MAYBE

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

Strict Trs:
  { isEmpty(empty()) -> true()
  , isEmpty(node(l, x, r)) -> false()
  , left(empty()) -> empty()
  , left(node(l, x, r)) -> l
  , right(empty()) -> empty()
  , right(node(l, x, r)) -> r
  , elem(node(l, x, r)) -> x
  , append(nil(), x) -> cons(x, nil())
  , append(cons(y(), ys), x) -> cons(y(), append(ys, x))
  , listify(n, xs) ->
    if(isEmpty(n),
       isEmpty(left(n)),
       right(n),
       node(left(left(n)),
            elem(left(n)),
            node(right(left(n)), elem(n), right(n))),
       xs,
       append(xs, n))
  , if(true(), b, n, m, xs, ys) -> xs
  , if(false(), true(), n, m, xs, ys) -> listify(n, ys)
  , if(false(), false(), n, m, xs, ys) -> listify(m, xs)
  , toList(n) -> listify(n, nil()) }
Obligation:
  runtime complexity
Answer:
  MAYBE

None of the processors succeeded.

Details of failed attempt(s):
-----------------------------
1) 'WithProblem (timeout of 60 seconds)' failed due to the
   following reason:
   
   Computation stopped due to timeout after 60.0 seconds.

2) 'Best' failed due to the following reason:
   
   None of the processors succeeded.
   
   Details of failed attempt(s):
   -----------------------------
   1) 'WithProblem (timeout of 30 seconds) (timeout of 60 seconds)'
      failed due to the following reason:
      
      Computation stopped due to timeout after 30.0 seconds.
   
   2) 'Best' failed due to the following reason:
      
      None of the processors succeeded.
      
      Details of failed attempt(s):
      -----------------------------
      1) 'Polynomial Path Order (PS) (timeout of 60 seconds)' failed due
         to the following reason:
         
         The processor is inapplicable, reason:
           Processor only applicable for innermost runtime complexity analysis
      
      2) 'bsearch-popstar (timeout of 60 seconds)' failed due to the
         following reason:
         
         The processor is inapplicable, reason:
           Processor only applicable for innermost runtime complexity analysis
      
   
   3) 'Fastest (timeout of 5 seconds) (timeout of 60 seconds)' failed
      due to the following reason:
      
      None of the processors succeeded.
      
      Details of failed attempt(s):
      -----------------------------
      1) 'Bounds with minimal-enrichment and initial automaton 'match''
         failed due to the following reason:
         
         match-boundness of the problem could not be verified.
      
      2) 'Bounds with perSymbol-enrichment and initial automaton 'match''
         failed due to the following reason:
         
         match-boundness of the problem could not be verified.
      
   

3) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed
   due to the following reason:
   
   We add the following weak dependency pairs:
   
   Strict DPs:
     { isEmpty^#(empty()) -> c_1()
     , isEmpty^#(node(l, x, r)) -> c_2()
     , left^#(empty()) -> c_3()
     , left^#(node(l, x, r)) -> c_4(l)
     , right^#(empty()) -> c_5()
     , right^#(node(l, x, r)) -> c_6(r)
     , elem^#(node(l, x, r)) -> c_7(x)
     , append^#(nil(), x) -> c_8(x)
     , append^#(cons(y(), ys), x) -> c_9(append^#(ys, x))
     , listify^#(n, xs) ->
       c_10(if^#(isEmpty(n),
                 isEmpty(left(n)),
                 right(n),
                 node(left(left(n)),
                      elem(left(n)),
                      node(right(left(n)), elem(n), right(n))),
                 xs,
                 append(xs, n)))
     , if^#(true(), b, n, m, xs, ys) -> c_11(xs)
     , if^#(false(), true(), n, m, xs, ys) -> c_12(listify^#(n, ys))
     , if^#(false(), false(), n, m, xs, ys) -> c_13(listify^#(m, xs))
     , toList^#(n) -> c_14(listify^#(n, nil())) }
   
   and mark the set of starting terms.
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { isEmpty^#(empty()) -> c_1()
     , isEmpty^#(node(l, x, r)) -> c_2()
     , left^#(empty()) -> c_3()
     , left^#(node(l, x, r)) -> c_4(l)
     , right^#(empty()) -> c_5()
     , right^#(node(l, x, r)) -> c_6(r)
     , elem^#(node(l, x, r)) -> c_7(x)
     , append^#(nil(), x) -> c_8(x)
     , append^#(cons(y(), ys), x) -> c_9(append^#(ys, x))
     , listify^#(n, xs) ->
       c_10(if^#(isEmpty(n),
                 isEmpty(left(n)),
                 right(n),
                 node(left(left(n)),
                      elem(left(n)),
                      node(right(left(n)), elem(n), right(n))),
                 xs,
                 append(xs, n)))
     , if^#(true(), b, n, m, xs, ys) -> c_11(xs)
     , if^#(false(), true(), n, m, xs, ys) -> c_12(listify^#(n, ys))
     , if^#(false(), false(), n, m, xs, ys) -> c_13(listify^#(m, xs))
     , toList^#(n) -> c_14(listify^#(n, nil())) }
   Strict Trs:
     { isEmpty(empty()) -> true()
     , isEmpty(node(l, x, r)) -> false()
     , left(empty()) -> empty()
     , left(node(l, x, r)) -> l
     , right(empty()) -> empty()
     , right(node(l, x, r)) -> r
     , elem(node(l, x, r)) -> x
     , append(nil(), x) -> cons(x, nil())
     , append(cons(y(), ys), x) -> cons(y(), append(ys, x))
     , listify(n, xs) ->
       if(isEmpty(n),
          isEmpty(left(n)),
          right(n),
          node(left(left(n)),
               elem(left(n)),
               node(right(left(n)), elem(n), right(n))),
          xs,
          append(xs, n))
     , if(true(), b, n, m, xs, ys) -> xs
     , if(false(), true(), n, m, xs, ys) -> listify(n, ys)
     , if(false(), false(), n, m, xs, ys) -> listify(m, xs)
     , toList(n) -> listify(n, nil()) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {1,2,3,5} by applications
   of Pre({1,2,3,5}) = {4,6,7,8,11}. Here rules are labeled as
   follows:
   
     DPs:
       { 1: isEmpty^#(empty()) -> c_1()
       , 2: isEmpty^#(node(l, x, r)) -> c_2()
       , 3: left^#(empty()) -> c_3()
       , 4: left^#(node(l, x, r)) -> c_4(l)
       , 5: right^#(empty()) -> c_5()
       , 6: right^#(node(l, x, r)) -> c_6(r)
       , 7: elem^#(node(l, x, r)) -> c_7(x)
       , 8: append^#(nil(), x) -> c_8(x)
       , 9: append^#(cons(y(), ys), x) -> c_9(append^#(ys, x))
       , 10: listify^#(n, xs) ->
             c_10(if^#(isEmpty(n),
                       isEmpty(left(n)),
                       right(n),
                       node(left(left(n)),
                            elem(left(n)),
                            node(right(left(n)), elem(n), right(n))),
                       xs,
                       append(xs, n)))
       , 11: if^#(true(), b, n, m, xs, ys) -> c_11(xs)
       , 12: if^#(false(), true(), n, m, xs, ys) -> c_12(listify^#(n, ys))
       , 13: if^#(false(), false(), n, m, xs, ys) ->
             c_13(listify^#(m, xs))
       , 14: toList^#(n) -> c_14(listify^#(n, nil())) }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { left^#(node(l, x, r)) -> c_4(l)
     , right^#(node(l, x, r)) -> c_6(r)
     , elem^#(node(l, x, r)) -> c_7(x)
     , append^#(nil(), x) -> c_8(x)
     , append^#(cons(y(), ys), x) -> c_9(append^#(ys, x))
     , listify^#(n, xs) ->
       c_10(if^#(isEmpty(n),
                 isEmpty(left(n)),
                 right(n),
                 node(left(left(n)),
                      elem(left(n)),
                      node(right(left(n)), elem(n), right(n))),
                 xs,
                 append(xs, n)))
     , if^#(true(), b, n, m, xs, ys) -> c_11(xs)
     , if^#(false(), true(), n, m, xs, ys) -> c_12(listify^#(n, ys))
     , if^#(false(), false(), n, m, xs, ys) -> c_13(listify^#(m, xs))
     , toList^#(n) -> c_14(listify^#(n, nil())) }
   Strict Trs:
     { isEmpty(empty()) -> true()
     , isEmpty(node(l, x, r)) -> false()
     , left(empty()) -> empty()
     , left(node(l, x, r)) -> l
     , right(empty()) -> empty()
     , right(node(l, x, r)) -> r
     , elem(node(l, x, r)) -> x
     , append(nil(), x) -> cons(x, nil())
     , append(cons(y(), ys), x) -> cons(y(), append(ys, x))
     , listify(n, xs) ->
       if(isEmpty(n),
          isEmpty(left(n)),
          right(n),
          node(left(left(n)),
               elem(left(n)),
               node(right(left(n)), elem(n), right(n))),
          xs,
          append(xs, n))
     , if(true(), b, n, m, xs, ys) -> xs
     , if(false(), true(), n, m, xs, ys) -> listify(n, ys)
     , if(false(), false(), n, m, xs, ys) -> listify(m, xs)
     , toList(n) -> listify(n, nil()) }
   Weak DPs:
     { isEmpty^#(empty()) -> c_1()
     , isEmpty^#(node(l, x, r)) -> c_2()
     , left^#(empty()) -> c_3()
     , right^#(empty()) -> c_5() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Empty strict component of the problem is NOT empty.


Arrrr..