MAYBE

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

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

We add following dependency tuples:

Strict DPs:
  { leq^#(0(), y) -> c_1()
  , leq^#(s(x), 0()) -> c_2()
  , leq^#(s(x), s(y)) -> c_3(leq^#(x, y))
  , if^#(true(), x, y) -> c_4()
  , if^#(false(), x, y) -> c_5()
  , -^#(x, 0()) -> c_6()
  , -^#(s(x), s(y)) -> c_7(-^#(x, y))
  , mod^#(0(), y) -> c_8()
  , mod^#(s(x), 0()) -> c_9()
  , mod^#(s(x), s(y)) ->
    c_10(if^#(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x)),
         leq^#(y, x),
         mod^#(-(s(x), s(y)), s(y)),
         -^#(s(x), s(y))) }

and mark the set of starting terms.

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

Strict DPs:
  { leq^#(0(), y) -> c_1()
  , leq^#(s(x), 0()) -> c_2()
  , leq^#(s(x), s(y)) -> c_3(leq^#(x, y))
  , if^#(true(), x, y) -> c_4()
  , if^#(false(), x, y) -> c_5()
  , -^#(x, 0()) -> c_6()
  , -^#(s(x), s(y)) -> c_7(-^#(x, y))
  , mod^#(0(), y) -> c_8()
  , mod^#(s(x), 0()) -> c_9()
  , mod^#(s(x), s(y)) ->
    c_10(if^#(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x)),
         leq^#(y, x),
         mod^#(-(s(x), s(y)), s(y)),
         -^#(s(x), s(y))) }
Weak Trs:
  { leq(0(), y) -> true()
  , leq(s(x), 0()) -> false()
  , leq(s(x), s(y)) -> leq(x, y)
  , if(true(), x, y) -> x
  , if(false(), x, y) -> y
  , -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y)
  , mod(0(), y) -> 0()
  , mod(s(x), 0()) -> 0()
  , mod(s(x), s(y)) ->
    if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: leq^#(0(), y) -> c_1()
    , 2: leq^#(s(x), 0()) -> c_2()
    , 3: leq^#(s(x), s(y)) -> c_3(leq^#(x, y))
    , 4: if^#(true(), x, y) -> c_4()
    , 5: if^#(false(), x, y) -> c_5()
    , 6: -^#(x, 0()) -> c_6()
    , 7: -^#(s(x), s(y)) -> c_7(-^#(x, y))
    , 8: mod^#(0(), y) -> c_8()
    , 9: mod^#(s(x), 0()) -> c_9()
    , 10: mod^#(s(x), s(y)) ->
          c_10(if^#(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x)),
               leq^#(y, x),
               mod^#(-(s(x), s(y)), s(y)),
               -^#(s(x), s(y))) }

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

Strict DPs:
  { leq^#(s(x), s(y)) -> c_3(leq^#(x, y))
  , -^#(s(x), s(y)) -> c_7(-^#(x, y))
  , mod^#(s(x), s(y)) ->
    c_10(if^#(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x)),
         leq^#(y, x),
         mod^#(-(s(x), s(y)), s(y)),
         -^#(s(x), s(y))) }
Weak DPs:
  { leq^#(0(), y) -> c_1()
  , leq^#(s(x), 0()) -> c_2()
  , if^#(true(), x, y) -> c_4()
  , if^#(false(), x, y) -> c_5()
  , -^#(x, 0()) -> c_6()
  , mod^#(0(), y) -> c_8()
  , mod^#(s(x), 0()) -> c_9() }
Weak Trs:
  { leq(0(), y) -> true()
  , leq(s(x), 0()) -> false()
  , leq(s(x), s(y)) -> leq(x, y)
  , if(true(), x, y) -> x
  , if(false(), x, y) -> y
  , -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y)
  , mod(0(), y) -> 0()
  , mod(s(x), 0()) -> 0()
  , mod(s(x), s(y)) ->
    if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x)) }
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.

{ leq^#(0(), y) -> c_1()
, leq^#(s(x), 0()) -> c_2()
, if^#(true(), x, y) -> c_4()
, if^#(false(), x, y) -> c_5()
, -^#(x, 0()) -> c_6()
, mod^#(0(), y) -> c_8()
, mod^#(s(x), 0()) -> c_9() }

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

Strict DPs:
  { leq^#(s(x), s(y)) -> c_3(leq^#(x, y))
  , -^#(s(x), s(y)) -> c_7(-^#(x, y))
  , mod^#(s(x), s(y)) ->
    c_10(if^#(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x)),
         leq^#(y, x),
         mod^#(-(s(x), s(y)), s(y)),
         -^#(s(x), s(y))) }
Weak Trs:
  { leq(0(), y) -> true()
  , leq(s(x), 0()) -> false()
  , leq(s(x), s(y)) -> leq(x, y)
  , if(true(), x, y) -> x
  , if(false(), x, y) -> y
  , -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y)
  , mod(0(), y) -> 0()
  , mod(s(x), 0()) -> 0()
  , mod(s(x), s(y)) ->
    if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  { mod^#(s(x), s(y)) ->
    c_10(if^#(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x)),
         leq^#(y, x),
         mod^#(-(s(x), s(y)), s(y)),
         -^#(s(x), s(y))) }

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

Strict DPs:
  { leq^#(s(x), s(y)) -> c_1(leq^#(x, y))
  , -^#(s(x), s(y)) -> c_2(-^#(x, y))
  , mod^#(s(x), s(y)) ->
    c_3(leq^#(y, x), mod^#(-(s(x), s(y)), s(y)), -^#(s(x), s(y))) }
Weak Trs:
  { leq(0(), y) -> true()
  , leq(s(x), 0()) -> false()
  , leq(s(x), s(y)) -> leq(x, y)
  , if(true(), x, y) -> x
  , if(false(), x, y) -> y
  , -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y)
  , mod(0(), y) -> 0()
  , mod(s(x), 0()) -> 0()
  , mod(s(x), s(y)) ->
    if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

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

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

Strict DPs:
  { leq^#(s(x), s(y)) -> c_1(leq^#(x, y))
  , -^#(s(x), s(y)) -> c_2(-^#(x, y))
  , mod^#(s(x), s(y)) ->
    c_3(leq^#(y, x), mod^#(-(s(x), s(y)), s(y)), -^#(s(x), s(y))) }
Weak Trs:
  { -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, 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:
      
      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..