MAYBE

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

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

We add following dependency tuples:

Strict DPs:
  { +^#(x, 0()) -> c_1()
  , +^#(x, s(y)) -> c_2(+^#(x, y))
  , *^#(x, 0()) -> c_3()
  , *^#(x, s(y)) -> c_4(+^#(*(x, y), x), *^#(x, y))
  , ge^#(x, 0()) -> c_5()
  , ge^#(0(), s(y)) -> c_6()
  , ge^#(s(x), s(y)) -> c_7(ge^#(x, y))
  , -^#(x, 0()) -> c_8()
  , -^#(s(x), s(y)) -> c_9(-^#(x, y))
  , fact^#(x) ->
    c_10(iffact^#(x, ge(x, s(s(0())))), ge^#(x, s(s(0()))))
  , iffact^#(x, true()) ->
    c_11(*^#(x, fact(-(x, s(0())))),
         fact^#(-(x, s(0()))),
         -^#(x, s(0())))
  , iffact^#(x, false()) -> c_12() }

and mark the set of starting terms.

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

Strict DPs:
  { +^#(x, 0()) -> c_1()
  , +^#(x, s(y)) -> c_2(+^#(x, y))
  , *^#(x, 0()) -> c_3()
  , *^#(x, s(y)) -> c_4(+^#(*(x, y), x), *^#(x, y))
  , ge^#(x, 0()) -> c_5()
  , ge^#(0(), s(y)) -> c_6()
  , ge^#(s(x), s(y)) -> c_7(ge^#(x, y))
  , -^#(x, 0()) -> c_8()
  , -^#(s(x), s(y)) -> c_9(-^#(x, y))
  , fact^#(x) ->
    c_10(iffact^#(x, ge(x, s(s(0())))), ge^#(x, s(s(0()))))
  , iffact^#(x, true()) ->
    c_11(*^#(x, fact(-(x, s(0())))),
         fact^#(-(x, s(0()))),
         -^#(x, s(0())))
  , iffact^#(x, false()) -> c_12() }
Weak Trs:
  { +(x, 0()) -> x
  , +(x, s(y)) -> s(+(x, y))
  , *(x, 0()) -> 0()
  , *(x, s(y)) -> +(*(x, y), x)
  , ge(x, 0()) -> true()
  , ge(0(), s(y)) -> false()
  , ge(s(x), s(y)) -> ge(x, y)
  , -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y)
  , fact(x) -> iffact(x, ge(x, s(s(0()))))
  , iffact(x, true()) -> *(x, fact(-(x, s(0()))))
  , iffact(x, false()) -> s(0()) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {1,3,5,6,8,12} by
applications of Pre({1,3,5,6,8,12}) = {2,4,7,9,10,11}. Here rules
are labeled as follows:

  DPs:
    { 1: +^#(x, 0()) -> c_1()
    , 2: +^#(x, s(y)) -> c_2(+^#(x, y))
    , 3: *^#(x, 0()) -> c_3()
    , 4: *^#(x, s(y)) -> c_4(+^#(*(x, y), x), *^#(x, y))
    , 5: ge^#(x, 0()) -> c_5()
    , 6: ge^#(0(), s(y)) -> c_6()
    , 7: ge^#(s(x), s(y)) -> c_7(ge^#(x, y))
    , 8: -^#(x, 0()) -> c_8()
    , 9: -^#(s(x), s(y)) -> c_9(-^#(x, y))
    , 10: fact^#(x) ->
          c_10(iffact^#(x, ge(x, s(s(0())))), ge^#(x, s(s(0()))))
    , 11: iffact^#(x, true()) ->
          c_11(*^#(x, fact(-(x, s(0())))),
               fact^#(-(x, s(0()))),
               -^#(x, s(0())))
    , 12: iffact^#(x, false()) -> c_12() }

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

Strict DPs:
  { +^#(x, s(y)) -> c_2(+^#(x, y))
  , *^#(x, s(y)) -> c_4(+^#(*(x, y), x), *^#(x, y))
  , ge^#(s(x), s(y)) -> c_7(ge^#(x, y))
  , -^#(s(x), s(y)) -> c_9(-^#(x, y))
  , fact^#(x) ->
    c_10(iffact^#(x, ge(x, s(s(0())))), ge^#(x, s(s(0()))))
  , iffact^#(x, true()) ->
    c_11(*^#(x, fact(-(x, s(0())))),
         fact^#(-(x, s(0()))),
         -^#(x, s(0()))) }
Weak DPs:
  { +^#(x, 0()) -> c_1()
  , *^#(x, 0()) -> c_3()
  , ge^#(x, 0()) -> c_5()
  , ge^#(0(), s(y)) -> c_6()
  , -^#(x, 0()) -> c_8()
  , iffact^#(x, false()) -> c_12() }
Weak Trs:
  { +(x, 0()) -> x
  , +(x, s(y)) -> s(+(x, y))
  , *(x, 0()) -> 0()
  , *(x, s(y)) -> +(*(x, y), x)
  , ge(x, 0()) -> true()
  , ge(0(), s(y)) -> false()
  , ge(s(x), s(y)) -> ge(x, y)
  , -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y)
  , fact(x) -> iffact(x, ge(x, s(s(0()))))
  , iffact(x, true()) -> *(x, fact(-(x, s(0()))))
  , iffact(x, false()) -> s(0()) }
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.

{ +^#(x, 0()) -> c_1()
, *^#(x, 0()) -> c_3()
, ge^#(x, 0()) -> c_5()
, ge^#(0(), s(y)) -> c_6()
, -^#(x, 0()) -> c_8()
, iffact^#(x, false()) -> c_12() }

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

Strict DPs:
  { +^#(x, s(y)) -> c_2(+^#(x, y))
  , *^#(x, s(y)) -> c_4(+^#(*(x, y), x), *^#(x, y))
  , ge^#(s(x), s(y)) -> c_7(ge^#(x, y))
  , -^#(s(x), s(y)) -> c_9(-^#(x, y))
  , fact^#(x) ->
    c_10(iffact^#(x, ge(x, s(s(0())))), ge^#(x, s(s(0()))))
  , iffact^#(x, true()) ->
    c_11(*^#(x, fact(-(x, s(0())))),
         fact^#(-(x, s(0()))),
         -^#(x, s(0()))) }
Weak Trs:
  { +(x, 0()) -> x
  , +(x, s(y)) -> s(+(x, y))
  , *(x, 0()) -> 0()
  , *(x, s(y)) -> +(*(x, y), x)
  , ge(x, 0()) -> true()
  , ge(0(), s(y)) -> false()
  , ge(s(x), s(y)) -> ge(x, y)
  , -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y)
  , fact(x) -> iffact(x, ge(x, s(s(0()))))
  , iffact(x, true()) -> *(x, fact(-(x, s(0()))))
  , iffact(x, false()) -> s(0()) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..