Input TRS:
    1: a__primes() -> a__sieve(a__from(s(s(0()))))
    2: a__from(X) -> cons(mark(X),from(s(X)))
    3: a__head(cons(X,Y)) -> mark(X)
    4: a__tail(cons(X,Y)) -> mark(Y)
    5: a__if(true(),X,Y) -> mark(X)
    6: a__if(false(),X,Y) -> mark(Y)
    7: a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y))))
    8: a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y)))
    9: mark(primes()) -> a__primes()
    10: mark(sieve(X)) -> a__sieve(mark(X))
    11: mark(from(X)) -> a__from(mark(X))
    12: mark(head(X)) -> a__head(mark(X))
    13: mark(tail(X)) -> a__tail(mark(X))
    14: mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
    15: mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2))
    16: mark(s(X)) -> s(mark(X))
    17: mark(0()) -> 0()
    18: mark(cons(X1,X2)) -> cons(mark(X1),X2)
    19: mark(true()) -> true()
    20: mark(false()) -> false()
    21: mark(divides(X1,X2)) -> divides(mark(X1),mark(X2))
    22: a__primes() -> primes()
    23: a__sieve(X) -> sieve(X)
    24: a__from(X) -> from(X)
    25: a__head(X) -> head(X)
    26: a__tail(X) -> tail(X)
    27: a__if(X1,X2,X3) -> if(X1,X2,X3)
    28: a__filter(X1,X2) -> filter(X1,X2)
Number of strict rules: 28
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #a__from(X) -> #mark(X)
   #2: #a__if(false(),X,Y) -> #mark(Y)
   #3: #mark(tail(X)) -> #a__tail(mark(X))
   #4: #mark(tail(X)) -> #mark(X)
   #5: #mark(primes()) -> #a__primes()
   #6: #mark(from(X)) -> #a__from(mark(X))
   #7: #mark(from(X)) -> #mark(X)
   #8: #mark(head(X)) -> #a__head(mark(X))
   #9: #mark(head(X)) -> #mark(X)
   #10: #mark(if(X1,X2,X3)) -> #a__if(mark(X1),X2,X3)
   #11: #mark(if(X1,X2,X3)) -> #mark(X1)
   #12: #a__filter(s(s(X)),cons(Y,Z)) -> #a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y))))
   #13: #a__filter(s(s(X)),cons(Y,Z)) -> #mark(X)
   #14: #a__filter(s(s(X)),cons(Y,Z)) -> #mark(Y)
   #15: #mark(sieve(X)) -> #a__sieve(mark(X))
   #16: #mark(sieve(X)) -> #mark(X)
   #17: #a__if(true(),X,Y) -> #mark(X)
   #18: #mark(divides(X1,X2)) -> #mark(X1)
   #19: #mark(divides(X1,X2)) -> #mark(X2)
   #20: #mark(s(X)) -> #mark(X)
   #21: #a__head(cons(X,Y)) -> #mark(X)
   #22: #a__primes() -> #a__sieve(a__from(s(s(0()))))
   #23: #a__primes() -> #a__from(s(s(0())))
   #24: #a__sieve(cons(X,Y)) -> #mark(X)
   #25: #mark(filter(X1,X2)) -> #a__filter(mark(X1),mark(X2))
   #26: #mark(filter(X1,X2)) -> #mark(X1)
   #27: #mark(filter(X1,X2)) -> #mark(X2)
   #28: #a__tail(cons(X,Y)) -> #mark(Y)
   #29: #mark(cons(X1,X2)) -> #mark(X1)
Number of SCCs: 1, DPs: 28, edges: 354
	SCC { #1..11 #13..29 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded.
     s(x1)	weight: x1
#a__head(x1)	weight: (/ 3 8) + x1
#a__from(x1)	weight: (/ 3 8) + x1
a__from(x1)	weight: (/ 1 4) + x1
 false()	weight: (/ 1 8)
#a__filter(x1,x2)	weight: max{(/ 1 8) + x2, (/ 1 4) + x1}
a__sieve(x1)	weight: (/ 1 4) + x1
  true()	weight: (/ 1 8)
  tail(x1)	weight: (/ 1 4) + x1
#mark(x1)	weight: (/ 1 4) + x1
     0()	weight: (/ 1 8)
    if(x1,x2,x3)	weight: max{x3, x2, x1}
#a__primes()	weight: (/ 7 8)
  from(x1)	weight: (/ 1 4) + x1
  mark(x1)	weight: x1
 sieve(x1)	weight: (/ 1 4) + x1
#a__if(x1,x2,x3)	weight: max{0, (/ 1 4) + x3, (/ 1 4) + x2}
  head(x1)	weight: (/ 1 4) + x1
  cons(x1,x2)	weight: max{x2, (/ 1 4) + x1}
a__primes()	weight: (/ 3 4)
filter(x1,x2)	weight: max{x2, x1}
primes()	weight: (/ 3 4)
#a__sieve(x1)	weight: (/ 3 8) + x1
divides(x1,x2)	weight: max{(/ 1 8) + x2, x1}
a__tail(x1)	weight: (/ 1 4) + x1
a__filter(x1,x2)	weight: max{x2, x1}
#a__tail(x1)	weight: (/ 3 8) + x1
 a__if(x1,x2,x3)	weight: max{x3, x2, x1}
a__head(x1)	weight: (/ 1 4) + x1
    Usable rules: { 1..28 }
    Removed DPs: #1 #3..9 #14..16 #19 #21..24 #28 #29
Number of SCCs: 1, DPs: 10, edges: 59
	SCC { #2 #10 #11 #13 #17 #18 #20 #25..27 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded.
     s(x1)	weight: x1
#a__head(x1)	weight: (/ 3 8) + x1
#a__from(x1)	weight: (/ 3 8) + x1
a__from(x1)	weight: (/ 3 8) + x1
 false()	weight: (/ 1 8)
#a__filter(x1,x2)	weight: max{(/ 1 8) + x2, (/ 3 8) + x1}
a__sieve(x1)	weight: (/ 3 8) + x1
  true()	weight: (/ 1 8)
  tail(x1)	weight: (/ 1 4) + x1
#mark(x1)	weight: (/ 1 4) + x1
     0()	weight: (/ 1 8)
    if(x1,x2,x3)	weight: max{x3, x2, (/ 1 8) + x1}
#a__primes()	weight: (/ 7 8)
  from(x1)	weight: (/ 3 8) + x1
  mark(x1)	weight: x1
 sieve(x1)	weight: (/ 3 8) + x1
#a__if(x1,x2,x3)	weight: max{0, (/ 1 4) + x3, (/ 1 4) + x2}
  head(x1)	weight: (/ 1 4) + x1
  cons(x1,x2)	weight: max{x2, (/ 3 8) + x1}
a__primes()	weight: (/ 7 8)
filter(x1,x2)	weight: max{x2, (/ 1 2) + x1}
primes()	weight: (/ 7 8)
#a__sieve(x1)	weight: (/ 3 8) + x1
divides(x1,x2)	weight: max{(/ 1 4) + x2, (/ 1 8) + x1}
a__tail(x1)	weight: (/ 1 4) + x1
a__filter(x1,x2)	weight: max{x2, (/ 1 2) + x1}
#a__tail(x1)	weight: (/ 3 8) + x1
 a__if(x1,x2,x3)	weight: max{x3, x2, (/ 1 8) + x1}
a__head(x1)	weight: (/ 1 4) + x1
    Usable rules: { 1..28 }
    Removed DPs: #11 #13 #18 #25 #26
Number of SCCs: 1, DPs: 5, edges: 14
	SCC { #2 #10 #17 #20 #27 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     s(x1)	weight: (/ 1 8) + x1
#a__head(x1)	weight: 0
#a__from(x1)	weight: (/ 1 8)
a__from(x1)	weight: (/ 1 4)
 false()	weight: 0
#a__filter(x1,x2)	weight: 0
a__sieve(x1)	weight: (/ 1 8) + x1
  true()	weight: 0
  tail(x1)	weight: (/ 3 8)
#mark(x1)	weight: x1
     0()	weight: 0
    if(x1,x2,x3)	weight: (/ 1 2) + x1 + x2 + x3
#a__primes()	weight: 0
  from(x1)	weight: (/ 3 8)
  mark(x1)	weight: (/ 1 8)
 sieve(x1)	weight: (/ 1 4)
#a__if(x1,x2,x3)	weight: (/ 1 8) + x2 + x3
  head(x1)	weight: (/ 3 8) + x1
  cons(x1,x2)	weight: (/ 3 8)
a__primes()	weight: 0
filter(x1,x2)	weight: (/ 3 8) + x2
primes()	weight: 0
#a__sieve(x1)	weight: 0
divides(x1,x2)	weight: (/ 1 4)
a__tail(x1)	weight: (/ 1 4)
a__filter(x1,x2)	weight: (/ 1 4)
#a__tail(x1)	weight: (/ 1 8)
 a__if(x1,x2,x3)	weight: (/ 3 8)
a__head(x1)	weight: (/ 1 4)
    Usable rules: { }
    Removed DPs: #2 #10 #17 #20 #27
Number of SCCs: 0, DPs: 0, edges: 0
YES