4.63/2.29 WORST_CASE(Omega(n^1), O(n^1)) 4.63/2.29 proof of /export/starexec/sandbox/benchmark/theBenchmark.koat 4.63/2.29 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.63/2.29 4.63/2.29 4.63/2.29 The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1). 4.63/2.29 4.63/2.29 (0) CpxIntTrs 4.63/2.29 (1) Koat Proof [FINISHED, 119 ms] 4.63/2.29 (2) BOUNDS(1, n^1) 4.63/2.29 (3) Loat Proof [FINISHED, 631 ms] 4.63/2.29 (4) BOUNDS(n^1, INF) 4.63/2.29 4.63/2.29 4.63/2.29 ---------------------------------------- 4.63/2.29 4.63/2.29 (0) 4.63/2.29 Obligation: 4.63/2.29 Complexity Int TRS consisting of the following rules: 4.63/2.29 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 4.63/2.29 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 4.63/2.29 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 4.63/2.29 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 4.63/2.29 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 4.63/2.29 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 4.63/2.29 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 4.63/2.29 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 4.63/2.29 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 4.63/2.29 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 4.63/2.29 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 4.63/2.29 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 4.63/2.29 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 4.63/2.29 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 4.63/2.29 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 4.63/2.29 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 4.63/2.29 4.63/2.29 The start-symbols are:[eval_start_start_5] 4.63/2.29 4.63/2.29 4.63/2.29 ---------------------------------------- 4.63/2.29 4.63/2.29 (1) Koat Proof (FINISHED) 4.63/2.29 YES(?, 5*ar_1 + 5*ar_3 + 11) 4.63/2.29 4.63/2.29 4.63/2.29 4.63/2.29 Initial complexity problem: 4.63/2.29 4.63/2.29 1: T: 4.63/2.29 4.63/2.29 (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)) 4.63/2.29 4.63/2.29 (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)) 4.63/2.29 4.63/2.29 (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)) 4.63/2.29 4.63/2.29 (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)) 4.63/2.29 4.63/2.29 (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)) 4.63/2.29 4.63/2.29 (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)) 4.63/2.29 4.63/2.29 (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)) 4.63/2.29 4.63/2.29 (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 ] 4.63/2.29 4.63/2.29 (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 ] 4.63/2.29 4.63/2.29 (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)) 4.63/2.29 4.63/2.29 (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, f)) 4.63/2.29 4.63/2.29 (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 ] 4.63/2.29 4.63/2.29 (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 ] 4.63/2.29 4.63/2.29 (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 ] 4.63/2.29 4.63/2.29 (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 ] 4.63/2.29 4.63/2.29 (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)) 4.63/2.29 4.63/2.29 (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 ] 4.63/2.29 4.63/2.29 start location: koat_start 4.63/2.29 4.63/2.29 leaf cost: 0 4.63/2.29 4.63/2.29 4.63/2.29 4.63/2.29 Testing for reachability in the complexity graph removes the following transitions from problem 1: 4.63/2.29 4.63/2.29 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 ] 4.63/2.29 4.63/2.29 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 ] 4.63/2.29 4.63/2.29 We thus obtain the following problem: 4.63/2.29 4.63/2.29 2: T: 4.63/2.29 4.63/2.29 (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 ] 4.63/2.30 4.63/2.30 (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 ] 4.63/2.30 4.63/2.30 (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, f)) 4.63/2.30 4.63/2.30 (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)) 4.63/2.30 4.63/2.30 (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)) 4.63/2.30 4.63/2.30 (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 ] 4.63/2.30 4.63/2.30 (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 ] 4.63/2.30 4.63/2.30 (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)) 4.63/2.30 4.63/2.30 (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)) 4.63/2.30 4.63/2.30 (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)) 4.63/2.30 4.63/2.30 (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)) 4.63/2.30 4.63/2.30 (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)) 4.63/2.30 4.63/2.30 (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)) 4.63/2.30 4.63/2.30 (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)) 4.63/2.30 4.63/2.30 (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 ] 4.63/2.30 4.63/2.30 start location: koat_start 4.63/2.30 4.63/2.30 leaf cost: 0 4.63/2.30 4.63/2.30 4.63/2.30 4.63/2.30 Repeatedly propagating knowledge in problem 2 produces the following problem: 4.63/2.30 4.63/2.30 3: T: 4.63/2.30 4.63/2.30 (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 ] 4.63/2.30 4.63/2.30 (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 ] 4.63/2.30 4.63/2.30 (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, f)) 4.63/2.30 4.63/2.30 (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)) 4.63/2.30 4.63/2.30 (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)) 4.63/2.30 4.63/2.30 (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 ] 4.63/2.30 4.63/2.30 (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 ] 4.63/2.30 4.63/2.30 (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)) 4.63/2.30 4.63/2.30 (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)) 4.63/2.30 4.63/2.30 (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)) 4.63/2.30 4.63/2.30 (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)) 4.63/2.30 4.63/2.30 (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)) 4.63/2.30 4.63/2.30 (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)) 4.63/2.30 4.63/2.30 (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)) 4.63/2.30 4.63/2.30 (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 ] 4.63/2.30 4.63/2.30 start location: koat_start 4.63/2.30 4.63/2.30 leaf cost: 0 4.63/2.30 4.63/2.30 4.63/2.30 4.63/2.30 A polynomial rank function with 4.63/2.30 4.63/2.30 Pol(evalstart6) = 2 4.63/2.30 4.63/2.30 Pol(evalstartbb1in) = 2 4.63/2.30 4.63/2.30 Pol(evalstart5) = 2 4.63/2.30 4.63/2.30 Pol(evalstartbb3in) = 1 4.63/2.30 4.63/2.30 Pol(evalstartstop) = 0 4.63/2.30 4.63/2.30 Pol(evalstartbb2in) = 2 4.63/2.30 4.63/2.30 Pol(evalstart4) = 2 4.63/2.30 4.63/2.30 Pol(evalstart3) = 2 4.63/2.30 4.63/2.30 Pol(evalstart2) = 2 4.63/2.30 4.63/2.30 Pol(evalstart1) = 2 4.63/2.30 4.63/2.30 Pol(evalstart0) = 2 4.63/2.30 4.63/2.30 Pol(evalstartbb0in) = 2 4.63/2.30 4.63/2.30 Pol(evalstartstart) = 2 4.63/2.30 4.63/2.30 Pol(koat_start) = 2 4.63/2.30 4.63/2.30 orients all transitions weakly and the transitions 4.63/2.30 4.63/2.30 evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3, ar_4)) 4.63/2.30 4.63/2.30 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 ] 4.63/2.30 4.63/2.30 strictly and produces the following problem: 4.63/2.30 4.63/2.30 4: T: 4.63/2.30 4.63/2.30 (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 ] 4.63/2.30 4.63/2.30 (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 ] 4.63/2.30 4.63/2.30 (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, f)) 4.63/2.30 4.63/2.30 (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)) 4.63/2.30 4.63/2.30 (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)) 4.63/2.30 4.63/2.30 (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 ] 4.63/2.30 4.63/2.30 (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 ] 4.63/2.30 4.63/2.30 (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)) 4.63/2.30 4.63/2.30 (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)) 4.63/2.30 4.63/2.30 (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)) 4.63/2.30 4.63/2.30 (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)) 4.63/2.30 4.63/2.30 (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)) 4.63/2.30 4.63/2.30 (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)) 4.63/2.30 4.63/2.30 (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)) 4.63/2.30 4.63/2.30 (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 ] 4.63/2.30 4.63/2.30 start location: koat_start 4.63/2.30 4.63/2.30 leaf cost: 0 4.63/2.30 4.63/2.30 4.63/2.30 4.63/2.30 A polynomial rank function with 4.63/2.30 4.63/2.30 Pol(evalstart6) = V_1 - V_3 - 1 4.63/2.30 4.63/2.30 Pol(evalstartbb1in) = V_1 - V_3 4.63/2.30 4.63/2.30 Pol(evalstart5) = V_1 - V_3 - 1 4.63/2.30 4.63/2.30 Pol(evalstartbb3in) = V_1 - V_3 4.63/2.30 4.63/2.30 Pol(evalstartstop) = V_1 - V_3 4.63/2.30 4.63/2.30 Pol(evalstartbb2in) = V_1 - V_3 - 1 4.63/2.30 4.63/2.30 Pol(evalstart4) = V_2 - V_4 4.63/2.30 4.63/2.30 Pol(evalstart3) = V_2 - V_4 4.63/2.30 4.63/2.30 Pol(evalstart2) = V_2 - V_4 4.63/2.30 4.63/2.30 Pol(evalstart1) = V_2 - V_4 4.63/2.30 4.63/2.30 Pol(evalstart0) = V_2 - V_4 4.63/2.30 4.63/2.30 Pol(evalstartbb0in) = V_2 - V_4 4.63/2.30 4.63/2.30 Pol(evalstartstart) = V_2 - V_4 4.63/2.30 4.63/2.30 Pol(koat_start) = V_2 - V_4 4.63/2.30 4.63/2.30 orients all transitions weakly and the transition 4.63/2.30 4.63/2.30 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 ] 4.63/2.30 4.63/2.30 strictly and produces the following problem: 4.63/2.30 4.63/2.30 5: T: 4.63/2.30 4.63/2.30 (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 ] 4.63/2.30 4.63/2.30 (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 ] 4.63/2.30 4.63/2.30 (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, f)) 4.63/2.30 4.63/2.30 (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)) 4.63/2.30 4.63/2.30 (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)) 4.63/2.30 4.63/2.30 (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 ] 4.63/2.30 4.63/2.30 (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 ] 4.79/2.30 4.79/2.30 (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)) 4.79/2.30 4.79/2.30 (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)) 4.79/2.30 4.79/2.30 (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)) 4.79/2.30 4.79/2.30 (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)) 4.79/2.30 4.79/2.30 (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)) 4.79/2.30 4.79/2.30 (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)) 4.79/2.30 4.79/2.30 (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)) 4.79/2.30 4.79/2.30 (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 ] 4.79/2.30 4.79/2.30 start location: koat_start 4.79/2.30 4.79/2.30 leaf cost: 0 4.79/2.30 4.79/2.30 4.79/2.30 4.79/2.30 Repeatedly propagating knowledge in problem 5 produces the following problem: 4.79/2.30 4.79/2.30 6: T: 4.79/2.30 4.79/2.30 (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 ] 4.79/2.30 4.79/2.30 (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 ] 4.79/2.30 4.79/2.30 (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, f)) 4.79/2.30 4.79/2.30 (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)) 4.79/2.30 4.79/2.30 (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)) 4.79/2.30 4.79/2.30 (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 ] 4.79/2.30 4.79/2.30 (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 ] 4.79/2.30 4.79/2.30 (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)) 4.79/2.30 4.79/2.30 (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)) 4.79/2.30 4.79/2.30 (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)) 4.79/2.30 4.79/2.30 (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)) 4.79/2.30 4.79/2.30 (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)) 4.79/2.30 4.79/2.30 (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)) 4.79/2.30 4.79/2.30 (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)) 4.79/2.30 4.79/2.30 (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 ] 4.79/2.30 4.79/2.30 start location: koat_start 4.79/2.30 4.79/2.30 leaf cost: 0 4.79/2.30 4.79/2.30 4.79/2.30 4.79/2.30 Complexity upper bound 5*ar_1 + 5*ar_3 + 11 4.79/2.30 4.79/2.30 4.79/2.30 4.79/2.30 Time: 0.145 sec (SMT: 0.121 sec) 4.79/2.30 4.79/2.30 4.79/2.30 ---------------------------------------- 4.79/2.30 4.79/2.30 (2) 4.79/2.30 BOUNDS(1, n^1) 4.79/2.30 4.79/2.30 ---------------------------------------- 4.79/2.30 4.79/2.30 (3) Loat Proof (FINISHED) 4.79/2.30 4.79/2.30 4.79/2.30 ### Pre-processing the ITS problem ### 4.79/2.30 4.79/2.30 4.79/2.30 4.79/2.30 Initial linear ITS problem 4.79/2.30 4.79/2.30 Start location: evalstartstart 4.79/2.30 4.79/2.30 0: evalstartstart -> evalstartbb0in : [], cost: 1 4.79/2.30 4.79/2.30 1: evalstartbb0in -> evalstart0 : [], cost: 1 4.79/2.30 4.79/2.30 2: evalstart0 -> evalstart1 : [], cost: 1 4.79/2.30 4.79/2.30 3: evalstart1 -> evalstart2 : [], cost: 1 4.79/2.30 4.79/2.30 4: evalstart2 -> evalstart3 : [], cost: 1 4.79/2.30 4.79/2.30 5: evalstart3 -> evalstart4 : [], cost: 1 4.79/2.30 4.79/2.30 6: evalstart4 -> evalstartbb1in : A'=B, C'=D, [], cost: 1 4.79/2.30 4.79/2.30 7: evalstartbb1in -> evalstartbb2in : [ A>=1+C ], cost: 1 4.79/2.30 4.79/2.30 8: evalstartbb1in -> evalstartbb3in : [ C>=A ], cost: 1 4.79/2.30 4.79/2.30 9: evalstartbb2in -> evalstart5 : [], cost: 1 4.79/2.30 4.79/2.30 10: evalstart5 -> evalstart6 : E'=free, [], cost: 1 4.79/2.30 4.79/2.30 11: evalstart6 -> evalstartbb1in : C'=1+C, [ E>=1 ], cost: 1 4.79/2.30 4.79/2.30 12: evalstart6 -> evalstartbb1in : A'=-1+A, C'=1+C, [ E>=1 && 0>=E ], cost: 1 4.79/2.30 4.79/2.30 13: evalstart6 -> evalstartbb1in : [ 0>=E && E>=1 ], cost: 1 4.79/2.30 4.79/2.30 14: evalstart6 -> evalstartbb1in : A'=-1+A, [ 0>=E ], cost: 1 4.79/2.30 4.79/2.30 15: evalstartbb3in -> evalstartstop : [], cost: 1 4.79/2.30 4.79/2.30 4.79/2.30 4.79/2.30 Removed unreachable and leaf rules: 4.79/2.30 4.79/2.30 Start location: evalstartstart 4.79/2.30 4.79/2.30 0: evalstartstart -> evalstartbb0in : [], cost: 1 4.79/2.30 4.79/2.30 1: evalstartbb0in -> evalstart0 : [], cost: 1 4.79/2.30 4.79/2.30 2: evalstart0 -> evalstart1 : [], cost: 1 4.79/2.30 4.79/2.30 3: evalstart1 -> evalstart2 : [], cost: 1 4.79/2.30 4.79/2.30 4: evalstart2 -> evalstart3 : [], cost: 1 4.79/2.30 4.79/2.30 5: evalstart3 -> evalstart4 : [], cost: 1 4.79/2.30 4.79/2.30 6: evalstart4 -> evalstartbb1in : A'=B, C'=D, [], cost: 1 4.79/2.30 4.79/2.30 7: evalstartbb1in -> evalstartbb2in : [ A>=1+C ], cost: 1 4.79/2.30 4.79/2.30 9: evalstartbb2in -> evalstart5 : [], cost: 1 4.79/2.30 4.79/2.30 10: evalstart5 -> evalstart6 : E'=free, [], cost: 1 4.79/2.30 4.79/2.30 11: evalstart6 -> evalstartbb1in : C'=1+C, [ E>=1 ], cost: 1 4.79/2.30 4.79/2.30 12: evalstart6 -> evalstartbb1in : A'=-1+A, C'=1+C, [ E>=1 && 0>=E ], cost: 1 4.79/2.30 4.79/2.30 13: evalstart6 -> evalstartbb1in : [ 0>=E && E>=1 ], cost: 1 4.79/2.30 4.79/2.30 14: evalstart6 -> evalstartbb1in : A'=-1+A, [ 0>=E ], cost: 1 4.79/2.30 4.79/2.30 4.79/2.30 4.79/2.30 Removed rules with unsatisfiable guard: 4.79/2.30 4.79/2.30 Start location: evalstartstart 4.79/2.30 4.79/2.30 0: evalstartstart -> evalstartbb0in : [], cost: 1 4.79/2.30 4.79/2.30 1: evalstartbb0in -> evalstart0 : [], cost: 1 4.79/2.30 4.79/2.30 2: evalstart0 -> evalstart1 : [], cost: 1 4.79/2.30 4.79/2.30 3: evalstart1 -> evalstart2 : [], cost: 1 4.79/2.30 4.79/2.30 4: evalstart2 -> evalstart3 : [], cost: 1 4.79/2.30 4.79/2.30 5: evalstart3 -> evalstart4 : [], cost: 1 4.79/2.30 4.79/2.30 6: evalstart4 -> evalstartbb1in : A'=B, C'=D, [], cost: 1 4.79/2.30 4.79/2.30 7: evalstartbb1in -> evalstartbb2in : [ A>=1+C ], cost: 1 4.79/2.30 4.79/2.30 9: evalstartbb2in -> evalstart5 : [], cost: 1 4.79/2.30 4.79/2.30 10: evalstart5 -> evalstart6 : E'=free, [], cost: 1 4.79/2.30 4.79/2.30 11: evalstart6 -> evalstartbb1in : C'=1+C, [ E>=1 ], cost: 1 4.79/2.30 4.79/2.30 14: evalstart6 -> evalstartbb1in : A'=-1+A, [ 0>=E ], cost: 1 4.79/2.30 4.79/2.30 4.79/2.30 4.79/2.30 ### Simplification by acceleration and chaining ### 4.79/2.30 4.79/2.30 4.79/2.30 4.79/2.30 Eliminated locations (on linear paths): 4.79/2.30 4.79/2.30 Start location: evalstartstart 4.79/2.30 4.79/2.30 21: evalstartstart -> evalstartbb1in : A'=B, C'=D, [], cost: 7 4.79/2.30 4.79/2.30 23: evalstartbb1in -> evalstart6 : E'=free, [ A>=1+C ], cost: 3 4.79/2.30 4.79/2.30 11: evalstart6 -> evalstartbb1in : C'=1+C, [ E>=1 ], cost: 1 4.79/2.30 4.79/2.30 14: evalstart6 -> evalstartbb1in : A'=-1+A, [ 0>=E ], cost: 1 4.79/2.30 4.79/2.30 4.79/2.30 4.79/2.30 Eliminated locations (on tree-shaped paths): 4.79/2.30 4.79/2.30 Start location: evalstartstart 4.79/2.30 4.79/2.30 21: evalstartstart -> evalstartbb1in : A'=B, C'=D, [], cost: 7 4.79/2.30 4.79/2.30 24: evalstartbb1in -> evalstartbb1in : C'=1+C, E'=free, [ A>=1+C && free>=1 ], cost: 4 4.79/2.30 4.79/2.30 25: evalstartbb1in -> evalstartbb1in : A'=-1+A, E'=free, [ A>=1+C && 0>=free ], cost: 4 4.79/2.30 4.79/2.30 4.79/2.30 4.79/2.30 Accelerating simple loops of location 7. 4.79/2.30 4.79/2.30 Accelerating the following rules: 4.79/2.30 4.79/2.30 24: evalstartbb1in -> evalstartbb1in : C'=1+C, E'=free, [ A>=1+C && free>=1 ], cost: 4 4.79/2.30 4.79/2.30 25: evalstartbb1in -> evalstartbb1in : A'=-1+A, E'=free, [ A>=1+C && 0>=free ], cost: 4 4.79/2.30 4.79/2.30 4.79/2.30 4.79/2.30 Accelerated rule 24 with metering function -C+A, yielding the new rule 26. 4.79/2.30 4.79/2.30 Accelerated rule 25 with metering function -C+A, yielding the new rule 27. 4.79/2.30 4.79/2.30 Removing the simple loops: 24 25. 4.79/2.30 4.79/2.30 4.79/2.30 4.79/2.30 Accelerated all simple loops using metering functions (where possible): 4.79/2.30 4.79/2.30 Start location: evalstartstart 4.79/2.30 4.79/2.30 21: evalstartstart -> evalstartbb1in : A'=B, C'=D, [], cost: 7 4.79/2.30 4.79/2.30 26: evalstartbb1in -> evalstartbb1in : C'=A, E'=free, [ A>=1+C && free>=1 ], cost: -4*C+4*A 4.79/2.30 4.79/2.30 27: evalstartbb1in -> evalstartbb1in : A'=C, E'=free, [ A>=1+C && 0>=free ], cost: -4*C+4*A 4.79/2.30 4.79/2.30 4.79/2.30 4.79/2.30 Chained accelerated rules (with incoming rules): 4.79/2.30 4.79/2.30 Start location: evalstartstart 4.79/2.30 4.79/2.30 21: evalstartstart -> evalstartbb1in : A'=B, C'=D, [], cost: 7 4.79/2.30 4.79/2.30 28: evalstartstart -> evalstartbb1in : A'=B, C'=B, E'=free, [ B>=1+D && free>=1 ], cost: 7-4*D+4*B 4.79/2.30 4.79/2.30 29: evalstartstart -> evalstartbb1in : A'=D, C'=D, E'=free, [ B>=1+D && 0>=free ], cost: 7-4*D+4*B 4.79/2.30 4.79/2.30 4.79/2.30 4.79/2.30 Removed unreachable locations (and leaf rules with constant cost): 4.79/2.30 4.79/2.30 Start location: evalstartstart 4.79/2.30 4.79/2.30 28: evalstartstart -> evalstartbb1in : A'=B, C'=B, E'=free, [ B>=1+D && free>=1 ], cost: 7-4*D+4*B 4.79/2.30 4.79/2.30 29: evalstartstart -> evalstartbb1in : A'=D, C'=D, E'=free, [ B>=1+D && 0>=free ], cost: 7-4*D+4*B 4.79/2.30 4.79/2.30 4.79/2.30 4.79/2.30 ### Computing asymptotic complexity ### 4.79/2.30 4.79/2.30 4.79/2.30 4.79/2.30 Fully simplified ITS problem 4.79/2.30 4.79/2.30 Start location: evalstartstart 4.79/2.30 4.79/2.30 28: evalstartstart -> evalstartbb1in : A'=B, C'=B, E'=free, [ B>=1+D && free>=1 ], cost: 7-4*D+4*B 4.79/2.30 4.79/2.30 29: evalstartstart -> evalstartbb1in : A'=D, C'=D, E'=free, [ B>=1+D && 0>=free ], cost: 7-4*D+4*B 4.79/2.30 4.79/2.30 4.79/2.30 4.79/2.30 Computing asymptotic complexity for rule 28 4.79/2.30 4.79/2.30 Solved the limit problem by the following transformations: 4.79/2.30 4.79/2.30 Created initial limit problem: 4.79/2.30 4.79/2.30 7-4*D+4*B (+), -D+B (+/+!), free (+/+!) [not solved] 4.79/2.30 4.79/2.30 4.79/2.30 4.79/2.30 removing all constraints (solved by SMT) 4.79/2.30 4.79/2.30 resulting limit problem: [solved] 4.79/2.30 4.79/2.30 4.79/2.30 4.79/2.30 applying transformation rule (C) using substitution {free==n,D==0,B==n} 4.79/2.30 4.79/2.30 resulting limit problem: 4.79/2.30 4.79/2.30 [solved] 4.79/2.30 4.79/2.30 4.79/2.30 4.79/2.30 Solution: 4.79/2.30 4.79/2.30 free / n 4.79/2.30 4.79/2.30 D / 0 4.79/2.30 4.79/2.30 B / n 4.79/2.30 4.79/2.30 Resulting cost 7+4*n has complexity: Poly(n^1) 4.79/2.30 4.79/2.30 4.79/2.30 4.79/2.30 Found new complexity Poly(n^1). 4.79/2.30 4.79/2.30 4.79/2.30 4.79/2.30 Obtained the following overall complexity (w.r.t. the length of the input n): 4.79/2.30 4.79/2.30 Complexity: Poly(n^1) 4.79/2.30 4.79/2.30 Cpx degree: 1 4.79/2.30 4.79/2.30 Solved cost: 7+4*n 4.79/2.30 4.79/2.30 Rule cost: 7-4*D+4*B 4.79/2.30 4.79/2.30 Rule guard: [ B>=1+D && free>=1 ] 4.79/2.30 4.79/2.30 4.79/2.30 4.79/2.30 WORST_CASE(Omega(n^1),?) 4.79/2.30 4.79/2.30 4.79/2.30 ---------------------------------------- 4.79/2.30 4.79/2.30 (4) 4.79/2.30 BOUNDS(n^1, INF) 4.79/2.31 EOF