MAYBE

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

Strict Trs:
  { a__U11(X1, X2, X3) -> U11(X1, X2, X3)
  , a__U11(tt(), M, N) -> a__U12(tt(), M, N)
  , a__U12(X1, X2, X3) -> U12(X1, X2, X3)
  , a__U12(tt(), M, N) -> s(a__plus(mark(N), mark(M)))
  , a__plus(X1, X2) -> plus(X1, X2)
  , a__plus(N, s(M)) -> a__U11(tt(), M, N)
  , a__plus(N, 0()) -> mark(N)
  , mark(tt()) -> tt()
  , mark(s(X)) -> s(mark(X))
  , mark(0()) -> 0()
  , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
  , mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3)
  , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { a__U11^#(X1, X2, X3) -> c_1()
  , a__U11^#(tt(), M, N) -> c_2(a__U12^#(tt(), M, N))
  , a__U12^#(X1, X2, X3) -> c_3()
  , a__U12^#(tt(), M, N) ->
    c_4(a__plus^#(mark(N), mark(M)), mark^#(N), mark^#(M))
  , a__plus^#(X1, X2) -> c_5()
  , a__plus^#(N, s(M)) -> c_6(a__U11^#(tt(), M, N))
  , a__plus^#(N, 0()) -> c_7(mark^#(N))
  , mark^#(tt()) -> c_8()
  , mark^#(s(X)) -> c_9(mark^#(X))
  , mark^#(0()) -> c_10()
  , mark^#(U11(X1, X2, X3)) ->
    c_11(a__U11^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U12(X1, X2, X3)) ->
    c_12(a__U12^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(plus(X1, X2)) ->
    c_13(a__plus^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2)) }

and mark the set of starting terms.

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

Strict DPs:
  { a__U11^#(X1, X2, X3) -> c_1()
  , a__U11^#(tt(), M, N) -> c_2(a__U12^#(tt(), M, N))
  , a__U12^#(X1, X2, X3) -> c_3()
  , a__U12^#(tt(), M, N) ->
    c_4(a__plus^#(mark(N), mark(M)), mark^#(N), mark^#(M))
  , a__plus^#(X1, X2) -> c_5()
  , a__plus^#(N, s(M)) -> c_6(a__U11^#(tt(), M, N))
  , a__plus^#(N, 0()) -> c_7(mark^#(N))
  , mark^#(tt()) -> c_8()
  , mark^#(s(X)) -> c_9(mark^#(X))
  , mark^#(0()) -> c_10()
  , mark^#(U11(X1, X2, X3)) ->
    c_11(a__U11^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U12(X1, X2, X3)) ->
    c_12(a__U12^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(plus(X1, X2)) ->
    c_13(a__plus^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2)) }
Weak Trs:
  { a__U11(X1, X2, X3) -> U11(X1, X2, X3)
  , a__U11(tt(), M, N) -> a__U12(tt(), M, N)
  , a__U12(X1, X2, X3) -> U12(X1, X2, X3)
  , a__U12(tt(), M, N) -> s(a__plus(mark(N), mark(M)))
  , a__plus(X1, X2) -> plus(X1, X2)
  , a__plus(N, s(M)) -> a__U11(tt(), M, N)
  , a__plus(N, 0()) -> mark(N)
  , mark(tt()) -> tt()
  , mark(s(X)) -> s(mark(X))
  , mark(0()) -> 0()
  , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
  , mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3)
  , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {1,3,5,8,10} by
applications of Pre({1,3,5,8,10}) = {2,4,6,7,9,11,12,13}. Here
rules are labeled as follows:

  DPs:
    { 1: a__U11^#(X1, X2, X3) -> c_1()
    , 2: a__U11^#(tt(), M, N) -> c_2(a__U12^#(tt(), M, N))
    , 3: a__U12^#(X1, X2, X3) -> c_3()
    , 4: a__U12^#(tt(), M, N) ->
         c_4(a__plus^#(mark(N), mark(M)), mark^#(N), mark^#(M))
    , 5: a__plus^#(X1, X2) -> c_5()
    , 6: a__plus^#(N, s(M)) -> c_6(a__U11^#(tt(), M, N))
    , 7: a__plus^#(N, 0()) -> c_7(mark^#(N))
    , 8: mark^#(tt()) -> c_8()
    , 9: mark^#(s(X)) -> c_9(mark^#(X))
    , 10: mark^#(0()) -> c_10()
    , 11: mark^#(U11(X1, X2, X3)) ->
          c_11(a__U11^#(mark(X1), X2, X3), mark^#(X1))
    , 12: mark^#(U12(X1, X2, X3)) ->
          c_12(a__U12^#(mark(X1), X2, X3), mark^#(X1))
    , 13: mark^#(plus(X1, X2)) ->
          c_13(a__plus^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2)) }

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

Strict DPs:
  { a__U11^#(tt(), M, N) -> c_2(a__U12^#(tt(), M, N))
  , a__U12^#(tt(), M, N) ->
    c_4(a__plus^#(mark(N), mark(M)), mark^#(N), mark^#(M))
  , a__plus^#(N, s(M)) -> c_6(a__U11^#(tt(), M, N))
  , a__plus^#(N, 0()) -> c_7(mark^#(N))
  , mark^#(s(X)) -> c_9(mark^#(X))
  , mark^#(U11(X1, X2, X3)) ->
    c_11(a__U11^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U12(X1, X2, X3)) ->
    c_12(a__U12^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(plus(X1, X2)) ->
    c_13(a__plus^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2)) }
Weak DPs:
  { a__U11^#(X1, X2, X3) -> c_1()
  , a__U12^#(X1, X2, X3) -> c_3()
  , a__plus^#(X1, X2) -> c_5()
  , mark^#(tt()) -> c_8()
  , mark^#(0()) -> c_10() }
Weak Trs:
  { a__U11(X1, X2, X3) -> U11(X1, X2, X3)
  , a__U11(tt(), M, N) -> a__U12(tt(), M, N)
  , a__U12(X1, X2, X3) -> U12(X1, X2, X3)
  , a__U12(tt(), M, N) -> s(a__plus(mark(N), mark(M)))
  , a__plus(X1, X2) -> plus(X1, X2)
  , a__plus(N, s(M)) -> a__U11(tt(), M, N)
  , a__plus(N, 0()) -> mark(N)
  , mark(tt()) -> tt()
  , mark(s(X)) -> s(mark(X))
  , mark(0()) -> 0()
  , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
  , mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3)
  , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) }
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.

{ a__U11^#(X1, X2, X3) -> c_1()
, a__U12^#(X1, X2, X3) -> c_3()
, a__plus^#(X1, X2) -> c_5()
, mark^#(tt()) -> c_8()
, mark^#(0()) -> c_10() }

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

Strict DPs:
  { a__U11^#(tt(), M, N) -> c_2(a__U12^#(tt(), M, N))
  , a__U12^#(tt(), M, N) ->
    c_4(a__plus^#(mark(N), mark(M)), mark^#(N), mark^#(M))
  , a__plus^#(N, s(M)) -> c_6(a__U11^#(tt(), M, N))
  , a__plus^#(N, 0()) -> c_7(mark^#(N))
  , mark^#(s(X)) -> c_9(mark^#(X))
  , mark^#(U11(X1, X2, X3)) ->
    c_11(a__U11^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U12(X1, X2, X3)) ->
    c_12(a__U12^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(plus(X1, X2)) ->
    c_13(a__plus^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2)) }
Weak Trs:
  { a__U11(X1, X2, X3) -> U11(X1, X2, X3)
  , a__U11(tt(), M, N) -> a__U12(tt(), M, N)
  , a__U12(X1, X2, X3) -> U12(X1, X2, X3)
  , a__U12(tt(), M, N) -> s(a__plus(mark(N), mark(M)))
  , a__plus(X1, X2) -> plus(X1, X2)
  , a__plus(N, s(M)) -> a__U11(tt(), M, N)
  , a__plus(N, 0()) -> mark(N)
  , mark(tt()) -> tt()
  , mark(s(X)) -> s(mark(X))
  , mark(0()) -> 0()
  , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
  , mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3)
  , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) }
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..