MAYBE

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

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

We add following weak dependency pairs:

Strict DPs:
  { f^#(0()) -> c_1()
  , f^#(s(x)) -> c_2(g^#(s(s(x))))
  , g^#(0()) -> c_3()
  , g^#(s(0())) -> c_4()
  , g^#(s(s(x))) -> c_5(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^#(0()) -> c_1()
  , f^#(s(x)) -> c_2(g^#(s(s(x))))
  , g^#(0()) -> c_3()
  , g^#(s(0())) -> c_4()
  , g^#(s(s(x))) -> c_5(f^#(x)) }
Strict Trs:
  { f(0()) -> s(0())
  , f(s(x)) -> g(s(s(x)))
  , g(0()) -> s(0())
  , g(s(0())) -> s(0())
  , g(s(s(x))) -> f(x) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

No rule is usable, rules are removed from the input problem.

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

Strict DPs:
  { f^#(0()) -> c_1()
  , f^#(s(x)) -> c_2(g^#(s(s(x))))
  , g^#(0()) -> c_3()
  , g^#(s(0())) -> c_4()
  , g^#(s(s(x))) -> c_5(f^#(x)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The weightgap principle applies (using the following constant
growth matrix-interpretation)

The following argument positions are usable:
  Uargs(c_2) = {1}, Uargs(c_5) = {1}

TcT has computed following constructor-restricted matrix
interpretation.

        [0] = [0]         
                          
    [s](x1) = [0]         
                          
  [f^#](x1) = [1]         
                          
      [c_1] = [0]         
                          
  [c_2](x1) = [1] x1 + [1]
                          
  [g^#](x1) = [0]         
                          
      [c_3] = [1]         
                          
      [c_4] = [1]         
                          
  [c_5](x1) = [1] x1 + [1]

This order satisfies following ordering constraints:


Further, it can be verified that all rules not oriented are covered by the weightgap condition.

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

Strict DPs:
  { f^#(s(x)) -> c_2(g^#(s(s(x))))
  , g^#(0()) -> c_3()
  , g^#(s(0())) -> c_4()
  , g^#(s(s(x))) -> c_5(f^#(x)) }
Weak DPs: { f^#(0()) -> c_1() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: f^#(s(x)) -> c_2(g^#(s(s(x))))
    , 2: g^#(0()) -> c_3()
    , 3: g^#(s(0())) -> c_4()
    , 4: g^#(s(s(x))) -> c_5(f^#(x))
    , 5: f^#(0()) -> c_1() }

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

Strict DPs:
  { f^#(s(x)) -> c_2(g^#(s(s(x))))
  , g^#(s(s(x))) -> c_5(f^#(x)) }
Weak DPs:
  { f^#(0()) -> c_1()
  , g^#(0()) -> c_3()
  , g^#(s(0())) -> c_4() }
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^#(0()) -> c_1()
, g^#(0()) -> c_3()
, g^#(s(0())) -> c_4() }

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

Strict DPs:
  { f^#(s(x)) -> c_2(g^#(s(s(x))))
  , g^#(s(s(x))) -> c_5(f^#(x)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..