YES

We show the termination of the TRS R:

  active(filter(cons(X,Y),|0|(),M)) -> mark(cons(|0|(),filter(Y,M,M)))
  active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M)))
  active(sieve(cons(|0|(),Y))) -> mark(cons(|0|(),sieve(Y)))
  active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N))))
  active(nats(N)) -> mark(cons(N,nats(s(N))))
  active(zprimes()) -> mark(sieve(nats(s(s(|0|())))))
  active(filter(X1,X2,X3)) -> filter(active(X1),X2,X3)
  active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3)
  active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3))
  active(cons(X1,X2)) -> cons(active(X1),X2)
  active(s(X)) -> s(active(X))
  active(sieve(X)) -> sieve(active(X))
  active(nats(X)) -> nats(active(X))
  filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3))
  filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3))
  filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3))
  cons(mark(X1),X2) -> mark(cons(X1,X2))
  s(mark(X)) -> mark(s(X))
  sieve(mark(X)) -> mark(sieve(X))
  nats(mark(X)) -> mark(nats(X))
  proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3))
  proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
  proper(|0|()) -> ok(|0|())
  proper(s(X)) -> s(proper(X))
  proper(sieve(X)) -> sieve(proper(X))
  proper(nats(X)) -> nats(proper(X))
  proper(zprimes()) -> ok(zprimes())
  filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3))
  cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
  s(ok(X)) -> ok(s(X))
  sieve(ok(X)) -> ok(sieve(X))
  nats(ok(X)) -> ok(nats(X))
  top(mark(X)) -> top(proper(X))
  top(ok(X)) -> top(active(X))

-- SCC decomposition.

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

p1: active#(filter(cons(X,Y),|0|(),M)) -> cons#(|0|(),filter(Y,M,M))
p2: active#(filter(cons(X,Y),|0|(),M)) -> filter#(Y,M,M)
p3: active#(filter(cons(X,Y),s(N),M)) -> cons#(X,filter(Y,N,M))
p4: active#(filter(cons(X,Y),s(N),M)) -> filter#(Y,N,M)
p5: active#(sieve(cons(|0|(),Y))) -> cons#(|0|(),sieve(Y))
p6: active#(sieve(cons(|0|(),Y))) -> sieve#(Y)
p7: active#(sieve(cons(s(N),Y))) -> cons#(s(N),sieve(filter(Y,N,N)))
p8: active#(sieve(cons(s(N),Y))) -> sieve#(filter(Y,N,N))
p9: active#(sieve(cons(s(N),Y))) -> filter#(Y,N,N)
p10: active#(nats(N)) -> cons#(N,nats(s(N)))
p11: active#(nats(N)) -> nats#(s(N))
p12: active#(nats(N)) -> s#(N)
p13: active#(zprimes()) -> sieve#(nats(s(s(|0|()))))
p14: active#(zprimes()) -> nats#(s(s(|0|())))
p15: active#(zprimes()) -> s#(s(|0|()))
p16: active#(zprimes()) -> s#(|0|())
p17: active#(filter(X1,X2,X3)) -> filter#(active(X1),X2,X3)
p18: active#(filter(X1,X2,X3)) -> active#(X1)
p19: active#(filter(X1,X2,X3)) -> filter#(X1,active(X2),X3)
p20: active#(filter(X1,X2,X3)) -> active#(X2)
p21: active#(filter(X1,X2,X3)) -> filter#(X1,X2,active(X3))
p22: active#(filter(X1,X2,X3)) -> active#(X3)
p23: active#(cons(X1,X2)) -> cons#(active(X1),X2)
p24: active#(cons(X1,X2)) -> active#(X1)
p25: active#(s(X)) -> s#(active(X))
p26: active#(s(X)) -> active#(X)
p27: active#(sieve(X)) -> sieve#(active(X))
p28: active#(sieve(X)) -> active#(X)
p29: active#(nats(X)) -> nats#(active(X))
p30: active#(nats(X)) -> active#(X)
p31: filter#(mark(X1),X2,X3) -> filter#(X1,X2,X3)
p32: filter#(X1,mark(X2),X3) -> filter#(X1,X2,X3)
p33: filter#(X1,X2,mark(X3)) -> filter#(X1,X2,X3)
p34: cons#(mark(X1),X2) -> cons#(X1,X2)
p35: s#(mark(X)) -> s#(X)
p36: sieve#(mark(X)) -> sieve#(X)
p37: nats#(mark(X)) -> nats#(X)
p38: proper#(filter(X1,X2,X3)) -> filter#(proper(X1),proper(X2),proper(X3))
p39: proper#(filter(X1,X2,X3)) -> proper#(X1)
p40: proper#(filter(X1,X2,X3)) -> proper#(X2)
p41: proper#(filter(X1,X2,X3)) -> proper#(X3)
p42: proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
p43: proper#(cons(X1,X2)) -> proper#(X1)
p44: proper#(cons(X1,X2)) -> proper#(X2)
p45: proper#(s(X)) -> s#(proper(X))
p46: proper#(s(X)) -> proper#(X)
p47: proper#(sieve(X)) -> sieve#(proper(X))
p48: proper#(sieve(X)) -> proper#(X)
p49: proper#(nats(X)) -> nats#(proper(X))
p50: proper#(nats(X)) -> proper#(X)
p51: filter#(ok(X1),ok(X2),ok(X3)) -> filter#(X1,X2,X3)
p52: cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
p53: s#(ok(X)) -> s#(X)
p54: sieve#(ok(X)) -> sieve#(X)
p55: nats#(ok(X)) -> nats#(X)
p56: top#(mark(X)) -> top#(proper(X))
p57: top#(mark(X)) -> proper#(X)
p58: top#(ok(X)) -> top#(active(X))
p59: top#(ok(X)) -> active#(X)

and R consists of:

r1: active(filter(cons(X,Y),|0|(),M)) -> mark(cons(|0|(),filter(Y,M,M)))
r2: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M)))
r3: active(sieve(cons(|0|(),Y))) -> mark(cons(|0|(),sieve(Y)))
r4: active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N))))
r5: active(nats(N)) -> mark(cons(N,nats(s(N))))
r6: active(zprimes()) -> mark(sieve(nats(s(s(|0|())))))
r7: active(filter(X1,X2,X3)) -> filter(active(X1),X2,X3)
r8: active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3)
r9: active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3))
r10: active(cons(X1,X2)) -> cons(active(X1),X2)
r11: active(s(X)) -> s(active(X))
r12: active(sieve(X)) -> sieve(active(X))
r13: active(nats(X)) -> nats(active(X))
r14: filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3))
r15: filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3))
r16: filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3))
r17: cons(mark(X1),X2) -> mark(cons(X1,X2))
r18: s(mark(X)) -> mark(s(X))
r19: sieve(mark(X)) -> mark(sieve(X))
r20: nats(mark(X)) -> mark(nats(X))
r21: proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3))
r22: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
r23: proper(|0|()) -> ok(|0|())
r24: proper(s(X)) -> s(proper(X))
r25: proper(sieve(X)) -> sieve(proper(X))
r26: proper(nats(X)) -> nats(proper(X))
r27: proper(zprimes()) -> ok(zprimes())
r28: filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3))
r29: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
r30: s(ok(X)) -> ok(s(X))
r31: sieve(ok(X)) -> ok(sieve(X))
r32: nats(ok(X)) -> ok(nats(X))
r33: top(mark(X)) -> top(proper(X))
r34: top(ok(X)) -> top(active(X))

The estimated dependency graph contains the following SCCs:

  {p56, p58}
  {p18, p20, p22, p24, p26, p28, p30}
  {p39, p40, p41, p43, p44, p46, p48, p50}
  {p31, p32, p33, p51}
  {p34, p52}
  {p36, p54}
  {p37, p55}
  {p35, p53}


-- Reduction pair.

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

p1: top#(ok(X)) -> top#(active(X))
p2: top#(mark(X)) -> top#(proper(X))

and R consists of:

r1: active(filter(cons(X,Y),|0|(),M)) -> mark(cons(|0|(),filter(Y,M,M)))
r2: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M)))
r3: active(sieve(cons(|0|(),Y))) -> mark(cons(|0|(),sieve(Y)))
r4: active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N))))
r5: active(nats(N)) -> mark(cons(N,nats(s(N))))
r6: active(zprimes()) -> mark(sieve(nats(s(s(|0|())))))
r7: active(filter(X1,X2,X3)) -> filter(active(X1),X2,X3)
r8: active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3)
r9: active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3))
r10: active(cons(X1,X2)) -> cons(active(X1),X2)
r11: active(s(X)) -> s(active(X))
r12: active(sieve(X)) -> sieve(active(X))
r13: active(nats(X)) -> nats(active(X))
r14: filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3))
r15: filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3))
r16: filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3))
r17: cons(mark(X1),X2) -> mark(cons(X1,X2))
r18: s(mark(X)) -> mark(s(X))
r19: sieve(mark(X)) -> mark(sieve(X))
r20: nats(mark(X)) -> mark(nats(X))
r21: proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3))
r22: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
r23: proper(|0|()) -> ok(|0|())
r24: proper(s(X)) -> s(proper(X))
r25: proper(sieve(X)) -> sieve(proper(X))
r26: proper(nats(X)) -> nats(proper(X))
r27: proper(zprimes()) -> ok(zprimes())
r28: filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3))
r29: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
r30: s(ok(X)) -> ok(s(X))
r31: sieve(ok(X)) -> ok(sieve(X))
r32: nats(ok(X)) -> ok(nats(X))
r33: top(mark(X)) -> top(proper(X))
r34: top(ok(X)) -> top(active(X))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            top#_A(x1) = ((0,1),(0,0)) x1 + (0,2)
            ok_A(x1) = x1
            active_A(x1) = ((0,1),(0,1)) x1 + (2,0)
            mark_A(x1) = ((0,0),(0,1)) x1 + (2,1)
            proper_A(x1) = ((0,1),(0,1)) x1
            filter_A(x1,x2,x3) = ((0,0),(0,1)) x1 + ((0,0),(0,1)) x2 + ((0,0),(0,1)) x3 + (2,13)
            cons_A(x1,x2) = ((0,0),(0,1)) x1 + (4,5)
            s_A(x1) = x1
            sieve_A(x1) = x1 + (1,7)
            nats_A(x1) = x1 + (6,6)
            |0|_A() = (0,6)
            zprimes_A() = (1,20)
    
      precedence:
      
        active = proper = zprimes > sieve = nats > |0| > filter = cons > s > top# = ok = mark
      
      partial status:
    
        pi(top#) = []
        pi(ok) = []
        pi(active) = []
        pi(mark) = []
        pi(proper) = []
        pi(filter) = []
        pi(cons) = []
        pi(s) = []
        pi(sieve) = []
        pi(nats) = []
        pi(|0|) = []
        pi(zprimes) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            top#_A(x1) = (8,0)
            ok_A(x1) = (0,0)
            active_A(x1) = (5,2)
            mark_A(x1) = (0,0)
            proper_A(x1) = (7,2)
            filter_A(x1,x2,x3) = (2,0)
            cons_A(x1,x2) = (3,2)
            s_A(x1) = (0,2)
            sieve_A(x1) = (4,2)
            nats_A(x1) = (5,0)
            |0|_A() = (1,1)
            zprimes_A() = (6,2)
    
      precedence:
      
        zprimes > |0| > cons > s > active > sieve > ok = filter > mark > top# > proper > nats
      
      partial status:
    
        pi(top#) = []
        pi(ok) = []
        pi(active) = []
        pi(mark) = []
        pi(proper) = []
        pi(filter) = []
        pi(cons) = []
        pi(s) = []
        pi(sieve) = []
        pi(nats) = []
        pi(|0|) = []
        pi(zprimes) = []
    

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: top#(ok(X)) -> top#(active(X))

and R consists of:

r1: active(filter(cons(X,Y),|0|(),M)) -> mark(cons(|0|(),filter(Y,M,M)))
r2: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M)))
r3: active(sieve(cons(|0|(),Y))) -> mark(cons(|0|(),sieve(Y)))
r4: active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N))))
r5: active(nats(N)) -> mark(cons(N,nats(s(N))))
r6: active(zprimes()) -> mark(sieve(nats(s(s(|0|())))))
r7: active(filter(X1,X2,X3)) -> filter(active(X1),X2,X3)
r8: active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3)
r9: active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3))
r10: active(cons(X1,X2)) -> cons(active(X1),X2)
r11: active(s(X)) -> s(active(X))
r12: active(sieve(X)) -> sieve(active(X))
r13: active(nats(X)) -> nats(active(X))
r14: filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3))
r15: filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3))
r16: filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3))
r17: cons(mark(X1),X2) -> mark(cons(X1,X2))
r18: s(mark(X)) -> mark(s(X))
r19: sieve(mark(X)) -> mark(sieve(X))
r20: nats(mark(X)) -> mark(nats(X))
r21: proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3))
r22: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
r23: proper(|0|()) -> ok(|0|())
r24: proper(s(X)) -> s(proper(X))
r25: proper(sieve(X)) -> sieve(proper(X))
r26: proper(nats(X)) -> nats(proper(X))
r27: proper(zprimes()) -> ok(zprimes())
r28: filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3))
r29: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
r30: s(ok(X)) -> ok(s(X))
r31: sieve(ok(X)) -> ok(sieve(X))
r32: nats(ok(X)) -> ok(nats(X))
r33: top(mark(X)) -> top(proper(X))
r34: top(ok(X)) -> top(active(X))

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: top#(ok(X)) -> top#(active(X))

and R consists of:

r1: active(filter(cons(X,Y),|0|(),M)) -> mark(cons(|0|(),filter(Y,M,M)))
r2: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M)))
r3: active(sieve(cons(|0|(),Y))) -> mark(cons(|0|(),sieve(Y)))
r4: active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N))))
r5: active(nats(N)) -> mark(cons(N,nats(s(N))))
r6: active(zprimes()) -> mark(sieve(nats(s(s(|0|())))))
r7: active(filter(X1,X2,X3)) -> filter(active(X1),X2,X3)
r8: active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3)
r9: active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3))
r10: active(cons(X1,X2)) -> cons(active(X1),X2)
r11: active(s(X)) -> s(active(X))
r12: active(sieve(X)) -> sieve(active(X))
r13: active(nats(X)) -> nats(active(X))
r14: filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3))
r15: filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3))
r16: filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3))
r17: cons(mark(X1),X2) -> mark(cons(X1,X2))
r18: s(mark(X)) -> mark(s(X))
r19: sieve(mark(X)) -> mark(sieve(X))
r20: nats(mark(X)) -> mark(nats(X))
r21: proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3))
r22: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
r23: proper(|0|()) -> ok(|0|())
r24: proper(s(X)) -> s(proper(X))
r25: proper(sieve(X)) -> sieve(proper(X))
r26: proper(nats(X)) -> nats(proper(X))
r27: proper(zprimes()) -> ok(zprimes())
r28: filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3))
r29: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
r30: s(ok(X)) -> ok(s(X))
r31: sieve(ok(X)) -> ok(sieve(X))
r32: nats(ok(X)) -> ok(nats(X))
r33: top(mark(X)) -> top(proper(X))
r34: top(ok(X)) -> top(active(X))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r28, r29, r30, r31, r32

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          top#_A(x1) = max{206, x1 + 205}
          ok_A(x1) = max{205, x1 + 126}
          active_A(x1) = x1 + 23
          filter_A(x1,x2,x3) = max{78, x1 + 4, x2 + 6, x3 + 6}
          mark_A(x1) = max{79, x1}
          cons_A(x1,x2) = max{x1 + 65, x2 + 23}
          s_A(x1) = max{70, x1}
          sieve_A(x1) = x1 + 35
          nats_A(x1) = max{125, x1 + 46}
          |0|_A = 3
          zprimes_A = 161
    
      precedence:
      
        top# = active > filter = cons > ok = s = sieve = nats > mark = |0| = zprimes
      
      partial status:
    
        pi(top#) = [1]
        pi(ok) = [1]
        pi(active) = [1]
        pi(filter) = [1, 2, 3]
        pi(mark) = [1]
        pi(cons) = [1, 2]
        pi(s) = [1]
        pi(sieve) = [1]
        pi(nats) = [1]
        pi(|0|) = []
        pi(zprimes) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          top#_A(x1) = x1 + 62
          ok_A(x1) = max{60, x1 + 40}
          active_A(x1) = x1
          filter_A(x1,x2,x3) = max{x1 - 18, x2 + 9, x3 + 9}
          mark_A(x1) = x1 + 150
          cons_A(x1,x2) = max{x1 + 36, x2 + 51}
          s_A(x1) = x1 + 24
          sieve_A(x1) = x1 + 39
          nats_A(x1) = x1 + 25
          |0|_A = 35
          zprimes_A = 148
    
      precedence:
      
        top# = ok = active = filter = mark = cons = s = sieve = nats = |0| = zprimes
      
      partial status:
    
        pi(top#) = []
        pi(ok) = [1]
        pi(active) = [1]
        pi(filter) = [3]
        pi(mark) = [1]
        pi(cons) = [1, 2]
        pi(s) = [1]
        pi(sieve) = [1]
        pi(nats) = [1]
        pi(|0|) = []
        pi(zprimes) = []
    

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

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

p1: active#(nats(X)) -> active#(X)
p2: active#(sieve(X)) -> active#(X)
p3: active#(s(X)) -> active#(X)
p4: active#(cons(X1,X2)) -> active#(X1)
p5: active#(filter(X1,X2,X3)) -> active#(X3)
p6: active#(filter(X1,X2,X3)) -> active#(X2)
p7: active#(filter(X1,X2,X3)) -> active#(X1)

and R consists of:

r1: active(filter(cons(X,Y),|0|(),M)) -> mark(cons(|0|(),filter(Y,M,M)))
r2: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M)))
r3: active(sieve(cons(|0|(),Y))) -> mark(cons(|0|(),sieve(Y)))
r4: active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N))))
r5: active(nats(N)) -> mark(cons(N,nats(s(N))))
r6: active(zprimes()) -> mark(sieve(nats(s(s(|0|())))))
r7: active(filter(X1,X2,X3)) -> filter(active(X1),X2,X3)
r8: active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3)
r9: active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3))
r10: active(cons(X1,X2)) -> cons(active(X1),X2)
r11: active(s(X)) -> s(active(X))
r12: active(sieve(X)) -> sieve(active(X))
r13: active(nats(X)) -> nats(active(X))
r14: filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3))
r15: filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3))
r16: filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3))
r17: cons(mark(X1),X2) -> mark(cons(X1,X2))
r18: s(mark(X)) -> mark(s(X))
r19: sieve(mark(X)) -> mark(sieve(X))
r20: nats(mark(X)) -> mark(nats(X))
r21: proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3))
r22: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
r23: proper(|0|()) -> ok(|0|())
r24: proper(s(X)) -> s(proper(X))
r25: proper(sieve(X)) -> sieve(proper(X))
r26: proper(nats(X)) -> nats(proper(X))
r27: proper(zprimes()) -> ok(zprimes())
r28: filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3))
r29: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
r30: s(ok(X)) -> ok(s(X))
r31: sieve(ok(X)) -> ok(sieve(X))
r32: nats(ok(X)) -> ok(nats(X))
r33: top(mark(X)) -> top(proper(X))
r34: top(ok(X)) -> top(active(X))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          active#_A(x1) = max{2, x1 + 1}
          nats_A(x1) = max{1, x1}
          sieve_A(x1) = max{1, x1}
          s_A(x1) = max{1, x1}
          cons_A(x1,x2) = max{x1, x2}
          filter_A(x1,x2,x3) = max{1, x1, x2, x3}
    
      precedence:
      
        active# = nats = sieve = s = cons = filter
      
      partial status:
    
        pi(active#) = [1]
        pi(nats) = [1]
        pi(sieve) = [1]
        pi(s) = [1]
        pi(cons) = [1, 2]
        pi(filter) = [1, 2, 3]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          active#_A(x1) = x1 + 3
          nats_A(x1) = x1 + 1
          sieve_A(x1) = x1 + 1
          s_A(x1) = x1 + 1
          cons_A(x1,x2) = max{x1 + 1, x2}
          filter_A(x1,x2,x3) = max{x1 + 1, x2 + 1, x3 + 1}
    
      precedence:
      
        active# = nats = sieve = s = cons = filter
      
      partial status:
    
        pi(active#) = []
        pi(nats) = [1]
        pi(sieve) = [1]
        pi(s) = [1]
        pi(cons) = [2]
        pi(filter) = [3]
    

The next rules are strictly ordered:

  p1, p2, p3, p4, p5, p6, p7

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

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

p1: proper#(nats(X)) -> proper#(X)
p2: proper#(sieve(X)) -> proper#(X)
p3: proper#(s(X)) -> proper#(X)
p4: proper#(cons(X1,X2)) -> proper#(X2)
p5: proper#(cons(X1,X2)) -> proper#(X1)
p6: proper#(filter(X1,X2,X3)) -> proper#(X3)
p7: proper#(filter(X1,X2,X3)) -> proper#(X2)
p8: proper#(filter(X1,X2,X3)) -> proper#(X1)

and R consists of:

r1: active(filter(cons(X,Y),|0|(),M)) -> mark(cons(|0|(),filter(Y,M,M)))
r2: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M)))
r3: active(sieve(cons(|0|(),Y))) -> mark(cons(|0|(),sieve(Y)))
r4: active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N))))
r5: active(nats(N)) -> mark(cons(N,nats(s(N))))
r6: active(zprimes()) -> mark(sieve(nats(s(s(|0|())))))
r7: active(filter(X1,X2,X3)) -> filter(active(X1),X2,X3)
r8: active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3)
r9: active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3))
r10: active(cons(X1,X2)) -> cons(active(X1),X2)
r11: active(s(X)) -> s(active(X))
r12: active(sieve(X)) -> sieve(active(X))
r13: active(nats(X)) -> nats(active(X))
r14: filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3))
r15: filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3))
r16: filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3))
r17: cons(mark(X1),X2) -> mark(cons(X1,X2))
r18: s(mark(X)) -> mark(s(X))
r19: sieve(mark(X)) -> mark(sieve(X))
r20: nats(mark(X)) -> mark(nats(X))
r21: proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3))
r22: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
r23: proper(|0|()) -> ok(|0|())
r24: proper(s(X)) -> s(proper(X))
r25: proper(sieve(X)) -> sieve(proper(X))
r26: proper(nats(X)) -> nats(proper(X))
r27: proper(zprimes()) -> ok(zprimes())
r28: filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3))
r29: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
r30: s(ok(X)) -> ok(s(X))
r31: sieve(ok(X)) -> ok(sieve(X))
r32: nats(ok(X)) -> ok(nats(X))
r33: top(mark(X)) -> top(proper(X))
r34: top(ok(X)) -> top(active(X))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          proper#_A(x1) = max{2, x1 + 1}
          nats_A(x1) = max{1, x1}
          sieve_A(x1) = max{1, x1}
          s_A(x1) = max{1, x1}
          cons_A(x1,x2) = max{x1, x2}
          filter_A(x1,x2,x3) = max{1, x1, x2, x3}
    
      precedence:
      
        proper# = nats = sieve = s = cons = filter
      
      partial status:
    
        pi(proper#) = [1]
        pi(nats) = [1]
        pi(sieve) = [1]
        pi(s) = [1]
        pi(cons) = [1, 2]
        pi(filter) = [1, 2, 3]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          proper#_A(x1) = x1 + 2
          nats_A(x1) = x1 + 1
          sieve_A(x1) = x1 + 1
          s_A(x1) = x1 + 1
          cons_A(x1,x2) = max{x1 + 1, x2 + 1}
          filter_A(x1,x2,x3) = max{x1 + 1, x2 + 1, x3 + 1}
    
      precedence:
      
        proper# = nats = sieve = s = cons = filter
      
      partial status:
    
        pi(proper#) = []
        pi(nats) = [1]
        pi(sieve) = [1]
        pi(s) = [1]
        pi(cons) = [2]
        pi(filter) = [3]
    

The next rules are strictly ordered:

  p1, p2, p3, p4, p5, p6, p7, p8

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

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

p1: filter#(mark(X1),X2,X3) -> filter#(X1,X2,X3)
p2: filter#(ok(X1),ok(X2),ok(X3)) -> filter#(X1,X2,X3)
p3: filter#(X1,X2,mark(X3)) -> filter#(X1,X2,X3)
p4: filter#(X1,mark(X2),X3) -> filter#(X1,X2,X3)

and R consists of:

r1: active(filter(cons(X,Y),|0|(),M)) -> mark(cons(|0|(),filter(Y,M,M)))
r2: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M)))
r3: active(sieve(cons(|0|(),Y))) -> mark(cons(|0|(),sieve(Y)))
r4: active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N))))
r5: active(nats(N)) -> mark(cons(N,nats(s(N))))
r6: active(zprimes()) -> mark(sieve(nats(s(s(|0|())))))
r7: active(filter(X1,X2,X3)) -> filter(active(X1),X2,X3)
r8: active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3)
r9: active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3))
r10: active(cons(X1,X2)) -> cons(active(X1),X2)
r11: active(s(X)) -> s(active(X))
r12: active(sieve(X)) -> sieve(active(X))
r13: active(nats(X)) -> nats(active(X))
r14: filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3))
r15: filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3))
r16: filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3))
r17: cons(mark(X1),X2) -> mark(cons(X1,X2))
r18: s(mark(X)) -> mark(s(X))
r19: sieve(mark(X)) -> mark(sieve(X))
r20: nats(mark(X)) -> mark(nats(X))
r21: proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3))
r22: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
r23: proper(|0|()) -> ok(|0|())
r24: proper(s(X)) -> s(proper(X))
r25: proper(sieve(X)) -> sieve(proper(X))
r26: proper(nats(X)) -> nats(proper(X))
r27: proper(zprimes()) -> ok(zprimes())
r28: filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3))
r29: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
r30: s(ok(X)) -> ok(s(X))
r31: sieve(ok(X)) -> ok(sieve(X))
r32: nats(ok(X)) -> ok(nats(X))
r33: top(mark(X)) -> top(proper(X))
r34: top(ok(X)) -> top(active(X))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          filter#_A(x1,x2,x3) = max{x1 + 1, x2 + 1, x3 + 3}
          mark_A(x1) = max{1, x1}
          ok_A(x1) = max{3, x1 + 2}
    
      precedence:
      
        filter# = mark = ok
      
      partial status:
    
        pi(filter#) = [1, 2, 3]
        pi(mark) = [1]
        pi(ok) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          filter#_A(x1,x2,x3) = max{x1 + 1, x2 + 1}
          mark_A(x1) = x1
          ok_A(x1) = x1
    
      precedence:
      
        filter# = mark = ok
      
      partial status:
    
        pi(filter#) = [1, 2]
        pi(mark) = [1]
        pi(ok) = [1]
    

The next rules are strictly ordered:

  p1, p2, p3, p4

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

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

p1: cons#(mark(X1),X2) -> cons#(X1,X2)
p2: cons#(ok(X1),ok(X2)) -> cons#(X1,X2)

and R consists of:

r1: active(filter(cons(X,Y),|0|(),M)) -> mark(cons(|0|(),filter(Y,M,M)))
r2: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M)))
r3: active(sieve(cons(|0|(),Y))) -> mark(cons(|0|(),sieve(Y)))
r4: active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N))))
r5: active(nats(N)) -> mark(cons(N,nats(s(N))))
r6: active(zprimes()) -> mark(sieve(nats(s(s(|0|())))))
r7: active(filter(X1,X2,X3)) -> filter(active(X1),X2,X3)
r8: active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3)
r9: active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3))
r10: active(cons(X1,X2)) -> cons(active(X1),X2)
r11: active(s(X)) -> s(active(X))
r12: active(sieve(X)) -> sieve(active(X))
r13: active(nats(X)) -> nats(active(X))
r14: filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3))
r15: filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3))
r16: filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3))
r17: cons(mark(X1),X2) -> mark(cons(X1,X2))
r18: s(mark(X)) -> mark(s(X))
r19: sieve(mark(X)) -> mark(sieve(X))
r20: nats(mark(X)) -> mark(nats(X))
r21: proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3))
r22: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
r23: proper(|0|()) -> ok(|0|())
r24: proper(s(X)) -> s(proper(X))
r25: proper(sieve(X)) -> sieve(proper(X))
r26: proper(nats(X)) -> nats(proper(X))
r27: proper(zprimes()) -> ok(zprimes())
r28: filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3))
r29: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
r30: s(ok(X)) -> ok(s(X))
r31: sieve(ok(X)) -> ok(sieve(X))
r32: nats(ok(X)) -> ok(nats(X))
r33: top(mark(X)) -> top(proper(X))
r34: top(ok(X)) -> top(active(X))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          cons#_A(x1,x2) = max{2, x1 + 1, x2 + 1}
          mark_A(x1) = max{1, x1}
          ok_A(x1) = max{1, x1}
    
      precedence:
      
        cons# = mark = ok
      
      partial status:
    
        pi(cons#) = [1, 2]
        pi(mark) = [1]
        pi(ok) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          cons#_A(x1,x2) = max{x1 + 1, x2 - 2}
          mark_A(x1) = x1
          ok_A(x1) = max{4, x1 + 1}
    
      precedence:
      
        cons# = mark = ok
      
      partial status:
    
        pi(cons#) = [1]
        pi(mark) = [1]
        pi(ok) = [1]
    

The next rules are strictly ordered:

  p1, p2

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

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

p1: sieve#(mark(X)) -> sieve#(X)
p2: sieve#(ok(X)) -> sieve#(X)

and R consists of:

r1: active(filter(cons(X,Y),|0|(),M)) -> mark(cons(|0|(),filter(Y,M,M)))
r2: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M)))
r3: active(sieve(cons(|0|(),Y))) -> mark(cons(|0|(),sieve(Y)))
r4: active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N))))
r5: active(nats(N)) -> mark(cons(N,nats(s(N))))
r6: active(zprimes()) -> mark(sieve(nats(s(s(|0|())))))
r7: active(filter(X1,X2,X3)) -> filter(active(X1),X2,X3)
r8: active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3)
r9: active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3))
r10: active(cons(X1,X2)) -> cons(active(X1),X2)
r11: active(s(X)) -> s(active(X))
r12: active(sieve(X)) -> sieve(active(X))
r13: active(nats(X)) -> nats(active(X))
r14: filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3))
r15: filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3))
r16: filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3))
r17: cons(mark(X1),X2) -> mark(cons(X1,X2))
r18: s(mark(X)) -> mark(s(X))
r19: sieve(mark(X)) -> mark(sieve(X))
r20: nats(mark(X)) -> mark(nats(X))
r21: proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3))
r22: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
r23: proper(|0|()) -> ok(|0|())
r24: proper(s(X)) -> s(proper(X))
r25: proper(sieve(X)) -> sieve(proper(X))
r26: proper(nats(X)) -> nats(proper(X))
r27: proper(zprimes()) -> ok(zprimes())
r28: filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3))
r29: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
r30: s(ok(X)) -> ok(s(X))
r31: sieve(ok(X)) -> ok(sieve(X))
r32: nats(ok(X)) -> ok(nats(X))
r33: top(mark(X)) -> top(proper(X))
r34: top(ok(X)) -> top(active(X))

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          sieve#_A(x1) = max{6, x1 + 4}
          mark_A(x1) = max{4, x1 + 3}
          ok_A(x1) = max{2, x1 + 1}
    
      precedence:
      
        sieve# = mark = ok
      
      partial status:
    
        pi(sieve#) = [1]
        pi(mark) = [1]
        pi(ok) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          sieve#_A(x1) = x1 + 1
          mark_A(x1) = x1
          ok_A(x1) = x1
    
      precedence:
      
        sieve# = mark = ok
      
      partial status:
    
        pi(sieve#) = [1]
        pi(mark) = [1]
        pi(ok) = [1]
    

The next rules are strictly ordered:

  p1, p2

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

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

p1: nats#(mark(X)) -> nats#(X)
p2: nats#(ok(X)) -> nats#(X)

and R consists of:

r1: active(filter(cons(X,Y),|0|(),M)) -> mark(cons(|0|(),filter(Y,M,M)))
r2: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M)))
r3: active(sieve(cons(|0|(),Y))) -> mark(cons(|0|(),sieve(Y)))
r4: active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N))))
r5: active(nats(N)) -> mark(cons(N,nats(s(N))))
r6: active(zprimes()) -> mark(sieve(nats(s(s(|0|())))))
r7: active(filter(X1,X2,X3)) -> filter(active(X1),X2,X3)
r8: active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3)
r9: active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3))
r10: active(cons(X1,X2)) -> cons(active(X1),X2)
r11: active(s(X)) -> s(active(X))
r12: active(sieve(X)) -> sieve(active(X))
r13: active(nats(X)) -> nats(active(X))
r14: filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3))
r15: filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3))
r16: filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3))
r17: cons(mark(X1),X2) -> mark(cons(X1,X2))
r18: s(mark(X)) -> mark(s(X))
r19: sieve(mark(X)) -> mark(sieve(X))
r20: nats(mark(X)) -> mark(nats(X))
r21: proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3))
r22: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
r23: proper(|0|()) -> ok(|0|())
r24: proper(s(X)) -> s(proper(X))
r25: proper(sieve(X)) -> sieve(proper(X))
r26: proper(nats(X)) -> nats(proper(X))
r27: proper(zprimes()) -> ok(zprimes())
r28: filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3))
r29: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
r30: s(ok(X)) -> ok(s(X))
r31: sieve(ok(X)) -> ok(sieve(X))
r32: nats(ok(X)) -> ok(nats(X))
r33: top(mark(X)) -> top(proper(X))
r34: top(ok(X)) -> top(active(X))

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          nats#_A(x1) = x1 + 2
          mark_A(x1) = x1
          ok_A(x1) = x1 + 2
    
      precedence:
      
        mark > ok > nats#
      
      partial status:
    
        pi(nats#) = [1]
        pi(mark) = [1]
        pi(ok) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          nats#_A(x1) = x1 + 1
          mark_A(x1) = x1
          ok_A(x1) = x1
    
      precedence:
      
        nats# = mark = ok
      
      partial status:
    
        pi(nats#) = [1]
        pi(mark) = [1]
        pi(ok) = [1]
    

The next rules are strictly ordered:

  p1, p2

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

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

p1: s#(mark(X)) -> s#(X)
p2: s#(ok(X)) -> s#(X)

and R consists of:

r1: active(filter(cons(X,Y),|0|(),M)) -> mark(cons(|0|(),filter(Y,M,M)))
r2: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M)))
r3: active(sieve(cons(|0|(),Y))) -> mark(cons(|0|(),sieve(Y)))
r4: active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N))))
r5: active(nats(N)) -> mark(cons(N,nats(s(N))))
r6: active(zprimes()) -> mark(sieve(nats(s(s(|0|())))))
r7: active(filter(X1,X2,X3)) -> filter(active(X1),X2,X3)
r8: active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3)
r9: active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3))
r10: active(cons(X1,X2)) -> cons(active(X1),X2)
r11: active(s(X)) -> s(active(X))
r12: active(sieve(X)) -> sieve(active(X))
r13: active(nats(X)) -> nats(active(X))
r14: filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3))
r15: filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3))
r16: filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3))
r17: cons(mark(X1),X2) -> mark(cons(X1,X2))
r18: s(mark(X)) -> mark(s(X))
r19: sieve(mark(X)) -> mark(sieve(X))
r20: nats(mark(X)) -> mark(nats(X))
r21: proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3))
r22: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
r23: proper(|0|()) -> ok(|0|())
r24: proper(s(X)) -> s(proper(X))
r25: proper(sieve(X)) -> sieve(proper(X))
r26: proper(nats(X)) -> nats(proper(X))
r27: proper(zprimes()) -> ok(zprimes())
r28: filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3))
r29: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
r30: s(ok(X)) -> ok(s(X))
r31: sieve(ok(X)) -> ok(sieve(X))
r32: nats(ok(X)) -> ok(nats(X))
r33: top(mark(X)) -> top(proper(X))
r34: top(ok(X)) -> top(active(X))

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          s#_A(x1) = max{6, x1 + 4}
          mark_A(x1) = max{4, x1 + 3}
          ok_A(x1) = max{2, x1 + 1}
    
      precedence:
      
        s# = mark = ok
      
      partial status:
    
        pi(s#) = [1]
        pi(mark) = [1]
        pi(ok) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          s#_A(x1) = x1 + 1
          mark_A(x1) = x1
          ok_A(x1) = x1
    
      precedence:
      
        s# = mark = ok
      
      partial status:
    
        pi(s#) = [1]
        pi(mark) = [1]
        pi(ok) = [1]
    

The next rules are strictly ordered:

  p1, p2

We remove them from the problem.  Then no dependency pair remains.