YES

Problem:
 filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M))
 filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M))
 sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y)))
 sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(filter(activate(Y),N,N)))
 nats(N) -> cons(N,n__nats(s(N)))
 zprimes() -> sieve(nats(s(s(0()))))
 filter(X1,X2,X3) -> n__filter(X1,X2,X3)
 sieve(X) -> n__sieve(X)
 nats(X) -> n__nats(X)
 activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3)
 activate(n__sieve(X)) -> sieve(X)
 activate(n__nats(X)) -> nats(X)
 activate(X) -> X

Proof:
 DP Processor:
  DPs:
   filter#(cons(X,Y),0(),M) -> activate#(Y)
   filter#(cons(X,Y),s(N),M) -> activate#(Y)
   sieve#(cons(0(),Y)) -> activate#(Y)
   sieve#(cons(s(N),Y)) -> activate#(Y)
   sieve#(cons(s(N),Y)) -> filter#(activate(Y),N,N)
   zprimes#() -> nats#(s(s(0())))
   zprimes#() -> sieve#(nats(s(s(0()))))
   activate#(n__filter(X1,X2,X3)) -> filter#(X1,X2,X3)
   activate#(n__sieve(X)) -> sieve#(X)
   activate#(n__nats(X)) -> nats#(X)
  TRS:
   filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M))
   filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M))
   sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y)))
   sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(filter(activate(Y),N,N)))
   nats(N) -> cons(N,n__nats(s(N)))
   zprimes() -> sieve(nats(s(s(0()))))
   filter(X1,X2,X3) -> n__filter(X1,X2,X3)
   sieve(X) -> n__sieve(X)
   nats(X) -> n__nats(X)
   activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3)
   activate(n__sieve(X)) -> sieve(X)
   activate(n__nats(X)) -> nats(X)
   activate(X) -> X
  TDG Processor:
   DPs:
    filter#(cons(X,Y),0(),M) -> activate#(Y)
    filter#(cons(X,Y),s(N),M) -> activate#(Y)
    sieve#(cons(0(),Y)) -> activate#(Y)
    sieve#(cons(s(N),Y)) -> activate#(Y)
    sieve#(cons(s(N),Y)) -> filter#(activate(Y),N,N)
    zprimes#() -> nats#(s(s(0())))
    zprimes#() -> sieve#(nats(s(s(0()))))
    activate#(n__filter(X1,X2,X3)) -> filter#(X1,X2,X3)
    activate#(n__sieve(X)) -> sieve#(X)
    activate#(n__nats(X)) -> nats#(X)
   TRS:
    filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M))
    filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M))
    sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y)))
    sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(filter(activate(Y),N,N)))
    nats(N) -> cons(N,n__nats(s(N)))
    zprimes() -> sieve(nats(s(s(0()))))
    filter(X1,X2,X3) -> n__filter(X1,X2,X3)
    sieve(X) -> n__sieve(X)
    nats(X) -> n__nats(X)
    activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3)
    activate(n__sieve(X)) -> sieve(X)
    activate(n__nats(X)) -> nats(X)
    activate(X) -> X
   graph:
    zprimes#() -> sieve#(nats(s(s(0())))) ->
    sieve#(cons(s(N),Y)) -> filter#(activate(Y),N,N)
    zprimes#() -> sieve#(nats(s(s(0())))) ->
    sieve#(cons(s(N),Y)) -> activate#(Y)
    zprimes#() -> sieve#(nats(s(s(0())))) ->
    sieve#(cons(0(),Y)) -> activate#(Y)
    sieve#(cons(s(N),Y)) -> activate#(Y) ->
    activate#(n__nats(X)) -> nats#(X)
    sieve#(cons(s(N),Y)) -> activate#(Y) ->
    activate#(n__sieve(X)) -> sieve#(X)
    sieve#(cons(s(N),Y)) -> activate#(Y) ->
    activate#(n__filter(X1,X2,X3)) -> filter#(X1,X2,X3)
    sieve#(cons(s(N),Y)) -> filter#(activate(Y),N,N) ->
    filter#(cons(X,Y),s(N),M) -> activate#(Y)
    sieve#(cons(s(N),Y)) -> filter#(activate(Y),N,N) ->
    filter#(cons(X,Y),0(),M) -> activate#(Y)
    sieve#(cons(0(),Y)) -> activate#(Y) ->
    activate#(n__nats(X)) -> nats#(X)
    sieve#(cons(0(),Y)) -> activate#(Y) ->
    activate#(n__sieve(X)) -> sieve#(X)
    sieve#(cons(0(),Y)) -> activate#(Y) ->
    activate#(n__filter(X1,X2,X3)) -> filter#(X1,X2,X3)
    activate#(n__sieve(X)) -> sieve#(X) ->
    sieve#(cons(s(N),Y)) -> filter#(activate(Y),N,N)
    activate#(n__sieve(X)) -> sieve#(X) ->
    sieve#(cons(s(N),Y)) -> activate#(Y)
    activate#(n__sieve(X)) -> sieve#(X) ->
    sieve#(cons(0(),Y)) -> activate#(Y)
    activate#(n__filter(X1,X2,X3)) -> filter#(X1,X2,X3) ->
    filter#(cons(X,Y),s(N),M) -> activate#(Y)
    activate#(n__filter(X1,X2,X3)) -> filter#(X1,X2,X3) ->
    filter#(cons(X,Y),0(),M) -> activate#(Y)
    filter#(cons(X,Y),s(N),M) -> activate#(Y) ->
    activate#(n__nats(X)) -> nats#(X)
    filter#(cons(X,Y),s(N),M) -> activate#(Y) ->
    activate#(n__sieve(X)) -> sieve#(X)
    filter#(cons(X,Y),s(N),M) -> activate#(Y) ->
    activate#(n__filter(X1,X2,X3)) -> filter#(X1,X2,X3)
    filter#(cons(X,Y),0(),M) -> activate#(Y) ->
    activate#(n__nats(X)) -> nats#(X)
    filter#(cons(X,Y),0(),M) -> activate#(Y) ->
    activate#(n__sieve(X)) -> sieve#(X)
    filter#(cons(X,Y),0(),M) -> activate#(Y) -> activate#(n__filter(X1,X2,X3)) -> filter#(X1,X2,X3)
   SCC Processor:
    #sccs: 1
    #rules: 7
    #arcs: 22/100
    DPs:
     sieve#(cons(0(),Y)) -> activate#(Y)
     activate#(n__filter(X1,X2,X3)) -> filter#(X1,X2,X3)
     filter#(cons(X,Y),0(),M) -> activate#(Y)
     activate#(n__sieve(X)) -> sieve#(X)
     sieve#(cons(s(N),Y)) -> activate#(Y)
     sieve#(cons(s(N),Y)) -> filter#(activate(Y),N,N)
     filter#(cons(X,Y),s(N),M) -> activate#(Y)
    TRS:
     filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M))
     filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M))
     sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y)))
     sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(filter(activate(Y),N,N)))
     nats(N) -> cons(N,n__nats(s(N)))
     zprimes() -> sieve(nats(s(s(0()))))
     filter(X1,X2,X3) -> n__filter(X1,X2,X3)
     sieve(X) -> n__sieve(X)
     nats(X) -> n__nats(X)
     activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3)
     activate(n__sieve(X)) -> sieve(X)
     activate(n__nats(X)) -> nats(X)
     activate(X) -> X
    Arctic Interpretation Processor:
     dimension: 1
     interpretation:
      [sieve#](x0) = x0,
      
      [activate#](x0) = x0,
      
      [filter#](x0, x1, x2) = x0 + x1 + x2,
      
      [zprimes] = 6,
      
      [n__nats](x0) = x0 + 4,
      
      [nats](x0) = x0 + 4,
      
      [n__sieve](x0) = 2x0 + 0,
      
      [sieve](x0) = 2x0 + 0,
      
      [s](x0) = x0 + 0,
      
      [n__filter](x0, x1, x2) = x0 + x1 + x2 + 0,
      
      [activate](x0) = x0 + 0,
      
      [filter](x0, x1, x2) = x0 + x1 + x2 + 0,
      
      [0] = 0,
      
      [cons](x0, x1) = x0 + x1 + 0
     orientation:
      sieve#(cons(0(),Y)) = Y + 0 >= Y = activate#(Y)
      
      activate#(n__filter(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 = filter#(X1,X2,X3)
      
      filter#(cons(X,Y),0(),M) = M + X + Y + 0 >= Y = activate#(Y)
      
      activate#(n__sieve(X)) = 2X + 0 >= X = sieve#(X)
      
      sieve#(cons(s(N),Y)) = N + Y + 0 >= Y = activate#(Y)
      
      sieve#(cons(s(N),Y)) = N + Y + 0 >= N + Y + 0 = filter#(activate(Y),N,N)
      
      filter#(cons(X,Y),s(N),M) = M + N + X + Y + 0 >= Y = activate#(Y)
      
      filter(cons(X,Y),0(),M) = M + X + Y + 0 >= M + Y + 0 = cons(0(),n__filter(activate(Y),M,M))
      
      filter(cons(X,Y),s(N),M) = M + N + X + Y + 0 >= M + N + X + Y + 0 = cons(X,n__filter(activate(Y),N,M))
      
      sieve(cons(0(),Y)) = 2Y + 2 >= 2Y + 2 = cons(0(),n__sieve(activate(Y)))
      
      sieve(cons(s(N),Y)) = 2N + 2Y + 2 >= 2N + 2Y + 2 = cons(s(N),n__sieve(filter(activate(Y),N,N)))
      
      nats(N) = N + 4 >= N + 4 = cons(N,n__nats(s(N)))
      
      zprimes() = 6 >= 6 = sieve(nats(s(s(0()))))
      
      filter(X1,X2,X3) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = n__filter(X1,X2,X3)
      
      sieve(X) = 2X + 0 >= 2X + 0 = n__sieve(X)
      
      nats(X) = X + 4 >= X + 4 = n__nats(X)
      
      activate(n__filter(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = filter(X1,X2,X3)
      
      activate(n__sieve(X)) = 2X + 0 >= 2X + 0 = sieve(X)
      
      activate(n__nats(X)) = X + 4 >= X + 4 = nats(X)
      
      activate(X) = X + 0 >= X = X
     problem:
      DPs:
       sieve#(cons(0(),Y)) -> activate#(Y)
       activate#(n__filter(X1,X2,X3)) -> filter#(X1,X2,X3)
       filter#(cons(X,Y),0(),M) -> activate#(Y)
       sieve#(cons(s(N),Y)) -> activate#(Y)
       sieve#(cons(s(N),Y)) -> filter#(activate(Y),N,N)
       filter#(cons(X,Y),s(N),M) -> activate#(Y)
      TRS:
       filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M))
       filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M))
       sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y)))
       sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(filter(activate(Y),N,N)))
       nats(N) -> cons(N,n__nats(s(N)))
       zprimes() -> sieve(nats(s(s(0()))))
       filter(X1,X2,X3) -> n__filter(X1,X2,X3)
       sieve(X) -> n__sieve(X)
       nats(X) -> n__nats(X)
       activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3)
       activate(n__sieve(X)) -> sieve(X)
       activate(n__nats(X)) -> nats(X)
       activate(X) -> X
     SCC Processor:
      #sccs: 1
      #rules: 3
      #arcs: 15/36
      DPs:
       activate#(n__filter(X1,X2,X3)) -> filter#(X1,X2,X3)
       filter#(cons(X,Y),0(),M) -> activate#(Y)
       filter#(cons(X,Y),s(N),M) -> activate#(Y)
      TRS:
       filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M))
       filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M))
       sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y)))
       sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(filter(activate(Y),N,N)))
       nats(N) -> cons(N,n__nats(s(N)))
       zprimes() -> sieve(nats(s(s(0()))))
       filter(X1,X2,X3) -> n__filter(X1,X2,X3)
       sieve(X) -> n__sieve(X)
       nats(X) -> n__nats(X)
       activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3)
       activate(n__sieve(X)) -> sieve(X)
       activate(n__nats(X)) -> nats(X)
       activate(X) -> X
      Subterm Criterion Processor:
       simple projection:
        pi(filter#) = 0
        pi(activate#) = 0
       problem:
        DPs:
         
        TRS:
         filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M))
         filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M))
         sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y)))
         sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(filter(activate(Y),N,N)))
         nats(N) -> cons(N,n__nats(s(N)))
         zprimes() -> sieve(nats(s(s(0()))))
         filter(X1,X2,X3) -> n__filter(X1,X2,X3)
         sieve(X) -> n__sieve(X)
         nats(X) -> n__nats(X)
         activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3)
         activate(n__sieve(X)) -> sieve(X)
         activate(n__nats(X)) -> nats(X)
         activate(X) -> X
       Qed