MAYBE

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

Strict Trs:
  { sub(0(), 0()) -> 0()
  , sub(0(), s(x)) -> 0()
  , sub(s(x), 0()) -> s(x)
  , sub(s(x), s(y)) -> sub(x, y)
  , zero(nil()) -> zero2(0(), nil())
  , zero(cons(x, xs)) -> zero2(sub(x, x), cons(x, xs))
  , zero2(0(), nil()) -> nil()
  , zero2(0(), cons(x, xs)) -> cons(sub(x, x), zero(xs))
  , zero2(s(y), nil()) -> zero(nil())
  , zero2(s(y), cons(x, xs)) -> zero(cons(x, xs)) }
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 perSymbol-enrichment and initial automaton 'match''
         failed due to the following reason:
         
         match-boundness of the problem could not be verified.
      
      2) 'Bounds with minimal-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:
     { sub^#(0(), 0()) -> c_1()
     , sub^#(0(), s(x)) -> c_2()
     , sub^#(s(x), 0()) -> c_3(x)
     , sub^#(s(x), s(y)) -> c_4(sub^#(x, y))
     , zero^#(nil()) -> c_5(zero2^#(0(), nil()))
     , zero^#(cons(x, xs)) -> c_6(zero2^#(sub(x, x), cons(x, xs)))
     , zero2^#(0(), nil()) -> c_7()
     , zero2^#(0(), cons(x, xs)) -> c_8(sub^#(x, x), zero^#(xs))
     , zero2^#(s(y), nil()) -> c_9(zero^#(nil()))
     , zero2^#(s(y), cons(x, xs)) -> c_10(zero^#(cons(x, xs))) }
   
   and mark the set of starting terms.
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { sub^#(0(), 0()) -> c_1()
     , sub^#(0(), s(x)) -> c_2()
     , sub^#(s(x), 0()) -> c_3(x)
     , sub^#(s(x), s(y)) -> c_4(sub^#(x, y))
     , zero^#(nil()) -> c_5(zero2^#(0(), nil()))
     , zero^#(cons(x, xs)) -> c_6(zero2^#(sub(x, x), cons(x, xs)))
     , zero2^#(0(), nil()) -> c_7()
     , zero2^#(0(), cons(x, xs)) -> c_8(sub^#(x, x), zero^#(xs))
     , zero2^#(s(y), nil()) -> c_9(zero^#(nil()))
     , zero2^#(s(y), cons(x, xs)) -> c_10(zero^#(cons(x, xs))) }
   Strict Trs:
     { sub(0(), 0()) -> 0()
     , sub(0(), s(x)) -> 0()
     , sub(s(x), 0()) -> s(x)
     , sub(s(x), s(y)) -> sub(x, y)
     , zero(nil()) -> zero2(0(), nil())
     , zero(cons(x, xs)) -> zero2(sub(x, x), cons(x, xs))
     , zero2(0(), nil()) -> nil()
     , zero2(0(), cons(x, xs)) -> cons(sub(x, x), zero(xs))
     , zero2(s(y), nil()) -> zero(nil())
     , zero2(s(y), cons(x, xs)) -> zero(cons(x, xs)) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {1,2,7} by applications of
   Pre({1,2,7}) = {3,4,5,8}. Here rules are labeled as follows:
   
     DPs:
       { 1: sub^#(0(), 0()) -> c_1()
       , 2: sub^#(0(), s(x)) -> c_2()
       , 3: sub^#(s(x), 0()) -> c_3(x)
       , 4: sub^#(s(x), s(y)) -> c_4(sub^#(x, y))
       , 5: zero^#(nil()) -> c_5(zero2^#(0(), nil()))
       , 6: zero^#(cons(x, xs)) -> c_6(zero2^#(sub(x, x), cons(x, xs)))
       , 7: zero2^#(0(), nil()) -> c_7()
       , 8: zero2^#(0(), cons(x, xs)) -> c_8(sub^#(x, x), zero^#(xs))
       , 9: zero2^#(s(y), nil()) -> c_9(zero^#(nil()))
       , 10: zero2^#(s(y), cons(x, xs)) -> c_10(zero^#(cons(x, xs))) }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { sub^#(s(x), 0()) -> c_3(x)
     , sub^#(s(x), s(y)) -> c_4(sub^#(x, y))
     , zero^#(nil()) -> c_5(zero2^#(0(), nil()))
     , zero^#(cons(x, xs)) -> c_6(zero2^#(sub(x, x), cons(x, xs)))
     , zero2^#(0(), cons(x, xs)) -> c_8(sub^#(x, x), zero^#(xs))
     , zero2^#(s(y), nil()) -> c_9(zero^#(nil()))
     , zero2^#(s(y), cons(x, xs)) -> c_10(zero^#(cons(x, xs))) }
   Strict Trs:
     { sub(0(), 0()) -> 0()
     , sub(0(), s(x)) -> 0()
     , sub(s(x), 0()) -> s(x)
     , sub(s(x), s(y)) -> sub(x, y)
     , zero(nil()) -> zero2(0(), nil())
     , zero(cons(x, xs)) -> zero2(sub(x, x), cons(x, xs))
     , zero2(0(), nil()) -> nil()
     , zero2(0(), cons(x, xs)) -> cons(sub(x, x), zero(xs))
     , zero2(s(y), nil()) -> zero(nil())
     , zero2(s(y), cons(x, xs)) -> zero(cons(x, xs)) }
   Weak DPs:
     { sub^#(0(), 0()) -> c_1()
     , sub^#(0(), s(x)) -> c_2()
     , zero2^#(0(), nil()) -> c_7() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {3} by applications of
   Pre({3}) = {1,5,6}. Here rules are labeled as follows:
   
     DPs:
       { 1: sub^#(s(x), 0()) -> c_3(x)
       , 2: sub^#(s(x), s(y)) -> c_4(sub^#(x, y))
       , 3: zero^#(nil()) -> c_5(zero2^#(0(), nil()))
       , 4: zero^#(cons(x, xs)) -> c_6(zero2^#(sub(x, x), cons(x, xs)))
       , 5: zero2^#(0(), cons(x, xs)) -> c_8(sub^#(x, x), zero^#(xs))
       , 6: zero2^#(s(y), nil()) -> c_9(zero^#(nil()))
       , 7: zero2^#(s(y), cons(x, xs)) -> c_10(zero^#(cons(x, xs)))
       , 8: sub^#(0(), 0()) -> c_1()
       , 9: sub^#(0(), s(x)) -> c_2()
       , 10: zero2^#(0(), nil()) -> c_7() }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { sub^#(s(x), 0()) -> c_3(x)
     , sub^#(s(x), s(y)) -> c_4(sub^#(x, y))
     , zero^#(cons(x, xs)) -> c_6(zero2^#(sub(x, x), cons(x, xs)))
     , zero2^#(0(), cons(x, xs)) -> c_8(sub^#(x, x), zero^#(xs))
     , zero2^#(s(y), nil()) -> c_9(zero^#(nil()))
     , zero2^#(s(y), cons(x, xs)) -> c_10(zero^#(cons(x, xs))) }
   Strict Trs:
     { sub(0(), 0()) -> 0()
     , sub(0(), s(x)) -> 0()
     , sub(s(x), 0()) -> s(x)
     , sub(s(x), s(y)) -> sub(x, y)
     , zero(nil()) -> zero2(0(), nil())
     , zero(cons(x, xs)) -> zero2(sub(x, x), cons(x, xs))
     , zero2(0(), nil()) -> nil()
     , zero2(0(), cons(x, xs)) -> cons(sub(x, x), zero(xs))
     , zero2(s(y), nil()) -> zero(nil())
     , zero2(s(y), cons(x, xs)) -> zero(cons(x, xs)) }
   Weak DPs:
     { sub^#(0(), 0()) -> c_1()
     , sub^#(0(), s(x)) -> c_2()
     , zero^#(nil()) -> c_5(zero2^#(0(), nil()))
     , zero2^#(0(), nil()) -> c_7() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {5} by applications of
   Pre({5}) = {1}. Here rules are labeled as follows:
   
     DPs:
       { 1: sub^#(s(x), 0()) -> c_3(x)
       , 2: sub^#(s(x), s(y)) -> c_4(sub^#(x, y))
       , 3: zero^#(cons(x, xs)) -> c_6(zero2^#(sub(x, x), cons(x, xs)))
       , 4: zero2^#(0(), cons(x, xs)) -> c_8(sub^#(x, x), zero^#(xs))
       , 5: zero2^#(s(y), nil()) -> c_9(zero^#(nil()))
       , 6: zero2^#(s(y), cons(x, xs)) -> c_10(zero^#(cons(x, xs)))
       , 7: sub^#(0(), 0()) -> c_1()
       , 8: sub^#(0(), s(x)) -> c_2()
       , 9: zero^#(nil()) -> c_5(zero2^#(0(), nil()))
       , 10: zero2^#(0(), nil()) -> c_7() }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { sub^#(s(x), 0()) -> c_3(x)
     , sub^#(s(x), s(y)) -> c_4(sub^#(x, y))
     , zero^#(cons(x, xs)) -> c_6(zero2^#(sub(x, x), cons(x, xs)))
     , zero2^#(0(), cons(x, xs)) -> c_8(sub^#(x, x), zero^#(xs))
     , zero2^#(s(y), cons(x, xs)) -> c_10(zero^#(cons(x, xs))) }
   Strict Trs:
     { sub(0(), 0()) -> 0()
     , sub(0(), s(x)) -> 0()
     , sub(s(x), 0()) -> s(x)
     , sub(s(x), s(y)) -> sub(x, y)
     , zero(nil()) -> zero2(0(), nil())
     , zero(cons(x, xs)) -> zero2(sub(x, x), cons(x, xs))
     , zero2(0(), nil()) -> nil()
     , zero2(0(), cons(x, xs)) -> cons(sub(x, x), zero(xs))
     , zero2(s(y), nil()) -> zero(nil())
     , zero2(s(y), cons(x, xs)) -> zero(cons(x, xs)) }
   Weak DPs:
     { sub^#(0(), 0()) -> c_1()
     , sub^#(0(), s(x)) -> c_2()
     , zero^#(nil()) -> c_5(zero2^#(0(), nil()))
     , zero2^#(0(), nil()) -> c_7()
     , zero2^#(s(y), nil()) -> c_9(zero^#(nil())) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Empty strict component of the problem is NOT empty.


Arrrr..