MAYBE

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

Strict Trs:
  { nthtail(n, l) -> cond(ge(n, length(l)), n, l)
  , cond(true(), n, l) -> l
  , cond(false(), n, l) -> tail(nthtail(s(n), l))
  , ge(u, 0()) -> true()
  , ge(s(u), s(v)) -> ge(u, v)
  , ge(0(), s(v)) -> false()
  , length(nil()) -> 0()
  , length(cons(x, l)) -> s(length(l))
  , tail(nil()) -> nil()
  , tail(cons(x, l)) -> l }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { nthtail^#(n, l) ->
    c_1(cond^#(ge(n, length(l)), n, l),
        ge^#(n, length(l)),
        length^#(l))
  , cond^#(true(), n, l) -> c_2()
  , cond^#(false(), n, l) ->
    c_3(tail^#(nthtail(s(n), l)), nthtail^#(s(n), l))
  , ge^#(u, 0()) -> c_4()
  , ge^#(s(u), s(v)) -> c_5(ge^#(u, v))
  , ge^#(0(), s(v)) -> c_6()
  , length^#(nil()) -> c_7()
  , length^#(cons(x, l)) -> c_8(length^#(l))
  , tail^#(nil()) -> c_9()
  , tail^#(cons(x, l)) -> c_10() }

and mark the set of starting terms.

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

Strict DPs:
  { nthtail^#(n, l) ->
    c_1(cond^#(ge(n, length(l)), n, l),
        ge^#(n, length(l)),
        length^#(l))
  , cond^#(true(), n, l) -> c_2()
  , cond^#(false(), n, l) ->
    c_3(tail^#(nthtail(s(n), l)), nthtail^#(s(n), l))
  , ge^#(u, 0()) -> c_4()
  , ge^#(s(u), s(v)) -> c_5(ge^#(u, v))
  , ge^#(0(), s(v)) -> c_6()
  , length^#(nil()) -> c_7()
  , length^#(cons(x, l)) -> c_8(length^#(l))
  , tail^#(nil()) -> c_9()
  , tail^#(cons(x, l)) -> c_10() }
Weak Trs:
  { nthtail(n, l) -> cond(ge(n, length(l)), n, l)
  , cond(true(), n, l) -> l
  , cond(false(), n, l) -> tail(nthtail(s(n), l))
  , ge(u, 0()) -> true()
  , ge(s(u), s(v)) -> ge(u, v)
  , ge(0(), s(v)) -> false()
  , length(nil()) -> 0()
  , length(cons(x, l)) -> s(length(l))
  , tail(nil()) -> nil()
  , tail(cons(x, l)) -> l }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {2,4,6,7,9,10} by
applications of Pre({2,4,6,7,9,10}) = {1,3,5,8}. Here rules are
labeled as follows:

  DPs:
    { 1: nthtail^#(n, l) ->
         c_1(cond^#(ge(n, length(l)), n, l),
             ge^#(n, length(l)),
             length^#(l))
    , 2: cond^#(true(), n, l) -> c_2()
    , 3: cond^#(false(), n, l) ->
         c_3(tail^#(nthtail(s(n), l)), nthtail^#(s(n), l))
    , 4: ge^#(u, 0()) -> c_4()
    , 5: ge^#(s(u), s(v)) -> c_5(ge^#(u, v))
    , 6: ge^#(0(), s(v)) -> c_6()
    , 7: length^#(nil()) -> c_7()
    , 8: length^#(cons(x, l)) -> c_8(length^#(l))
    , 9: tail^#(nil()) -> c_9()
    , 10: tail^#(cons(x, l)) -> c_10() }

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

Strict DPs:
  { nthtail^#(n, l) ->
    c_1(cond^#(ge(n, length(l)), n, l),
        ge^#(n, length(l)),
        length^#(l))
  , cond^#(false(), n, l) ->
    c_3(tail^#(nthtail(s(n), l)), nthtail^#(s(n), l))
  , ge^#(s(u), s(v)) -> c_5(ge^#(u, v))
  , length^#(cons(x, l)) -> c_8(length^#(l)) }
Weak DPs:
  { cond^#(true(), n, l) -> c_2()
  , ge^#(u, 0()) -> c_4()
  , ge^#(0(), s(v)) -> c_6()
  , length^#(nil()) -> c_7()
  , tail^#(nil()) -> c_9()
  , tail^#(cons(x, l)) -> c_10() }
Weak Trs:
  { nthtail(n, l) -> cond(ge(n, length(l)), n, l)
  , cond(true(), n, l) -> l
  , cond(false(), n, l) -> tail(nthtail(s(n), l))
  , ge(u, 0()) -> true()
  , ge(s(u), s(v)) -> ge(u, v)
  , ge(0(), s(v)) -> false()
  , length(nil()) -> 0()
  , length(cons(x, l)) -> s(length(l))
  , tail(nil()) -> nil()
  , tail(cons(x, l)) -> l }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ cond^#(true(), n, l) -> c_2()
, ge^#(u, 0()) -> c_4()
, ge^#(0(), s(v)) -> c_6()
, length^#(nil()) -> c_7()
, tail^#(nil()) -> c_9()
, tail^#(cons(x, l)) -> c_10() }

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

Strict DPs:
  { nthtail^#(n, l) ->
    c_1(cond^#(ge(n, length(l)), n, l),
        ge^#(n, length(l)),
        length^#(l))
  , cond^#(false(), n, l) ->
    c_3(tail^#(nthtail(s(n), l)), nthtail^#(s(n), l))
  , ge^#(s(u), s(v)) -> c_5(ge^#(u, v))
  , length^#(cons(x, l)) -> c_8(length^#(l)) }
Weak Trs:
  { nthtail(n, l) -> cond(ge(n, length(l)), n, l)
  , cond(true(), n, l) -> l
  , cond(false(), n, l) -> tail(nthtail(s(n), l))
  , ge(u, 0()) -> true()
  , ge(s(u), s(v)) -> ge(u, v)
  , ge(0(), s(v)) -> false()
  , length(nil()) -> 0()
  , length(cons(x, l)) -> s(length(l))
  , tail(nil()) -> nil()
  , tail(cons(x, l)) -> l }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

Due to missing edges in the dependency-graph, the right-hand sides
of following rules could be simplified:

  { cond^#(false(), n, l) ->
    c_3(tail^#(nthtail(s(n), l)), nthtail^#(s(n), l)) }

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

Strict DPs:
  { nthtail^#(n, l) ->
    c_1(cond^#(ge(n, length(l)), n, l),
        ge^#(n, length(l)),
        length^#(l))
  , cond^#(false(), n, l) -> c_2(nthtail^#(s(n), l))
  , ge^#(s(u), s(v)) -> c_3(ge^#(u, v))
  , length^#(cons(x, l)) -> c_4(length^#(l)) }
Weak Trs:
  { nthtail(n, l) -> cond(ge(n, length(l)), n, l)
  , cond(true(), n, l) -> l
  , cond(false(), n, l) -> tail(nthtail(s(n), l))
  , ge(u, 0()) -> true()
  , ge(s(u), s(v)) -> ge(u, v)
  , ge(0(), s(v)) -> false()
  , length(nil()) -> 0()
  , length(cons(x, l)) -> s(length(l))
  , tail(nil()) -> nil()
  , tail(cons(x, l)) -> l }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { ge(u, 0()) -> true()
    , ge(s(u), s(v)) -> ge(u, v)
    , ge(0(), s(v)) -> false()
    , length(nil()) -> 0()
    , length(cons(x, l)) -> s(length(l)) }

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

Strict DPs:
  { nthtail^#(n, l) ->
    c_1(cond^#(ge(n, length(l)), n, l),
        ge^#(n, length(l)),
        length^#(l))
  , cond^#(false(), n, l) -> c_2(nthtail^#(s(n), l))
  , ge^#(s(u), s(v)) -> c_3(ge^#(u, v))
  , length^#(cons(x, l)) -> c_4(length^#(l)) }
Weak Trs:
  { ge(u, 0()) -> true()
  , ge(s(u), s(v)) -> ge(u, v)
  , ge(0(), s(v)) -> false()
  , length(nil()) -> 0()
  , length(cons(x, l)) -> s(length(l)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

None of the processors succeeded.

Details of failed attempt(s):
-----------------------------
1) 'matrices' failed due to the following reason:
   
   None of the processors succeeded.
   
   Details of failed attempt(s):
   -----------------------------
   1) 'matrix interpretation of dimension 4' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   2) 'matrix interpretation of dimension 3' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   3) 'matrix interpretation of dimension 3' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   4) 'matrix interpretation of dimension 2' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   5) 'matrix interpretation of dimension 2' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   6) 'matrix interpretation of dimension 1' failed due to the
      following reason:
      
      The input cannot be shown compatible
   

2) 'empty' failed due to the following reason:
   
   Empty strict component of the problem is NOT empty.


Arrrr..