/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 366 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: qsort(nil) -> nil qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs))))) filterlow(n, nil) -> nil filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) if1(true, n, x, xs) -> filterlow(n, xs) if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) filterhigh(n, nil) -> nil filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) if2(true, n, x, xs) -> filterhigh(n, xs) if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) ge(x, 0) -> true ge(0, s(x)) -> false ge(s(x), s(y)) -> ge(x, y) append(nil, ys) -> ys append(cons(x, xs), ys) -> cons(x, append(xs, ys)) last(nil) -> 0 last(cons(x, nil)) -> x last(cons(x, cons(y, xs))) -> last(cons(y, xs)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(true) -> true encArg(false) -> false encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(ys) -> ys encArg(cons_qsort(x_1)) -> qsort(encArg(x_1)) encArg(cons_filterlow(x_1, x_2)) -> filterlow(encArg(x_1), encArg(x_2)) encArg(cons_if1(x_1, x_2, x_3, x_4)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_filterhigh(x_1, x_2)) -> filterhigh(encArg(x_1), encArg(x_2)) encArg(cons_if2(x_1, x_2, x_3, x_4)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_last(x_1)) -> last(encArg(x_1)) encode_qsort(x_1) -> qsort(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2)) encode_filterlow(x_1, x_2) -> filterlow(encArg(x_1), encArg(x_2)) encode_last(x_1) -> last(encArg(x_1)) encode_filterhigh(x_1, x_2) -> filterhigh(encArg(x_1), encArg(x_2)) encode_if1(x_1, x_2, x_3, x_4) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_if2(x_1, x_2, x_3, x_4) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_ys -> ys ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: qsort(nil) -> nil qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs))))) filterlow(n, nil) -> nil filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) if1(true, n, x, xs) -> filterlow(n, xs) if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) filterhigh(n, nil) -> nil filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) if2(true, n, x, xs) -> filterhigh(n, xs) if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) ge(x, 0) -> true ge(0, s(x)) -> false ge(s(x), s(y)) -> ge(x, y) append(nil, ys) -> ys append(cons(x, xs), ys) -> cons(x, append(xs, ys)) last(nil) -> 0 last(cons(x, nil)) -> x last(cons(x, cons(y, xs))) -> last(cons(y, xs)) The (relative) TRS S consists of the following rules: encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(true) -> true encArg(false) -> false encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(ys) -> ys encArg(cons_qsort(x_1)) -> qsort(encArg(x_1)) encArg(cons_filterlow(x_1, x_2)) -> filterlow(encArg(x_1), encArg(x_2)) encArg(cons_if1(x_1, x_2, x_3, x_4)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_filterhigh(x_1, x_2)) -> filterhigh(encArg(x_1), encArg(x_2)) encArg(cons_if2(x_1, x_2, x_3, x_4)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_last(x_1)) -> last(encArg(x_1)) encode_qsort(x_1) -> qsort(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2)) encode_filterlow(x_1, x_2) -> filterlow(encArg(x_1), encArg(x_2)) encode_last(x_1) -> last(encArg(x_1)) encode_filterhigh(x_1, x_2) -> filterhigh(encArg(x_1), encArg(x_2)) encode_if1(x_1, x_2, x_3, x_4) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_if2(x_1, x_2, x_3, x_4) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_ys -> ys Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: qsort(nil) -> nil qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs))))) filterlow(n, nil) -> nil filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) if1(true, n, x, xs) -> filterlow(n, xs) if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) filterhigh(n, nil) -> nil filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) if2(true, n, x, xs) -> filterhigh(n, xs) if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) ge(x, 0) -> true ge(0, s(x)) -> false ge(s(x), s(y)) -> ge(x, y) append(nil, ys) -> ys append(cons(x, xs), ys) -> cons(x, append(xs, ys)) last(nil) -> 0 last(cons(x, nil)) -> x last(cons(x, cons(y, xs))) -> last(cons(y, xs)) The (relative) TRS S consists of the following rules: encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(true) -> true encArg(false) -> false encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(ys) -> ys encArg(cons_qsort(x_1)) -> qsort(encArg(x_1)) encArg(cons_filterlow(x_1, x_2)) -> filterlow(encArg(x_1), encArg(x_2)) encArg(cons_if1(x_1, x_2, x_3, x_4)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_filterhigh(x_1, x_2)) -> filterhigh(encArg(x_1), encArg(x_2)) encArg(cons_if2(x_1, x_2, x_3, x_4)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_last(x_1)) -> last(encArg(x_1)) encode_qsort(x_1) -> qsort(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2)) encode_filterlow(x_1, x_2) -> filterlow(encArg(x_1), encArg(x_2)) encode_last(x_1) -> last(encArg(x_1)) encode_filterhigh(x_1, x_2) -> filterhigh(encArg(x_1), encArg(x_2)) encode_if1(x_1, x_2, x_3, x_4) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_if2(x_1, x_2, x_3, x_4) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_ys -> ys Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (6) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: qsort(nil) -> nil qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs))))) filterlow(n, nil) -> nil filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) if1(true, n, x, xs) -> filterlow(n, xs) if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) filterhigh(n, nil) -> nil filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) if2(true, n, x, xs) -> filterhigh(n, xs) if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) ge(x, 0) -> true ge(0, s(x)) -> false ge(s(x), s(y)) -> ge(x, y) append(nil, ys) -> ys append(cons(x, xs), ys) -> cons(x, append(xs, ys)) last(nil) -> 0 last(cons(x, nil)) -> x last(cons(x, cons(y, xs))) -> last(cons(y, xs)) The (relative) TRS S consists of the following rules: encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(true) -> true encArg(false) -> false encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(ys) -> ys encArg(cons_qsort(x_1)) -> qsort(encArg(x_1)) encArg(cons_filterlow(x_1, x_2)) -> filterlow(encArg(x_1), encArg(x_2)) encArg(cons_if1(x_1, x_2, x_3, x_4)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_filterhigh(x_1, x_2)) -> filterhigh(encArg(x_1), encArg(x_2)) encArg(cons_if2(x_1, x_2, x_3, x_4)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_last(x_1)) -> last(encArg(x_1)) encode_qsort(x_1) -> qsort(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2)) encode_filterlow(x_1, x_2) -> filterlow(encArg(x_1), encArg(x_2)) encode_last(x_1) -> last(encArg(x_1)) encode_filterhigh(x_1, x_2) -> filterhigh(encArg(x_1), encArg(x_2)) encode_if1(x_1, x_2, x_3, x_4) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_if2(x_1, x_2, x_3, x_4) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_ys -> ys Rewrite Strategy: INNERMOST ---------------------------------------- (7) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence append(cons(x, xs), ys) ->^+ cons(x, append(xs, ys)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [xs / cons(x, xs)]. The result substitution is [ ]. ---------------------------------------- (8) Complex Obligation (BEST) ---------------------------------------- (9) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: qsort(nil) -> nil qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs))))) filterlow(n, nil) -> nil filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) if1(true, n, x, xs) -> filterlow(n, xs) if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) filterhigh(n, nil) -> nil filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) if2(true, n, x, xs) -> filterhigh(n, xs) if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) ge(x, 0) -> true ge(0, s(x)) -> false ge(s(x), s(y)) -> ge(x, y) append(nil, ys) -> ys append(cons(x, xs), ys) -> cons(x, append(xs, ys)) last(nil) -> 0 last(cons(x, nil)) -> x last(cons(x, cons(y, xs))) -> last(cons(y, xs)) The (relative) TRS S consists of the following rules: encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(true) -> true encArg(false) -> false encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(ys) -> ys encArg(cons_qsort(x_1)) -> qsort(encArg(x_1)) encArg(cons_filterlow(x_1, x_2)) -> filterlow(encArg(x_1), encArg(x_2)) encArg(cons_if1(x_1, x_2, x_3, x_4)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_filterhigh(x_1, x_2)) -> filterhigh(encArg(x_1), encArg(x_2)) encArg(cons_if2(x_1, x_2, x_3, x_4)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_last(x_1)) -> last(encArg(x_1)) encode_qsort(x_1) -> qsort(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2)) encode_filterlow(x_1, x_2) -> filterlow(encArg(x_1), encArg(x_2)) encode_last(x_1) -> last(encArg(x_1)) encode_filterhigh(x_1, x_2) -> filterhigh(encArg(x_1), encArg(x_2)) encode_if1(x_1, x_2, x_3, x_4) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_if2(x_1, x_2, x_3, x_4) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_ys -> ys Rewrite Strategy: INNERMOST ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: qsort(nil) -> nil qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs))))) filterlow(n, nil) -> nil filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) if1(true, n, x, xs) -> filterlow(n, xs) if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) filterhigh(n, nil) -> nil filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) if2(true, n, x, xs) -> filterhigh(n, xs) if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) ge(x, 0) -> true ge(0, s(x)) -> false ge(s(x), s(y)) -> ge(x, y) append(nil, ys) -> ys append(cons(x, xs), ys) -> cons(x, append(xs, ys)) last(nil) -> 0 last(cons(x, nil)) -> x last(cons(x, cons(y, xs))) -> last(cons(y, xs)) The (relative) TRS S consists of the following rules: encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(true) -> true encArg(false) -> false encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(ys) -> ys encArg(cons_qsort(x_1)) -> qsort(encArg(x_1)) encArg(cons_filterlow(x_1, x_2)) -> filterlow(encArg(x_1), encArg(x_2)) encArg(cons_if1(x_1, x_2, x_3, x_4)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_filterhigh(x_1, x_2)) -> filterhigh(encArg(x_1), encArg(x_2)) encArg(cons_if2(x_1, x_2, x_3, x_4)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_last(x_1)) -> last(encArg(x_1)) encode_qsort(x_1) -> qsort(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2)) encode_filterlow(x_1, x_2) -> filterlow(encArg(x_1), encArg(x_2)) encode_last(x_1) -> last(encArg(x_1)) encode_filterhigh(x_1, x_2) -> filterhigh(encArg(x_1), encArg(x_2)) encode_if1(x_1, x_2, x_3, x_4) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_if2(x_1, x_2, x_3, x_4) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_ys -> ys Rewrite Strategy: INNERMOST