MAYBE

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

Strict Trs:
  { ge(x, 0()) -> true()
  , ge(0(), s(x)) -> false()
  , ge(s(x), s(y)) -> ge(x, y)
  , minus(x, 0()) -> x
  , minus(s(x), s(y)) -> minus(x, y)
  , div(x, y) -> ify(ge(y, s(0())), x, y)
  , ify(true(), x, y) -> if(ge(x, y), x, y)
  , ify(false(), x, y) -> divByZeroError()
  , if(true(), x, y) -> s(div(minus(x, y), y))
  , if(false(), x, y) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { ge^#(x, 0()) -> c_1()
  , ge^#(0(), s(x)) -> c_2()
  , ge^#(s(x), s(y)) -> c_3(ge^#(x, y))
  , minus^#(x, 0()) -> c_4()
  , minus^#(s(x), s(y)) -> c_5(minus^#(x, y))
  , div^#(x, y) -> c_6(ify^#(ge(y, s(0())), x, y), ge^#(y, s(0())))
  , ify^#(true(), x, y) -> c_7(if^#(ge(x, y), x, y), ge^#(x, y))
  , ify^#(false(), x, y) -> c_8()
  , if^#(true(), x, y) -> c_9(div^#(minus(x, y), y), minus^#(x, y))
  , if^#(false(), x, y) -> c_10() }

and mark the set of starting terms.

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

Strict DPs:
  { ge^#(x, 0()) -> c_1()
  , ge^#(0(), s(x)) -> c_2()
  , ge^#(s(x), s(y)) -> c_3(ge^#(x, y))
  , minus^#(x, 0()) -> c_4()
  , minus^#(s(x), s(y)) -> c_5(minus^#(x, y))
  , div^#(x, y) -> c_6(ify^#(ge(y, s(0())), x, y), ge^#(y, s(0())))
  , ify^#(true(), x, y) -> c_7(if^#(ge(x, y), x, y), ge^#(x, y))
  , ify^#(false(), x, y) -> c_8()
  , if^#(true(), x, y) -> c_9(div^#(minus(x, y), y), minus^#(x, y))
  , if^#(false(), x, y) -> c_10() }
Weak Trs:
  { ge(x, 0()) -> true()
  , ge(0(), s(x)) -> false()
  , ge(s(x), s(y)) -> ge(x, y)
  , minus(x, 0()) -> x
  , minus(s(x), s(y)) -> minus(x, y)
  , div(x, y) -> ify(ge(y, s(0())), x, y)
  , ify(true(), x, y) -> if(ge(x, y), x, y)
  , ify(false(), x, y) -> divByZeroError()
  , if(true(), x, y) -> s(div(minus(x, y), y))
  , if(false(), x, y) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: ge^#(x, 0()) -> c_1()
    , 2: ge^#(0(), s(x)) -> c_2()
    , 3: ge^#(s(x), s(y)) -> c_3(ge^#(x, y))
    , 4: minus^#(x, 0()) -> c_4()
    , 5: minus^#(s(x), s(y)) -> c_5(minus^#(x, y))
    , 6: div^#(x, y) ->
         c_6(ify^#(ge(y, s(0())), x, y), ge^#(y, s(0())))
    , 7: ify^#(true(), x, y) -> c_7(if^#(ge(x, y), x, y), ge^#(x, y))
    , 8: ify^#(false(), x, y) -> c_8()
    , 9: if^#(true(), x, y) ->
         c_9(div^#(minus(x, y), y), minus^#(x, y))
    , 10: if^#(false(), x, y) -> c_10() }

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

Strict DPs:
  { ge^#(s(x), s(y)) -> c_3(ge^#(x, y))
  , minus^#(s(x), s(y)) -> c_5(minus^#(x, y))
  , div^#(x, y) -> c_6(ify^#(ge(y, s(0())), x, y), ge^#(y, s(0())))
  , ify^#(true(), x, y) -> c_7(if^#(ge(x, y), x, y), ge^#(x, y))
  , if^#(true(), x, y) -> c_9(div^#(minus(x, y), y), minus^#(x, y)) }
Weak DPs:
  { ge^#(x, 0()) -> c_1()
  , ge^#(0(), s(x)) -> c_2()
  , minus^#(x, 0()) -> c_4()
  , ify^#(false(), x, y) -> c_8()
  , if^#(false(), x, y) -> c_10() }
Weak Trs:
  { ge(x, 0()) -> true()
  , ge(0(), s(x)) -> false()
  , ge(s(x), s(y)) -> ge(x, y)
  , minus(x, 0()) -> x
  , minus(s(x), s(y)) -> minus(x, y)
  , div(x, y) -> ify(ge(y, s(0())), x, y)
  , ify(true(), x, y) -> if(ge(x, y), x, y)
  , ify(false(), x, y) -> divByZeroError()
  , if(true(), x, y) -> s(div(minus(x, y), y))
  , if(false(), x, y) -> 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.

{ ge^#(x, 0()) -> c_1()
, ge^#(0(), s(x)) -> c_2()
, minus^#(x, 0()) -> c_4()
, ify^#(false(), x, y) -> c_8()
, if^#(false(), x, y) -> c_10() }

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

Strict DPs:
  { ge^#(s(x), s(y)) -> c_3(ge^#(x, y))
  , minus^#(s(x), s(y)) -> c_5(minus^#(x, y))
  , div^#(x, y) -> c_6(ify^#(ge(y, s(0())), x, y), ge^#(y, s(0())))
  , ify^#(true(), x, y) -> c_7(if^#(ge(x, y), x, y), ge^#(x, y))
  , if^#(true(), x, y) -> c_9(div^#(minus(x, y), y), minus^#(x, y)) }
Weak Trs:
  { ge(x, 0()) -> true()
  , ge(0(), s(x)) -> false()
  , ge(s(x), s(y)) -> ge(x, y)
  , minus(x, 0()) -> x
  , minus(s(x), s(y)) -> minus(x, y)
  , div(x, y) -> ify(ge(y, s(0())), x, y)
  , ify(true(), x, y) -> if(ge(x, y), x, y)
  , ify(false(), x, y) -> divByZeroError()
  , if(true(), x, y) -> s(div(minus(x, y), y))
  , if(false(), x, y) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { ge(x, 0()) -> true()
    , ge(0(), s(x)) -> false()
    , ge(s(x), s(y)) -> ge(x, y)
    , minus(x, 0()) -> x
    , minus(s(x), s(y)) -> minus(x, y) }

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

Strict DPs:
  { ge^#(s(x), s(y)) -> c_3(ge^#(x, y))
  , minus^#(s(x), s(y)) -> c_5(minus^#(x, y))
  , div^#(x, y) -> c_6(ify^#(ge(y, s(0())), x, y), ge^#(y, s(0())))
  , ify^#(true(), x, y) -> c_7(if^#(ge(x, y), x, y), ge^#(x, y))
  , if^#(true(), x, y) -> c_9(div^#(minus(x, y), y), minus^#(x, y)) }
Weak Trs:
  { ge(x, 0()) -> true()
  , ge(0(), s(x)) -> false()
  , ge(s(x), s(y)) -> ge(x, y)
  , minus(x, 0()) -> x
  , minus(s(x), s(y)) -> minus(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..