MAYBE

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

Strict Trs:
  { ge(x, 0()) -> true()
  , ge(0(), s(y)) -> false()
  , ge(s(x), s(y)) -> ge(x, y)
  , minus(x, 0()) -> x
  , minus(0(), y) -> 0()
  , minus(s(x), s(y)) -> minus(x, y)
  , id_inc(x) -> x
  , id_inc(x) -> s(x)
  , quot(x, y) -> div(x, y, 0())
  , div(x, y, z) -> if(ge(y, s(0())), ge(x, y), x, y, z)
  , if(true(), true(), x, y, z) -> div(minus(x, y), y, id_inc(z))
  , if(true(), false(), x, y, z) -> z
  , if(false(), b, x, y, z) -> div_by_zero() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { ge^#(x, 0()) -> c_1()
  , ge^#(0(), s(y)) -> c_2()
  , ge^#(s(x), s(y)) -> c_3(ge^#(x, y))
  , minus^#(x, 0()) -> c_4()
  , minus^#(0(), y) -> c_5()
  , minus^#(s(x), s(y)) -> c_6(minus^#(x, y))
  , id_inc^#(x) -> c_7()
  , id_inc^#(x) -> c_8()
  , quot^#(x, y) -> c_9(div^#(x, y, 0()))
  , div^#(x, y, z) ->
    c_10(if^#(ge(y, s(0())), ge(x, y), x, y, z),
         ge^#(y, s(0())),
         ge^#(x, y))
  , if^#(true(), true(), x, y, z) ->
    c_11(div^#(minus(x, y), y, id_inc(z)), minus^#(x, y), id_inc^#(z))
  , if^#(true(), false(), x, y, z) -> c_12()
  , if^#(false(), b, x, y, z) -> c_13() }

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(y)) -> c_2()
  , ge^#(s(x), s(y)) -> c_3(ge^#(x, y))
  , minus^#(x, 0()) -> c_4()
  , minus^#(0(), y) -> c_5()
  , minus^#(s(x), s(y)) -> c_6(minus^#(x, y))
  , id_inc^#(x) -> c_7()
  , id_inc^#(x) -> c_8()
  , quot^#(x, y) -> c_9(div^#(x, y, 0()))
  , div^#(x, y, z) ->
    c_10(if^#(ge(y, s(0())), ge(x, y), x, y, z),
         ge^#(y, s(0())),
         ge^#(x, y))
  , if^#(true(), true(), x, y, z) ->
    c_11(div^#(minus(x, y), y, id_inc(z)), minus^#(x, y), id_inc^#(z))
  , if^#(true(), false(), x, y, z) -> c_12()
  , if^#(false(), b, x, y, z) -> c_13() }
Weak Trs:
  { ge(x, 0()) -> true()
  , ge(0(), s(y)) -> false()
  , ge(s(x), s(y)) -> ge(x, y)
  , minus(x, 0()) -> x
  , minus(0(), y) -> 0()
  , minus(s(x), s(y)) -> minus(x, y)
  , id_inc(x) -> x
  , id_inc(x) -> s(x)
  , quot(x, y) -> div(x, y, 0())
  , div(x, y, z) -> if(ge(y, s(0())), ge(x, y), x, y, z)
  , if(true(), true(), x, y, z) -> div(minus(x, y), y, id_inc(z))
  , if(true(), false(), x, y, z) -> z
  , if(false(), b, x, y, z) -> div_by_zero() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: ge^#(x, 0()) -> c_1()
    , 2: ge^#(0(), s(y)) -> c_2()
    , 3: ge^#(s(x), s(y)) -> c_3(ge^#(x, y))
    , 4: minus^#(x, 0()) -> c_4()
    , 5: minus^#(0(), y) -> c_5()
    , 6: minus^#(s(x), s(y)) -> c_6(minus^#(x, y))
    , 7: id_inc^#(x) -> c_7()
    , 8: id_inc^#(x) -> c_8()
    , 9: quot^#(x, y) -> c_9(div^#(x, y, 0()))
    , 10: div^#(x, y, z) ->
          c_10(if^#(ge(y, s(0())), ge(x, y), x, y, z),
               ge^#(y, s(0())),
               ge^#(x, y))
    , 11: if^#(true(), true(), x, y, z) ->
          c_11(div^#(minus(x, y), y, id_inc(z)), minus^#(x, y), id_inc^#(z))
    , 12: if^#(true(), false(), x, y, z) -> c_12()
    , 13: if^#(false(), b, x, y, z) -> c_13() }

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_6(minus^#(x, y))
  , quot^#(x, y) -> c_9(div^#(x, y, 0()))
  , div^#(x, y, z) ->
    c_10(if^#(ge(y, s(0())), ge(x, y), x, y, z),
         ge^#(y, s(0())),
         ge^#(x, y))
  , if^#(true(), true(), x, y, z) ->
    c_11(div^#(minus(x, y), y, id_inc(z)),
         minus^#(x, y),
         id_inc^#(z)) }
Weak DPs:
  { ge^#(x, 0()) -> c_1()
  , ge^#(0(), s(y)) -> c_2()
  , minus^#(x, 0()) -> c_4()
  , minus^#(0(), y) -> c_5()
  , id_inc^#(x) -> c_7()
  , id_inc^#(x) -> c_8()
  , if^#(true(), false(), x, y, z) -> c_12()
  , if^#(false(), b, x, y, z) -> c_13() }
Weak Trs:
  { ge(x, 0()) -> true()
  , ge(0(), s(y)) -> false()
  , ge(s(x), s(y)) -> ge(x, y)
  , minus(x, 0()) -> x
  , minus(0(), y) -> 0()
  , minus(s(x), s(y)) -> minus(x, y)
  , id_inc(x) -> x
  , id_inc(x) -> s(x)
  , quot(x, y) -> div(x, y, 0())
  , div(x, y, z) -> if(ge(y, s(0())), ge(x, y), x, y, z)
  , if(true(), true(), x, y, z) -> div(minus(x, y), y, id_inc(z))
  , if(true(), false(), x, y, z) -> z
  , if(false(), b, x, y, z) -> div_by_zero() }
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(y)) -> c_2()
, minus^#(x, 0()) -> c_4()
, minus^#(0(), y) -> c_5()
, id_inc^#(x) -> c_7()
, id_inc^#(x) -> c_8()
, if^#(true(), false(), x, y, z) -> c_12()
, if^#(false(), b, x, y, z) -> c_13() }

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_6(minus^#(x, y))
  , quot^#(x, y) -> c_9(div^#(x, y, 0()))
  , div^#(x, y, z) ->
    c_10(if^#(ge(y, s(0())), ge(x, y), x, y, z),
         ge^#(y, s(0())),
         ge^#(x, y))
  , if^#(true(), true(), x, y, z) ->
    c_11(div^#(minus(x, y), y, id_inc(z)),
         minus^#(x, y),
         id_inc^#(z)) }
Weak Trs:
  { ge(x, 0()) -> true()
  , ge(0(), s(y)) -> false()
  , ge(s(x), s(y)) -> ge(x, y)
  , minus(x, 0()) -> x
  , minus(0(), y) -> 0()
  , minus(s(x), s(y)) -> minus(x, y)
  , id_inc(x) -> x
  , id_inc(x) -> s(x)
  , quot(x, y) -> div(x, y, 0())
  , div(x, y, z) -> if(ge(y, s(0())), ge(x, y), x, y, z)
  , if(true(), true(), x, y, z) -> div(minus(x, y), y, id_inc(z))
  , if(true(), false(), x, y, z) -> z
  , if(false(), b, x, y, z) -> div_by_zero() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

Due to missing edges in the dependency-graph, the right-hand sides
of following rules could be simplified:

  { if^#(true(), true(), x, y, z) ->
    c_11(div^#(minus(x, y), y, id_inc(z)),
         minus^#(x, y),
         id_inc^#(z)) }

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

Strict DPs:
  { ge^#(s(x), s(y)) -> c_1(ge^#(x, y))
  , minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
  , quot^#(x, y) -> c_3(div^#(x, y, 0()))
  , div^#(x, y, z) ->
    c_4(if^#(ge(y, s(0())), ge(x, y), x, y, z),
        ge^#(y, s(0())),
        ge^#(x, y))
  , if^#(true(), true(), x, y, z) ->
    c_5(div^#(minus(x, y), y, id_inc(z)), minus^#(x, y)) }
Weak Trs:
  { ge(x, 0()) -> true()
  , ge(0(), s(y)) -> false()
  , ge(s(x), s(y)) -> ge(x, y)
  , minus(x, 0()) -> x
  , minus(0(), y) -> 0()
  , minus(s(x), s(y)) -> minus(x, y)
  , id_inc(x) -> x
  , id_inc(x) -> s(x)
  , quot(x, y) -> div(x, y, 0())
  , div(x, y, z) -> if(ge(y, s(0())), ge(x, y), x, y, z)
  , if(true(), true(), x, y, z) -> div(minus(x, y), y, id_inc(z))
  , if(true(), false(), x, y, z) -> z
  , if(false(), b, x, y, z) -> div_by_zero() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

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

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

Strict DPs:
  { ge^#(s(x), s(y)) -> c_1(ge^#(x, y))
  , minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
  , quot^#(x, y) -> c_3(div^#(x, y, 0()))
  , div^#(x, y, z) ->
    c_4(if^#(ge(y, s(0())), ge(x, y), x, y, z),
        ge^#(y, s(0())),
        ge^#(x, y))
  , if^#(true(), true(), x, y, z) ->
    c_5(div^#(minus(x, y), y, id_inc(z)), minus^#(x, y)) }
Weak Trs:
  { ge(x, 0()) -> true()
  , ge(0(), s(y)) -> false()
  , ge(s(x), s(y)) -> ge(x, y)
  , minus(x, 0()) -> x
  , minus(0(), y) -> 0()
  , minus(s(x), s(y)) -> minus(x, y)
  , id_inc(x) -> x
  , id_inc(x) -> s(x) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

Consider the dependency graph

  1: ge^#(s(x), s(y)) -> c_1(ge^#(x, y))
     -->_1 ge^#(s(x), s(y)) -> c_1(ge^#(x, y)) :1
  
  2: minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
     -->_1 minus^#(s(x), s(y)) -> c_2(minus^#(x, y)) :2
  
  3: quot^#(x, y) -> c_3(div^#(x, y, 0()))
     -->_1 div^#(x, y, z) ->
           c_4(if^#(ge(y, s(0())), ge(x, y), x, y, z),
               ge^#(y, s(0())),
               ge^#(x, y)) :4
  
  4: div^#(x, y, z) ->
     c_4(if^#(ge(y, s(0())), ge(x, y), x, y, z),
         ge^#(y, s(0())),
         ge^#(x, y))
     -->_1 if^#(true(), true(), x, y, z) ->
           c_5(div^#(minus(x, y), y, id_inc(z)), minus^#(x, y)) :5
     -->_3 ge^#(s(x), s(y)) -> c_1(ge^#(x, y)) :1
     -->_2 ge^#(s(x), s(y)) -> c_1(ge^#(x, y)) :1
  
  5: if^#(true(), true(), x, y, z) ->
     c_5(div^#(minus(x, y), y, id_inc(z)), minus^#(x, y))
     -->_1 div^#(x, y, z) ->
           c_4(if^#(ge(y, s(0())), ge(x, y), x, y, z),
               ge^#(y, s(0())),
               ge^#(x, y)) :4
     -->_2 minus^#(s(x), s(y)) -> c_2(minus^#(x, y)) :2
  

Following roots of the dependency graph are removed, as the
considered set of starting terms is closed under reduction with
respect to these rules (modulo compound contexts).

  { quot^#(x, y) -> c_3(div^#(x, y, 0())) }


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

Strict DPs:
  { ge^#(s(x), s(y)) -> c_1(ge^#(x, y))
  , minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
  , div^#(x, y, z) ->
    c_4(if^#(ge(y, s(0())), ge(x, y), x, y, z),
        ge^#(y, s(0())),
        ge^#(x, y))
  , if^#(true(), true(), x, y, z) ->
    c_5(div^#(minus(x, y), y, id_inc(z)), minus^#(x, y)) }
Weak Trs:
  { ge(x, 0()) -> true()
  , ge(0(), s(y)) -> false()
  , ge(s(x), s(y)) -> ge(x, y)
  , minus(x, 0()) -> x
  , minus(0(), y) -> 0()
  , minus(s(x), s(y)) -> minus(x, y)
  , id_inc(x) -> x
  , id_inc(x) -> s(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:
      
      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..