MAYBE

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

Strict Trs:
  { rev(nil()) -> nil()
  , rev(.(x, y)) -> ++(rev(y), .(x, nil()))
  , ++(nil(), y) -> y
  , ++(.(x, y), z) -> .(x, ++(y, z))
  , car(.(x, y)) -> x
  , cdr(.(x, y)) -> y
  , null(nil()) -> true()
  , null(.(x, y)) -> false() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { rev^#(nil()) -> c_1()
  , rev^#(.(x, y)) -> c_2(++^#(rev(y), .(x, nil())), rev^#(y))
  , ++^#(nil(), y) -> c_3()
  , ++^#(.(x, y), z) -> c_4(++^#(y, z))
  , car^#(.(x, y)) -> c_5()
  , cdr^#(.(x, y)) -> c_6()
  , null^#(nil()) -> c_7()
  , null^#(.(x, y)) -> c_8() }

and mark the set of starting terms.

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

Strict DPs:
  { rev^#(nil()) -> c_1()
  , rev^#(.(x, y)) -> c_2(++^#(rev(y), .(x, nil())), rev^#(y))
  , ++^#(nil(), y) -> c_3()
  , ++^#(.(x, y), z) -> c_4(++^#(y, z))
  , car^#(.(x, y)) -> c_5()
  , cdr^#(.(x, y)) -> c_6()
  , null^#(nil()) -> c_7()
  , null^#(.(x, y)) -> c_8() }
Weak Trs:
  { rev(nil()) -> nil()
  , rev(.(x, y)) -> ++(rev(y), .(x, nil()))
  , ++(nil(), y) -> y
  , ++(.(x, y), z) -> .(x, ++(y, z))
  , car(.(x, y)) -> x
  , cdr(.(x, y)) -> y
  , null(nil()) -> true()
  , null(.(x, y)) -> false() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: rev^#(nil()) -> c_1()
    , 2: rev^#(.(x, y)) -> c_2(++^#(rev(y), .(x, nil())), rev^#(y))
    , 3: ++^#(nil(), y) -> c_3()
    , 4: ++^#(.(x, y), z) -> c_4(++^#(y, z))
    , 5: car^#(.(x, y)) -> c_5()
    , 6: cdr^#(.(x, y)) -> c_6()
    , 7: null^#(nil()) -> c_7()
    , 8: null^#(.(x, y)) -> c_8() }

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

Strict DPs:
  { rev^#(.(x, y)) -> c_2(++^#(rev(y), .(x, nil())), rev^#(y))
  , ++^#(.(x, y), z) -> c_4(++^#(y, z)) }
Weak DPs:
  { rev^#(nil()) -> c_1()
  , ++^#(nil(), y) -> c_3()
  , car^#(.(x, y)) -> c_5()
  , cdr^#(.(x, y)) -> c_6()
  , null^#(nil()) -> c_7()
  , null^#(.(x, y)) -> c_8() }
Weak Trs:
  { rev(nil()) -> nil()
  , rev(.(x, y)) -> ++(rev(y), .(x, nil()))
  , ++(nil(), y) -> y
  , ++(.(x, y), z) -> .(x, ++(y, z))
  , car(.(x, y)) -> x
  , cdr(.(x, y)) -> y
  , null(nil()) -> true()
  , null(.(x, y)) -> false() }
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.

{ rev^#(nil()) -> c_1()
, ++^#(nil(), y) -> c_3()
, car^#(.(x, y)) -> c_5()
, cdr^#(.(x, y)) -> c_6()
, null^#(nil()) -> c_7()
, null^#(.(x, y)) -> c_8() }

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

Strict DPs:
  { rev^#(.(x, y)) -> c_2(++^#(rev(y), .(x, nil())), rev^#(y))
  , ++^#(.(x, y), z) -> c_4(++^#(y, z)) }
Weak Trs:
  { rev(nil()) -> nil()
  , rev(.(x, y)) -> ++(rev(y), .(x, nil()))
  , ++(nil(), y) -> y
  , ++(.(x, y), z) -> .(x, ++(y, z))
  , car(.(x, y)) -> x
  , cdr(.(x, y)) -> y
  , null(nil()) -> true()
  , null(.(x, y)) -> false() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { rev(nil()) -> nil()
    , rev(.(x, y)) -> ++(rev(y), .(x, nil()))
    , ++(nil(), y) -> y
    , ++(.(x, y), z) -> .(x, ++(y, z)) }

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

Strict DPs:
  { rev^#(.(x, y)) -> c_2(++^#(rev(y), .(x, nil())), rev^#(y))
  , ++^#(.(x, y), z) -> c_4(++^#(y, z)) }
Weak Trs:
  { rev(nil()) -> nil()
  , rev(.(x, y)) -> ++(rev(y), .(x, nil()))
  , ++(nil(), y) -> y
  , ++(.(x, y), z) -> .(x, ++(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..