MAYBE

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

Strict Trs:
  { battle(H(H(H(0(), x), y), z), n) ->
    battle(H(f(x, y, n), z), s(n))
  , battle(H(H(0(), x), y), n) -> battle(f(x, y, n), s(n))
  , battle(H(0(), x), n) -> battle(x, s(n))
  , f(x, y, s(n)) -> H(x, f(x, y, n))
  , f(x, y, o()) -> y }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { battle^#(H(H(H(0(), x), y), z), n) ->
    c_1(battle^#(H(f(x, y, n), z), s(n)), f^#(x, y, n))
  , battle^#(H(H(0(), x), y), n) ->
    c_2(battle^#(f(x, y, n), s(n)), f^#(x, y, n))
  , battle^#(H(0(), x), n) -> c_3(battle^#(x, s(n)))
  , f^#(x, y, s(n)) -> c_4(f^#(x, y, n))
  , f^#(x, y, o()) -> c_5() }

and mark the set of starting terms.

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

Strict DPs:
  { battle^#(H(H(H(0(), x), y), z), n) ->
    c_1(battle^#(H(f(x, y, n), z), s(n)), f^#(x, y, n))
  , battle^#(H(H(0(), x), y), n) ->
    c_2(battle^#(f(x, y, n), s(n)), f^#(x, y, n))
  , battle^#(H(0(), x), n) -> c_3(battle^#(x, s(n)))
  , f^#(x, y, s(n)) -> c_4(f^#(x, y, n))
  , f^#(x, y, o()) -> c_5() }
Weak Trs:
  { battle(H(H(H(0(), x), y), z), n) ->
    battle(H(f(x, y, n), z), s(n))
  , battle(H(H(0(), x), y), n) -> battle(f(x, y, n), s(n))
  , battle(H(0(), x), n) -> battle(x, s(n))
  , f(x, y, s(n)) -> H(x, f(x, y, n))
  , f(x, y, o()) -> y }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: battle^#(H(H(H(0(), x), y), z), n) ->
         c_1(battle^#(H(f(x, y, n), z), s(n)), f^#(x, y, n))
    , 2: battle^#(H(H(0(), x), y), n) ->
         c_2(battle^#(f(x, y, n), s(n)), f^#(x, y, n))
    , 3: battle^#(H(0(), x), n) -> c_3(battle^#(x, s(n)))
    , 4: f^#(x, y, s(n)) -> c_4(f^#(x, y, n))
    , 5: f^#(x, y, o()) -> c_5() }

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

Strict DPs:
  { battle^#(H(H(H(0(), x), y), z), n) ->
    c_1(battle^#(H(f(x, y, n), z), s(n)), f^#(x, y, n))
  , battle^#(H(H(0(), x), y), n) ->
    c_2(battle^#(f(x, y, n), s(n)), f^#(x, y, n))
  , battle^#(H(0(), x), n) -> c_3(battle^#(x, s(n)))
  , f^#(x, y, s(n)) -> c_4(f^#(x, y, n)) }
Weak DPs: { f^#(x, y, o()) -> c_5() }
Weak Trs:
  { battle(H(H(H(0(), x), y), z), n) ->
    battle(H(f(x, y, n), z), s(n))
  , battle(H(H(0(), x), y), n) -> battle(f(x, y, n), s(n))
  , battle(H(0(), x), n) -> battle(x, s(n))
  , f(x, y, s(n)) -> H(x, f(x, y, n))
  , f(x, y, o()) -> 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.

{ f^#(x, y, o()) -> c_5() }

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

Strict DPs:
  { battle^#(H(H(H(0(), x), y), z), n) ->
    c_1(battle^#(H(f(x, y, n), z), s(n)), f^#(x, y, n))
  , battle^#(H(H(0(), x), y), n) ->
    c_2(battle^#(f(x, y, n), s(n)), f^#(x, y, n))
  , battle^#(H(0(), x), n) -> c_3(battle^#(x, s(n)))
  , f^#(x, y, s(n)) -> c_4(f^#(x, y, n)) }
Weak Trs:
  { battle(H(H(H(0(), x), y), z), n) ->
    battle(H(f(x, y, n), z), s(n))
  , battle(H(H(0(), x), y), n) -> battle(f(x, y, n), s(n))
  , battle(H(0(), x), n) -> battle(x, s(n))
  , f(x, y, s(n)) -> H(x, f(x, y, n))
  , f(x, y, o()) -> y }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { f(x, y, s(n)) -> H(x, f(x, y, n))
    , f(x, y, o()) -> y }

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

Strict DPs:
  { battle^#(H(H(H(0(), x), y), z), n) ->
    c_1(battle^#(H(f(x, y, n), z), s(n)), f^#(x, y, n))
  , battle^#(H(H(0(), x), y), n) ->
    c_2(battle^#(f(x, y, n), s(n)), f^#(x, y, n))
  , battle^#(H(0(), x), n) -> c_3(battle^#(x, s(n)))
  , f^#(x, y, s(n)) -> c_4(f^#(x, y, n)) }
Weak Trs:
  { f(x, y, s(n)) -> H(x, f(x, y, n))
  , f(x, y, o()) -> 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..