/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). (0) CpxTRS (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (2) TRS for Loop Detection (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (4) BEST (5) proven lower bound (6) LowerBoundPropagationProof [FINISHED, 0 ms] (7) BOUNDS(n^1, INF) (8) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: fibs_2#1(zipwith_l, plus, tail_l, x3) -> ConsL(S(0), zipwith_l#3(fibs, fibs_2(zipwith_l, plus, tail_l))) cond_take_l_n_xs(ConsL(x16, x18), 0) -> Nil cond_take_l_n_xs(ConsL(x7, fibs_2(x4, x8, x12)), S(x16)) -> Cons(x7, cond_take_l_n_xs(fibs_2#1(x4, x8, x12, bot[0]), x16)) cond_take_l_n_xs(ConsL(x7, zipwith_l_f_xs_ys(x4, x8, x12)), S(x16)) -> Cons(x7, cond_take_l_n_xs(zipwith_l_f_xs_ys#1(x4, x8, x12, bot[0]), x16)) plus#2(0, x12) -> x12 plus#2(S(x4), x2) -> S(plus#2(x4, x2)) cond_zipwith_l_f_xs_ys_1(ConsL(x4, x3), x2, x1) -> ConsL(plus#2(x2, x4), zipwith_l#3(x1, x3)) cond_zipwith_l_f_xs_ys(ConsL(x5, x4), zipwith_l_f_xs_ys(x1, x2, x3)) -> cond_zipwith_l_f_xs_ys_1(zipwith_l_f_xs_ys#1(x1, x2, x3, bot[6]), x5, x4) cond_zipwith_l_f_xs_ys(ConsL(x5, x4), fibs_2(x1, x2, x3)) -> cond_zipwith_l_f_xs_ys_1(fibs_2#1(x1, x2, x3, bot[6]), x5, x4) zipwith_l_f_xs_ys#1(plus, fibs, x5, x3) -> cond_zipwith_l_f_xs_ys(ConsL(0, fibs_2(zipwith_l, plus, tail_l)), x5) zipwith_l_f_xs_ys#1(plus, fibs_2(x3, x4, x5), x2, x1) -> cond_zipwith_l_f_xs_ys(fibs_2#1(x3, x4, x5, bot[7]), x2) zipwith_l_f_xs_ys#1(plus, zipwith_l_f_xs_ys(x3, x4, x5), x2, x1) -> cond_zipwith_l_f_xs_ys(zipwith_l_f_xs_ys#1(x3, x4, x5, bot[7]), x2) zipwith_l#3(x8, x4) -> zipwith_l_f_xs_ys(plus, x8, x4) main(x12) -> cond_take_l_n_xs(ConsL(0, fibs_2(zipwith_l, plus, tail_l)), x12) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (2) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: fibs_2#1(zipwith_l, plus, tail_l, x3) -> ConsL(S(0), zipwith_l#3(fibs, fibs_2(zipwith_l, plus, tail_l))) cond_take_l_n_xs(ConsL(x16, x18), 0) -> Nil cond_take_l_n_xs(ConsL(x7, fibs_2(x4, x8, x12)), S(x16)) -> Cons(x7, cond_take_l_n_xs(fibs_2#1(x4, x8, x12, bot[0]), x16)) cond_take_l_n_xs(ConsL(x7, zipwith_l_f_xs_ys(x4, x8, x12)), S(x16)) -> Cons(x7, cond_take_l_n_xs(zipwith_l_f_xs_ys#1(x4, x8, x12, bot[0]), x16)) plus#2(0, x12) -> x12 plus#2(S(x4), x2) -> S(plus#2(x4, x2)) cond_zipwith_l_f_xs_ys_1(ConsL(x4, x3), x2, x1) -> ConsL(plus#2(x2, x4), zipwith_l#3(x1, x3)) cond_zipwith_l_f_xs_ys(ConsL(x5, x4), zipwith_l_f_xs_ys(x1, x2, x3)) -> cond_zipwith_l_f_xs_ys_1(zipwith_l_f_xs_ys#1(x1, x2, x3, bot[6]), x5, x4) cond_zipwith_l_f_xs_ys(ConsL(x5, x4), fibs_2(x1, x2, x3)) -> cond_zipwith_l_f_xs_ys_1(fibs_2#1(x1, x2, x3, bot[6]), x5, x4) zipwith_l_f_xs_ys#1(plus, fibs, x5, x3) -> cond_zipwith_l_f_xs_ys(ConsL(0, fibs_2(zipwith_l, plus, tail_l)), x5) zipwith_l_f_xs_ys#1(plus, fibs_2(x3, x4, x5), x2, x1) -> cond_zipwith_l_f_xs_ys(fibs_2#1(x3, x4, x5, bot[7]), x2) zipwith_l_f_xs_ys#1(plus, zipwith_l_f_xs_ys(x3, x4, x5), x2, x1) -> cond_zipwith_l_f_xs_ys(zipwith_l_f_xs_ys#1(x3, x4, x5, bot[7]), x2) zipwith_l#3(x8, x4) -> zipwith_l_f_xs_ys(plus, x8, x4) main(x12) -> cond_take_l_n_xs(ConsL(0, fibs_2(zipwith_l, plus, tail_l)), x12) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (3) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence plus#2(S(x4), x2) ->^+ S(plus#2(x4, x2)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [x4 / S(x4)]. The result substitution is [ ]. ---------------------------------------- (4) Complex Obligation (BEST) ---------------------------------------- (5) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: fibs_2#1(zipwith_l, plus, tail_l, x3) -> ConsL(S(0), zipwith_l#3(fibs, fibs_2(zipwith_l, plus, tail_l))) cond_take_l_n_xs(ConsL(x16, x18), 0) -> Nil cond_take_l_n_xs(ConsL(x7, fibs_2(x4, x8, x12)), S(x16)) -> Cons(x7, cond_take_l_n_xs(fibs_2#1(x4, x8, x12, bot[0]), x16)) cond_take_l_n_xs(ConsL(x7, zipwith_l_f_xs_ys(x4, x8, x12)), S(x16)) -> Cons(x7, cond_take_l_n_xs(zipwith_l_f_xs_ys#1(x4, x8, x12, bot[0]), x16)) plus#2(0, x12) -> x12 plus#2(S(x4), x2) -> S(plus#2(x4, x2)) cond_zipwith_l_f_xs_ys_1(ConsL(x4, x3), x2, x1) -> ConsL(plus#2(x2, x4), zipwith_l#3(x1, x3)) cond_zipwith_l_f_xs_ys(ConsL(x5, x4), zipwith_l_f_xs_ys(x1, x2, x3)) -> cond_zipwith_l_f_xs_ys_1(zipwith_l_f_xs_ys#1(x1, x2, x3, bot[6]), x5, x4) cond_zipwith_l_f_xs_ys(ConsL(x5, x4), fibs_2(x1, x2, x3)) -> cond_zipwith_l_f_xs_ys_1(fibs_2#1(x1, x2, x3, bot[6]), x5, x4) zipwith_l_f_xs_ys#1(plus, fibs, x5, x3) -> cond_zipwith_l_f_xs_ys(ConsL(0, fibs_2(zipwith_l, plus, tail_l)), x5) zipwith_l_f_xs_ys#1(plus, fibs_2(x3, x4, x5), x2, x1) -> cond_zipwith_l_f_xs_ys(fibs_2#1(x3, x4, x5, bot[7]), x2) zipwith_l_f_xs_ys#1(plus, zipwith_l_f_xs_ys(x3, x4, x5), x2, x1) -> cond_zipwith_l_f_xs_ys(zipwith_l_f_xs_ys#1(x3, x4, x5, bot[7]), x2) zipwith_l#3(x8, x4) -> zipwith_l_f_xs_ys(plus, x8, x4) main(x12) -> cond_take_l_n_xs(ConsL(0, fibs_2(zipwith_l, plus, tail_l)), x12) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (6) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (7) BOUNDS(n^1, INF) ---------------------------------------- (8) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: fibs_2#1(zipwith_l, plus, tail_l, x3) -> ConsL(S(0), zipwith_l#3(fibs, fibs_2(zipwith_l, plus, tail_l))) cond_take_l_n_xs(ConsL(x16, x18), 0) -> Nil cond_take_l_n_xs(ConsL(x7, fibs_2(x4, x8, x12)), S(x16)) -> Cons(x7, cond_take_l_n_xs(fibs_2#1(x4, x8, x12, bot[0]), x16)) cond_take_l_n_xs(ConsL(x7, zipwith_l_f_xs_ys(x4, x8, x12)), S(x16)) -> Cons(x7, cond_take_l_n_xs(zipwith_l_f_xs_ys#1(x4, x8, x12, bot[0]), x16)) plus#2(0, x12) -> x12 plus#2(S(x4), x2) -> S(plus#2(x4, x2)) cond_zipwith_l_f_xs_ys_1(ConsL(x4, x3), x2, x1) -> ConsL(plus#2(x2, x4), zipwith_l#3(x1, x3)) cond_zipwith_l_f_xs_ys(ConsL(x5, x4), zipwith_l_f_xs_ys(x1, x2, x3)) -> cond_zipwith_l_f_xs_ys_1(zipwith_l_f_xs_ys#1(x1, x2, x3, bot[6]), x5, x4) cond_zipwith_l_f_xs_ys(ConsL(x5, x4), fibs_2(x1, x2, x3)) -> cond_zipwith_l_f_xs_ys_1(fibs_2#1(x1, x2, x3, bot[6]), x5, x4) zipwith_l_f_xs_ys#1(plus, fibs, x5, x3) -> cond_zipwith_l_f_xs_ys(ConsL(0, fibs_2(zipwith_l, plus, tail_l)), x5) zipwith_l_f_xs_ys#1(plus, fibs_2(x3, x4, x5), x2, x1) -> cond_zipwith_l_f_xs_ys(fibs_2#1(x3, x4, x5, bot[7]), x2) zipwith_l_f_xs_ys#1(plus, zipwith_l_f_xs_ys(x3, x4, x5), x2, x1) -> cond_zipwith_l_f_xs_ys(zipwith_l_f_xs_ys#1(x3, x4, x5, bot[7]), x2) zipwith_l#3(x8, x4) -> zipwith_l_f_xs_ys(plus, x8, x4) main(x12) -> cond_take_l_n_xs(ConsL(0, fibs_2(zipwith_l, plus, tail_l)), x12) S is empty. Rewrite Strategy: INNERMOST