MAYBE

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

Strict Trs:
  { p(s(x)) -> x
  , fact(s(x)) -> *(s(x), fact(p(s(x))))
  , fact(0()) -> s(0())
  , *(s(x), y) -> +(*(x, y), y)
  , *(0(), y) -> 0()
  , +(x, s(y)) -> s(+(x, y))
  , +(x, 0()) -> x }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { p^#(s(x)) -> c_1()
  , fact^#(s(x)) ->
    c_2(*^#(s(x), fact(p(s(x)))), fact^#(p(s(x))), p^#(s(x)))
  , fact^#(0()) -> c_3()
  , *^#(s(x), y) -> c_4(+^#(*(x, y), y), *^#(x, y))
  , *^#(0(), y) -> c_5()
  , +^#(x, s(y)) -> c_6(+^#(x, y))
  , +^#(x, 0()) -> c_7() }

and mark the set of starting terms.

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

Strict DPs:
  { p^#(s(x)) -> c_1()
  , fact^#(s(x)) ->
    c_2(*^#(s(x), fact(p(s(x)))), fact^#(p(s(x))), p^#(s(x)))
  , fact^#(0()) -> c_3()
  , *^#(s(x), y) -> c_4(+^#(*(x, y), y), *^#(x, y))
  , *^#(0(), y) -> c_5()
  , +^#(x, s(y)) -> c_6(+^#(x, y))
  , +^#(x, 0()) -> c_7() }
Weak Trs:
  { p(s(x)) -> x
  , fact(s(x)) -> *(s(x), fact(p(s(x))))
  , fact(0()) -> s(0())
  , *(s(x), y) -> +(*(x, y), y)
  , *(0(), y) -> 0()
  , +(x, s(y)) -> s(+(x, y))
  , +(x, 0()) -> x }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: p^#(s(x)) -> c_1()
    , 2: fact^#(s(x)) ->
         c_2(*^#(s(x), fact(p(s(x)))), fact^#(p(s(x))), p^#(s(x)))
    , 3: fact^#(0()) -> c_3()
    , 4: *^#(s(x), y) -> c_4(+^#(*(x, y), y), *^#(x, y))
    , 5: *^#(0(), y) -> c_5()
    , 6: +^#(x, s(y)) -> c_6(+^#(x, y))
    , 7: +^#(x, 0()) -> c_7() }

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

Strict DPs:
  { fact^#(s(x)) ->
    c_2(*^#(s(x), fact(p(s(x)))), fact^#(p(s(x))), p^#(s(x)))
  , *^#(s(x), y) -> c_4(+^#(*(x, y), y), *^#(x, y))
  , +^#(x, s(y)) -> c_6(+^#(x, y)) }
Weak DPs:
  { p^#(s(x)) -> c_1()
  , fact^#(0()) -> c_3()
  , *^#(0(), y) -> c_5()
  , +^#(x, 0()) -> c_7() }
Weak Trs:
  { p(s(x)) -> x
  , fact(s(x)) -> *(s(x), fact(p(s(x))))
  , fact(0()) -> s(0())
  , *(s(x), y) -> +(*(x, y), y)
  , *(0(), y) -> 0()
  , +(x, s(y)) -> s(+(x, y))
  , +(x, 0()) -> 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.

{ p^#(s(x)) -> c_1()
, fact^#(0()) -> c_3()
, *^#(0(), y) -> c_5()
, +^#(x, 0()) -> c_7() }

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

Strict DPs:
  { fact^#(s(x)) ->
    c_2(*^#(s(x), fact(p(s(x)))), fact^#(p(s(x))), p^#(s(x)))
  , *^#(s(x), y) -> c_4(+^#(*(x, y), y), *^#(x, y))
  , +^#(x, s(y)) -> c_6(+^#(x, y)) }
Weak Trs:
  { p(s(x)) -> x
  , fact(s(x)) -> *(s(x), fact(p(s(x))))
  , fact(0()) -> s(0())
  , *(s(x), y) -> +(*(x, y), y)
  , *(0(), y) -> 0()
  , +(x, s(y)) -> s(+(x, y))
  , +(x, 0()) -> x }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

Due to missing edges in the dependency-graph, the right-hand sides
of following rules could be simplified:

  { fact^#(s(x)) ->
    c_2(*^#(s(x), fact(p(s(x)))), fact^#(p(s(x))), p^#(s(x))) }

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

Strict DPs:
  { fact^#(s(x)) -> c_1(*^#(s(x), fact(p(s(x)))), fact^#(p(s(x))))
  , *^#(s(x), y) -> c_2(+^#(*(x, y), y), *^#(x, y))
  , +^#(x, s(y)) -> c_3(+^#(x, y)) }
Weak Trs:
  { p(s(x)) -> x
  , fact(s(x)) -> *(s(x), fact(p(s(x))))
  , fact(0()) -> s(0())
  , *(s(x), y) -> +(*(x, y), y)
  , *(0(), y) -> 0()
  , +(x, s(y)) -> s(+(x, y))
  , +(x, 0()) -> x }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..