MAYBE

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

Strict Trs:
  { plus(x, y) -> ifPlus(isZero(x), x, inc(y))
  , ifPlus(true(), x, y) -> p(y)
  , ifPlus(false(), x, y) -> plus(p(x), y)
  , isZero(0()) -> true()
  , isZero(s(0())) -> false()
  , isZero(s(s(x))) -> isZero(s(x))
  , inc(x) -> s(x)
  , inc(0()) -> s(0())
  , inc(s(x)) -> s(inc(x))
  , p(0()) -> 0()
  , p(s(x)) -> x
  , p(s(s(x))) -> s(p(s(x)))
  , times(x, y) -> timesIter(0(), x, y, 0())
  , timesIter(i, x, y, z) -> ifTimes(ge(i, x), i, x, y, z)
  , ifTimes(true(), i, x, y, z) -> z
  , ifTimes(false(), i, x, y, z) ->
    timesIter(inc(i), x, y, plus(z, y))
  , ge(x, 0()) -> true()
  , ge(0(), s(y)) -> false()
  , ge(s(x), s(y)) -> ge(x, y)
  , f0(x, y, z) -> d()
  , f0(0(), y, x) -> f1(x, y, x)
  , f1(x, y, z) -> f2(x, y, z)
  , f1(x, y, z) -> c()
  , f2(x, 1(), z) -> f0(x, z, 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:
     { plus^#(x, y) -> c_1(ifPlus^#(isZero(x), x, inc(y)))
     , ifPlus^#(true(), x, y) -> c_2(p^#(y))
     , ifPlus^#(false(), x, y) -> c_3(plus^#(p(x), y))
     , p^#(0()) -> c_10()
     , p^#(s(x)) -> c_11(x)
     , p^#(s(s(x))) -> c_12(p^#(s(x)))
     , isZero^#(0()) -> c_4()
     , isZero^#(s(0())) -> c_5()
     , isZero^#(s(s(x))) -> c_6(isZero^#(s(x)))
     , inc^#(x) -> c_7(x)
     , inc^#(0()) -> c_8()
     , inc^#(s(x)) -> c_9(inc^#(x))
     , times^#(x, y) -> c_13(timesIter^#(0(), x, y, 0()))
     , timesIter^#(i, x, y, z) -> c_14(ifTimes^#(ge(i, x), i, x, y, z))
     , ifTimes^#(true(), i, x, y, z) -> c_15(z)
     , ifTimes^#(false(), i, x, y, z) ->
       c_16(timesIter^#(inc(i), x, y, plus(z, y)))
     , ge^#(x, 0()) -> c_17()
     , ge^#(0(), s(y)) -> c_18()
     , ge^#(s(x), s(y)) -> c_19(ge^#(x, y))
     , f0^#(x, y, z) -> c_20()
     , f0^#(0(), y, x) -> c_21(f1^#(x, y, x))
     , f1^#(x, y, z) -> c_22(f2^#(x, y, z))
     , f1^#(x, y, z) -> c_23()
     , f2^#(x, 1(), z) -> c_24(f0^#(x, z, z)) }
   
   and mark the set of starting terms.
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { plus^#(x, y) -> c_1(ifPlus^#(isZero(x), x, inc(y)))
     , ifPlus^#(true(), x, y) -> c_2(p^#(y))
     , ifPlus^#(false(), x, y) -> c_3(plus^#(p(x), y))
     , p^#(0()) -> c_10()
     , p^#(s(x)) -> c_11(x)
     , p^#(s(s(x))) -> c_12(p^#(s(x)))
     , isZero^#(0()) -> c_4()
     , isZero^#(s(0())) -> c_5()
     , isZero^#(s(s(x))) -> c_6(isZero^#(s(x)))
     , inc^#(x) -> c_7(x)
     , inc^#(0()) -> c_8()
     , inc^#(s(x)) -> c_9(inc^#(x))
     , times^#(x, y) -> c_13(timesIter^#(0(), x, y, 0()))
     , timesIter^#(i, x, y, z) -> c_14(ifTimes^#(ge(i, x), i, x, y, z))
     , ifTimes^#(true(), i, x, y, z) -> c_15(z)
     , ifTimes^#(false(), i, x, y, z) ->
       c_16(timesIter^#(inc(i), x, y, plus(z, y)))
     , ge^#(x, 0()) -> c_17()
     , ge^#(0(), s(y)) -> c_18()
     , ge^#(s(x), s(y)) -> c_19(ge^#(x, y))
     , f0^#(x, y, z) -> c_20()
     , f0^#(0(), y, x) -> c_21(f1^#(x, y, x))
     , f1^#(x, y, z) -> c_22(f2^#(x, y, z))
     , f1^#(x, y, z) -> c_23()
     , f2^#(x, 1(), z) -> c_24(f0^#(x, z, z)) }
   Strict Trs:
     { plus(x, y) -> ifPlus(isZero(x), x, inc(y))
     , ifPlus(true(), x, y) -> p(y)
     , ifPlus(false(), x, y) -> plus(p(x), y)
     , isZero(0()) -> true()
     , isZero(s(0())) -> false()
     , isZero(s(s(x))) -> isZero(s(x))
     , inc(x) -> s(x)
     , inc(0()) -> s(0())
     , inc(s(x)) -> s(inc(x))
     , p(0()) -> 0()
     , p(s(x)) -> x
     , p(s(s(x))) -> s(p(s(x)))
     , times(x, y) -> timesIter(0(), x, y, 0())
     , timesIter(i, x, y, z) -> ifTimes(ge(i, x), i, x, y, z)
     , ifTimes(true(), i, x, y, z) -> z
     , ifTimes(false(), i, x, y, z) ->
       timesIter(inc(i), x, y, plus(z, y))
     , ge(x, 0()) -> true()
     , ge(0(), s(y)) -> false()
     , ge(s(x), s(y)) -> ge(x, y)
     , f0(x, y, z) -> d()
     , f0(0(), y, x) -> f1(x, y, x)
     , f1(x, y, z) -> f2(x, y, z)
     , f1(x, y, z) -> c()
     , f2(x, 1(), z) -> f0(x, z, z) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {4,7,8,11,17,18,20,23} by
   applications of Pre({4,7,8,11,17,18,20,23}) =
   {2,5,9,10,12,15,19,21,24}. Here rules are labeled as follows:
   
     DPs:
       { 1: plus^#(x, y) -> c_1(ifPlus^#(isZero(x), x, inc(y)))
       , 2: ifPlus^#(true(), x, y) -> c_2(p^#(y))
       , 3: ifPlus^#(false(), x, y) -> c_3(plus^#(p(x), y))
       , 4: p^#(0()) -> c_10()
       , 5: p^#(s(x)) -> c_11(x)
       , 6: p^#(s(s(x))) -> c_12(p^#(s(x)))
       , 7: isZero^#(0()) -> c_4()
       , 8: isZero^#(s(0())) -> c_5()
       , 9: isZero^#(s(s(x))) -> c_6(isZero^#(s(x)))
       , 10: inc^#(x) -> c_7(x)
       , 11: inc^#(0()) -> c_8()
       , 12: inc^#(s(x)) -> c_9(inc^#(x))
       , 13: times^#(x, y) -> c_13(timesIter^#(0(), x, y, 0()))
       , 14: timesIter^#(i, x, y, z) ->
             c_14(ifTimes^#(ge(i, x), i, x, y, z))
       , 15: ifTimes^#(true(), i, x, y, z) -> c_15(z)
       , 16: ifTimes^#(false(), i, x, y, z) ->
             c_16(timesIter^#(inc(i), x, y, plus(z, y)))
       , 17: ge^#(x, 0()) -> c_17()
       , 18: ge^#(0(), s(y)) -> c_18()
       , 19: ge^#(s(x), s(y)) -> c_19(ge^#(x, y))
       , 20: f0^#(x, y, z) -> c_20()
       , 21: f0^#(0(), y, x) -> c_21(f1^#(x, y, x))
       , 22: f1^#(x, y, z) -> c_22(f2^#(x, y, z))
       , 23: f1^#(x, y, z) -> c_23()
       , 24: f2^#(x, 1(), z) -> c_24(f0^#(x, z, z)) }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { plus^#(x, y) -> c_1(ifPlus^#(isZero(x), x, inc(y)))
     , ifPlus^#(true(), x, y) -> c_2(p^#(y))
     , ifPlus^#(false(), x, y) -> c_3(plus^#(p(x), y))
     , p^#(s(x)) -> c_11(x)
     , p^#(s(s(x))) -> c_12(p^#(s(x)))
     , isZero^#(s(s(x))) -> c_6(isZero^#(s(x)))
     , inc^#(x) -> c_7(x)
     , inc^#(s(x)) -> c_9(inc^#(x))
     , times^#(x, y) -> c_13(timesIter^#(0(), x, y, 0()))
     , timesIter^#(i, x, y, z) -> c_14(ifTimes^#(ge(i, x), i, x, y, z))
     , ifTimes^#(true(), i, x, y, z) -> c_15(z)
     , ifTimes^#(false(), i, x, y, z) ->
       c_16(timesIter^#(inc(i), x, y, plus(z, y)))
     , ge^#(s(x), s(y)) -> c_19(ge^#(x, y))
     , f0^#(0(), y, x) -> c_21(f1^#(x, y, x))
     , f1^#(x, y, z) -> c_22(f2^#(x, y, z))
     , f2^#(x, 1(), z) -> c_24(f0^#(x, z, z)) }
   Strict Trs:
     { plus(x, y) -> ifPlus(isZero(x), x, inc(y))
     , ifPlus(true(), x, y) -> p(y)
     , ifPlus(false(), x, y) -> plus(p(x), y)
     , isZero(0()) -> true()
     , isZero(s(0())) -> false()
     , isZero(s(s(x))) -> isZero(s(x))
     , inc(x) -> s(x)
     , inc(0()) -> s(0())
     , inc(s(x)) -> s(inc(x))
     , p(0()) -> 0()
     , p(s(x)) -> x
     , p(s(s(x))) -> s(p(s(x)))
     , times(x, y) -> timesIter(0(), x, y, 0())
     , timesIter(i, x, y, z) -> ifTimes(ge(i, x), i, x, y, z)
     , ifTimes(true(), i, x, y, z) -> z
     , ifTimes(false(), i, x, y, z) ->
       timesIter(inc(i), x, y, plus(z, y))
     , ge(x, 0()) -> true()
     , ge(0(), s(y)) -> false()
     , ge(s(x), s(y)) -> ge(x, y)
     , f0(x, y, z) -> d()
     , f0(0(), y, x) -> f1(x, y, x)
     , f1(x, y, z) -> f2(x, y, z)
     , f1(x, y, z) -> c()
     , f2(x, 1(), z) -> f0(x, z, z) }
   Weak DPs:
     { p^#(0()) -> c_10()
     , isZero^#(0()) -> c_4()
     , isZero^#(s(0())) -> c_5()
     , inc^#(0()) -> c_8()
     , ge^#(x, 0()) -> c_17()
     , ge^#(0(), s(y)) -> c_18()
     , f0^#(x, y, z) -> c_20()
     , f1^#(x, y, z) -> c_23() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Empty strict component of the problem is NOT empty.


Arrrr..