MAYBE

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

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

We add following dependency tuples:

Strict DPs:
  { f^#(x, 0(), 0()) -> c_1()
  , f^#(0(), y, 0()) -> c_2()
  , f^#(0(), 0(), z) -> c_3()
  , f^#(0(), s(0()), s(0())) -> c_4()
  , f^#(0(), s(0()), s(s(z))) -> c_5(f^#(0(), s(0()), z))
  , f^#(0(), s(s(y)), s(0())) -> c_6(f^#(0(), y, s(0())))
  , f^#(0(), s(s(y)), s(s(z))) ->
    c_7(f^#(0(), y, f(0(), s(s(y)), s(z))), f^#(0(), s(s(y)), s(z)))
  , f^#(s(x), 0(), s(z)) -> c_8(f^#(x, s(0()), z))
  , f^#(s(x), s(y), 0()) -> c_9(f^#(x, y, s(0())))
  , f^#(s(x), s(y), s(z)) ->
    c_10(f^#(x, y, f(s(x), s(y), z)), f^#(s(x), s(y), z))
  , f^#(s(0()), y, z) -> c_11(f^#(0(), s(y), s(z))) }

and mark the set of starting terms.

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

Strict DPs:
  { f^#(x, 0(), 0()) -> c_1()
  , f^#(0(), y, 0()) -> c_2()
  , f^#(0(), 0(), z) -> c_3()
  , f^#(0(), s(0()), s(0())) -> c_4()
  , f^#(0(), s(0()), s(s(z))) -> c_5(f^#(0(), s(0()), z))
  , f^#(0(), s(s(y)), s(0())) -> c_6(f^#(0(), y, s(0())))
  , f^#(0(), s(s(y)), s(s(z))) ->
    c_7(f^#(0(), y, f(0(), s(s(y)), s(z))), f^#(0(), s(s(y)), s(z)))
  , f^#(s(x), 0(), s(z)) -> c_8(f^#(x, s(0()), z))
  , f^#(s(x), s(y), 0()) -> c_9(f^#(x, y, s(0())))
  , f^#(s(x), s(y), s(z)) ->
    c_10(f^#(x, y, f(s(x), s(y), z)), f^#(s(x), s(y), z))
  , f^#(s(0()), y, z) -> c_11(f^#(0(), s(y), s(z))) }
Weak Trs:
  { f(x, 0(), 0()) -> s(x)
  , f(0(), y, 0()) -> s(y)
  , f(0(), 0(), z) -> s(z)
  , f(0(), s(0()), s(0())) -> s(s(0()))
  , f(0(), s(0()), s(s(z))) -> f(0(), s(0()), z)
  , f(0(), s(s(y)), s(0())) -> f(0(), y, s(0()))
  , f(0(), s(s(y)), s(s(z))) -> f(0(), y, f(0(), s(s(y)), s(z)))
  , f(s(x), 0(), s(z)) -> f(x, s(0()), z)
  , f(s(x), s(y), 0()) -> f(x, y, s(0()))
  , f(s(x), s(y), s(z)) -> f(x, y, f(s(x), s(y), z))
  , f(s(0()), y, z) -> f(0(), s(y), s(z)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: f^#(x, 0(), 0()) -> c_1()
    , 2: f^#(0(), y, 0()) -> c_2()
    , 3: f^#(0(), 0(), z) -> c_3()
    , 4: f^#(0(), s(0()), s(0())) -> c_4()
    , 5: f^#(0(), s(0()), s(s(z))) -> c_5(f^#(0(), s(0()), z))
    , 6: f^#(0(), s(s(y)), s(0())) -> c_6(f^#(0(), y, s(0())))
    , 7: f^#(0(), s(s(y)), s(s(z))) ->
         c_7(f^#(0(), y, f(0(), s(s(y)), s(z))), f^#(0(), s(s(y)), s(z)))
    , 8: f^#(s(x), 0(), s(z)) -> c_8(f^#(x, s(0()), z))
    , 9: f^#(s(x), s(y), 0()) -> c_9(f^#(x, y, s(0())))
    , 10: f^#(s(x), s(y), s(z)) ->
          c_10(f^#(x, y, f(s(x), s(y), z)), f^#(s(x), s(y), z))
    , 11: f^#(s(0()), y, z) -> c_11(f^#(0(), s(y), s(z))) }

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

Strict DPs:
  { f^#(0(), s(0()), s(s(z))) -> c_5(f^#(0(), s(0()), z))
  , f^#(0(), s(s(y)), s(0())) -> c_6(f^#(0(), y, s(0())))
  , f^#(0(), s(s(y)), s(s(z))) ->
    c_7(f^#(0(), y, f(0(), s(s(y)), s(z))), f^#(0(), s(s(y)), s(z)))
  , f^#(s(x), 0(), s(z)) -> c_8(f^#(x, s(0()), z))
  , f^#(s(x), s(y), 0()) -> c_9(f^#(x, y, s(0())))
  , f^#(s(x), s(y), s(z)) ->
    c_10(f^#(x, y, f(s(x), s(y), z)), f^#(s(x), s(y), z))
  , f^#(s(0()), y, z) -> c_11(f^#(0(), s(y), s(z))) }
Weak DPs:
  { f^#(x, 0(), 0()) -> c_1()
  , f^#(0(), y, 0()) -> c_2()
  , f^#(0(), 0(), z) -> c_3()
  , f^#(0(), s(0()), s(0())) -> c_4() }
Weak Trs:
  { f(x, 0(), 0()) -> s(x)
  , f(0(), y, 0()) -> s(y)
  , f(0(), 0(), z) -> s(z)
  , f(0(), s(0()), s(0())) -> s(s(0()))
  , f(0(), s(0()), s(s(z))) -> f(0(), s(0()), z)
  , f(0(), s(s(y)), s(0())) -> f(0(), y, s(0()))
  , f(0(), s(s(y)), s(s(z))) -> f(0(), y, f(0(), s(s(y)), s(z)))
  , f(s(x), 0(), s(z)) -> f(x, s(0()), z)
  , f(s(x), s(y), 0()) -> f(x, y, s(0()))
  , f(s(x), s(y), s(z)) -> f(x, y, f(s(x), s(y), z))
  , f(s(0()), y, z) -> f(0(), s(y), s(z)) }
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, 0(), 0()) -> c_1()
, f^#(0(), y, 0()) -> c_2()
, f^#(0(), 0(), z) -> c_3()
, f^#(0(), s(0()), s(0())) -> c_4() }

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

Strict DPs:
  { f^#(0(), s(0()), s(s(z))) -> c_5(f^#(0(), s(0()), z))
  , f^#(0(), s(s(y)), s(0())) -> c_6(f^#(0(), y, s(0())))
  , f^#(0(), s(s(y)), s(s(z))) ->
    c_7(f^#(0(), y, f(0(), s(s(y)), s(z))), f^#(0(), s(s(y)), s(z)))
  , f^#(s(x), 0(), s(z)) -> c_8(f^#(x, s(0()), z))
  , f^#(s(x), s(y), 0()) -> c_9(f^#(x, y, s(0())))
  , f^#(s(x), s(y), s(z)) ->
    c_10(f^#(x, y, f(s(x), s(y), z)), f^#(s(x), s(y), z))
  , f^#(s(0()), y, z) -> c_11(f^#(0(), s(y), s(z))) }
Weak Trs:
  { f(x, 0(), 0()) -> s(x)
  , f(0(), y, 0()) -> s(y)
  , f(0(), 0(), z) -> s(z)
  , f(0(), s(0()), s(0())) -> s(s(0()))
  , f(0(), s(0()), s(s(z))) -> f(0(), s(0()), z)
  , f(0(), s(s(y)), s(0())) -> f(0(), y, s(0()))
  , f(0(), s(s(y)), s(s(z))) -> f(0(), y, f(0(), s(s(y)), s(z)))
  , f(s(x), 0(), s(z)) -> f(x, s(0()), z)
  , f(s(x), s(y), 0()) -> f(x, y, s(0()))
  , f(s(x), s(y), s(z)) -> f(x, y, f(s(x), s(y), z))
  , f(s(0()), y, z) -> f(0(), s(y), s(z)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..