MAYBE

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

Strict Trs:
  { fib(0()) -> 0()
  , fib(s(0())) -> s(0())
  , fib(s(s(x))) -> sp(g(x))
  , fib(s(s(0()))) -> s(0())
  , sp(pair(x, y)) -> +(x, y)
  , g(0()) -> pair(s(0()), 0())
  , g(s(x)) -> np(g(x))
  , g(s(0())) -> pair(s(0()), s(0()))
  , np(pair(x, y)) -> pair(+(x, y), x)
  , +(x, 0()) -> x
  , +(x, s(y)) -> s(+(x, y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { fib^#(0()) -> c_1()
  , fib^#(s(0())) -> c_2()
  , fib^#(s(s(x))) -> c_3(sp^#(g(x)), g^#(x))
  , fib^#(s(s(0()))) -> c_4()
  , sp^#(pair(x, y)) -> c_5(+^#(x, y))
  , g^#(0()) -> c_6()
  , g^#(s(x)) -> c_7(np^#(g(x)), g^#(x))
  , g^#(s(0())) -> c_8()
  , +^#(x, 0()) -> c_10()
  , +^#(x, s(y)) -> c_11(+^#(x, y))
  , np^#(pair(x, y)) -> c_9(+^#(x, y)) }

and mark the set of starting terms.

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

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

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

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

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

Strict DPs:
  { fib^#(s(s(x))) -> c_3(sp^#(g(x)), g^#(x))
  , sp^#(pair(x, y)) -> c_5(+^#(x, y))
  , g^#(s(x)) -> c_7(np^#(g(x)), g^#(x))
  , +^#(x, s(y)) -> c_11(+^#(x, y))
  , np^#(pair(x, y)) -> c_9(+^#(x, y)) }
Weak DPs:
  { fib^#(0()) -> c_1()
  , fib^#(s(0())) -> c_2()
  , fib^#(s(s(0()))) -> c_4()
  , g^#(0()) -> c_6()
  , g^#(s(0())) -> c_8()
  , +^#(x, 0()) -> c_10() }
Weak Trs:
  { fib(0()) -> 0()
  , fib(s(0())) -> s(0())
  , fib(s(s(x))) -> sp(g(x))
  , fib(s(s(0()))) -> s(0())
  , sp(pair(x, y)) -> +(x, y)
  , g(0()) -> pair(s(0()), 0())
  , g(s(x)) -> np(g(x))
  , g(s(0())) -> pair(s(0()), s(0()))
  , np(pair(x, y)) -> pair(+(x, y), x)
  , +(x, 0()) -> x
  , +(x, s(y)) -> s(+(x, y)) }
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.

{ fib^#(0()) -> c_1()
, fib^#(s(0())) -> c_2()
, fib^#(s(s(0()))) -> c_4()
, g^#(0()) -> c_6()
, g^#(s(0())) -> c_8()
, +^#(x, 0()) -> c_10() }

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

Strict DPs:
  { fib^#(s(s(x))) -> c_3(sp^#(g(x)), g^#(x))
  , sp^#(pair(x, y)) -> c_5(+^#(x, y))
  , g^#(s(x)) -> c_7(np^#(g(x)), g^#(x))
  , +^#(x, s(y)) -> c_11(+^#(x, y))
  , np^#(pair(x, y)) -> c_9(+^#(x, y)) }
Weak Trs:
  { fib(0()) -> 0()
  , fib(s(0())) -> s(0())
  , fib(s(s(x))) -> sp(g(x))
  , fib(s(s(0()))) -> s(0())
  , sp(pair(x, y)) -> +(x, y)
  , g(0()) -> pair(s(0()), 0())
  , g(s(x)) -> np(g(x))
  , g(s(0())) -> pair(s(0()), s(0()))
  , np(pair(x, y)) -> pair(+(x, y), x)
  , +(x, 0()) -> x
  , +(x, s(y)) -> s(+(x, y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { g(0()) -> pair(s(0()), 0())
    , g(s(x)) -> np(g(x))
    , g(s(0())) -> pair(s(0()), s(0()))
    , np(pair(x, y)) -> pair(+(x, y), x)
    , +(x, 0()) -> x
    , +(x, s(y)) -> s(+(x, y)) }

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

Strict DPs:
  { fib^#(s(s(x))) -> c_3(sp^#(g(x)), g^#(x))
  , sp^#(pair(x, y)) -> c_5(+^#(x, y))
  , g^#(s(x)) -> c_7(np^#(g(x)), g^#(x))
  , +^#(x, s(y)) -> c_11(+^#(x, y))
  , np^#(pair(x, y)) -> c_9(+^#(x, y)) }
Weak Trs:
  { g(0()) -> pair(s(0()), 0())
  , g(s(x)) -> np(g(x))
  , g(s(0())) -> pair(s(0()), s(0()))
  , np(pair(x, y)) -> pair(+(x, y), x)
  , +(x, 0()) -> x
  , +(x, s(y)) -> s(+(x, y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..