MAYBE

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

Strict Trs:
  { ack_in(0(), n) -> ack_out(s(n))
  , ack_in(s(m), 0()) -> u11(ack_in(m, s(0())))
  , ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
  , u11(ack_out(n)) -> ack_out(n)
  , u21(ack_out(n), m) -> u22(ack_in(m, n))
  , u22(ack_out(n)) -> ack_out(n) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { ack_in^#(0(), n) -> c_1()
  , ack_in^#(s(m), 0()) ->
    c_2(u11^#(ack_in(m, s(0()))), ack_in^#(m, s(0())))
  , ack_in^#(s(m), s(n)) ->
    c_3(u21^#(ack_in(s(m), n), m), ack_in^#(s(m), n))
  , u11^#(ack_out(n)) -> c_4()
  , u21^#(ack_out(n), m) -> c_5(u22^#(ack_in(m, n)), ack_in^#(m, n))
  , u22^#(ack_out(n)) -> c_6() }

and mark the set of starting terms.

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

Strict DPs:
  { ack_in^#(0(), n) -> c_1()
  , ack_in^#(s(m), 0()) ->
    c_2(u11^#(ack_in(m, s(0()))), ack_in^#(m, s(0())))
  , ack_in^#(s(m), s(n)) ->
    c_3(u21^#(ack_in(s(m), n), m), ack_in^#(s(m), n))
  , u11^#(ack_out(n)) -> c_4()
  , u21^#(ack_out(n), m) -> c_5(u22^#(ack_in(m, n)), ack_in^#(m, n))
  , u22^#(ack_out(n)) -> c_6() }
Weak Trs:
  { ack_in(0(), n) -> ack_out(s(n))
  , ack_in(s(m), 0()) -> u11(ack_in(m, s(0())))
  , ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
  , u11(ack_out(n)) -> ack_out(n)
  , u21(ack_out(n), m) -> u22(ack_in(m, n))
  , u22(ack_out(n)) -> ack_out(n) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: ack_in^#(0(), n) -> c_1()
    , 2: ack_in^#(s(m), 0()) ->
         c_2(u11^#(ack_in(m, s(0()))), ack_in^#(m, s(0())))
    , 3: ack_in^#(s(m), s(n)) ->
         c_3(u21^#(ack_in(s(m), n), m), ack_in^#(s(m), n))
    , 4: u11^#(ack_out(n)) -> c_4()
    , 5: u21^#(ack_out(n), m) ->
         c_5(u22^#(ack_in(m, n)), ack_in^#(m, n))
    , 6: u22^#(ack_out(n)) -> c_6() }

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

Strict DPs:
  { ack_in^#(s(m), 0()) ->
    c_2(u11^#(ack_in(m, s(0()))), ack_in^#(m, s(0())))
  , ack_in^#(s(m), s(n)) ->
    c_3(u21^#(ack_in(s(m), n), m), ack_in^#(s(m), n))
  , u21^#(ack_out(n), m) ->
    c_5(u22^#(ack_in(m, n)), ack_in^#(m, n)) }
Weak DPs:
  { ack_in^#(0(), n) -> c_1()
  , u11^#(ack_out(n)) -> c_4()
  , u22^#(ack_out(n)) -> c_6() }
Weak Trs:
  { ack_in(0(), n) -> ack_out(s(n))
  , ack_in(s(m), 0()) -> u11(ack_in(m, s(0())))
  , ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
  , u11(ack_out(n)) -> ack_out(n)
  , u21(ack_out(n), m) -> u22(ack_in(m, n))
  , u22(ack_out(n)) -> ack_out(n) }
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.

{ ack_in^#(0(), n) -> c_1()
, u11^#(ack_out(n)) -> c_4()
, u22^#(ack_out(n)) -> c_6() }

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

Strict DPs:
  { ack_in^#(s(m), 0()) ->
    c_2(u11^#(ack_in(m, s(0()))), ack_in^#(m, s(0())))
  , ack_in^#(s(m), s(n)) ->
    c_3(u21^#(ack_in(s(m), n), m), ack_in^#(s(m), n))
  , u21^#(ack_out(n), m) ->
    c_5(u22^#(ack_in(m, n)), ack_in^#(m, n)) }
Weak Trs:
  { ack_in(0(), n) -> ack_out(s(n))
  , ack_in(s(m), 0()) -> u11(ack_in(m, s(0())))
  , ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
  , u11(ack_out(n)) -> ack_out(n)
  , u21(ack_out(n), m) -> u22(ack_in(m, n))
  , u22(ack_out(n)) -> ack_out(n) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  { ack_in^#(s(m), 0()) ->
    c_2(u11^#(ack_in(m, s(0()))), ack_in^#(m, s(0())))
  , u21^#(ack_out(n), m) ->
    c_5(u22^#(ack_in(m, n)), ack_in^#(m, n)) }

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

Strict DPs:
  { ack_in^#(s(m), 0()) -> c_1(ack_in^#(m, s(0())))
  , ack_in^#(s(m), s(n)) ->
    c_2(u21^#(ack_in(s(m), n), m), ack_in^#(s(m), n))
  , u21^#(ack_out(n), m) -> c_3(ack_in^#(m, n)) }
Weak Trs:
  { ack_in(0(), n) -> ack_out(s(n))
  , ack_in(s(m), 0()) -> u11(ack_in(m, s(0())))
  , ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
  , u11(ack_out(n)) -> ack_out(n)
  , u21(ack_out(n), m) -> u22(ack_in(m, n))
  , u22(ack_out(n)) -> ack_out(n) }
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..