MAYBE

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

Strict Trs:
  { plus(x, 0()) -> x
  , plus(x, s(y)) -> s(plus(x, y))
  , times(x, 0()) -> 0()
  , times(0(), y) -> 0()
  , times(s(x), y) -> plus(times(x, y), y)
  , p(s(0())) -> 0()
  , p(s(s(x))) -> s(p(s(x)))
  , fac(s(x)) -> times(fac(p(s(x))), s(x)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { plus^#(x, 0()) -> c_1()
  , plus^#(x, s(y)) -> c_2(plus^#(x, y))
  , times^#(x, 0()) -> c_3()
  , times^#(0(), y) -> c_4()
  , times^#(s(x), y) -> c_5(plus^#(times(x, y), y), times^#(x, y))
  , p^#(s(0())) -> c_6()
  , p^#(s(s(x))) -> c_7(p^#(s(x)))
  , fac^#(s(x)) ->
    c_8(times^#(fac(p(s(x))), s(x)), fac^#(p(s(x))), p^#(s(x))) }

and mark the set of starting terms.

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

Strict DPs:
  { plus^#(x, 0()) -> c_1()
  , plus^#(x, s(y)) -> c_2(plus^#(x, y))
  , times^#(x, 0()) -> c_3()
  , times^#(0(), y) -> c_4()
  , times^#(s(x), y) -> c_5(plus^#(times(x, y), y), times^#(x, y))
  , p^#(s(0())) -> c_6()
  , p^#(s(s(x))) -> c_7(p^#(s(x)))
  , fac^#(s(x)) ->
    c_8(times^#(fac(p(s(x))), s(x)), fac^#(p(s(x))), p^#(s(x))) }
Weak Trs:
  { plus(x, 0()) -> x
  , plus(x, s(y)) -> s(plus(x, y))
  , times(x, 0()) -> 0()
  , times(0(), y) -> 0()
  , times(s(x), y) -> plus(times(x, y), y)
  , p(s(0())) -> 0()
  , p(s(s(x))) -> s(p(s(x)))
  , fac(s(x)) -> times(fac(p(s(x))), s(x)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: plus^#(x, 0()) -> c_1()
    , 2: plus^#(x, s(y)) -> c_2(plus^#(x, y))
    , 3: times^#(x, 0()) -> c_3()
    , 4: times^#(0(), y) -> c_4()
    , 5: times^#(s(x), y) -> c_5(plus^#(times(x, y), y), times^#(x, y))
    , 6: p^#(s(0())) -> c_6()
    , 7: p^#(s(s(x))) -> c_7(p^#(s(x)))
    , 8: fac^#(s(x)) ->
         c_8(times^#(fac(p(s(x))), s(x)), fac^#(p(s(x))), p^#(s(x))) }

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

Strict DPs:
  { plus^#(x, s(y)) -> c_2(plus^#(x, y))
  , times^#(s(x), y) -> c_5(plus^#(times(x, y), y), times^#(x, y))
  , p^#(s(s(x))) -> c_7(p^#(s(x)))
  , fac^#(s(x)) ->
    c_8(times^#(fac(p(s(x))), s(x)), fac^#(p(s(x))), p^#(s(x))) }
Weak DPs:
  { plus^#(x, 0()) -> c_1()
  , times^#(x, 0()) -> c_3()
  , times^#(0(), y) -> c_4()
  , p^#(s(0())) -> c_6() }
Weak Trs:
  { plus(x, 0()) -> x
  , plus(x, s(y)) -> s(plus(x, y))
  , times(x, 0()) -> 0()
  , times(0(), y) -> 0()
  , times(s(x), y) -> plus(times(x, y), y)
  , p(s(0())) -> 0()
  , p(s(s(x))) -> s(p(s(x)))
  , fac(s(x)) -> times(fac(p(s(x))), s(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.

{ plus^#(x, 0()) -> c_1()
, times^#(x, 0()) -> c_3()
, times^#(0(), y) -> c_4()
, p^#(s(0())) -> c_6() }

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

Strict DPs:
  { plus^#(x, s(y)) -> c_2(plus^#(x, y))
  , times^#(s(x), y) -> c_5(plus^#(times(x, y), y), times^#(x, y))
  , p^#(s(s(x))) -> c_7(p^#(s(x)))
  , fac^#(s(x)) ->
    c_8(times^#(fac(p(s(x))), s(x)), fac^#(p(s(x))), p^#(s(x))) }
Weak Trs:
  { plus(x, 0()) -> x
  , plus(x, s(y)) -> s(plus(x, y))
  , times(x, 0()) -> 0()
  , times(0(), y) -> 0()
  , times(s(x), y) -> plus(times(x, y), y)
  , p(s(0())) -> 0()
  , p(s(s(x))) -> s(p(s(x)))
  , fac(s(x)) -> times(fac(p(s(x))), s(x)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..