MAYBE

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

Strict Trs:
  { fac(0()) -> 1()
  , fac(0()) -> s(0())
  , fac(s(x)) -> *(s(x), fac(x))
  , 1() -> s(0())
  , *(x, 0()) -> 0()
  , *(x, s(y)) -> +(*(x, y), x)
  , floop(0(), y) -> y
  , floop(s(x), y) -> floop(x, *(s(x), y))
  , +(x, 0()) -> x
  , +(x, s(y)) -> s(+(x, y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { fac^#(0()) -> c_1(1^#())
  , fac^#(0()) -> c_2()
  , fac^#(s(x)) -> c_3(*^#(s(x), fac(x)), fac^#(x))
  , 1^#() -> c_4()
  , *^#(x, 0()) -> c_5()
  , *^#(x, s(y)) -> c_6(+^#(*(x, y), x), *^#(x, y))
  , +^#(x, 0()) -> c_9()
  , +^#(x, s(y)) -> c_10(+^#(x, y))
  , floop^#(0(), y) -> c_7()
  , floop^#(s(x), y) -> c_8(floop^#(x, *(s(x), y)), *^#(s(x), y)) }

and mark the set of starting terms.

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

Strict DPs:
  { fac^#(0()) -> c_1(1^#())
  , fac^#(0()) -> c_2()
  , fac^#(s(x)) -> c_3(*^#(s(x), fac(x)), fac^#(x))
  , 1^#() -> c_4()
  , *^#(x, 0()) -> c_5()
  , *^#(x, s(y)) -> c_6(+^#(*(x, y), x), *^#(x, y))
  , +^#(x, 0()) -> c_9()
  , +^#(x, s(y)) -> c_10(+^#(x, y))
  , floop^#(0(), y) -> c_7()
  , floop^#(s(x), y) -> c_8(floop^#(x, *(s(x), y)), *^#(s(x), y)) }
Weak Trs:
  { fac(0()) -> 1()
  , fac(0()) -> s(0())
  , fac(s(x)) -> *(s(x), fac(x))
  , 1() -> s(0())
  , *(x, 0()) -> 0()
  , *(x, s(y)) -> +(*(x, y), x)
  , floop(0(), y) -> y
  , floop(s(x), y) -> floop(x, *(s(x), y))
  , +(x, 0()) -> x
  , +(x, s(y)) -> s(+(x, y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

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

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

Strict DPs:
  { fac^#(0()) -> c_1(1^#())
  , fac^#(s(x)) -> c_3(*^#(s(x), fac(x)), fac^#(x))
  , *^#(x, s(y)) -> c_6(+^#(*(x, y), x), *^#(x, y))
  , +^#(x, s(y)) -> c_10(+^#(x, y))
  , floop^#(s(x), y) -> c_8(floop^#(x, *(s(x), y)), *^#(s(x), y)) }
Weak DPs:
  { fac^#(0()) -> c_2()
  , 1^#() -> c_4()
  , *^#(x, 0()) -> c_5()
  , +^#(x, 0()) -> c_9()
  , floop^#(0(), y) -> c_7() }
Weak Trs:
  { fac(0()) -> 1()
  , fac(0()) -> s(0())
  , fac(s(x)) -> *(s(x), fac(x))
  , 1() -> s(0())
  , *(x, 0()) -> 0()
  , *(x, s(y)) -> +(*(x, y), x)
  , floop(0(), y) -> y
  , floop(s(x), y) -> floop(x, *(s(x), y))
  , +(x, 0()) -> x
  , +(x, s(y)) -> s(+(x, y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

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

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

Strict DPs:
  { fac^#(s(x)) -> c_3(*^#(s(x), fac(x)), fac^#(x))
  , *^#(x, s(y)) -> c_6(+^#(*(x, y), x), *^#(x, y))
  , +^#(x, s(y)) -> c_10(+^#(x, y))
  , floop^#(s(x), y) -> c_8(floop^#(x, *(s(x), y)), *^#(s(x), y)) }
Weak DPs:
  { fac^#(0()) -> c_1(1^#())
  , fac^#(0()) -> c_2()
  , 1^#() -> c_4()
  , *^#(x, 0()) -> c_5()
  , +^#(x, 0()) -> c_9()
  , floop^#(0(), y) -> c_7() }
Weak Trs:
  { fac(0()) -> 1()
  , fac(0()) -> s(0())
  , fac(s(x)) -> *(s(x), fac(x))
  , 1() -> s(0())
  , *(x, 0()) -> 0()
  , *(x, s(y)) -> +(*(x, y), x)
  , floop(0(), y) -> y
  , floop(s(x), y) -> floop(x, *(s(x), y))
  , +(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.

{ fac^#(0()) -> c_1(1^#())
, fac^#(0()) -> c_2()
, 1^#() -> c_4()
, *^#(x, 0()) -> c_5()
, +^#(x, 0()) -> c_9()
, floop^#(0(), y) -> c_7() }

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

Strict DPs:
  { fac^#(s(x)) -> c_3(*^#(s(x), fac(x)), fac^#(x))
  , *^#(x, s(y)) -> c_6(+^#(*(x, y), x), *^#(x, y))
  , +^#(x, s(y)) -> c_10(+^#(x, y))
  , floop^#(s(x), y) -> c_8(floop^#(x, *(s(x), y)), *^#(s(x), y)) }
Weak Trs:
  { fac(0()) -> 1()
  , fac(0()) -> s(0())
  , fac(s(x)) -> *(s(x), fac(x))
  , 1() -> s(0())
  , *(x, 0()) -> 0()
  , *(x, s(y)) -> +(*(x, y), x)
  , floop(0(), y) -> y
  , floop(s(x), y) -> floop(x, *(s(x), y))
  , +(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:
    { fac(0()) -> 1()
    , fac(0()) -> s(0())
    , fac(s(x)) -> *(s(x), fac(x))
    , 1() -> s(0())
    , *(x, 0()) -> 0()
    , *(x, s(y)) -> +(*(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:
  { fac^#(s(x)) -> c_3(*^#(s(x), fac(x)), fac^#(x))
  , *^#(x, s(y)) -> c_6(+^#(*(x, y), x), *^#(x, y))
  , +^#(x, s(y)) -> c_10(+^#(x, y))
  , floop^#(s(x), y) -> c_8(floop^#(x, *(s(x), y)), *^#(s(x), y)) }
Weak Trs:
  { fac(0()) -> 1()
  , fac(0()) -> s(0())
  , fac(s(x)) -> *(s(x), fac(x))
  , 1() -> s(0())
  , *(x, 0()) -> 0()
  , *(x, s(y)) -> +(*(x, y), x)
  , +(x, 0()) -> x
  , +(x, s(y)) -> s(+(x, y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

None of the processors succeeded.

Details of failed attempt(s):
-----------------------------
1) 'matrices' failed due to the following reason:
   
   None of the processors succeeded.
   
   Details of failed attempt(s):
   -----------------------------
   1) 'matrix interpretation of dimension 4' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   2) 'matrix interpretation of dimension 3' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   3) 'matrix interpretation of dimension 3' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   4) 'matrix interpretation of dimension 2' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   5) 'matrix interpretation of dimension 2' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   6) 'matrix interpretation of dimension 1' failed due to the
      following reason:
      
      The input cannot be shown compatible
   

2) 'empty' failed due to the following reason:
   
   Empty strict component of the problem is NOT empty.


Arrrr..