MAYBE

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

Strict Trs:
  { f(X) -> cons(X, n__f(g(X)))
  , f(X) -> n__f(X)
  , g(0()) -> s(0())
  , g(s(X)) -> s(s(g(X)))
  , sel(0(), cons(X, Y)) -> X
  , sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
  , activate(X) -> X
  , activate(n__f(X)) -> f(X) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { f^#(X) -> c_1(g^#(X))
  , f^#(X) -> c_2()
  , g^#(0()) -> c_3()
  , g^#(s(X)) -> c_4(g^#(X))
  , sel^#(0(), cons(X, Y)) -> c_5()
  , sel^#(s(X), cons(Y, Z)) ->
    c_6(sel^#(X, activate(Z)), activate^#(Z))
  , activate^#(X) -> c_7()
  , activate^#(n__f(X)) -> c_8(f^#(X)) }

and mark the set of starting terms.

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

Strict DPs:
  { f^#(X) -> c_1(g^#(X))
  , f^#(X) -> c_2()
  , g^#(0()) -> c_3()
  , g^#(s(X)) -> c_4(g^#(X))
  , sel^#(0(), cons(X, Y)) -> c_5()
  , sel^#(s(X), cons(Y, Z)) ->
    c_6(sel^#(X, activate(Z)), activate^#(Z))
  , activate^#(X) -> c_7()
  , activate^#(n__f(X)) -> c_8(f^#(X)) }
Weak Trs:
  { f(X) -> cons(X, n__f(g(X)))
  , f(X) -> n__f(X)
  , g(0()) -> s(0())
  , g(s(X)) -> s(s(g(X)))
  , sel(0(), cons(X, Y)) -> X
  , sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
  , activate(X) -> X
  , activate(n__f(X)) -> f(X) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: f^#(X) -> c_1(g^#(X))
    , 2: f^#(X) -> c_2()
    , 3: g^#(0()) -> c_3()
    , 4: g^#(s(X)) -> c_4(g^#(X))
    , 5: sel^#(0(), cons(X, Y)) -> c_5()
    , 6: sel^#(s(X), cons(Y, Z)) ->
         c_6(sel^#(X, activate(Z)), activate^#(Z))
    , 7: activate^#(X) -> c_7()
    , 8: activate^#(n__f(X)) -> c_8(f^#(X)) }

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

Strict DPs:
  { f^#(X) -> c_1(g^#(X))
  , g^#(s(X)) -> c_4(g^#(X))
  , sel^#(s(X), cons(Y, Z)) ->
    c_6(sel^#(X, activate(Z)), activate^#(Z))
  , activate^#(n__f(X)) -> c_8(f^#(X)) }
Weak DPs:
  { f^#(X) -> c_2()
  , g^#(0()) -> c_3()
  , sel^#(0(), cons(X, Y)) -> c_5()
  , activate^#(X) -> c_7() }
Weak Trs:
  { f(X) -> cons(X, n__f(g(X)))
  , f(X) -> n__f(X)
  , g(0()) -> s(0())
  , g(s(X)) -> s(s(g(X)))
  , sel(0(), cons(X, Y)) -> X
  , sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
  , activate(X) -> X
  , activate(n__f(X)) -> f(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.

{ f^#(X) -> c_2()
, g^#(0()) -> c_3()
, sel^#(0(), cons(X, Y)) -> c_5()
, activate^#(X) -> c_7() }

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

Strict DPs:
  { f^#(X) -> c_1(g^#(X))
  , g^#(s(X)) -> c_4(g^#(X))
  , sel^#(s(X), cons(Y, Z)) ->
    c_6(sel^#(X, activate(Z)), activate^#(Z))
  , activate^#(n__f(X)) -> c_8(f^#(X)) }
Weak Trs:
  { f(X) -> cons(X, n__f(g(X)))
  , f(X) -> n__f(X)
  , g(0()) -> s(0())
  , g(s(X)) -> s(s(g(X)))
  , sel(0(), cons(X, Y)) -> X
  , sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
  , activate(X) -> X
  , activate(n__f(X)) -> f(X) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { f(X) -> cons(X, n__f(g(X)))
    , f(X) -> n__f(X)
    , g(0()) -> s(0())
    , g(s(X)) -> s(s(g(X)))
    , activate(X) -> X
    , activate(n__f(X)) -> f(X) }

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

Strict DPs:
  { f^#(X) -> c_1(g^#(X))
  , g^#(s(X)) -> c_4(g^#(X))
  , sel^#(s(X), cons(Y, Z)) ->
    c_6(sel^#(X, activate(Z)), activate^#(Z))
  , activate^#(n__f(X)) -> c_8(f^#(X)) }
Weak Trs:
  { f(X) -> cons(X, n__f(g(X)))
  , f(X) -> n__f(X)
  , g(0()) -> s(0())
  , g(s(X)) -> s(s(g(X)))
  , activate(X) -> X
  , activate(n__f(X)) -> f(X) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..