MAYBE

Problem:
 a__U11(tt(),M,N) -> a__U12(tt(),M,N)
 a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
 a__plus(N,0()) -> mark(N)
 a__plus(N,s(M)) -> a__U11(tt(),M,N)
 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))
 mark(tt()) -> tt()
 mark(s(X)) -> s(mark(X))
 mark(0()) -> 0()
 a__U11(X1,X2,X3) -> U11(X1,X2,X3)
 a__U12(X1,X2,X3) -> U12(X1,X2,X3)
 a__plus(X1,X2) -> plus(X1,X2)

Proof:
 DP Processor:
  DPs:
   a__U11#(tt(),M,N) -> a__U12#(tt(),M,N)
   a__U12#(tt(),M,N) -> mark#(M)
   a__U12#(tt(),M,N) -> mark#(N)
   a__U12#(tt(),M,N) -> a__plus#(mark(N),mark(M))
   a__plus#(N,0()) -> mark#(N)
   a__plus#(N,s(M)) -> a__U11#(tt(),M,N)
   mark#(U11(X1,X2,X3)) -> mark#(X1)
   mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)
   mark#(U12(X1,X2,X3)) -> mark#(X1)
   mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3)
   mark#(plus(X1,X2)) -> mark#(X2)
   mark#(plus(X1,X2)) -> mark#(X1)
   mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
   mark#(s(X)) -> mark#(X)
  TRS:
   a__U11(tt(),M,N) -> a__U12(tt(),M,N)
   a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
   a__plus(N,0()) -> mark(N)
   a__plus(N,s(M)) -> a__U11(tt(),M,N)
   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))
   mark(tt()) -> tt()
   mark(s(X)) -> s(mark(X))
   mark(0()) -> 0()
   a__U11(X1,X2,X3) -> U11(X1,X2,X3)
   a__U12(X1,X2,X3) -> U12(X1,X2,X3)
   a__plus(X1,X2) -> plus(X1,X2)
  ADG Processor:
   DPs:
    a__U11#(tt(),M,N) -> a__U12#(tt(),M,N)
    a__U12#(tt(),M,N) -> mark#(M)
    a__U12#(tt(),M,N) -> mark#(N)
    a__U12#(tt(),M,N) -> a__plus#(mark(N),mark(M))
    a__plus#(N,0()) -> mark#(N)
    a__plus#(N,s(M)) -> a__U11#(tt(),M,N)
    mark#(U11(X1,X2,X3)) -> mark#(X1)
    mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)
    mark#(U12(X1,X2,X3)) -> mark#(X1)
    mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3)
    mark#(plus(X1,X2)) -> mark#(X2)
    mark#(plus(X1,X2)) -> mark#(X1)
    mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
    mark#(s(X)) -> mark#(X)
   TRS:
    a__U11(tt(),M,N) -> a__U12(tt(),M,N)
    a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
    a__plus(N,0()) -> mark(N)
    a__plus(N,s(M)) -> a__U11(tt(),M,N)
    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))
    mark(tt()) -> tt()
    mark(s(X)) -> s(mark(X))
    mark(0()) -> 0()
    a__U11(X1,X2,X3) -> U11(X1,X2,X3)
    a__U12(X1,X2,X3) -> U12(X1,X2,X3)
    a__plus(X1,X2) -> plus(X1,X2)
   graph:
    a__plus#(N,0()) -> mark#(N) -> mark#(U11(X1,X2,X3)) -> mark#(X1)
    a__plus#(N,0()) -> mark#(N) ->
    mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)
    a__plus#(N,0()) -> mark#(N) -> mark#(U12(X1,X2,X3)) -> mark#(X1)
    a__plus#(N,0()) -> mark#(N) ->
    mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3)
    a__plus#(N,0()) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X2)
    a__plus#(N,0()) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X1)
    a__plus#(N,0()) -> mark#(N) ->
    mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
    a__plus#(N,0()) -> mark#(N) -> mark#(s(X)) -> mark#(X)
    a__plus#(N,s(M)) -> a__U11#(tt(),M,N) ->
    a__U11#(tt(),M,N) -> a__U12#(tt(),M,N)
    mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) ->
    a__plus#(N,0()) -> mark#(N)
    mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) ->
    a__plus#(N,s(M)) -> a__U11#(tt(),M,N)
    mark#(plus(X1,X2)) -> mark#(X2) ->
    mark#(U11(X1,X2,X3)) -> mark#(X1)
    mark#(plus(X1,X2)) -> mark#(X2) ->
    mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)
    mark#(plus(X1,X2)) -> mark#(X2) ->
    mark#(U12(X1,X2,X3)) -> mark#(X1)
    mark#(plus(X1,X2)) -> mark#(X2) ->
    mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3)
    mark#(plus(X1,X2)) -> mark#(X2) ->
    mark#(plus(X1,X2)) -> mark#(X2)
    mark#(plus(X1,X2)) -> mark#(X2) ->
    mark#(plus(X1,X2)) -> mark#(X1)
    mark#(plus(X1,X2)) -> mark#(X2) ->
    mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
    mark#(plus(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X)
    mark#(plus(X1,X2)) -> mark#(X1) ->
    mark#(U11(X1,X2,X3)) -> mark#(X1)
    mark#(plus(X1,X2)) -> mark#(X1) ->
    mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)
    mark#(plus(X1,X2)) -> mark#(X1) ->
    mark#(U12(X1,X2,X3)) -> mark#(X1)
    mark#(plus(X1,X2)) -> mark#(X1) ->
    mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3)
    mark#(plus(X1,X2)) -> mark#(X1) ->
    mark#(plus(X1,X2)) -> mark#(X2)
    mark#(plus(X1,X2)) -> mark#(X1) ->
    mark#(plus(X1,X2)) -> mark#(X1)
    mark#(plus(X1,X2)) -> mark#(X1) ->
    mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
    mark#(plus(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X)
    mark#(U12(X1,X2,X3)) -> mark#(X1) ->
    mark#(U11(X1,X2,X3)) -> mark#(X1)
    mark#(U12(X1,X2,X3)) -> mark#(X1) ->
    mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)
    mark#(U12(X1,X2,X3)) -> mark#(X1) ->
    mark#(U12(X1,X2,X3)) -> mark#(X1)
    mark#(U12(X1,X2,X3)) -> mark#(X1) ->
    mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3)
    mark#(U12(X1,X2,X3)) -> mark#(X1) ->
    mark#(plus(X1,X2)) -> mark#(X2)
    mark#(U12(X1,X2,X3)) -> mark#(X1) ->
    mark#(plus(X1,X2)) -> mark#(X1)
    mark#(U12(X1,X2,X3)) -> mark#(X1) ->
    mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
    mark#(U12(X1,X2,X3)) -> mark#(X1) ->
    mark#(s(X)) -> mark#(X)
    mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3) ->
    a__U12#(tt(),M,N) -> mark#(M)
    mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3) ->
    a__U12#(tt(),M,N) -> mark#(N)
    mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3) ->
    a__U12#(tt(),M,N) -> a__plus#(mark(N),mark(M))
    mark#(U11(X1,X2,X3)) -> mark#(X1) ->
    mark#(U11(X1,X2,X3)) -> mark#(X1)
    mark#(U11(X1,X2,X3)) -> mark#(X1) ->
    mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)
    mark#(U11(X1,X2,X3)) -> mark#(X1) ->
    mark#(U12(X1,X2,X3)) -> mark#(X1)
    mark#(U11(X1,X2,X3)) -> mark#(X1) ->
    mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3)
    mark#(U11(X1,X2,X3)) -> mark#(X1) ->
    mark#(plus(X1,X2)) -> mark#(X2)
    mark#(U11(X1,X2,X3)) -> mark#(X1) ->
    mark#(plus(X1,X2)) -> mark#(X1)
    mark#(U11(X1,X2,X3)) -> mark#(X1) ->
    mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
    mark#(U11(X1,X2,X3)) -> mark#(X1) ->
    mark#(s(X)) -> mark#(X)
    mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) ->
    a__U11#(tt(),M,N) -> a__U12#(tt(),M,N)
    mark#(s(X)) -> mark#(X) -> mark#(U11(X1,X2,X3)) -> mark#(X1)
    mark#(s(X)) -> mark#(X) ->
    mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)
    mark#(s(X)) -> mark#(X) -> mark#(U12(X1,X2,X3)) -> mark#(X1)
    mark#(s(X)) -> mark#(X) ->
    mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3)
    mark#(s(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X2)
    mark#(s(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X1)
    mark#(s(X)) -> mark#(X) ->
    mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
    mark#(s(X)) -> mark#(X) ->
    mark#(s(X)) -> mark#(X)
    a__U12#(tt(),M,N) -> a__plus#(mark(N),mark(M)) ->
    a__plus#(N,0()) -> mark#(N)
    a__U12#(tt(),M,N) -> a__plus#(mark(N),mark(M)) ->
    a__plus#(N,s(M)) -> a__U11#(tt(),M,N)
    a__U12#(tt(),M,N) -> mark#(N) -> mark#(U11(X1,X2,X3)) -> mark#(X1)
    a__U12#(tt(),M,N) -> mark#(N) ->
    mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)
    a__U12#(tt(),M,N) -> mark#(N) -> mark#(U12(X1,X2,X3)) -> mark#(X1)
    a__U12#(tt(),M,N) -> mark#(N) ->
    mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3)
    a__U12#(tt(),M,N) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X2)
    a__U12#(tt(),M,N) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X1)
    a__U12#(tt(),M,N) -> mark#(N) ->
    mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
    a__U12#(tt(),M,N) -> mark#(N) -> mark#(s(X)) -> mark#(X)
    a__U12#(tt(),M,N) -> mark#(M) -> mark#(U11(X1,X2,X3)) -> mark#(X1)
    a__U12#(tt(),M,N) -> mark#(M) ->
    mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)
    a__U12#(tt(),M,N) -> mark#(M) -> mark#(U12(X1,X2,X3)) -> mark#(X1)
    a__U12#(tt(),M,N) -> mark#(M) ->
    mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3)
    a__U12#(tt(),M,N) -> mark#(M) -> mark#(plus(X1,X2)) -> mark#(X2)
    a__U12#(tt(),M,N) -> mark#(M) -> mark#(plus(X1,X2)) -> mark#(X1)
    a__U12#(tt(),M,N) -> mark#(M) ->
    mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
    a__U12#(tt(),M,N) -> mark#(M) -> mark#(s(X)) -> mark#(X)
    a__U11#(tt(),M,N) -> a__U12#(tt(),M,N) ->
    a__U12#(tt(),M,N) -> mark#(M)
    a__U11#(tt(),M,N) -> a__U12#(tt(),M,N) ->
    a__U12#(tt(),M,N) -> mark#(N)
    a__U11#(tt(),M,N) -> a__U12#(tt(),M,N) -> a__U12#(tt(),M,N) -> a__plus#(mark(N),mark(M))
   Restore Modifier:
    DPs:
     a__U11#(tt(),M,N) -> a__U12#(tt(),M,N)
     a__U12#(tt(),M,N) -> mark#(M)
     a__U12#(tt(),M,N) -> mark#(N)
     a__U12#(tt(),M,N) -> a__plus#(mark(N),mark(M))
     a__plus#(N,0()) -> mark#(N)
     a__plus#(N,s(M)) -> a__U11#(tt(),M,N)
     mark#(U11(X1,X2,X3)) -> mark#(X1)
     mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)
     mark#(U12(X1,X2,X3)) -> mark#(X1)
     mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3)
     mark#(plus(X1,X2)) -> mark#(X2)
     mark#(plus(X1,X2)) -> mark#(X1)
     mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
     mark#(s(X)) -> mark#(X)
    TRS:
     a__U11(tt(),M,N) -> a__U12(tt(),M,N)
     a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
     a__plus(N,0()) -> mark(N)
     a__plus(N,s(M)) -> a__U11(tt(),M,N)
     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))
     mark(tt()) -> tt()
     mark(s(X)) -> s(mark(X))
     mark(0()) -> 0()
     a__U11(X1,X2,X3) -> U11(X1,X2,X3)
     a__U12(X1,X2,X3) -> U12(X1,X2,X3)
     a__plus(X1,X2) -> plus(X1,X2)
    SCC Processor:
     #sccs: 1
     #rules: 14
     #arcs: 76/196
     DPs:
      a__plus#(N,0()) -> mark#(N)
      mark#(s(X)) -> mark#(X)
      mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
      a__plus#(N,s(M)) -> a__U11#(tt(),M,N)
      a__U11#(tt(),M,N) -> a__U12#(tt(),M,N)
      a__U12#(tt(),M,N) -> a__plus#(mark(N),mark(M))
      a__U12#(tt(),M,N) -> mark#(N)
      mark#(plus(X1,X2)) -> mark#(X1)
      mark#(plus(X1,X2)) -> mark#(X2)
      mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3)
      a__U12#(tt(),M,N) -> mark#(M)
      mark#(U12(X1,X2,X3)) -> mark#(X1)
      mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)
      mark#(U11(X1,X2,X3)) -> mark#(X1)
     TRS:
      a__U11(tt(),M,N) -> a__U12(tt(),M,N)
      a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
      a__plus(N,0()) -> mark(N)
      a__plus(N,s(M)) -> a__U11(tt(),M,N)
      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))
      mark(tt()) -> tt()
      mark(s(X)) -> s(mark(X))
      mark(0()) -> 0()
      a__U11(X1,X2,X3) -> U11(X1,X2,X3)
      a__U12(X1,X2,X3) -> U12(X1,X2,X3)
      a__plus(X1,X2) -> plus(X1,X2)
     Open