5.42/2.40 WORST_CASE(Omega(n^1), O(n^1)) 5.42/2.41 proof of /export/starexec/sandbox/benchmark/theBenchmark.koat 5.42/2.41 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.42/2.41 5.42/2.41 5.42/2.41 The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1). 5.42/2.41 5.42/2.41 (0) CpxIntTrs 5.42/2.41 (1) Koat Proof [FINISHED, 417 ms] 5.42/2.41 (2) BOUNDS(1, n^1) 5.42/2.41 (3) Loat Proof [FINISHED, 716 ms] 5.42/2.41 (4) BOUNDS(n^1, INF) 5.42/2.41 5.42/2.41 5.42/2.41 ---------------------------------------- 5.42/2.41 5.42/2.41 (0) 5.42/2.41 Obligation: 5.42/2.41 Complexity Int TRS consisting of the following rules: 5.42/2.41 eval_start_start(v__0, v__01, v_m, v_n, v_x, v_y) -> Com_1(eval_start_bb0_in(v__0, v__01, v_m, v_n, v_x, v_y)) :|: TRUE 5.42/2.41 eval_start_bb0_in(v__0, v__01, v_m, v_n, v_x, v_y) -> Com_1(eval_start_0(v__0, v__01, v_m, v_n, v_x, v_y)) :|: TRUE 5.42/2.41 eval_start_0(v__0, v__01, v_m, v_n, v_x, v_y) -> Com_1(eval_start_1(v__0, v__01, v_m, v_n, v_x, v_y)) :|: TRUE 5.42/2.41 eval_start_1(v__0, v__01, v_m, v_n, v_x, v_y) -> Com_1(eval_start_2(v__0, v__01, v_m, v_n, v_x, v_y)) :|: TRUE 5.42/2.41 eval_start_2(v__0, v__01, v_m, v_n, v_x, v_y) -> Com_1(eval_start_3(v__0, v__01, v_m, v_n, v_x, v_y)) :|: TRUE 5.42/2.41 eval_start_3(v__0, v__01, v_m, v_n, v_x, v_y) -> Com_1(eval_start_4(v__0, v__01, v_m, v_n, v_x, v_y)) :|: TRUE 5.42/2.41 eval_start_4(v__0, v__01, v_m, v_n, v_x, v_y) -> Com_1(eval_start_5(v__0, v__01, v_m, v_n, v_x, v_y)) :|: TRUE 5.42/2.41 eval_start_5(v__0, v__01, v_m, v_n, v_x, v_y) -> Com_1(eval_start_6(v__0, v__01, v_m, v_n, v_x, v_y)) :|: TRUE 5.42/2.41 eval_start_6(v__0, v__01, v_m, v_n, v_x, v_y) -> Com_1(eval_start_bb1_in(v_x, v_y, v_m, v_n, v_x, v_y)) :|: TRUE 5.42/2.41 eval_start_bb1_in(v__0, v__01, v_m, v_n, v_x, v_y) -> Com_1(eval_start_bb2_in(v__0, v__01, v_m, v_n, v_x, v_y)) :|: v_n > v__0 5.42/2.41 eval_start_bb1_in(v__0, v__01, v_m, v_n, v_x, v_y) -> Com_1(eval_start_bb3_in(v__0, v__01, v_m, v_n, v_x, v_y)) :|: v_n <= v__0 5.42/2.41 eval_start_bb2_in(v__0, v__01, v_m, v_n, v_x, v_y) -> Com_1(eval_start_bb1_in(v__0, v__01 + 1, v_m, v_n, v_x, v_y)) :|: v_m > v__01 5.42/2.41 eval_start_bb2_in(v__0, v__01, v_m, v_n, v_x, v_y) -> Com_1(eval_start_bb1_in(v__0 + 1, v__01 + 1, v_m, v_n, v_x, v_y)) :|: v_m > v__01 && v_m <= v__01 5.42/2.41 eval_start_bb2_in(v__0, v__01, v_m, v_n, v_x, v_y) -> Com_1(eval_start_bb1_in(v__0, v__01, v_m, v_n, v_x, v_y)) :|: v_m <= v__01 && v_m > v__01 5.42/2.41 eval_start_bb2_in(v__0, v__01, v_m, v_n, v_x, v_y) -> Com_1(eval_start_bb1_in(v__0 + 1, v__01, v_m, v_n, v_x, v_y)) :|: v_m <= v__01 5.42/2.41 eval_start_bb3_in(v__0, v__01, v_m, v_n, v_x, v_y) -> Com_1(eval_start_stop(v__0, v__01, v_m, v_n, v_x, v_y)) :|: TRUE 5.42/2.41 5.42/2.41 The start-symbols are:[eval_start_start_6] 5.42/2.41 5.42/2.41 5.42/2.41 ---------------------------------------- 5.42/2.41 5.42/2.41 (1) Koat Proof (FINISHED) 5.42/2.41 YES(?, 2*ar_1 + 2*ar_4 + 2*ar_3 + 2*ar_5 + 14) 5.42/2.41 5.42/2.41 5.42/2.41 5.42/2.41 Initial complexity problem: 5.42/2.41 5.42/2.41 1: T: 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstartstart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstart0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstart1(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstart2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart4(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstart4(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart5(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstart5(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart6(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstart6(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb1in(ar_1, ar_1, ar_3, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 >= ar_0 + 1 ] 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= ar_4 ] 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb1in(ar_0, ar_1, ar_2 + 1, ar_3, ar_4, ar_5)) [ ar_5 >= ar_2 + 1 ] 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2 + 1, ar_3, ar_4, ar_5)) [ ar_5 >= ar_2 + 1 /\ ar_2 >= ar_5 ] 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_5 /\ ar_5 >= ar_2 + 1 ] 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_5 ] 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartstart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ] 5.42/2.41 5.42/2.41 start location: koat_start 5.42/2.41 5.42/2.41 leaf cost: 0 5.42/2.41 5.42/2.41 5.42/2.41 5.42/2.41 Testing for reachability in the complexity graph removes the following transitions from problem 1: 5.42/2.41 5.42/2.41 evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2 + 1, ar_3, ar_4, ar_5)) [ ar_5 >= ar_2 + 1 /\ ar_2 >= ar_5 ] 5.42/2.41 5.42/2.41 evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_5 /\ ar_5 >= ar_2 + 1 ] 5.42/2.41 5.42/2.41 We thus obtain the following problem: 5.42/2.41 5.42/2.41 2: T: 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_5 ] 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb1in(ar_0, ar_1, ar_2 + 1, ar_3, ar_4, ar_5)) [ ar_5 >= ar_2 + 1 ] 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= ar_4 ] 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 >= ar_0 + 1 ] 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstart6(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb1in(ar_1, ar_1, ar_3, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstart5(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart6(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstart4(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart5(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart4(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstart2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstart1(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstart0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstartstart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartstart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ] 5.42/2.41 5.42/2.41 start location: koat_start 5.42/2.41 5.42/2.41 leaf cost: 0 5.42/2.41 5.42/2.41 5.42/2.41 5.42/2.41 Repeatedly propagating knowledge in problem 2 produces the following problem: 5.42/2.41 5.42/2.41 3: T: 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_5 ] 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb1in(ar_0, ar_1, ar_2 + 1, ar_3, ar_4, ar_5)) [ ar_5 >= ar_2 + 1 ] 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= ar_4 ] 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 >= ar_0 + 1 ] 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart6(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb1in(ar_1, ar_1, ar_3, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart5(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart6(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart4(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart5(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart4(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart1(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstartstart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartstart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ] 5.42/2.41 5.42/2.41 start location: koat_start 5.42/2.41 5.42/2.41 leaf cost: 0 5.42/2.41 5.42/2.41 5.42/2.41 5.42/2.41 A polynomial rank function with 5.42/2.41 5.42/2.41 Pol(evalstartbb3in) = 1 5.42/2.41 5.42/2.41 Pol(evalstartstop) = 0 5.42/2.41 5.42/2.41 Pol(evalstartbb2in) = 2 5.42/2.41 5.42/2.41 Pol(evalstartbb1in) = 2 5.42/2.41 5.42/2.41 Pol(evalstart6) = 2 5.42/2.41 5.42/2.41 Pol(evalstart5) = 2 5.42/2.41 5.42/2.41 Pol(evalstart4) = 2 5.42/2.41 5.42/2.41 Pol(evalstart3) = 2 5.42/2.41 5.42/2.41 Pol(evalstart2) = 2 5.42/2.41 5.42/2.41 Pol(evalstart1) = 2 5.42/2.41 5.42/2.41 Pol(evalstart0) = 2 5.42/2.41 5.42/2.41 Pol(evalstartbb0in) = 2 5.42/2.41 5.42/2.41 Pol(evalstartstart) = 2 5.42/2.41 5.42/2.41 Pol(koat_start) = 2 5.42/2.41 5.42/2.41 orients all transitions weakly and the transitions 5.42/2.41 5.42/2.41 evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= ar_4 ] 5.42/2.41 5.42/2.41 strictly and produces the following problem: 5.42/2.41 5.42/2.41 4: T: 5.42/2.41 5.42/2.41 (Comp: 2, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_5 ] 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb1in(ar_0, ar_1, ar_2 + 1, ar_3, ar_4, ar_5)) [ ar_5 >= ar_2 + 1 ] 5.42/2.41 5.42/2.41 (Comp: 2, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= ar_4 ] 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 >= ar_0 + 1 ] 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart6(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb1in(ar_1, ar_1, ar_3, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart5(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart6(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart4(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart5(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart4(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart1(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstartstart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartstart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ] 5.42/2.41 5.42/2.41 start location: koat_start 5.42/2.41 5.42/2.41 leaf cost: 0 5.42/2.41 5.42/2.41 5.42/2.41 5.42/2.41 A polynomial rank function with 5.42/2.41 5.42/2.41 Pol(evalstartbb3in) = -V_3 + V_6 5.42/2.41 5.42/2.41 Pol(evalstartstop) = -V_3 + V_6 5.42/2.41 5.42/2.41 Pol(evalstartbb2in) = -V_3 + V_6 5.42/2.41 5.42/2.41 Pol(evalstartbb1in) = -V_3 + V_6 5.42/2.41 5.42/2.41 Pol(evalstart6) = -V_4 + V_6 5.42/2.41 5.42/2.41 Pol(evalstart5) = -V_4 + V_6 5.42/2.41 5.42/2.41 Pol(evalstart4) = -V_4 + V_6 5.42/2.41 5.42/2.41 Pol(evalstart3) = -V_4 + V_6 5.42/2.41 5.42/2.41 Pol(evalstart2) = -V_4 + V_6 5.42/2.41 5.42/2.41 Pol(evalstart1) = -V_4 + V_6 5.42/2.41 5.42/2.41 Pol(evalstart0) = -V_4 + V_6 5.42/2.41 5.42/2.41 Pol(evalstartbb0in) = -V_4 + V_6 5.42/2.41 5.42/2.41 Pol(evalstartstart) = -V_4 + V_6 5.42/2.41 5.42/2.41 Pol(koat_start) = -V_4 + V_6 5.42/2.41 5.42/2.41 orients all transitions weakly and the transition 5.42/2.41 5.42/2.41 evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb1in(ar_0, ar_1, ar_2 + 1, ar_3, ar_4, ar_5)) [ ar_5 >= ar_2 + 1 ] 5.42/2.41 5.42/2.41 strictly and produces the following problem: 5.42/2.41 5.42/2.41 5: T: 5.42/2.41 5.42/2.41 (Comp: 2, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 >= ar_5 ] 5.42/2.41 5.42/2.41 (Comp: ar_3 + ar_5, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb1in(ar_0, ar_1, ar_2 + 1, ar_3, ar_4, ar_5)) [ ar_5 >= ar_2 + 1 ] 5.42/2.41 5.42/2.41 (Comp: 2, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 >= ar_4 ] 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_4 >= ar_0 + 1 ] 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart6(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb1in(ar_1, ar_1, ar_3, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart5(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart6(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart4(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart5(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart4(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart1(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstartstart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartstart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ] 5.42/2.41 5.42/2.41 start location: koat_start 5.42/2.41 5.42/2.41 leaf cost: 0 5.42/2.41 5.42/2.41 5.42/2.41 5.42/2.41 Applied AI with 'oct' on problem 5 to obtain the following invariants: 5.42/2.41 5.42/2.41 For symbol evalstartbb1in: X_3 - X_4 >= 0 /\ X_1 - X_2 >= 0 5.42/2.41 5.42/2.41 For symbol evalstartbb2in: -X_2 + X_5 - 1 >= 0 /\ -X_1 + X_5 - 1 >= 0 /\ X_3 - X_4 >= 0 /\ X_1 - X_2 >= 0 5.42/2.41 5.42/2.41 For symbol evalstartbb3in: X_1 - X_5 >= 0 /\ X_3 - X_4 >= 0 /\ X_1 - X_2 >= 0 5.42/2.41 5.42/2.41 5.42/2.41 5.42/2.41 5.42/2.41 5.42/2.41 This yielded the following problem: 5.42/2.41 5.42/2.41 6: T: 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartstart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ] 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstartstart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart1(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart4(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart4(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart5(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart5(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart6(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart6(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb1in(ar_1, ar_1, ar_3, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_4 >= ar_0 + 1 ] 5.42/2.41 5.42/2.41 (Comp: 2, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_0 >= ar_4 ] 5.42/2.41 5.42/2.41 (Comp: ar_3 + ar_5, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb1in(ar_0, ar_1, ar_2 + 1, ar_3, ar_4, ar_5)) [ -ar_1 + ar_4 - 1 >= 0 /\ -ar_0 + ar_4 - 1 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_5 >= ar_2 + 1 ] 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3, ar_4, ar_5)) [ -ar_1 + ar_4 - 1 >= 0 /\ -ar_0 + ar_4 - 1 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_2 >= ar_5 ] 5.42/2.41 5.42/2.41 (Comp: 2, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 - ar_4 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 ] 5.42/2.41 5.42/2.41 start location: koat_start 5.42/2.41 5.42/2.41 leaf cost: 0 5.42/2.41 5.42/2.41 5.42/2.41 5.42/2.41 A polynomial rank function with 5.42/2.41 5.42/2.41 Pol(koat_start) = -V_2 + V_5 5.42/2.41 5.42/2.41 Pol(evalstartstart) = -V_2 + V_5 5.42/2.41 5.42/2.41 Pol(evalstartbb0in) = -V_2 + V_5 5.42/2.41 5.42/2.41 Pol(evalstart0) = -V_2 + V_5 5.42/2.41 5.42/2.41 Pol(evalstart1) = -V_2 + V_5 5.42/2.41 5.42/2.41 Pol(evalstart2) = -V_2 + V_5 5.42/2.41 5.42/2.41 Pol(evalstart3) = -V_2 + V_5 5.42/2.41 5.42/2.41 Pol(evalstart4) = -V_2 + V_5 5.42/2.41 5.42/2.41 Pol(evalstart5) = -V_2 + V_5 5.42/2.41 5.42/2.41 Pol(evalstart6) = -V_2 + V_5 5.42/2.41 5.42/2.41 Pol(evalstartbb1in) = -V_1 + V_5 5.42/2.41 5.42/2.41 Pol(evalstartbb2in) = -V_1 + V_5 5.42/2.41 5.42/2.41 Pol(evalstartbb3in) = -V_1 + V_5 5.42/2.41 5.42/2.41 Pol(evalstartstop) = -V_1 + V_5 5.42/2.41 5.42/2.41 orients all transitions weakly and the transition 5.42/2.41 5.42/2.41 evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3, ar_4, ar_5)) [ -ar_1 + ar_4 - 1 >= 0 /\ -ar_0 + ar_4 - 1 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_2 >= ar_5 ] 5.42/2.41 5.42/2.41 strictly and produces the following problem: 5.42/2.41 5.42/2.41 7: T: 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartstart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ] 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstartstart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart1(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart4(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart4(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart5(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart5(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart6(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart6(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb1in(ar_1, ar_1, ar_3, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_4 >= ar_0 + 1 ] 5.42/2.41 5.42/2.41 (Comp: 2, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_0 >= ar_4 ] 5.42/2.41 5.42/2.41 (Comp: ar_3 + ar_5, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb1in(ar_0, ar_1, ar_2 + 1, ar_3, ar_4, ar_5)) [ -ar_1 + ar_4 - 1 >= 0 /\ -ar_0 + ar_4 - 1 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_5 >= ar_2 + 1 ] 5.42/2.41 5.42/2.41 (Comp: ar_1 + ar_4, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3, ar_4, ar_5)) [ -ar_1 + ar_4 - 1 >= 0 /\ -ar_0 + ar_4 - 1 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_2 >= ar_5 ] 5.42/2.41 5.42/2.41 (Comp: 2, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 - ar_4 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 ] 5.42/2.41 5.42/2.41 start location: koat_start 5.42/2.41 5.42/2.41 leaf cost: 0 5.42/2.41 5.42/2.41 5.42/2.41 5.42/2.41 Repeatedly propagating knowledge in problem 7 produces the following problem: 5.42/2.41 5.42/2.41 8: T: 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartstart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ 0 <= 0 ] 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstartstart(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart0(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart1(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart2(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart4(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart4(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart5(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart5(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstart6(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: 1, Cost: 1) evalstart6(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb1in(ar_1, ar_1, ar_3, ar_3, ar_4, ar_5)) 5.42/2.41 5.42/2.41 (Comp: ar_1 + ar_4 + ar_3 + ar_5 + 1, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_4 >= ar_0 + 1 ] 5.42/2.41 5.42/2.41 (Comp: 2, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_0 >= ar_4 ] 5.42/2.41 5.42/2.41 (Comp: ar_3 + ar_5, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb1in(ar_0, ar_1, ar_2 + 1, ar_3, ar_4, ar_5)) [ -ar_1 + ar_4 - 1 >= 0 /\ -ar_0 + ar_4 - 1 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_5 >= ar_2 + 1 ] 5.42/2.41 5.42/2.41 (Comp: ar_1 + ar_4, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3, ar_4, ar_5)) [ -ar_1 + ar_4 - 1 >= 0 /\ -ar_0 + ar_4 - 1 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_2 >= ar_5 ] 5.42/2.41 5.42/2.41 (Comp: 2, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3, ar_4, ar_5)) [ ar_0 - ar_4 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 ] 5.42/2.41 5.42/2.41 start location: koat_start 5.42/2.41 5.42/2.41 leaf cost: 0 5.42/2.41 5.42/2.41 5.42/2.41 5.42/2.41 Complexity upper bound 2*ar_1 + 2*ar_4 + 2*ar_3 + 2*ar_5 + 14 5.42/2.41 5.42/2.41 5.42/2.41 5.42/2.41 Time: 0.394 sec (SMT: 0.289 sec) 5.42/2.41 5.42/2.41 5.42/2.41 ---------------------------------------- 5.42/2.41 5.42/2.41 (2) 5.42/2.41 BOUNDS(1, n^1) 5.42/2.41 5.42/2.41 ---------------------------------------- 5.42/2.41 5.42/2.41 (3) Loat Proof (FINISHED) 5.42/2.41 5.42/2.41 5.42/2.41 ### Pre-processing the ITS problem ### 5.42/2.41 5.42/2.41 5.42/2.41 5.42/2.41 Initial linear ITS problem 5.42/2.41 5.42/2.41 Start location: evalstartstart 5.42/2.41 5.42/2.41 0: evalstartstart -> evalstartbb0in : [], cost: 1 5.42/2.41 5.42/2.41 1: evalstartbb0in -> evalstart0 : [], cost: 1 5.42/2.41 5.42/2.41 2: evalstart0 -> evalstart1 : [], cost: 1 5.42/2.41 5.42/2.41 3: evalstart1 -> evalstart2 : [], cost: 1 5.42/2.41 5.42/2.41 4: evalstart2 -> evalstart3 : [], cost: 1 5.42/2.41 5.42/2.41 5: evalstart3 -> evalstart4 : [], cost: 1 5.42/2.41 5.42/2.41 6: evalstart4 -> evalstart5 : [], cost: 1 5.42/2.41 5.42/2.41 7: evalstart5 -> evalstart6 : [], cost: 1 5.42/2.41 5.42/2.41 8: evalstart6 -> evalstartbb1in : A'=B, C'=D, [], cost: 1 5.42/2.41 5.42/2.41 9: evalstartbb1in -> evalstartbb2in : [ E>=1+A ], cost: 1 5.42/2.41 5.42/2.41 10: evalstartbb1in -> evalstartbb3in : [ A>=E ], cost: 1 5.42/2.41 5.42/2.41 11: evalstartbb2in -> evalstartbb1in : C'=1+C, [ F>=1+C ], cost: 1 5.42/2.41 5.42/2.41 12: evalstartbb2in -> evalstartbb1in : A'=1+A, C'=1+C, [ F>=1+C && C>=F ], cost: 1 5.42/2.41 5.42/2.41 13: evalstartbb2in -> evalstartbb1in : [ C>=F && F>=1+C ], cost: 1 5.42/2.41 5.42/2.41 14: evalstartbb2in -> evalstartbb1in : A'=1+A, [ C>=F ], cost: 1 5.42/2.41 5.42/2.41 15: evalstartbb3in -> evalstartstop : [], cost: 1 5.42/2.41 5.42/2.41 5.42/2.41 5.42/2.41 Removed unreachable and leaf rules: 5.42/2.41 5.42/2.41 Start location: evalstartstart 5.42/2.41 5.42/2.41 0: evalstartstart -> evalstartbb0in : [], cost: 1 5.42/2.41 5.42/2.41 1: evalstartbb0in -> evalstart0 : [], cost: 1 5.42/2.41 5.42/2.41 2: evalstart0 -> evalstart1 : [], cost: 1 5.42/2.41 5.42/2.41 3: evalstart1 -> evalstart2 : [], cost: 1 5.42/2.41 5.42/2.41 4: evalstart2 -> evalstart3 : [], cost: 1 5.42/2.41 5.42/2.41 5: evalstart3 -> evalstart4 : [], cost: 1 5.42/2.41 5.42/2.41 6: evalstart4 -> evalstart5 : [], cost: 1 5.42/2.41 5.42/2.41 7: evalstart5 -> evalstart6 : [], cost: 1 5.42/2.41 5.42/2.41 8: evalstart6 -> evalstartbb1in : A'=B, C'=D, [], cost: 1 5.42/2.41 5.42/2.41 9: evalstartbb1in -> evalstartbb2in : [ E>=1+A ], cost: 1 5.42/2.41 5.42/2.41 11: evalstartbb2in -> evalstartbb1in : C'=1+C, [ F>=1+C ], cost: 1 5.42/2.41 5.42/2.41 12: evalstartbb2in -> evalstartbb1in : A'=1+A, C'=1+C, [ F>=1+C && C>=F ], cost: 1 5.42/2.41 5.42/2.41 13: evalstartbb2in -> evalstartbb1in : [ C>=F && F>=1+C ], cost: 1 5.42/2.41 5.42/2.41 14: evalstartbb2in -> evalstartbb1in : A'=1+A, [ C>=F ], cost: 1 5.42/2.41 5.42/2.41 5.42/2.41 5.42/2.41 Removed rules with unsatisfiable guard: 5.42/2.41 5.42/2.41 Start location: evalstartstart 5.42/2.41 5.42/2.41 0: evalstartstart -> evalstartbb0in : [], cost: 1 5.42/2.41 5.42/2.41 1: evalstartbb0in -> evalstart0 : [], cost: 1 5.42/2.41 5.42/2.41 2: evalstart0 -> evalstart1 : [], cost: 1 5.42/2.41 5.42/2.41 3: evalstart1 -> evalstart2 : [], cost: 1 5.42/2.41 5.42/2.41 4: evalstart2 -> evalstart3 : [], cost: 1 5.42/2.41 5.42/2.41 5: evalstart3 -> evalstart4 : [], cost: 1 5.42/2.41 5.42/2.41 6: evalstart4 -> evalstart5 : [], cost: 1 5.42/2.41 5.42/2.41 7: evalstart5 -> evalstart6 : [], cost: 1 5.42/2.41 5.42/2.41 8: evalstart6 -> evalstartbb1in : A'=B, C'=D, [], cost: 1 5.42/2.41 5.42/2.41 9: evalstartbb1in -> evalstartbb2in : [ E>=1+A ], cost: 1 5.42/2.41 5.42/2.41 11: evalstartbb2in -> evalstartbb1in : C'=1+C, [ F>=1+C ], cost: 1 5.42/2.41 5.42/2.41 14: evalstartbb2in -> evalstartbb1in : A'=1+A, [ C>=F ], cost: 1 5.42/2.41 5.42/2.41 5.42/2.41 5.42/2.41 ### Simplification by acceleration and chaining ### 5.42/2.41 5.42/2.41 5.42/2.41 5.42/2.41 Eliminated locations (on linear paths): 5.42/2.41 5.42/2.41 Start location: evalstartstart 5.42/2.41 5.42/2.41 23: evalstartstart -> evalstartbb1in : A'=B, C'=D, [], cost: 9 5.42/2.41 5.42/2.41 9: evalstartbb1in -> evalstartbb2in : [ E>=1+A ], cost: 1 5.42/2.41 5.42/2.41 11: evalstartbb2in -> evalstartbb1in : C'=1+C, [ F>=1+C ], cost: 1 5.42/2.41 5.42/2.41 14: evalstartbb2in -> evalstartbb1in : A'=1+A, [ C>=F ], cost: 1 5.42/2.41 5.42/2.41 5.42/2.41 5.42/2.41 Eliminated locations (on tree-shaped paths): 5.42/2.41 5.42/2.41 Start location: evalstartstart 5.42/2.41 5.42/2.41 23: evalstartstart -> evalstartbb1in : A'=B, C'=D, [], cost: 9 5.42/2.41 5.42/2.41 24: evalstartbb1in -> evalstartbb1in : C'=1+C, [ E>=1+A && F>=1+C ], cost: 2 5.42/2.41 5.42/2.41 25: evalstartbb1in -> evalstartbb1in : A'=1+A, [ E>=1+A && C>=F ], cost: 2 5.42/2.41 5.42/2.41 5.42/2.41 5.42/2.41 Accelerating simple loops of location 9. 5.42/2.41 5.42/2.41 Accelerating the following rules: 5.42/2.41 5.42/2.41 24: evalstartbb1in -> evalstartbb1in : C'=1+C, [ E>=1+A && F>=1+C ], cost: 2 5.42/2.41 5.42/2.41 25: evalstartbb1in -> evalstartbb1in : A'=1+A, [ E>=1+A && C>=F ], cost: 2 5.42/2.41 5.42/2.41 5.42/2.41 5.42/2.41 Accelerated rule 24 with metering function F-C, yielding the new rule 26. 5.42/2.41 5.42/2.41 Accelerated rule 25 with metering function -A+E, yielding the new rule 27. 5.42/2.41 5.42/2.41 Removing the simple loops: 24 25. 5.42/2.41 5.42/2.41 5.42/2.41 5.42/2.41 Accelerated all simple loops using metering functions (where possible): 5.42/2.41 5.42/2.41 Start location: evalstartstart 5.42/2.41 5.42/2.41 23: evalstartstart -> evalstartbb1in : A'=B, C'=D, [], cost: 9 5.42/2.41 5.42/2.41 26: evalstartbb1in -> evalstartbb1in : C'=F, [ E>=1+A && F>=1+C ], cost: 2*F-2*C 5.42/2.41 5.42/2.41 27: evalstartbb1in -> evalstartbb1in : A'=E, [ E>=1+A && C>=F ], cost: -2*A+2*E 5.42/2.41 5.42/2.41 5.42/2.41 5.42/2.41 Chained accelerated rules (with incoming rules): 5.42/2.41 5.42/2.41 Start location: evalstartstart 5.42/2.41 5.42/2.41 23: evalstartstart -> evalstartbb1in : A'=B, C'=D, [], cost: 9 5.42/2.41 5.42/2.41 28: evalstartstart -> evalstartbb1in : A'=B, C'=F, [ E>=1+B && F>=1+D ], cost: 9+2*F-2*D 5.42/2.41 5.42/2.41 29: evalstartstart -> evalstartbb1in : A'=E, C'=D, [ E>=1+B && D>=F ], cost: 9+2*E-2*B 5.42/2.41 5.42/2.41 5.42/2.41 5.42/2.41 Removed unreachable locations (and leaf rules with constant cost): 5.42/2.41 5.42/2.41 Start location: evalstartstart 5.42/2.41 5.42/2.41 28: evalstartstart -> evalstartbb1in : A'=B, C'=F, [ E>=1+B && F>=1+D ], cost: 9+2*F-2*D 5.42/2.41 5.42/2.41 29: evalstartstart -> evalstartbb1in : A'=E, C'=D, [ E>=1+B && D>=F ], cost: 9+2*E-2*B 5.42/2.41 5.42/2.41 5.42/2.41 5.42/2.41 ### Computing asymptotic complexity ### 5.42/2.41 5.42/2.41 5.42/2.41 5.42/2.41 Fully simplified ITS problem 5.42/2.41 5.42/2.41 Start location: evalstartstart 5.42/2.41 5.42/2.41 28: evalstartstart -> evalstartbb1in : A'=B, C'=F, [ E>=1+B && F>=1+D ], cost: 9+2*F-2*D 5.42/2.41 5.42/2.41 29: evalstartstart -> evalstartbb1in : A'=E, C'=D, [ E>=1+B && D>=F ], cost: 9+2*E-2*B 5.42/2.41 5.42/2.41 5.42/2.41 5.42/2.41 Computing asymptotic complexity for rule 28 5.42/2.41 5.42/2.41 Solved the limit problem by the following transformations: 5.42/2.41 5.42/2.41 Created initial limit problem: 5.42/2.42 5.42/2.42 9+2*F-2*D (+), F-D (+/+!), E-B (+/+!) [not solved] 5.42/2.42 5.42/2.42 5.42/2.42 5.42/2.42 removing all constraints (solved by SMT) 5.42/2.42 5.42/2.42 resulting limit problem: [solved] 5.42/2.42 5.42/2.42 5.42/2.42 5.42/2.42 applying transformation rule (C) using substitution {F==0,D==-n,E==1,B==0} 5.42/2.42 5.42/2.42 resulting limit problem: 5.42/2.42 5.42/2.42 [solved] 5.42/2.42 5.42/2.42 5.42/2.42 5.42/2.42 Solution: 5.42/2.42 5.42/2.42 F / 0 5.42/2.42 5.42/2.42 D / -n 5.42/2.42 5.42/2.42 E / 1 5.42/2.42 5.42/2.42 B / 0 5.42/2.42 5.42/2.42 Resulting cost 9+2*n has complexity: Poly(n^1) 5.42/2.42 5.42/2.42 5.42/2.42 5.42/2.42 Found new complexity Poly(n^1). 5.42/2.42 5.42/2.42 5.42/2.42 5.42/2.42 Obtained the following overall complexity (w.r.t. the length of the input n): 5.42/2.42 5.42/2.42 Complexity: Poly(n^1) 5.42/2.42 5.42/2.42 Cpx degree: 1 5.42/2.42 5.42/2.42 Solved cost: 9+2*n 5.42/2.42 5.42/2.42 Rule cost: 9+2*F-2*D 5.42/2.42 5.42/2.42 Rule guard: [ E>=1+B && F>=1+D ] 5.42/2.42 5.42/2.42 5.42/2.42 5.42/2.42 WORST_CASE(Omega(n^1),?) 5.42/2.42 5.42/2.42 5.42/2.42 ---------------------------------------- 5.42/2.42 5.42/2.42 (4) 5.42/2.42 BOUNDS(n^1, INF) 5.49/2.44 EOF