MAYBE

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict Trs:
  { 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)))))
  , append(nil(), ys()) -> ys()
  , append(cons(x, xs), ys()) -> cons(x, append(xs, ys()))
  , filterlow(n, nil()) -> nil()
  , filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs)
  , last(nil()) -> 0()
  , last(cons(x, nil())) -> x
  , last(cons(x, cons(y, xs))) -> last(cons(y, xs))
  , filterhigh(n, nil()) -> nil()
  , filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs)
  , if1(true(), n, x, xs) -> filterlow(n, xs)
  , if1(false(), n, x, xs) -> cons(x, filterlow(n, xs))
  , ge(x, 0()) -> true()
  , ge(0(), s(x)) -> false()
  , ge(s(x), s(y)) -> ge(x, y)
  , if2(true(), n, x, xs) -> filterhigh(n, xs)
  , if2(false(), n, x, xs) -> cons(x, filterhigh(n, xs)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { qsort^#(nil()) -> c_1()
  , qsort^#(cons(x, xs)) ->
    c_2(append^#(qsort(filterlow(last(cons(x, xs)), cons(x, xs))),
                 cons(last(cons(x, xs)),
                      qsort(filterhigh(last(cons(x, xs)), cons(x, xs))))),
        qsort^#(filterlow(last(cons(x, xs)), cons(x, xs))),
        filterlow^#(last(cons(x, xs)), cons(x, xs)),
        last^#(cons(x, xs)),
        last^#(cons(x, xs)),
        qsort^#(filterhigh(last(cons(x, xs)), cons(x, xs))),
        filterhigh^#(last(cons(x, xs)), cons(x, xs)),
        last^#(cons(x, xs)))
  , append^#(nil(), ys()) -> c_3()
  , append^#(cons(x, xs), ys()) -> c_4(append^#(xs, ys()))
  , filterlow^#(n, nil()) -> c_5()
  , filterlow^#(n, cons(x, xs)) ->
    c_6(if1^#(ge(n, x), n, x, xs), ge^#(n, x))
  , last^#(nil()) -> c_7()
  , last^#(cons(x, nil())) -> c_8()
  , last^#(cons(x, cons(y, xs))) -> c_9(last^#(cons(y, xs)))
  , filterhigh^#(n, nil()) -> c_10()
  , filterhigh^#(n, cons(x, xs)) ->
    c_11(if2^#(ge(x, n), n, x, xs), ge^#(x, n))
  , if1^#(true(), n, x, xs) -> c_12(filterlow^#(n, xs))
  , if1^#(false(), n, x, xs) -> c_13(filterlow^#(n, xs))
  , ge^#(x, 0()) -> c_14()
  , ge^#(0(), s(x)) -> c_15()
  , ge^#(s(x), s(y)) -> c_16(ge^#(x, y))
  , if2^#(true(), n, x, xs) -> c_17(filterhigh^#(n, xs))
  , if2^#(false(), n, x, xs) -> c_18(filterhigh^#(n, xs)) }

and mark the set of starting terms.

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { qsort^#(nil()) -> c_1()
  , qsort^#(cons(x, xs)) ->
    c_2(append^#(qsort(filterlow(last(cons(x, xs)), cons(x, xs))),
                 cons(last(cons(x, xs)),
                      qsort(filterhigh(last(cons(x, xs)), cons(x, xs))))),
        qsort^#(filterlow(last(cons(x, xs)), cons(x, xs))),
        filterlow^#(last(cons(x, xs)), cons(x, xs)),
        last^#(cons(x, xs)),
        last^#(cons(x, xs)),
        qsort^#(filterhigh(last(cons(x, xs)), cons(x, xs))),
        filterhigh^#(last(cons(x, xs)), cons(x, xs)),
        last^#(cons(x, xs)))
  , append^#(nil(), ys()) -> c_3()
  , append^#(cons(x, xs), ys()) -> c_4(append^#(xs, ys()))
  , filterlow^#(n, nil()) -> c_5()
  , filterlow^#(n, cons(x, xs)) ->
    c_6(if1^#(ge(n, x), n, x, xs), ge^#(n, x))
  , last^#(nil()) -> c_7()
  , last^#(cons(x, nil())) -> c_8()
  , last^#(cons(x, cons(y, xs))) -> c_9(last^#(cons(y, xs)))
  , filterhigh^#(n, nil()) -> c_10()
  , filterhigh^#(n, cons(x, xs)) ->
    c_11(if2^#(ge(x, n), n, x, xs), ge^#(x, n))
  , if1^#(true(), n, x, xs) -> c_12(filterlow^#(n, xs))
  , if1^#(false(), n, x, xs) -> c_13(filterlow^#(n, xs))
  , ge^#(x, 0()) -> c_14()
  , ge^#(0(), s(x)) -> c_15()
  , ge^#(s(x), s(y)) -> c_16(ge^#(x, y))
  , if2^#(true(), n, x, xs) -> c_17(filterhigh^#(n, xs))
  , if2^#(false(), n, x, xs) -> c_18(filterhigh^#(n, xs)) }
Weak Trs:
  { 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)))))
  , append(nil(), ys()) -> ys()
  , append(cons(x, xs), ys()) -> cons(x, append(xs, ys()))
  , filterlow(n, nil()) -> nil()
  , filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs)
  , last(nil()) -> 0()
  , last(cons(x, nil())) -> x
  , last(cons(x, cons(y, xs))) -> last(cons(y, xs))
  , filterhigh(n, nil()) -> nil()
  , filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs)
  , if1(true(), n, x, xs) -> filterlow(n, xs)
  , if1(false(), n, x, xs) -> cons(x, filterlow(n, xs))
  , ge(x, 0()) -> true()
  , ge(0(), s(x)) -> false()
  , ge(s(x), s(y)) -> ge(x, y)
  , if2(true(), n, x, xs) -> filterhigh(n, xs)
  , if2(false(), n, x, xs) -> cons(x, filterhigh(n, xs)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {1,3,5,7,8,10,14,15} by
applications of Pre({1,3,5,7,8,10,14,15}) =
{2,4,6,9,11,12,13,16,17,18}. Here rules are labeled as follows:

  DPs:
    { 1: qsort^#(nil()) -> c_1()
    , 2: qsort^#(cons(x, xs)) ->
         c_2(append^#(qsort(filterlow(last(cons(x, xs)), cons(x, xs))),
                      cons(last(cons(x, xs)),
                           qsort(filterhigh(last(cons(x, xs)), cons(x, xs))))),
             qsort^#(filterlow(last(cons(x, xs)), cons(x, xs))),
             filterlow^#(last(cons(x, xs)), cons(x, xs)),
             last^#(cons(x, xs)),
             last^#(cons(x, xs)),
             qsort^#(filterhigh(last(cons(x, xs)), cons(x, xs))),
             filterhigh^#(last(cons(x, xs)), cons(x, xs)),
             last^#(cons(x, xs)))
    , 3: append^#(nil(), ys()) -> c_3()
    , 4: append^#(cons(x, xs), ys()) -> c_4(append^#(xs, ys()))
    , 5: filterlow^#(n, nil()) -> c_5()
    , 6: filterlow^#(n, cons(x, xs)) ->
         c_6(if1^#(ge(n, x), n, x, xs), ge^#(n, x))
    , 7: last^#(nil()) -> c_7()
    , 8: last^#(cons(x, nil())) -> c_8()
    , 9: last^#(cons(x, cons(y, xs))) -> c_9(last^#(cons(y, xs)))
    , 10: filterhigh^#(n, nil()) -> c_10()
    , 11: filterhigh^#(n, cons(x, xs)) ->
          c_11(if2^#(ge(x, n), n, x, xs), ge^#(x, n))
    , 12: if1^#(true(), n, x, xs) -> c_12(filterlow^#(n, xs))
    , 13: if1^#(false(), n, x, xs) -> c_13(filterlow^#(n, xs))
    , 14: ge^#(x, 0()) -> c_14()
    , 15: ge^#(0(), s(x)) -> c_15()
    , 16: ge^#(s(x), s(y)) -> c_16(ge^#(x, y))
    , 17: if2^#(true(), n, x, xs) -> c_17(filterhigh^#(n, xs))
    , 18: if2^#(false(), n, x, xs) -> c_18(filterhigh^#(n, xs)) }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { qsort^#(cons(x, xs)) ->
    c_2(append^#(qsort(filterlow(last(cons(x, xs)), cons(x, xs))),
                 cons(last(cons(x, xs)),
                      qsort(filterhigh(last(cons(x, xs)), cons(x, xs))))),
        qsort^#(filterlow(last(cons(x, xs)), cons(x, xs))),
        filterlow^#(last(cons(x, xs)), cons(x, xs)),
        last^#(cons(x, xs)),
        last^#(cons(x, xs)),
        qsort^#(filterhigh(last(cons(x, xs)), cons(x, xs))),
        filterhigh^#(last(cons(x, xs)), cons(x, xs)),
        last^#(cons(x, xs)))
  , append^#(cons(x, xs), ys()) -> c_4(append^#(xs, ys()))
  , filterlow^#(n, cons(x, xs)) ->
    c_6(if1^#(ge(n, x), n, x, xs), ge^#(n, x))
  , last^#(cons(x, cons(y, xs))) -> c_9(last^#(cons(y, xs)))
  , filterhigh^#(n, cons(x, xs)) ->
    c_11(if2^#(ge(x, n), n, x, xs), ge^#(x, n))
  , if1^#(true(), n, x, xs) -> c_12(filterlow^#(n, xs))
  , if1^#(false(), n, x, xs) -> c_13(filterlow^#(n, xs))
  , ge^#(s(x), s(y)) -> c_16(ge^#(x, y))
  , if2^#(true(), n, x, xs) -> c_17(filterhigh^#(n, xs))
  , if2^#(false(), n, x, xs) -> c_18(filterhigh^#(n, xs)) }
Weak DPs:
  { qsort^#(nil()) -> c_1()
  , append^#(nil(), ys()) -> c_3()
  , filterlow^#(n, nil()) -> c_5()
  , last^#(nil()) -> c_7()
  , last^#(cons(x, nil())) -> c_8()
  , filterhigh^#(n, nil()) -> c_10()
  , ge^#(x, 0()) -> c_14()
  , ge^#(0(), s(x)) -> c_15() }
Weak Trs:
  { 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)))))
  , append(nil(), ys()) -> ys()
  , append(cons(x, xs), ys()) -> cons(x, append(xs, ys()))
  , filterlow(n, nil()) -> nil()
  , filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs)
  , last(nil()) -> 0()
  , last(cons(x, nil())) -> x
  , last(cons(x, cons(y, xs))) -> last(cons(y, xs))
  , filterhigh(n, nil()) -> nil()
  , filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs)
  , if1(true(), n, x, xs) -> filterlow(n, xs)
  , if1(false(), n, x, xs) -> cons(x, filterlow(n, xs))
  , ge(x, 0()) -> true()
  , ge(0(), s(x)) -> false()
  , ge(s(x), s(y)) -> ge(x, y)
  , if2(true(), n, x, xs) -> filterhigh(n, xs)
  , if2(false(), n, x, xs) -> cons(x, filterhigh(n, xs)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ qsort^#(nil()) -> c_1()
, append^#(nil(), ys()) -> c_3()
, filterlow^#(n, nil()) -> c_5()
, last^#(nil()) -> c_7()
, last^#(cons(x, nil())) -> c_8()
, filterhigh^#(n, nil()) -> c_10()
, ge^#(x, 0()) -> c_14()
, ge^#(0(), s(x)) -> c_15() }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { qsort^#(cons(x, xs)) ->
    c_2(append^#(qsort(filterlow(last(cons(x, xs)), cons(x, xs))),
                 cons(last(cons(x, xs)),
                      qsort(filterhigh(last(cons(x, xs)), cons(x, xs))))),
        qsort^#(filterlow(last(cons(x, xs)), cons(x, xs))),
        filterlow^#(last(cons(x, xs)), cons(x, xs)),
        last^#(cons(x, xs)),
        last^#(cons(x, xs)),
        qsort^#(filterhigh(last(cons(x, xs)), cons(x, xs))),
        filterhigh^#(last(cons(x, xs)), cons(x, xs)),
        last^#(cons(x, xs)))
  , append^#(cons(x, xs), ys()) -> c_4(append^#(xs, ys()))
  , filterlow^#(n, cons(x, xs)) ->
    c_6(if1^#(ge(n, x), n, x, xs), ge^#(n, x))
  , last^#(cons(x, cons(y, xs))) -> c_9(last^#(cons(y, xs)))
  , filterhigh^#(n, cons(x, xs)) ->
    c_11(if2^#(ge(x, n), n, x, xs), ge^#(x, n))
  , if1^#(true(), n, x, xs) -> c_12(filterlow^#(n, xs))
  , if1^#(false(), n, x, xs) -> c_13(filterlow^#(n, xs))
  , ge^#(s(x), s(y)) -> c_16(ge^#(x, y))
  , if2^#(true(), n, x, xs) -> c_17(filterhigh^#(n, xs))
  , if2^#(false(), n, x, xs) -> c_18(filterhigh^#(n, xs)) }
Weak Trs:
  { 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)))))
  , append(nil(), ys()) -> ys()
  , append(cons(x, xs), ys()) -> cons(x, append(xs, ys()))
  , filterlow(n, nil()) -> nil()
  , filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs)
  , last(nil()) -> 0()
  , last(cons(x, nil())) -> x
  , last(cons(x, cons(y, xs))) -> last(cons(y, xs))
  , filterhigh(n, nil()) -> nil()
  , filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs)
  , if1(true(), n, x, xs) -> filterlow(n, xs)
  , if1(false(), n, x, xs) -> cons(x, filterlow(n, xs))
  , ge(x, 0()) -> true()
  , ge(0(), s(x)) -> false()
  , ge(s(x), s(y)) -> ge(x, y)
  , if2(true(), n, x, xs) -> filterhigh(n, xs)
  , if2(false(), n, x, xs) -> cons(x, filterhigh(n, xs)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

Due to missing edges in the dependency-graph, the right-hand sides
of following rules could be simplified:

  { qsort^#(cons(x, xs)) ->
    c_2(append^#(qsort(filterlow(last(cons(x, xs)), cons(x, xs))),
                 cons(last(cons(x, xs)),
                      qsort(filterhigh(last(cons(x, xs)), cons(x, xs))))),
        qsort^#(filterlow(last(cons(x, xs)), cons(x, xs))),
        filterlow^#(last(cons(x, xs)), cons(x, xs)),
        last^#(cons(x, xs)),
        last^#(cons(x, xs)),
        qsort^#(filterhigh(last(cons(x, xs)), cons(x, xs))),
        filterhigh^#(last(cons(x, xs)), cons(x, xs)),
        last^#(cons(x, xs))) }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { qsort^#(cons(x, xs)) ->
    c_1(qsort^#(filterlow(last(cons(x, xs)), cons(x, xs))),
        filterlow^#(last(cons(x, xs)), cons(x, xs)),
        last^#(cons(x, xs)),
        last^#(cons(x, xs)),
        qsort^#(filterhigh(last(cons(x, xs)), cons(x, xs))),
        filterhigh^#(last(cons(x, xs)), cons(x, xs)),
        last^#(cons(x, xs)))
  , append^#(cons(x, xs), ys()) -> c_2(append^#(xs, ys()))
  , filterlow^#(n, cons(x, xs)) ->
    c_3(if1^#(ge(n, x), n, x, xs), ge^#(n, x))
  , last^#(cons(x, cons(y, xs))) -> c_4(last^#(cons(y, xs)))
  , filterhigh^#(n, cons(x, xs)) ->
    c_5(if2^#(ge(x, n), n, x, xs), ge^#(x, n))
  , if1^#(true(), n, x, xs) -> c_6(filterlow^#(n, xs))
  , if1^#(false(), n, x, xs) -> c_7(filterlow^#(n, xs))
  , ge^#(s(x), s(y)) -> c_8(ge^#(x, y))
  , if2^#(true(), n, x, xs) -> c_9(filterhigh^#(n, xs))
  , if2^#(false(), n, x, xs) -> c_10(filterhigh^#(n, xs)) }
Weak Trs:
  { 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)))))
  , append(nil(), ys()) -> ys()
  , append(cons(x, xs), ys()) -> cons(x, append(xs, ys()))
  , filterlow(n, nil()) -> nil()
  , filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs)
  , last(nil()) -> 0()
  , last(cons(x, nil())) -> x
  , last(cons(x, cons(y, xs))) -> last(cons(y, xs))
  , filterhigh(n, nil()) -> nil()
  , filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs)
  , if1(true(), n, x, xs) -> filterlow(n, xs)
  , if1(false(), n, x, xs) -> cons(x, filterlow(n, xs))
  , ge(x, 0()) -> true()
  , ge(0(), s(x)) -> false()
  , ge(s(x), s(y)) -> ge(x, y)
  , if2(true(), n, x, xs) -> filterhigh(n, xs)
  , if2(false(), n, x, xs) -> cons(x, filterhigh(n, xs)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { filterlow(n, nil()) -> nil()
    , filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs)
    , last(nil()) -> 0()
    , last(cons(x, nil())) -> x
    , last(cons(x, cons(y, xs))) -> last(cons(y, xs))
    , filterhigh(n, nil()) -> nil()
    , filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs)
    , if1(true(), n, x, xs) -> filterlow(n, xs)
    , if1(false(), n, x, xs) -> cons(x, filterlow(n, xs))
    , ge(x, 0()) -> true()
    , ge(0(), s(x)) -> false()
    , ge(s(x), s(y)) -> ge(x, y)
    , if2(true(), n, x, xs) -> filterhigh(n, xs)
    , if2(false(), n, x, xs) -> cons(x, filterhigh(n, xs)) }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { qsort^#(cons(x, xs)) ->
    c_1(qsort^#(filterlow(last(cons(x, xs)), cons(x, xs))),
        filterlow^#(last(cons(x, xs)), cons(x, xs)),
        last^#(cons(x, xs)),
        last^#(cons(x, xs)),
        qsort^#(filterhigh(last(cons(x, xs)), cons(x, xs))),
        filterhigh^#(last(cons(x, xs)), cons(x, xs)),
        last^#(cons(x, xs)))
  , append^#(cons(x, xs), ys()) -> c_2(append^#(xs, ys()))
  , filterlow^#(n, cons(x, xs)) ->
    c_3(if1^#(ge(n, x), n, x, xs), ge^#(n, x))
  , last^#(cons(x, cons(y, xs))) -> c_4(last^#(cons(y, xs)))
  , filterhigh^#(n, cons(x, xs)) ->
    c_5(if2^#(ge(x, n), n, x, xs), ge^#(x, n))
  , if1^#(true(), n, x, xs) -> c_6(filterlow^#(n, xs))
  , if1^#(false(), n, x, xs) -> c_7(filterlow^#(n, xs))
  , ge^#(s(x), s(y)) -> c_8(ge^#(x, y))
  , if2^#(true(), n, x, xs) -> c_9(filterhigh^#(n, xs))
  , if2^#(false(), n, x, xs) -> c_10(filterhigh^#(n, xs)) }
Weak Trs:
  { filterlow(n, nil()) -> nil()
  , filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs)
  , last(nil()) -> 0()
  , last(cons(x, nil())) -> x
  , last(cons(x, cons(y, xs))) -> last(cons(y, xs))
  , filterhigh(n, nil()) -> nil()
  , filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs)
  , if1(true(), n, x, xs) -> filterlow(n, xs)
  , if1(false(), n, x, xs) -> cons(x, filterlow(n, xs))
  , ge(x, 0()) -> true()
  , ge(0(), s(x)) -> false()
  , ge(s(x), s(y)) -> ge(x, y)
  , if2(true(), n, x, xs) -> filterhigh(n, xs)
  , if2(false(), n, x, xs) -> cons(x, filterhigh(n, xs)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..