MAYBE

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

Strict Trs:
  { eq(0(), 0()) -> true()
  , eq(0(), s(m)) -> false()
  , eq(s(n), 0()) -> false()
  , eq(s(n), s(m)) -> eq(n, m)
  , le(0(), m) -> true()
  , le(s(n), 0()) -> false()
  , le(s(n), s(m)) -> le(n, m)
  , min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x)))
  , min(cons(x, nil())) -> x
  , if_min(true(), cons(n, cons(m, x))) -> min(cons(n, x))
  , if_min(false(), cons(n, cons(m, x))) -> min(cons(m, x))
  , replace(n, m, cons(k, x)) ->
    if_replace(eq(n, k), n, m, cons(k, x))
  , replace(n, m, nil()) -> nil()
  , if_replace(true(), n, m, cons(k, x)) -> cons(m, x)
  , if_replace(false(), n, m, cons(k, x)) ->
    cons(k, replace(n, m, x))
  , empty(cons(n, x)) -> false()
  , empty(nil()) -> true()
  , head(cons(n, x)) -> n
  , tail(cons(n, x)) -> x
  , tail(nil()) -> nil()
  , sort(x) -> sortIter(x, nil())
  , sortIter(x, y) ->
    if(empty(x), x, y, append(y, cons(min(x), nil())))
  , if(true(), x, y, z) -> y
  , if(false(), x, y, z) ->
    sortIter(replace(min(x), head(x), tail(x)), 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) '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) '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) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed
   due to the following reason:
   
   We add the following weak dependency pairs:
   
   Strict DPs:
     { eq^#(0(), 0()) -> c_1()
     , eq^#(0(), s(m)) -> c_2()
     , eq^#(s(n), 0()) -> c_3()
     , eq^#(s(n), s(m)) -> c_4(eq^#(n, m))
     , le^#(0(), m) -> c_5()
     , le^#(s(n), 0()) -> c_6()
     , le^#(s(n), s(m)) -> c_7(le^#(n, m))
     , min^#(cons(n, cons(m, x))) ->
       c_8(if_min^#(le(n, m), cons(n, cons(m, x))))
     , min^#(cons(x, nil())) -> c_9(x)
     , if_min^#(true(), cons(n, cons(m, x))) -> c_10(min^#(cons(n, x)))
     , if_min^#(false(), cons(n, cons(m, x))) -> c_11(min^#(cons(m, x)))
     , replace^#(n, m, cons(k, x)) ->
       c_12(if_replace^#(eq(n, k), n, m, cons(k, x)))
     , replace^#(n, m, nil()) -> c_13()
     , if_replace^#(true(), n, m, cons(k, x)) -> c_14(m, x)
     , if_replace^#(false(), n, m, cons(k, x)) ->
       c_15(k, replace^#(n, m, x))
     , empty^#(cons(n, x)) -> c_16()
     , empty^#(nil()) -> c_17()
     , head^#(cons(n, x)) -> c_18(n)
     , tail^#(cons(n, x)) -> c_19(x)
     , tail^#(nil()) -> c_20()
     , sort^#(x) -> c_21(sortIter^#(x, nil()))
     , sortIter^#(x, y) ->
       c_22(if^#(empty(x), x, y, append(y, cons(min(x), nil()))))
     , if^#(true(), x, y, z) -> c_23(y)
     , if^#(false(), x, y, z) ->
       c_24(sortIter^#(replace(min(x), head(x), tail(x)), z)) }
   
   and mark the set of starting terms.
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { eq^#(0(), 0()) -> c_1()
     , eq^#(0(), s(m)) -> c_2()
     , eq^#(s(n), 0()) -> c_3()
     , eq^#(s(n), s(m)) -> c_4(eq^#(n, m))
     , le^#(0(), m) -> c_5()
     , le^#(s(n), 0()) -> c_6()
     , le^#(s(n), s(m)) -> c_7(le^#(n, m))
     , min^#(cons(n, cons(m, x))) ->
       c_8(if_min^#(le(n, m), cons(n, cons(m, x))))
     , min^#(cons(x, nil())) -> c_9(x)
     , if_min^#(true(), cons(n, cons(m, x))) -> c_10(min^#(cons(n, x)))
     , if_min^#(false(), cons(n, cons(m, x))) -> c_11(min^#(cons(m, x)))
     , replace^#(n, m, cons(k, x)) ->
       c_12(if_replace^#(eq(n, k), n, m, cons(k, x)))
     , replace^#(n, m, nil()) -> c_13()
     , if_replace^#(true(), n, m, cons(k, x)) -> c_14(m, x)
     , if_replace^#(false(), n, m, cons(k, x)) ->
       c_15(k, replace^#(n, m, x))
     , empty^#(cons(n, x)) -> c_16()
     , empty^#(nil()) -> c_17()
     , head^#(cons(n, x)) -> c_18(n)
     , tail^#(cons(n, x)) -> c_19(x)
     , tail^#(nil()) -> c_20()
     , sort^#(x) -> c_21(sortIter^#(x, nil()))
     , sortIter^#(x, y) ->
       c_22(if^#(empty(x), x, y, append(y, cons(min(x), nil()))))
     , if^#(true(), x, y, z) -> c_23(y)
     , if^#(false(), x, y, z) ->
       c_24(sortIter^#(replace(min(x), head(x), tail(x)), z)) }
   Strict Trs:
     { eq(0(), 0()) -> true()
     , eq(0(), s(m)) -> false()
     , eq(s(n), 0()) -> false()
     , eq(s(n), s(m)) -> eq(n, m)
     , le(0(), m) -> true()
     , le(s(n), 0()) -> false()
     , le(s(n), s(m)) -> le(n, m)
     , min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x)))
     , min(cons(x, nil())) -> x
     , if_min(true(), cons(n, cons(m, x))) -> min(cons(n, x))
     , if_min(false(), cons(n, cons(m, x))) -> min(cons(m, x))
     , replace(n, m, cons(k, x)) ->
       if_replace(eq(n, k), n, m, cons(k, x))
     , replace(n, m, nil()) -> nil()
     , if_replace(true(), n, m, cons(k, x)) -> cons(m, x)
     , if_replace(false(), n, m, cons(k, x)) ->
       cons(k, replace(n, m, x))
     , empty(cons(n, x)) -> false()
     , empty(nil()) -> true()
     , head(cons(n, x)) -> n
     , tail(cons(n, x)) -> x
     , tail(nil()) -> nil()
     , sort(x) -> sortIter(x, nil())
     , sortIter(x, y) ->
       if(empty(x), x, y, append(y, cons(min(x), nil())))
     , if(true(), x, y, z) -> y
     , if(false(), x, y, z) ->
       sortIter(replace(min(x), head(x), tail(x)), z) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {1,2,3,5,6,13,16,17,20} by
   applications of Pre({1,2,3,5,6,13,16,17,20}) =
   {4,7,9,14,15,18,19,23}. Here rules are labeled as follows:
   
     DPs:
       { 1: eq^#(0(), 0()) -> c_1()
       , 2: eq^#(0(), s(m)) -> c_2()
       , 3: eq^#(s(n), 0()) -> c_3()
       , 4: eq^#(s(n), s(m)) -> c_4(eq^#(n, m))
       , 5: le^#(0(), m) -> c_5()
       , 6: le^#(s(n), 0()) -> c_6()
       , 7: le^#(s(n), s(m)) -> c_7(le^#(n, m))
       , 8: min^#(cons(n, cons(m, x))) ->
            c_8(if_min^#(le(n, m), cons(n, cons(m, x))))
       , 9: min^#(cons(x, nil())) -> c_9(x)
       , 10: if_min^#(true(), cons(n, cons(m, x))) ->
             c_10(min^#(cons(n, x)))
       , 11: if_min^#(false(), cons(n, cons(m, x))) ->
             c_11(min^#(cons(m, x)))
       , 12: replace^#(n, m, cons(k, x)) ->
             c_12(if_replace^#(eq(n, k), n, m, cons(k, x)))
       , 13: replace^#(n, m, nil()) -> c_13()
       , 14: if_replace^#(true(), n, m, cons(k, x)) -> c_14(m, x)
       , 15: if_replace^#(false(), n, m, cons(k, x)) ->
             c_15(k, replace^#(n, m, x))
       , 16: empty^#(cons(n, x)) -> c_16()
       , 17: empty^#(nil()) -> c_17()
       , 18: head^#(cons(n, x)) -> c_18(n)
       , 19: tail^#(cons(n, x)) -> c_19(x)
       , 20: tail^#(nil()) -> c_20()
       , 21: sort^#(x) -> c_21(sortIter^#(x, nil()))
       , 22: sortIter^#(x, y) ->
             c_22(if^#(empty(x), x, y, append(y, cons(min(x), nil()))))
       , 23: if^#(true(), x, y, z) -> c_23(y)
       , 24: if^#(false(), x, y, z) ->
             c_24(sortIter^#(replace(min(x), head(x), tail(x)), z)) }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { eq^#(s(n), s(m)) -> c_4(eq^#(n, m))
     , le^#(s(n), s(m)) -> c_7(le^#(n, m))
     , min^#(cons(n, cons(m, x))) ->
       c_8(if_min^#(le(n, m), cons(n, cons(m, x))))
     , min^#(cons(x, nil())) -> c_9(x)
     , if_min^#(true(), cons(n, cons(m, x))) -> c_10(min^#(cons(n, x)))
     , if_min^#(false(), cons(n, cons(m, x))) -> c_11(min^#(cons(m, x)))
     , replace^#(n, m, cons(k, x)) ->
       c_12(if_replace^#(eq(n, k), n, m, cons(k, x)))
     , if_replace^#(true(), n, m, cons(k, x)) -> c_14(m, x)
     , if_replace^#(false(), n, m, cons(k, x)) ->
       c_15(k, replace^#(n, m, x))
     , head^#(cons(n, x)) -> c_18(n)
     , tail^#(cons(n, x)) -> c_19(x)
     , sort^#(x) -> c_21(sortIter^#(x, nil()))
     , sortIter^#(x, y) ->
       c_22(if^#(empty(x), x, y, append(y, cons(min(x), nil()))))
     , if^#(true(), x, y, z) -> c_23(y)
     , if^#(false(), x, y, z) ->
       c_24(sortIter^#(replace(min(x), head(x), tail(x)), z)) }
   Strict Trs:
     { eq(0(), 0()) -> true()
     , eq(0(), s(m)) -> false()
     , eq(s(n), 0()) -> false()
     , eq(s(n), s(m)) -> eq(n, m)
     , le(0(), m) -> true()
     , le(s(n), 0()) -> false()
     , le(s(n), s(m)) -> le(n, m)
     , min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x)))
     , min(cons(x, nil())) -> x
     , if_min(true(), cons(n, cons(m, x))) -> min(cons(n, x))
     , if_min(false(), cons(n, cons(m, x))) -> min(cons(m, x))
     , replace(n, m, cons(k, x)) ->
       if_replace(eq(n, k), n, m, cons(k, x))
     , replace(n, m, nil()) -> nil()
     , if_replace(true(), n, m, cons(k, x)) -> cons(m, x)
     , if_replace(false(), n, m, cons(k, x)) ->
       cons(k, replace(n, m, x))
     , empty(cons(n, x)) -> false()
     , empty(nil()) -> true()
     , head(cons(n, x)) -> n
     , tail(cons(n, x)) -> x
     , tail(nil()) -> nil()
     , sort(x) -> sortIter(x, nil())
     , sortIter(x, y) ->
       if(empty(x), x, y, append(y, cons(min(x), nil())))
     , if(true(), x, y, z) -> y
     , if(false(), x, y, z) ->
       sortIter(replace(min(x), head(x), tail(x)), z) }
   Weak DPs:
     { eq^#(0(), 0()) -> c_1()
     , eq^#(0(), s(m)) -> c_2()
     , eq^#(s(n), 0()) -> c_3()
     , le^#(0(), m) -> c_5()
     , le^#(s(n), 0()) -> c_6()
     , replace^#(n, m, nil()) -> c_13()
     , empty^#(cons(n, x)) -> c_16()
     , empty^#(nil()) -> c_17()
     , tail^#(nil()) -> c_20() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Empty strict component of the problem is NOT empty.


Arrrr..