MAYBE

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

Strict Trs:
  { le(0(), Y) -> true()
  , le(s(X), 0()) -> false()
  , le(s(X), s(Y)) -> le(X, Y)
  , app(nil(), Y) -> Y
  , app(cons(N, L), Y) -> cons(N, app(L, Y))
  , low(N, nil()) -> nil()
  , low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
  , iflow(true(), N, cons(M, L)) -> cons(M, low(N, L))
  , iflow(false(), N, cons(M, L)) -> low(N, L)
  , high(N, nil()) -> nil()
  , high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
  , ifhigh(true(), N, cons(M, L)) -> high(N, L)
  , ifhigh(false(), N, cons(M, L)) -> cons(M, high(N, L))
  , quicksort(nil()) -> nil()
  , quicksort(cons(N, L)) ->
    app(quicksort(low(N, L)), cons(N, quicksort(high(N, L)))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { le^#(0(), Y) -> c_1()
  , le^#(s(X), 0()) -> c_2()
  , le^#(s(X), s(Y)) -> c_3(le^#(X, Y))
  , app^#(nil(), Y) -> c_4()
  , app^#(cons(N, L), Y) -> c_5(app^#(L, Y))
  , low^#(N, nil()) -> c_6()
  , low^#(N, cons(M, L)) ->
    c_7(iflow^#(le(M, N), N, cons(M, L)), le^#(M, N))
  , iflow^#(true(), N, cons(M, L)) -> c_8(low^#(N, L))
  , iflow^#(false(), N, cons(M, L)) -> c_9(low^#(N, L))
  , high^#(N, nil()) -> c_10()
  , high^#(N, cons(M, L)) ->
    c_11(ifhigh^#(le(M, N), N, cons(M, L)), le^#(M, N))
  , ifhigh^#(true(), N, cons(M, L)) -> c_12(high^#(N, L))
  , ifhigh^#(false(), N, cons(M, L)) -> c_13(high^#(N, L))
  , quicksort^#(nil()) -> c_14()
  , quicksort^#(cons(N, L)) ->
    c_15(app^#(quicksort(low(N, L)), cons(N, quicksort(high(N, L)))),
         quicksort^#(low(N, L)),
         low^#(N, L),
         quicksort^#(high(N, L)),
         high^#(N, L)) }

and mark the set of starting terms.

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

Strict DPs:
  { le^#(0(), Y) -> c_1()
  , le^#(s(X), 0()) -> c_2()
  , le^#(s(X), s(Y)) -> c_3(le^#(X, Y))
  , app^#(nil(), Y) -> c_4()
  , app^#(cons(N, L), Y) -> c_5(app^#(L, Y))
  , low^#(N, nil()) -> c_6()
  , low^#(N, cons(M, L)) ->
    c_7(iflow^#(le(M, N), N, cons(M, L)), le^#(M, N))
  , iflow^#(true(), N, cons(M, L)) -> c_8(low^#(N, L))
  , iflow^#(false(), N, cons(M, L)) -> c_9(low^#(N, L))
  , high^#(N, nil()) -> c_10()
  , high^#(N, cons(M, L)) ->
    c_11(ifhigh^#(le(M, N), N, cons(M, L)), le^#(M, N))
  , ifhigh^#(true(), N, cons(M, L)) -> c_12(high^#(N, L))
  , ifhigh^#(false(), N, cons(M, L)) -> c_13(high^#(N, L))
  , quicksort^#(nil()) -> c_14()
  , quicksort^#(cons(N, L)) ->
    c_15(app^#(quicksort(low(N, L)), cons(N, quicksort(high(N, L)))),
         quicksort^#(low(N, L)),
         low^#(N, L),
         quicksort^#(high(N, L)),
         high^#(N, L)) }
Weak Trs:
  { le(0(), Y) -> true()
  , le(s(X), 0()) -> false()
  , le(s(X), s(Y)) -> le(X, Y)
  , app(nil(), Y) -> Y
  , app(cons(N, L), Y) -> cons(N, app(L, Y))
  , low(N, nil()) -> nil()
  , low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
  , iflow(true(), N, cons(M, L)) -> cons(M, low(N, L))
  , iflow(false(), N, cons(M, L)) -> low(N, L)
  , high(N, nil()) -> nil()
  , high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
  , ifhigh(true(), N, cons(M, L)) -> high(N, L)
  , ifhigh(false(), N, cons(M, L)) -> cons(M, high(N, L))
  , quicksort(nil()) -> nil()
  , quicksort(cons(N, L)) ->
    app(quicksort(low(N, L)), cons(N, quicksort(high(N, L)))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: le^#(0(), Y) -> c_1()
    , 2: le^#(s(X), 0()) -> c_2()
    , 3: le^#(s(X), s(Y)) -> c_3(le^#(X, Y))
    , 4: app^#(nil(), Y) -> c_4()
    , 5: app^#(cons(N, L), Y) -> c_5(app^#(L, Y))
    , 6: low^#(N, nil()) -> c_6()
    , 7: low^#(N, cons(M, L)) ->
         c_7(iflow^#(le(M, N), N, cons(M, L)), le^#(M, N))
    , 8: iflow^#(true(), N, cons(M, L)) -> c_8(low^#(N, L))
    , 9: iflow^#(false(), N, cons(M, L)) -> c_9(low^#(N, L))
    , 10: high^#(N, nil()) -> c_10()
    , 11: high^#(N, cons(M, L)) ->
          c_11(ifhigh^#(le(M, N), N, cons(M, L)), le^#(M, N))
    , 12: ifhigh^#(true(), N, cons(M, L)) -> c_12(high^#(N, L))
    , 13: ifhigh^#(false(), N, cons(M, L)) -> c_13(high^#(N, L))
    , 14: quicksort^#(nil()) -> c_14()
    , 15: quicksort^#(cons(N, L)) ->
          c_15(app^#(quicksort(low(N, L)), cons(N, quicksort(high(N, L)))),
               quicksort^#(low(N, L)),
               low^#(N, L),
               quicksort^#(high(N, L)),
               high^#(N, L)) }

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

Strict DPs:
  { le^#(s(X), s(Y)) -> c_3(le^#(X, Y))
  , app^#(cons(N, L), Y) -> c_5(app^#(L, Y))
  , low^#(N, cons(M, L)) ->
    c_7(iflow^#(le(M, N), N, cons(M, L)), le^#(M, N))
  , iflow^#(true(), N, cons(M, L)) -> c_8(low^#(N, L))
  , iflow^#(false(), N, cons(M, L)) -> c_9(low^#(N, L))
  , high^#(N, cons(M, L)) ->
    c_11(ifhigh^#(le(M, N), N, cons(M, L)), le^#(M, N))
  , ifhigh^#(true(), N, cons(M, L)) -> c_12(high^#(N, L))
  , ifhigh^#(false(), N, cons(M, L)) -> c_13(high^#(N, L))
  , quicksort^#(cons(N, L)) ->
    c_15(app^#(quicksort(low(N, L)), cons(N, quicksort(high(N, L)))),
         quicksort^#(low(N, L)),
         low^#(N, L),
         quicksort^#(high(N, L)),
         high^#(N, L)) }
Weak DPs:
  { le^#(0(), Y) -> c_1()
  , le^#(s(X), 0()) -> c_2()
  , app^#(nil(), Y) -> c_4()
  , low^#(N, nil()) -> c_6()
  , high^#(N, nil()) -> c_10()
  , quicksort^#(nil()) -> c_14() }
Weak Trs:
  { le(0(), Y) -> true()
  , le(s(X), 0()) -> false()
  , le(s(X), s(Y)) -> le(X, Y)
  , app(nil(), Y) -> Y
  , app(cons(N, L), Y) -> cons(N, app(L, Y))
  , low(N, nil()) -> nil()
  , low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
  , iflow(true(), N, cons(M, L)) -> cons(M, low(N, L))
  , iflow(false(), N, cons(M, L)) -> low(N, L)
  , high(N, nil()) -> nil()
  , high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
  , ifhigh(true(), N, cons(M, L)) -> high(N, L)
  , ifhigh(false(), N, cons(M, L)) -> cons(M, high(N, L))
  , quicksort(nil()) -> nil()
  , quicksort(cons(N, L)) ->
    app(quicksort(low(N, L)), cons(N, quicksort(high(N, L)))) }
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.

{ le^#(0(), Y) -> c_1()
, le^#(s(X), 0()) -> c_2()
, app^#(nil(), Y) -> c_4()
, low^#(N, nil()) -> c_6()
, high^#(N, nil()) -> c_10()
, quicksort^#(nil()) -> c_14() }

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

Strict DPs:
  { le^#(s(X), s(Y)) -> c_3(le^#(X, Y))
  , app^#(cons(N, L), Y) -> c_5(app^#(L, Y))
  , low^#(N, cons(M, L)) ->
    c_7(iflow^#(le(M, N), N, cons(M, L)), le^#(M, N))
  , iflow^#(true(), N, cons(M, L)) -> c_8(low^#(N, L))
  , iflow^#(false(), N, cons(M, L)) -> c_9(low^#(N, L))
  , high^#(N, cons(M, L)) ->
    c_11(ifhigh^#(le(M, N), N, cons(M, L)), le^#(M, N))
  , ifhigh^#(true(), N, cons(M, L)) -> c_12(high^#(N, L))
  , ifhigh^#(false(), N, cons(M, L)) -> c_13(high^#(N, L))
  , quicksort^#(cons(N, L)) ->
    c_15(app^#(quicksort(low(N, L)), cons(N, quicksort(high(N, L)))),
         quicksort^#(low(N, L)),
         low^#(N, L),
         quicksort^#(high(N, L)),
         high^#(N, L)) }
Weak Trs:
  { le(0(), Y) -> true()
  , le(s(X), 0()) -> false()
  , le(s(X), s(Y)) -> le(X, Y)
  , app(nil(), Y) -> Y
  , app(cons(N, L), Y) -> cons(N, app(L, Y))
  , low(N, nil()) -> nil()
  , low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
  , iflow(true(), N, cons(M, L)) -> cons(M, low(N, L))
  , iflow(false(), N, cons(M, L)) -> low(N, L)
  , high(N, nil()) -> nil()
  , high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
  , ifhigh(true(), N, cons(M, L)) -> high(N, L)
  , ifhigh(false(), N, cons(M, L)) -> cons(M, high(N, L))
  , quicksort(nil()) -> nil()
  , quicksort(cons(N, L)) ->
    app(quicksort(low(N, L)), cons(N, quicksort(high(N, L)))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..