MAYBE

Problem:
 plus(s(s(x)),y) -> s(plus(x,s(y)))
 plus(x,s(s(y))) -> s(plus(s(x),y))
 plus(s(0()),y) -> s(y)
 plus(0(),y) -> y
 ack(0(),y) -> s(y)
 ack(s(x),0()) -> ack(x,s(0()))
 ack(s(x),s(y)) -> ack(x,plus(y,ack(s(x),y)))

Proof:
 DP Processor:
  DPs:
   plus#(s(s(x)),y) -> plus#(x,s(y))
   plus#(x,s(s(y))) -> plus#(s(x),y)
   ack#(s(x),0()) -> ack#(x,s(0()))
   ack#(s(x),s(y)) -> ack#(s(x),y)
   ack#(s(x),s(y)) -> plus#(y,ack(s(x),y))
   ack#(s(x),s(y)) -> ack#(x,plus(y,ack(s(x),y)))
  TRS:
   plus(s(s(x)),y) -> s(plus(x,s(y)))
   plus(x,s(s(y))) -> s(plus(s(x),y))
   plus(s(0()),y) -> s(y)
   plus(0(),y) -> y
   ack(0(),y) -> s(y)
   ack(s(x),0()) -> ack(x,s(0()))
   ack(s(x),s(y)) -> ack(x,plus(y,ack(s(x),y)))
  TDG Processor:
   DPs:
    plus#(s(s(x)),y) -> plus#(x,s(y))
    plus#(x,s(s(y))) -> plus#(s(x),y)
    ack#(s(x),0()) -> ack#(x,s(0()))
    ack#(s(x),s(y)) -> ack#(s(x),y)
    ack#(s(x),s(y)) -> plus#(y,ack(s(x),y))
    ack#(s(x),s(y)) -> ack#(x,plus(y,ack(s(x),y)))
   TRS:
    plus(s(s(x)),y) -> s(plus(x,s(y)))
    plus(x,s(s(y))) -> s(plus(s(x),y))
    plus(s(0()),y) -> s(y)
    plus(0(),y) -> y
    ack(0(),y) -> s(y)
    ack(s(x),0()) -> ack(x,s(0()))
    ack(s(x),s(y)) -> ack(x,plus(y,ack(s(x),y)))
   graph:
    ack#(s(x),0()) -> ack#(x,s(0())) ->
    ack#(s(x),s(y)) -> ack#(x,plus(y,ack(s(x),y)))
    ack#(s(x),0()) -> ack#(x,s(0())) ->
    ack#(s(x),s(y)) -> plus#(y,ack(s(x),y))
    ack#(s(x),0()) -> ack#(x,s(0())) ->
    ack#(s(x),s(y)) -> ack#(s(x),y)
    ack#(s(x),0()) -> ack#(x,s(0())) ->
    ack#(s(x),0()) -> ack#(x,s(0()))
    ack#(s(x),s(y)) -> ack#(s(x),y) ->
    ack#(s(x),s(y)) -> ack#(x,plus(y,ack(s(x),y)))
    ack#(s(x),s(y)) -> ack#(s(x),y) ->
    ack#(s(x),s(y)) -> plus#(y,ack(s(x),y))
    ack#(s(x),s(y)) -> ack#(s(x),y) ->
    ack#(s(x),s(y)) -> ack#(s(x),y)
    ack#(s(x),s(y)) -> ack#(s(x),y) ->
    ack#(s(x),0()) -> ack#(x,s(0()))
    ack#(s(x),s(y)) -> ack#(x,plus(y,ack(s(x),y))) ->
    ack#(s(x),s(y)) -> ack#(x,plus(y,ack(s(x),y)))
    ack#(s(x),s(y)) -> ack#(x,plus(y,ack(s(x),y))) ->
    ack#(s(x),s(y)) -> plus#(y,ack(s(x),y))
    ack#(s(x),s(y)) -> ack#(x,plus(y,ack(s(x),y))) ->
    ack#(s(x),s(y)) -> ack#(s(x),y)
    ack#(s(x),s(y)) -> ack#(x,plus(y,ack(s(x),y))) ->
    ack#(s(x),0()) -> ack#(x,s(0()))
    ack#(s(x),s(y)) -> plus#(y,ack(s(x),y)) ->
    plus#(x,s(s(y))) -> plus#(s(x),y)
    ack#(s(x),s(y)) -> plus#(y,ack(s(x),y)) ->
    plus#(s(s(x)),y) -> plus#(x,s(y))
    plus#(s(s(x)),y) -> plus#(x,s(y)) ->
    plus#(x,s(s(y))) -> plus#(s(x),y)
    plus#(s(s(x)),y) -> plus#(x,s(y)) ->
    plus#(s(s(x)),y) -> plus#(x,s(y))
    plus#(x,s(s(y))) -> plus#(s(x),y) ->
    plus#(x,s(s(y))) -> plus#(s(x),y)
    plus#(x,s(s(y))) -> plus#(s(x),y) -> plus#(s(s(x)),y) -> plus#(x,s(y))
   SCC Processor:
    #sccs: 2
    #rules: 5
    #arcs: 18/36
    DPs:
     ack#(s(x),0()) -> ack#(x,s(0()))
     ack#(s(x),s(y)) -> ack#(s(x),y)
     ack#(s(x),s(y)) -> ack#(x,plus(y,ack(s(x),y)))
    TRS:
     plus(s(s(x)),y) -> s(plus(x,s(y)))
     plus(x,s(s(y))) -> s(plus(s(x),y))
     plus(s(0()),y) -> s(y)
     plus(0(),y) -> y
     ack(0(),y) -> s(y)
     ack(s(x),0()) -> ack(x,s(0()))
     ack(s(x),s(y)) -> ack(x,plus(y,ack(s(x),y)))
    Subterm Criterion Processor:
     simple projection:
      pi(ack#) = 0
     problem:
      DPs:
       ack#(s(x),s(y)) -> ack#(s(x),y)
      TRS:
       plus(s(s(x)),y) -> s(plus(x,s(y)))
       plus(x,s(s(y))) -> s(plus(s(x),y))
       plus(s(0()),y) -> s(y)
       plus(0(),y) -> y
       ack(0(),y) -> s(y)
       ack(s(x),0()) -> ack(x,s(0()))
       ack(s(x),s(y)) -> ack(x,plus(y,ack(s(x),y)))
     Subterm Criterion Processor:
      simple projection:
       pi(ack#) = 1
      problem:
       DPs:
        
       TRS:
        plus(s(s(x)),y) -> s(plus(x,s(y)))
        plus(x,s(s(y))) -> s(plus(s(x),y))
        plus(s(0()),y) -> s(y)
        plus(0(),y) -> y
        ack(0(),y) -> s(y)
        ack(s(x),0()) -> ack(x,s(0()))
        ack(s(x),s(y)) -> ack(x,plus(y,ack(s(x),y)))
      Qed
    
    DPs:
     plus#(s(s(x)),y) -> plus#(x,s(y))
     plus#(x,s(s(y))) -> plus#(s(x),y)
    TRS:
     plus(s(s(x)),y) -> s(plus(x,s(y)))
     plus(x,s(s(y))) -> s(plus(s(x),y))
     plus(s(0()),y) -> s(y)
     plus(0(),y) -> y
     ack(0(),y) -> s(y)
     ack(s(x),0()) -> ack(x,s(0()))
     ack(s(x),s(y)) -> ack(x,plus(y,ack(s(x),y)))
    Open