MAYBE

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

Strict Trs:
  { f(s(x)) -> f(id_inc(c(x, x)))
  , f(c(s(x), y)) -> g(c(x, y))
  , id_inc(s(x)) -> s(id_inc(x))
  , id_inc(c(x, y)) -> c(id_inc(x), id_inc(y))
  , id_inc(0()) -> s(0())
  , id_inc(0()) -> 0()
  , g(c(x, x)) -> f(x)
  , g(c(x, s(y))) -> g(c(y, x))
  , g(c(s(x), y)) -> g(c(y, x)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { f^#(s(x)) -> c_1(f^#(id_inc(c(x, x))), id_inc^#(c(x, x)))
  , f^#(c(s(x), y)) -> c_2(g^#(c(x, y)))
  , id_inc^#(s(x)) -> c_3(id_inc^#(x))
  , id_inc^#(c(x, y)) -> c_4(id_inc^#(x), id_inc^#(y))
  , id_inc^#(0()) -> c_5()
  , id_inc^#(0()) -> c_6()
  , g^#(c(x, x)) -> c_7(f^#(x))
  , g^#(c(x, s(y))) -> c_8(g^#(c(y, x)))
  , g^#(c(s(x), y)) -> c_9(g^#(c(y, x))) }

and mark the set of starting terms.

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

Strict DPs:
  { f^#(s(x)) -> c_1(f^#(id_inc(c(x, x))), id_inc^#(c(x, x)))
  , f^#(c(s(x), y)) -> c_2(g^#(c(x, y)))
  , id_inc^#(s(x)) -> c_3(id_inc^#(x))
  , id_inc^#(c(x, y)) -> c_4(id_inc^#(x), id_inc^#(y))
  , id_inc^#(0()) -> c_5()
  , id_inc^#(0()) -> c_6()
  , g^#(c(x, x)) -> c_7(f^#(x))
  , g^#(c(x, s(y))) -> c_8(g^#(c(y, x)))
  , g^#(c(s(x), y)) -> c_9(g^#(c(y, x))) }
Weak Trs:
  { f(s(x)) -> f(id_inc(c(x, x)))
  , f(c(s(x), y)) -> g(c(x, y))
  , id_inc(s(x)) -> s(id_inc(x))
  , id_inc(c(x, y)) -> c(id_inc(x), id_inc(y))
  , id_inc(0()) -> s(0())
  , id_inc(0()) -> 0()
  , g(c(x, x)) -> f(x)
  , g(c(x, s(y))) -> g(c(y, x))
  , g(c(s(x), y)) -> g(c(y, x)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: f^#(s(x)) -> c_1(f^#(id_inc(c(x, x))), id_inc^#(c(x, x)))
    , 2: f^#(c(s(x), y)) -> c_2(g^#(c(x, y)))
    , 3: id_inc^#(s(x)) -> c_3(id_inc^#(x))
    , 4: id_inc^#(c(x, y)) -> c_4(id_inc^#(x), id_inc^#(y))
    , 5: id_inc^#(0()) -> c_5()
    , 6: id_inc^#(0()) -> c_6()
    , 7: g^#(c(x, x)) -> c_7(f^#(x))
    , 8: g^#(c(x, s(y))) -> c_8(g^#(c(y, x)))
    , 9: g^#(c(s(x), y)) -> c_9(g^#(c(y, x))) }

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

Strict DPs:
  { f^#(s(x)) -> c_1(f^#(id_inc(c(x, x))), id_inc^#(c(x, x)))
  , f^#(c(s(x), y)) -> c_2(g^#(c(x, y)))
  , id_inc^#(s(x)) -> c_3(id_inc^#(x))
  , id_inc^#(c(x, y)) -> c_4(id_inc^#(x), id_inc^#(y))
  , g^#(c(x, x)) -> c_7(f^#(x))
  , g^#(c(x, s(y))) -> c_8(g^#(c(y, x)))
  , g^#(c(s(x), y)) -> c_9(g^#(c(y, x))) }
Weak DPs:
  { id_inc^#(0()) -> c_5()
  , id_inc^#(0()) -> c_6() }
Weak Trs:
  { f(s(x)) -> f(id_inc(c(x, x)))
  , f(c(s(x), y)) -> g(c(x, y))
  , id_inc(s(x)) -> s(id_inc(x))
  , id_inc(c(x, y)) -> c(id_inc(x), id_inc(y))
  , id_inc(0()) -> s(0())
  , id_inc(0()) -> 0()
  , g(c(x, x)) -> f(x)
  , g(c(x, s(y))) -> g(c(y, x))
  , g(c(s(x), y)) -> g(c(y, 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.

{ id_inc^#(0()) -> c_5()
, id_inc^#(0()) -> c_6() }

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

Strict DPs:
  { f^#(s(x)) -> c_1(f^#(id_inc(c(x, x))), id_inc^#(c(x, x)))
  , f^#(c(s(x), y)) -> c_2(g^#(c(x, y)))
  , id_inc^#(s(x)) -> c_3(id_inc^#(x))
  , id_inc^#(c(x, y)) -> c_4(id_inc^#(x), id_inc^#(y))
  , g^#(c(x, x)) -> c_7(f^#(x))
  , g^#(c(x, s(y))) -> c_8(g^#(c(y, x)))
  , g^#(c(s(x), y)) -> c_9(g^#(c(y, x))) }
Weak Trs:
  { f(s(x)) -> f(id_inc(c(x, x)))
  , f(c(s(x), y)) -> g(c(x, y))
  , id_inc(s(x)) -> s(id_inc(x))
  , id_inc(c(x, y)) -> c(id_inc(x), id_inc(y))
  , id_inc(0()) -> s(0())
  , id_inc(0()) -> 0()
  , g(c(x, x)) -> f(x)
  , g(c(x, s(y))) -> g(c(y, x))
  , g(c(s(x), y)) -> g(c(y, x)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { id_inc(s(x)) -> s(id_inc(x))
    , id_inc(c(x, y)) -> c(id_inc(x), id_inc(y))
    , id_inc(0()) -> s(0())
    , id_inc(0()) -> 0() }

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

Strict DPs:
  { f^#(s(x)) -> c_1(f^#(id_inc(c(x, x))), id_inc^#(c(x, x)))
  , f^#(c(s(x), y)) -> c_2(g^#(c(x, y)))
  , id_inc^#(s(x)) -> c_3(id_inc^#(x))
  , id_inc^#(c(x, y)) -> c_4(id_inc^#(x), id_inc^#(y))
  , g^#(c(x, x)) -> c_7(f^#(x))
  , g^#(c(x, s(y))) -> c_8(g^#(c(y, x)))
  , g^#(c(s(x), y)) -> c_9(g^#(c(y, x))) }
Weak Trs:
  { id_inc(s(x)) -> s(id_inc(x))
  , id_inc(c(x, y)) -> c(id_inc(x), id_inc(y))
  , id_inc(0()) -> s(0())
  , id_inc(0()) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..