MAYBE

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

Strict Trs:
  { numbers() -> d(0())
  , d(x) -> if(le(x, nr()), x)
  , if(true(), x) -> cons(x, d(s(x)))
  , if(false(), x) -> nil()
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , nr() -> ack(s(s(s(s(s(s(0())))))), 0())
  , ack(0(), x) -> s(x)
  , ack(s(x), 0()) -> ack(x, s(0()))
  , ack(s(x), s(y)) -> ack(x, ack(s(x), y)) }
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) '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) '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) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed
   due to the following reason:
   
   We add the following weak dependency pairs:
   
   Strict DPs:
     { numbers^#() -> c_1(d^#(0()))
     , d^#(x) -> c_2(if^#(le(x, nr()), x))
     , if^#(true(), x) -> c_3(x, d^#(s(x)))
     , if^#(false(), x) -> c_4()
     , le^#(0(), y) -> c_5()
     , le^#(s(x), 0()) -> c_6()
     , le^#(s(x), s(y)) -> c_7(le^#(x, y))
     , nr^#() -> c_8(ack^#(s(s(s(s(s(s(0())))))), 0()))
     , ack^#(0(), x) -> c_9(x)
     , ack^#(s(x), 0()) -> c_10(ack^#(x, s(0())))
     , ack^#(s(x), s(y)) -> c_11(ack^#(x, ack(s(x), y))) }
   
   and mark the set of starting terms.
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { numbers^#() -> c_1(d^#(0()))
     , d^#(x) -> c_2(if^#(le(x, nr()), x))
     , if^#(true(), x) -> c_3(x, d^#(s(x)))
     , if^#(false(), x) -> c_4()
     , le^#(0(), y) -> c_5()
     , le^#(s(x), 0()) -> c_6()
     , le^#(s(x), s(y)) -> c_7(le^#(x, y))
     , nr^#() -> c_8(ack^#(s(s(s(s(s(s(0())))))), 0()))
     , ack^#(0(), x) -> c_9(x)
     , ack^#(s(x), 0()) -> c_10(ack^#(x, s(0())))
     , ack^#(s(x), s(y)) -> c_11(ack^#(x, ack(s(x), y))) }
   Strict Trs:
     { numbers() -> d(0())
     , d(x) -> if(le(x, nr()), x)
     , if(true(), x) -> cons(x, d(s(x)))
     , if(false(), x) -> nil()
     , le(0(), y) -> true()
     , le(s(x), 0()) -> false()
     , le(s(x), s(y)) -> le(x, y)
     , nr() -> ack(s(s(s(s(s(s(0())))))), 0())
     , ack(0(), x) -> s(x)
     , ack(s(x), 0()) -> ack(x, s(0()))
     , ack(s(x), s(y)) -> ack(x, ack(s(x), y)) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {4,5,6} by applications of
   Pre({4,5,6}) = {2,3,7,9}. Here rules are labeled as follows:
   
     DPs:
       { 1: numbers^#() -> c_1(d^#(0()))
       , 2: d^#(x) -> c_2(if^#(le(x, nr()), x))
       , 3: if^#(true(), x) -> c_3(x, d^#(s(x)))
       , 4: if^#(false(), x) -> c_4()
       , 5: le^#(0(), y) -> c_5()
       , 6: le^#(s(x), 0()) -> c_6()
       , 7: le^#(s(x), s(y)) -> c_7(le^#(x, y))
       , 8: nr^#() -> c_8(ack^#(s(s(s(s(s(s(0())))))), 0()))
       , 9: ack^#(0(), x) -> c_9(x)
       , 10: ack^#(s(x), 0()) -> c_10(ack^#(x, s(0())))
       , 11: ack^#(s(x), s(y)) -> c_11(ack^#(x, ack(s(x), y))) }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { numbers^#() -> c_1(d^#(0()))
     , d^#(x) -> c_2(if^#(le(x, nr()), x))
     , if^#(true(), x) -> c_3(x, d^#(s(x)))
     , le^#(s(x), s(y)) -> c_7(le^#(x, y))
     , nr^#() -> c_8(ack^#(s(s(s(s(s(s(0())))))), 0()))
     , ack^#(0(), x) -> c_9(x)
     , ack^#(s(x), 0()) -> c_10(ack^#(x, s(0())))
     , ack^#(s(x), s(y)) -> c_11(ack^#(x, ack(s(x), y))) }
   Strict Trs:
     { numbers() -> d(0())
     , d(x) -> if(le(x, nr()), x)
     , if(true(), x) -> cons(x, d(s(x)))
     , if(false(), x) -> nil()
     , le(0(), y) -> true()
     , le(s(x), 0()) -> false()
     , le(s(x), s(y)) -> le(x, y)
     , nr() -> ack(s(s(s(s(s(s(0())))))), 0())
     , ack(0(), x) -> s(x)
     , ack(s(x), 0()) -> ack(x, s(0()))
     , ack(s(x), s(y)) -> ack(x, ack(s(x), y)) }
   Weak DPs:
     { if^#(false(), x) -> c_4()
     , le^#(0(), y) -> c_5()
     , le^#(s(x), 0()) -> c_6() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Empty strict component of the problem is NOT empty.


Arrrr..