5.72/3.00 WORST_CASE(Omega(n^1), O(n^1)) 5.72/3.01 proof of /export/starexec/sandbox/benchmark/theBenchmark.koat 5.72/3.01 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.72/3.01 5.72/3.01 5.72/3.01 The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1). 5.72/3.01 5.72/3.01 (0) CpxIntTrs 5.72/3.01 (1) Koat Proof [FINISHED, 114 ms] 5.72/3.01 (2) BOUNDS(1, n^1) 5.72/3.01 (3) Loat Proof [FINISHED, 1020 ms] 5.72/3.01 (4) BOUNDS(n^1, INF) 5.72/3.01 5.72/3.01 5.72/3.01 ---------------------------------------- 5.72/3.01 5.72/3.01 (0) 5.72/3.01 Obligation: 5.72/3.01 Complexity Int TRS consisting of the following rules: 5.72/3.01 eval_start_start(v_1, v_3, v_n, v_x_0, v_x_0_sink) -> Com_1(eval_start_bb0_in(v_1, v_3, v_n, v_x_0, v_x_0_sink)) :|: TRUE 5.72/3.01 eval_start_bb0_in(v_1, v_3, v_n, v_x_0, v_x_0_sink) -> Com_1(eval_start_0(v_1, v_3, v_n, v_x_0, v_x_0_sink)) :|: TRUE 5.72/3.01 eval_start_0(v_1, v_3, v_n, v_x_0, v_x_0_sink) -> Com_1(eval_start_1(v_1, v_3, v_n, v_x_0, v_x_0_sink)) :|: TRUE 5.72/3.01 eval_start_1(v_1, v_3, v_n, v_x_0, v_x_0_sink) -> Com_1(eval_start_2(v_1, v_3, v_n, v_x_0, v_x_0_sink)) :|: TRUE 5.72/3.01 eval_start_2(v_1, v_3, v_n, v_x_0, v_x_0_sink) -> Com_1(eval_start_3(v_1, v_3, v_n, v_x_0, v_x_0_sink)) :|: TRUE 5.72/3.01 eval_start_3(v_1, v_3, v_n, v_x_0, v_x_0_sink) -> Com_1(eval_start_bb1_in(v_1, v_3, v_n, 0, v_x_0_sink)) :|: TRUE 5.72/3.01 eval_start_bb1_in(v_1, v_3, v_n, v_x_0, v_x_0_sink) -> Com_1(eval_start_bb2_in(v_1, v_3, v_n, v_x_0, v_x_0)) :|: v_x_0 < v_n 5.72/3.01 eval_start_bb1_in(v_1, v_3, v_n, v_x_0, v_x_0_sink) -> Com_1(eval_start_bb4_in(v_1, v_3, v_n, v_x_0, v_x_0_sink)) :|: v_x_0 >= v_n 5.72/3.01 eval_start_bb2_in(v_1, v_3, v_n, v_x_0, v_x_0_sink) -> Com_1(eval_start_bb3_in(v_x_0_sink + 1, v_3, v_n, v_x_0, v_x_0_sink)) :|: v_x_0_sink + 1 < v_n 5.72/3.01 eval_start_bb2_in(v_1, v_3, v_n, v_x_0, v_x_0_sink) -> Com_1(eval_start_bb1_in(v_1, v_3, v_n, v_x_0_sink + 1, v_x_0_sink)) :|: v_x_0_sink + 1 >= v_n 5.72/3.01 eval_start_bb3_in(v_1, v_3, v_n, v_x_0, v_x_0_sink) -> Com_1(eval_start_6(v_1, v_3, v_n, v_x_0, v_x_0_sink)) :|: TRUE 5.72/3.01 eval_start_6(v_1, v_3, v_n, v_x_0, v_x_0_sink) -> Com_1(eval_start_7(v_1, nondef_0, v_n, v_x_0, v_x_0_sink)) :|: TRUE 5.72/3.01 eval_start_7(v_1, v_3, v_n, v_x_0, v_x_0_sink) -> Com_1(eval_start_bb1_in(v_1, v_3, v_n, v_1, v_x_0_sink)) :|: v_3 > 0 5.72/3.01 eval_start_7(v_1, v_3, v_n, v_x_0, v_x_0_sink) -> Com_1(eval_start_bb2_in(v_1, v_3, v_n, v_x_0, v_1)) :|: v_3 <= 0 5.72/3.01 eval_start_bb4_in(v_1, v_3, v_n, v_x_0, v_x_0_sink) -> Com_1(eval_start_stop(v_1, v_3, v_n, v_x_0, v_x_0_sink)) :|: TRUE 5.72/3.01 5.72/3.01 The start-symbols are:[eval_start_start_5] 5.72/3.01 5.72/3.01 5.72/3.01 ---------------------------------------- 5.72/3.01 5.72/3.01 (1) Koat Proof (FINISHED) 5.72/3.01 YES(?, 16*ar_1 + 10) 5.72/3.01 5.72/3.01 5.72/3.01 5.72/3.01 Initial complexity problem: 5.72/3.01 5.72/3.01 1: T: 5.72/3.01 5.72/3.01 (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)) 5.72/3.01 5.72/3.01 (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)) 5.72/3.01 5.72/3.01 (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)) 5.72/3.01 5.72/3.01 (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)) 5.72/3.01 5.72/3.01 (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)) 5.72/3.01 5.72/3.01 (Comp: ?, Cost: 1) evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(0, ar_1, ar_2, ar_3, ar_4)) 5.72/3.01 5.72/3.01 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_0, ar_3, ar_4)) [ ar_1 >= ar_0 + 1 ] 5.72/3.01 5.72/3.01 (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)) [ ar_0 >= ar_1 ] 5.72/3.01 5.72/3.01 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_2 + 1, ar_4)) [ ar_1 >= ar_2 + 2 ] 5.72/3.01 5.72/3.01 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_2 + 1, ar_1, ar_2, ar_3, ar_4)) [ ar_2 + 1 >= ar_1 ] 5.72/3.01 5.72/3.01 (Comp: ?, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart6(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.72/3.01 5.72/3.01 (Comp: ?, Cost: 1) evalstart6(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart7(ar_0, ar_1, ar_2, ar_3, f)) 5.72/3.01 5.72/3.01 (Comp: ?, Cost: 1) evalstart7(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_3, ar_1, ar_2, ar_3, ar_4)) [ ar_4 >= 1 ] 5.72/3.01 5.72/3.01 (Comp: ?, Cost: 1) evalstart7(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_3, ar_3, ar_4)) [ 0 >= ar_4 ] 5.72/3.01 5.72/3.01 (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)) 5.72/3.01 5.72/3.01 (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 ] 5.72/3.01 5.72/3.01 start location: koat_start 5.72/3.01 5.72/3.01 leaf cost: 0 5.72/3.01 5.72/3.01 5.72/3.01 5.72/3.01 Repeatedly propagating knowledge in problem 1 produces the following problem: 5.72/3.01 5.72/3.01 2: T: 5.72/3.01 5.72/3.01 (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)) 5.72/3.01 5.72/3.01 (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)) 5.72/3.01 5.72/3.01 (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)) 5.72/3.01 5.72/3.01 (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)) 5.72/3.01 5.72/3.01 (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)) 5.72/3.01 5.72/3.01 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(0, ar_1, ar_2, ar_3, ar_4)) 5.72/3.01 5.72/3.01 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_0, ar_3, ar_4)) [ ar_1 >= ar_0 + 1 ] 5.72/3.01 5.72/3.01 (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)) [ ar_0 >= ar_1 ] 5.72/3.01 5.72/3.01 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_2 + 1, ar_4)) [ ar_1 >= ar_2 + 2 ] 5.72/3.01 5.72/3.01 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_2 + 1, ar_1, ar_2, ar_3, ar_4)) [ ar_2 + 1 >= ar_1 ] 5.72/3.01 5.72/3.01 (Comp: ?, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart6(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.72/3.01 5.72/3.01 (Comp: ?, Cost: 1) evalstart6(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart7(ar_0, ar_1, ar_2, ar_3, f)) 5.72/3.01 5.72/3.01 (Comp: ?, Cost: 1) evalstart7(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_3, ar_1, ar_2, ar_3, ar_4)) [ ar_4 >= 1 ] 5.72/3.01 5.72/3.01 (Comp: ?, Cost: 1) evalstart7(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_3, ar_3, ar_4)) [ 0 >= ar_4 ] 5.72/3.01 5.72/3.01 (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)) 5.72/3.01 5.72/3.01 (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 ] 5.72/3.01 5.72/3.01 start location: koat_start 5.72/3.01 5.72/3.01 leaf cost: 0 5.72/3.01 5.72/3.01 5.72/3.01 5.72/3.01 A polynomial rank function with 5.72/3.01 5.72/3.01 Pol(evalstartstart) = 2 5.72/3.01 5.72/3.01 Pol(evalstartbb0in) = 2 5.72/3.01 5.72/3.01 Pol(evalstart0) = 2 5.72/3.01 5.72/3.01 Pol(evalstart1) = 2 5.72/3.01 5.72/3.01 Pol(evalstart2) = 2 5.72/3.01 5.72/3.01 Pol(evalstart3) = 2 5.72/3.01 5.72/3.01 Pol(evalstartbb1in) = 2 5.72/3.01 5.72/3.01 Pol(evalstartbb2in) = 2 5.72/3.01 5.72/3.01 Pol(evalstartbb4in) = 1 5.72/3.01 5.72/3.01 Pol(evalstartbb3in) = 2 5.72/3.01 5.72/3.01 Pol(evalstart6) = 2 5.72/3.01 5.72/3.01 Pol(evalstart7) = 2 5.72/3.01 5.72/3.01 Pol(evalstartstop) = 0 5.72/3.01 5.72/3.01 Pol(koat_start) = 2 5.72/3.01 5.72/3.01 orients all transitions weakly and the transitions 5.72/3.01 5.72/3.01 evalstartbb4in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.72/3.01 5.72/3.01 evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 >= ar_1 ] 5.72/3.01 5.72/3.01 strictly and produces the following problem: 5.72/3.01 5.72/3.01 3: T: 5.72/3.01 5.72/3.01 (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)) 5.72/3.01 5.72/3.01 (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)) 5.72/3.01 5.72/3.01 (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)) 5.72/3.01 5.72/3.01 (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)) 5.72/3.01 5.72/3.01 (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)) 5.72/3.01 5.72/3.01 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(0, ar_1, ar_2, ar_3, ar_4)) 5.72/3.01 5.72/3.01 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_0, ar_3, ar_4)) [ ar_1 >= ar_0 + 1 ] 5.72/3.01 5.72/3.01 (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)) [ ar_0 >= ar_1 ] 5.72/3.01 5.72/3.01 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_2 + 1, ar_4)) [ ar_1 >= ar_2 + 2 ] 5.72/3.01 5.72/3.01 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_2 + 1, ar_1, ar_2, ar_3, ar_4)) [ ar_2 + 1 >= ar_1 ] 5.72/3.01 5.72/3.01 (Comp: ?, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart6(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.72/3.01 5.72/3.01 (Comp: ?, Cost: 1) evalstart6(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart7(ar_0, ar_1, ar_2, ar_3, f)) 5.72/3.01 5.72/3.01 (Comp: ?, Cost: 1) evalstart7(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_3, ar_1, ar_2, ar_3, ar_4)) [ ar_4 >= 1 ] 5.72/3.01 5.72/3.01 (Comp: ?, Cost: 1) evalstart7(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_3, ar_3, ar_4)) [ 0 >= ar_4 ] 5.72/3.01 5.72/3.01 (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)) 5.72/3.01 5.72/3.01 (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 ] 5.72/3.01 5.72/3.01 start location: koat_start 5.72/3.01 5.72/3.01 leaf cost: 0 5.72/3.01 5.72/3.01 5.72/3.01 5.72/3.01 A polynomial rank function with 5.72/3.01 5.72/3.01 Pol(evalstartstart) = 2*V_2 5.72/3.01 5.72/3.01 Pol(evalstartbb0in) = 2*V_2 5.72/3.01 5.72/3.01 Pol(evalstart0) = 2*V_2 5.72/3.01 5.72/3.01 Pol(evalstart1) = 2*V_2 5.72/3.01 5.72/3.01 Pol(evalstart2) = 2*V_2 5.72/3.01 5.72/3.01 Pol(evalstart3) = 2*V_2 5.72/3.01 5.72/3.01 Pol(evalstartbb1in) = -2*V_1 + 2*V_2 5.72/3.01 5.72/3.01 Pol(evalstartbb2in) = 2*V_2 - 2*V_3 - 1 5.72/3.01 5.72/3.01 Pol(evalstartbb4in) = -2*V_1 + 2*V_2 5.72/3.01 5.72/3.01 Pol(evalstartbb3in) = 2*V_2 - 2*V_4 5.72/3.01 5.72/3.01 Pol(evalstart6) = 2*V_2 - 2*V_4 5.72/3.01 5.72/3.01 Pol(evalstart7) = 2*V_2 - 2*V_4 5.72/3.01 5.72/3.01 Pol(evalstartstop) = -2*V_1 + 2*V_2 5.72/3.01 5.72/3.01 Pol(koat_start) = 2*V_2 5.72/3.01 5.72/3.01 orients all transitions weakly and the transitions 5.72/3.01 5.72/3.01 evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_2 + 1, ar_4)) [ ar_1 >= ar_2 + 2 ] 5.72/3.01 5.72/3.01 evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_0, ar_3, ar_4)) [ ar_1 >= ar_0 + 1 ] 5.72/3.01 5.72/3.01 strictly and produces the following problem: 5.72/3.01 5.72/3.01 4: T: 5.72/3.01 5.72/3.01 (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)) 5.72/3.01 5.72/3.01 (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)) 5.72/3.01 5.72/3.01 (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)) 5.72/3.01 5.72/3.01 (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)) 5.72/3.01 5.72/3.01 (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)) 5.72/3.01 5.72/3.01 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(0, ar_1, ar_2, ar_3, ar_4)) 5.72/3.01 5.72/3.01 (Comp: 2*ar_1, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_0, ar_3, ar_4)) [ ar_1 >= ar_0 + 1 ] 5.72/3.01 5.72/3.01 (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)) [ ar_0 >= ar_1 ] 5.72/3.01 5.72/3.01 (Comp: 2*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_2 + 1, ar_4)) [ ar_1 >= ar_2 + 2 ] 5.72/3.01 5.72/3.01 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_2 + 1, ar_1, ar_2, ar_3, ar_4)) [ ar_2 + 1 >= ar_1 ] 5.72/3.01 5.72/3.01 (Comp: ?, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart6(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.72/3.01 5.72/3.01 (Comp: ?, Cost: 1) evalstart6(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart7(ar_0, ar_1, ar_2, ar_3, f)) 5.72/3.01 5.72/3.01 (Comp: ?, Cost: 1) evalstart7(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_3, ar_1, ar_2, ar_3, ar_4)) [ ar_4 >= 1 ] 5.72/3.01 5.72/3.01 (Comp: ?, Cost: 1) evalstart7(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_3, ar_3, ar_4)) [ 0 >= ar_4 ] 5.72/3.01 5.72/3.01 (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)) 5.72/3.01 5.72/3.01 (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 ] 5.72/3.01 5.72/3.01 start location: koat_start 5.72/3.01 5.72/3.01 leaf cost: 0 5.72/3.01 5.72/3.01 5.72/3.01 5.72/3.01 Repeatedly propagating knowledge in problem 4 produces the following problem: 5.72/3.01 5.72/3.01 5: T: 5.72/3.01 5.72/3.01 (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)) 5.72/3.01 5.72/3.01 (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)) 5.72/3.01 5.72/3.01 (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)) 5.72/3.01 5.72/3.01 (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)) 5.72/3.01 5.72/3.01 (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)) 5.72/3.01 5.72/3.01 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(0, ar_1, ar_2, ar_3, ar_4)) 5.72/3.01 5.72/3.01 (Comp: 2*ar_1, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_0, ar_3, ar_4)) [ ar_1 >= ar_0 + 1 ] 5.72/3.01 5.72/3.01 (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)) [ ar_0 >= ar_1 ] 5.72/3.01 5.72/3.01 (Comp: 2*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_2 + 1, ar_4)) [ ar_1 >= ar_2 + 2 ] 5.72/3.01 5.72/3.01 (Comp: 4*ar_1, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_2 + 1, ar_1, ar_2, ar_3, ar_4)) [ ar_2 + 1 >= ar_1 ] 5.72/3.01 5.72/3.01 (Comp: 2*ar_1, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart6(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.72/3.01 5.72/3.01 (Comp: 2*ar_1, Cost: 1) evalstart6(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart7(ar_0, ar_1, ar_2, ar_3, f)) 5.72/3.01 5.72/3.01 (Comp: 2*ar_1, Cost: 1) evalstart7(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_3, ar_1, ar_2, ar_3, ar_4)) [ ar_4 >= 1 ] 5.72/3.01 5.72/3.01 (Comp: 2*ar_1, Cost: 1) evalstart7(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_3, ar_3, ar_4)) [ 0 >= ar_4 ] 5.72/3.01 5.72/3.01 (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)) 5.72/3.01 5.72/3.01 (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 ] 5.72/3.01 5.72/3.01 start location: koat_start 5.72/3.01 5.72/3.01 leaf cost: 0 5.72/3.01 5.72/3.01 5.72/3.01 5.72/3.01 Complexity upper bound 16*ar_1 + 10 5.72/3.01 5.72/3.01 5.72/3.01 5.72/3.01 Time: 0.161 sec (SMT: 0.134 sec) 5.72/3.01 5.72/3.01 5.72/3.01 ---------------------------------------- 5.72/3.01 5.72/3.01 (2) 5.72/3.01 BOUNDS(1, n^1) 5.72/3.01 5.72/3.01 ---------------------------------------- 5.72/3.01 5.72/3.01 (3) Loat Proof (FINISHED) 5.72/3.01 5.72/3.01 5.72/3.01 ### Pre-processing the ITS problem ### 5.72/3.01 5.72/3.01 5.72/3.01 5.72/3.01 Initial linear ITS problem 5.72/3.01 5.72/3.01 Start location: evalstartstart 5.72/3.01 5.72/3.01 0: evalstartstart -> evalstartbb0in : [], cost: 1 5.72/3.01 5.72/3.01 1: evalstartbb0in -> evalstart0 : [], cost: 1 5.72/3.01 5.72/3.01 2: evalstart0 -> evalstart1 : [], cost: 1 5.72/3.01 5.72/3.01 3: evalstart1 -> evalstart2 : [], cost: 1 5.72/3.01 5.72/3.01 4: evalstart2 -> evalstart3 : [], cost: 1 5.72/3.01 5.72/3.01 5: evalstart3 -> evalstartbb1in : A'=0, [], cost: 1 5.72/3.01 5.72/3.01 6: evalstartbb1in -> evalstartbb2in : C'=A, [ B>=1+A ], cost: 1 5.72/3.01 5.72/3.01 7: evalstartbb1in -> evalstartbb4in : [ A>=B ], cost: 1 5.72/3.01 5.72/3.01 8: evalstartbb2in -> evalstartbb3in : D'=1+C, [ B>=2+C ], cost: 1 5.72/3.01 5.72/3.01 9: evalstartbb2in -> evalstartbb1in : A'=1+C, [ 1+C>=B ], cost: 1 5.72/3.01 5.72/3.01 10: evalstartbb3in -> evalstart6 : [], cost: 1 5.72/3.01 5.72/3.01 11: evalstart6 -> evalstart7 : E'=free, [], cost: 1 5.72/3.01 5.72/3.01 12: evalstart7 -> evalstartbb1in : A'=D, [ E>=1 ], cost: 1 5.72/3.01 5.72/3.01 13: evalstart7 -> evalstartbb2in : C'=D, [ 0>=E ], cost: 1 5.72/3.01 5.72/3.01 14: evalstartbb4in -> evalstartstop : [], cost: 1 5.72/3.01 5.72/3.01 5.72/3.01 5.72/3.01 Removed unreachable and leaf rules: 5.72/3.01 5.72/3.01 Start location: evalstartstart 5.72/3.01 5.72/3.01 0: evalstartstart -> evalstartbb0in : [], cost: 1 5.72/3.01 5.72/3.01 1: evalstartbb0in -> evalstart0 : [], cost: 1 5.72/3.01 5.72/3.01 2: evalstart0 -> evalstart1 : [], cost: 1 5.72/3.01 5.72/3.01 3: evalstart1 -> evalstart2 : [], cost: 1 5.72/3.01 5.72/3.01 4: evalstart2 -> evalstart3 : [], cost: 1 5.72/3.01 5.72/3.01 5: evalstart3 -> evalstartbb1in : A'=0, [], cost: 1 5.72/3.01 5.72/3.01 6: evalstartbb1in -> evalstartbb2in : C'=A, [ B>=1+A ], cost: 1 5.72/3.01 5.72/3.01 8: evalstartbb2in -> evalstartbb3in : D'=1+C, [ B>=2+C ], cost: 1 5.72/3.01 5.72/3.01 9: evalstartbb2in -> evalstartbb1in : A'=1+C, [ 1+C>=B ], cost: 1 5.72/3.01 5.72/3.01 10: evalstartbb3in -> evalstart6 : [], cost: 1 5.72/3.01 5.72/3.01 11: evalstart6 -> evalstart7 : E'=free, [], cost: 1 5.72/3.01 5.72/3.01 12: evalstart7 -> evalstartbb1in : A'=D, [ E>=1 ], cost: 1 5.72/3.01 5.72/3.01 13: evalstart7 -> evalstartbb2in : C'=D, [ 0>=E ], cost: 1 5.72/3.01 5.72/3.01 5.72/3.01 5.72/3.01 ### Simplification by acceleration and chaining ### 5.72/3.01 5.72/3.01 5.72/3.01 5.72/3.01 Eliminated locations (on linear paths): 5.72/3.01 5.72/3.01 Start location: evalstartstart 5.72/3.01 5.72/3.01 19: evalstartstart -> evalstartbb1in : A'=0, [], cost: 6 5.72/3.01 5.72/3.01 6: evalstartbb1in -> evalstartbb2in : C'=A, [ B>=1+A ], cost: 1 5.72/3.01 5.72/3.01 9: evalstartbb2in -> evalstartbb1in : A'=1+C, [ 1+C>=B ], cost: 1 5.72/3.01 5.72/3.01 21: evalstartbb2in -> evalstart7 : D'=1+C, E'=free, [ B>=2+C ], cost: 3 5.72/3.01 5.72/3.01 12: evalstart7 -> evalstartbb1in : A'=D, [ E>=1 ], cost: 1 5.72/3.01 5.72/3.01 13: evalstart7 -> evalstartbb2in : C'=D, [ 0>=E ], cost: 1 5.72/3.01 5.72/3.01 5.72/3.01 5.72/3.01 Eliminated locations (on tree-shaped paths): 5.72/3.01 5.72/3.01 Start location: evalstartstart 5.72/3.01 5.72/3.01 19: evalstartstart -> evalstartbb1in : A'=0, [], cost: 6 5.72/3.01 5.72/3.01 6: evalstartbb1in -> evalstartbb2in : C'=A, [ B>=1+A ], cost: 1 5.72/3.01 5.72/3.01 9: evalstartbb2in -> evalstartbb1in : A'=1+C, [ 1+C>=B ], cost: 1 5.72/3.01 5.72/3.01 22: evalstartbb2in -> evalstartbb1in : A'=1+C, D'=1+C, E'=free, [ B>=2+C && free>=1 ], cost: 4 5.72/3.01 5.72/3.01 23: evalstartbb2in -> evalstartbb2in : C'=1+C, D'=1+C, E'=free, [ B>=2+C && 0>=free ], cost: 4 5.72/3.01 5.72/3.01 5.72/3.01 5.72/3.01 Accelerating simple loops of location 7. 5.72/3.01 5.72/3.01 Accelerating the following rules: 5.72/3.01 5.72/3.01 23: evalstartbb2in -> evalstartbb2in : C'=1+C, D'=1+C, E'=free, [ B>=2+C && 0>=free ], cost: 4 5.72/3.01 5.72/3.01 5.72/3.01 5.72/3.01 Accelerated rule 23 with metering function -1-C+B, yielding the new rule 24. 5.72/3.01 5.72/3.01 Removing the simple loops: 23. 5.72/3.01 5.72/3.01 5.72/3.01 5.72/3.01 Accelerated all simple loops using metering functions (where possible): 5.72/3.01 5.72/3.01 Start location: evalstartstart 5.72/3.01 5.72/3.01 19: evalstartstart -> evalstartbb1in : A'=0, [], cost: 6 5.72/3.01 5.72/3.01 6: evalstartbb1in -> evalstartbb2in : C'=A, [ B>=1+A ], cost: 1 5.72/3.01 5.72/3.01 9: evalstartbb2in -> evalstartbb1in : A'=1+C, [ 1+C>=B ], cost: 1 5.72/3.01 5.72/3.01 22: evalstartbb2in -> evalstartbb1in : A'=1+C, D'=1+C, E'=free, [ B>=2+C && free>=1 ], cost: 4 5.72/3.01 5.72/3.01 24: evalstartbb2in -> evalstartbb2in : C'=-1+B, D'=-1+B, E'=free, [ B>=2+C && 0>=free ], cost: -4-4*C+4*B 5.72/3.01 5.72/3.01 5.72/3.01 5.72/3.01 Chained accelerated rules (with incoming rules): 5.72/3.01 5.72/3.01 Start location: evalstartstart 5.72/3.01 5.72/3.01 19: evalstartstart -> evalstartbb1in : A'=0, [], cost: 6 5.72/3.01 5.72/3.01 6: evalstartbb1in -> evalstartbb2in : C'=A, [ B>=1+A ], cost: 1 5.72/3.01 5.72/3.01 25: evalstartbb1in -> evalstartbb2in : C'=-1+B, D'=-1+B, E'=free, [ B>=2+A && 0>=free ], cost: -3-4*A+4*B 5.72/3.01 5.72/3.01 9: evalstartbb2in -> evalstartbb1in : A'=1+C, [ 1+C>=B ], cost: 1 5.72/3.01 5.72/3.01 22: evalstartbb2in -> evalstartbb1in : A'=1+C, D'=1+C, E'=free, [ B>=2+C && free>=1 ], cost: 4 5.72/3.01 5.72/3.01 5.72/3.01 5.72/3.01 Eliminated locations (on tree-shaped paths): 5.72/3.01 5.72/3.01 Start location: evalstartstart 5.72/3.01 5.72/3.01 19: evalstartstart -> evalstartbb1in : A'=0, [], cost: 6 5.72/3.01 5.72/3.01 26: evalstartbb1in -> evalstartbb1in : A'=1+A, C'=A, [ B>=1+A && 1+A>=B ], cost: 2 5.72/3.01 5.72/3.01 27: evalstartbb1in -> evalstartbb1in : A'=1+A, C'=A, D'=1+A, E'=free, [ B>=2+A && free>=1 ], cost: 5 5.72/3.01 5.72/3.01 28: evalstartbb1in -> evalstartbb1in : A'=B, C'=-1+B, D'=-1+B, E'=free, [ B>=2+A && 0>=free ], cost: -2-4*A+4*B 5.72/3.01 5.72/3.01 29: evalstartbb1in -> [14] : [ B>=2+A && 0>=free ], cost: -3-4*A+4*B 5.72/3.01 5.72/3.01 5.72/3.01 5.72/3.01 Accelerating simple loops of location 6. 5.72/3.01 5.72/3.01 Simplified some of the simple loops (and removed duplicate rules). 5.72/3.01 5.72/3.01 Accelerating the following rules: 5.72/3.01 5.72/3.01 26: evalstartbb1in -> evalstartbb1in : A'=1+A, C'=A, [ 1+A-B==0 ], cost: 2 5.72/3.01 5.72/3.01 27: evalstartbb1in -> evalstartbb1in : A'=1+A, C'=A, D'=1+A, E'=free, [ B>=2+A && free>=1 ], cost: 5 5.72/3.01 5.72/3.01 28: evalstartbb1in -> evalstartbb1in : A'=B, C'=-1+B, D'=-1+B, E'=free, [ B>=2+A && 0>=free ], cost: -2-4*A+4*B 5.72/3.01 5.72/3.01 5.72/3.01 5.72/3.01 Accelerated rule 26 with metering function -A+B, yielding the new rule 30. 5.72/3.01 5.72/3.01 Accelerated rule 27 with metering function -1-A+B, yielding the new rule 31. 5.72/3.01 5.72/3.01 Found no metering function for rule 28. 5.72/3.01 5.72/3.01 Removing the simple loops: 26 27. 5.72/3.01 5.72/3.01 5.72/3.01 5.72/3.01 Accelerated all simple loops using metering functions (where possible): 5.72/3.01 5.72/3.01 Start location: evalstartstart 5.72/3.01 5.72/3.01 19: evalstartstart -> evalstartbb1in : A'=0, [], cost: 6 5.72/3.01 5.72/3.01 28: evalstartbb1in -> evalstartbb1in : A'=B, C'=-1+B, D'=-1+B, E'=free, [ B>=2+A && 0>=free ], cost: -2-4*A+4*B 5.72/3.01 5.72/3.01 29: evalstartbb1in -> [14] : [ B>=2+A && 0>=free ], cost: -3-4*A+4*B 5.72/3.01 5.72/3.01 30: evalstartbb1in -> evalstartbb1in : A'=B, C'=-1+B, [ 1+A-B==0 ], cost: -2*A+2*B 5.72/3.01 5.72/3.01 31: evalstartbb1in -> evalstartbb1in : A'=-1+B, C'=-2+B, D'=-1+B, E'=free, [ B>=2+A && free>=1 ], cost: -5-5*A+5*B 5.72/3.01 5.72/3.01 5.72/3.01 5.72/3.01 Chained accelerated rules (with incoming rules): 5.72/3.01 5.72/3.01 Start location: evalstartstart 5.72/3.01 5.72/3.01 19: evalstartstart -> evalstartbb1in : A'=0, [], cost: 6 5.72/3.01 5.72/3.01 32: evalstartstart -> evalstartbb1in : A'=B, C'=-1+B, D'=-1+B, E'=free, [ B>=2 && 0>=free ], cost: 4+4*B 5.72/3.01 5.72/3.01 33: evalstartstart -> evalstartbb1in : A'=B, C'=-1+B, [ 1-B==0 ], cost: 6+2*B 5.72/3.01 5.72/3.01 34: evalstartstart -> evalstartbb1in : A'=-1+B, C'=-2+B, D'=-1+B, E'=free, [ B>=2 && free>=1 ], cost: 1+5*B 5.72/3.01 5.72/3.01 29: evalstartbb1in -> [14] : [ B>=2+A && 0>=free ], cost: -3-4*A+4*B 5.72/3.01 5.72/3.01 5.72/3.01 5.72/3.01 Eliminated locations (on tree-shaped paths): 5.72/3.01 5.72/3.01 Start location: evalstartstart 5.72/3.01 5.72/3.01 35: evalstartstart -> [14] : A'=0, [ B>=2 && 0>=free ], cost: 3+4*B 5.72/3.01 5.72/3.01 36: evalstartstart -> [16] : [ B>=2 && 0>=free ], cost: 4+4*B 5.72/3.01 5.72/3.01 37: evalstartstart -> [16] : [ 1-B==0 ], cost: 6+2*B 5.72/3.01 5.72/3.01 38: evalstartstart -> [16] : [ B>=2 && free>=1 ], cost: 1+5*B 5.72/3.01 5.72/3.01 5.72/3.01 5.72/3.01 ### Computing asymptotic complexity ### 5.72/3.01 5.72/3.01 5.72/3.01 5.72/3.01 Fully simplified ITS problem 5.72/3.01 5.72/3.01 Start location: evalstartstart 5.72/3.01 5.72/3.01 36: evalstartstart -> [16] : [ B>=2 && 0>=free ], cost: 4+4*B 5.72/3.01 5.72/3.01 37: evalstartstart -> [16] : [ 1-B==0 ], cost: 6+2*B 5.72/3.01 5.72/3.01 38: evalstartstart -> [16] : [ B>=2 && free>=1 ], cost: 1+5*B 5.72/3.01 5.72/3.01 5.72/3.01 5.72/3.01 Computing asymptotic complexity for rule 36 5.72/3.01 5.72/3.01 Solved the limit problem by the following transformations: 5.72/3.01 5.72/3.01 Created initial limit problem: 5.72/3.01 5.72/3.01 4+4*B (+), 1-free (+/+!), -1+B (+/+!) [not solved] 5.72/3.01 5.72/3.01 5.72/3.01 5.72/3.01 removing all constraints (solved by SMT) 5.72/3.01 5.72/3.01 resulting limit problem: [solved] 5.72/3.01 5.72/3.01 5.72/3.01 5.72/3.01 applying transformation rule (C) using substitution {free==-n,B==n} 5.72/3.01 5.72/3.01 resulting limit problem: 5.72/3.01 5.72/3.01 [solved] 5.72/3.01 5.72/3.01 5.72/3.01 5.72/3.01 Solution: 5.72/3.01 5.72/3.01 free / -n 5.72/3.01 5.72/3.01 B / n 5.72/3.01 5.72/3.01 Resulting cost 4+4*n has complexity: Poly(n^1) 5.72/3.01 5.72/3.01 5.72/3.01 5.72/3.01 Found new complexity Poly(n^1). 5.72/3.01 5.72/3.01 5.72/3.01 5.72/3.01 Obtained the following overall complexity (w.r.t. the length of the input n): 5.72/3.01 5.72/3.01 Complexity: Poly(n^1) 5.72/3.01 5.72/3.01 Cpx degree: 1 5.72/3.01 5.72/3.01 Solved cost: 4+4*n 5.72/3.01 5.72/3.01 Rule cost: 4+4*B 5.72/3.01 5.72/3.01 Rule guard: [ B>=2 && 0>=free ] 5.72/3.01 5.72/3.01 5.72/3.01 5.72/3.01 WORST_CASE(Omega(n^1),?) 5.72/3.01 5.72/3.01 5.72/3.01 ---------------------------------------- 5.72/3.01 5.72/3.01 (4) 5.72/3.01 BOUNDS(n^1, INF) 5.91/3.04 EOF