MAYBE

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

Strict Trs:
  { cond1(true(), x, y) -> cond2(gr(x, y), x, y)
  , cond2(true(), x, y) -> cond3(gr(x, 0()), x, y)
  , cond2(false(), x, y) -> cond4(gr(y, 0()), x, y)
  , gr(0(), x) -> false()
  , gr(s(x), 0()) -> true()
  , gr(s(x), s(y)) -> gr(x, y)
  , cond3(true(), x, y) -> cond3(gr(x, 0()), p(x), y)
  , cond3(false(), x, y) -> cond1(and(gr(x, 0()), gr(y, 0())), x, y)
  , cond4(true(), x, y) -> cond4(gr(y, 0()), x, p(y))
  , cond4(false(), x, y) -> cond1(and(gr(x, 0()), gr(y, 0())), x, y)
  , p(0()) -> 0()
  , p(s(x)) -> x
  , and(x, false()) -> false()
  , and(true(), true()) -> true()
  , and(false(), x) -> false() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { cond1^#(true(), x, y) -> c_1(cond2^#(gr(x, y), x, y), gr^#(x, y))
  , cond2^#(true(), x, y) ->
    c_2(cond3^#(gr(x, 0()), x, y), gr^#(x, 0()))
  , cond2^#(false(), x, y) ->
    c_3(cond4^#(gr(y, 0()), x, y), gr^#(y, 0()))
  , gr^#(0(), x) -> c_4()
  , gr^#(s(x), 0()) -> c_5()
  , gr^#(s(x), s(y)) -> c_6(gr^#(x, y))
  , cond3^#(true(), x, y) ->
    c_7(cond3^#(gr(x, 0()), p(x), y), gr^#(x, 0()), p^#(x))
  , cond3^#(false(), x, y) ->
    c_8(cond1^#(and(gr(x, 0()), gr(y, 0())), x, y),
        and^#(gr(x, 0()), gr(y, 0())),
        gr^#(x, 0()),
        gr^#(y, 0()))
  , cond4^#(true(), x, y) ->
    c_9(cond4^#(gr(y, 0()), x, p(y)), gr^#(y, 0()), p^#(y))
  , cond4^#(false(), x, y) ->
    c_10(cond1^#(and(gr(x, 0()), gr(y, 0())), x, y),
         and^#(gr(x, 0()), gr(y, 0())),
         gr^#(x, 0()),
         gr^#(y, 0()))
  , p^#(0()) -> c_11()
  , p^#(s(x)) -> c_12()
  , and^#(x, false()) -> c_13()
  , and^#(true(), true()) -> c_14()
  , and^#(false(), x) -> c_15() }

and mark the set of starting terms.

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

Strict DPs:
  { cond1^#(true(), x, y) -> c_1(cond2^#(gr(x, y), x, y), gr^#(x, y))
  , cond2^#(true(), x, y) ->
    c_2(cond3^#(gr(x, 0()), x, y), gr^#(x, 0()))
  , cond2^#(false(), x, y) ->
    c_3(cond4^#(gr(y, 0()), x, y), gr^#(y, 0()))
  , gr^#(0(), x) -> c_4()
  , gr^#(s(x), 0()) -> c_5()
  , gr^#(s(x), s(y)) -> c_6(gr^#(x, y))
  , cond3^#(true(), x, y) ->
    c_7(cond3^#(gr(x, 0()), p(x), y), gr^#(x, 0()), p^#(x))
  , cond3^#(false(), x, y) ->
    c_8(cond1^#(and(gr(x, 0()), gr(y, 0())), x, y),
        and^#(gr(x, 0()), gr(y, 0())),
        gr^#(x, 0()),
        gr^#(y, 0()))
  , cond4^#(true(), x, y) ->
    c_9(cond4^#(gr(y, 0()), x, p(y)), gr^#(y, 0()), p^#(y))
  , cond4^#(false(), x, y) ->
    c_10(cond1^#(and(gr(x, 0()), gr(y, 0())), x, y),
         and^#(gr(x, 0()), gr(y, 0())),
         gr^#(x, 0()),
         gr^#(y, 0()))
  , p^#(0()) -> c_11()
  , p^#(s(x)) -> c_12()
  , and^#(x, false()) -> c_13()
  , and^#(true(), true()) -> c_14()
  , and^#(false(), x) -> c_15() }
Weak Trs:
  { cond1(true(), x, y) -> cond2(gr(x, y), x, y)
  , cond2(true(), x, y) -> cond3(gr(x, 0()), x, y)
  , cond2(false(), x, y) -> cond4(gr(y, 0()), x, y)
  , gr(0(), x) -> false()
  , gr(s(x), 0()) -> true()
  , gr(s(x), s(y)) -> gr(x, y)
  , cond3(true(), x, y) -> cond3(gr(x, 0()), p(x), y)
  , cond3(false(), x, y) -> cond1(and(gr(x, 0()), gr(y, 0())), x, y)
  , cond4(true(), x, y) -> cond4(gr(y, 0()), x, p(y))
  , cond4(false(), x, y) -> cond1(and(gr(x, 0()), gr(y, 0())), x, y)
  , p(0()) -> 0()
  , p(s(x)) -> x
  , and(x, false()) -> false()
  , and(true(), true()) -> true()
  , and(false(), x) -> false() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {4,5,11,12,13,14,15} by
applications of Pre({4,5,11,12,13,14,15}) = {1,2,3,6,7,8,9,10}.
Here rules are labeled as follows:

  DPs:
    { 1: cond1^#(true(), x, y) ->
         c_1(cond2^#(gr(x, y), x, y), gr^#(x, y))
    , 2: cond2^#(true(), x, y) ->
         c_2(cond3^#(gr(x, 0()), x, y), gr^#(x, 0()))
    , 3: cond2^#(false(), x, y) ->
         c_3(cond4^#(gr(y, 0()), x, y), gr^#(y, 0()))
    , 4: gr^#(0(), x) -> c_4()
    , 5: gr^#(s(x), 0()) -> c_5()
    , 6: gr^#(s(x), s(y)) -> c_6(gr^#(x, y))
    , 7: cond3^#(true(), x, y) ->
         c_7(cond3^#(gr(x, 0()), p(x), y), gr^#(x, 0()), p^#(x))
    , 8: cond3^#(false(), x, y) ->
         c_8(cond1^#(and(gr(x, 0()), gr(y, 0())), x, y),
             and^#(gr(x, 0()), gr(y, 0())),
             gr^#(x, 0()),
             gr^#(y, 0()))
    , 9: cond4^#(true(), x, y) ->
         c_9(cond4^#(gr(y, 0()), x, p(y)), gr^#(y, 0()), p^#(y))
    , 10: cond4^#(false(), x, y) ->
          c_10(cond1^#(and(gr(x, 0()), gr(y, 0())), x, y),
               and^#(gr(x, 0()), gr(y, 0())),
               gr^#(x, 0()),
               gr^#(y, 0()))
    , 11: p^#(0()) -> c_11()
    , 12: p^#(s(x)) -> c_12()
    , 13: and^#(x, false()) -> c_13()
    , 14: and^#(true(), true()) -> c_14()
    , 15: and^#(false(), x) -> c_15() }

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

Strict DPs:
  { cond1^#(true(), x, y) -> c_1(cond2^#(gr(x, y), x, y), gr^#(x, y))
  , cond2^#(true(), x, y) ->
    c_2(cond3^#(gr(x, 0()), x, y), gr^#(x, 0()))
  , cond2^#(false(), x, y) ->
    c_3(cond4^#(gr(y, 0()), x, y), gr^#(y, 0()))
  , gr^#(s(x), s(y)) -> c_6(gr^#(x, y))
  , cond3^#(true(), x, y) ->
    c_7(cond3^#(gr(x, 0()), p(x), y), gr^#(x, 0()), p^#(x))
  , cond3^#(false(), x, y) ->
    c_8(cond1^#(and(gr(x, 0()), gr(y, 0())), x, y),
        and^#(gr(x, 0()), gr(y, 0())),
        gr^#(x, 0()),
        gr^#(y, 0()))
  , cond4^#(true(), x, y) ->
    c_9(cond4^#(gr(y, 0()), x, p(y)), gr^#(y, 0()), p^#(y))
  , cond4^#(false(), x, y) ->
    c_10(cond1^#(and(gr(x, 0()), gr(y, 0())), x, y),
         and^#(gr(x, 0()), gr(y, 0())),
         gr^#(x, 0()),
         gr^#(y, 0())) }
Weak DPs:
  { gr^#(0(), x) -> c_4()
  , gr^#(s(x), 0()) -> c_5()
  , p^#(0()) -> c_11()
  , p^#(s(x)) -> c_12()
  , and^#(x, false()) -> c_13()
  , and^#(true(), true()) -> c_14()
  , and^#(false(), x) -> c_15() }
Weak Trs:
  { cond1(true(), x, y) -> cond2(gr(x, y), x, y)
  , cond2(true(), x, y) -> cond3(gr(x, 0()), x, y)
  , cond2(false(), x, y) -> cond4(gr(y, 0()), x, y)
  , gr(0(), x) -> false()
  , gr(s(x), 0()) -> true()
  , gr(s(x), s(y)) -> gr(x, y)
  , cond3(true(), x, y) -> cond3(gr(x, 0()), p(x), y)
  , cond3(false(), x, y) -> cond1(and(gr(x, 0()), gr(y, 0())), x, y)
  , cond4(true(), x, y) -> cond4(gr(y, 0()), x, p(y))
  , cond4(false(), x, y) -> cond1(and(gr(x, 0()), gr(y, 0())), x, y)
  , p(0()) -> 0()
  , p(s(x)) -> x
  , and(x, false()) -> false()
  , and(true(), true()) -> true()
  , and(false(), x) -> false() }
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.

{ gr^#(0(), x) -> c_4()
, gr^#(s(x), 0()) -> c_5()
, p^#(0()) -> c_11()
, p^#(s(x)) -> c_12()
, and^#(x, false()) -> c_13()
, and^#(true(), true()) -> c_14()
, and^#(false(), x) -> c_15() }

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

Strict DPs:
  { cond1^#(true(), x, y) -> c_1(cond2^#(gr(x, y), x, y), gr^#(x, y))
  , cond2^#(true(), x, y) ->
    c_2(cond3^#(gr(x, 0()), x, y), gr^#(x, 0()))
  , cond2^#(false(), x, y) ->
    c_3(cond4^#(gr(y, 0()), x, y), gr^#(y, 0()))
  , gr^#(s(x), s(y)) -> c_6(gr^#(x, y))
  , cond3^#(true(), x, y) ->
    c_7(cond3^#(gr(x, 0()), p(x), y), gr^#(x, 0()), p^#(x))
  , cond3^#(false(), x, y) ->
    c_8(cond1^#(and(gr(x, 0()), gr(y, 0())), x, y),
        and^#(gr(x, 0()), gr(y, 0())),
        gr^#(x, 0()),
        gr^#(y, 0()))
  , cond4^#(true(), x, y) ->
    c_9(cond4^#(gr(y, 0()), x, p(y)), gr^#(y, 0()), p^#(y))
  , cond4^#(false(), x, y) ->
    c_10(cond1^#(and(gr(x, 0()), gr(y, 0())), x, y),
         and^#(gr(x, 0()), gr(y, 0())),
         gr^#(x, 0()),
         gr^#(y, 0())) }
Weak Trs:
  { cond1(true(), x, y) -> cond2(gr(x, y), x, y)
  , cond2(true(), x, y) -> cond3(gr(x, 0()), x, y)
  , cond2(false(), x, y) -> cond4(gr(y, 0()), x, y)
  , gr(0(), x) -> false()
  , gr(s(x), 0()) -> true()
  , gr(s(x), s(y)) -> gr(x, y)
  , cond3(true(), x, y) -> cond3(gr(x, 0()), p(x), y)
  , cond3(false(), x, y) -> cond1(and(gr(x, 0()), gr(y, 0())), x, y)
  , cond4(true(), x, y) -> cond4(gr(y, 0()), x, p(y))
  , cond4(false(), x, y) -> cond1(and(gr(x, 0()), gr(y, 0())), x, y)
  , p(0()) -> 0()
  , p(s(x)) -> x
  , and(x, false()) -> false()
  , and(true(), true()) -> true()
  , and(false(), x) -> false() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  { cond2^#(true(), x, y) ->
    c_2(cond3^#(gr(x, 0()), x, y), gr^#(x, 0()))
  , cond2^#(false(), x, y) ->
    c_3(cond4^#(gr(y, 0()), x, y), gr^#(y, 0()))
  , cond3^#(true(), x, y) ->
    c_7(cond3^#(gr(x, 0()), p(x), y), gr^#(x, 0()), p^#(x))
  , cond3^#(false(), x, y) ->
    c_8(cond1^#(and(gr(x, 0()), gr(y, 0())), x, y),
        and^#(gr(x, 0()), gr(y, 0())),
        gr^#(x, 0()),
        gr^#(y, 0()))
  , cond4^#(true(), x, y) ->
    c_9(cond4^#(gr(y, 0()), x, p(y)), gr^#(y, 0()), p^#(y))
  , cond4^#(false(), x, y) ->
    c_10(cond1^#(and(gr(x, 0()), gr(y, 0())), x, y),
         and^#(gr(x, 0()), gr(y, 0())),
         gr^#(x, 0()),
         gr^#(y, 0())) }

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

Strict DPs:
  { cond1^#(true(), x, y) -> c_1(cond2^#(gr(x, y), x, y), gr^#(x, y))
  , cond2^#(true(), x, y) -> c_2(cond3^#(gr(x, 0()), x, y))
  , cond2^#(false(), x, y) -> c_3(cond4^#(gr(y, 0()), x, y))
  , gr^#(s(x), s(y)) -> c_4(gr^#(x, y))
  , cond3^#(true(), x, y) -> c_5(cond3^#(gr(x, 0()), p(x), y))
  , cond3^#(false(), x, y) ->
    c_6(cond1^#(and(gr(x, 0()), gr(y, 0())), x, y))
  , cond4^#(true(), x, y) -> c_7(cond4^#(gr(y, 0()), x, p(y)))
  , cond4^#(false(), x, y) ->
    c_8(cond1^#(and(gr(x, 0()), gr(y, 0())), x, y)) }
Weak Trs:
  { cond1(true(), x, y) -> cond2(gr(x, y), x, y)
  , cond2(true(), x, y) -> cond3(gr(x, 0()), x, y)
  , cond2(false(), x, y) -> cond4(gr(y, 0()), x, y)
  , gr(0(), x) -> false()
  , gr(s(x), 0()) -> true()
  , gr(s(x), s(y)) -> gr(x, y)
  , cond3(true(), x, y) -> cond3(gr(x, 0()), p(x), y)
  , cond3(false(), x, y) -> cond1(and(gr(x, 0()), gr(y, 0())), x, y)
  , cond4(true(), x, y) -> cond4(gr(y, 0()), x, p(y))
  , cond4(false(), x, y) -> cond1(and(gr(x, 0()), gr(y, 0())), x, y)
  , p(0()) -> 0()
  , p(s(x)) -> x
  , and(x, false()) -> false()
  , and(true(), true()) -> true()
  , and(false(), x) -> false() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { gr(0(), x) -> false()
    , gr(s(x), 0()) -> true()
    , gr(s(x), s(y)) -> gr(x, y)
    , p(0()) -> 0()
    , p(s(x)) -> x
    , and(x, false()) -> false()
    , and(true(), true()) -> true()
    , and(false(), x) -> false() }

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

Strict DPs:
  { cond1^#(true(), x, y) -> c_1(cond2^#(gr(x, y), x, y), gr^#(x, y))
  , cond2^#(true(), x, y) -> c_2(cond3^#(gr(x, 0()), x, y))
  , cond2^#(false(), x, y) -> c_3(cond4^#(gr(y, 0()), x, y))
  , gr^#(s(x), s(y)) -> c_4(gr^#(x, y))
  , cond3^#(true(), x, y) -> c_5(cond3^#(gr(x, 0()), p(x), y))
  , cond3^#(false(), x, y) ->
    c_6(cond1^#(and(gr(x, 0()), gr(y, 0())), x, y))
  , cond4^#(true(), x, y) -> c_7(cond4^#(gr(y, 0()), x, p(y)))
  , cond4^#(false(), x, y) ->
    c_8(cond1^#(and(gr(x, 0()), gr(y, 0())), x, y)) }
Weak Trs:
  { gr(0(), x) -> false()
  , gr(s(x), 0()) -> true()
  , gr(s(x), s(y)) -> gr(x, y)
  , p(0()) -> 0()
  , p(s(x)) -> x
  , and(x, false()) -> false()
  , and(true(), true()) -> true()
  , and(false(), x) -> false() }
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..