/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.koat /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox/benchmark/theBenchmark.koat # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1). (0) CpxIntTrs (1) Koat Proof [FINISHED, 118 ms] (2) BOUNDS(1, n^1) (3) Loat Proof [FINISHED, 619 ms] (4) BOUNDS(n^1, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_start_start(v__0, v__1, v_flag_0, v_flag_1, v_n) -> Com_1(eval_start_bb0_in(v__0, v__1, v_flag_0, v_flag_1, v_n)) :|: TRUE eval_start_bb0_in(v__0, v__1, v_flag_0, v_flag_1, v_n) -> Com_1(eval_start_0(v__0, v__1, v_flag_0, v_flag_1, v_n)) :|: TRUE eval_start_0(v__0, v__1, v_flag_0, v_flag_1, v_n) -> Com_1(eval_start_1(v__0, v__1, v_flag_0, v_flag_1, v_n)) :|: TRUE eval_start_1(v__0, v__1, v_flag_0, v_flag_1, v_n) -> Com_1(eval_start_2(v__0, v__1, v_flag_0, v_flag_1, v_n)) :|: TRUE eval_start_2(v__0, v__1, v_flag_0, v_flag_1, v_n) -> Com_1(eval_start_3(v__0, v__1, v_flag_0, v_flag_1, v_n)) :|: TRUE eval_start_3(v__0, v__1, v_flag_0, v_flag_1, v_n) -> Com_1(eval_start_4(v__0, v__1, v_flag_0, v_flag_1, v_n)) :|: TRUE eval_start_4(v__0, v__1, v_flag_0, v_flag_1, v_n) -> Com_1(eval_start_5(v__0, v__1, v_flag_0, v_flag_1, v_n)) :|: TRUE eval_start_5(v__0, v__1, v_flag_0, v_flag_1, v_n) -> Com_1(eval_start_bb1_in(v_n, v__1, 1, v_flag_1, v_n)) :|: TRUE eval_start_bb1_in(v__0, v__1, v_flag_0, v_flag_1, v_n) -> Com_1(eval_start_bb2_in(v__0, v__0, v_flag_0, 0, v_n)) :|: v_flag_0 > 0 eval_start_bb1_in(v__0, v__1, v_flag_0, v_flag_1, v_n) -> Com_1(eval_start_bb4_in(v__0, v__1, v_flag_0, v_flag_1, v_n)) :|: v_flag_0 <= 0 eval_start_bb2_in(v__0, v__1, v_flag_0, v_flag_1, v_n) -> Com_1(eval_start_bb3_in(v__0, v__1, v_flag_0, v_flag_1, v_n)) :|: v__1 > 0 eval_start_bb2_in(v__0, v__1, v_flag_0, v_flag_1, v_n) -> Com_1(eval_start_bb1_in(v__1, v__1, v_flag_1, v_flag_1, v_n)) :|: v__1 <= 0 eval_start_bb3_in(v__0, v__1, v_flag_0, v_flag_1, v_n) -> Com_1(eval_start_bb2_in(v__0, v__1 - 1, v_flag_0, 1, v_n)) :|: TRUE eval_start_bb4_in(v__0, v__1, v_flag_0, v_flag_1, v_n) -> Com_1(eval_start_stop(v__0, v__1, v_flag_0, v_flag_1, v_n)) :|: TRUE The start-symbols are:[eval_start_start_5] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 5*Ar_1 + 14) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalstart4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalstart5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_1, Ar_1, 1, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_0, 0)) [ Ar_2 >= 1 ] (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_2 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= 1 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_3, Ar_1, Ar_4, Ar_3, Ar_4)) [ 0 >= Ar_3 ] (Comp: ?, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1, 1)) (Comp: ?, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 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) evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_1, Ar_1, 1, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_0, 0)) [ Ar_2 >= 1 ] (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_2 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= 1 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_3, Ar_1, Ar_4, Ar_3, Ar_4)) [ 0 >= Ar_3 ] (Comp: ?, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1, 1)) (Comp: ?, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalstartstart) = 2 Pol(evalstartbb0in) = 2 Pol(evalstart0) = 2 Pol(evalstart1) = 2 Pol(evalstart2) = 2 Pol(evalstart3) = 2 Pol(evalstart4) = 2 Pol(evalstart5) = 2 Pol(evalstartbb1in) = 2 Pol(evalstartbb2in) = 2 Pol(evalstartbb4in) = 1 Pol(evalstartbb3in) = 2 Pol(evalstartstop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_2 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_1, Ar_1, 1, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_0, 0)) [ Ar_2 >= 1 ] (Comp: 2, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_2 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= 1 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_3, Ar_1, Ar_4, Ar_3, Ar_4)) [ 0 >= Ar_3 ] (Comp: ?, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1, 1)) (Comp: 2, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalstartstart) = V_2 Pol(evalstartbb0in) = V_2 Pol(evalstart0) = V_2 Pol(evalstart1) = V_2 Pol(evalstart2) = V_2 Pol(evalstart3) = V_2 Pol(evalstart4) = V_2 Pol(evalstart5) = V_2 Pol(evalstartbb1in) = V_1 Pol(evalstartbb2in) = V_4 Pol(evalstartbb4in) = V_1 Pol(evalstartbb3in) = V_4 - 1 Pol(evalstartstop) = V_1 Pol(koat_start) = V_2 orients all transitions weakly and the transition evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= 1 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_1, Ar_1, 1, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_0, 0)) [ Ar_2 >= 1 ] (Comp: 2, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_2 ] (Comp: Ar_1, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= 1 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_3, Ar_1, Ar_4, Ar_3, Ar_4)) [ 0 >= Ar_3 ] (Comp: ?, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1, 1)) (Comp: 2, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 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) evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_1, Ar_1, 1, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_0, 0)) [ Ar_2 >= 1 ] (Comp: 2, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_2 ] (Comp: Ar_1, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= 1 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_3, Ar_1, Ar_4, Ar_3, Ar_4)) [ 0 >= Ar_3 ] (Comp: Ar_1, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1, 1)) (Comp: 2, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalstartbb2in) = V_5 Pol(evalstartbb1in) = V_3 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]", 0-2) = Ar_2 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]", 0-3) = Ar_3 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ]", 0-4) = Ar_4 S("evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-0) = 2*Ar_1 S("evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-1) = Ar_1 S("evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-2) = 1 S("evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-3) = 2*Ar_1 S("evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-4) = 1 S("evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1, 1))", 0-0) = 2*Ar_1 S("evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1, 1))", 0-1) = Ar_1 S("evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1, 1))", 0-2) = 1 S("evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1, 1))", 0-3) = 2*Ar_1 S("evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1, 1))", 0-4) = 1 S("evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_3, Ar_1, Ar_4, Ar_3, Ar_4)) [ 0 >= Ar_3 ]", 0-0) = 2*Ar_1 S("evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_3, Ar_1, Ar_4, Ar_3, Ar_4)) [ 0 >= Ar_3 ]", 0-1) = Ar_1 S("evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_3, Ar_1, Ar_4, Ar_3, Ar_4)) [ 0 >= Ar_3 ]", 0-2) = 1 S("evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_3, Ar_1, Ar_4, Ar_3, Ar_4)) [ 0 >= Ar_3 ]", 0-3) = 2*Ar_1 S("evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_3, Ar_1, Ar_4, Ar_3, Ar_4)) [ 0 >= Ar_3 ]", 0-4) = 1 S("evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= 1 ]", 0-0) = 2*Ar_1 S("evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= 1 ]", 0-1) = Ar_1 S("evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= 1 ]", 0-2) = 1 S("evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= 1 ]", 0-3) = 2*Ar_1 S("evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= 1 ]", 0-4) = 1 S("evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_2 ]", 0-0) = 2*Ar_1 S("evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_2 ]", 0-1) = Ar_1 S("evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_2 ]", 0-2) = 1 S("evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_2 ]", 0-3) = 2*Ar_1 S("evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_2 ]", 0-4) = 1 S("evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_0, 0)) [ Ar_2 >= 1 ]", 0-0) = 2*Ar_1 S("evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_0, 0)) [ Ar_2 >= 1 ]", 0-1) = Ar_1 S("evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_0, 0)) [ Ar_2 >= 1 ]", 0-2) = 1 S("evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_0, 0)) [ Ar_2 >= 1 ]", 0-3) = 2*Ar_1 S("evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_0, 0)) [ Ar_2 >= 1 ]", 0-4) = 0 S("evalstart5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_1, Ar_1, 1, Ar_3, Ar_4))", 0-0) = Ar_1 S("evalstart5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_1, Ar_1, 1, Ar_3, Ar_4))", 0-1) = Ar_1 S("evalstart5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_1, Ar_1, 1, Ar_3, Ar_4))", 0-2) = 1 S("evalstart5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_1, Ar_1, 1, Ar_3, Ar_4))", 0-3) = Ar_3 S("evalstart5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_1, Ar_1, 1, Ar_3, Ar_4))", 0-4) = Ar_4 S("evalstart4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-0) = Ar_0 S("evalstart4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-1) = Ar_1 S("evalstart4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-2) = Ar_2 S("evalstart4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-3) = Ar_3 S("evalstart4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-4) = Ar_4 S("evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-0) = Ar_0 S("evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-1) = Ar_1 S("evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-2) = Ar_2 S("evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-3) = Ar_3 S("evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-4) = Ar_4 S("evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-0) = Ar_0 S("evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-1) = Ar_1 S("evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-2) = Ar_2 S("evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-3) = Ar_3 S("evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-4) = Ar_4 S("evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-0) = Ar_0 S("evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-1) = Ar_1 S("evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-2) = Ar_2 S("evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-3) = Ar_3 S("evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-4) = Ar_4 S("evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-0) = Ar_0 S("evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-1) = Ar_1 S("evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-2) = Ar_2 S("evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-3) = Ar_3 S("evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-4) = Ar_4 S("evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-0) = Ar_0 S("evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-1) = Ar_1 S("evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-2) = Ar_2 S("evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-3) = Ar_3 S("evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-4) = Ar_4 S("evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-0) = Ar_0 S("evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-1) = Ar_1 S("evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-2) = Ar_2 S("evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-3) = Ar_3 S("evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4))", 0-4) = Ar_4 orients the transitions evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_3, Ar_1, Ar_4, Ar_3, Ar_4)) [ 0 >= Ar_3 ] evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_0, 0)) [ Ar_2 >= 1 ] weakly and the transition evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_0, 0)) [ Ar_2 >= 1 ] strictly and produces the following problem: 6: T: (Comp: 1, Cost: 1) evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_1, Ar_1, 1, Ar_3, Ar_4)) (Comp: Ar_1 + 1, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_0, 0)) [ Ar_2 >= 1 ] (Comp: 2, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_2 ] (Comp: Ar_1, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= 1 ] (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_3, Ar_1, Ar_4, Ar_3, Ar_4)) [ 0 >= Ar_3 ] (Comp: Ar_1, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1, 1)) (Comp: 2, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 6 produces the following problem: 7: T: (Comp: 1, Cost: 1) evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstartbb0in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart0(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart1(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart2(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart3(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 1) evalstart5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_1, Ar_1, 1, Ar_3, Ar_4)) (Comp: Ar_1 + 1, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_0, 0)) [ Ar_2 >= 1 ] (Comp: 2, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_2 ] (Comp: Ar_1, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_3 >= 1 ] (Comp: 2*Ar_1 + 1, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_3, Ar_1, Ar_4, Ar_3, Ar_4)) [ 0 >= Ar_3 ] (Comp: Ar_1, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3 - 1, 1)) (Comp: 2, Cost: 1) evalstartbb4in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstart(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 5*Ar_1 + 14 Time: 0.137 sec (SMT: 0.089 sec) ---------------------------------------- (2) BOUNDS(1, n^1) ---------------------------------------- (3) Loat Proof (FINISHED) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: evalstartstart 0: evalstartstart -> evalstartbb0in : [], cost: 1 1: evalstartbb0in -> evalstart0 : [], cost: 1 2: evalstart0 -> evalstart1 : [], cost: 1 3: evalstart1 -> evalstart2 : [], cost: 1 4: evalstart2 -> evalstart3 : [], cost: 1 5: evalstart3 -> evalstart4 : [], cost: 1 6: evalstart4 -> evalstart5 : [], cost: 1 7: evalstart5 -> evalstartbb1in : A'=B, C'=1, [], cost: 1 8: evalstartbb1in -> evalstartbb2in : D'=A, E'=0, [ C>=1 ], cost: 1 9: evalstartbb1in -> evalstartbb4in : [ 0>=C ], cost: 1 10: evalstartbb2in -> evalstartbb3in : [ D>=1 ], cost: 1 11: evalstartbb2in -> evalstartbb1in : A'=D, C'=E, [ 0>=D ], cost: 1 12: evalstartbb3in -> evalstartbb2in : D'=-1+D, E'=1, [], cost: 1 13: evalstartbb4in -> evalstartstop : [], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 0: evalstartstart -> evalstartbb0in : [], cost: 1 Removed unreachable and leaf rules: Start location: evalstartstart 0: evalstartstart -> evalstartbb0in : [], cost: 1 1: evalstartbb0in -> evalstart0 : [], cost: 1 2: evalstart0 -> evalstart1 : [], cost: 1 3: evalstart1 -> evalstart2 : [], cost: 1 4: evalstart2 -> evalstart3 : [], cost: 1 5: evalstart3 -> evalstart4 : [], cost: 1 6: evalstart4 -> evalstart5 : [], cost: 1 7: evalstart5 -> evalstartbb1in : A'=B, C'=1, [], cost: 1 8: evalstartbb1in -> evalstartbb2in : D'=A, E'=0, [ C>=1 ], cost: 1 10: evalstartbb2in -> evalstartbb3in : [ D>=1 ], cost: 1 11: evalstartbb2in -> evalstartbb1in : A'=D, C'=E, [ 0>=D ], cost: 1 12: evalstartbb3in -> evalstartbb2in : D'=-1+D, E'=1, [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: evalstartstart 20: evalstartstart -> evalstartbb1in : A'=B, C'=1, [], cost: 8 8: evalstartbb1in -> evalstartbb2in : D'=A, E'=0, [ C>=1 ], cost: 1 11: evalstartbb2in -> evalstartbb1in : A'=D, C'=E, [ 0>=D ], cost: 1 21: evalstartbb2in -> evalstartbb2in : D'=-1+D, E'=1, [ D>=1 ], cost: 2 Accelerating simple loops of location 9. Accelerating the following rules: 21: evalstartbb2in -> evalstartbb2in : D'=-1+D, E'=1, [ D>=1 ], cost: 2 Accelerated rule 21 with metering function D, yielding the new rule 22. Removing the simple loops: 21. Accelerated all simple loops using metering functions (where possible): Start location: evalstartstart 20: evalstartstart -> evalstartbb1in : A'=B, C'=1, [], cost: 8 8: evalstartbb1in -> evalstartbb2in : D'=A, E'=0, [ C>=1 ], cost: 1 11: evalstartbb2in -> evalstartbb1in : A'=D, C'=E, [ 0>=D ], cost: 1 22: evalstartbb2in -> evalstartbb2in : D'=0, E'=1, [ D>=1 ], cost: 2*D Chained accelerated rules (with incoming rules): Start location: evalstartstart 20: evalstartstart -> evalstartbb1in : A'=B, C'=1, [], cost: 8 8: evalstartbb1in -> evalstartbb2in : D'=A, E'=0, [ C>=1 ], cost: 1 23: evalstartbb1in -> evalstartbb2in : D'=0, E'=1, [ C>=1 && A>=1 ], cost: 1+2*A 11: evalstartbb2in -> evalstartbb1in : A'=D, C'=E, [ 0>=D ], cost: 1 Eliminated locations (on tree-shaped paths): Start location: evalstartstart 20: evalstartstart -> evalstartbb1in : A'=B, C'=1, [], cost: 8 24: evalstartbb1in -> evalstartbb1in : A'=A, C'=0, D'=A, E'=0, [ C>=1 && 0>=A ], cost: 2 25: evalstartbb1in -> evalstartbb1in : A'=0, C'=1, D'=0, E'=1, [ C>=1 && A>=1 ], cost: 2+2*A Accelerating simple loops of location 8. Simplified some of the simple loops (and removed duplicate rules). Accelerating the following rules: 24: evalstartbb1in -> evalstartbb1in : C'=0, D'=A, E'=0, [ C>=1 && 0>=A ], cost: 2 25: evalstartbb1in -> evalstartbb1in : A'=0, C'=1, D'=0, E'=1, [ C>=1 && A>=1 ], cost: 2+2*A Found no metering function for rule 24. Found no metering function for rule 25. Removing the simple loops:. Accelerated all simple loops using metering functions (where possible): Start location: evalstartstart 20: evalstartstart -> evalstartbb1in : A'=B, C'=1, [], cost: 8 24: evalstartbb1in -> evalstartbb1in : C'=0, D'=A, E'=0, [ C>=1 && 0>=A ], cost: 2 25: evalstartbb1in -> evalstartbb1in : A'=0, C'=1, D'=0, E'=1, [ C>=1 && A>=1 ], cost: 2+2*A Chained accelerated rules (with incoming rules): Start location: evalstartstart 20: evalstartstart -> evalstartbb1in : A'=B, C'=1, [], cost: 8 26: evalstartstart -> evalstartbb1in : A'=B, C'=0, D'=B, E'=0, [ 0>=B ], cost: 10 27: evalstartstart -> evalstartbb1in : A'=0, C'=1, D'=0, E'=1, [ B>=1 ], cost: 10+2*B Removed unreachable locations (and leaf rules with constant cost): Start location: evalstartstart 27: evalstartstart -> evalstartbb1in : A'=0, C'=1, D'=0, E'=1, [ B>=1 ], cost: 10+2*B ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: evalstartstart 27: evalstartstart -> evalstartbb1in : A'=0, C'=1, D'=0, E'=1, [ B>=1 ], cost: 10+2*B Computing asymptotic complexity for rule 27 Solved the limit problem by the following transformations: Created initial limit problem: 10+2*B (+), B (+/+!) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {B==n} resulting limit problem: [solved] Solution: B / n Resulting cost 10+2*n has complexity: Poly(n^1) Found new complexity Poly(n^1). Obtained the following overall complexity (w.r.t. the length of the input n): Complexity: Poly(n^1) Cpx degree: 1 Solved cost: 10+2*n Rule cost: 10+2*B Rule guard: [ B>=1 ] WORST_CASE(Omega(n^1),?) ---------------------------------------- (4) BOUNDS(n^1, INF)