/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.koat /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1). (0) CpxIntTrs (1) Koat Proof [FINISHED, 132 ms] (2) BOUNDS(1, n^1) (3) Loat Proof [FINISHED, 625 ms] (4) BOUNDS(n^1, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_start_start(v__0, v__01, v_1, v_x, v_y) -> Com_1(eval_start_bb0_in(v__0, v__01, v_1, v_x, v_y)) :|: TRUE eval_start_bb0_in(v__0, v__01, v_1, v_x, v_y) -> Com_1(eval_start_0(v__0, v__01, v_1, v_x, v_y)) :|: TRUE eval_start_0(v__0, v__01, v_1, v_x, v_y) -> Com_1(eval_start_1(v__0, v__01, v_1, v_x, v_y)) :|: TRUE eval_start_1(v__0, v__01, v_1, v_x, v_y) -> Com_1(eval_start_2(v__0, v__01, v_1, v_x, v_y)) :|: TRUE eval_start_2(v__0, v__01, v_1, v_x, v_y) -> Com_1(eval_start_3(v__0, v__01, v_1, v_x, v_y)) :|: TRUE eval_start_3(v__0, v__01, v_1, v_x, v_y) -> Com_1(eval_start_4(v__0, v__01, v_1, v_x, v_y)) :|: TRUE eval_start_4(v__0, v__01, v_1, v_x, v_y) -> Com_1(eval_start_bb1_in(v_x, v_y, v_1, v_x, v_y)) :|: TRUE eval_start_bb1_in(v__0, v__01, v_1, v_x, v_y) -> Com_1(eval_start_bb2_in(v__0, v__01, v_1, v_x, v_y)) :|: v__0 > v__01 eval_start_bb1_in(v__0, v__01, v_1, v_x, v_y) -> Com_1(eval_start_bb3_in(v__0, v__01, v_1, v_x, v_y)) :|: v__0 <= v__01 eval_start_bb2_in(v__0, v__01, v_1, v_x, v_y) -> Com_1(eval_start_5(v__0, v__01, v_1, v_x, v_y)) :|: TRUE eval_start_5(v__0, v__01, v_1, v_x, v_y) -> Com_1(eval_start_6(v__0, v__01, nondef_0, v_x, v_y)) :|: TRUE eval_start_6(v__0, v__01, v_1, v_x, v_y) -> Com_1(eval_start_bb1_in(v__0, v__01 + 1, v_1, v_x, v_y)) :|: v_1 > 0 eval_start_6(v__0, v__01, v_1, v_x, v_y) -> Com_1(eval_start_bb1_in(v__0 - 1, v__01 + 1, v_1, v_x, v_y)) :|: v_1 > 0 && v_1 <= 0 eval_start_6(v__0, v__01, v_1, v_x, v_y) -> Com_1(eval_start_bb1_in(v__0, v__01, v_1, v_x, v_y)) :|: v_1 <= 0 && v_1 > 0 eval_start_6(v__0, v__01, v_1, v_x, v_y) -> Com_1(eval_start_bb1_in(v__0 - 1, v__01, v_1, v_x, v_y)) :|: v_1 <= 0 eval_start_bb3_in(v__0, v__01, v_1, v_x, v_y) -> Com_1(eval_start_stop(v__0, v__01, v_1, v_x, v_y)) :|: TRUE The start-symbols are:[eval_start_start_5] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 5*Ar_1 + 5*Ar_3 + 11) 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(evalstartbb1in(Ar_1, Ar_1, Ar_3, 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_3, Ar_4)) [ Ar_0 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_2 >= Ar_0 ] (Comp: ?, Cost: 1) evalstartbb2in(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(evalstart6(Ar_0, Ar_1, Ar_2, Ar_3, Fresh_0)) (Comp: ?, Cost: 1) evalstart6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3, Ar_4)) [ Ar_4 >= 1 ] (Comp: ?, Cost: 1) evalstart6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_0 - 1, Ar_1, Ar_2 + 1, Ar_3, Ar_4)) [ Ar_4 >= 1 /\ 0 >= Ar_4 ] (Comp: ?, Cost: 1) evalstart6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_4 /\ Ar_4 >= 1 ] (Comp: ?, Cost: 1) evalstart6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_4 ] (Comp: ?, Cost: 1) evalstartbb3in(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 Testing for reachability in the complexity graph removes the following transitions from problem 1: evalstart6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_0 - 1, Ar_1, Ar_2 + 1, Ar_3, Ar_4)) [ Ar_4 >= 1 /\ 0 >= Ar_4 ] evalstart6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_4 /\ Ar_4 >= 1 ] We thus obtain the following problem: 2: T: (Comp: ?, Cost: 1) evalstart6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_4 ] (Comp: ?, Cost: 1) evalstart6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3, Ar_4)) [ Ar_4 >= 1 ] (Comp: ?, Cost: 1) evalstart5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart6(Ar_0, Ar_1, Ar_2, Ar_3, Fresh_0)) (Comp: ?, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalstartbb2in(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) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_2 >= Ar_0 ] (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalstart4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_1, Ar_1, Ar_3, 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) 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) 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) 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) 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) 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: 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 2 produces the following problem: 3: T: (Comp: ?, Cost: 1) evalstart6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_4 ] (Comp: ?, Cost: 1) evalstart6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3, Ar_4)) [ Ar_4 >= 1 ] (Comp: ?, Cost: 1) evalstart5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart6(Ar_0, Ar_1, Ar_2, Ar_3, Fresh_0)) (Comp: ?, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalstartbb2in(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) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_2 >= Ar_0 ] (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_2 + 1 ] (Comp: 1, Cost: 1) evalstart4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_1, Ar_1, Ar_3, 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) 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) 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) 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) 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) 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: 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(evalstart6) = 2 Pol(evalstartbb1in) = 2 Pol(evalstart5) = 2 Pol(evalstartbb3in) = 1 Pol(evalstartstop) = 0 Pol(evalstartbb2in) = 2 Pol(evalstart4) = 2 Pol(evalstart3) = 2 Pol(evalstart2) = 2 Pol(evalstart1) = 2 Pol(evalstart0) = 2 Pol(evalstartbb0in) = 2 Pol(evalstartstart) = 2 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalstartbb3in(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(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_2 >= Ar_0 ] strictly and produces the following problem: 4: T: (Comp: ?, Cost: 1) evalstart6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_4 ] (Comp: ?, Cost: 1) evalstart6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3, Ar_4)) [ Ar_4 >= 1 ] (Comp: ?, Cost: 1) evalstart5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart6(Ar_0, Ar_1, Ar_2, Ar_3, Fresh_0)) (Comp: 2, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 2, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_2 >= Ar_0 ] (Comp: ?, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_2 + 1 ] (Comp: 1, Cost: 1) evalstart4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_1, Ar_1, Ar_3, 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) 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) 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) 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) 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) 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: 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(evalstart6) = V_1 - V_3 - 1 Pol(evalstartbb1in) = V_1 - V_3 Pol(evalstart5) = V_1 - V_3 - 1 Pol(evalstartbb3in) = V_1 - V_3 Pol(evalstartstop) = V_1 - V_3 Pol(evalstartbb2in) = V_1 - V_3 - 1 Pol(evalstart4) = V_2 - V_4 Pol(evalstart3) = V_2 - V_4 Pol(evalstart2) = V_2 - V_4 Pol(evalstart1) = V_2 - V_4 Pol(evalstart0) = V_2 - V_4 Pol(evalstartbb0in) = V_2 - V_4 Pol(evalstartstart) = V_2 - V_4 Pol(koat_start) = V_2 - V_4 orients all transitions 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_3, Ar_4)) [ Ar_0 >= Ar_2 + 1 ] strictly and produces the following problem: 5: T: (Comp: ?, Cost: 1) evalstart6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_4 ] (Comp: ?, Cost: 1) evalstart6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3, Ar_4)) [ Ar_4 >= 1 ] (Comp: ?, Cost: 1) evalstart5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart6(Ar_0, Ar_1, Ar_2, Ar_3, Fresh_0)) (Comp: 2, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: ?, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 2, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_2 >= Ar_0 ] (Comp: Ar_1 + Ar_3, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_2 + 1 ] (Comp: 1, Cost: 1) evalstart4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_1, Ar_1, Ar_3, 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) 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) 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) 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) 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) 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: 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 5 produces the following problem: 6: T: (Comp: Ar_1 + Ar_3, Cost: 1) evalstart6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3, Ar_4)) [ 0 >= Ar_4 ] (Comp: Ar_1 + Ar_3, Cost: 1) evalstart6(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3, Ar_4)) [ Ar_4 >= 1 ] (Comp: Ar_1 + Ar_3, Cost: 1) evalstart5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart6(Ar_0, Ar_1, Ar_2, Ar_3, Fresh_0)) (Comp: 2, Cost: 1) evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartstop(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: Ar_1 + Ar_3, Cost: 1) evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstart5(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) (Comp: 2, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb3in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_2 >= Ar_0 ] (Comp: Ar_1 + Ar_3, Cost: 1) evalstartbb1in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb2in(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4)) [ Ar_0 >= Ar_2 + 1 ] (Comp: 1, Cost: 1) evalstart4(Ar_0, Ar_1, Ar_2, Ar_3, Ar_4) -> Com_1(evalstartbb1in(Ar_1, Ar_1, Ar_3, 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) 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) 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) 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) 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) 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: 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 + 5*Ar_3 + 11 Time: 0.113 sec (SMT: 0.082 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 -> evalstartbb1in : A'=B, C'=D, [], cost: 1 7: evalstartbb1in -> evalstartbb2in : [ A>=1+C ], cost: 1 8: evalstartbb1in -> evalstartbb3in : [ C>=A ], cost: 1 9: evalstartbb2in -> evalstart5 : [], cost: 1 10: evalstart5 -> evalstart6 : E'=free, [], cost: 1 11: evalstart6 -> evalstartbb1in : C'=1+C, [ E>=1 ], cost: 1 12: evalstart6 -> evalstartbb1in : A'=-1+A, C'=1+C, [ E>=1 && 0>=E ], cost: 1 13: evalstart6 -> evalstartbb1in : [ 0>=E && E>=1 ], cost: 1 14: evalstart6 -> evalstartbb1in : A'=-1+A, [ 0>=E ], cost: 1 15: evalstartbb3in -> 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 -> evalstartbb1in : A'=B, C'=D, [], cost: 1 7: evalstartbb1in -> evalstartbb2in : [ A>=1+C ], cost: 1 9: evalstartbb2in -> evalstart5 : [], cost: 1 10: evalstart5 -> evalstart6 : E'=free, [], cost: 1 11: evalstart6 -> evalstartbb1in : C'=1+C, [ E>=1 ], cost: 1 12: evalstart6 -> evalstartbb1in : A'=-1+A, C'=1+C, [ E>=1 && 0>=E ], cost: 1 13: evalstart6 -> evalstartbb1in : [ 0>=E && E>=1 ], cost: 1 14: evalstart6 -> evalstartbb1in : A'=-1+A, [ 0>=E ], cost: 1 Removed rules with unsatisfiable guard: 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 -> evalstartbb1in : A'=B, C'=D, [], cost: 1 7: evalstartbb1in -> evalstartbb2in : [ A>=1+C ], cost: 1 9: evalstartbb2in -> evalstart5 : [], cost: 1 10: evalstart5 -> evalstart6 : E'=free, [], cost: 1 11: evalstart6 -> evalstartbb1in : C'=1+C, [ E>=1 ], cost: 1 14: evalstart6 -> evalstartbb1in : A'=-1+A, [ 0>=E ], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: evalstartstart 21: evalstartstart -> evalstartbb1in : A'=B, C'=D, [], cost: 7 23: evalstartbb1in -> evalstart6 : E'=free, [ A>=1+C ], cost: 3 11: evalstart6 -> evalstartbb1in : C'=1+C, [ E>=1 ], cost: 1 14: evalstart6 -> evalstartbb1in : A'=-1+A, [ 0>=E ], cost: 1 Eliminated locations (on tree-shaped paths): Start location: evalstartstart 21: evalstartstart -> evalstartbb1in : A'=B, C'=D, [], cost: 7 24: evalstartbb1in -> evalstartbb1in : C'=1+C, E'=free, [ A>=1+C && free>=1 ], cost: 4 25: evalstartbb1in -> evalstartbb1in : A'=-1+A, E'=free, [ A>=1+C && 0>=free ], cost: 4 Accelerating simple loops of location 7. Accelerating the following rules: 24: evalstartbb1in -> evalstartbb1in : C'=1+C, E'=free, [ A>=1+C && free>=1 ], cost: 4 25: evalstartbb1in -> evalstartbb1in : A'=-1+A, E'=free, [ A>=1+C && 0>=free ], cost: 4 Accelerated rule 24 with metering function -C+A, yielding the new rule 26. Accelerated rule 25 with metering function -C+A, yielding the new rule 27. Removing the simple loops: 24 25. Accelerated all simple loops using metering functions (where possible): Start location: evalstartstart 21: evalstartstart -> evalstartbb1in : A'=B, C'=D, [], cost: 7 26: evalstartbb1in -> evalstartbb1in : C'=A, E'=free, [ A>=1+C && free>=1 ], cost: -4*C+4*A 27: evalstartbb1in -> evalstartbb1in : A'=C, E'=free, [ A>=1+C && 0>=free ], cost: -4*C+4*A Chained accelerated rules (with incoming rules): Start location: evalstartstart 21: evalstartstart -> evalstartbb1in : A'=B, C'=D, [], cost: 7 28: evalstartstart -> evalstartbb1in : A'=B, C'=B, E'=free, [ B>=1+D && free>=1 ], cost: 7-4*D+4*B 29: evalstartstart -> evalstartbb1in : A'=D, C'=D, E'=free, [ B>=1+D && 0>=free ], cost: 7-4*D+4*B Removed unreachable locations (and leaf rules with constant cost): Start location: evalstartstart 28: evalstartstart -> evalstartbb1in : A'=B, C'=B, E'=free, [ B>=1+D && free>=1 ], cost: 7-4*D+4*B 29: evalstartstart -> evalstartbb1in : A'=D, C'=D, E'=free, [ B>=1+D && 0>=free ], cost: 7-4*D+4*B ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: evalstartstart 28: evalstartstart -> evalstartbb1in : A'=B, C'=B, E'=free, [ B>=1+D && free>=1 ], cost: 7-4*D+4*B 29: evalstartstart -> evalstartbb1in : A'=D, C'=D, E'=free, [ B>=1+D && 0>=free ], cost: 7-4*D+4*B Computing asymptotic complexity for rule 28 Solved the limit problem by the following transformations: Created initial limit problem: free (+/+!), -D+B (+/+!), 7-4*D+4*B (+) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {free==1,D==0,B==n} resulting limit problem: [solved] Solution: free / 1 D / 0 B / n Resulting cost 7+4*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: 7+4*n Rule cost: 7-4*D+4*B Rule guard: [ B>=1+D && free>=1 ] WORST_CASE(Omega(n^1),?) ---------------------------------------- (4) BOUNDS(n^1, INF)