Input TRS:
    1: filter(cons(X,Y),0(),M) -> cons(0(),n__filter(activate(Y),M,M))
    2: filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M))
    3: sieve(cons(0(),Y)) -> cons(0(),n__sieve(activate(Y)))
    4: sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(n__filter(activate(Y),N,N)))
    5: nats(N) -> cons(N,n__nats(n__s(N)))
    6: zprimes() -> sieve(nats(s(s(0()))))
    7: filter(X1,X2,X3) -> n__filter(X1,X2,X3)
    8: sieve(X) -> n__sieve(X)
    9: nats(X) -> n__nats(X)
    10: s(X) -> n__s(X)
    11: activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3))
    12: activate(n__sieve(X)) -> sieve(activate(X))
    13: activate(n__nats(X)) -> nats(activate(X))
    14: activate(n__s(X)) -> s(activate(X))
    15: activate(X) -> X
Number of strict rules: 15
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #filter(cons(X,Y),s(N),M) -> #activate(Y)
   #2: #zprimes() -> #sieve(nats(s(s(0()))))
   #3: #zprimes() -> #nats(s(s(0())))
   #4: #zprimes() -> #s(s(0()))
   #5: #zprimes() -> #s(0())
   #6: #activate(n__nats(X)) -> #nats(activate(X))
   #7: #activate(n__nats(X)) -> #activate(X)
   #8: #activate(n__filter(X1,X2,X3)) -> #filter(activate(X1),activate(X2),activate(X3))
   #9: #activate(n__filter(X1,X2,X3)) -> #activate(X1)
   #10: #activate(n__filter(X1,X2,X3)) -> #activate(X2)
   #11: #activate(n__filter(X1,X2,X3)) -> #activate(X3)
   #12: #activate(n__sieve(X)) -> #sieve(activate(X))
   #13: #activate(n__sieve(X)) -> #activate(X)
   #14: #activate(n__s(X)) -> #s(activate(X))
   #15: #activate(n__s(X)) -> #activate(X)
   #16: #sieve(cons(0(),Y)) -> #activate(Y)
   #17: #filter(cons(X,Y),0(),M) -> #activate(Y)
   #18: #sieve(cons(s(N),Y)) -> #activate(Y)
Number of SCCs: 1, DPs: 12, edges: 84
	SCC { #1 #7..13 #15..18 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded.
zprimes()	weight: 0
#nats(x1)	weight: 0
     s(x1)	weight: x1
activate(x1)	weight: x1
#filter(x1,x2,x3)	weight: max{(/ 3 8) + x3, (/ 1 4) + x2, (/ 1 8) + x1}
#activate(x1)	weight: (/ 1 8) + x1
#zprimes()	weight: 0
n__nats(x1)	weight: (/ 3 8) + x1
  n__s(x1)	weight: x1
     0()	weight: 0
   #s(x1)	weight: 0
n__filter(x1,x2,x3)	weight: max{(/ 1 4) + x3, (/ 1 8) + x2, x1}
 sieve(x1)	weight: (/ 1 4) + x1
n__sieve(x1)	weight: (/ 1 4) + x1
  nats(x1)	weight: (/ 3 8) + x1
  cons(x1,x2)	weight: max{x2, (/ 3 8) + x1}
filter(x1,x2,x3)	weight: max{(/ 1 4) + x3, (/ 1 8) + x2, x1}
#sieve(x1)	weight: (/ 1 4) + x1
    Usable rules: { 1..5 7..15 }
    Removed DPs: #7 #10..13 #16 #18
Number of SCCs: 1, DPs: 5, edges: 14
	SCC { #1 #8 #9 #15 #17 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
zprimes()	weight: 0
#nats(x1)	weight: 0
     s(x1)	weight: (/ 1 4) + x1
activate(x1)	weight: x1
#filter(x1,x2,x3)	weight: (/ 1 4) + x1
#activate(x1)	weight: x1
#zprimes()	weight: 0
n__nats(x1)	weight: 0
  n__s(x1)	weight: (/ 1 4) + x1
     0()	weight: 0
   #s(x1)	weight: 0
n__filter(x1,x2,x3)	weight: (/ 1 2) + x1
 sieve(x1)	weight: 0
n__sieve(x1)	weight: 0
  nats(x1)	weight: 0
  cons(x1,x2)	weight: x2
filter(x1,x2,x3)	weight: (/ 1 2) + x1
#sieve(x1)	weight: 0
    Usable rules: { 1..5 7..15 }
    Removed DPs: #1 #8 #9 #15 #17
Number of SCCs: 0, DPs: 0, edges: 0
YES