YES

We show the termination of the TRS R:

  filter(cons(X,Y),|0|(),M) -> cons(|0|(),n__filter(activate(Y),M,M))
  filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M))
  sieve(cons(|0|(),Y)) -> cons(|0|(),n__sieve(activate(Y)))
  sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(filter(activate(Y),N,N)))
  nats(N) -> cons(N,n__nats(s(N)))
  zprimes() -> sieve(nats(s(s(|0|()))))
  filter(X1,X2,X3) -> n__filter(X1,X2,X3)
  sieve(X) -> n__sieve(X)
  nats(X) -> n__nats(X)
  activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3)
  activate(n__sieve(X)) -> sieve(X)
  activate(n__nats(X)) -> nats(X)
  activate(X) -> X

-- SCC decomposition.

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

p1: filter#(cons(X,Y),|0|(),M) -> activate#(Y)
p2: filter#(cons(X,Y),s(N),M) -> activate#(Y)
p3: sieve#(cons(|0|(),Y)) -> activate#(Y)
p4: sieve#(cons(s(N),Y)) -> filter#(activate(Y),N,N)
p5: sieve#(cons(s(N),Y)) -> activate#(Y)
p6: zprimes#() -> sieve#(nats(s(s(|0|()))))
p7: zprimes#() -> nats#(s(s(|0|())))
p8: activate#(n__filter(X1,X2,X3)) -> filter#(X1,X2,X3)
p9: activate#(n__sieve(X)) -> sieve#(X)
p10: activate#(n__nats(X)) -> nats#(X)

and R consists of:

r1: filter(cons(X,Y),|0|(),M) -> cons(|0|(),n__filter(activate(Y),M,M))
r2: filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M))
r3: sieve(cons(|0|(),Y)) -> cons(|0|(),n__sieve(activate(Y)))
r4: sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(filter(activate(Y),N,N)))
r5: nats(N) -> cons(N,n__nats(s(N)))
r6: zprimes() -> sieve(nats(s(s(|0|()))))
r7: filter(X1,X2,X3) -> n__filter(X1,X2,X3)
r8: sieve(X) -> n__sieve(X)
r9: nats(X) -> n__nats(X)
r10: activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3)
r11: activate(n__sieve(X)) -> sieve(X)
r12: activate(n__nats(X)) -> nats(X)
r13: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p8, p9}


-- Reduction pair.

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

p1: filter#(cons(X,Y),|0|(),M) -> activate#(Y)
p2: activate#(n__sieve(X)) -> sieve#(X)
p3: sieve#(cons(s(N),Y)) -> activate#(Y)
p4: activate#(n__filter(X1,X2,X3)) -> filter#(X1,X2,X3)
p5: filter#(cons(X,Y),s(N),M) -> activate#(Y)
p6: sieve#(cons(s(N),Y)) -> filter#(activate(Y),N,N)
p7: sieve#(cons(|0|(),Y)) -> activate#(Y)

and R consists of:

r1: filter(cons(X,Y),|0|(),M) -> cons(|0|(),n__filter(activate(Y),M,M))
r2: filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M))
r3: sieve(cons(|0|(),Y)) -> cons(|0|(),n__sieve(activate(Y)))
r4: sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(filter(activate(Y),N,N)))
r5: nats(N) -> cons(N,n__nats(s(N)))
r6: zprimes() -> sieve(nats(s(s(|0|()))))
r7: filter(X1,X2,X3) -> n__filter(X1,X2,X3)
r8: sieve(X) -> n__sieve(X)
r9: nats(X) -> n__nats(X)
r10: activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3)
r11: activate(n__sieve(X)) -> sieve(X)
r12: activate(n__nats(X)) -> nats(X)
r13: activate(X) -> X

The set of usable rules consists of

  r1, r2, r3, r4, r5, r7, r8, r9, r10, r11, r12, r13

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          filter#_A(x1,x2,x3) = x1
          cons_A(x1,x2) = x2
          |0|_A() = 4
          activate#_A(x1) = x1
          n__sieve_A(x1) = x1 + 3
          sieve#_A(x1) = x1 + 1
          s_A(x1) = x1 + 2
          n__filter_A(x1,x2,x3) = x1
          activate_A(x1) = x1
          filter_A(x1,x2,x3) = x1
          sieve_A(x1) = x1 + 3
          nats_A(x1) = 1
          n__nats_A(x1) = 1
  
    precedence:
    
      |0| > activate = filter > n__sieve = sieve = nats > filter# = cons = activate# = n__filter > sieve# > s = n__nats
    
    partial status:
  
      pi(filter#) = [1]
      pi(cons) = [2]
      pi(|0|) = []
      pi(activate#) = [1]
      pi(n__sieve) = []
      pi(sieve#) = [1]
      pi(s) = []
      pi(n__filter) = [1]
      pi(activate) = [1]
      pi(filter) = [1]
      pi(sieve) = [1]
      pi(nats) = []
      pi(n__nats) = []

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

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

p1: filter#(cons(X,Y),|0|(),M) -> activate#(Y)
p2: activate#(n__sieve(X)) -> sieve#(X)
p3: sieve#(cons(s(N),Y)) -> activate#(Y)
p4: filter#(cons(X,Y),s(N),M) -> activate#(Y)
p5: sieve#(cons(s(N),Y)) -> filter#(activate(Y),N,N)
p6: sieve#(cons(|0|(),Y)) -> activate#(Y)

and R consists of:

r1: filter(cons(X,Y),|0|(),M) -> cons(|0|(),n__filter(activate(Y),M,M))
r2: filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M))
r3: sieve(cons(|0|(),Y)) -> cons(|0|(),n__sieve(activate(Y)))
r4: sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(filter(activate(Y),N,N)))
r5: nats(N) -> cons(N,n__nats(s(N)))
r6: zprimes() -> sieve(nats(s(s(|0|()))))
r7: filter(X1,X2,X3) -> n__filter(X1,X2,X3)
r8: sieve(X) -> n__sieve(X)
r9: nats(X) -> n__nats(X)
r10: activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3)
r11: activate(n__sieve(X)) -> sieve(X)
r12: activate(n__nats(X)) -> nats(X)
r13: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6}


-- Reduction pair.

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

p1: filter#(cons(X,Y),|0|(),M) -> activate#(Y)
p2: activate#(n__sieve(X)) -> sieve#(X)
p3: sieve#(cons(|0|(),Y)) -> activate#(Y)
p4: sieve#(cons(s(N),Y)) -> filter#(activate(Y),N,N)
p5: filter#(cons(X,Y),s(N),M) -> activate#(Y)
p6: sieve#(cons(s(N),Y)) -> activate#(Y)

and R consists of:

r1: filter(cons(X,Y),|0|(),M) -> cons(|0|(),n__filter(activate(Y),M,M))
r2: filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M))
r3: sieve(cons(|0|(),Y)) -> cons(|0|(),n__sieve(activate(Y)))
r4: sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(filter(activate(Y),N,N)))
r5: nats(N) -> cons(N,n__nats(s(N)))
r6: zprimes() -> sieve(nats(s(s(|0|()))))
r7: filter(X1,X2,X3) -> n__filter(X1,X2,X3)
r8: sieve(X) -> n__sieve(X)
r9: nats(X) -> n__nats(X)
r10: activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3)
r11: activate(n__sieve(X)) -> sieve(X)
r12: activate(n__nats(X)) -> nats(X)
r13: activate(X) -> X

The set of usable rules consists of

  r1, r2, r3, r4, r5, r7, r8, r9, r10, r11, r12, r13

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          filter#_A(x1,x2,x3) = x1 + 1
          cons_A(x1,x2) = x2 + 5
          |0|_A() = 1
          activate#_A(x1) = x1 + 2
          n__sieve_A(x1) = x1 + 9
          sieve#_A(x1) = x1 + 8
          s_A(x1) = x1 + 3
          activate_A(x1) = x1 + 11
          filter_A(x1,x2,x3) = 10
          n__filter_A(x1,x2,x3) = 4
          sieve_A(x1) = x1 + 20
          nats_A(x1) = 7
          n__nats_A(x1) = 1
  
    precedence:
    
      |0| = n__sieve = s = activate = filter = n__filter = sieve = nats = n__nats > filter# > activate# > sieve# > cons
    
    partial status:
  
      pi(filter#) = [1]
      pi(cons) = []
      pi(|0|) = []
      pi(activate#) = [1]
      pi(n__sieve) = []
      pi(sieve#) = [1]
      pi(s) = []
      pi(activate) = [1]
      pi(filter) = []
      pi(n__filter) = []
      pi(sieve) = []
      pi(nats) = []
      pi(n__nats) = []

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

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

p1: filter#(cons(X,Y),|0|(),M) -> activate#(Y)
p2: activate#(n__sieve(X)) -> sieve#(X)
p3: sieve#(cons(|0|(),Y)) -> activate#(Y)
p4: sieve#(cons(s(N),Y)) -> filter#(activate(Y),N,N)
p5: sieve#(cons(s(N),Y)) -> activate#(Y)

and R consists of:

r1: filter(cons(X,Y),|0|(),M) -> cons(|0|(),n__filter(activate(Y),M,M))
r2: filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M))
r3: sieve(cons(|0|(),Y)) -> cons(|0|(),n__sieve(activate(Y)))
r4: sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(filter(activate(Y),N,N)))
r5: nats(N) -> cons(N,n__nats(s(N)))
r6: zprimes() -> sieve(nats(s(s(|0|()))))
r7: filter(X1,X2,X3) -> n__filter(X1,X2,X3)
r8: sieve(X) -> n__sieve(X)
r9: nats(X) -> n__nats(X)
r10: activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3)
r11: activate(n__sieve(X)) -> sieve(X)
r12: activate(n__nats(X)) -> nats(X)
r13: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

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

p1: filter#(cons(X,Y),|0|(),M) -> activate#(Y)
p2: activate#(n__sieve(X)) -> sieve#(X)
p3: sieve#(cons(s(N),Y)) -> activate#(Y)
p4: sieve#(cons(s(N),Y)) -> filter#(activate(Y),N,N)
p5: sieve#(cons(|0|(),Y)) -> activate#(Y)

and R consists of:

r1: filter(cons(X,Y),|0|(),M) -> cons(|0|(),n__filter(activate(Y),M,M))
r2: filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M))
r3: sieve(cons(|0|(),Y)) -> cons(|0|(),n__sieve(activate(Y)))
r4: sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(filter(activate(Y),N,N)))
r5: nats(N) -> cons(N,n__nats(s(N)))
r6: zprimes() -> sieve(nats(s(s(|0|()))))
r7: filter(X1,X2,X3) -> n__filter(X1,X2,X3)
r8: sieve(X) -> n__sieve(X)
r9: nats(X) -> n__nats(X)
r10: activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3)
r11: activate(n__sieve(X)) -> sieve(X)
r12: activate(n__nats(X)) -> nats(X)
r13: activate(X) -> X

The set of usable rules consists of

  r1, r2, r3, r4, r5, r7, r8, r9, r10, r11, r12, r13

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          filter#_A(x1,x2,x3) = x1 + 1
          cons_A(x1,x2) = x2 + 3
          |0|_A() = 1
          activate#_A(x1) = x1 + 2
          n__sieve_A(x1) = x1 + 3
          sieve#_A(x1) = x1 + 4
          s_A(x1) = 12
          activate_A(x1) = x1 + 5
          filter_A(x1,x2,x3) = 4
          n__filter_A(x1,x2,x3) = 0
          sieve_A(x1) = x1 + 8
          nats_A(x1) = 13
          n__nats_A(x1) = 9
  
    precedence:
    
      filter# = cons = |0| = activate# = n__sieve = sieve# = s = activate = filter = n__filter = sieve = nats = n__nats
    
    partial status:
  
      pi(filter#) = []
      pi(cons) = []
      pi(|0|) = []
      pi(activate#) = []
      pi(n__sieve) = []
      pi(sieve#) = []
      pi(s) = []
      pi(activate) = [1]
      pi(filter) = []
      pi(n__filter) = []
      pi(sieve) = [1]
      pi(nats) = []
      pi(n__nats) = []

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

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

p1: filter#(cons(X,Y),|0|(),M) -> activate#(Y)
p2: activate#(n__sieve(X)) -> sieve#(X)
p3: sieve#(cons(s(N),Y)) -> activate#(Y)
p4: sieve#(cons(s(N),Y)) -> filter#(activate(Y),N,N)

and R consists of:

r1: filter(cons(X,Y),|0|(),M) -> cons(|0|(),n__filter(activate(Y),M,M))
r2: filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M))
r3: sieve(cons(|0|(),Y)) -> cons(|0|(),n__sieve(activate(Y)))
r4: sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(filter(activate(Y),N,N)))
r5: nats(N) -> cons(N,n__nats(s(N)))
r6: zprimes() -> sieve(nats(s(s(|0|()))))
r7: filter(X1,X2,X3) -> n__filter(X1,X2,X3)
r8: sieve(X) -> n__sieve(X)
r9: nats(X) -> n__nats(X)
r10: activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3)
r11: activate(n__sieve(X)) -> sieve(X)
r12: activate(n__nats(X)) -> nats(X)
r13: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

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

p1: filter#(cons(X,Y),|0|(),M) -> activate#(Y)
p2: activate#(n__sieve(X)) -> sieve#(X)
p3: sieve#(cons(s(N),Y)) -> filter#(activate(Y),N,N)
p4: sieve#(cons(s(N),Y)) -> activate#(Y)

and R consists of:

r1: filter(cons(X,Y),|0|(),M) -> cons(|0|(),n__filter(activate(Y),M,M))
r2: filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M))
r3: sieve(cons(|0|(),Y)) -> cons(|0|(),n__sieve(activate(Y)))
r4: sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(filter(activate(Y),N,N)))
r5: nats(N) -> cons(N,n__nats(s(N)))
r6: zprimes() -> sieve(nats(s(s(|0|()))))
r7: filter(X1,X2,X3) -> n__filter(X1,X2,X3)
r8: sieve(X) -> n__sieve(X)
r9: nats(X) -> n__nats(X)
r10: activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3)
r11: activate(n__sieve(X)) -> sieve(X)
r12: activate(n__nats(X)) -> nats(X)
r13: activate(X) -> X

The set of usable rules consists of

  r1, r2, r3, r4, r5, r7, r8, r9, r10, r11, r12, r13

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          filter#_A(x1,x2,x3) = x1 + 1
          cons_A(x1,x2) = x2 + 3
          |0|_A() = 4
          activate#_A(x1) = x1
          n__sieve_A(x1) = x1 + 6
          sieve#_A(x1) = x1 + 6
          s_A(x1) = 1
          activate_A(x1) = x1 + 7
          filter_A(x1,x2,x3) = 6
          n__filter_A(x1,x2,x3) = 2
          sieve_A(x1) = x1 + 13
          nats_A(x1) = x1 + 6
          n__nats_A(x1) = x1 + 1
  
    precedence:
    
      activate# = sieve# > filter# = activate > sieve > filter = n__filter > |0| = nats > cons = s > n__sieve = n__nats
    
    partial status:
  
      pi(filter#) = [1]
      pi(cons) = [2]
      pi(|0|) = []
      pi(activate#) = [1]
      pi(n__sieve) = []
      pi(sieve#) = [1]
      pi(s) = []
      pi(activate) = [1]
      pi(filter) = []
      pi(n__filter) = []
      pi(sieve) = [1]
      pi(nats) = [1]
      pi(n__nats) = []

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

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

p1: filter#(cons(X,Y),|0|(),M) -> activate#(Y)
p2: activate#(n__sieve(X)) -> sieve#(X)
p3: sieve#(cons(s(N),Y)) -> activate#(Y)

and R consists of:

r1: filter(cons(X,Y),|0|(),M) -> cons(|0|(),n__filter(activate(Y),M,M))
r2: filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M))
r3: sieve(cons(|0|(),Y)) -> cons(|0|(),n__sieve(activate(Y)))
r4: sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(filter(activate(Y),N,N)))
r5: nats(N) -> cons(N,n__nats(s(N)))
r6: zprimes() -> sieve(nats(s(s(|0|()))))
r7: filter(X1,X2,X3) -> n__filter(X1,X2,X3)
r8: sieve(X) -> n__sieve(X)
r9: nats(X) -> n__nats(X)
r10: activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3)
r11: activate(n__sieve(X)) -> sieve(X)
r12: activate(n__nats(X)) -> nats(X)
r13: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p2, p3}


-- Reduction pair.

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

p1: activate#(n__sieve(X)) -> sieve#(X)
p2: sieve#(cons(s(N),Y)) -> activate#(Y)

and R consists of:

r1: filter(cons(X,Y),|0|(),M) -> cons(|0|(),n__filter(activate(Y),M,M))
r2: filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M))
r3: sieve(cons(|0|(),Y)) -> cons(|0|(),n__sieve(activate(Y)))
r4: sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(filter(activate(Y),N,N)))
r5: nats(N) -> cons(N,n__nats(s(N)))
r6: zprimes() -> sieve(nats(s(s(|0|()))))
r7: filter(X1,X2,X3) -> n__filter(X1,X2,X3)
r8: sieve(X) -> n__sieve(X)
r9: nats(X) -> n__nats(X)
r10: activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3)
r11: activate(n__sieve(X)) -> sieve(X)
r12: activate(n__nats(X)) -> nats(X)
r13: activate(X) -> X

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          activate#_A(x1) = x1 + 2
          n__sieve_A(x1) = x1 + 1
          sieve#_A(x1) = x1
          cons_A(x1,x2) = x2 + 3
          s_A(x1) = x1 + 1
  
    precedence:
    
      activate# > n__sieve = s > sieve# = cons
    
    partial status:
  
      pi(activate#) = [1]
      pi(n__sieve) = []
      pi(sieve#) = [1]
      pi(cons) = [2]
      pi(s) = [1]

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: activate#(n__sieve(X)) -> sieve#(X)

and R consists of:

r1: filter(cons(X,Y),|0|(),M) -> cons(|0|(),n__filter(activate(Y),M,M))
r2: filter(cons(X,Y),s(N),M) -> cons(X,n__filter(activate(Y),N,M))
r3: sieve(cons(|0|(),Y)) -> cons(|0|(),n__sieve(activate(Y)))
r4: sieve(cons(s(N),Y)) -> cons(s(N),n__sieve(filter(activate(Y),N,N)))
r5: nats(N) -> cons(N,n__nats(s(N)))
r6: zprimes() -> sieve(nats(s(s(|0|()))))
r7: filter(X1,X2,X3) -> n__filter(X1,X2,X3)
r8: sieve(X) -> n__sieve(X)
r9: nats(X) -> n__nats(X)
r10: activate(n__filter(X1,X2,X3)) -> filter(X1,X2,X3)
r11: activate(n__sieve(X)) -> sieve(X)
r12: activate(n__nats(X)) -> nats(X)
r13: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  (no SCCs)