YES
We show the termination of the TRS R:

  a(a(divides(),|0|()),a(s(),y)) -> true()
  a(a(divides(),a(s(),x)),a(s(),y)) -> a(a(a(div2(),x),a(s(),y)),y)
  a(a(a(div2(),x),y),|0|()) -> a(a(divides(),x),y)
  a(a(a(div2(),|0|()),y),a(s(),z)) -> false()
  a(a(a(div2(),a(s(),x)),y),a(s(),z)) -> a(a(a(div2(),x),y),z)
  a(a(filter(),f),nil()) -> nil()
  a(a(filter(),f),a(a(cons(),x),xs)) -> a(a(a(if(),a(f,x)),x),a(a(filter(),f),xs))
  a(a(a(if(),true()),x),xs) -> a(a(cons(),x),xs)
  a(a(a(if(),false()),x),xs) -> xs
  a(a(not(),f),x) -> a(not2(),a(f,x))
  a(not2(),true()) -> false()
  a(not2(),false()) -> true()
  a(sieve(),nil()) -> nil()
  a(sieve(),a(a(cons(),x),xs)) -> a(a(cons(),x),a(sieve(),a(a(filter(),a(not(),a(divides(),x))),xs)))

Instead we prove termination of the uncurried system:

  a(cons_0(),y) -> cons_1(y)
  a(cons_1(x1),y) -> cons_2(x1,y)
  a(div2_0(),y) -> div2_1(y)
  a(div2_1(x1),y) -> div2_2(x1,y)
  a(div2_2(x1,x2),y) -> div2_3(x1,x2,y)
  a(divides_0(),y) -> divides_1(y)
  a(divides_1(x1),y) -> divides_2(x1,y)
  a(filter_0(),y) -> filter_1(y)
  a(filter_1(x1),y) -> filter_2(x1,y)
  a(if_0(),y) -> if_1(y)
  a(if_1(x1),y) -> if_2(x1,y)
  a(if_2(x1,x2),y) -> if_3(x1,x2,y)
  a(not_0(),y) -> not_1(y)
  a(not_1(x1),y) -> not_2(x1,y)
  a(not2_0(),y) -> not2_1(y)
  a(s_0(),y) -> s_1(y)
  a(sieve_0(),y) -> sieve_1(y)
  divides_2(|0|_0(),s_1(y)) -> true_0()
  divides_2(s_1(x),s_1(y)) -> div2_3(x,s_1(y),y)
  div2_3(x,y,|0|_0()) -> divides_2(x,y)
  div2_3(|0|_0(),y,s_1(z)) -> false_0()
  div2_3(s_1(x),y,s_1(z)) -> div2_3(x,y,z)
  filter_2(f,nil_0()) -> nil_0()
  filter_2(f,cons_2(x,xs)) -> if_3(a(f,x),x,filter_2(f,xs))
  if_3(true_0(),x,xs) -> cons_2(x,xs)
  if_3(false_0(),x,xs) -> xs
  not_2(f,x) -> not2_1(a(f,x))
  not2_1(true_0()) -> false_0()
  not2_1(false_0()) -> true_0()
  sieve_1(nil_0()) -> nil_0()
  sieve_1(cons_2(x,xs)) -> cons_2(x,sieve_1(filter_2(not_1(divides_1(x)),xs)))


-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: a#(div2_2(x1,x2),y) -> div2_3#(x1,x2,y)
p2: a#(divides_1(x1),y) -> divides_2#(x1,y)
p3: a#(filter_1(x1),y) -> filter_2#(x1,y)
p4: a#(if_2(x1,x2),y) -> if_3#(x1,x2,y)
p5: a#(not_1(x1),y) -> not_2#(x1,y)
p6: a#(not2_0(),y) -> not2_1#(y)
p7: a#(sieve_0(),y) -> sieve_1#(y)
p8: divides_2#(s_1(x),s_1(y)) -> div2_3#(x,s_1(y),y)
p9: div2_3#(x,y,|0|_0()) -> divides_2#(x,y)
p10: div2_3#(s_1(x),y,s_1(z)) -> div2_3#(x,y,z)
p11: filter_2#(f,cons_2(x,xs)) -> if_3#(a(f,x),x,filter_2(f,xs))
p12: filter_2#(f,cons_2(x,xs)) -> a#(f,x)
p13: filter_2#(f,cons_2(x,xs)) -> filter_2#(f,xs)
p14: not_2#(f,x) -> not2_1#(a(f,x))
p15: not_2#(f,x) -> a#(f,x)
p16: sieve_1#(cons_2(x,xs)) -> sieve_1#(filter_2(not_1(divides_1(x)),xs))
p17: sieve_1#(cons_2(x,xs)) -> filter_2#(not_1(divides_1(x)),xs)

and R consists of:

r1: a(cons_0(),y) -> cons_1(y)
r2: a(cons_1(x1),y) -> cons_2(x1,y)
r3: a(div2_0(),y) -> div2_1(y)
r4: a(div2_1(x1),y) -> div2_2(x1,y)
r5: a(div2_2(x1,x2),y) -> div2_3(x1,x2,y)
r6: a(divides_0(),y) -> divides_1(y)
r7: a(divides_1(x1),y) -> divides_2(x1,y)
r8: a(filter_0(),y) -> filter_1(y)
r9: a(filter_1(x1),y) -> filter_2(x1,y)
r10: a(if_0(),y) -> if_1(y)
r11: a(if_1(x1),y) -> if_2(x1,y)
r12: a(if_2(x1,x2),y) -> if_3(x1,x2,y)
r13: a(not_0(),y) -> not_1(y)
r14: a(not_1(x1),y) -> not_2(x1,y)
r15: a(not2_0(),y) -> not2_1(y)
r16: a(s_0(),y) -> s_1(y)
r17: a(sieve_0(),y) -> sieve_1(y)
r18: divides_2(|0|_0(),s_1(y)) -> true_0()
r19: divides_2(s_1(x),s_1(y)) -> div2_3(x,s_1(y),y)
r20: div2_3(x,y,|0|_0()) -> divides_2(x,y)
r21: div2_3(|0|_0(),y,s_1(z)) -> false_0()
r22: div2_3(s_1(x),y,s_1(z)) -> div2_3(x,y,z)
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> if_3(a(f,x),x,filter_2(f,xs))
r25: if_3(true_0(),x,xs) -> cons_2(x,xs)
r26: if_3(false_0(),x,xs) -> xs
r27: not_2(f,x) -> not2_1(a(f,x))
r28: not2_1(true_0()) -> false_0()
r29: not2_1(false_0()) -> true_0()
r30: sieve_1(nil_0()) -> nil_0()
r31: sieve_1(cons_2(x,xs)) -> cons_2(x,sieve_1(filter_2(not_1(divides_1(x)),xs)))

The estimated dependency graph contains the following SCCs:

  {p3, p5, p7, p12, p13, p15, p16, p17}
  {p8, p9, p10}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: not_2#(f,x) -> a#(f,x)
p2: a#(sieve_0(),y) -> sieve_1#(y)
p3: sieve_1#(cons_2(x,xs)) -> filter_2#(not_1(divides_1(x)),xs)
p4: filter_2#(f,cons_2(x,xs)) -> filter_2#(f,xs)
p5: filter_2#(f,cons_2(x,xs)) -> a#(f,x)
p6: a#(not_1(x1),y) -> not_2#(x1,y)
p7: a#(filter_1(x1),y) -> filter_2#(x1,y)
p8: sieve_1#(cons_2(x,xs)) -> sieve_1#(filter_2(not_1(divides_1(x)),xs))

and R consists of:

r1: a(cons_0(),y) -> cons_1(y)
r2: a(cons_1(x1),y) -> cons_2(x1,y)
r3: a(div2_0(),y) -> div2_1(y)
r4: a(div2_1(x1),y) -> div2_2(x1,y)
r5: a(div2_2(x1,x2),y) -> div2_3(x1,x2,y)
r6: a(divides_0(),y) -> divides_1(y)
r7: a(divides_1(x1),y) -> divides_2(x1,y)
r8: a(filter_0(),y) -> filter_1(y)
r9: a(filter_1(x1),y) -> filter_2(x1,y)
r10: a(if_0(),y) -> if_1(y)
r11: a(if_1(x1),y) -> if_2(x1,y)
r12: a(if_2(x1,x2),y) -> if_3(x1,x2,y)
r13: a(not_0(),y) -> not_1(y)
r14: a(not_1(x1),y) -> not_2(x1,y)
r15: a(not2_0(),y) -> not2_1(y)
r16: a(s_0(),y) -> s_1(y)
r17: a(sieve_0(),y) -> sieve_1(y)
r18: divides_2(|0|_0(),s_1(y)) -> true_0()
r19: divides_2(s_1(x),s_1(y)) -> div2_3(x,s_1(y),y)
r20: div2_3(x,y,|0|_0()) -> divides_2(x,y)
r21: div2_3(|0|_0(),y,s_1(z)) -> false_0()
r22: div2_3(s_1(x),y,s_1(z)) -> div2_3(x,y,z)
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> if_3(a(f,x),x,filter_2(f,xs))
r25: if_3(true_0(),x,xs) -> cons_2(x,xs)
r26: if_3(false_0(),x,xs) -> xs
r27: not_2(f,x) -> not2_1(a(f,x))
r28: not2_1(true_0()) -> false_0()
r29: not2_1(false_0()) -> true_0()
r30: sieve_1(nil_0()) -> nil_0()
r31: sieve_1(cons_2(x,xs)) -> cons_2(x,sieve_1(filter_2(not_1(divides_1(x)),xs)))

The set of usable rules consists of

  r1, r3, r8, r10, r13, r15, r16, r21, r23, r24, r25, r26, r28, r29

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        not_2#_A(x1,x2) = max{24, x1 + 14, x2 + 23}
        a#_A(x1,x2) = max{x1 + 14, x2 + 23}
        sieve_0_A = 60
        sieve_1#_A(x1) = max{74, x1 + 23}
        cons_2_A(x1,x2) = max{26, x1 + 7, x2}
        filter_2#_A(x1,x2) = max{x1 + 15, x2 + 17}
        not_1_A(x1) = max{10, x1}
        divides_1_A(x1) = x1 + 14
        filter_1_A(x1) = max{11, x1 + 7}
        filter_2_A(x1,x2) = max{50, x1 - 25, x2}
        a_A(x1,x2) = max{11, x2 + 8}
        cons_0_A = 1
        cons_1_A(x1) = max{9, x1 - 1}
        div2_0_A = 1
        div2_1_A(x1) = max{9, x1 - 12}
        filter_0_A = 12
        if_0_A = 1
        if_1_A(x1) = max{0, x1 - 12}
        not_0_A = 1
        not2_0_A = 1
        not2_1_A(x1) = max{11, x1 - 12}
        s_0_A = 1
        s_1_A(x1) = max{11, x1 - 14}
        div2_3_A(x1,x2,x3) = max{13, x1 - 13, x2 - 26, x3 - 13}
        |0|_0_A = 12
        false_0_A = 11
        nil_0_A = 0
        if_3_A(x1,x2,x3) = max{26, x2 + 7, x3}
        true_0_A = 11
  
    base order 2:
  
      precedence:
      
        a > filter_2 = div2_3 > not_1 = not2_1 = s_1 = |0|_0 = if_3 > cons_2 = filter_1 = if_0 = not_0 = not2_0 = s_0 = false_0 > not_2# = a# = cons_1 = div2_1 = filter_0 = if_1 = true_0 > cons_0 = div2_0 > divides_1 > sieve_1# > sieve_0 = filter_2# = nil_0
      
  
    partial status:
  
      pi(not_2#) = []
      pi(a#) = []
      pi(sieve_0) = []
      pi(sieve_1#) = []
      pi(cons_2) = [1, 2]
      pi(filter_2#) = []
      pi(not_1) = [1]
      pi(divides_1) = [1]
      pi(filter_1) = []
      pi(filter_2) = [2]
      pi(a) = [2]
      pi(cons_0) = []
      pi(cons_1) = []
      pi(div2_0) = []
      pi(div2_1) = []
      pi(filter_0) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(not_0) = []
      pi(not2_0) = []
      pi(not2_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(div2_3) = []
      pi(|0|_0) = []
      pi(false_0) = []
      pi(nil_0) = []
      pi(if_3) = [2, 3]
      pi(true_0) = []

The following rules are removed:

  p3



-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: not_2#(f,x) -> a#(f,x)
p2: a#(sieve_0(),y) -> sieve_1#(y)
p3: filter_2#(f,cons_2(x,xs)) -> filter_2#(f,xs)
p4: filter_2#(f,cons_2(x,xs)) -> a#(f,x)
p5: a#(not_1(x1),y) -> not_2#(x1,y)
p6: a#(filter_1(x1),y) -> filter_2#(x1,y)
p7: sieve_1#(cons_2(x,xs)) -> sieve_1#(filter_2(not_1(divides_1(x)),xs))

and R consists of:

r1: a(cons_0(),y) -> cons_1(y)
r2: a(cons_1(x1),y) -> cons_2(x1,y)
r3: a(div2_0(),y) -> div2_1(y)
r4: a(div2_1(x1),y) -> div2_2(x1,y)
r5: a(div2_2(x1,x2),y) -> div2_3(x1,x2,y)
r6: a(divides_0(),y) -> divides_1(y)
r7: a(divides_1(x1),y) -> divides_2(x1,y)
r8: a(filter_0(),y) -> filter_1(y)
r9: a(filter_1(x1),y) -> filter_2(x1,y)
r10: a(if_0(),y) -> if_1(y)
r11: a(if_1(x1),y) -> if_2(x1,y)
r12: a(if_2(x1,x2),y) -> if_3(x1,x2,y)
r13: a(not_0(),y) -> not_1(y)
r14: a(not_1(x1),y) -> not_2(x1,y)
r15: a(not2_0(),y) -> not2_1(y)
r16: a(s_0(),y) -> s_1(y)
r17: a(sieve_0(),y) -> sieve_1(y)
r18: divides_2(|0|_0(),s_1(y)) -> true_0()
r19: divides_2(s_1(x),s_1(y)) -> div2_3(x,s_1(y),y)
r20: div2_3(x,y,|0|_0()) -> divides_2(x,y)
r21: div2_3(|0|_0(),y,s_1(z)) -> false_0()
r22: div2_3(s_1(x),y,s_1(z)) -> div2_3(x,y,z)
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> if_3(a(f,x),x,filter_2(f,xs))
r25: if_3(true_0(),x,xs) -> cons_2(x,xs)
r26: if_3(false_0(),x,xs) -> xs
r27: not_2(f,x) -> not2_1(a(f,x))
r28: not2_1(true_0()) -> false_0()
r29: not2_1(false_0()) -> true_0()
r30: sieve_1(nil_0()) -> nil_0()
r31: sieve_1(cons_2(x,xs)) -> cons_2(x,sieve_1(filter_2(not_1(divides_1(x)),xs)))

The estimated dependency graph contains the following SCCs:

  {p1, p3, p4, p5, p6}
  {p7}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: not_2#(f,x) -> a#(f,x)
p2: a#(filter_1(x1),y) -> filter_2#(x1,y)
p3: filter_2#(f,cons_2(x,xs)) -> a#(f,x)
p4: a#(not_1(x1),y) -> not_2#(x1,y)
p5: filter_2#(f,cons_2(x,xs)) -> filter_2#(f,xs)

and R consists of:

r1: a(cons_0(),y) -> cons_1(y)
r2: a(cons_1(x1),y) -> cons_2(x1,y)
r3: a(div2_0(),y) -> div2_1(y)
r4: a(div2_1(x1),y) -> div2_2(x1,y)
r5: a(div2_2(x1,x2),y) -> div2_3(x1,x2,y)
r6: a(divides_0(),y) -> divides_1(y)
r7: a(divides_1(x1),y) -> divides_2(x1,y)
r8: a(filter_0(),y) -> filter_1(y)
r9: a(filter_1(x1),y) -> filter_2(x1,y)
r10: a(if_0(),y) -> if_1(y)
r11: a(if_1(x1),y) -> if_2(x1,y)
r12: a(if_2(x1,x2),y) -> if_3(x1,x2,y)
r13: a(not_0(),y) -> not_1(y)
r14: a(not_1(x1),y) -> not_2(x1,y)
r15: a(not2_0(),y) -> not2_1(y)
r16: a(s_0(),y) -> s_1(y)
r17: a(sieve_0(),y) -> sieve_1(y)
r18: divides_2(|0|_0(),s_1(y)) -> true_0()
r19: divides_2(s_1(x),s_1(y)) -> div2_3(x,s_1(y),y)
r20: div2_3(x,y,|0|_0()) -> divides_2(x,y)
r21: div2_3(|0|_0(),y,s_1(z)) -> false_0()
r22: div2_3(s_1(x),y,s_1(z)) -> div2_3(x,y,z)
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> if_3(a(f,x),x,filter_2(f,xs))
r25: if_3(true_0(),x,xs) -> cons_2(x,xs)
r26: if_3(false_0(),x,xs) -> xs
r27: not_2(f,x) -> not2_1(a(f,x))
r28: not2_1(true_0()) -> false_0()
r29: not2_1(false_0()) -> true_0()
r30: sieve_1(nil_0()) -> nil_0()
r31: sieve_1(cons_2(x,xs)) -> cons_2(x,sieve_1(filter_2(not_1(divides_1(x)),xs)))

The set of usable rules consists of

  r1, r3, r4, r6, r8, r11, r13, r15, r16, r18, r21, r28, r29, r30

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        not_2#_A(x1,x2) = max{26, x2 + 1}
        a#_A(x1,x2) = max{26, x2 + 1}
        filter_1_A(x1) = max{21, x1 - 25}
        filter_2#_A(x1,x2) = max{2, x2}
        cons_2_A(x1,x2) = max{26, x1 + 13, x2 + 3}
        not_1_A(x1) = max{4, x1 - 25}
        a_A(x1,x2) = max{31, x1 - 28, x2 + 20}
        cons_0_A = 1
        cons_1_A(x1) = max{27, x1 - 1}
        div2_0_A = 20
        div2_1_A(x1) = max{21, x1 - 1}
        div2_2_A(x1,x2) = max{23, x1 - 30, x2 - 1}
        divides_0_A = 23
        divides_1_A(x1) = max{24, x1}
        filter_0_A = 20
        if_1_A(x1) = max{32, x1 - 1}
        if_2_A(x1,x2) = max{29, x1 - 30, x2 - 1}
        not_0_A = 3
        not2_0_A = 21
        not2_1_A(x1) = max{22, x1 - 33}
        s_0_A = 20
        s_1_A(x1) = max{21, x1 - 10}
        divides_2_A(x1,x2) = max{25, x1 - 31, x2 - 22}
        |0|_0_A = 24
        true_0_A = 1
        div2_3_A(x1,x2,x3) = max{22, x1 - 59, x2 - 30, x3 - 33}
        false_0_A = 2
        sieve_1_A(x1) = max{24, x1 - 28}
        nil_0_A = 6
  
    base order 2:
  
      precedence:
      
        a > if_2 = s_1 = divides_2 = div2_3 > cons_2 = div2_2 = if_1 = not2_1 = s_0 = |0|_0 > cons_1 = div2_1 = not2_0 = true_0 = sieve_1 > not_1 = cons_0 = div2_0 = divides_1 = false_0 = nil_0 > not_2# = a# = divides_0 = not_0 > filter_2# > filter_1 > filter_0
      
  
    partial status:
  
      pi(not_2#) = []
      pi(a#) = []
      pi(filter_1) = []
      pi(filter_2#) = [2]
      pi(cons_2) = [1, 2]
      pi(not_1) = []
      pi(a) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(div2_0) = []
      pi(div2_1) = []
      pi(div2_2) = []
      pi(divides_0) = []
      pi(divides_1) = [1]
      pi(filter_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(not_0) = []
      pi(not2_0) = []
      pi(not2_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(divides_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(div2_3) = []
      pi(false_0) = []
      pi(sieve_1) = []
      pi(nil_0) = []

The following rules are removed:

  p3



-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: not_2#(f,x) -> a#(f,x)
p2: a#(filter_1(x1),y) -> filter_2#(x1,y)
p3: a#(not_1(x1),y) -> not_2#(x1,y)
p4: filter_2#(f,cons_2(x,xs)) -> filter_2#(f,xs)

and R consists of:

r1: a(cons_0(),y) -> cons_1(y)
r2: a(cons_1(x1),y) -> cons_2(x1,y)
r3: a(div2_0(),y) -> div2_1(y)
r4: a(div2_1(x1),y) -> div2_2(x1,y)
r5: a(div2_2(x1,x2),y) -> div2_3(x1,x2,y)
r6: a(divides_0(),y) -> divides_1(y)
r7: a(divides_1(x1),y) -> divides_2(x1,y)
r8: a(filter_0(),y) -> filter_1(y)
r9: a(filter_1(x1),y) -> filter_2(x1,y)
r10: a(if_0(),y) -> if_1(y)
r11: a(if_1(x1),y) -> if_2(x1,y)
r12: a(if_2(x1,x2),y) -> if_3(x1,x2,y)
r13: a(not_0(),y) -> not_1(y)
r14: a(not_1(x1),y) -> not_2(x1,y)
r15: a(not2_0(),y) -> not2_1(y)
r16: a(s_0(),y) -> s_1(y)
r17: a(sieve_0(),y) -> sieve_1(y)
r18: divides_2(|0|_0(),s_1(y)) -> true_0()
r19: divides_2(s_1(x),s_1(y)) -> div2_3(x,s_1(y),y)
r20: div2_3(x,y,|0|_0()) -> divides_2(x,y)
r21: div2_3(|0|_0(),y,s_1(z)) -> false_0()
r22: div2_3(s_1(x),y,s_1(z)) -> div2_3(x,y,z)
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> if_3(a(f,x),x,filter_2(f,xs))
r25: if_3(true_0(),x,xs) -> cons_2(x,xs)
r26: if_3(false_0(),x,xs) -> xs
r27: not_2(f,x) -> not2_1(a(f,x))
r28: not2_1(true_0()) -> false_0()
r29: not2_1(false_0()) -> true_0()
r30: sieve_1(nil_0()) -> nil_0()
r31: sieve_1(cons_2(x,xs)) -> cons_2(x,sieve_1(filter_2(not_1(divides_1(x)),xs)))

The estimated dependency graph contains the following SCCs:

  {p1, p3}
  {p4}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: not_2#(f,x) -> a#(f,x)
p2: a#(not_1(x1),y) -> not_2#(x1,y)

and R consists of:

r1: a(cons_0(),y) -> cons_1(y)
r2: a(cons_1(x1),y) -> cons_2(x1,y)
r3: a(div2_0(),y) -> div2_1(y)
r4: a(div2_1(x1),y) -> div2_2(x1,y)
r5: a(div2_2(x1,x2),y) -> div2_3(x1,x2,y)
r6: a(divides_0(),y) -> divides_1(y)
r7: a(divides_1(x1),y) -> divides_2(x1,y)
r8: a(filter_0(),y) -> filter_1(y)
r9: a(filter_1(x1),y) -> filter_2(x1,y)
r10: a(if_0(),y) -> if_1(y)
r11: a(if_1(x1),y) -> if_2(x1,y)
r12: a(if_2(x1,x2),y) -> if_3(x1,x2,y)
r13: a(not_0(),y) -> not_1(y)
r14: a(not_1(x1),y) -> not_2(x1,y)
r15: a(not2_0(),y) -> not2_1(y)
r16: a(s_0(),y) -> s_1(y)
r17: a(sieve_0(),y) -> sieve_1(y)
r18: divides_2(|0|_0(),s_1(y)) -> true_0()
r19: divides_2(s_1(x),s_1(y)) -> div2_3(x,s_1(y),y)
r20: div2_3(x,y,|0|_0()) -> divides_2(x,y)
r21: div2_3(|0|_0(),y,s_1(z)) -> false_0()
r22: div2_3(s_1(x),y,s_1(z)) -> div2_3(x,y,z)
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> if_3(a(f,x),x,filter_2(f,xs))
r25: if_3(true_0(),x,xs) -> cons_2(x,xs)
r26: if_3(false_0(),x,xs) -> xs
r27: not_2(f,x) -> not2_1(a(f,x))
r28: not2_1(true_0()) -> false_0()
r29: not2_1(false_0()) -> true_0()
r30: sieve_1(nil_0()) -> nil_0()
r31: sieve_1(cons_2(x,xs)) -> cons_2(x,sieve_1(filter_2(not_1(divides_1(x)),xs)))

The set of usable rules consists of

  r1, r3, r4, r8, r10, r11, r13, r15, r16, r18, r21, r25, r28, r29

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        not_2#_A(x1,x2) = max{0, x1 - 1}
        a#_A(x1,x2) = max{0, x1 - 1}
        not_1_A(x1) = max{9, x1 + 2}
        a_A(x1,x2) = max{12, x2 + 4}
        cons_0_A = 7
        cons_1_A(x1) = 6
        div2_0_A = 4
        div2_1_A(x1) = 11
        div2_2_A(x1,x2) = 10
        filter_0_A = 7
        filter_1_A(x1) = 6
        if_0_A = 7
        if_1_A(x1) = 6
        if_2_A(x1,x2) = max{7, x2 - 2}
        not_0_A = 6
        not2_0_A = 10
        not2_1_A(x1) = max{9, x1 + 3}
        s_0_A = 6
        s_1_A(x1) = max{11, x1 - 14}
        divides_2_A(x1,x2) = max{23, x1 - 15, x2 - 13}
        |0|_0_A = 13
        true_0_A = 8
        div2_3_A(x1,x2,x3) = max{24, x1 - 14, x2 - 12, x3 - 12}
        false_0_A = 10
        if_3_A(x1,x2,x3) = max{13, x1 - 30, x2 - 2, x3 - 10}
        cons_2_A(x1,x2) = max{13, x1 - 2, x2 - 10}
  
    base order 2:
  
      precedence:
      
        if_3 > not_2# = filter_0 = if_0 = div2_3 = cons_2 > a# = not_1 > a = cons_0 = not_0 > cons_1 = div2_1 = filter_1 = if_2 = s_1 > div2_0 = div2_2 = if_1 = s_0 = divides_2 > not2_1 = |0|_0 > not2_0 = false_0 > true_0
      
  
    partial status:
  
      pi(not_2#) = []
      pi(a#) = []
      pi(not_1) = []
      pi(a) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(div2_0) = []
      pi(div2_1) = []
      pi(div2_2) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(not_0) = []
      pi(not2_0) = []
      pi(not2_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(divides_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(div2_3) = []
      pi(false_0) = []
      pi(if_3) = []
      pi(cons_2) = []

The following rules are removed:

  p2



-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: not_2#(f,x) -> a#(f,x)

and R consists of:

r1: a(cons_0(),y) -> cons_1(y)
r2: a(cons_1(x1),y) -> cons_2(x1,y)
r3: a(div2_0(),y) -> div2_1(y)
r4: a(div2_1(x1),y) -> div2_2(x1,y)
r5: a(div2_2(x1,x2),y) -> div2_3(x1,x2,y)
r6: a(divides_0(),y) -> divides_1(y)
r7: a(divides_1(x1),y) -> divides_2(x1,y)
r8: a(filter_0(),y) -> filter_1(y)
r9: a(filter_1(x1),y) -> filter_2(x1,y)
r10: a(if_0(),y) -> if_1(y)
r11: a(if_1(x1),y) -> if_2(x1,y)
r12: a(if_2(x1,x2),y) -> if_3(x1,x2,y)
r13: a(not_0(),y) -> not_1(y)
r14: a(not_1(x1),y) -> not_2(x1,y)
r15: a(not2_0(),y) -> not2_1(y)
r16: a(s_0(),y) -> s_1(y)
r17: a(sieve_0(),y) -> sieve_1(y)
r18: divides_2(|0|_0(),s_1(y)) -> true_0()
r19: divides_2(s_1(x),s_1(y)) -> div2_3(x,s_1(y),y)
r20: div2_3(x,y,|0|_0()) -> divides_2(x,y)
r21: div2_3(|0|_0(),y,s_1(z)) -> false_0()
r22: div2_3(s_1(x),y,s_1(z)) -> div2_3(x,y,z)
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> if_3(a(f,x),x,filter_2(f,xs))
r25: if_3(true_0(),x,xs) -> cons_2(x,xs)
r26: if_3(false_0(),x,xs) -> xs
r27: not_2(f,x) -> not2_1(a(f,x))
r28: not2_1(true_0()) -> false_0()
r29: not2_1(false_0()) -> true_0()
r30: sieve_1(nil_0()) -> nil_0()
r31: sieve_1(cons_2(x,xs)) -> cons_2(x,sieve_1(filter_2(not_1(divides_1(x)),xs)))

The estimated dependency graph contains the following SCCs:

  (no SCCs)

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: filter_2#(f,cons_2(x,xs)) -> filter_2#(f,xs)

and R consists of:

r1: a(cons_0(),y) -> cons_1(y)
r2: a(cons_1(x1),y) -> cons_2(x1,y)
r3: a(div2_0(),y) -> div2_1(y)
r4: a(div2_1(x1),y) -> div2_2(x1,y)
r5: a(div2_2(x1,x2),y) -> div2_3(x1,x2,y)
r6: a(divides_0(),y) -> divides_1(y)
r7: a(divides_1(x1),y) -> divides_2(x1,y)
r8: a(filter_0(),y) -> filter_1(y)
r9: a(filter_1(x1),y) -> filter_2(x1,y)
r10: a(if_0(),y) -> if_1(y)
r11: a(if_1(x1),y) -> if_2(x1,y)
r12: a(if_2(x1,x2),y) -> if_3(x1,x2,y)
r13: a(not_0(),y) -> not_1(y)
r14: a(not_1(x1),y) -> not_2(x1,y)
r15: a(not2_0(),y) -> not2_1(y)
r16: a(s_0(),y) -> s_1(y)
r17: a(sieve_0(),y) -> sieve_1(y)
r18: divides_2(|0|_0(),s_1(y)) -> true_0()
r19: divides_2(s_1(x),s_1(y)) -> div2_3(x,s_1(y),y)
r20: div2_3(x,y,|0|_0()) -> divides_2(x,y)
r21: div2_3(|0|_0(),y,s_1(z)) -> false_0()
r22: div2_3(s_1(x),y,s_1(z)) -> div2_3(x,y,z)
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> if_3(a(f,x),x,filter_2(f,xs))
r25: if_3(true_0(),x,xs) -> cons_2(x,xs)
r26: if_3(false_0(),x,xs) -> xs
r27: not_2(f,x) -> not2_1(a(f,x))
r28: not2_1(true_0()) -> false_0()
r29: not2_1(false_0()) -> true_0()
r30: sieve_1(nil_0()) -> nil_0()
r31: sieve_1(cons_2(x,xs)) -> cons_2(x,sieve_1(filter_2(not_1(divides_1(x)),xs)))

The set of usable rules consists of

  r1, r3, r6, r8, r10, r13, r15, r16, r18, r21, r28, r29, r30

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        filter_2#_A(x1,x2) = max{x1 + 11, x2 + 26}
        cons_2_A(x1,x2) = max{9, x1 - 15, x2 + 2}
        a_A(x1,x2) = max{8, x2 + 3}
        cons_0_A = 5
        cons_1_A(x1) = max{4, x1 - 29}
        div2_0_A = 1
        div2_1_A(x1) = max{2, x1 - 29}
        divides_0_A = 1
        divides_1_A(x1) = max{7, x1 - 29}
        filter_0_A = 1
        filter_1_A(x1) = max{4, x1 - 29}
        if_0_A = 1
        if_1_A(x1) = max{4, x1 - 29}
        not_0_A = 5
        not_1_A(x1) = max{4, x1 - 29}
        not2_0_A = 1
        not2_1_A(x1) = max{4, x1 - 30}
        s_0_A = 1
        s_1_A(x1) = max{4, x1 - 31}
        divides_2_A(x1,x2) = max{9, x1 - 31, x2 - 68}
        |0|_0_A = 3
        true_0_A = 1
        div2_3_A(x1,x2,x3) = max{10, x1 - 30, x2 - 67, x3 - 5}
        false_0_A = 2
        sieve_1_A(x1) = max{8, x1 - 13}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        filter_2# > cons_2 = a = cons_0 = div2_3 > cons_1 = div2_1 = not2_1 = s_1 = sieve_1 > div2_0 = not_1 = not2_0 = s_0 = divides_2 > divides_1 = filter_1 = not_0 = |0|_0 = nil_0 > divides_0 = filter_0 = false_0 > if_1 = true_0 > if_0
      
  
    partial status:
  
      pi(filter_2#) = [2]
      pi(cons_2) = [2]
      pi(a) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(div2_0) = []
      pi(div2_1) = []
      pi(divides_0) = []
      pi(divides_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(not_0) = []
      pi(not_1) = []
      pi(not2_0) = []
      pi(not2_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(divides_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(div2_3) = []
      pi(false_0) = []
      pi(sieve_1) = []
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: sieve_1#(cons_2(x,xs)) -> sieve_1#(filter_2(not_1(divides_1(x)),xs))

and R consists of:

r1: a(cons_0(),y) -> cons_1(y)
r2: a(cons_1(x1),y) -> cons_2(x1,y)
r3: a(div2_0(),y) -> div2_1(y)
r4: a(div2_1(x1),y) -> div2_2(x1,y)
r5: a(div2_2(x1,x2),y) -> div2_3(x1,x2,y)
r6: a(divides_0(),y) -> divides_1(y)
r7: a(divides_1(x1),y) -> divides_2(x1,y)
r8: a(filter_0(),y) -> filter_1(y)
r9: a(filter_1(x1),y) -> filter_2(x1,y)
r10: a(if_0(),y) -> if_1(y)
r11: a(if_1(x1),y) -> if_2(x1,y)
r12: a(if_2(x1,x2),y) -> if_3(x1,x2,y)
r13: a(not_0(),y) -> not_1(y)
r14: a(not_1(x1),y) -> not_2(x1,y)
r15: a(not2_0(),y) -> not2_1(y)
r16: a(s_0(),y) -> s_1(y)
r17: a(sieve_0(),y) -> sieve_1(y)
r18: divides_2(|0|_0(),s_1(y)) -> true_0()
r19: divides_2(s_1(x),s_1(y)) -> div2_3(x,s_1(y),y)
r20: div2_3(x,y,|0|_0()) -> divides_2(x,y)
r21: div2_3(|0|_0(),y,s_1(z)) -> false_0()
r22: div2_3(s_1(x),y,s_1(z)) -> div2_3(x,y,z)
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> if_3(a(f,x),x,filter_2(f,xs))
r25: if_3(true_0(),x,xs) -> cons_2(x,xs)
r26: if_3(false_0(),x,xs) -> xs
r27: not_2(f,x) -> not2_1(a(f,x))
r28: not2_1(true_0()) -> false_0()
r29: not2_1(false_0()) -> true_0()
r30: sieve_1(nil_0()) -> nil_0()
r31: sieve_1(cons_2(x,xs)) -> cons_2(x,sieve_1(filter_2(not_1(divides_1(x)),xs)))

The set of usable rules consists of

  r1, r3, r4, r8, r15, r16, r18, r21, r23, r24, r25, r26, r28, r29, r30

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        sieve_1#_A(x1) = x1 + 3
        cons_2_A(x1,x2) = max{x1 + 1, x2 + 9}
        filter_2_A(x1,x2) = x2
        not_1_A(x1) = max{10, x1 - 1}
        divides_1_A(x1) = max{20, x1 + 11}
        a_A(x1,x2) = max{7, x1 - 21, x2 - 21}
        cons_0_A = 5
        cons_1_A(x1) = max{6, x1 - 22}
        div2_0_A = 20
        div2_1_A(x1) = max{4, x1 - 21}
        div2_2_A(x1,x2) = max{5, x1 - 43, x2 - 22}
        filter_0_A = 1
        filter_1_A(x1) = max{2, x1 - 22}
        not2_0_A = 2
        not2_1_A(x1) = max{3, x1 - 26}
        s_0_A = 4
        s_1_A(x1) = max{5, x1 - 66}
        divides_2_A(x1,x2) = max{4, x1 - 66, x2 - 45}
        |0|_0_A = 3
        true_0_A = 1
        div2_3_A(x1,x2,x3) = max{6, x1 - 65, x2 - 44, x3 - 22}
        false_0_A = 2
        nil_0_A = 1
        if_3_A(x1,x2,x3) = max{x2 + 1, x3 + 9}
        sieve_1_A(x1) = max{8, x1 - 10}
  
    base order 2:
  
      precedence:
      
        not_1 > a > filter_2 = div2_3 > div2_2 = filter_1 = s_1 = if_3 > div2_1 = filter_0 = s_0 = divides_2 > divides_1 = div2_0 = not2_1 = |0|_0 > cons_2 = not2_0 = false_0 > sieve_1# = cons_1 = true_0 = sieve_1 > cons_0 = nil_0
      
  
    partial status:
  
      pi(sieve_1#) = [1]
      pi(cons_2) = [2]
      pi(filter_2) = [2]
      pi(not_1) = []
      pi(divides_1) = [1]
      pi(a) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(div2_0) = []
      pi(div2_1) = []
      pi(div2_2) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(not2_0) = []
      pi(not2_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(divides_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(div2_3) = []
      pi(false_0) = []
      pi(nil_0) = []
      pi(if_3) = [3]
      pi(sieve_1) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: div2_3#(x,y,|0|_0()) -> divides_2#(x,y)
p2: divides_2#(s_1(x),s_1(y)) -> div2_3#(x,s_1(y),y)
p3: div2_3#(s_1(x),y,s_1(z)) -> div2_3#(x,y,z)

and R consists of:

r1: a(cons_0(),y) -> cons_1(y)
r2: a(cons_1(x1),y) -> cons_2(x1,y)
r3: a(div2_0(),y) -> div2_1(y)
r4: a(div2_1(x1),y) -> div2_2(x1,y)
r5: a(div2_2(x1,x2),y) -> div2_3(x1,x2,y)
r6: a(divides_0(),y) -> divides_1(y)
r7: a(divides_1(x1),y) -> divides_2(x1,y)
r8: a(filter_0(),y) -> filter_1(y)
r9: a(filter_1(x1),y) -> filter_2(x1,y)
r10: a(if_0(),y) -> if_1(y)
r11: a(if_1(x1),y) -> if_2(x1,y)
r12: a(if_2(x1,x2),y) -> if_3(x1,x2,y)
r13: a(not_0(),y) -> not_1(y)
r14: a(not_1(x1),y) -> not_2(x1,y)
r15: a(not2_0(),y) -> not2_1(y)
r16: a(s_0(),y) -> s_1(y)
r17: a(sieve_0(),y) -> sieve_1(y)
r18: divides_2(|0|_0(),s_1(y)) -> true_0()
r19: divides_2(s_1(x),s_1(y)) -> div2_3(x,s_1(y),y)
r20: div2_3(x,y,|0|_0()) -> divides_2(x,y)
r21: div2_3(|0|_0(),y,s_1(z)) -> false_0()
r22: div2_3(s_1(x),y,s_1(z)) -> div2_3(x,y,z)
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> if_3(a(f,x),x,filter_2(f,xs))
r25: if_3(true_0(),x,xs) -> cons_2(x,xs)
r26: if_3(false_0(),x,xs) -> xs
r27: not_2(f,x) -> not2_1(a(f,x))
r28: not2_1(true_0()) -> false_0()
r29: not2_1(false_0()) -> true_0()
r30: sieve_1(nil_0()) -> nil_0()
r31: sieve_1(cons_2(x,xs)) -> cons_2(x,sieve_1(filter_2(not_1(divides_1(x)),xs)))

The set of usable rules consists of

  r1, r3, r4, r6, r8, r10, r11, r13, r15, r18, r21, r28, r29, r30

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        div2_3#_A(x1,x2,x3) = x1 + 4
        |0|_0_A = 29
        divides_2#_A(x1,x2) = x1 + 3
        s_1_A(x1) = x1 + 7
        a_A(x1,x2) = max{32, x1 - 6, x2 - 7}
        cons_0_A = 5
        cons_1_A(x1) = max{3, x1 - 8}
        div2_0_A = 5
        div2_1_A(x1) = max{7, x1 - 8}
        div2_2_A(x1,x2) = max{5, x1 - 14, x2 - 7}
        divides_0_A = 2
        divides_1_A(x1) = max{3, x1 - 21}
        filter_0_A = 5
        filter_1_A(x1) = max{7, x1 - 7}
        if_0_A = 1
        if_1_A(x1) = max{2, x1 - 8}
        if_2_A(x1,x2) = max{3, x2 - 8}
        not_0_A = 4
        not_1_A(x1) = max{5, x1 - 8}
        not2_0_A = 7
        not2_1_A(x1) = max{8, x1 - 34}
        divides_2_A(x1,x2) = max{30, x1 - 28, x2 - 38}
        true_0_A = 5
        div2_3_A(x1,x2,x3) = max{31, x1 - 24, x2 - 29, x3 - 30}
        false_0_A = 6
        sieve_1_A(x1) = max{6, x1 - 35}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        divides_2 > div2_3 > div2_3# = s_1 > divides_2# = a > |0|_0 = if_1 = if_2 = not2_1 > cons_0 = div2_2 = if_0 = not2_0 = false_0 > cons_1 = div2_1 = filter_1 = true_0 = sieve_1 > div2_0 = divides_1 = filter_0 = not_1 = nil_0 > divides_0 = not_0
      
  
    partial status:
  
      pi(div2_3#) = []
      pi(|0|_0) = []
      pi(divides_2#) = []
      pi(s_1) = [1]
      pi(a) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(div2_0) = []
      pi(div2_1) = []
      pi(div2_2) = []
      pi(divides_0) = []
      pi(divides_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(not_0) = []
      pi(not_1) = []
      pi(not2_0) = []
      pi(not2_1) = []
      pi(divides_2) = []
      pi(true_0) = []
      pi(div2_3) = []
      pi(false_0) = []
      pi(sieve_1) = []
      pi(nil_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: divides_2#(s_1(x),s_1(y)) -> div2_3#(x,s_1(y),y)
p2: div2_3#(s_1(x),y,s_1(z)) -> div2_3#(x,y,z)

and R consists of:

r1: a(cons_0(),y) -> cons_1(y)
r2: a(cons_1(x1),y) -> cons_2(x1,y)
r3: a(div2_0(),y) -> div2_1(y)
r4: a(div2_1(x1),y) -> div2_2(x1,y)
r5: a(div2_2(x1,x2),y) -> div2_3(x1,x2,y)
r6: a(divides_0(),y) -> divides_1(y)
r7: a(divides_1(x1),y) -> divides_2(x1,y)
r8: a(filter_0(),y) -> filter_1(y)
r9: a(filter_1(x1),y) -> filter_2(x1,y)
r10: a(if_0(),y) -> if_1(y)
r11: a(if_1(x1),y) -> if_2(x1,y)
r12: a(if_2(x1,x2),y) -> if_3(x1,x2,y)
r13: a(not_0(),y) -> not_1(y)
r14: a(not_1(x1),y) -> not_2(x1,y)
r15: a(not2_0(),y) -> not2_1(y)
r16: a(s_0(),y) -> s_1(y)
r17: a(sieve_0(),y) -> sieve_1(y)
r18: divides_2(|0|_0(),s_1(y)) -> true_0()
r19: divides_2(s_1(x),s_1(y)) -> div2_3(x,s_1(y),y)
r20: div2_3(x,y,|0|_0()) -> divides_2(x,y)
r21: div2_3(|0|_0(),y,s_1(z)) -> false_0()
r22: div2_3(s_1(x),y,s_1(z)) -> div2_3(x,y,z)
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> if_3(a(f,x),x,filter_2(f,xs))
r25: if_3(true_0(),x,xs) -> cons_2(x,xs)
r26: if_3(false_0(),x,xs) -> xs
r27: not_2(f,x) -> not2_1(a(f,x))
r28: not2_1(true_0()) -> false_0()
r29: not2_1(false_0()) -> true_0()
r30: sieve_1(nil_0()) -> nil_0()
r31: sieve_1(cons_2(x,xs)) -> cons_2(x,sieve_1(filter_2(not_1(divides_1(x)),xs)))

The estimated dependency graph contains the following SCCs:

  {p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: div2_3#(s_1(x),y,s_1(z)) -> div2_3#(x,y,z)

and R consists of:

r1: a(cons_0(),y) -> cons_1(y)
r2: a(cons_1(x1),y) -> cons_2(x1,y)
r3: a(div2_0(),y) -> div2_1(y)
r4: a(div2_1(x1),y) -> div2_2(x1,y)
r5: a(div2_2(x1,x2),y) -> div2_3(x1,x2,y)
r6: a(divides_0(),y) -> divides_1(y)
r7: a(divides_1(x1),y) -> divides_2(x1,y)
r8: a(filter_0(),y) -> filter_1(y)
r9: a(filter_1(x1),y) -> filter_2(x1,y)
r10: a(if_0(),y) -> if_1(y)
r11: a(if_1(x1),y) -> if_2(x1,y)
r12: a(if_2(x1,x2),y) -> if_3(x1,x2,y)
r13: a(not_0(),y) -> not_1(y)
r14: a(not_1(x1),y) -> not_2(x1,y)
r15: a(not2_0(),y) -> not2_1(y)
r16: a(s_0(),y) -> s_1(y)
r17: a(sieve_0(),y) -> sieve_1(y)
r18: divides_2(|0|_0(),s_1(y)) -> true_0()
r19: divides_2(s_1(x),s_1(y)) -> div2_3(x,s_1(y),y)
r20: div2_3(x,y,|0|_0()) -> divides_2(x,y)
r21: div2_3(|0|_0(),y,s_1(z)) -> false_0()
r22: div2_3(s_1(x),y,s_1(z)) -> div2_3(x,y,z)
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> if_3(a(f,x),x,filter_2(f,xs))
r25: if_3(true_0(),x,xs) -> cons_2(x,xs)
r26: if_3(false_0(),x,xs) -> xs
r27: not_2(f,x) -> not2_1(a(f,x))
r28: not2_1(true_0()) -> false_0()
r29: not2_1(false_0()) -> true_0()
r30: sieve_1(nil_0()) -> nil_0()
r31: sieve_1(cons_2(x,xs)) -> cons_2(x,sieve_1(filter_2(not_1(divides_1(x)),xs)))

The set of usable rules consists of

  r1, r2, r3, r4, r6, r8, r10, r11, r13, r15, r18, r19, r20, r21, r22, r25, r28, r29, r30

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        div2_3#_A(x1,x2,x3) = max{1, x1 - 1, x3 - 4}
        s_1_A(x1) = x1 + 6
        a_A(x1,x2) = max{10, x1 - 7, x2 - 8}
        cons_0_A = 6
        cons_1_A(x1) = max{10, x1 - 8}
        cons_2_A(x1,x2) = max{5, x1 - 104, x2 - 53}
        div2_0_A = 6
        div2_1_A(x1) = max{9, x1 - 9}
        div2_2_A(x1,x2) = max{1, x1 - 16, x2 - 8}
        divides_0_A = 5
        divides_1_A(x1) = max{6, x1 - 25}
        filter_0_A = 2
        filter_1_A(x1) = max{3, x1 - 9}
        if_0_A = 14
        if_1_A(x1) = max{6, x1 - 9}
        if_2_A(x1,x2) = max{1, x1 - 16, x2 - 8}
        not_0_A = 5
        not_1_A(x1) = max{6, x1 - 9}
        not2_0_A = 8
        not2_1_A(x1) = max{9, x1 - 11}
        divides_2_A(x1,x2) = max{11, x1 - 24, x2 - 7}
        |0|_0_A = 6
        true_0_A = 4
        div2_3_A(x1,x2,x3) = max{11, x1 - 24, x2 - 7, x3 - 8}
        false_0_A = 5
        if_3_A(x1,x2,x3) = max{6, x1 - 64, x2 - 103, x3 - 9}
        sieve_1_A(x1) = max{3, x1 - 56}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        div2_3# = divides_1 = divides_2 = div2_3 > s_1 > a = cons_0 = divides_0 = not_0 = |0|_0 > div2_1 = div2_2 = not_1 = not2_1 = if_3 > div2_0 = filter_1 = if_2 = not2_0 = false_0 > cons_1 = cons_2 = filter_0 = if_1 > if_0 = true_0 = sieve_1 > nil_0
      
  
    partial status:
  
      pi(div2_3#) = []
      pi(s_1) = [1]
      pi(a) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      pi(div2_0) = []
      pi(div2_1) = []
      pi(div2_2) = []
      pi(divides_0) = []
      pi(divides_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(not_0) = []
      pi(not_1) = []
      pi(not2_0) = []
      pi(not2_1) = []
      pi(divides_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(div2_3) = []
      pi(false_0) = []
      pi(if_3) = []
      pi(sieve_1) = []
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.