MAYBE

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

Strict Trs:
  { app(x, y) -> helpa(0(), plus(length(x), length(y)), x, y)
  , helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
  , plus(x, 0()) -> x
  , plus(x, s(y)) -> s(plus(x, y))
  , length(nil()) -> 0()
  , length(cons(x, y)) -> s(length(y))
  , if(true(), c, l, ys, zs) -> nil()
  , if(false(), c, l, ys, zs) ->
    helpb(c, l, greater(ys, zs), smaller(ys, zs))
  , ge(x, 0()) -> true()
  , ge(0(), s(x)) -> false()
  , ge(s(x), s(y)) -> ge(x, y)
  , helpb(c, l, cons(y, ys), zs) -> cons(y, helpa(s(c), l, ys, zs))
  , greater(ys, zs) -> helpc(ge(length(ys), length(zs)), ys, zs)
  , smaller(ys, zs) -> helpc(ge(length(ys), length(zs)), zs, ys)
  , helpc(true(), ys, zs) -> ys
  , helpc(false(), ys, zs) -> zs }
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:
     { app^#(x, y) ->
       c_1(helpa^#(0(), plus(length(x), length(y)), x, y))
     , helpa^#(c, l, ys, zs) -> c_2(if^#(ge(c, l), c, l, ys, zs))
     , if^#(true(), c, l, ys, zs) -> c_7()
     , if^#(false(), c, l, ys, zs) ->
       c_8(helpb^#(c, l, greater(ys, zs), smaller(ys, zs)))
     , plus^#(x, 0()) -> c_3(x)
     , plus^#(x, s(y)) -> c_4(plus^#(x, y))
     , length^#(nil()) -> c_5()
     , length^#(cons(x, y)) -> c_6(length^#(y))
     , helpb^#(c, l, cons(y, ys), zs) ->
       c_12(y, helpa^#(s(c), l, ys, zs))
     , ge^#(x, 0()) -> c_9()
     , ge^#(0(), s(x)) -> c_10()
     , ge^#(s(x), s(y)) -> c_11(ge^#(x, y))
     , greater^#(ys, zs) ->
       c_13(helpc^#(ge(length(ys), length(zs)), ys, zs))
     , helpc^#(true(), ys, zs) -> c_15(ys)
     , helpc^#(false(), ys, zs) -> c_16(zs)
     , smaller^#(ys, zs) ->
       c_14(helpc^#(ge(length(ys), length(zs)), zs, ys)) }
   
   and mark the set of starting terms.
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { app^#(x, y) ->
       c_1(helpa^#(0(), plus(length(x), length(y)), x, y))
     , helpa^#(c, l, ys, zs) -> c_2(if^#(ge(c, l), c, l, ys, zs))
     , if^#(true(), c, l, ys, zs) -> c_7()
     , if^#(false(), c, l, ys, zs) ->
       c_8(helpb^#(c, l, greater(ys, zs), smaller(ys, zs)))
     , plus^#(x, 0()) -> c_3(x)
     , plus^#(x, s(y)) -> c_4(plus^#(x, y))
     , length^#(nil()) -> c_5()
     , length^#(cons(x, y)) -> c_6(length^#(y))
     , helpb^#(c, l, cons(y, ys), zs) ->
       c_12(y, helpa^#(s(c), l, ys, zs))
     , ge^#(x, 0()) -> c_9()
     , ge^#(0(), s(x)) -> c_10()
     , ge^#(s(x), s(y)) -> c_11(ge^#(x, y))
     , greater^#(ys, zs) ->
       c_13(helpc^#(ge(length(ys), length(zs)), ys, zs))
     , helpc^#(true(), ys, zs) -> c_15(ys)
     , helpc^#(false(), ys, zs) -> c_16(zs)
     , smaller^#(ys, zs) ->
       c_14(helpc^#(ge(length(ys), length(zs)), zs, ys)) }
   Strict Trs:
     { app(x, y) -> helpa(0(), plus(length(x), length(y)), x, y)
     , helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
     , plus(x, 0()) -> x
     , plus(x, s(y)) -> s(plus(x, y))
     , length(nil()) -> 0()
     , length(cons(x, y)) -> s(length(y))
     , if(true(), c, l, ys, zs) -> nil()
     , if(false(), c, l, ys, zs) ->
       helpb(c, l, greater(ys, zs), smaller(ys, zs))
     , ge(x, 0()) -> true()
     , ge(0(), s(x)) -> false()
     , ge(s(x), s(y)) -> ge(x, y)
     , helpb(c, l, cons(y, ys), zs) -> cons(y, helpa(s(c), l, ys, zs))
     , greater(ys, zs) -> helpc(ge(length(ys), length(zs)), ys, zs)
     , smaller(ys, zs) -> helpc(ge(length(ys), length(zs)), zs, ys)
     , helpc(true(), ys, zs) -> ys
     , helpc(false(), ys, zs) -> zs }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {3,7,10,11} by
   applications of Pre({3,7,10,11}) = {2,5,8,9,12,14,15}. Here rules
   are labeled as follows:
   
     DPs:
       { 1: app^#(x, y) ->
            c_1(helpa^#(0(), plus(length(x), length(y)), x, y))
       , 2: helpa^#(c, l, ys, zs) -> c_2(if^#(ge(c, l), c, l, ys, zs))
       , 3: if^#(true(), c, l, ys, zs) -> c_7()
       , 4: if^#(false(), c, l, ys, zs) ->
            c_8(helpb^#(c, l, greater(ys, zs), smaller(ys, zs)))
       , 5: plus^#(x, 0()) -> c_3(x)
       , 6: plus^#(x, s(y)) -> c_4(plus^#(x, y))
       , 7: length^#(nil()) -> c_5()
       , 8: length^#(cons(x, y)) -> c_6(length^#(y))
       , 9: helpb^#(c, l, cons(y, ys), zs) ->
            c_12(y, helpa^#(s(c), l, ys, zs))
       , 10: ge^#(x, 0()) -> c_9()
       , 11: ge^#(0(), s(x)) -> c_10()
       , 12: ge^#(s(x), s(y)) -> c_11(ge^#(x, y))
       , 13: greater^#(ys, zs) ->
             c_13(helpc^#(ge(length(ys), length(zs)), ys, zs))
       , 14: helpc^#(true(), ys, zs) -> c_15(ys)
       , 15: helpc^#(false(), ys, zs) -> c_16(zs)
       , 16: smaller^#(ys, zs) ->
             c_14(helpc^#(ge(length(ys), length(zs)), zs, ys)) }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { app^#(x, y) ->
       c_1(helpa^#(0(), plus(length(x), length(y)), x, y))
     , helpa^#(c, l, ys, zs) -> c_2(if^#(ge(c, l), c, l, ys, zs))
     , if^#(false(), c, l, ys, zs) ->
       c_8(helpb^#(c, l, greater(ys, zs), smaller(ys, zs)))
     , plus^#(x, 0()) -> c_3(x)
     , plus^#(x, s(y)) -> c_4(plus^#(x, y))
     , length^#(cons(x, y)) -> c_6(length^#(y))
     , helpb^#(c, l, cons(y, ys), zs) ->
       c_12(y, helpa^#(s(c), l, ys, zs))
     , ge^#(s(x), s(y)) -> c_11(ge^#(x, y))
     , greater^#(ys, zs) ->
       c_13(helpc^#(ge(length(ys), length(zs)), ys, zs))
     , helpc^#(true(), ys, zs) -> c_15(ys)
     , helpc^#(false(), ys, zs) -> c_16(zs)
     , smaller^#(ys, zs) ->
       c_14(helpc^#(ge(length(ys), length(zs)), zs, ys)) }
   Strict Trs:
     { app(x, y) -> helpa(0(), plus(length(x), length(y)), x, y)
     , helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
     , plus(x, 0()) -> x
     , plus(x, s(y)) -> s(plus(x, y))
     , length(nil()) -> 0()
     , length(cons(x, y)) -> s(length(y))
     , if(true(), c, l, ys, zs) -> nil()
     , if(false(), c, l, ys, zs) ->
       helpb(c, l, greater(ys, zs), smaller(ys, zs))
     , ge(x, 0()) -> true()
     , ge(0(), s(x)) -> false()
     , ge(s(x), s(y)) -> ge(x, y)
     , helpb(c, l, cons(y, ys), zs) -> cons(y, helpa(s(c), l, ys, zs))
     , greater(ys, zs) -> helpc(ge(length(ys), length(zs)), ys, zs)
     , smaller(ys, zs) -> helpc(ge(length(ys), length(zs)), zs, ys)
     , helpc(true(), ys, zs) -> ys
     , helpc(false(), ys, zs) -> zs }
   Weak DPs:
     { if^#(true(), c, l, ys, zs) -> c_7()
     , length^#(nil()) -> c_5()
     , ge^#(x, 0()) -> c_9()
     , ge^#(0(), s(x)) -> c_10() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Empty strict component of the problem is NOT empty.


Arrrr..