MAYBE

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

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

We add following dependency tuples:

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

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()
  , le^#(0(), y) -> c_3()
  , le^#(s(x), 0()) -> c_4()
  , le^#(s(x), s(y)) -> c_5(le^#(x, y))
  , minus^#(x, 0()) -> c_6()
  , minus^#(x, s(y)) ->
    c_7(if^#(le(x, s(y)), 0(), p(minus(x, p(s(y))))),
        le^#(x, s(y)),
        p^#(minus(x, p(s(y)))),
        minus^#(x, p(s(y))),
        p^#(s(y)))
  , if^#(true(), x, y) -> c_8()
  , if^#(false(), x, y) -> c_9() }
Weak Trs:
  { p(0()) -> 0()
  , p(s(x)) -> x
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , minus(x, 0()) -> x
  , minus(x, s(y)) -> if(le(x, s(y)), 0(), p(minus(x, p(s(y)))))
  , if(true(), x, y) -> x
  , if(false(), x, y) -> y }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

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

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

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

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

Strict DPs:
  { le^#(s(x), s(y)) -> c_5(le^#(x, y))
  , minus^#(x, s(y)) ->
    c_7(if^#(le(x, s(y)), 0(), p(minus(x, p(s(y))))),
        le^#(x, s(y)),
        p^#(minus(x, p(s(y)))),
        minus^#(x, p(s(y))),
        p^#(s(y))) }
Weak Trs:
  { p(0()) -> 0()
  , p(s(x)) -> x
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , minus(x, 0()) -> x
  , minus(x, s(y)) -> if(le(x, s(y)), 0(), p(minus(x, p(s(y)))))
  , if(true(), x, y) -> x
  , if(false(), x, y) -> 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_7(if^#(le(x, s(y)), 0(), p(minus(x, p(s(y))))),
        le^#(x, s(y)),
        p^#(minus(x, p(s(y)))),
        minus^#(x, p(s(y))),
        p^#(s(y))) }

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

Strict DPs:
  { le^#(s(x), s(y)) -> c_1(le^#(x, y))
  , minus^#(x, s(y)) -> c_2(le^#(x, s(y)), minus^#(x, p(s(y)))) }
Weak Trs:
  { p(0()) -> 0()
  , p(s(x)) -> x
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , minus(x, 0()) -> x
  , minus(x, s(y)) -> if(le(x, s(y)), 0(), p(minus(x, p(s(y)))))
  , if(true(), x, y) -> x
  , if(false(), x, y) -> y }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

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

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

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