MAYBE

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

Strict Trs:
  { minus(x, x) -> 0()
  , minus(x, 0()) -> x
  , minus(0(), x) -> 0()
  , minus(s(x), s(y)) -> minus(x, y)
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , quot(x, y) -> if_quot(minus(x, y), y, le(y, 0()), le(y, x))
  , if_quot(x, y, true(), z) -> divByZeroError()
  , if_quot(x, y, false(), true()) -> s(quot(x, y))
  , if_quot(x, y, false(), false()) -> 0() }
Obligation:
  runtime complexity
Answer:
  MAYBE

None of the processors succeeded.

Details of failed attempt(s):
-----------------------------
1) '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
      
   

2) '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, x) -> c_1()
     , minus^#(x, 0()) -> c_2(x)
     , minus^#(0(), x) -> c_3()
     , minus^#(s(x), s(y)) -> c_4(minus^#(x, y))
     , le^#(0(), y) -> c_5()
     , le^#(s(x), 0()) -> c_6()
     , le^#(s(x), s(y)) -> c_7(le^#(x, y))
     , quot^#(x, y) ->
       c_8(if_quot^#(minus(x, y), y, le(y, 0()), le(y, x)))
     , if_quot^#(x, y, true(), z) -> c_9()
     , if_quot^#(x, y, false(), true()) -> c_10(quot^#(x, y))
     , if_quot^#(x, y, false(), false()) -> c_11() }
   
   and mark the set of starting terms.
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { minus^#(x, x) -> c_1()
     , minus^#(x, 0()) -> c_2(x)
     , minus^#(0(), x) -> c_3()
     , minus^#(s(x), s(y)) -> c_4(minus^#(x, y))
     , le^#(0(), y) -> c_5()
     , le^#(s(x), 0()) -> c_6()
     , le^#(s(x), s(y)) -> c_7(le^#(x, y))
     , quot^#(x, y) ->
       c_8(if_quot^#(minus(x, y), y, le(y, 0()), le(y, x)))
     , if_quot^#(x, y, true(), z) -> c_9()
     , if_quot^#(x, y, false(), true()) -> c_10(quot^#(x, y))
     , if_quot^#(x, y, false(), false()) -> c_11() }
   Strict Trs:
     { minus(x, x) -> 0()
     , minus(x, 0()) -> x
     , minus(0(), x) -> 0()
     , minus(s(x), s(y)) -> minus(x, y)
     , le(0(), y) -> true()
     , le(s(x), 0()) -> false()
     , le(s(x), s(y)) -> le(x, y)
     , quot(x, y) -> if_quot(minus(x, y), y, le(y, 0()), le(y, x))
     , if_quot(x, y, true(), z) -> divByZeroError()
     , if_quot(x, y, false(), true()) -> s(quot(x, y))
     , if_quot(x, y, false(), false()) -> 0() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {1,3,5,6,9,11} by
   applications of Pre({1,3,5,6,9,11}) = {2,4,7,8}. Here rules are
   labeled as follows:
   
     DPs:
       { 1: minus^#(x, x) -> c_1()
       , 2: minus^#(x, 0()) -> c_2(x)
       , 3: minus^#(0(), x) -> c_3()
       , 4: minus^#(s(x), s(y)) -> c_4(minus^#(x, y))
       , 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: quot^#(x, y) ->
            c_8(if_quot^#(minus(x, y), y, le(y, 0()), le(y, x)))
       , 9: if_quot^#(x, y, true(), z) -> c_9()
       , 10: if_quot^#(x, y, false(), true()) -> c_10(quot^#(x, y))
       , 11: if_quot^#(x, y, false(), false()) -> c_11() }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { minus^#(x, 0()) -> c_2(x)
     , minus^#(s(x), s(y)) -> c_4(minus^#(x, y))
     , le^#(s(x), s(y)) -> c_7(le^#(x, y))
     , quot^#(x, y) ->
       c_8(if_quot^#(minus(x, y), y, le(y, 0()), le(y, x)))
     , if_quot^#(x, y, false(), true()) -> c_10(quot^#(x, y)) }
   Strict Trs:
     { minus(x, x) -> 0()
     , minus(x, 0()) -> x
     , minus(0(), x) -> 0()
     , minus(s(x), s(y)) -> minus(x, y)
     , le(0(), y) -> true()
     , le(s(x), 0()) -> false()
     , le(s(x), s(y)) -> le(x, y)
     , quot(x, y) -> if_quot(minus(x, y), y, le(y, 0()), le(y, x))
     , if_quot(x, y, true(), z) -> divByZeroError()
     , if_quot(x, y, false(), true()) -> s(quot(x, y))
     , if_quot(x, y, false(), false()) -> 0() }
   Weak DPs:
     { minus^#(x, x) -> c_1()
     , minus^#(0(), x) -> c_3()
     , le^#(0(), y) -> c_5()
     , le^#(s(x), 0()) -> c_6()
     , if_quot^#(x, y, true(), z) -> c_9()
     , if_quot^#(x, y, false(), false()) -> c_11() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Empty strict component of the problem is NOT empty.


Arrrr..