MAYBE

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

Strict Trs:
  { p(0()) -> 0()
  , p(s(x)) -> x
  , minus(x, 0()) -> x
  , minus(x, s(y)) -> p(minus(x, y))
  , minus(s(x), s(y)) -> minus(x, y)
  , div(0(), s(y)) -> 0()
  , div(s(x), s(y)) -> s(div(minus(s(x), s(y)), s(y)))
  , log(s(0()), s(s(y))) -> 0()
  , log(s(s(x)), s(s(y))) ->
    s(log(div(minus(x, y), s(s(y))), s(s(y)))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { p^#(0()) -> c_1()
  , p^#(s(x)) -> c_2()
  , minus^#(x, 0()) -> c_3()
  , minus^#(x, s(y)) -> c_4(p^#(minus(x, y)), minus^#(x, y))
  , minus^#(s(x), s(y)) -> c_5(minus^#(x, y))
  , div^#(0(), s(y)) -> c_6()
  , div^#(s(x), s(y)) ->
    c_7(div^#(minus(s(x), s(y)), s(y)), minus^#(s(x), s(y)))
  , log^#(s(0()), s(s(y))) -> c_8()
  , log^#(s(s(x)), s(s(y))) ->
    c_9(log^#(div(minus(x, y), s(s(y))), s(s(y))),
        div^#(minus(x, y), s(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:
  { p^#(0()) -> c_1()
  , p^#(s(x)) -> c_2()
  , minus^#(x, 0()) -> c_3()
  , minus^#(x, s(y)) -> c_4(p^#(minus(x, y)), minus^#(x, y))
  , minus^#(s(x), s(y)) -> c_5(minus^#(x, y))
  , div^#(0(), s(y)) -> c_6()
  , div^#(s(x), s(y)) ->
    c_7(div^#(minus(s(x), s(y)), s(y)), minus^#(s(x), s(y)))
  , log^#(s(0()), s(s(y))) -> c_8()
  , log^#(s(s(x)), s(s(y))) ->
    c_9(log^#(div(minus(x, y), s(s(y))), s(s(y))),
        div^#(minus(x, y), s(s(y))),
        minus^#(x, y)) }
Weak Trs:
  { p(0()) -> 0()
  , p(s(x)) -> x
  , minus(x, 0()) -> x
  , minus(x, s(y)) -> p(minus(x, y))
  , minus(s(x), s(y)) -> minus(x, y)
  , div(0(), s(y)) -> 0()
  , div(s(x), s(y)) -> s(div(minus(s(x), s(y)), s(y)))
  , log(s(0()), s(s(y))) -> 0()
  , log(s(s(x)), s(s(y))) ->
    s(log(div(minus(x, y), s(s(y))), s(s(y)))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: p^#(0()) -> c_1()
    , 2: p^#(s(x)) -> c_2()
    , 3: minus^#(x, 0()) -> c_3()
    , 4: minus^#(x, s(y)) -> c_4(p^#(minus(x, y)), minus^#(x, y))
    , 5: minus^#(s(x), s(y)) -> c_5(minus^#(x, y))
    , 6: div^#(0(), s(y)) -> c_6()
    , 7: div^#(s(x), s(y)) ->
         c_7(div^#(minus(s(x), s(y)), s(y)), minus^#(s(x), s(y)))
    , 8: log^#(s(0()), s(s(y))) -> c_8()
    , 9: log^#(s(s(x)), s(s(y))) ->
         c_9(log^#(div(minus(x, y), s(s(y))), s(s(y))),
             div^#(minus(x, y), s(s(y))),
             minus^#(x, y)) }

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

Strict DPs:
  { minus^#(x, s(y)) -> c_4(p^#(minus(x, y)), minus^#(x, y))
  , minus^#(s(x), s(y)) -> c_5(minus^#(x, y))
  , div^#(s(x), s(y)) ->
    c_7(div^#(minus(s(x), s(y)), s(y)), minus^#(s(x), s(y)))
  , log^#(s(s(x)), s(s(y))) ->
    c_9(log^#(div(minus(x, y), s(s(y))), s(s(y))),
        div^#(minus(x, y), s(s(y))),
        minus^#(x, y)) }
Weak DPs:
  { p^#(0()) -> c_1()
  , p^#(s(x)) -> c_2()
  , minus^#(x, 0()) -> c_3()
  , div^#(0(), s(y)) -> c_6()
  , log^#(s(0()), s(s(y))) -> c_8() }
Weak Trs:
  { p(0()) -> 0()
  , p(s(x)) -> x
  , minus(x, 0()) -> x
  , minus(x, s(y)) -> p(minus(x, y))
  , minus(s(x), s(y)) -> minus(x, y)
  , div(0(), s(y)) -> 0()
  , div(s(x), s(y)) -> s(div(minus(s(x), s(y)), s(y)))
  , log(s(0()), s(s(y))) -> 0()
  , log(s(s(x)), s(s(y))) ->
    s(log(div(minus(x, y), s(s(y))), s(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.

{ p^#(0()) -> c_1()
, p^#(s(x)) -> c_2()
, minus^#(x, 0()) -> c_3()
, div^#(0(), s(y)) -> c_6()
, log^#(s(0()), s(s(y))) -> c_8() }

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

Strict DPs:
  { minus^#(x, s(y)) -> c_4(p^#(minus(x, y)), minus^#(x, y))
  , minus^#(s(x), s(y)) -> c_5(minus^#(x, y))
  , div^#(s(x), s(y)) ->
    c_7(div^#(minus(s(x), s(y)), s(y)), minus^#(s(x), s(y)))
  , log^#(s(s(x)), s(s(y))) ->
    c_9(log^#(div(minus(x, y), s(s(y))), s(s(y))),
        div^#(minus(x, y), s(s(y))),
        minus^#(x, y)) }
Weak Trs:
  { p(0()) -> 0()
  , p(s(x)) -> x
  , minus(x, 0()) -> x
  , minus(x, s(y)) -> p(minus(x, y))
  , minus(s(x), s(y)) -> minus(x, y)
  , div(0(), s(y)) -> 0()
  , div(s(x), s(y)) -> s(div(minus(s(x), s(y)), s(y)))
  , log(s(0()), s(s(y))) -> 0()
  , log(s(s(x)), s(s(y))) ->
    s(log(div(minus(x, y), s(s(y))), s(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, s(y)) -> c_4(p^#(minus(x, y)), minus^#(x, y)) }

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

Strict DPs:
  { minus^#(x, s(y)) -> c_1(minus^#(x, y))
  , minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
  , div^#(s(x), s(y)) ->
    c_3(div^#(minus(s(x), s(y)), s(y)), minus^#(s(x), s(y)))
  , log^#(s(s(x)), s(s(y))) ->
    c_4(log^#(div(minus(x, y), s(s(y))), s(s(y))),
        div^#(minus(x, y), s(s(y))),
        minus^#(x, y)) }
Weak Trs:
  { p(0()) -> 0()
  , p(s(x)) -> x
  , minus(x, 0()) -> x
  , minus(x, s(y)) -> p(minus(x, y))
  , minus(s(x), s(y)) -> minus(x, y)
  , div(0(), s(y)) -> 0()
  , div(s(x), s(y)) -> s(div(minus(s(x), s(y)), s(y)))
  , log(s(0()), s(s(y))) -> 0()
  , log(s(s(x)), s(s(y))) ->
    s(log(div(minus(x, y), s(s(y))), s(s(y)))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { p(0()) -> 0()
    , p(s(x)) -> x
    , minus(x, 0()) -> x
    , minus(x, s(y)) -> p(minus(x, y))
    , minus(s(x), s(y)) -> minus(x, y)
    , div(0(), s(y)) -> 0()
    , div(s(x), s(y)) -> s(div(minus(s(x), s(y)), s(y))) }

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

Strict DPs:
  { minus^#(x, s(y)) -> c_1(minus^#(x, y))
  , minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
  , div^#(s(x), s(y)) ->
    c_3(div^#(minus(s(x), s(y)), s(y)), minus^#(s(x), s(y)))
  , log^#(s(s(x)), s(s(y))) ->
    c_4(log^#(div(minus(x, y), s(s(y))), s(s(y))),
        div^#(minus(x, y), s(s(y))),
        minus^#(x, y)) }
Weak Trs:
  { p(0()) -> 0()
  , p(s(x)) -> x
  , minus(x, 0()) -> x
  , minus(x, s(y)) -> p(minus(x, y))
  , minus(s(x), s(y)) -> minus(x, y)
  , div(0(), s(y)) -> 0()
  , div(s(x), s(y)) -> s(div(minus(s(x), s(y)), s(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:
      
      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..