MAYBE

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

Strict Trs:
  { plus(x, y) -> plusIter(x, y, 0())
  , plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
  , ifPlus(true(), x, y, z) -> y
  , ifPlus(false(), x, y, z) -> plusIter(x, s(y), s(z))
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , sum(xs) -> sumIter(xs, 0())
  , sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs)))
  , ifSum(true(), xs, x, y) -> x
  , ifSum(false(), xs, x, y) -> sumIter(tail(xs), y)
  , isempty(nil()) -> true()
  , isempty(cons(x, xs)) -> false()
  , head(nil()) -> error()
  , head(cons(x, xs)) -> x
  , tail(nil()) -> nil()
  , tail(cons(x, xs)) -> xs
  , a() -> b()
  , a() -> c() }
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:
     { plus^#(x, y) -> c_1(plusIter^#(x, y, 0()))
     , plusIter^#(x, y, z) -> c_2(ifPlus^#(le(x, z), x, y, z))
     , ifPlus^#(true(), x, y, z) -> c_3(y)
     , ifPlus^#(false(), x, y, z) -> c_4(plusIter^#(x, s(y), s(z)))
     , le^#(0(), y) -> c_5()
     , le^#(s(x), 0()) -> c_6()
     , le^#(s(x), s(y)) -> c_7(le^#(x, y))
     , sum^#(xs) -> c_8(sumIter^#(xs, 0()))
     , sumIter^#(xs, x) ->
       c_9(ifSum^#(isempty(xs), xs, x, plus(x, head(xs))))
     , ifSum^#(true(), xs, x, y) -> c_10(x)
     , ifSum^#(false(), xs, x, y) -> c_11(sumIter^#(tail(xs), y))
     , isempty^#(nil()) -> c_12()
     , isempty^#(cons(x, xs)) -> c_13()
     , head^#(nil()) -> c_14()
     , head^#(cons(x, xs)) -> c_15(x)
     , tail^#(nil()) -> c_16()
     , tail^#(cons(x, xs)) -> c_17(xs)
     , a^#() -> c_18()
     , a^#() -> c_19() }
   
   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(plusIter^#(x, y, 0()))
     , plusIter^#(x, y, z) -> c_2(ifPlus^#(le(x, z), x, y, z))
     , ifPlus^#(true(), x, y, z) -> c_3(y)
     , ifPlus^#(false(), x, y, z) -> c_4(plusIter^#(x, s(y), s(z)))
     , le^#(0(), y) -> c_5()
     , le^#(s(x), 0()) -> c_6()
     , le^#(s(x), s(y)) -> c_7(le^#(x, y))
     , sum^#(xs) -> c_8(sumIter^#(xs, 0()))
     , sumIter^#(xs, x) ->
       c_9(ifSum^#(isempty(xs), xs, x, plus(x, head(xs))))
     , ifSum^#(true(), xs, x, y) -> c_10(x)
     , ifSum^#(false(), xs, x, y) -> c_11(sumIter^#(tail(xs), y))
     , isempty^#(nil()) -> c_12()
     , isempty^#(cons(x, xs)) -> c_13()
     , head^#(nil()) -> c_14()
     , head^#(cons(x, xs)) -> c_15(x)
     , tail^#(nil()) -> c_16()
     , tail^#(cons(x, xs)) -> c_17(xs)
     , a^#() -> c_18()
     , a^#() -> c_19() }
   Strict Trs:
     { plus(x, y) -> plusIter(x, y, 0())
     , plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
     , ifPlus(true(), x, y, z) -> y
     , ifPlus(false(), x, y, z) -> plusIter(x, s(y), s(z))
     , le(0(), y) -> true()
     , le(s(x), 0()) -> false()
     , le(s(x), s(y)) -> le(x, y)
     , sum(xs) -> sumIter(xs, 0())
     , sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs)))
     , ifSum(true(), xs, x, y) -> x
     , ifSum(false(), xs, x, y) -> sumIter(tail(xs), y)
     , isempty(nil()) -> true()
     , isempty(cons(x, xs)) -> false()
     , head(nil()) -> error()
     , head(cons(x, xs)) -> x
     , tail(nil()) -> nil()
     , tail(cons(x, xs)) -> xs
     , a() -> b()
     , a() -> c() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {5,6,12,13,14,16,18,19} by
   applications of Pre({5,6,12,13,14,16,18,19}) = {3,7,10,15,17}. Here
   rules are labeled as follows:
   
     DPs:
       { 1: plus^#(x, y) -> c_1(plusIter^#(x, y, 0()))
       , 2: plusIter^#(x, y, z) -> c_2(ifPlus^#(le(x, z), x, y, z))
       , 3: ifPlus^#(true(), x, y, z) -> c_3(y)
       , 4: ifPlus^#(false(), x, y, z) -> c_4(plusIter^#(x, s(y), s(z)))
       , 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: sum^#(xs) -> c_8(sumIter^#(xs, 0()))
       , 9: sumIter^#(xs, x) ->
            c_9(ifSum^#(isempty(xs), xs, x, plus(x, head(xs))))
       , 10: ifSum^#(true(), xs, x, y) -> c_10(x)
       , 11: ifSum^#(false(), xs, x, y) -> c_11(sumIter^#(tail(xs), y))
       , 12: isempty^#(nil()) -> c_12()
       , 13: isempty^#(cons(x, xs)) -> c_13()
       , 14: head^#(nil()) -> c_14()
       , 15: head^#(cons(x, xs)) -> c_15(x)
       , 16: tail^#(nil()) -> c_16()
       , 17: tail^#(cons(x, xs)) -> c_17(xs)
       , 18: a^#() -> c_18()
       , 19: a^#() -> c_19() }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { plus^#(x, y) -> c_1(plusIter^#(x, y, 0()))
     , plusIter^#(x, y, z) -> c_2(ifPlus^#(le(x, z), x, y, z))
     , ifPlus^#(true(), x, y, z) -> c_3(y)
     , ifPlus^#(false(), x, y, z) -> c_4(plusIter^#(x, s(y), s(z)))
     , le^#(s(x), s(y)) -> c_7(le^#(x, y))
     , sum^#(xs) -> c_8(sumIter^#(xs, 0()))
     , sumIter^#(xs, x) ->
       c_9(ifSum^#(isempty(xs), xs, x, plus(x, head(xs))))
     , ifSum^#(true(), xs, x, y) -> c_10(x)
     , ifSum^#(false(), xs, x, y) -> c_11(sumIter^#(tail(xs), y))
     , head^#(cons(x, xs)) -> c_15(x)
     , tail^#(cons(x, xs)) -> c_17(xs) }
   Strict Trs:
     { plus(x, y) -> plusIter(x, y, 0())
     , plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
     , ifPlus(true(), x, y, z) -> y
     , ifPlus(false(), x, y, z) -> plusIter(x, s(y), s(z))
     , le(0(), y) -> true()
     , le(s(x), 0()) -> false()
     , le(s(x), s(y)) -> le(x, y)
     , sum(xs) -> sumIter(xs, 0())
     , sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs)))
     , ifSum(true(), xs, x, y) -> x
     , ifSum(false(), xs, x, y) -> sumIter(tail(xs), y)
     , isempty(nil()) -> true()
     , isempty(cons(x, xs)) -> false()
     , head(nil()) -> error()
     , head(cons(x, xs)) -> x
     , tail(nil()) -> nil()
     , tail(cons(x, xs)) -> xs
     , a() -> b()
     , a() -> c() }
   Weak DPs:
     { le^#(0(), y) -> c_5()
     , le^#(s(x), 0()) -> c_6()
     , isempty^#(nil()) -> c_12()
     , isempty^#(cons(x, xs)) -> c_13()
     , head^#(nil()) -> c_14()
     , tail^#(nil()) -> c_16()
     , a^#() -> c_18()
     , a^#() -> c_19() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Empty strict component of the problem is NOT empty.


Arrrr..