MAYBE

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

Strict Trs:
  { merge(x, nil()) -> x
  , merge(nil(), y) -> y
  , merge(.(x, y), .(u, v)) ->
    if(<(x, u), .(x, merge(y, .(u, v))), .(u, merge(.(x, y), v)))
  , if(true(), x, y) -> x
  , if(false(), x, y) -> x
  , ++(nil(), y) -> y
  , ++(.(x, y), z) -> .(x, ++(y, z)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { merge^#(x, nil()) -> c_1()
  , merge^#(nil(), y) -> c_2()
  , merge^#(.(x, y), .(u, v)) ->
    c_3(if^#(<(x, u),
             .(x, merge(y, .(u, v))),
             .(u, merge(.(x, y), v))),
        merge^#(y, .(u, v)),
        merge^#(.(x, y), v))
  , if^#(true(), x, y) -> c_4()
  , if^#(false(), x, y) -> c_5()
  , ++^#(nil(), y) -> c_6()
  , ++^#(.(x, y), z) -> c_7(++^#(y, z)) }

and mark the set of starting terms.

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

Strict DPs:
  { merge^#(x, nil()) -> c_1()
  , merge^#(nil(), y) -> c_2()
  , merge^#(.(x, y), .(u, v)) ->
    c_3(if^#(<(x, u),
             .(x, merge(y, .(u, v))),
             .(u, merge(.(x, y), v))),
        merge^#(y, .(u, v)),
        merge^#(.(x, y), v))
  , if^#(true(), x, y) -> c_4()
  , if^#(false(), x, y) -> c_5()
  , ++^#(nil(), y) -> c_6()
  , ++^#(.(x, y), z) -> c_7(++^#(y, z)) }
Weak Trs:
  { merge(x, nil()) -> x
  , merge(nil(), y) -> y
  , merge(.(x, y), .(u, v)) ->
    if(<(x, u), .(x, merge(y, .(u, v))), .(u, merge(.(x, y), v)))
  , if(true(), x, y) -> x
  , if(false(), x, y) -> x
  , ++(nil(), y) -> y
  , ++(.(x, y), z) -> .(x, ++(y, z)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: merge^#(x, nil()) -> c_1()
    , 2: merge^#(nil(), y) -> c_2()
    , 3: merge^#(.(x, y), .(u, v)) ->
         c_3(if^#(<(x, u),
                  .(x, merge(y, .(u, v))),
                  .(u, merge(.(x, y), v))),
             merge^#(y, .(u, v)),
             merge^#(.(x, y), v))
    , 4: if^#(true(), x, y) -> c_4()
    , 5: if^#(false(), x, y) -> c_5()
    , 6: ++^#(nil(), y) -> c_6()
    , 7: ++^#(.(x, y), z) -> c_7(++^#(y, z)) }

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

Strict DPs:
  { merge^#(.(x, y), .(u, v)) ->
    c_3(if^#(<(x, u),
             .(x, merge(y, .(u, v))),
             .(u, merge(.(x, y), v))),
        merge^#(y, .(u, v)),
        merge^#(.(x, y), v))
  , ++^#(.(x, y), z) -> c_7(++^#(y, z)) }
Weak DPs:
  { merge^#(x, nil()) -> c_1()
  , merge^#(nil(), y) -> c_2()
  , if^#(true(), x, y) -> c_4()
  , if^#(false(), x, y) -> c_5()
  , ++^#(nil(), y) -> c_6() }
Weak Trs:
  { merge(x, nil()) -> x
  , merge(nil(), y) -> y
  , merge(.(x, y), .(u, v)) ->
    if(<(x, u), .(x, merge(y, .(u, v))), .(u, merge(.(x, y), v)))
  , if(true(), x, y) -> x
  , if(false(), x, y) -> x
  , ++(nil(), y) -> y
  , ++(.(x, y), z) -> .(x, ++(y, z)) }
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.

{ merge^#(x, nil()) -> c_1()
, merge^#(nil(), y) -> c_2()
, if^#(true(), x, y) -> c_4()
, if^#(false(), x, y) -> c_5()
, ++^#(nil(), y) -> c_6() }

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

Strict DPs:
  { merge^#(.(x, y), .(u, v)) ->
    c_3(if^#(<(x, u),
             .(x, merge(y, .(u, v))),
             .(u, merge(.(x, y), v))),
        merge^#(y, .(u, v)),
        merge^#(.(x, y), v))
  , ++^#(.(x, y), z) -> c_7(++^#(y, z)) }
Weak Trs:
  { merge(x, nil()) -> x
  , merge(nil(), y) -> y
  , merge(.(x, y), .(u, v)) ->
    if(<(x, u), .(x, merge(y, .(u, v))), .(u, merge(.(x, y), v)))
  , if(true(), x, y) -> x
  , if(false(), x, y) -> x
  , ++(nil(), y) -> y
  , ++(.(x, y), z) -> .(x, ++(y, z)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  { merge^#(.(x, y), .(u, v)) ->
    c_3(if^#(<(x, u),
             .(x, merge(y, .(u, v))),
             .(u, merge(.(x, y), v))),
        merge^#(y, .(u, v)),
        merge^#(.(x, y), v)) }

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

Strict DPs:
  { merge^#(.(x, y), .(u, v)) ->
    c_1(merge^#(y, .(u, v)), merge^#(.(x, y), v))
  , ++^#(.(x, y), z) -> c_2(++^#(y, z)) }
Weak Trs:
  { merge(x, nil()) -> x
  , merge(nil(), y) -> y
  , merge(.(x, y), .(u, v)) ->
    if(<(x, u), .(x, merge(y, .(u, v))), .(u, merge(.(x, y), v)))
  , if(true(), x, y) -> x
  , if(false(), x, y) -> x
  , ++(nil(), y) -> y
  , ++(.(x, y), z) -> .(x, ++(y, z)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

No rule is usable, rules are removed from the input problem.

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

Strict DPs:
  { merge^#(.(x, y), .(u, v)) ->
    c_1(merge^#(y, .(u, v)), merge^#(.(x, y), v))
  , ++^#(.(x, y), z) -> c_2(++^#(y, z)) }
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..