MAYBE

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

Strict Trs:
  { minus(x, 0()) -> x
  , minus(s(x), s(y)) -> minus(x, y)
  , quot(0(), s(y)) -> 0()
  , quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , app(nil(), y) -> y
  , app(add(n, x), y) -> add(n, app(x, y))
  , low(n, nil()) -> nil()
  , low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
  , if_low(true(), n, add(m, x)) -> add(m, low(n, x))
  , if_low(false(), n, add(m, x)) -> low(n, x)
  , high(n, nil()) -> nil()
  , high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
  , if_high(true(), n, add(m, x)) -> high(n, x)
  , if_high(false(), n, add(m, x)) -> add(m, high(n, x))
  , quicksort(nil()) -> nil()
  , quicksort(add(n, x)) ->
    app(quicksort(low(n, x)), add(n, quicksort(high(n, x)))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { minus^#(x, 0()) -> c_1()
  , minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
  , quot^#(0(), s(y)) -> c_3()
  , quot^#(s(x), s(y)) ->
    c_4(quot^#(minus(x, y), s(y)), minus^#(x, y))
  , le^#(0(), y) -> c_5()
  , le^#(s(x), 0()) -> c_6()
  , le^#(s(x), s(y)) -> c_7(le^#(x, y))
  , app^#(nil(), y) -> c_8()
  , app^#(add(n, x), y) -> c_9(app^#(x, y))
  , low^#(n, nil()) -> c_10()
  , low^#(n, add(m, x)) ->
    c_11(if_low^#(le(m, n), n, add(m, x)), le^#(m, n))
  , if_low^#(true(), n, add(m, x)) -> c_12(low^#(n, x))
  , if_low^#(false(), n, add(m, x)) -> c_13(low^#(n, x))
  , high^#(n, nil()) -> c_14()
  , high^#(n, add(m, x)) ->
    c_15(if_high^#(le(m, n), n, add(m, x)), le^#(m, n))
  , if_high^#(true(), n, add(m, x)) -> c_16(high^#(n, x))
  , if_high^#(false(), n, add(m, x)) -> c_17(high^#(n, x))
  , quicksort^#(nil()) -> c_18()
  , quicksort^#(add(n, x)) ->
    c_19(app^#(quicksort(low(n, x)), add(n, quicksort(high(n, x)))),
         quicksort^#(low(n, x)),
         low^#(n, x),
         quicksort^#(high(n, x)),
         high^#(n, x)) }

and mark the set of starting terms.

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

Strict DPs:
  { minus^#(x, 0()) -> c_1()
  , minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
  , quot^#(0(), s(y)) -> c_3()
  , quot^#(s(x), s(y)) ->
    c_4(quot^#(minus(x, y), s(y)), minus^#(x, y))
  , le^#(0(), y) -> c_5()
  , le^#(s(x), 0()) -> c_6()
  , le^#(s(x), s(y)) -> c_7(le^#(x, y))
  , app^#(nil(), y) -> c_8()
  , app^#(add(n, x), y) -> c_9(app^#(x, y))
  , low^#(n, nil()) -> c_10()
  , low^#(n, add(m, x)) ->
    c_11(if_low^#(le(m, n), n, add(m, x)), le^#(m, n))
  , if_low^#(true(), n, add(m, x)) -> c_12(low^#(n, x))
  , if_low^#(false(), n, add(m, x)) -> c_13(low^#(n, x))
  , high^#(n, nil()) -> c_14()
  , high^#(n, add(m, x)) ->
    c_15(if_high^#(le(m, n), n, add(m, x)), le^#(m, n))
  , if_high^#(true(), n, add(m, x)) -> c_16(high^#(n, x))
  , if_high^#(false(), n, add(m, x)) -> c_17(high^#(n, x))
  , quicksort^#(nil()) -> c_18()
  , quicksort^#(add(n, x)) ->
    c_19(app^#(quicksort(low(n, x)), add(n, quicksort(high(n, x)))),
         quicksort^#(low(n, x)),
         low^#(n, x),
         quicksort^#(high(n, x)),
         high^#(n, x)) }
Weak Trs:
  { minus(x, 0()) -> x
  , minus(s(x), s(y)) -> minus(x, y)
  , quot(0(), s(y)) -> 0()
  , quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , app(nil(), y) -> y
  , app(add(n, x), y) -> add(n, app(x, y))
  , low(n, nil()) -> nil()
  , low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
  , if_low(true(), n, add(m, x)) -> add(m, low(n, x))
  , if_low(false(), n, add(m, x)) -> low(n, x)
  , high(n, nil()) -> nil()
  , high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
  , if_high(true(), n, add(m, x)) -> high(n, x)
  , if_high(false(), n, add(m, x)) -> add(m, high(n, x))
  , quicksort(nil()) -> nil()
  , quicksort(add(n, x)) ->
    app(quicksort(low(n, x)), add(n, quicksort(high(n, x)))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {1,3,5,6,8,10,14,18} by
applications of Pre({1,3,5,6,8,10,14,18}) =
{2,4,7,9,11,12,13,15,16,17,19}. Here rules are labeled as follows:

  DPs:
    { 1: minus^#(x, 0()) -> c_1()
    , 2: minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
    , 3: quot^#(0(), s(y)) -> c_3()
    , 4: quot^#(s(x), s(y)) ->
         c_4(quot^#(minus(x, y), s(y)), minus^#(x, y))
    , 5: le^#(0(), y) -> c_5()
    , 6: le^#(s(x), 0()) -> c_6()
    , 7: le^#(s(x), s(y)) -> c_7(le^#(x, y))
    , 8: app^#(nil(), y) -> c_8()
    , 9: app^#(add(n, x), y) -> c_9(app^#(x, y))
    , 10: low^#(n, nil()) -> c_10()
    , 11: low^#(n, add(m, x)) ->
          c_11(if_low^#(le(m, n), n, add(m, x)), le^#(m, n))
    , 12: if_low^#(true(), n, add(m, x)) -> c_12(low^#(n, x))
    , 13: if_low^#(false(), n, add(m, x)) -> c_13(low^#(n, x))
    , 14: high^#(n, nil()) -> c_14()
    , 15: high^#(n, add(m, x)) ->
          c_15(if_high^#(le(m, n), n, add(m, x)), le^#(m, n))
    , 16: if_high^#(true(), n, add(m, x)) -> c_16(high^#(n, x))
    , 17: if_high^#(false(), n, add(m, x)) -> c_17(high^#(n, x))
    , 18: quicksort^#(nil()) -> c_18()
    , 19: quicksort^#(add(n, x)) ->
          c_19(app^#(quicksort(low(n, x)), add(n, quicksort(high(n, x)))),
               quicksort^#(low(n, x)),
               low^#(n, x),
               quicksort^#(high(n, x)),
               high^#(n, x)) }

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

Strict DPs:
  { minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
  , quot^#(s(x), s(y)) ->
    c_4(quot^#(minus(x, y), s(y)), minus^#(x, y))
  , le^#(s(x), s(y)) -> c_7(le^#(x, y))
  , app^#(add(n, x), y) -> c_9(app^#(x, y))
  , low^#(n, add(m, x)) ->
    c_11(if_low^#(le(m, n), n, add(m, x)), le^#(m, n))
  , if_low^#(true(), n, add(m, x)) -> c_12(low^#(n, x))
  , if_low^#(false(), n, add(m, x)) -> c_13(low^#(n, x))
  , high^#(n, add(m, x)) ->
    c_15(if_high^#(le(m, n), n, add(m, x)), le^#(m, n))
  , if_high^#(true(), n, add(m, x)) -> c_16(high^#(n, x))
  , if_high^#(false(), n, add(m, x)) -> c_17(high^#(n, x))
  , quicksort^#(add(n, x)) ->
    c_19(app^#(quicksort(low(n, x)), add(n, quicksort(high(n, x)))),
         quicksort^#(low(n, x)),
         low^#(n, x),
         quicksort^#(high(n, x)),
         high^#(n, x)) }
Weak DPs:
  { minus^#(x, 0()) -> c_1()
  , quot^#(0(), s(y)) -> c_3()
  , le^#(0(), y) -> c_5()
  , le^#(s(x), 0()) -> c_6()
  , app^#(nil(), y) -> c_8()
  , low^#(n, nil()) -> c_10()
  , high^#(n, nil()) -> c_14()
  , quicksort^#(nil()) -> c_18() }
Weak Trs:
  { minus(x, 0()) -> x
  , minus(s(x), s(y)) -> minus(x, y)
  , quot(0(), s(y)) -> 0()
  , quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , app(nil(), y) -> y
  , app(add(n, x), y) -> add(n, app(x, y))
  , low(n, nil()) -> nil()
  , low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
  , if_low(true(), n, add(m, x)) -> add(m, low(n, x))
  , if_low(false(), n, add(m, x)) -> low(n, x)
  , high(n, nil()) -> nil()
  , high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
  , if_high(true(), n, add(m, x)) -> high(n, x)
  , if_high(false(), n, add(m, x)) -> add(m, high(n, x))
  , quicksort(nil()) -> nil()
  , quicksort(add(n, x)) ->
    app(quicksort(low(n, x)), add(n, quicksort(high(n, x)))) }
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.

{ minus^#(x, 0()) -> c_1()
, quot^#(0(), s(y)) -> c_3()
, le^#(0(), y) -> c_5()
, le^#(s(x), 0()) -> c_6()
, app^#(nil(), y) -> c_8()
, low^#(n, nil()) -> c_10()
, high^#(n, nil()) -> c_14()
, quicksort^#(nil()) -> c_18() }

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

Strict DPs:
  { minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
  , quot^#(s(x), s(y)) ->
    c_4(quot^#(minus(x, y), s(y)), minus^#(x, y))
  , le^#(s(x), s(y)) -> c_7(le^#(x, y))
  , app^#(add(n, x), y) -> c_9(app^#(x, y))
  , low^#(n, add(m, x)) ->
    c_11(if_low^#(le(m, n), n, add(m, x)), le^#(m, n))
  , if_low^#(true(), n, add(m, x)) -> c_12(low^#(n, x))
  , if_low^#(false(), n, add(m, x)) -> c_13(low^#(n, x))
  , high^#(n, add(m, x)) ->
    c_15(if_high^#(le(m, n), n, add(m, x)), le^#(m, n))
  , if_high^#(true(), n, add(m, x)) -> c_16(high^#(n, x))
  , if_high^#(false(), n, add(m, x)) -> c_17(high^#(n, x))
  , quicksort^#(add(n, x)) ->
    c_19(app^#(quicksort(low(n, x)), add(n, quicksort(high(n, x)))),
         quicksort^#(low(n, x)),
         low^#(n, x),
         quicksort^#(high(n, x)),
         high^#(n, x)) }
Weak Trs:
  { minus(x, 0()) -> x
  , minus(s(x), s(y)) -> minus(x, y)
  , quot(0(), s(y)) -> 0()
  , quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , app(nil(), y) -> y
  , app(add(n, x), y) -> add(n, app(x, y))
  , low(n, nil()) -> nil()
  , low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
  , if_low(true(), n, add(m, x)) -> add(m, low(n, x))
  , if_low(false(), n, add(m, x)) -> low(n, x)
  , high(n, nil()) -> nil()
  , high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
  , if_high(true(), n, add(m, x)) -> high(n, x)
  , if_high(false(), n, add(m, x)) -> add(m, high(n, x))
  , quicksort(nil()) -> nil()
  , quicksort(add(n, x)) ->
    app(quicksort(low(n, x)), add(n, quicksort(high(n, x)))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { minus(x, 0()) -> x
    , minus(s(x), s(y)) -> minus(x, y)
    , le(0(), y) -> true()
    , le(s(x), 0()) -> false()
    , le(s(x), s(y)) -> le(x, y)
    , app(nil(), y) -> y
    , app(add(n, x), y) -> add(n, app(x, y))
    , low(n, nil()) -> nil()
    , low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
    , if_low(true(), n, add(m, x)) -> add(m, low(n, x))
    , if_low(false(), n, add(m, x)) -> low(n, x)
    , high(n, nil()) -> nil()
    , high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
    , if_high(true(), n, add(m, x)) -> high(n, x)
    , if_high(false(), n, add(m, x)) -> add(m, high(n, x))
    , quicksort(nil()) -> nil()
    , quicksort(add(n, x)) ->
      app(quicksort(low(n, x)), add(n, quicksort(high(n, x)))) }

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

Strict DPs:
  { minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
  , quot^#(s(x), s(y)) ->
    c_4(quot^#(minus(x, y), s(y)), minus^#(x, y))
  , le^#(s(x), s(y)) -> c_7(le^#(x, y))
  , app^#(add(n, x), y) -> c_9(app^#(x, y))
  , low^#(n, add(m, x)) ->
    c_11(if_low^#(le(m, n), n, add(m, x)), le^#(m, n))
  , if_low^#(true(), n, add(m, x)) -> c_12(low^#(n, x))
  , if_low^#(false(), n, add(m, x)) -> c_13(low^#(n, x))
  , high^#(n, add(m, x)) ->
    c_15(if_high^#(le(m, n), n, add(m, x)), le^#(m, n))
  , if_high^#(true(), n, add(m, x)) -> c_16(high^#(n, x))
  , if_high^#(false(), n, add(m, x)) -> c_17(high^#(n, x))
  , quicksort^#(add(n, x)) ->
    c_19(app^#(quicksort(low(n, x)), add(n, quicksort(high(n, x)))),
         quicksort^#(low(n, x)),
         low^#(n, x),
         quicksort^#(high(n, x)),
         high^#(n, x)) }
Weak Trs:
  { minus(x, 0()) -> x
  , minus(s(x), s(y)) -> minus(x, y)
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , app(nil(), y) -> y
  , app(add(n, x), y) -> add(n, app(x, y))
  , low(n, nil()) -> nil()
  , low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
  , if_low(true(), n, add(m, x)) -> add(m, low(n, x))
  , if_low(false(), n, add(m, x)) -> low(n, x)
  , high(n, nil()) -> nil()
  , high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
  , if_high(true(), n, add(m, x)) -> high(n, x)
  , if_high(false(), n, add(m, x)) -> add(m, high(n, x))
  , quicksort(nil()) -> nil()
  , quicksort(add(n, x)) ->
    app(quicksort(low(n, x)), add(n, quicksort(high(n, x)))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..