Ar_1 + 1)) [ Ar_0 = 0 ] (Comp: ?, Cost: 1) evaleasy1bb2in(Ar_0, Ar_1) -> Com_1(evaleasy1bb1in(Ar_0, Ar_1 + 2)) [ 0 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evaleasy1bb2in(Ar_0, Ar_1) -> Com_1(evaleasy1bb1in(Ar_0, Ar_1 + 2)) [ Ar_0 >= 1 ] (Comp: ?, Cost: 1) evaleasy1bb3in(Ar_0, Ar_1) -> Com_1(evaleasy1stop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evaleasy1start(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) evaleasy1start(Ar_0, Ar_1) -> Com_1(evaleasy1bb0in(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy1bb0in(Ar_0, Ar_1) -> Com_1(evaleasy11(Fresh_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy11(Ar_0, Ar_1) -> Com_1(evaleasy12(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy12(Ar_0, Ar_1) -> Com_1(evaleasy13(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy13(Ar_0, Ar_1) -> Com_1(evaleasy14(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy14(Ar_0, Ar_1) -> Com_1(evaleasy15(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy15(Ar_0, Ar_1) -> Com_1(evaleasy16(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy16(Ar_0, Ar_1) -> Com_1(evaleasy1bb1in(Ar_0, 0)) (Comp: ?, Cost: 1) evaleasy1bb1in(Ar_0, Ar_1) -> Com_1(evaleasy1bb2in(Ar_0, Ar_1)) [ 39 >= Ar_1 ] (Comp: ?, Cost: 1) evaleasy1bb1in(Ar_0, Ar_1) -> Com_1(evaleasy1bb3in(Ar_0, Ar_1)) [ Ar_1 >= 40 ] (Comp: ?, Cost: 1) evaleasy1bb2in(Ar_0, Ar_1) -> Com_1(evaleasy1bb1in(Ar_0, Ar_1 + 1)) [ Ar_0 = 0 ] (Comp: ?, Cost: 1) evaleasy1bb2in(Ar_0, Ar_1) -> Com_1(evaleasy1bb1in(Ar_0, Ar_1 + 2)) [ 0 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evaleasy1bb2in(Ar_0, Ar_1) -> Com_1(evaleasy1bb1in(Ar_0, Ar_1 + 2)) [ Ar_0 >= 1 ] (Comp: ?, Cost: 1) evaleasy1bb3in(Ar_0, Ar_1) -> Com_1(evaleasy1stop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evaleasy1start(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evaleasy1start) = 2 Pol(evaleasy1bb0in) = 2 Pol(evaleasy11) = 2 Pol(evaleasy12) = 2 Pol(evaleasy13) = 2 Pol(evaleasy14) = 2 Pol(evaleasy15) = 2 Pol(evaleasy16) = 2 Pol(evaleasy1bb1in) = 2 Pol(evaleasy1bb2in) = 2 Pol(evaleasy1bb3in) = 1 Pol(evaleasy1stop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evaleasy1bb3in(Ar_0, Ar_1) -> Com_1(evaleasy1stop(Ar_0, Ar_1)) evaleasy1bb1in(Ar_0, Ar_1) -> Com_1(evaleasy1bb3in(Ar_0, Ar_1)) [ Ar_1 >= 40 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evaleasy1start(Ar_0, Ar_1) -> Com_1(evaleasy1bb0in(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy1bb0in(Ar_0, Ar_1) -> Com_1(evaleasy11(Fresh_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy11(Ar_0, Ar_1) -> Com_1(evaleasy12(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy12(Ar_0, Ar_1) -> Com_1(evaleasy13(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy13(Ar_0, Ar_1) -> Com_1(evaleasy14(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy14(Ar_0, Ar_1) -> Com_1(evaleasy15(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy15(Ar_0, Ar_1) -> Com_1(evaleasy16(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy16(Ar_0, Ar_1) -> Com_1(evaleasy1bb1in(Ar_0, 0)) (Comp: ?, Cost: 1) evaleasy1bb1in(Ar_0, Ar_1) -> Com_1(evaleasy1bb2in(Ar_0, Ar_1)) [ 39 >= Ar_1 ] (Comp: 2, Cost: 1) evaleasy1bb1in(Ar_0, Ar_1) -> Com_1(evaleasy1bb3in(Ar_0, Ar_1)) [ Ar_1 >= 40 ] (Comp: ?, Cost: 1) evaleasy1bb2in(Ar_0, Ar_1) -> Com_1(evaleasy1bb1in(Ar_0, Ar_1 + 1)) [ Ar_0 = 0 ] (Comp: ?, Cost: 1) evaleasy1bb2in(Ar_0, Ar_1) -> Com_1(evaleasy1bb1in(Ar_0, Ar_1 + 2)) [ 0 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evaleasy1bb2in(Ar_0, Ar_1) -> Com_1(evaleasy1bb1in(Ar_0, Ar_1 + 2)) [ Ar_0 >= 1 ] (Comp: 2, Cost: 1) evaleasy1bb3in(Ar_0, Ar_1) -> Com_1(evaleasy1stop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evaleasy1start(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evaleasy1start) = 40 Pol(evaleasy1bb0in) = 40 Pol(evaleasy11) = 40 Pol(evaleasy12) = 40 Pol(evaleasy13) = 40 Pol(evaleasy14) = 40 Pol(evaleasy15) = 40 Pol(evaleasy16) = 40 Pol(evaleasy1bb1in) = -V_2 + 40 Pol(evaleasy1bb2in) = -V_2 + 39 Pol(evaleasy1bb3in) = -V_2 Pol(evaleasy1stop) = -V_2 Pol(koat_start) = 40 orients all transitions weakly and the transition evaleasy1bb1in(Ar_0, Ar_1) -> Com_1(evaleasy1bb2in(Ar_0, Ar_1)) [ 39 >= Ar_1 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) evaleasy1start(Ar_0, Ar_1) -> Com_1(evaleasy1bb0in(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy1bb0in(Ar_0, Ar_1) -> Com_1(evaleasy11(Fresh_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy11(Ar_0, Ar_1) -> Com_1(evaleasy12(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy12(Ar_0, Ar_1) -> Com_1(evaleasy13(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy13(Ar_0, Ar_1) -> Com_1(evaleasy14(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy14(Ar_0, Ar_1) -> Com_1(evaleasy15(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy15(Ar_0, Ar_1) -> Com_1(evaleasy16(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy16(Ar_0, Ar_1) -> Com_1(evaleasy1bb1in(Ar_0, 0)) (Comp: 40, Cost: 1) evaleasy1bb1in(Ar_0, Ar_1) -> Com_1(evaleasy1bb2in(Ar_0, Ar_1)) [ 39 >= Ar_1 ] (Comp: 2, Cost: 1) evaleasy1bb1in(Ar_0, Ar_1) -> Com_1(evaleasy1bb3in(Ar_0, Ar_1)) [ Ar_1 >= 40 ] (Comp: ?, Cost: 1) evaleasy1bb2in(Ar_0, Ar_1) -> Com_1(evaleasy1bb1in(Ar_0, Ar_1 + 1)) [ Ar_0 = 0 ] (Comp: ?, Cost: 1) evaleasy1bb2in(Ar_0, Ar_1) -> Com_1(evaleasy1bb1in(Ar_0, Ar_1 + 2)) [ 0 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evaleasy1bb2in(Ar_0, Ar_1) -> Com_1(evaleasy1bb1in(Ar_0, Ar_1 + 2)) [ Ar_0 >= 1 ] (Comp: 2, Cost: 1) evaleasy1bb3in(Ar_0, Ar_1) -> Com_1(evaleasy1stop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evaleasy1start(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 4 produces the following problem: 5: T: (Comp: 1, Cost: 1) evaleasy1start(Ar_0, Ar_1) -> Com_1(evaleasy1bb0in(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy1bb0in(Ar_0, Ar_1) -> Com_1(evaleasy11(Fresh_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy11(Ar_0, Ar_1) -> Com_1(evaleasy12(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy12(Ar_0, Ar_1) -> Com_1(evaleasy13(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy13(Ar_0, Ar_1) -> Com_1(evaleasy14(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy14(Ar_0, Ar_1) -> Com_1(evaleasy15(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy15(Ar_0, Ar_1) -> Com_1(evaleasy16(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evaleasy16(Ar_0, Ar_1) -> Com_1(evaleasy1bb1in(Ar_0, 0)) (Comp: 40, Cost: 1) evaleasy1bb1in(Ar_0, Ar_1) -> Com_1(evaleasy1bb2in(Ar_0, Ar_1)) [ 39 >= Ar_1 ] (Comp: 2, Cost: 1) evaleasy1bb1in(Ar_0, Ar_1) -> Com_1(evaleasy1bb3in(Ar_0, Ar_1)) [ Ar_1 >= 40 ] (Comp: 40, Cost: 1) evaleasy1bb2in(Ar_0, Ar_1) -> Com_1(evaleasy1bb1in(Ar_0, Ar_1 + 1)) [ Ar_0 = 0 ] (Comp: 40, Cost: 1) evaleasy1bb2in(Ar_0, Ar_1) -> Com_1(evaleasy1bb1in(Ar_0, Ar_1 + 2)) [ 0 >= Ar_0 + 1 ] (Comp: 40, Cost: 1) evaleasy1bb2in(Ar_0, Ar_1) -> Com_1(evaleasy1bb1in(Ar_0, Ar_1 + 2)) [ Ar_0 >= 1 ] (Comp: 2, Cost: 1) evaleasy1bb3in(Ar_0, Ar_1) -> Com_1(evaleasy1stop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evaleasy1start(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 172 Time: 0.087 sec (SMT: 0.069 sec) ---------------------------------------- (2) BOUNDS(1, 1)