MAYBE

Problem:
 a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
 a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M))
 a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y))
 a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N)))
 a__nats(N) -> cons(mark(N),nats(s(N)))
 a__zprimes() -> a__sieve(a__nats(s(s(0()))))
 mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3))
 mark(sieve(X)) -> a__sieve(mark(X))
 mark(nats(X)) -> a__nats(mark(X))
 mark(zprimes()) -> a__zprimes()
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(0()) -> 0()
 mark(s(X)) -> s(mark(X))
 a__filter(X1,X2,X3) -> filter(X1,X2,X3)
 a__sieve(X) -> sieve(X)
 a__nats(X) -> nats(X)
 a__zprimes() -> zprimes()

Proof:
 DP Processor:
  DPs:
   a__filter#(cons(X,Y),s(N),M) -> mark#(X)
   a__sieve#(cons(s(N),Y)) -> mark#(N)
   a__nats#(N) -> mark#(N)
   a__zprimes#() -> a__nats#(s(s(0())))
   a__zprimes#() -> a__sieve#(a__nats(s(s(0()))))
   mark#(filter(X1,X2,X3)) -> mark#(X3)
   mark#(filter(X1,X2,X3)) -> mark#(X2)
   mark#(filter(X1,X2,X3)) -> mark#(X1)
   mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3))
   mark#(sieve(X)) -> mark#(X)
   mark#(sieve(X)) -> a__sieve#(mark(X))
   mark#(nats(X)) -> mark#(X)
   mark#(nats(X)) -> a__nats#(mark(X))
   mark#(zprimes()) -> a__zprimes#()
   mark#(cons(X1,X2)) -> mark#(X1)
   mark#(s(X)) -> mark#(X)
  TRS:
   a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
   a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M))
   a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y))
   a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N)))
   a__nats(N) -> cons(mark(N),nats(s(N)))
   a__zprimes() -> a__sieve(a__nats(s(s(0()))))
   mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3))
   mark(sieve(X)) -> a__sieve(mark(X))
   mark(nats(X)) -> a__nats(mark(X))
   mark(zprimes()) -> a__zprimes()
   mark(cons(X1,X2)) -> cons(mark(X1),X2)
   mark(0()) -> 0()
   mark(s(X)) -> s(mark(X))
   a__filter(X1,X2,X3) -> filter(X1,X2,X3)
   a__sieve(X) -> sieve(X)
   a__nats(X) -> nats(X)
   a__zprimes() -> zprimes()
  ADG Processor:
   DPs:
    a__filter#(cons(X,Y),s(N),M) -> mark#(X)
    a__sieve#(cons(s(N),Y)) -> mark#(N)
    a__nats#(N) -> mark#(N)
    a__zprimes#() -> a__nats#(s(s(0())))
    a__zprimes#() -> a__sieve#(a__nats(s(s(0()))))
    mark#(filter(X1,X2,X3)) -> mark#(X3)
    mark#(filter(X1,X2,X3)) -> mark#(X2)
    mark#(filter(X1,X2,X3)) -> mark#(X1)
    mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3))
    mark#(sieve(X)) -> mark#(X)
    mark#(sieve(X)) -> a__sieve#(mark(X))
    mark#(nats(X)) -> mark#(X)
    mark#(nats(X)) -> a__nats#(mark(X))
    mark#(zprimes()) -> a__zprimes#()
    mark#(cons(X1,X2)) -> mark#(X1)
    mark#(s(X)) -> mark#(X)
   TRS:
    a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
    a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M))
    a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y))
    a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N)))
    a__nats(N) -> cons(mark(N),nats(s(N)))
    a__zprimes() -> a__sieve(a__nats(s(s(0()))))
    mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3))
    mark(sieve(X)) -> a__sieve(mark(X))
    mark(nats(X)) -> a__nats(mark(X))
    mark(zprimes()) -> a__zprimes()
    mark(cons(X1,X2)) -> cons(mark(X1),X2)
    mark(0()) -> 0()
    mark(s(X)) -> s(mark(X))
    a__filter(X1,X2,X3) -> filter(X1,X2,X3)
    a__sieve(X) -> sieve(X)
    a__nats(X) -> nats(X)
    a__zprimes() -> zprimes()
   graph:
    a__zprimes#() -> a__nats#(s(s(0()))) ->
    a__nats#(N) -> mark#(N)
    a__zprimes#() -> a__sieve#(a__nats(s(s(0())))) ->
    a__sieve#(cons(s(N),Y)) -> mark#(N)
    a__nats#(N) -> mark#(N) -> mark#(filter(X1,X2,X3)) -> mark#(X3)
    a__nats#(N) -> mark#(N) -> mark#(filter(X1,X2,X3)) -> mark#(X2)
    a__nats#(N) -> mark#(N) -> mark#(filter(X1,X2,X3)) -> mark#(X1)
    a__nats#(N) -> mark#(N) ->
    mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3))
    a__nats#(N) -> mark#(N) -> mark#(sieve(X)) -> mark#(X)
    a__nats#(N) -> mark#(N) -> mark#(sieve(X)) -> a__sieve#(mark(X))
    a__nats#(N) -> mark#(N) -> mark#(nats(X)) -> mark#(X)
    a__nats#(N) -> mark#(N) -> mark#(nats(X)) -> a__nats#(mark(X))
    a__nats#(N) -> mark#(N) -> mark#(zprimes()) -> a__zprimes#()
    a__nats#(N) -> mark#(N) -> mark#(cons(X1,X2)) -> mark#(X1)
    a__nats#(N) -> mark#(N) -> mark#(s(X)) -> mark#(X)
    a__sieve#(cons(s(N),Y)) -> mark#(N) ->
    mark#(filter(X1,X2,X3)) -> mark#(X3)
    a__sieve#(cons(s(N),Y)) -> mark#(N) ->
    mark#(filter(X1,X2,X3)) -> mark#(X2)
    a__sieve#(cons(s(N),Y)) -> mark#(N) ->
    mark#(filter(X1,X2,X3)) -> mark#(X1)
    a__sieve#(cons(s(N),Y)) -> mark#(N) ->
    mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3))
    a__sieve#(cons(s(N),Y)) -> mark#(N) ->
    mark#(sieve(X)) -> mark#(X)
    a__sieve#(cons(s(N),Y)) -> mark#(N) ->
    mark#(sieve(X)) -> a__sieve#(mark(X))
    a__sieve#(cons(s(N),Y)) -> mark#(N) ->
    mark#(nats(X)) -> mark#(X)
    a__sieve#(cons(s(N),Y)) -> mark#(N) ->
    mark#(nats(X)) -> a__nats#(mark(X))
    a__sieve#(cons(s(N),Y)) -> mark#(N) ->
    mark#(zprimes()) -> a__zprimes#()
    a__sieve#(cons(s(N),Y)) -> mark#(N) ->
    mark#(cons(X1,X2)) -> mark#(X1)
    a__sieve#(cons(s(N),Y)) -> mark#(N) -> mark#(s(X)) -> mark#(X)
    mark#(zprimes()) -> a__zprimes#() ->
    a__zprimes#() -> a__nats#(s(s(0())))
    mark#(zprimes()) -> a__zprimes#() ->
    a__zprimes#() -> a__sieve#(a__nats(s(s(0()))))
    mark#(nats(X)) -> a__nats#(mark(X)) -> a__nats#(N) -> mark#(N)
    mark#(nats(X)) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> mark#(X3)
    mark#(nats(X)) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> mark#(X2)
    mark#(nats(X)) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> mark#(X1)
    mark#(nats(X)) -> mark#(X) ->
    mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3))
    mark#(nats(X)) -> mark#(X) -> mark#(sieve(X)) -> mark#(X)
    mark#(nats(X)) -> mark#(X) -> mark#(sieve(X)) -> a__sieve#(mark(X))
    mark#(nats(X)) -> mark#(X) -> mark#(nats(X)) -> mark#(X)
    mark#(nats(X)) -> mark#(X) -> mark#(nats(X)) -> a__nats#(mark(X))
    mark#(nats(X)) -> mark#(X) -> mark#(zprimes()) -> a__zprimes#()
    mark#(nats(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1)
    mark#(nats(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X)
    mark#(sieve(X)) -> a__sieve#(mark(X)) ->
    a__sieve#(cons(s(N),Y)) -> mark#(N)
    mark#(sieve(X)) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> mark#(X3)
    mark#(sieve(X)) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> mark#(X2)
    mark#(sieve(X)) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> mark#(X1)
    mark#(sieve(X)) -> mark#(X) ->
    mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3))
    mark#(sieve(X)) -> mark#(X) -> mark#(sieve(X)) -> mark#(X)
    mark#(sieve(X)) -> mark#(X) -> mark#(sieve(X)) -> a__sieve#(mark(X))
    mark#(sieve(X)) -> mark#(X) -> mark#(nats(X)) -> mark#(X)
    mark#(sieve(X)) -> mark#(X) -> mark#(nats(X)) -> a__nats#(mark(X))
    mark#(sieve(X)) -> mark#(X) -> mark#(zprimes()) -> a__zprimes#()
    mark#(sieve(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1)
    mark#(sieve(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X)
    mark#(s(X)) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> mark#(X3)
    mark#(s(X)) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> mark#(X2)
    mark#(s(X)) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> mark#(X1)
    mark#(s(X)) -> mark#(X) ->
    mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3))
    mark#(s(X)) -> mark#(X) -> mark#(sieve(X)) -> mark#(X)
    mark#(s(X)) -> mark#(X) -> mark#(sieve(X)) -> a__sieve#(mark(X))
    mark#(s(X)) -> mark#(X) -> mark#(nats(X)) -> mark#(X)
    mark#(s(X)) -> mark#(X) -> mark#(nats(X)) -> a__nats#(mark(X))
    mark#(s(X)) -> mark#(X) -> mark#(zprimes()) -> a__zprimes#()
    mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1)
    mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X)
    mark#(filter(X1,X2,X3)) -> mark#(X3) ->
    mark#(filter(X1,X2,X3)) -> mark#(X3)
    mark#(filter(X1,X2,X3)) -> mark#(X3) ->
    mark#(filter(X1,X2,X3)) -> mark#(X2)
    mark#(filter(X1,X2,X3)) -> mark#(X3) ->
    mark#(filter(X1,X2,X3)) -> mark#(X1)
    mark#(filter(X1,X2,X3)) -> mark#(X3) ->
    mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3))
    mark#(filter(X1,X2,X3)) -> mark#(X3) ->
    mark#(sieve(X)) -> mark#(X)
    mark#(filter(X1,X2,X3)) -> mark#(X3) ->
    mark#(sieve(X)) -> a__sieve#(mark(X))
    mark#(filter(X1,X2,X3)) -> mark#(X3) ->
    mark#(nats(X)) -> mark#(X)
    mark#(filter(X1,X2,X3)) -> mark#(X3) ->
    mark#(nats(X)) -> a__nats#(mark(X))
    mark#(filter(X1,X2,X3)) -> mark#(X3) ->
    mark#(zprimes()) -> a__zprimes#()
    mark#(filter(X1,X2,X3)) -> mark#(X3) ->
    mark#(cons(X1,X2)) -> mark#(X1)
    mark#(filter(X1,X2,X3)) -> mark#(X3) ->
    mark#(s(X)) -> mark#(X)
    mark#(filter(X1,X2,X3)) -> mark#(X2) ->
    mark#(filter(X1,X2,X3)) -> mark#(X3)
    mark#(filter(X1,X2,X3)) -> mark#(X2) ->
    mark#(filter(X1,X2,X3)) -> mark#(X2)
    mark#(filter(X1,X2,X3)) -> mark#(X2) ->
    mark#(filter(X1,X2,X3)) -> mark#(X1)
    mark#(filter(X1,X2,X3)) -> mark#(X2) ->
    mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3))
    mark#(filter(X1,X2,X3)) -> mark#(X2) ->
    mark#(sieve(X)) -> mark#(X)
    mark#(filter(X1,X2,X3)) -> mark#(X2) ->
    mark#(sieve(X)) -> a__sieve#(mark(X))
    mark#(filter(X1,X2,X3)) -> mark#(X2) ->
    mark#(nats(X)) -> mark#(X)
    mark#(filter(X1,X2,X3)) -> mark#(X2) ->
    mark#(nats(X)) -> a__nats#(mark(X))
    mark#(filter(X1,X2,X3)) -> mark#(X2) ->
    mark#(zprimes()) -> a__zprimes#()
    mark#(filter(X1,X2,X3)) -> mark#(X2) ->
    mark#(cons(X1,X2)) -> mark#(X1)
    mark#(filter(X1,X2,X3)) -> mark#(X2) ->
    mark#(s(X)) -> mark#(X)
    mark#(filter(X1,X2,X3)) -> mark#(X1) ->
    mark#(filter(X1,X2,X3)) -> mark#(X3)
    mark#(filter(X1,X2,X3)) -> mark#(X1) ->
    mark#(filter(X1,X2,X3)) -> mark#(X2)
    mark#(filter(X1,X2,X3)) -> mark#(X1) ->
    mark#(filter(X1,X2,X3)) -> mark#(X1)
    mark#(filter(X1,X2,X3)) -> mark#(X1) ->
    mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3))
    mark#(filter(X1,X2,X3)) -> mark#(X1) ->
    mark#(sieve(X)) -> mark#(X)
    mark#(filter(X1,X2,X3)) -> mark#(X1) ->
    mark#(sieve(X)) -> a__sieve#(mark(X))
    mark#(filter(X1,X2,X3)) -> mark#(X1) ->
    mark#(nats(X)) -> mark#(X)
    mark#(filter(X1,X2,X3)) -> mark#(X1) ->
    mark#(nats(X)) -> a__nats#(mark(X))
    mark#(filter(X1,X2,X3)) -> mark#(X1) ->
    mark#(zprimes()) -> a__zprimes#()
    mark#(filter(X1,X2,X3)) -> mark#(X1) ->
    mark#(cons(X1,X2)) -> mark#(X1)
    mark#(filter(X1,X2,X3)) -> mark#(X1) ->
    mark#(s(X)) -> mark#(X)
    mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3)) ->
    a__filter#(cons(X,Y),s(N),M) -> mark#(X)
    mark#(cons(X1,X2)) -> mark#(X1) ->
    mark#(filter(X1,X2,X3)) -> mark#(X3)
    mark#(cons(X1,X2)) -> mark#(X1) ->
    mark#(filter(X1,X2,X3)) -> mark#(X2)
    mark#(cons(X1,X2)) -> mark#(X1) ->
    mark#(filter(X1,X2,X3)) -> mark#(X1)
    mark#(cons(X1,X2)) -> mark#(X1) ->
    mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3))
    mark#(cons(X1,X2)) -> mark#(X1) -> mark#(sieve(X)) -> mark#(X)
    mark#(cons(X1,X2)) -> mark#(X1) ->
    mark#(sieve(X)) -> a__sieve#(mark(X))
    mark#(cons(X1,X2)) -> mark#(X1) -> mark#(nats(X)) -> mark#(X)
    mark#(cons(X1,X2)) -> mark#(X1) ->
    mark#(nats(X)) -> a__nats#(mark(X))
    mark#(cons(X1,X2)) -> mark#(X1) ->
    mark#(zprimes()) -> a__zprimes#()
    mark#(cons(X1,X2)) -> mark#(X1) ->
    mark#(cons(X1,X2)) -> mark#(X1)
    mark#(cons(X1,X2)) -> mark#(X1) ->
    mark#(s(X)) -> mark#(X)
    a__filter#(cons(X,Y),s(N),M) -> mark#(X) ->
    mark#(filter(X1,X2,X3)) -> mark#(X3)
    a__filter#(cons(X,Y),s(N),M) -> mark#(X) ->
    mark#(filter(X1,X2,X3)) -> mark#(X2)
    a__filter#(cons(X,Y),s(N),M) -> mark#(X) ->
    mark#(filter(X1,X2,X3)) -> mark#(X1)
    a__filter#(cons(X,Y),s(N),M) -> mark#(X) ->
    mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3))
    a__filter#(cons(X,Y),s(N),M) -> mark#(X) ->
    mark#(sieve(X)) -> mark#(X)
    a__filter#(cons(X,Y),s(N),M) -> mark#(X) ->
    mark#(sieve(X)) -> a__sieve#(mark(X))
    a__filter#(cons(X,Y),s(N),M) -> mark#(X) ->
    mark#(nats(X)) -> mark#(X)
    a__filter#(cons(X,Y),s(N),M) -> mark#(X) ->
    mark#(nats(X)) -> a__nats#(mark(X))
    a__filter#(cons(X,Y),s(N),M) -> mark#(X) ->
    mark#(zprimes()) -> a__zprimes#()
    a__filter#(cons(X,Y),s(N),M) -> mark#(X) ->
    mark#(cons(X1,X2)) -> mark#(X1)
    a__filter#(cons(X,Y),s(N),M) -> mark#(X) -> mark#(s(X)) -> mark#(X)
   SCC Processor:
    #sccs: 1
    #rules: 16
    #arcs: 117/256
    DPs:
     a__zprimes#() -> a__nats#(s(s(0())))
     a__nats#(N) -> mark#(N)
     mark#(s(X)) -> mark#(X)
     mark#(cons(X1,X2)) -> mark#(X1)
     mark#(zprimes()) -> a__zprimes#()
     a__zprimes#() -> a__sieve#(a__nats(s(s(0()))))
     a__sieve#(cons(s(N),Y)) -> mark#(N)
     mark#(nats(X)) -> a__nats#(mark(X))
     mark#(nats(X)) -> mark#(X)
     mark#(sieve(X)) -> a__sieve#(mark(X))
     mark#(sieve(X)) -> mark#(X)
     mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3))
     a__filter#(cons(X,Y),s(N),M) -> mark#(X)
     mark#(filter(X1,X2,X3)) -> mark#(X1)
     mark#(filter(X1,X2,X3)) -> mark#(X2)
     mark#(filter(X1,X2,X3)) -> mark#(X3)
    TRS:
     a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
     a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M))
     a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y))
     a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N)))
     a__nats(N) -> cons(mark(N),nats(s(N)))
     a__zprimes() -> a__sieve(a__nats(s(s(0()))))
     mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3))
     mark(sieve(X)) -> a__sieve(mark(X))
     mark(nats(X)) -> a__nats(mark(X))
     mark(zprimes()) -> a__zprimes()
     mark(cons(X1,X2)) -> cons(mark(X1),X2)
     mark(0()) -> 0()
     mark(s(X)) -> s(mark(X))
     a__filter(X1,X2,X3) -> filter(X1,X2,X3)
     a__sieve(X) -> sieve(X)
     a__nats(X) -> nats(X)
     a__zprimes() -> zprimes()
    Open