MAYBE

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

Strict Trs:
  { minus(x, 0()) -> x
  , minus(0(), y) -> 0()
  , minus(s(x), s(y)) -> minus(x, y)
  , plus(0(), y) -> y
  , plus(s(x), y) -> plus(x, s(y))
  , zero(0()) -> true()
  , zero(s(x)) -> false()
  , p(s(x)) -> x
  , div(x, y) -> quot(x, y, 0())
  , quot(x, y, z) -> if(zero(x), x, y, plus(z, s(0())))
  , if(false(), x, s(y), z) -> quot(minus(x, s(y)), s(y), z)
  , if(true(), x, y, z) -> p(z) }
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) '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
      
      2) '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
      
   
   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:
     { minus^#(x, 0()) -> c_1(x)
     , minus^#(0(), y) -> c_2()
     , minus^#(s(x), s(y)) -> c_3(minus^#(x, y))
     , plus^#(0(), y) -> c_4(y)
     , plus^#(s(x), y) -> c_5(plus^#(x, s(y)))
     , zero^#(0()) -> c_6()
     , zero^#(s(x)) -> c_7()
     , p^#(s(x)) -> c_8(x)
     , div^#(x, y) -> c_9(quot^#(x, y, 0()))
     , quot^#(x, y, z) -> c_10(if^#(zero(x), x, y, plus(z, s(0()))))
     , if^#(false(), x, s(y), z) ->
       c_11(quot^#(minus(x, s(y)), s(y), z))
     , if^#(true(), x, y, z) -> c_12(p^#(z)) }
   
   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^#(0(), y) -> c_2()
     , minus^#(s(x), s(y)) -> c_3(minus^#(x, y))
     , plus^#(0(), y) -> c_4(y)
     , plus^#(s(x), y) -> c_5(plus^#(x, s(y)))
     , zero^#(0()) -> c_6()
     , zero^#(s(x)) -> c_7()
     , p^#(s(x)) -> c_8(x)
     , div^#(x, y) -> c_9(quot^#(x, y, 0()))
     , quot^#(x, y, z) -> c_10(if^#(zero(x), x, y, plus(z, s(0()))))
     , if^#(false(), x, s(y), z) ->
       c_11(quot^#(minus(x, s(y)), s(y), z))
     , if^#(true(), x, y, z) -> c_12(p^#(z)) }
   Strict Trs:
     { minus(x, 0()) -> x
     , minus(0(), y) -> 0()
     , minus(s(x), s(y)) -> minus(x, y)
     , plus(0(), y) -> y
     , plus(s(x), y) -> plus(x, s(y))
     , zero(0()) -> true()
     , zero(s(x)) -> false()
     , p(s(x)) -> x
     , div(x, y) -> quot(x, y, 0())
     , quot(x, y, z) -> if(zero(x), x, y, plus(z, s(0())))
     , if(false(), x, s(y), z) -> quot(minus(x, s(y)), s(y), z)
     , if(true(), x, y, z) -> p(z) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {2,6,7} by applications of
   Pre({2,6,7}) = {1,3,4,8}. Here rules are labeled as follows:
   
     DPs:
       { 1: minus^#(x, 0()) -> c_1(x)
       , 2: minus^#(0(), y) -> c_2()
       , 3: minus^#(s(x), s(y)) -> c_3(minus^#(x, y))
       , 4: plus^#(0(), y) -> c_4(y)
       , 5: plus^#(s(x), y) -> c_5(plus^#(x, s(y)))
       , 6: zero^#(0()) -> c_6()
       , 7: zero^#(s(x)) -> c_7()
       , 8: p^#(s(x)) -> c_8(x)
       , 9: div^#(x, y) -> c_9(quot^#(x, y, 0()))
       , 10: quot^#(x, y, z) -> c_10(if^#(zero(x), x, y, plus(z, s(0()))))
       , 11: if^#(false(), x, s(y), z) ->
             c_11(quot^#(minus(x, s(y)), s(y), z))
       , 12: if^#(true(), x, y, z) -> c_12(p^#(z)) }
   
   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_3(minus^#(x, y))
     , plus^#(0(), y) -> c_4(y)
     , plus^#(s(x), y) -> c_5(plus^#(x, s(y)))
     , p^#(s(x)) -> c_8(x)
     , div^#(x, y) -> c_9(quot^#(x, y, 0()))
     , quot^#(x, y, z) -> c_10(if^#(zero(x), x, y, plus(z, s(0()))))
     , if^#(false(), x, s(y), z) ->
       c_11(quot^#(minus(x, s(y)), s(y), z))
     , if^#(true(), x, y, z) -> c_12(p^#(z)) }
   Strict Trs:
     { minus(x, 0()) -> x
     , minus(0(), y) -> 0()
     , minus(s(x), s(y)) -> minus(x, y)
     , plus(0(), y) -> y
     , plus(s(x), y) -> plus(x, s(y))
     , zero(0()) -> true()
     , zero(s(x)) -> false()
     , p(s(x)) -> x
     , div(x, y) -> quot(x, y, 0())
     , quot(x, y, z) -> if(zero(x), x, y, plus(z, s(0())))
     , if(false(), x, s(y), z) -> quot(minus(x, s(y)), s(y), z)
     , if(true(), x, y, z) -> p(z) }
   Weak DPs:
     { minus^#(0(), y) -> c_2()
     , zero^#(0()) -> c_6()
     , zero^#(s(x)) -> c_7() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Empty strict component of the problem is NOT empty.


Arrrr..