MAYBE

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

Strict Trs:
  { lcs(x, e()) -> 0()
  , lcs(e(), y) -> 0()
  , lcs(a(x), a(y)) -> s(lcs(x, y))
  , lcs(a(x), b(y)) -> max(lcs(x, b(y)), lcs(a(x), y))
  , lcs(b(x), a(y)) -> max(lcs(x, a(y)), lcs(b(x), y))
  , lcs(b(x), b(y)) -> s(lcs(x, y))
  , max(x, 0()) -> 0()
  , max(0(), y) -> 0()
  , max(s(x), s(y)) -> max(x, y) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { lcs^#(x, e()) -> c_1()
  , lcs^#(e(), y) -> c_2()
  , lcs^#(a(x), a(y)) -> c_3(lcs^#(x, y))
  , lcs^#(a(x), b(y)) ->
    c_4(max^#(lcs(x, b(y)), lcs(a(x), y)),
        lcs^#(x, b(y)),
        lcs^#(a(x), y))
  , lcs^#(b(x), a(y)) ->
    c_5(max^#(lcs(x, a(y)), lcs(b(x), y)),
        lcs^#(x, a(y)),
        lcs^#(b(x), y))
  , lcs^#(b(x), b(y)) -> c_6(lcs^#(x, y))
  , max^#(x, 0()) -> c_7()
  , max^#(0(), y) -> c_8()
  , max^#(s(x), s(y)) -> c_9(max^#(x, y)) }

and mark the set of starting terms.

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

Strict DPs:
  { lcs^#(x, e()) -> c_1()
  , lcs^#(e(), y) -> c_2()
  , lcs^#(a(x), a(y)) -> c_3(lcs^#(x, y))
  , lcs^#(a(x), b(y)) ->
    c_4(max^#(lcs(x, b(y)), lcs(a(x), y)),
        lcs^#(x, b(y)),
        lcs^#(a(x), y))
  , lcs^#(b(x), a(y)) ->
    c_5(max^#(lcs(x, a(y)), lcs(b(x), y)),
        lcs^#(x, a(y)),
        lcs^#(b(x), y))
  , lcs^#(b(x), b(y)) -> c_6(lcs^#(x, y))
  , max^#(x, 0()) -> c_7()
  , max^#(0(), y) -> c_8()
  , max^#(s(x), s(y)) -> c_9(max^#(x, y)) }
Weak Trs:
  { lcs(x, e()) -> 0()
  , lcs(e(), y) -> 0()
  , lcs(a(x), a(y)) -> s(lcs(x, y))
  , lcs(a(x), b(y)) -> max(lcs(x, b(y)), lcs(a(x), y))
  , lcs(b(x), a(y)) -> max(lcs(x, a(y)), lcs(b(x), y))
  , lcs(b(x), b(y)) -> s(lcs(x, y))
  , max(x, 0()) -> 0()
  , max(0(), y) -> 0()
  , max(s(x), s(y)) -> max(x, y) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: lcs^#(x, e()) -> c_1()
    , 2: lcs^#(e(), y) -> c_2()
    , 3: lcs^#(a(x), a(y)) -> c_3(lcs^#(x, y))
    , 4: lcs^#(a(x), b(y)) ->
         c_4(max^#(lcs(x, b(y)), lcs(a(x), y)),
             lcs^#(x, b(y)),
             lcs^#(a(x), y))
    , 5: lcs^#(b(x), a(y)) ->
         c_5(max^#(lcs(x, a(y)), lcs(b(x), y)),
             lcs^#(x, a(y)),
             lcs^#(b(x), y))
    , 6: lcs^#(b(x), b(y)) -> c_6(lcs^#(x, y))
    , 7: max^#(x, 0()) -> c_7()
    , 8: max^#(0(), y) -> c_8()
    , 9: max^#(s(x), s(y)) -> c_9(max^#(x, y)) }

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

Strict DPs:
  { lcs^#(a(x), a(y)) -> c_3(lcs^#(x, y))
  , lcs^#(a(x), b(y)) ->
    c_4(max^#(lcs(x, b(y)), lcs(a(x), y)),
        lcs^#(x, b(y)),
        lcs^#(a(x), y))
  , lcs^#(b(x), a(y)) ->
    c_5(max^#(lcs(x, a(y)), lcs(b(x), y)),
        lcs^#(x, a(y)),
        lcs^#(b(x), y))
  , lcs^#(b(x), b(y)) -> c_6(lcs^#(x, y))
  , max^#(s(x), s(y)) -> c_9(max^#(x, y)) }
Weak DPs:
  { lcs^#(x, e()) -> c_1()
  , lcs^#(e(), y) -> c_2()
  , max^#(x, 0()) -> c_7()
  , max^#(0(), y) -> c_8() }
Weak Trs:
  { lcs(x, e()) -> 0()
  , lcs(e(), y) -> 0()
  , lcs(a(x), a(y)) -> s(lcs(x, y))
  , lcs(a(x), b(y)) -> max(lcs(x, b(y)), lcs(a(x), y))
  , lcs(b(x), a(y)) -> max(lcs(x, a(y)), lcs(b(x), y))
  , lcs(b(x), b(y)) -> s(lcs(x, y))
  , max(x, 0()) -> 0()
  , max(0(), y) -> 0()
  , max(s(x), s(y)) -> max(x, y) }
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.

{ lcs^#(x, e()) -> c_1()
, lcs^#(e(), y) -> c_2()
, max^#(x, 0()) -> c_7()
, max^#(0(), y) -> c_8() }

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

Strict DPs:
  { lcs^#(a(x), a(y)) -> c_3(lcs^#(x, y))
  , lcs^#(a(x), b(y)) ->
    c_4(max^#(lcs(x, b(y)), lcs(a(x), y)),
        lcs^#(x, b(y)),
        lcs^#(a(x), y))
  , lcs^#(b(x), a(y)) ->
    c_5(max^#(lcs(x, a(y)), lcs(b(x), y)),
        lcs^#(x, a(y)),
        lcs^#(b(x), y))
  , lcs^#(b(x), b(y)) -> c_6(lcs^#(x, y))
  , max^#(s(x), s(y)) -> c_9(max^#(x, y)) }
Weak Trs:
  { lcs(x, e()) -> 0()
  , lcs(e(), y) -> 0()
  , lcs(a(x), a(y)) -> s(lcs(x, y))
  , lcs(a(x), b(y)) -> max(lcs(x, b(y)), lcs(a(x), y))
  , lcs(b(x), a(y)) -> max(lcs(x, a(y)), lcs(b(x), y))
  , lcs(b(x), b(y)) -> s(lcs(x, y))
  , max(x, 0()) -> 0()
  , max(0(), y) -> 0()
  , max(s(x), s(y)) -> max(x, y) }
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:
      
      Following exception was raised:
        stack overflow
   
   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..