Input TRS:
    1: a__filter(cons(X,Y),|0|(),M) -> cons(|0|(),filter(Y,M,M))
    2: a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M))
    3: a__sieve(cons(|0|(),Y)) -> cons(|0|(),sieve(Y))
    4: a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N)))
    5: a__nats(N) -> cons(mark(N),nats(s(N)))
    6: a__zprimes() -> a__sieve(a__nats(s(s(|0|()))))
    7: mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3))
    8: mark(sieve(X)) -> a__sieve(mark(X))
    9: mark(nats(X)) -> a__nats(mark(X))
    10: mark(zprimes()) -> a__zprimes()
    11: mark(cons(X1,X2)) -> cons(mark(X1),X2)
    12: mark(|0|()) -> |0|()
    13: mark(s(X)) -> s(mark(X))
    14: a__filter(X1,X2,X3) -> filter(X1,X2,X3)
    15: a__sieve(X) -> sieve(X)
    16: a__nats(X) -> nats(X)
    17: a__zprimes() -> zprimes()
Number of strict rules: 17
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #a__filter(cons(X,Y),s(N),M) -> #mark(X)
   #2: #a__zprimes() -> #a__sieve(a__nats(s(s(|0|()))))
   #3: #a__zprimes() -> #a__nats(s(s(|0|())))
   #4: #mark(s(X)) -> #mark(X)
   #5: #mark(nats(X)) -> #a__nats(mark(X))
   #6: #mark(nats(X)) -> #mark(X)
   #7: #mark(cons(X1,X2)) -> #mark(X1)
   #8: #mark(filter(X1,X2,X3)) -> #a__filter(mark(X1),mark(X2),mark(X3))
   #9: #mark(filter(X1,X2,X3)) -> #mark(X1)
   #10: #mark(filter(X1,X2,X3)) -> #mark(X2)
   #11: #mark(filter(X1,X2,X3)) -> #mark(X3)
   #12: #mark(zprimes()) -> #a__zprimes()
   #13: #a__nats(N) -> #mark(N)
   #14: #mark(sieve(X)) -> #a__sieve(mark(X))
   #15: #mark(sieve(X)) -> #mark(X)
   #16: #a__sieve(cons(s(N),Y)) -> #mark(N)
Number of SCCs: 1, DPs: 16, edges: 117
	SCC { #1..16 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   |0|()	weight: 0
zprimes()	weight: 0
     s(x1)	weight: x1
a__zprimes()	weight: 0
a__nats(x1)	weight: x1
#a__filter(x1,x2,x3)	weight: (/ 1 4) + x1
a__sieve(x1)	weight: x1
#a__zprimes()	weight: 0
#mark(x1)	weight: x1
#a__nats(x1)	weight: x1
  mark(x1)	weight: x1
 sieve(x1)	weight: x1
  nats(x1)	weight: x1
  cons(x1,x2)	weight: x1
filter(x1,x2,x3)	weight: (/ 1 2) + x1 + x2 + x3
#a__sieve(x1)	weight: x1
a__filter(x1,x2,x3)	weight: (/ 1 2) + x1 + x2 + x3
    Usable rules: { 1..17 }
    Removed DPs: #1 #8..11
Number of SCCs: 1, DPs: 11, edges: 48
	SCC { #2..7 #12..16 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded.
   |0|()	weight: (/ 1 4)
zprimes()	weight: (/ 5 4)
     s(x1)	weight: (/ 1 8) + x1
a__zprimes()	weight: (/ 11 8)
a__nats(x1)	weight: (/ 3 8) + x1
#a__filter(x1,x2,x3)	weight: 0
a__sieve(x1)	weight: (/ 3 8) + x1
#a__zprimes()	weight: (/ 9 8)
#mark(x1)	weight: x1
#a__nats(x1)	weight: (/ 1 8) + x1
  mark(x1)	weight: (/ 1 8) + x1
 sieve(x1)	weight: (/ 3 8) + x1
  nats(x1)	weight: (/ 3 8) + x1
  cons(x1,x2)	weight: max{0, (/ 1 8) + x1}
filter(x1,x2,x3)	weight: max{0, (/ 3 8) + x3, (/ 1 8) + x1}
#a__sieve(x1)	weight: (/ 1 8) + x1
a__filter(x1,x2,x3)	weight: max{0, (/ 3 8) + x3, (/ 1 8) + x1}
    Usable rules: { 1..17 }
    Removed DPs: #2..7 #12..16
Number of SCCs: 0, DPs: 0, edges: 0
YES