4.66/2.39 WORST_CASE(Omega(n^1), O(n^1)) 4.66/2.40 proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat 4.66/2.40 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.66/2.40 4.66/2.40 4.66/2.40 The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1). 4.66/2.40 4.66/2.40 (0) CpxIntTrs 4.66/2.40 (1) Koat Proof [FINISHED, 132 ms] 4.66/2.40 (2) BOUNDS(1, n^1) 4.66/2.40 (3) Loat Proof [FINISHED, 499 ms] 4.66/2.40 (4) BOUNDS(n^1, INF) 4.66/2.40 4.66/2.40 4.66/2.40 ---------------------------------------- 4.66/2.40 4.66/2.40 (0) 4.66/2.40 Obligation: 4.66/2.40 Complexity Int TRS consisting of the following rules: 4.66/2.40 eval_start_start(v__0, v__01, v_x, v_y) -> Com_1(eval_start_bb0_in(v__0, v__01, v_x, v_y)) :|: TRUE 4.66/2.40 eval_start_bb0_in(v__0, v__01, v_x, v_y) -> Com_1(eval_start_0(v__0, v__01, v_x, v_y)) :|: TRUE 4.66/2.40 eval_start_0(v__0, v__01, v_x, v_y) -> Com_1(eval_start_1(v__0, v__01, v_x, v_y)) :|: TRUE 4.66/2.40 eval_start_1(v__0, v__01, v_x, v_y) -> Com_1(eval_start_2(v__0, v__01, v_x, v_y)) :|: TRUE 4.66/2.40 eval_start_2(v__0, v__01, v_x, v_y) -> Com_1(eval_start_3(v__0, v__01, v_x, v_y)) :|: TRUE 4.66/2.40 eval_start_3(v__0, v__01, v_x, v_y) -> Com_1(eval_start_bb1_in(v_x, v__01, v_x, v_y)) :|: TRUE 4.66/2.40 eval_start_bb1_in(v__0, v__01, v_x, v_y) -> Com_1(eval_start_bb2_in(v__0, v__01, v_x, v_y)) :|: v__0 < v_y 4.66/2.40 eval_start_bb1_in(v__0, v__01, v_x, v_y) -> Com_1(eval_start_bb3_in(v__0, v_y, v_x, v_y)) :|: v__0 >= v_y 4.66/2.40 eval_start_bb2_in(v__0, v__01, v_x, v_y) -> Com_1(eval_start_bb1_in(v__0 + 1, v__01, v_x, v_y)) :|: TRUE 4.66/2.40 eval_start_bb3_in(v__0, v__01, v_x, v_y) -> Com_1(eval_start_bb4_in(v__0, v__01, v_x, v_y)) :|: v__01 < v__0 4.66/2.40 eval_start_bb3_in(v__0, v__01, v_x, v_y) -> Com_1(eval_start_bb5_in(v__0, v__01, v_x, v_y)) :|: v__01 >= v__0 4.66/2.40 eval_start_bb4_in(v__0, v__01, v_x, v_y) -> Com_1(eval_start_bb3_in(v__0, v__01 + 1, v_x, v_y)) :|: TRUE 4.66/2.40 eval_start_bb5_in(v__0, v__01, v_x, v_y) -> Com_1(eval_start_stop(v__0, v__01, v_x, v_y)) :|: TRUE 4.66/2.40 4.66/2.40 The start-symbols are:[eval_start_start_4] 4.66/2.40 4.66/2.40 4.66/2.40 ---------------------------------------- 4.66/2.40 4.66/2.40 (1) Koat Proof (FINISHED) 4.66/2.40 YES(?, 14*ar_1 + 20*ar_2 + 21) 4.66/2.40 4.66/2.40 4.66/2.40 4.66/2.40 Initial complexity problem: 4.66/2.40 4.66/2.40 1: T: 4.66/2.40 4.66/2.40 (Comp: ?, Cost: 1) evalstartstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: ?, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: ?, Cost: 1) evalstart0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: ?, Cost: 1) evalstart1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: ?, Cost: 1) evalstart2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: ?, Cost: 1) evalstart3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_1, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ] 4.66/2.40 4.66/2.40 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_2)) [ ar_0 >= ar_2 ] 4.66/2.40 4.66/2.40 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: ?, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 + 1 ] 4.66/2.40 4.66/2.40 (Comp: ?, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 ] 4.66/2.40 4.66/2.40 (Comp: ?, Cost: 1) evalstartbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3 + 1)) 4.66/2.40 4.66/2.40 (Comp: ?, Cost: 1) evalstartbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] 4.66/2.40 4.66/2.40 start location: koat_start 4.66/2.40 4.66/2.40 leaf cost: 0 4.66/2.40 4.66/2.40 4.66/2.40 4.66/2.40 Repeatedly propagating knowledge in problem 1 produces the following problem: 4.66/2.40 4.66/2.40 2: T: 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstartstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstart0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstart1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstart2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_1, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ] 4.66/2.40 4.66/2.40 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_2)) [ ar_0 >= ar_2 ] 4.66/2.40 4.66/2.40 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: ?, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 + 1 ] 4.66/2.40 4.66/2.40 (Comp: ?, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 ] 4.66/2.40 4.66/2.40 (Comp: ?, Cost: 1) evalstartbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3 + 1)) 4.66/2.40 4.66/2.40 (Comp: ?, Cost: 1) evalstartbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] 4.66/2.40 4.66/2.40 start location: koat_start 4.66/2.40 4.66/2.40 leaf cost: 0 4.66/2.40 4.66/2.40 4.66/2.40 4.66/2.40 A polynomial rank function with 4.66/2.40 4.66/2.40 Pol(evalstartstart) = 3 4.66/2.40 4.66/2.40 Pol(evalstartbb0in) = 3 4.66/2.40 4.66/2.40 Pol(evalstart0) = 3 4.66/2.40 4.66/2.40 Pol(evalstart1) = 3 4.66/2.40 4.66/2.40 Pol(evalstart2) = 3 4.66/2.40 4.66/2.40 Pol(evalstart3) = 3 4.66/2.40 4.66/2.40 Pol(evalstartbb1in) = 3 4.66/2.40 4.66/2.40 Pol(evalstartbb2in) = 3 4.66/2.40 4.66/2.40 Pol(evalstartbb3in) = 2 4.66/2.40 4.66/2.40 Pol(evalstartbb4in) = 2 4.66/2.40 4.66/2.40 Pol(evalstartbb5in) = 1 4.66/2.40 4.66/2.40 Pol(evalstartstop) = 0 4.66/2.40 4.66/2.40 Pol(koat_start) = 3 4.66/2.40 4.66/2.40 orients all transitions weakly and the transitions 4.66/2.40 4.66/2.40 evalstartbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 ] 4.66/2.40 4.66/2.40 evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_2)) [ ar_0 >= ar_2 ] 4.66/2.40 4.66/2.40 strictly and produces the following problem: 4.66/2.40 4.66/2.40 3: T: 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstartstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstart0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstart1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstart2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_1, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ] 4.66/2.40 4.66/2.40 (Comp: 3, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_2)) [ ar_0 >= ar_2 ] 4.66/2.40 4.66/2.40 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: ?, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 + 1 ] 4.66/2.40 4.66/2.40 (Comp: 3, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 ] 4.66/2.40 4.66/2.40 (Comp: ?, Cost: 1) evalstartbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3 + 1)) 4.66/2.40 4.66/2.40 (Comp: 3, Cost: 1) evalstartbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] 4.66/2.40 4.66/2.40 start location: koat_start 4.66/2.40 4.66/2.40 leaf cost: 0 4.66/2.40 4.66/2.40 4.66/2.40 4.66/2.40 A polynomial rank function with 4.66/2.40 4.66/2.40 Pol(evalstartstart) = -V_2 + V_3 4.66/2.40 4.66/2.40 Pol(evalstartbb0in) = -V_2 + V_3 4.66/2.40 4.66/2.40 Pol(evalstart0) = -V_2 + V_3 4.66/2.40 4.66/2.40 Pol(evalstart1) = -V_2 + V_3 4.66/2.40 4.66/2.40 Pol(evalstart2) = -V_2 + V_3 4.66/2.40 4.66/2.40 Pol(evalstart3) = -V_2 + V_3 4.66/2.40 4.66/2.40 Pol(evalstartbb1in) = -V_1 + V_3 4.66/2.40 4.66/2.40 Pol(evalstartbb2in) = -V_1 + V_3 - 1 4.66/2.40 4.66/2.40 Pol(evalstartbb3in) = -V_1 + V_3 4.66/2.40 4.66/2.40 Pol(evalstartbb4in) = -V_1 + V_3 4.66/2.40 4.66/2.40 Pol(evalstartbb5in) = -V_1 + V_3 4.66/2.40 4.66/2.40 Pol(evalstartstop) = -V_1 + V_3 4.66/2.40 4.66/2.40 Pol(koat_start) = -V_2 + V_3 4.66/2.40 4.66/2.40 orients all transitions weakly and the transition 4.66/2.40 4.66/2.40 evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ] 4.66/2.40 4.66/2.40 strictly and produces the following problem: 4.66/2.40 4.66/2.40 4: T: 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstartstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstart0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstart1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstart2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_1, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: ar_1 + ar_2, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ] 4.66/2.40 4.66/2.40 (Comp: 3, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_2)) [ ar_0 >= ar_2 ] 4.66/2.40 4.66/2.40 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: ?, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 + 1 ] 4.66/2.40 4.66/2.40 (Comp: 3, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 ] 4.66/2.40 4.66/2.40 (Comp: ?, Cost: 1) evalstartbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3 + 1)) 4.66/2.40 4.66/2.40 (Comp: 3, Cost: 1) evalstartbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] 4.66/2.40 4.66/2.40 start location: koat_start 4.66/2.40 4.66/2.40 leaf cost: 0 4.66/2.40 4.66/2.40 4.66/2.40 4.66/2.40 Repeatedly propagating knowledge in problem 4 produces the following problem: 4.66/2.40 4.66/2.40 5: T: 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstartstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstart0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstart1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstart2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_1, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: ar_1 + ar_2, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ] 4.66/2.40 4.66/2.40 (Comp: 3, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_2)) [ ar_0 >= ar_2 ] 4.66/2.40 4.66/2.40 (Comp: ar_1 + ar_2, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: ?, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 + 1 ] 4.66/2.40 4.66/2.40 (Comp: 3, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 ] 4.66/2.40 4.66/2.40 (Comp: ?, Cost: 1) evalstartbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3 + 1)) 4.66/2.40 4.66/2.40 (Comp: 3, Cost: 1) evalstartbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] 4.66/2.40 4.66/2.40 start location: koat_start 4.66/2.40 4.66/2.40 leaf cost: 0 4.66/2.40 4.66/2.40 4.66/2.40 4.66/2.40 A polynomial rank function with 4.66/2.40 4.66/2.40 Pol(evalstartbb4in) = V_1 - V_4 4.66/2.40 4.66/2.40 Pol(evalstartbb3in) = V_1 - V_4 + 1 4.66/2.40 4.66/2.40 and size complexities 4.66/2.40 4.66/2.40 S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-0) = ar_0 4.66/2.40 4.66/2.40 S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-1) = ar_1 4.66/2.40 4.66/2.40 S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-2) = ar_2 4.66/2.40 4.66/2.40 S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-3) = ar_3 4.66/2.40 4.66/2.40 S("evalstartbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3))", 0-0) = 2*ar_1 + 2*ar_2 4.66/2.40 4.66/2.40 S("evalstartbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 4.66/2.40 4.66/2.40 S("evalstartbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 4.66/2.40 4.66/2.40 S("evalstartbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3))", 0-3) = ? 4.66/2.40 4.66/2.40 S("evalstartbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3 + 1))", 0-0) = 2*ar_1 + 2*ar_2 4.66/2.40 4.66/2.40 S("evalstartbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3 + 1))", 0-1) = ar_1 4.66/2.40 4.66/2.40 S("evalstartbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3 + 1))", 0-2) = ar_2 4.66/2.40 4.66/2.40 S("evalstartbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3 + 1))", 0-3) = ? 4.66/2.40 4.66/2.40 S("evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 ]", 0-0) = 2*ar_1 + 2*ar_2 4.66/2.40 4.66/2.40 S("evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 ]", 0-1) = ar_1 4.66/2.40 4.66/2.40 S("evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 ]", 0-2) = ar_2 4.66/2.40 4.66/2.40 S("evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 ]", 0-3) = ? 4.66/2.40 4.66/2.40 S("evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 + 1 ]", 0-0) = 2*ar_1 + 2*ar_2 4.66/2.40 4.66/2.40 S("evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 + 1 ]", 0-1) = ar_1 4.66/2.40 4.66/2.40 S("evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 + 1 ]", 0-2) = ar_2 4.66/2.40 4.66/2.40 S("evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 + 1 ]", 0-3) = ? 4.66/2.40 4.66/2.40 S("evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-0) = 2*ar_1 + 2*ar_2 4.66/2.40 4.66/2.40 S("evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-1) = ar_1 4.66/2.40 4.66/2.40 S("evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-2) = ar_2 4.66/2.40 4.66/2.40 S("evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3))", 0-3) = ar_3 4.66/2.40 4.66/2.40 S("evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_2)) [ ar_0 >= ar_2 ]", 0-0) = 2*ar_1 + 2*ar_2 4.66/2.40 4.66/2.40 S("evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_2)) [ ar_0 >= ar_2 ]", 0-1) = ar_1 4.66/2.40 4.66/2.40 S("evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_2)) [ ar_0 >= ar_2 ]", 0-2) = ar_2 4.66/2.40 4.66/2.40 S("evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_2)) [ ar_0 >= ar_2 ]", 0-3) = ar_2 4.66/2.40 4.66/2.40 S("evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ]", 0-0) = 2*ar_1 + 2*ar_2 4.66/2.40 4.66/2.40 S("evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ]", 0-1) = ar_1 4.66/2.40 4.66/2.40 S("evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ]", 0-2) = ar_2 4.66/2.40 4.66/2.40 S("evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ]", 0-3) = ar_3 4.66/2.40 4.66/2.40 S("evalstart3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_1, ar_1, ar_2, ar_3))", 0-0) = ar_1 4.66/2.40 4.66/2.40 S("evalstart3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_1, ar_1, ar_2, ar_3))", 0-1) = ar_1 4.66/2.40 4.66/2.40 S("evalstart3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_1, ar_1, ar_2, ar_3))", 0-2) = ar_2 4.66/2.40 4.66/2.40 S("evalstart3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_1, ar_1, ar_2, ar_3))", 0-3) = ar_3 4.66/2.40 4.66/2.40 S("evalstart2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0 4.66/2.40 4.66/2.40 S("evalstart2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 4.66/2.40 4.66/2.40 S("evalstart2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 4.66/2.40 4.66/2.40 S("evalstart2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 4.66/2.40 4.66/2.40 S("evalstart1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0 4.66/2.40 4.66/2.40 S("evalstart1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 4.66/2.40 4.66/2.40 S("evalstart1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 4.66/2.40 4.66/2.40 S("evalstart1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 4.66/2.40 4.66/2.40 S("evalstart0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0 4.66/2.40 4.66/2.40 S("evalstart0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 4.66/2.40 4.66/2.40 S("evalstart0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 4.66/2.40 4.66/2.40 S("evalstart0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 4.66/2.40 4.66/2.40 S("evalstartbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0 4.66/2.40 4.66/2.40 S("evalstartbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 4.66/2.40 4.66/2.40 S("evalstartbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 4.66/2.40 4.66/2.40 S("evalstartbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 4.66/2.40 4.66/2.40 S("evalstartstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0 4.66/2.40 4.66/2.40 S("evalstartstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 4.66/2.40 4.66/2.40 S("evalstartstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 4.66/2.40 4.66/2.40 S("evalstartstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 4.66/2.40 4.66/2.40 orients the transitions 4.66/2.40 4.66/2.40 evalstartbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3 + 1)) 4.66/2.40 4.66/2.40 evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 + 1 ] 4.66/2.40 4.66/2.40 weakly and the transition 4.66/2.40 4.66/2.40 evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 + 1 ] 4.66/2.40 4.66/2.40 strictly and produces the following problem: 4.66/2.40 4.66/2.40 6: T: 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstartstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstart0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstart1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstart2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_1, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: ar_1 + ar_2, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ] 4.66/2.40 4.66/2.40 (Comp: 3, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_2)) [ ar_0 >= ar_2 ] 4.66/2.40 4.66/2.40 (Comp: ar_1 + ar_2, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 6*ar_1 + 9*ar_2 + 3, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 + 1 ] 4.66/2.40 4.66/2.40 (Comp: 3, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 ] 4.66/2.40 4.66/2.40 (Comp: ?, Cost: 1) evalstartbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3 + 1)) 4.66/2.40 4.66/2.40 (Comp: 3, Cost: 1) evalstartbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] 4.66/2.40 4.66/2.40 start location: koat_start 4.66/2.40 4.66/2.40 leaf cost: 0 4.66/2.40 4.66/2.40 4.66/2.40 4.66/2.40 Repeatedly propagating knowledge in problem 6 produces the following problem: 4.66/2.40 4.66/2.40 7: T: 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstartstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstart0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstart1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstart2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_1, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: ar_1 + ar_2, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ] 4.66/2.40 4.66/2.40 (Comp: 3, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_2)) [ ar_0 >= ar_2 ] 4.66/2.40 4.66/2.40 (Comp: ar_1 + ar_2, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 6*ar_1 + 9*ar_2 + 3, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_3 + 1 ] 4.66/2.40 4.66/2.40 (Comp: 3, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb5in(ar_0, ar_1, ar_2, ar_3)) [ ar_3 >= ar_0 ] 4.66/2.40 4.66/2.40 (Comp: 6*ar_1 + 9*ar_2 + 3, Cost: 1) evalstartbb4in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3 + 1)) 4.66/2.40 4.66/2.40 (Comp: 3, Cost: 1) evalstartbb5in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3)) 4.66/2.40 4.66/2.40 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalstartstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] 4.66/2.40 4.66/2.40 start location: koat_start 4.66/2.40 4.66/2.40 leaf cost: 0 4.66/2.40 4.66/2.40 4.66/2.40 4.66/2.40 Complexity upper bound 14*ar_1 + 20*ar_2 + 21 4.66/2.40 4.66/2.40 4.66/2.40 4.66/2.40 Time: 0.136 sec (SMT: 0.111 sec) 4.66/2.40 4.66/2.40 4.66/2.40 ---------------------------------------- 4.66/2.40 4.66/2.40 (2) 4.66/2.40 BOUNDS(1, n^1) 4.66/2.40 4.66/2.40 ---------------------------------------- 4.66/2.40 4.66/2.40 (3) Loat Proof (FINISHED) 4.66/2.40 4.66/2.40 4.66/2.40 ### Pre-processing the ITS problem ### 4.66/2.40 4.66/2.40 4.66/2.40 4.66/2.40 Initial linear ITS problem 4.66/2.40 4.66/2.40 Start location: evalstartstart 4.66/2.40 4.66/2.40 0: evalstartstart -> evalstartbb0in : [], cost: 1 4.66/2.40 4.66/2.40 1: evalstartbb0in -> evalstart0 : [], cost: 1 4.66/2.40 4.66/2.40 2: evalstart0 -> evalstart1 : [], cost: 1 4.66/2.40 4.66/2.40 3: evalstart1 -> evalstart2 : [], cost: 1 4.66/2.40 4.66/2.40 4: evalstart2 -> evalstart3 : [], cost: 1 4.66/2.40 4.66/2.40 5: evalstart3 -> evalstartbb1in : A'=B, [], cost: 1 4.66/2.40 4.66/2.40 6: evalstartbb1in -> evalstartbb2in : [ C>=1+A ], cost: 1 4.66/2.40 4.66/2.40 7: evalstartbb1in -> evalstartbb3in : D'=C, [ A>=C ], cost: 1 4.66/2.40 4.66/2.40 8: evalstartbb2in -> evalstartbb1in : A'=1+A, [], cost: 1 4.66/2.40 4.66/2.40 9: evalstartbb3in -> evalstartbb4in : [ A>=1+D ], cost: 1 4.66/2.40 4.66/2.40 10: evalstartbb3in -> evalstartbb5in : [ D>=A ], cost: 1 4.66/2.40 4.66/2.40 11: evalstartbb4in -> evalstartbb3in : D'=1+D, [], cost: 1 4.66/2.40 4.66/2.40 12: evalstartbb5in -> evalstartstop : [], cost: 1 4.66/2.40 4.66/2.40 4.66/2.40 4.66/2.40 Removed unreachable and leaf rules: 4.66/2.40 4.66/2.40 Start location: evalstartstart 4.66/2.40 4.66/2.40 0: evalstartstart -> evalstartbb0in : [], cost: 1 4.66/2.40 4.66/2.40 1: evalstartbb0in -> evalstart0 : [], cost: 1 4.66/2.40 4.66/2.40 2: evalstart0 -> evalstart1 : [], cost: 1 4.66/2.40 4.66/2.40 3: evalstart1 -> evalstart2 : [], cost: 1 4.66/2.40 4.66/2.40 4: evalstart2 -> evalstart3 : [], cost: 1 4.66/2.40 4.66/2.40 5: evalstart3 -> evalstartbb1in : A'=B, [], cost: 1 4.66/2.40 4.66/2.40 6: evalstartbb1in -> evalstartbb2in : [ C>=1+A ], cost: 1 4.66/2.40 4.66/2.40 7: evalstartbb1in -> evalstartbb3in : D'=C, [ A>=C ], cost: 1 4.66/2.40 4.66/2.40 8: evalstartbb2in -> evalstartbb1in : A'=1+A, [], cost: 1 4.66/2.40 4.66/2.40 9: evalstartbb3in -> evalstartbb4in : [ A>=1+D ], cost: 1 4.66/2.40 4.66/2.40 11: evalstartbb4in -> evalstartbb3in : D'=1+D, [], cost: 1 4.66/2.40 4.66/2.40 4.66/2.40 4.66/2.40 ### Simplification by acceleration and chaining ### 4.66/2.40 4.66/2.40 4.66/2.40 4.66/2.40 Eliminated locations (on linear paths): 4.66/2.40 4.66/2.40 Start location: evalstartstart 4.66/2.40 4.66/2.40 17: evalstartstart -> evalstartbb1in : A'=B, [], cost: 6 4.66/2.40 4.66/2.40 7: evalstartbb1in -> evalstartbb3in : D'=C, [ A>=C ], cost: 1 4.66/2.40 4.66/2.40 18: evalstartbb1in -> evalstartbb1in : A'=1+A, [ C>=1+A ], cost: 2 4.66/2.40 4.66/2.40 19: evalstartbb3in -> evalstartbb3in : D'=1+D, [ A>=1+D ], cost: 2 4.66/2.40 4.66/2.40 4.66/2.40 4.66/2.40 Accelerating simple loops of location 6. 4.66/2.40 4.66/2.40 Accelerating the following rules: 4.66/2.40 4.66/2.40 18: evalstartbb1in -> evalstartbb1in : A'=1+A, [ C>=1+A ], cost: 2 4.66/2.40 4.66/2.40 4.66/2.40 4.66/2.40 Accelerated rule 18 with metering function C-A, yielding the new rule 20. 4.66/2.40 4.66/2.40 Removing the simple loops: 18. 4.66/2.40 4.66/2.40 4.66/2.40 4.66/2.40 Accelerating simple loops of location 8. 4.66/2.40 4.66/2.40 Accelerating the following rules: 4.66/2.40 4.66/2.40 19: evalstartbb3in -> evalstartbb3in : D'=1+D, [ A>=1+D ], cost: 2 4.66/2.40 4.66/2.40 4.66/2.40 4.66/2.40 Accelerated rule 19 with metering function -D+A, yielding the new rule 21. 4.66/2.40 4.66/2.40 Removing the simple loops: 19. 4.66/2.40 4.66/2.40 4.66/2.40 4.66/2.40 Accelerated all simple loops using metering functions (where possible): 4.66/2.40 4.66/2.40 Start location: evalstartstart 4.66/2.40 4.66/2.40 17: evalstartstart -> evalstartbb1in : A'=B, [], cost: 6 4.66/2.40 4.66/2.40 7: evalstartbb1in -> evalstartbb3in : D'=C, [ A>=C ], cost: 1 4.66/2.40 4.66/2.40 20: evalstartbb1in -> evalstartbb1in : A'=C, [ C>=1+A ], cost: 2*C-2*A 4.66/2.40 4.66/2.40 21: evalstartbb3in -> evalstartbb3in : D'=A, [ A>=1+D ], cost: -2*D+2*A 4.66/2.40 4.66/2.40 4.66/2.40 4.66/2.40 Chained accelerated rules (with incoming rules): 4.66/2.40 4.66/2.40 Start location: evalstartstart 4.66/2.40 4.66/2.40 17: evalstartstart -> evalstartbb1in : A'=B, [], cost: 6 4.66/2.40 4.66/2.40 22: evalstartstart -> evalstartbb1in : A'=C, [ C>=1+B ], cost: 6+2*C-2*B 4.66/2.40 4.66/2.40 7: evalstartbb1in -> evalstartbb3in : D'=C, [ A>=C ], cost: 1 4.66/2.40 4.66/2.40 23: evalstartbb1in -> evalstartbb3in : D'=A, [ A>=1+C ], cost: 1-2*C+2*A 4.66/2.40 4.66/2.40 4.66/2.40 4.66/2.40 Removed unreachable locations (and leaf rules with constant cost): 4.66/2.40 4.66/2.40 Start location: evalstartstart 4.66/2.40 4.66/2.40 17: evalstartstart -> evalstartbb1in : A'=B, [], cost: 6 4.66/2.40 4.66/2.40 22: evalstartstart -> evalstartbb1in : A'=C, [ C>=1+B ], cost: 6+2*C-2*B 4.66/2.40 4.66/2.40 23: evalstartbb1in -> evalstartbb3in : D'=A, [ A>=1+C ], cost: 1-2*C+2*A 4.66/2.40 4.66/2.40 4.66/2.40 4.66/2.40 Eliminated locations (on tree-shaped paths): 4.66/2.40 4.66/2.40 Start location: evalstartstart 4.66/2.40 4.66/2.40 24: evalstartstart -> evalstartbb3in : A'=B, D'=B, [ B>=1+C ], cost: 7-2*C+2*B 4.66/2.40 4.66/2.40 25: evalstartstart -> [14] : [ C>=1+B ], cost: 6+2*C-2*B 4.66/2.40 4.66/2.40 4.66/2.40 4.66/2.40 ### Computing asymptotic complexity ### 4.66/2.40 4.66/2.40 4.66/2.40 4.66/2.40 Fully simplified ITS problem 4.66/2.40 4.66/2.40 Start location: evalstartstart 4.66/2.40 4.66/2.40 24: evalstartstart -> evalstartbb3in : A'=B, D'=B, [ B>=1+C ], cost: 7-2*C+2*B 4.66/2.40 4.66/2.40 25: evalstartstart -> [14] : [ C>=1+B ], cost: 6+2*C-2*B 4.66/2.40 4.66/2.40 4.66/2.40 4.66/2.40 Computing asymptotic complexity for rule 24 4.66/2.40 4.66/2.40 Solved the limit problem by the following transformations: 4.66/2.40 4.66/2.40 Created initial limit problem: 4.66/2.40 4.66/2.40 7-2*C+2*B (+), -C+B (+/+!) [not solved] 4.66/2.40 4.66/2.40 4.66/2.40 4.66/2.40 removing all constraints (solved by SMT) 4.66/2.40 4.66/2.40 resulting limit problem: [solved] 4.66/2.40 4.66/2.40 4.66/2.40 4.66/2.40 applying transformation rule (C) using substitution {C==0,B==n} 4.66/2.40 4.66/2.40 resulting limit problem: 4.66/2.40 4.66/2.40 [solved] 4.66/2.40 4.66/2.40 4.66/2.40 4.66/2.40 Solution: 4.66/2.40 4.66/2.40 C / 0 4.66/2.40 4.66/2.40 B / n 4.66/2.40 4.66/2.40 Resulting cost 7+2*n has complexity: Poly(n^1) 4.66/2.40 4.66/2.40 4.66/2.40 4.66/2.40 Found new complexity Poly(n^1). 4.66/2.40 4.66/2.40 4.66/2.40 4.66/2.40 Obtained the following overall complexity (w.r.t. the length of the input n): 4.66/2.40 4.66/2.40 Complexity: Poly(n^1) 4.66/2.40 4.66/2.40 Cpx degree: 1 4.66/2.40 4.66/2.40 Solved cost: 7+2*n 4.66/2.40 4.66/2.40 Rule cost: 7-2*C+2*B 4.66/2.40 4.66/2.40 Rule guard: [ B>=1+C ] 4.66/2.40 4.66/2.40 4.66/2.40 4.66/2.40 WORST_CASE(Omega(n^1),?) 4.66/2.40 4.66/2.40 4.66/2.40 ---------------------------------------- 4.66/2.40 4.66/2.40 (4) 4.66/2.40 BOUNDS(n^1, INF) 4.79/2.42 EOF