MAYBE

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

Strict Trs:
  { gt(0(), Y) -> false()
  , gt(s(X), 0()) -> true()
  , gt(s(X), s(Y)) -> gt(X, Y)
  , p(0()) -> 0()
  , p(s(X)) -> X
  , if(false(), X, Y) -> Y
  , if(true(), X, Y) -> X
  , minus(X, Y) -> if(gt(Y, 0()), minus(p(X), p(Y)), X)
  , div(0(), s(Y)) -> 0()
  , div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { gt^#(0(), Y) -> c_1()
  , gt^#(s(X), 0()) -> c_2()
  , gt^#(s(X), s(Y)) -> c_3(gt^#(X, Y))
  , p^#(0()) -> c_4()
  , p^#(s(X)) -> c_5()
  , if^#(false(), X, Y) -> c_6()
  , if^#(true(), X, Y) -> c_7()
  , minus^#(X, Y) ->
    c_8(if^#(gt(Y, 0()), minus(p(X), p(Y)), X),
        gt^#(Y, 0()),
        minus^#(p(X), p(Y)),
        p^#(X),
        p^#(Y))
  , div^#(0(), s(Y)) -> c_9()
  , div^#(s(X), s(Y)) ->
    c_10(div^#(minus(X, Y), s(Y)), minus^#(X, Y)) }

and mark the set of starting terms.

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

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

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

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

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

Strict DPs:
  { gt^#(s(X), s(Y)) -> c_3(gt^#(X, Y))
  , minus^#(X, Y) ->
    c_8(if^#(gt(Y, 0()), minus(p(X), p(Y)), X),
        gt^#(Y, 0()),
        minus^#(p(X), p(Y)),
        p^#(X),
        p^#(Y))
  , div^#(s(X), s(Y)) ->
    c_10(div^#(minus(X, Y), s(Y)), minus^#(X, Y)) }
Weak DPs:
  { gt^#(0(), Y) -> c_1()
  , gt^#(s(X), 0()) -> c_2()
  , p^#(0()) -> c_4()
  , p^#(s(X)) -> c_5()
  , if^#(false(), X, Y) -> c_6()
  , if^#(true(), X, Y) -> c_7()
  , div^#(0(), s(Y)) -> c_9() }
Weak Trs:
  { gt(0(), Y) -> false()
  , gt(s(X), 0()) -> true()
  , gt(s(X), s(Y)) -> gt(X, Y)
  , p(0()) -> 0()
  , p(s(X)) -> X
  , if(false(), X, Y) -> Y
  , if(true(), X, Y) -> X
  , minus(X, Y) -> if(gt(Y, 0()), minus(p(X), p(Y)), X)
  , div(0(), s(Y)) -> 0()
  , div(s(X), s(Y)) -> s(div(minus(X, Y), s(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.

{ gt^#(0(), Y) -> c_1()
, gt^#(s(X), 0()) -> c_2()
, p^#(0()) -> c_4()
, p^#(s(X)) -> c_5()
, if^#(false(), X, Y) -> c_6()
, if^#(true(), X, Y) -> c_7()
, div^#(0(), s(Y)) -> c_9() }

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

Strict DPs:
  { gt^#(s(X), s(Y)) -> c_3(gt^#(X, Y))
  , minus^#(X, Y) ->
    c_8(if^#(gt(Y, 0()), minus(p(X), p(Y)), X),
        gt^#(Y, 0()),
        minus^#(p(X), p(Y)),
        p^#(X),
        p^#(Y))
  , div^#(s(X), s(Y)) ->
    c_10(div^#(minus(X, Y), s(Y)), minus^#(X, Y)) }
Weak Trs:
  { gt(0(), Y) -> false()
  , gt(s(X), 0()) -> true()
  , gt(s(X), s(Y)) -> gt(X, Y)
  , p(0()) -> 0()
  , p(s(X)) -> X
  , if(false(), X, Y) -> Y
  , if(true(), X, Y) -> X
  , minus(X, Y) -> if(gt(Y, 0()), minus(p(X), p(Y)), X)
  , div(0(), s(Y)) -> 0()
  , div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  { minus^#(X, Y) ->
    c_8(if^#(gt(Y, 0()), minus(p(X), p(Y)), X),
        gt^#(Y, 0()),
        minus^#(p(X), p(Y)),
        p^#(X),
        p^#(Y)) }

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

Strict DPs:
  { gt^#(s(X), s(Y)) -> c_1(gt^#(X, Y))
  , minus^#(X, Y) -> c_2(minus^#(p(X), p(Y)))
  , div^#(s(X), s(Y)) ->
    c_3(div^#(minus(X, Y), s(Y)), minus^#(X, Y)) }
Weak Trs:
  { gt(0(), Y) -> false()
  , gt(s(X), 0()) -> true()
  , gt(s(X), s(Y)) -> gt(X, Y)
  , p(0()) -> 0()
  , p(s(X)) -> X
  , if(false(), X, Y) -> Y
  , if(true(), X, Y) -> X
  , minus(X, Y) -> if(gt(Y, 0()), minus(p(X), p(Y)), X)
  , div(0(), s(Y)) -> 0()
  , div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { gt(0(), Y) -> false()
    , gt(s(X), 0()) -> true()
    , gt(s(X), s(Y)) -> gt(X, Y)
    , p(0()) -> 0()
    , p(s(X)) -> X
    , if(false(), X, Y) -> Y
    , if(true(), X, Y) -> X
    , minus(X, Y) -> if(gt(Y, 0()), minus(p(X), p(Y)), X) }

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

Strict DPs:
  { gt^#(s(X), s(Y)) -> c_1(gt^#(X, Y))
  , minus^#(X, Y) -> c_2(minus^#(p(X), p(Y)))
  , div^#(s(X), s(Y)) ->
    c_3(div^#(minus(X, Y), s(Y)), minus^#(X, Y)) }
Weak Trs:
  { gt(0(), Y) -> false()
  , gt(s(X), 0()) -> true()
  , gt(s(X), s(Y)) -> gt(X, Y)
  , p(0()) -> 0()
  , p(s(X)) -> X
  , if(false(), X, Y) -> Y
  , if(true(), X, Y) -> X
  , minus(X, Y) -> if(gt(Y, 0()), minus(p(X), p(Y)), 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..