MAYBE

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

Strict Trs:
  { le(0(), y, z) -> greater(y, z)
  , le(s(x), 0(), z) -> false()
  , le(s(x), s(y), 0()) -> false()
  , le(s(x), s(y), s(z)) -> le(x, y, z)
  , greater(x, 0()) -> first()
  , greater(0(), s(y)) -> second()
  , greater(s(x), s(y)) -> greater(x, y)
  , double(0()) -> 0()
  , double(s(x)) -> s(s(double(x)))
  , triple(x) -> if(le(x, x, double(x)), x, 0(), 0())
  , if(false(), x, y, z) -> true()
  , if(first(), x, y, z) -> if(le(s(x), y, s(z)), s(x), y, s(z))
  , if(second(), x, y, z) -> if(le(s(x), s(y), z), s(x), s(y), z) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { le^#(0(), y, z) -> c_1(greater^#(y, z))
  , le^#(s(x), 0(), z) -> c_2()
  , le^#(s(x), s(y), 0()) -> c_3()
  , le^#(s(x), s(y), s(z)) -> c_4(le^#(x, y, z))
  , greater^#(x, 0()) -> c_5()
  , greater^#(0(), s(y)) -> c_6()
  , greater^#(s(x), s(y)) -> c_7(greater^#(x, y))
  , double^#(0()) -> c_8()
  , double^#(s(x)) -> c_9(double^#(x))
  , triple^#(x) ->
    c_10(if^#(le(x, x, double(x)), x, 0(), 0()),
         le^#(x, x, double(x)),
         double^#(x))
  , if^#(false(), x, y, z) -> c_11()
  , if^#(first(), x, y, z) ->
    c_12(if^#(le(s(x), y, s(z)), s(x), y, s(z)), le^#(s(x), y, s(z)))
  , if^#(second(), x, y, z) ->
    c_13(if^#(le(s(x), s(y), z), s(x), s(y), z), le^#(s(x), s(y), z)) }

and mark the set of starting terms.

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

Strict DPs:
  { le^#(0(), y, z) -> c_1(greater^#(y, z))
  , le^#(s(x), 0(), z) -> c_2()
  , le^#(s(x), s(y), 0()) -> c_3()
  , le^#(s(x), s(y), s(z)) -> c_4(le^#(x, y, z))
  , greater^#(x, 0()) -> c_5()
  , greater^#(0(), s(y)) -> c_6()
  , greater^#(s(x), s(y)) -> c_7(greater^#(x, y))
  , double^#(0()) -> c_8()
  , double^#(s(x)) -> c_9(double^#(x))
  , triple^#(x) ->
    c_10(if^#(le(x, x, double(x)), x, 0(), 0()),
         le^#(x, x, double(x)),
         double^#(x))
  , if^#(false(), x, y, z) -> c_11()
  , if^#(first(), x, y, z) ->
    c_12(if^#(le(s(x), y, s(z)), s(x), y, s(z)), le^#(s(x), y, s(z)))
  , if^#(second(), x, y, z) ->
    c_13(if^#(le(s(x), s(y), z), s(x), s(y), z), le^#(s(x), s(y), z)) }
Weak Trs:
  { le(0(), y, z) -> greater(y, z)
  , le(s(x), 0(), z) -> false()
  , le(s(x), s(y), 0()) -> false()
  , le(s(x), s(y), s(z)) -> le(x, y, z)
  , greater(x, 0()) -> first()
  , greater(0(), s(y)) -> second()
  , greater(s(x), s(y)) -> greater(x, y)
  , double(0()) -> 0()
  , double(s(x)) -> s(s(double(x)))
  , triple(x) -> if(le(x, x, double(x)), x, 0(), 0())
  , if(false(), x, y, z) -> true()
  , if(first(), x, y, z) -> if(le(s(x), y, s(z)), s(x), y, s(z))
  , if(second(), x, y, z) -> if(le(s(x), s(y), z), s(x), s(y), z) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {2,3,5,6,8,11} by
applications of Pre({2,3,5,6,8,11}) = {1,4,7,9,10,12,13}. Here
rules are labeled as follows:

  DPs:
    { 1: le^#(0(), y, z) -> c_1(greater^#(y, z))
    , 2: le^#(s(x), 0(), z) -> c_2()
    , 3: le^#(s(x), s(y), 0()) -> c_3()
    , 4: le^#(s(x), s(y), s(z)) -> c_4(le^#(x, y, z))
    , 5: greater^#(x, 0()) -> c_5()
    , 6: greater^#(0(), s(y)) -> c_6()
    , 7: greater^#(s(x), s(y)) -> c_7(greater^#(x, y))
    , 8: double^#(0()) -> c_8()
    , 9: double^#(s(x)) -> c_9(double^#(x))
    , 10: triple^#(x) ->
          c_10(if^#(le(x, x, double(x)), x, 0(), 0()),
               le^#(x, x, double(x)),
               double^#(x))
    , 11: if^#(false(), x, y, z) -> c_11()
    , 12: if^#(first(), x, y, z) ->
          c_12(if^#(le(s(x), y, s(z)), s(x), y, s(z)), le^#(s(x), y, s(z)))
    , 13: if^#(second(), x, y, z) ->
          c_13(if^#(le(s(x), s(y), z), s(x), s(y), z), le^#(s(x), s(y), z)) }

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

Strict DPs:
  { le^#(0(), y, z) -> c_1(greater^#(y, z))
  , le^#(s(x), s(y), s(z)) -> c_4(le^#(x, y, z))
  , greater^#(s(x), s(y)) -> c_7(greater^#(x, y))
  , double^#(s(x)) -> c_9(double^#(x))
  , triple^#(x) ->
    c_10(if^#(le(x, x, double(x)), x, 0(), 0()),
         le^#(x, x, double(x)),
         double^#(x))
  , if^#(first(), x, y, z) ->
    c_12(if^#(le(s(x), y, s(z)), s(x), y, s(z)), le^#(s(x), y, s(z)))
  , if^#(second(), x, y, z) ->
    c_13(if^#(le(s(x), s(y), z), s(x), s(y), z), le^#(s(x), s(y), z)) }
Weak DPs:
  { le^#(s(x), 0(), z) -> c_2()
  , le^#(s(x), s(y), 0()) -> c_3()
  , greater^#(x, 0()) -> c_5()
  , greater^#(0(), s(y)) -> c_6()
  , double^#(0()) -> c_8()
  , if^#(false(), x, y, z) -> c_11() }
Weak Trs:
  { le(0(), y, z) -> greater(y, z)
  , le(s(x), 0(), z) -> false()
  , le(s(x), s(y), 0()) -> false()
  , le(s(x), s(y), s(z)) -> le(x, y, z)
  , greater(x, 0()) -> first()
  , greater(0(), s(y)) -> second()
  , greater(s(x), s(y)) -> greater(x, y)
  , double(0()) -> 0()
  , double(s(x)) -> s(s(double(x)))
  , triple(x) -> if(le(x, x, double(x)), x, 0(), 0())
  , if(false(), x, y, z) -> true()
  , if(first(), x, y, z) -> if(le(s(x), y, s(z)), s(x), y, s(z))
  , if(second(), x, y, z) -> if(le(s(x), s(y), z), s(x), s(y), z) }
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.

{ le^#(s(x), 0(), z) -> c_2()
, le^#(s(x), s(y), 0()) -> c_3()
, greater^#(x, 0()) -> c_5()
, greater^#(0(), s(y)) -> c_6()
, double^#(0()) -> c_8()
, if^#(false(), x, y, z) -> c_11() }

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

Strict DPs:
  { le^#(0(), y, z) -> c_1(greater^#(y, z))
  , le^#(s(x), s(y), s(z)) -> c_4(le^#(x, y, z))
  , greater^#(s(x), s(y)) -> c_7(greater^#(x, y))
  , double^#(s(x)) -> c_9(double^#(x))
  , triple^#(x) ->
    c_10(if^#(le(x, x, double(x)), x, 0(), 0()),
         le^#(x, x, double(x)),
         double^#(x))
  , if^#(first(), x, y, z) ->
    c_12(if^#(le(s(x), y, s(z)), s(x), y, s(z)), le^#(s(x), y, s(z)))
  , if^#(second(), x, y, z) ->
    c_13(if^#(le(s(x), s(y), z), s(x), s(y), z), le^#(s(x), s(y), z)) }
Weak Trs:
  { le(0(), y, z) -> greater(y, z)
  , le(s(x), 0(), z) -> false()
  , le(s(x), s(y), 0()) -> false()
  , le(s(x), s(y), s(z)) -> le(x, y, z)
  , greater(x, 0()) -> first()
  , greater(0(), s(y)) -> second()
  , greater(s(x), s(y)) -> greater(x, y)
  , double(0()) -> 0()
  , double(s(x)) -> s(s(double(x)))
  , triple(x) -> if(le(x, x, double(x)), x, 0(), 0())
  , if(false(), x, y, z) -> true()
  , if(first(), x, y, z) -> if(le(s(x), y, s(z)), s(x), y, s(z))
  , if(second(), x, y, z) -> if(le(s(x), s(y), z), s(x), s(y), z) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { le(0(), y, z) -> greater(y, z)
    , le(s(x), 0(), z) -> false()
    , le(s(x), s(y), 0()) -> false()
    , le(s(x), s(y), s(z)) -> le(x, y, z)
    , greater(x, 0()) -> first()
    , greater(0(), s(y)) -> second()
    , greater(s(x), s(y)) -> greater(x, y)
    , double(0()) -> 0()
    , double(s(x)) -> s(s(double(x))) }

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

Strict DPs:
  { le^#(0(), y, z) -> c_1(greater^#(y, z))
  , le^#(s(x), s(y), s(z)) -> c_4(le^#(x, y, z))
  , greater^#(s(x), s(y)) -> c_7(greater^#(x, y))
  , double^#(s(x)) -> c_9(double^#(x))
  , triple^#(x) ->
    c_10(if^#(le(x, x, double(x)), x, 0(), 0()),
         le^#(x, x, double(x)),
         double^#(x))
  , if^#(first(), x, y, z) ->
    c_12(if^#(le(s(x), y, s(z)), s(x), y, s(z)), le^#(s(x), y, s(z)))
  , if^#(second(), x, y, z) ->
    c_13(if^#(le(s(x), s(y), z), s(x), s(y), z), le^#(s(x), s(y), z)) }
Weak Trs:
  { le(0(), y, z) -> greater(y, z)
  , le(s(x), 0(), z) -> false()
  , le(s(x), s(y), 0()) -> false()
  , le(s(x), s(y), s(z)) -> le(x, y, z)
  , greater(x, 0()) -> first()
  , greater(0(), s(y)) -> second()
  , greater(s(x), s(y)) -> greater(x, y)
  , double(0()) -> 0()
  , double(s(x)) -> s(s(double(x))) }
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:
      
      Following exception was raised:
        stack overflow
   
   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..