MAYBE

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

Strict Trs:
  { minus(x, 0()) -> x
  , minus(s(x), s(y)) -> minus(x, y)
  , quot(0(), s(y)) -> 0()
  , quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
  , plus(minus(x, s(0())), minus(y, s(s(z)))) ->
    plus(minus(y, s(s(z))), minus(x, s(0())))
  , plus(0(), y) -> y
  , plus(s(x), y) -> s(plus(x, y))
  , plus(plus(x, s(0())), plus(y, s(s(z)))) ->
    plus(plus(y, s(s(z))), plus(x, s(0()))) }
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:
     { minus^#(x, 0()) -> c_1(x)
     , minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
     , quot^#(0(), s(y)) -> c_3()
     , quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y)))
     , plus^#(minus(x, s(0())), minus(y, s(s(z)))) ->
       c_5(plus^#(minus(y, s(s(z))), minus(x, s(0()))))
     , plus^#(0(), y) -> c_6(y)
     , plus^#(s(x), y) -> c_7(plus^#(x, y))
     , plus^#(plus(x, s(0())), plus(y, s(s(z)))) ->
       c_8(plus^#(plus(y, s(s(z))), plus(x, s(0())))) }
   
   and mark the set of starting terms.
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { minus^#(x, 0()) -> c_1(x)
     , minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
     , quot^#(0(), s(y)) -> c_3()
     , quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y)))
     , plus^#(minus(x, s(0())), minus(y, s(s(z)))) ->
       c_5(plus^#(minus(y, s(s(z))), minus(x, s(0()))))
     , plus^#(0(), y) -> c_6(y)
     , plus^#(s(x), y) -> c_7(plus^#(x, y))
     , plus^#(plus(x, s(0())), plus(y, s(s(z)))) ->
       c_8(plus^#(plus(y, s(s(z))), plus(x, s(0())))) }
   Strict Trs:
     { minus(x, 0()) -> x
     , minus(s(x), s(y)) -> minus(x, y)
     , quot(0(), s(y)) -> 0()
     , quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
     , plus(minus(x, s(0())), minus(y, s(s(z)))) ->
       plus(minus(y, s(s(z))), minus(x, s(0())))
     , plus(0(), y) -> y
     , plus(s(x), y) -> s(plus(x, y))
     , plus(plus(x, s(0())), plus(y, s(s(z)))) ->
       plus(plus(y, s(s(z))), plus(x, s(0()))) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {3} by applications of
   Pre({3}) = {1,4,6}. Here rules are labeled as follows:
   
     DPs:
       { 1: minus^#(x, 0()) -> c_1(x)
       , 2: minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
       , 3: quot^#(0(), s(y)) -> c_3()
       , 4: quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y)))
       , 5: plus^#(minus(x, s(0())), minus(y, s(s(z)))) ->
            c_5(plus^#(minus(y, s(s(z))), minus(x, s(0()))))
       , 6: plus^#(0(), y) -> c_6(y)
       , 7: plus^#(s(x), y) -> c_7(plus^#(x, y))
       , 8: plus^#(plus(x, s(0())), plus(y, s(s(z)))) ->
            c_8(plus^#(plus(y, s(s(z))), plus(x, s(0())))) }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { minus^#(x, 0()) -> c_1(x)
     , minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
     , quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y)))
     , plus^#(minus(x, s(0())), minus(y, s(s(z)))) ->
       c_5(plus^#(minus(y, s(s(z))), minus(x, s(0()))))
     , plus^#(0(), y) -> c_6(y)
     , plus^#(s(x), y) -> c_7(plus^#(x, y))
     , plus^#(plus(x, s(0())), plus(y, s(s(z)))) ->
       c_8(plus^#(plus(y, s(s(z))), plus(x, s(0())))) }
   Strict Trs:
     { minus(x, 0()) -> x
     , minus(s(x), s(y)) -> minus(x, y)
     , quot(0(), s(y)) -> 0()
     , quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
     , plus(minus(x, s(0())), minus(y, s(s(z)))) ->
       plus(minus(y, s(s(z))), minus(x, s(0())))
     , plus(0(), y) -> y
     , plus(s(x), y) -> s(plus(x, y))
     , plus(plus(x, s(0())), plus(y, s(s(z)))) ->
       plus(plus(y, s(s(z))), plus(x, s(0()))) }
   Weak DPs: { quot^#(0(), s(y)) -> c_3() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Empty strict component of the problem is NOT empty.


Arrrr..