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, r4, r10, r11, r18, r21, r25, r26, r28, r29, r30

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          not_2#_A(x1,x2) = x1 + 1
          a#_A(x1,x2) = x1
          sieve_0_A() = 7
          sieve_1#_A(x1) = 6
          cons_2_A(x1,x2) = x1 + x2
          filter_2#_A(x1,x2) = x1 + 1
          not_1_A(x1) = x1 + 4
          divides_1_A(x1) = 0
          filter_1_A(x1) = x1 + 4
          filter_2_A(x1,x2) = 0
          a_A(x1,x2) = 3
          cons_0_A() = 1
          cons_1_A(x1) = 1
          div2_1_A(x1) = x1 + 2
          div2_2_A(x1,x2) = 1
          if_0_A() = 3
          if_1_A(x1) = 2
          if_2_A(x1,x2) = 1
          divides_2_A(x1,x2) = x1 + x2 + 4
          |0|_0_A() = 3
          s_1_A(x1) = x1 + 4
          true_0_A() = 1
          div2_3_A(x1,x2,x3) = x1 + x2 + x3 + 9
          false_0_A() = 2
          if_3_A(x1,x2,x3) = x2 + x3 + 4
          not2_1_A(x1) = 4
          sieve_1_A(x1) = x1 + 8
          nil_0_A() = 0
  
    base order 2:
  
      precedence:
      
        div2_3 = not2_1 > divides_2 > cons_2 = not_1 = div2_1 = |0|_0 = s_1 = sieve_1 > not_2# = sieve_0 = false_0 > a# = a = if_0 = true_0 > sieve_1# = filter_2 = cons_1 = div2_2 = if_1 > filter_2# = divides_1 = cons_0 = if_3 = nil_0 > filter_1 = if_2
      
  
    partial status:
  
      pi(not_2#) = []
      pi(a#) = [1]
      pi(sieve_0) = []
      pi(sieve_1#) = []
      pi(cons_2) = [2]
      pi(filter_2#) = []
      pi(not_1) = []
      pi(divides_1) = []
      pi(filter_1) = [1]
      pi(filter_2) = []
      pi(a) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(div2_1) = [1]
      pi(div2_2) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(divides_2) = [2]
      pi(|0|_0) = []
      pi(s_1) = [1]
      pi(true_0) = []
      pi(div2_3) = []
      pi(false_0) = []
      pi(if_3) = [3]
      pi(not2_1) = []
      pi(sieve_1) = []
      pi(nil_0) = []

The following rules are removed:

  p6



-- 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: 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#(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:

  {p2, p3, p4, p5, p6, p7}


-- Reduction pair.

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

p1: a#(sieve_0(),y) -> sieve_1#(y)
p2: sieve_1#(cons_2(x,xs)) -> sieve_1#(filter_2(not_1(divides_1(x)),xs))
p3: sieve_1#(cons_2(x,xs)) -> filter_2#(not_1(divides_1(x)),xs)
p4: filter_2#(f,cons_2(x,xs)) -> a#(f,x)
p5: a#(filter_1(x1),y) -> filter_2#(x1,y)
p6: 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, r10, r11, r13, r15, r18, r21, r23, r24, r25, r26, r28, r29, r30

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          a#_A(x1,x2) = x1 + x2 + 18
          sieve_0_A() = 1
          sieve_1#_A(x1) = x1 + 2
          cons_2_A(x1,x2) = x1 + x2 + 17
          filter_2_A(x1,x2) = x1 + x2 + 7
          not_1_A(x1) = x1 + 5
          divides_1_A(x1) = x1 + 2
          filter_2#_A(x1,x2) = x1 + x2 + 9
          filter_1_A(x1) = x1 + 2
          a_A(x1,x2) = x2 + 6
          cons_0_A() = 2
          cons_1_A(x1) = 1
          div2_0_A() = 3
          div2_1_A(x1) = 2
          div2_2_A(x1,x2) = 1
          divides_0_A() = 1
          filter_0_A() = 1
          if_0_A() = 3
          if_1_A(x1) = 2
          if_2_A(x1,x2) = 1
          not_0_A() = 1
          not2_0_A() = 4
          not2_1_A(x1) = 3
          divides_2_A(x1,x2) = x1 + x2 + 8
          |0|_0_A() = 3
          s_1_A(x1) = x1 + 8
          true_0_A() = 1
          div2_3_A(x1,x2,x3) = x1 + 7
          false_0_A() = 2
          nil_0_A() = 1
          if_3_A(x1,x2,x3) = x2 + x3 + 17
          sieve_1_A(x1) = x1 + 2
  
    base order 2:
  
      precedence:
      
        a# > sieve_1# > filter_2 > filter_2# = if_3 > cons_2 = not2_0 = divides_2 > a = cons_0 = div2_0 = if_0 = |0|_0 = div2_3 > filter_1 = cons_1 = div2_1 = if_1 = s_1 = sieve_1 > sieve_0 = not_1 = divides_1 = div2_2 = filter_0 = if_2 = nil_0 > divides_0 = not_0 = not2_1 > false_0 > true_0
      
  
    partial status:
  
      pi(a#) = [2]
      pi(sieve_0) = []
      pi(sieve_1#) = [1]
      pi(cons_2) = [1]
      pi(filter_2) = [2]
      pi(not_1) = [1]
      pi(divides_1) = [1]
      pi(filter_2#) = []
      pi(filter_1) = [1]
      pi(a) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(div2_0) = []
      pi(div2_1) = []
      pi(div2_2) = []
      pi(divides_0) = []
      pi(filter_0) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(not_0) = []
      pi(not2_0) = []
      pi(not2_1) = []
      pi(divides_2) = [2]
      pi(|0|_0) = []
      pi(s_1) = [1]
      pi(true_0) = []
      pi(div2_3) = []
      pi(false_0) = []
      pi(nil_0) = []
      pi(if_3) = [2, 3]
      pi(sieve_1) = []

The following rules are removed:

  p1



-- SCC decomposition.

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))
p2: sieve_1#(cons_2(x,xs)) -> filter_2#(not_1(divides_1(x)),xs)
p3: filter_2#(f,cons_2(x,xs)) -> a#(f,x)
p4: a#(filter_1(x1),y) -> filter_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 estimated dependency graph contains the following SCCs:

  {p1}
  {p3, p4, p5}


-- 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, r6, r8, r10, r11, r13, r15, r16, r18, r21, r23, r24, r25, r26, r28, r29, r30

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          sieve_1#_A(x1) = x1
          cons_2_A(x1,x2) = x1 + x2 + 19
          filter_2_A(x1,x2) = x1 + x2 + 8
          not_1_A(x1) = x1 + 6
          divides_1_A(x1) = x1 + 2
          a_A(x1,x2) = x2 + 7
          cons_0_A() = 2
          cons_1_A(x1) = 1
          div2_0_A() = 3
          div2_1_A(x1) = 2
          div2_2_A(x1,x2) = 1
          divides_0_A() = 1
          filter_0_A() = 2
          filter_1_A(x1) = 1
          if_0_A() = 3
          if_1_A(x1) = 2
          if_2_A(x1,x2) = 1
          not_0_A() = 1
          not2_0_A() = 1
          not2_1_A(x1) = 4
          s_0_A() = 1
          s_1_A(x1) = x1 + 4
          divides_2_A(x1,x2) = x1 + x2 + 5
          |0|_0_A() = 1
          true_0_A() = 2
          div2_3_A(x1,x2,x3) = x2 + x3 + 8
          false_0_A() = 3
          nil_0_A() = 1
          if_3_A(x1,x2,x3) = x2 + x3 + 19
          sieve_1_A(x1) = x1 + 2
  
    base order 2:
  
      precedence:
      
        sieve_1# = a = nil_0 > filter_2 = if_0 > if_1 = if_3 > cons_2 = if_2 > sieve_1 > not_1 > div2_0 = not_0 = not2_0 > div2_1 = not2_1 > cons_0 = div2_2 = filter_0 = false_0 > cons_1 = filter_1 = s_1 = |0|_0 = div2_3 > s_0 = true_0 > divides_2 > divides_1 > divides_0
      
  
    partial status:
  
      pi(sieve_1#) = []
      pi(cons_2) = [2]
      pi(filter_2) = [2]
      pi(not_1) = [1]
      pi(divides_1) = []
      pi(a) = [2]
      pi(cons_0) = []
      pi(cons_1) = []
      pi(div2_0) = []
      pi(div2_1) = []
      pi(div2_2) = []
      pi(divides_0) = []
      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) = [1]
      pi(divides_2) = [2]
      pi(|0|_0) = []
      pi(true_0) = []
      pi(div2_3) = []
      pi(false_0) = []
      pi(nil_0) = []
      pi(if_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: filter_2#(f,cons_2(x,xs)) -> a#(f,x)
p2: a#(filter_1(x1),y) -> filter_2#(x1,y)
p3: 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, r10, r11, r13, r15, r16, r18, r21, r26, r28, r29, r30

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          filter_2#_A(x1,x2) = x1 + x2 + 3
          cons_2_A(x1,x2) = x1 + x2 + 17
          a#_A(x1,x2) = x1 + x2 + 2
          filter_1_A(x1) = x1 + 2
          a_A(x1,x2) = x2 + 11
          cons_0_A() = 2
          cons_1_A(x1) = 1
          div2_0_A() = 11
          div2_1_A(x1) = 10
          div2_2_A(x1,x2) = 9
          divides_0_A() = 1
          divides_1_A(x1) = x1 + 2
          filter_0_A() = 1
          if_0_A() = 3
          if_1_A(x1) = 2
          if_2_A(x1,x2) = 1
          not_0_A() = 1
          not_1_A(x1) = x1 + 2
          not2_0_A() = 1
          not2_1_A(x1) = 3
          s_0_A() = 1
          s_1_A(x1) = x1 + 3
          divides_2_A(x1,x2) = x1 + x2 + 4
          |0|_0_A() = 3
          true_0_A() = 1
          div2_3_A(x1,x2,x3) = x1 + x2 + x3 + 8
          false_0_A() = 2
          if_3_A(x1,x2,x3) = x3 + 6
          sieve_1_A(x1) = x1 + 2
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        not2_0 > not2_1 = |0|_0 > filter_2# = div2_3 = false_0 > a# = s_1 > cons_2 = divides_2 = sieve_1 > a = cons_0 = div2_0 = divides_0 = if_0 = s_0 = true_0 > cons_1 = div2_1 = divides_1 = if_1 = not_1 > filter_1 = div2_2 = not_0 = if_3 = nil_0 > filter_0 = if_2
      
  
    partial status:
  
      pi(filter_2#) = [2]
      pi(cons_2) = []
      pi(a#) = [2]
      pi(filter_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(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(not_0) = []
      pi(not_1) = [1]
      pi(not2_0) = []
      pi(not2_1) = []
      pi(s_0) = []
      pi(s_1) = [1]
      pi(divides_2) = [1, 2]
      pi(|0|_0) = []
      pi(true_0) = []
      pi(div2_3) = []
      pi(false_0) = []
      pi(if_3) = [3]
      pi(sieve_1) = [1]
      pi(nil_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: a#(filter_1(x1),y) -> filter_2#(x1,y)
p2: 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:

  {p2}


-- 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, r2, r3, r6, r8, r10, r11, r13, r15, r16, r18, r21, r28, r29, r30

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          filter_2#_A(x1,x2) = x1 + x2
          cons_2_A(x1,x2) = x1 + x2 + 8
          a_A(x1,x2) = x1 + x2 + 16
          cons_0_A() = 1
          cons_1_A(x1) = x1 + 2
          div2_0_A() = 2
          div2_1_A(x1) = 1
          divides_0_A() = 2
          divides_1_A(x1) = 1
          filter_0_A() = 2
          filter_1_A(x1) = 1
          if_0_A() = 1
          if_1_A(x1) = x1 + 2
          if_2_A(x1,x2) = x1 + x2 + 3
          not_0_A() = 6
          not_1_A(x1) = 5
          not2_0_A() = 4
          not2_1_A(x1) = 3
          s_0_A() = 1
          s_1_A(x1) = x1 + 3
          divides_2_A(x1,x2) = x1 + x2 + 18
          |0|_0_A() = 3
          true_0_A() = 1
          div2_3_A(x1,x2,x3) = x1 + x2 + x3 + 22
          false_0_A() = 2
          sieve_1_A(x1) = 6
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        filter_0 > filter_1 > filter_2# = a = div2_0 = divides_0 = not_0 = not2_0 = div2_3 > cons_2 = div2_1 = if_2 = s_1 = divides_2 > cons_1 = if_1 = s_0 = |0|_0 = sieve_1 > cons_0 = divides_1 = if_0 = not_1 = nil_0 > not2_1 > false_0 > true_0
      
  
    partial status:
  
      pi(filter_2#) = [1]
      pi(cons_2) = [2]
      pi(a) = [1]
      pi(cons_0) = []
      pi(cons_1) = [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) = [1]
      pi(if_2) = [2]
      pi(not_0) = []
      pi(not_1) = []
      pi(not2_0) = []
      pi(not2_1) = []
      pi(s_0) = []
      pi(s_1) = [1]
      pi(divides_2) = [2]
      pi(|0|_0) = []
      pi(true_0) = []
      pi(div2_3) = [1]
      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: 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, r2, r3, r4, r6, r8, r10, r11, r13, r15, r16, r18, r21, r26, r28, r29, r30

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          div2_3#_A(x1,x2,x3) = x1 + 2
          |0|_0_A() = 4
          divides_2#_A(x1,x2) = x1 + 1
          s_1_A(x1) = x1 + 4
          a_A(x1,x2) = x1 + x2 + 15
          cons_0_A() = 1
          cons_1_A(x1) = x1 + 2
          cons_2_A(x1,x2) = x1 + x2 + 11
          div2_0_A() = 1
          div2_1_A(x1) = x1 + 2
          div2_2_A(x1,x2) = x1 + x2 + 3
          divides_0_A() = 1
          divides_1_A(x1) = x1 + 4
          filter_0_A() = 1
          filter_1_A(x1) = x1 + 2
          if_0_A() = 1
          if_1_A(x1) = x1 + 2
          if_2_A(x1,x2) = x1 + x2 + 3
          not_0_A() = 1
          not_1_A(x1) = x1 + 2
          not2_0_A() = 4
          not2_1_A(x1) = 3
          s_0_A() = 1
          divides_2_A(x1,x2) = 3
          true_0_A() = 2
          div2_3_A(x1,x2,x3) = 2
          false_0_A() = 1
          if_3_A(x1,x2,x3) = x3 + 9
          sieve_1_A(x1) = x1 + 1
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        s_1 > a = not2_0 = s_0 > cons_2 = div2_2 = if_3 = nil_0 > cons_1 = div2_1 = if_2 = sieve_1 > cons_0 = div2_0 = divides_1 = if_1 = not_1 > |0|_0 = divides_0 = filter_1 = if_0 = not_0 > divides_2# = filter_0 = not2_1 = true_0 > div2_3# = divides_2 > div2_3 > false_0
      
  
    partial status:
  
      pi(div2_3#) = [1]
      pi(|0|_0) = []
      pi(divides_2#) = [1]
      pi(s_1) = [1]
      pi(a) = [2]
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(cons_2) = [2]
      pi(div2_0) = []
      pi(div2_1) = [1]
      pi(div2_2) = [2]
      pi(divides_0) = []
      pi(divides_1) = []
      pi(filter_0) = []
      pi(filter_1) = [1]
      pi(if_0) = []
      pi(if_1) = [1]
      pi(if_2) = [1, 2]
      pi(not_0) = []
      pi(not_1) = []
      pi(not2_0) = []
      pi(not2_1) = []
      pi(s_0) = []
      pi(divides_2) = []
      pi(true_0) = []
      pi(div2_3) = []
      pi(false_0) = []
      pi(if_3) = [3]
      pi(sieve_1) = [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, r3, r4, r6, r8, r10, r11, r13, r15, r16, r18, r19, r20, r21, r22, r26, r28, r29, r30

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          div2_3#_A(x1,x2,x3) = x1 + x2 + x3 + 4
          s_1_A(x1) = x1 + 3
          a_A(x1,x2) = x2 + 12
          cons_0_A() = 2
          cons_1_A(x1) = 1
          div2_0_A() = 3
          div2_1_A(x1) = 2
          div2_2_A(x1,x2) = 1
          divides_0_A() = 1
          divides_1_A(x1) = x1 + 2
          filter_0_A() = 2
          filter_1_A(x1) = 1
          if_0_A() = 3
          if_1_A(x1) = 2
          if_2_A(x1,x2) = 1
          not_0_A() = 1
          not_1_A(x1) = x1 + 5
          not2_0_A() = 4
          not2_1_A(x1) = 3
          s_0_A() = 1
          divides_2_A(x1,x2) = x1 + x2 + 13
          |0|_0_A() = 3
          true_0_A() = 1
          div2_3_A(x1,x2,x3) = x1 + x2 + 14
          false_0_A() = 2
          if_3_A(x1,x2,x3) = x3 + 11
          sieve_1_A(x1) = x1 + 2
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        div2_3# > s_1 > cons_0 = filter_0 = s_0 = divides_2 > cons_1 = |0|_0 = div2_3 > filter_1 = not2_1 = false_0 = if_3 > a = div2_0 = if_0 = not_1 = not2_0 = true_0 > div2_1 = if_1 = not_0 = sieve_1 > div2_2 = divides_1 = if_2 = nil_0 > divides_0
      
  
    partial status:
  
      pi(div2_3#) = [1]
      pi(s_1) = [1]
      pi(a) = [2]
      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(filter_1) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(not_0) = []
      pi(not_1) = [1]
      pi(not2_0) = []
      pi(not2_1) = []
      pi(s_0) = []
      pi(divides_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(div2_3) = [1]
      pi(false_0) = []
      pi(if_3) = [3]
      pi(sieve_1) = [1]
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.