MAYBE

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

Strict Trs:
  { from(X) -> cons(X, n__from(s(X)))
  , from(X) -> n__from(X)
  , sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
  , sel(0(), cons(X, XS)) -> X
  , activate(X) -> X
  , activate(n__from(X)) -> from(X)
  , activate(n__zWquot(X1, X2)) -> zWquot(X1, X2)
  , minus(X, 0()) -> 0()
  , minus(s(X), s(Y)) -> minus(X, Y)
  , quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
  , quot(0(), s(Y)) -> 0()
  , zWquot(X1, X2) -> n__zWquot(X1, X2)
  , zWquot(XS, nil()) -> nil()
  , zWquot(cons(X, XS), cons(Y, YS)) ->
    cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
  , zWquot(nil(), XS) -> nil() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { from^#(X) -> c_1()
  , from^#(X) -> c_2()
  , sel^#(s(N), cons(X, XS)) ->
    c_3(sel^#(N, activate(XS)), activate^#(XS))
  , sel^#(0(), cons(X, XS)) -> c_4()
  , activate^#(X) -> c_5()
  , activate^#(n__from(X)) -> c_6(from^#(X))
  , activate^#(n__zWquot(X1, X2)) -> c_7(zWquot^#(X1, X2))
  , zWquot^#(X1, X2) -> c_12()
  , zWquot^#(XS, nil()) -> c_13()
  , zWquot^#(cons(X, XS), cons(Y, YS)) ->
    c_14(quot^#(X, Y), activate^#(XS), activate^#(YS))
  , zWquot^#(nil(), XS) -> c_15()
  , minus^#(X, 0()) -> c_8()
  , minus^#(s(X), s(Y)) -> c_9(minus^#(X, Y))
  , quot^#(s(X), s(Y)) ->
    c_10(quot^#(minus(X, Y), s(Y)), minus^#(X, Y))
  , quot^#(0(), s(Y)) -> c_11() }

and mark the set of starting terms.

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

Strict DPs:
  { from^#(X) -> c_1()
  , from^#(X) -> c_2()
  , sel^#(s(N), cons(X, XS)) ->
    c_3(sel^#(N, activate(XS)), activate^#(XS))
  , sel^#(0(), cons(X, XS)) -> c_4()
  , activate^#(X) -> c_5()
  , activate^#(n__from(X)) -> c_6(from^#(X))
  , activate^#(n__zWquot(X1, X2)) -> c_7(zWquot^#(X1, X2))
  , zWquot^#(X1, X2) -> c_12()
  , zWquot^#(XS, nil()) -> c_13()
  , zWquot^#(cons(X, XS), cons(Y, YS)) ->
    c_14(quot^#(X, Y), activate^#(XS), activate^#(YS))
  , zWquot^#(nil(), XS) -> c_15()
  , minus^#(X, 0()) -> c_8()
  , minus^#(s(X), s(Y)) -> c_9(minus^#(X, Y))
  , quot^#(s(X), s(Y)) ->
    c_10(quot^#(minus(X, Y), s(Y)), minus^#(X, Y))
  , quot^#(0(), s(Y)) -> c_11() }
Weak Trs:
  { from(X) -> cons(X, n__from(s(X)))
  , from(X) -> n__from(X)
  , sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
  , sel(0(), cons(X, XS)) -> X
  , activate(X) -> X
  , activate(n__from(X)) -> from(X)
  , activate(n__zWquot(X1, X2)) -> zWquot(X1, X2)
  , minus(X, 0()) -> 0()
  , minus(s(X), s(Y)) -> minus(X, Y)
  , quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
  , quot(0(), s(Y)) -> 0()
  , zWquot(X1, X2) -> n__zWquot(X1, X2)
  , zWquot(XS, nil()) -> nil()
  , zWquot(cons(X, XS), cons(Y, YS)) ->
    cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
  , zWquot(nil(), XS) -> nil() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: from^#(X) -> c_1()
    , 2: from^#(X) -> c_2()
    , 3: sel^#(s(N), cons(X, XS)) ->
         c_3(sel^#(N, activate(XS)), activate^#(XS))
    , 4: sel^#(0(), cons(X, XS)) -> c_4()
    , 5: activate^#(X) -> c_5()
    , 6: activate^#(n__from(X)) -> c_6(from^#(X))
    , 7: activate^#(n__zWquot(X1, X2)) -> c_7(zWquot^#(X1, X2))
    , 8: zWquot^#(X1, X2) -> c_12()
    , 9: zWquot^#(XS, nil()) -> c_13()
    , 10: zWquot^#(cons(X, XS), cons(Y, YS)) ->
          c_14(quot^#(X, Y), activate^#(XS), activate^#(YS))
    , 11: zWquot^#(nil(), XS) -> c_15()
    , 12: minus^#(X, 0()) -> c_8()
    , 13: minus^#(s(X), s(Y)) -> c_9(minus^#(X, Y))
    , 14: quot^#(s(X), s(Y)) ->
          c_10(quot^#(minus(X, Y), s(Y)), minus^#(X, Y))
    , 15: quot^#(0(), s(Y)) -> c_11() }

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

Strict DPs:
  { sel^#(s(N), cons(X, XS)) ->
    c_3(sel^#(N, activate(XS)), activate^#(XS))
  , activate^#(n__from(X)) -> c_6(from^#(X))
  , activate^#(n__zWquot(X1, X2)) -> c_7(zWquot^#(X1, X2))
  , zWquot^#(cons(X, XS), cons(Y, YS)) ->
    c_14(quot^#(X, Y), activate^#(XS), activate^#(YS))
  , minus^#(s(X), s(Y)) -> c_9(minus^#(X, Y))
  , quot^#(s(X), s(Y)) ->
    c_10(quot^#(minus(X, Y), s(Y)), minus^#(X, Y)) }
Weak DPs:
  { from^#(X) -> c_1()
  , from^#(X) -> c_2()
  , sel^#(0(), cons(X, XS)) -> c_4()
  , activate^#(X) -> c_5()
  , zWquot^#(X1, X2) -> c_12()
  , zWquot^#(XS, nil()) -> c_13()
  , zWquot^#(nil(), XS) -> c_15()
  , minus^#(X, 0()) -> c_8()
  , quot^#(0(), s(Y)) -> c_11() }
Weak Trs:
  { from(X) -> cons(X, n__from(s(X)))
  , from(X) -> n__from(X)
  , sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
  , sel(0(), cons(X, XS)) -> X
  , activate(X) -> X
  , activate(n__from(X)) -> from(X)
  , activate(n__zWquot(X1, X2)) -> zWquot(X1, X2)
  , minus(X, 0()) -> 0()
  , minus(s(X), s(Y)) -> minus(X, Y)
  , quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
  , quot(0(), s(Y)) -> 0()
  , zWquot(X1, X2) -> n__zWquot(X1, X2)
  , zWquot(XS, nil()) -> nil()
  , zWquot(cons(X, XS), cons(Y, YS)) ->
    cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
  , zWquot(nil(), XS) -> nil() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {2} by applications of
Pre({2}) = {1,4}. Here rules are labeled as follows:

  DPs:
    { 1: sel^#(s(N), cons(X, XS)) ->
         c_3(sel^#(N, activate(XS)), activate^#(XS))
    , 2: activate^#(n__from(X)) -> c_6(from^#(X))
    , 3: activate^#(n__zWquot(X1, X2)) -> c_7(zWquot^#(X1, X2))
    , 4: zWquot^#(cons(X, XS), cons(Y, YS)) ->
         c_14(quot^#(X, Y), activate^#(XS), activate^#(YS))
    , 5: minus^#(s(X), s(Y)) -> c_9(minus^#(X, Y))
    , 6: quot^#(s(X), s(Y)) ->
         c_10(quot^#(minus(X, Y), s(Y)), minus^#(X, Y))
    , 7: from^#(X) -> c_1()
    , 8: from^#(X) -> c_2()
    , 9: sel^#(0(), cons(X, XS)) -> c_4()
    , 10: activate^#(X) -> c_5()
    , 11: zWquot^#(X1, X2) -> c_12()
    , 12: zWquot^#(XS, nil()) -> c_13()
    , 13: zWquot^#(nil(), XS) -> c_15()
    , 14: minus^#(X, 0()) -> c_8()
    , 15: quot^#(0(), s(Y)) -> c_11() }

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

Strict DPs:
  { sel^#(s(N), cons(X, XS)) ->
    c_3(sel^#(N, activate(XS)), activate^#(XS))
  , activate^#(n__zWquot(X1, X2)) -> c_7(zWquot^#(X1, X2))
  , zWquot^#(cons(X, XS), cons(Y, YS)) ->
    c_14(quot^#(X, Y), activate^#(XS), activate^#(YS))
  , minus^#(s(X), s(Y)) -> c_9(minus^#(X, Y))
  , quot^#(s(X), s(Y)) ->
    c_10(quot^#(minus(X, Y), s(Y)), minus^#(X, Y)) }
Weak DPs:
  { from^#(X) -> c_1()
  , from^#(X) -> c_2()
  , sel^#(0(), cons(X, XS)) -> c_4()
  , activate^#(X) -> c_5()
  , activate^#(n__from(X)) -> c_6(from^#(X))
  , zWquot^#(X1, X2) -> c_12()
  , zWquot^#(XS, nil()) -> c_13()
  , zWquot^#(nil(), XS) -> c_15()
  , minus^#(X, 0()) -> c_8()
  , quot^#(0(), s(Y)) -> c_11() }
Weak Trs:
  { from(X) -> cons(X, n__from(s(X)))
  , from(X) -> n__from(X)
  , sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
  , sel(0(), cons(X, XS)) -> X
  , activate(X) -> X
  , activate(n__from(X)) -> from(X)
  , activate(n__zWquot(X1, X2)) -> zWquot(X1, X2)
  , minus(X, 0()) -> 0()
  , minus(s(X), s(Y)) -> minus(X, Y)
  , quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
  , quot(0(), s(Y)) -> 0()
  , zWquot(X1, X2) -> n__zWquot(X1, X2)
  , zWquot(XS, nil()) -> nil()
  , zWquot(cons(X, XS), cons(Y, YS)) ->
    cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
  , zWquot(nil(), XS) -> nil() }
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.

{ from^#(X) -> c_1()
, from^#(X) -> c_2()
, sel^#(0(), cons(X, XS)) -> c_4()
, activate^#(X) -> c_5()
, activate^#(n__from(X)) -> c_6(from^#(X))
, zWquot^#(X1, X2) -> c_12()
, zWquot^#(XS, nil()) -> c_13()
, zWquot^#(nil(), XS) -> c_15()
, minus^#(X, 0()) -> c_8()
, quot^#(0(), s(Y)) -> c_11() }

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

Strict DPs:
  { sel^#(s(N), cons(X, XS)) ->
    c_3(sel^#(N, activate(XS)), activate^#(XS))
  , activate^#(n__zWquot(X1, X2)) -> c_7(zWquot^#(X1, X2))
  , zWquot^#(cons(X, XS), cons(Y, YS)) ->
    c_14(quot^#(X, Y), activate^#(XS), activate^#(YS))
  , minus^#(s(X), s(Y)) -> c_9(minus^#(X, Y))
  , quot^#(s(X), s(Y)) ->
    c_10(quot^#(minus(X, Y), s(Y)), minus^#(X, Y)) }
Weak Trs:
  { from(X) -> cons(X, n__from(s(X)))
  , from(X) -> n__from(X)
  , sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
  , sel(0(), cons(X, XS)) -> X
  , activate(X) -> X
  , activate(n__from(X)) -> from(X)
  , activate(n__zWquot(X1, X2)) -> zWquot(X1, X2)
  , minus(X, 0()) -> 0()
  , minus(s(X), s(Y)) -> minus(X, Y)
  , quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
  , quot(0(), s(Y)) -> 0()
  , zWquot(X1, X2) -> n__zWquot(X1, X2)
  , zWquot(XS, nil()) -> nil()
  , zWquot(cons(X, XS), cons(Y, YS)) ->
    cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
  , zWquot(nil(), XS) -> nil() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { from(X) -> cons(X, n__from(s(X)))
    , from(X) -> n__from(X)
    , activate(X) -> X
    , activate(n__from(X)) -> from(X)
    , activate(n__zWquot(X1, X2)) -> zWquot(X1, X2)
    , minus(X, 0()) -> 0()
    , minus(s(X), s(Y)) -> minus(X, Y)
    , quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
    , quot(0(), s(Y)) -> 0()
    , zWquot(X1, X2) -> n__zWquot(X1, X2)
    , zWquot(XS, nil()) -> nil()
    , zWquot(cons(X, XS), cons(Y, YS)) ->
      cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
    , zWquot(nil(), XS) -> nil() }

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

Strict DPs:
  { sel^#(s(N), cons(X, XS)) ->
    c_3(sel^#(N, activate(XS)), activate^#(XS))
  , activate^#(n__zWquot(X1, X2)) -> c_7(zWquot^#(X1, X2))
  , zWquot^#(cons(X, XS), cons(Y, YS)) ->
    c_14(quot^#(X, Y), activate^#(XS), activate^#(YS))
  , minus^#(s(X), s(Y)) -> c_9(minus^#(X, Y))
  , quot^#(s(X), s(Y)) ->
    c_10(quot^#(minus(X, Y), s(Y)), minus^#(X, Y)) }
Weak Trs:
  { from(X) -> cons(X, n__from(s(X)))
  , from(X) -> n__from(X)
  , activate(X) -> X
  , activate(n__from(X)) -> from(X)
  , activate(n__zWquot(X1, X2)) -> zWquot(X1, X2)
  , minus(X, 0()) -> 0()
  , minus(s(X), s(Y)) -> minus(X, Y)
  , quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
  , quot(0(), s(Y)) -> 0()
  , zWquot(X1, X2) -> n__zWquot(X1, X2)
  , zWquot(XS, nil()) -> nil()
  , zWquot(cons(X, XS), cons(Y, YS)) ->
    cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
  , zWquot(nil(), XS) -> nil() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..