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(n__filter(activate(Y),N,N)))
  nats(N) -> cons(N,n__nats(n__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)
  s(X) -> n__s(X)
  activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3))
  activate(n__sieve(X)) -> sieve(activate(X))
  activate(n__nats(X)) -> nats(activate(X))
  activate(n__s(X)) -> s(activate(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)) -> activate#(Y)
p5: zprimes#() -> sieve#(nats(s(s(|0|()))))
p6: zprimes#() -> nats#(s(s(|0|())))
p7: zprimes#() -> s#(s(|0|()))
p8: zprimes#() -> s#(|0|())
p9: activate#(n__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3))
p10: activate#(n__filter(X1,X2,X3)) -> activate#(X1)
p11: activate#(n__filter(X1,X2,X3)) -> activate#(X2)
p12: activate#(n__filter(X1,X2,X3)) -> activate#(X3)
p13: activate#(n__sieve(X)) -> sieve#(activate(X))
p14: activate#(n__sieve(X)) -> activate#(X)
p15: activate#(n__nats(X)) -> nats#(activate(X))
p16: activate#(n__nats(X)) -> activate#(X)
p17: activate#(n__s(X)) -> s#(activate(X))
p18: activate#(n__s(X)) -> activate#(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(n__filter(activate(Y),N,N)))
r5: nats(N) -> cons(N,n__nats(n__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: s(X) -> n__s(X)
r11: activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3))
r12: activate(n__sieve(X)) -> sieve(activate(X))
r13: activate(n__nats(X)) -> nats(activate(X))
r14: activate(n__s(X)) -> s(activate(X))
r15: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p9, p10, p11, p12, p13, p14, p16, p18}


-- 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__s(X)) -> activate#(X)
p3: activate#(n__nats(X)) -> activate#(X)
p4: activate#(n__sieve(X)) -> activate#(X)
p5: activate#(n__sieve(X)) -> sieve#(activate(X))
p6: sieve#(cons(s(N),Y)) -> activate#(Y)
p7: activate#(n__filter(X1,X2,X3)) -> activate#(X3)
p8: activate#(n__filter(X1,X2,X3)) -> activate#(X2)
p9: activate#(n__filter(X1,X2,X3)) -> activate#(X1)
p10: activate#(n__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3))
p11: filter#(cons(X,Y),s(N),M) -> activate#(Y)
p12: 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(n__filter(activate(Y),N,N)))
r5: nats(N) -> cons(N,n__nats(n__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: s(X) -> n__s(X)
r11: activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3))
r12: activate(n__sieve(X)) -> sieve(activate(X))
r13: activate(n__nats(X)) -> nats(activate(X))
r14: activate(n__s(X)) -> s(activate(X))
r15: activate(X) -> X

The set of usable rules consists of

  r1, r2, r3, r4, r5, r7, r8, r9, r10, r11, r12, r13, r14, r15

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 + 6, x2 + 5, x3 + 4}
          cons_A(x1,x2) = max{2, x1 + 1, x2}
          |0|_A = 0
          activate#_A(x1) = x1 + 6
          n__s_A(x1) = x1
          n__nats_A(x1) = x1 + 6
          n__sieve_A(x1) = max{9, x1 + 6}
          sieve#_A(x1) = x1 + 6
          activate_A(x1) = x1
          s_A(x1) = x1
          n__filter_A(x1,x2,x3) = max{x1, x2, x3}
          filter_A(x1,x2,x3) = max{x1, x2, x3}
          sieve_A(x1) = max{9, x1 + 6}
          nats_A(x1) = x1 + 6
    
      precedence:
      
        n__sieve = sieve > n__nats = nats > n__filter = filter > cons > filter# = activate# = sieve# > activate > s > n__s > |0|
      
      partial status:
    
        pi(filter#) = []
        pi(cons) = [1]
        pi(|0|) = []
        pi(activate#) = []
        pi(n__s) = [1]
        pi(n__nats) = []
        pi(n__sieve) = []
        pi(sieve#) = [1]
        pi(activate) = [1]
        pi(s) = [1]
        pi(n__filter) = []
        pi(filter) = []
        pi(sieve) = []
        pi(nats) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          filter#_A(x1,x2,x3) = 50
          cons_A(x1,x2) = max{49, x1 + 28}
          |0|_A = 23
          activate#_A(x1) = 50
          n__s_A(x1) = 12
          n__nats_A(x1) = 0
          n__sieve_A(x1) = 14
          sieve#_A(x1) = max{49, x1 + 36}
          activate_A(x1) = 13
          s_A(x1) = 13
          n__filter_A(x1,x2,x3) = 13
          filter_A(x1,x2,x3) = 13
          sieve_A(x1) = 49
          nats_A(x1) = 14
    
      precedence:
      
        n__sieve = sieve > filter# = cons = activate# > |0| = sieve# = activate > n__nats = s = filter = nats > n__s = n__filter
      
      partial status:
    
        pi(filter#) = []
        pi(cons) = []
        pi(|0|) = []
        pi(activate#) = []
        pi(n__s) = []
        pi(n__nats) = []
        pi(n__sieve) = []
        pi(sieve#) = [1]
        pi(activate) = []
        pi(s) = []
        pi(n__filter) = []
        pi(filter) = []
        pi(sieve) = []
        pi(nats) = []
    

The next rules are strictly ordered:

  p5, p6, p12

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__s(X)) -> activate#(X)
p3: activate#(n__nats(X)) -> activate#(X)
p4: activate#(n__sieve(X)) -> activate#(X)
p5: activate#(n__filter(X1,X2,X3)) -> activate#(X3)
p6: activate#(n__filter(X1,X2,X3)) -> activate#(X2)
p7: activate#(n__filter(X1,X2,X3)) -> activate#(X1)
p8: activate#(n__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3))
p9: filter#(cons(X,Y),s(N),M) -> 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(n__filter(activate(Y),N,N)))
r5: nats(N) -> cons(N,n__nats(n__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: s(X) -> n__s(X)
r11: activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3))
r12: activate(n__sieve(X)) -> sieve(activate(X))
r13: activate(n__nats(X)) -> nats(activate(X))
r14: activate(n__s(X)) -> s(activate(X))
r15: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, 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__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3))
p3: filter#(cons(X,Y),s(N),M) -> activate#(Y)
p4: activate#(n__filter(X1,X2,X3)) -> activate#(X1)
p5: activate#(n__filter(X1,X2,X3)) -> activate#(X2)
p6: activate#(n__filter(X1,X2,X3)) -> activate#(X3)
p7: activate#(n__sieve(X)) -> activate#(X)
p8: activate#(n__nats(X)) -> activate#(X)
p9: activate#(n__s(X)) -> activate#(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(n__filter(activate(Y),N,N)))
r5: nats(N) -> cons(N,n__nats(n__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: s(X) -> n__s(X)
r11: activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3))
r12: activate(n__sieve(X)) -> sieve(activate(X))
r13: activate(n__nats(X)) -> nats(activate(X))
r14: activate(n__s(X)) -> s(activate(X))
r15: activate(X) -> X

The set of usable rules consists of

  r1, r2, r3, r4, r5, r7, r8, r9, r10, r11, r12, r13, r14, r15

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{0, x1 - 1, x2 - 1, x3 - 2}
          cons_A(x1,x2) = max{x1 + 3, x2}
          |0|_A = 3
          activate#_A(x1) = max{2, x1 - 1}
          n__filter_A(x1,x2,x3) = max{4, x1, x2 + 3, x3 + 3}
          activate_A(x1) = x1
          s_A(x1) = max{4, x1}
          n__sieve_A(x1) = max{7, x1}
          n__nats_A(x1) = max{9, x1 + 5}
          n__s_A(x1) = max{4, x1}
          filter_A(x1,x2,x3) = max{4, x1, x2 + 3, x3 + 3}
          sieve_A(x1) = max{7, x1}
          nats_A(x1) = max{9, x1 + 5}
    
      precedence:
      
        n__sieve = sieve > n__filter = filter > cons = n__nats = nats > |0| > filter# = activate# > activate = s = n__s
      
      partial status:
    
        pi(filter#) = []
        pi(cons) = []
        pi(|0|) = []
        pi(activate#) = []
        pi(n__filter) = []
        pi(activate) = [1]
        pi(s) = []
        pi(n__sieve) = []
        pi(n__nats) = []
        pi(n__s) = []
        pi(filter) = []
        pi(sieve) = []
        pi(nats) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          filter#_A(x1,x2,x3) = 5
          cons_A(x1,x2) = 24
          |0|_A = 107
          activate#_A(x1) = 5
          n__filter_A(x1,x2,x3) = 28
          activate_A(x1) = 28
          s_A(x1) = 28
          n__sieve_A(x1) = 97
          n__nats_A(x1) = 98
          n__s_A(x1) = 3
          filter_A(x1,x2,x3) = 28
          sieve_A(x1) = 341
          nats_A(x1) = 135
    
      precedence:
      
        filter# = cons = |0| = activate# = activate = n__s > s = nats > n__nats > filter > n__filter = sieve > n__sieve
      
      partial status:
    
        pi(filter#) = []
        pi(cons) = []
        pi(|0|) = []
        pi(activate#) = []
        pi(n__filter) = []
        pi(activate) = []
        pi(s) = []
        pi(n__sieve) = []
        pi(n__nats) = []
        pi(n__s) = []
        pi(filter) = []
        pi(sieve) = []
        pi(nats) = []
    

The next rules are strictly ordered:

  p6

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__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3))
p3: filter#(cons(X,Y),s(N),M) -> activate#(Y)
p4: activate#(n__filter(X1,X2,X3)) -> activate#(X1)
p5: activate#(n__filter(X1,X2,X3)) -> activate#(X2)
p6: activate#(n__sieve(X)) -> activate#(X)
p7: activate#(n__nats(X)) -> activate#(X)
p8: activate#(n__s(X)) -> activate#(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(n__filter(activate(Y),N,N)))
r5: nats(N) -> cons(N,n__nats(n__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: s(X) -> n__s(X)
r11: activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3))
r12: activate(n__sieve(X)) -> sieve(activate(X))
r13: activate(n__nats(X)) -> nats(activate(X))
r14: activate(n__s(X)) -> s(activate(X))
r15: activate(X) -> X

The estimated dependency graph contains the following SCCs:

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


-- 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__s(X)) -> activate#(X)
p3: activate#(n__nats(X)) -> activate#(X)
p4: activate#(n__sieve(X)) -> activate#(X)
p5: activate#(n__filter(X1,X2,X3)) -> activate#(X2)
p6: activate#(n__filter(X1,X2,X3)) -> activate#(X1)
p7: activate#(n__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3))
p8: filter#(cons(X,Y),s(N),M) -> 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(n__filter(activate(Y),N,N)))
r5: nats(N) -> cons(N,n__nats(n__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: s(X) -> n__s(X)
r11: activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3))
r12: activate(n__sieve(X)) -> sieve(activate(X))
r13: activate(n__nats(X)) -> nats(activate(X))
r14: activate(n__s(X)) -> s(activate(X))
r15: activate(X) -> X

The set of usable rules consists of

  r1, r2, r3, r4, r5, r7, r8, r9, r10, r11, r12, r13, r14, r15

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 + 3, x2 + 1, x3 + 2}
          cons_A(x1,x2) = max{7, x1 + 6, x2}
          |0|_A = 1
          activate#_A(x1) = x1 + 3
          n__s_A(x1) = x1
          n__nats_A(x1) = max{9, x1 + 8}
          n__sieve_A(x1) = x1
          n__filter_A(x1,x2,x3) = max{x1, x2 + 5, x3 + 5}
          activate_A(x1) = x1
          s_A(x1) = x1
          filter_A(x1,x2,x3) = max{x1, x2 + 5, x3 + 5}
          sieve_A(x1) = x1
          nats_A(x1) = max{9, x1 + 8}
    
      precedence:
      
        filter# = cons = |0| = activate# = n__s = n__nats = n__sieve = n__filter = activate = s = filter = sieve = nats
      
      partial status:
    
        pi(filter#) = []
        pi(cons) = []
        pi(|0|) = []
        pi(activate#) = []
        pi(n__s) = []
        pi(n__nats) = []
        pi(n__sieve) = []
        pi(n__filter) = []
        pi(activate) = []
        pi(s) = []
        pi(filter) = []
        pi(sieve) = []
        pi(nats) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          filter#_A(x1,x2,x3) = 18
          cons_A(x1,x2) = 21
          |0|_A = 6
          activate#_A(x1) = 18
          n__s_A(x1) = 17
          n__nats_A(x1) = 22
          n__sieve_A(x1) = max{154, x1 + 97}
          n__filter_A(x1,x2,x3) = 17
          activate_A(x1) = max{21, x1}
          s_A(x1) = 21
          filter_A(x1,x2,x3) = 21
          sieve_A(x1) = max{154, x1 + 97}
          nats_A(x1) = 22
    
      precedence:
      
        filter# = |0| = activate# > activate = nats > filter > s > n__s > sieve > cons = n__nats = n__sieve = n__filter
      
      partial status:
    
        pi(filter#) = []
        pi(cons) = []
        pi(|0|) = []
        pi(activate#) = []
        pi(n__s) = []
        pi(n__nats) = []
        pi(n__sieve) = [1]
        pi(n__filter) = []
        pi(activate) = [1]
        pi(s) = []
        pi(filter) = []
        pi(sieve) = []
        pi(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__s(X)) -> activate#(X)
p3: activate#(n__sieve(X)) -> activate#(X)
p4: activate#(n__filter(X1,X2,X3)) -> activate#(X2)
p5: activate#(n__filter(X1,X2,X3)) -> activate#(X1)
p6: activate#(n__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3))
p7: filter#(cons(X,Y),s(N),M) -> 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(n__filter(activate(Y),N,N)))
r5: nats(N) -> cons(N,n__nats(n__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: s(X) -> n__s(X)
r11: activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3))
r12: activate(n__sieve(X)) -> sieve(activate(X))
r13: activate(n__nats(X)) -> nats(activate(X))
r14: activate(n__s(X)) -> s(activate(X))
r15: activate(X) -> X

The estimated dependency graph contains the following SCCs:

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


-- 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__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3))
p3: filter#(cons(X,Y),s(N),M) -> activate#(Y)
p4: activate#(n__filter(X1,X2,X3)) -> activate#(X1)
p5: activate#(n__filter(X1,X2,X3)) -> activate#(X2)
p6: activate#(n__sieve(X)) -> activate#(X)
p7: activate#(n__s(X)) -> activate#(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(n__filter(activate(Y),N,N)))
r5: nats(N) -> cons(N,n__nats(n__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: s(X) -> n__s(X)
r11: activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3))
r12: activate(n__sieve(X)) -> sieve(activate(X))
r13: activate(n__nats(X)) -> nats(activate(X))
r14: activate(n__s(X)) -> s(activate(X))
r15: activate(X) -> X

The set of usable rules consists of

  r1, r2, r3, r4, r5, r7, r8, r9, r10, r11, r12, r13, r14, r15

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{3, x1 - 2, x2 + 1, x3 - 3}
          cons_A(x1,x2) = max{x1 + 14, x2}
          |0|_A = 0
          activate#_A(x1) = max{0, x1 - 2}
          n__filter_A(x1,x2,x3) = max{x1, x2 + 5, x3 + 5}
          activate_A(x1) = x1
          s_A(x1) = x1
          n__sieve_A(x1) = max{15, x1 + 5}
          n__s_A(x1) = x1
          filter_A(x1,x2,x3) = max{x1, x2 + 5, x3 + 5}
          sieve_A(x1) = max{15, x1 + 5}
          nats_A(x1) = x1 + 14
          n__nats_A(x1) = x1 + 14
    
      precedence:
      
        activate = s = n__s = sieve = nats > n__nats > filter > cons = n__sieve > filter# = |0| = activate# = n__filter
      
      partial status:
    
        pi(filter#) = []
        pi(cons) = [2]
        pi(|0|) = []
        pi(activate#) = []
        pi(n__filter) = [2, 3]
        pi(activate) = [1]
        pi(s) = []
        pi(n__sieve) = []
        pi(n__s) = []
        pi(filter) = [2]
        pi(sieve) = []
        pi(nats) = []
        pi(n__nats) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          filter#_A(x1,x2,x3) = 100
          cons_A(x1,x2) = 31
          |0|_A = 60
          activate#_A(x1) = 100
          n__filter_A(x1,x2,x3) = x2 + 163
          activate_A(x1) = max{28, x1}
          s_A(x1) = 32
          n__sieve_A(x1) = 0
          n__s_A(x1) = 32
          filter_A(x1,x2,x3) = x2 + 283
          sieve_A(x1) = 31
          nats_A(x1) = 30
          n__nats_A(x1) = max{29, x1 - 1}
    
      precedence:
      
        n__sieve > sieve > filter# = cons = |0| = activate# = n__filter = filter = n__nats > nats > activate = s > n__s
      
      partial status:
    
        pi(filter#) = []
        pi(cons) = []
        pi(|0|) = []
        pi(activate#) = []
        pi(n__filter) = []
        pi(activate) = [1]
        pi(s) = []
        pi(n__sieve) = []
        pi(n__s) = []
        pi(filter) = [2]
        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__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3))
p3: filter#(cons(X,Y),s(N),M) -> activate#(Y)
p4: activate#(n__filter(X1,X2,X3)) -> activate#(X1)
p5: activate#(n__sieve(X)) -> activate#(X)
p6: activate#(n__s(X)) -> activate#(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(n__filter(activate(Y),N,N)))
r5: nats(N) -> cons(N,n__nats(n__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: s(X) -> n__s(X)
r11: activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3))
r12: activate(n__sieve(X)) -> sieve(activate(X))
r13: activate(n__nats(X)) -> nats(activate(X))
r14: activate(n__s(X)) -> s(activate(X))
r15: 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__s(X)) -> activate#(X)
p3: activate#(n__sieve(X)) -> activate#(X)
p4: activate#(n__filter(X1,X2,X3)) -> activate#(X1)
p5: activate#(n__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3))
p6: filter#(cons(X,Y),s(N),M) -> 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(n__filter(activate(Y),N,N)))
r5: nats(N) -> cons(N,n__nats(n__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: s(X) -> n__s(X)
r11: activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3))
r12: activate(n__sieve(X)) -> sieve(activate(X))
r13: activate(n__nats(X)) -> nats(activate(X))
r14: activate(n__s(X)) -> s(activate(X))
r15: activate(X) -> X

The set of usable rules consists of

  r1, r2, r3, r4, r5, r7, r8, r9, r10, r11, r12, r13, r14, r15

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{4, x1 - 1}
          cons_A(x1,x2) = max{11, x2}
          |0|_A = 0
          activate#_A(x1) = max{0, x1 - 1}
          n__s_A(x1) = max{14, x1 + 8}
          n__sieve_A(x1) = max{24, x1 + 6}
          n__filter_A(x1,x2,x3) = max{18, x1}
          activate_A(x1) = max{6, x1}
          s_A(x1) = max{14, x1 + 8}
          filter_A(x1,x2,x3) = max{18, x1}
          sieve_A(x1) = max{24, x1 + 6}
          nats_A(x1) = 22
          n__nats_A(x1) = 22
    
      precedence:
      
        activate > sieve > filter > n__sieve = nats > cons > s > filter# = |0| = activate# = n__s = n__filter = n__nats
      
      partial status:
    
        pi(filter#) = []
        pi(cons) = []
        pi(|0|) = []
        pi(activate#) = []
        pi(n__s) = [1]
        pi(n__sieve) = [1]
        pi(n__filter) = [1]
        pi(activate) = [1]
        pi(s) = []
        pi(filter) = [1]
        pi(sieve) = [1]
        pi(nats) = []
        pi(n__nats) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          filter#_A(x1,x2,x3) = 13
          cons_A(x1,x2) = 199
          |0|_A = 6
          activate#_A(x1) = 13
          n__s_A(x1) = max{197, x1 + 17}
          n__sieve_A(x1) = x1 + 31
          n__filter_A(x1,x2,x3) = max{54, x1}
          activate_A(x1) = max{198, x1 + 32}
          s_A(x1) = 48
          filter_A(x1,x2,x3) = max{198, x1}
          sieve_A(x1) = max{47, x1 + 13}
          nats_A(x1) = 199
          n__nats_A(x1) = 200
    
      precedence:
      
        n__sieve = activate = s = sieve > filter# = activate# = n__s = filter > n__nats > n__filter = nats > cons > |0|
      
      partial status:
    
        pi(filter#) = []
        pi(cons) = []
        pi(|0|) = []
        pi(activate#) = []
        pi(n__s) = [1]
        pi(n__sieve) = []
        pi(n__filter) = [1]
        pi(activate) = [1]
        pi(s) = []
        pi(filter) = [1]
        pi(sieve) = [1]
        pi(nats) = []
        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__s(X)) -> activate#(X)
p3: activate#(n__filter(X1,X2,X3)) -> activate#(X1)
p4: activate#(n__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3))
p5: filter#(cons(X,Y),s(N),M) -> 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(n__filter(activate(Y),N,N)))
r5: nats(N) -> cons(N,n__nats(n__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: s(X) -> n__s(X)
r11: activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3))
r12: activate(n__sieve(X)) -> sieve(activate(X))
r13: activate(n__nats(X)) -> nats(activate(X))
r14: activate(n__s(X)) -> s(activate(X))
r15: 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__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3))
p3: filter#(cons(X,Y),s(N),M) -> activate#(Y)
p4: activate#(n__filter(X1,X2,X3)) -> activate#(X1)
p5: activate#(n__s(X)) -> activate#(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(n__filter(activate(Y),N,N)))
r5: nats(N) -> cons(N,n__nats(n__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: s(X) -> n__s(X)
r11: activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3))
r12: activate(n__sieve(X)) -> sieve(activate(X))
r13: activate(n__nats(X)) -> nats(activate(X))
r14: activate(n__s(X)) -> s(activate(X))
r15: activate(X) -> X

The set of usable rules consists of

  r1, r2, r3, r4, r5, r7, r8, r9, r10, r11, r12, r13, r14, r15

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 + 14, x2 + 70, x3 + 14}
          cons_A(x1,x2) = max{40, x2 - 12}
          |0|_A = 0
          activate#_A(x1) = max{1, x1}
          n__filter_A(x1,x2,x3) = max{x1 + 69, x2 + 70, x3 + 59}
          activate_A(x1) = x1
          s_A(x1) = max{11, x1 + 10}
          n__s_A(x1) = max{11, x1 + 10}
          filter_A(x1,x2,x3) = max{x1 + 69, x2 + 70, x3 + 59}
          sieve_A(x1) = 40
          n__sieve_A(x1) = 40
          nats_A(x1) = x1 + 41
          n__nats_A(x1) = x1 + 41
    
      precedence:
      
        |0| = n__filter = activate = filter = sieve = nats > s > cons > filter# > activate# = n__s = n__sieve = n__nats
      
      partial status:
    
        pi(filter#) = [2]
        pi(cons) = []
        pi(|0|) = []
        pi(activate#) = [1]
        pi(n__filter) = [2, 3]
        pi(activate) = [1]
        pi(s) = [1]
        pi(n__s) = [1]
        pi(filter) = [2, 3]
        pi(sieve) = []
        pi(n__sieve) = []
        pi(nats) = []
        pi(n__nats) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          filter#_A(x1,x2,x3) = max{40, x2 + 39}
          cons_A(x1,x2) = 43
          |0|_A = 62
          activate#_A(x1) = 85
          n__filter_A(x1,x2,x3) = max{0, x3 - 46}
          activate_A(x1) = 45
          s_A(x1) = x1 + 78
          n__s_A(x1) = x1 + 77
          filter_A(x1,x2,x3) = max{x2, x3 - 46}
          sieve_A(x1) = 45
          n__sieve_A(x1) = 1
          nats_A(x1) = 43
          n__nats_A(x1) = 43
    
      precedence:
      
        s > activate > n__s = filter = sieve = n__sieve = nats > |0| > filter# = cons = activate# = n__nats > n__filter
      
      partial status:
    
        pi(filter#) = []
        pi(cons) = []
        pi(|0|) = []
        pi(activate#) = []
        pi(n__filter) = []
        pi(activate) = []
        pi(s) = []
        pi(n__s) = [1]
        pi(filter) = []
        pi(sieve) = []
        pi(n__sieve) = []
        pi(nats) = []
        pi(n__nats) = []
    

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: activate#(n__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3))
p2: filter#(cons(X,Y),s(N),M) -> activate#(Y)
p3: activate#(n__filter(X1,X2,X3)) -> activate#(X1)
p4: activate#(n__s(X)) -> activate#(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(n__filter(activate(Y),N,N)))
r5: nats(N) -> cons(N,n__nats(n__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: s(X) -> n__s(X)
r11: activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3))
r12: activate(n__sieve(X)) -> sieve(activate(X))
r13: activate(n__nats(X)) -> nats(activate(X))
r14: activate(n__s(X)) -> s(activate(X))
r15: 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: activate#(n__filter(X1,X2,X3)) -> filter#(activate(X1),activate(X2),activate(X3))
p2: filter#(cons(X,Y),s(N),M) -> activate#(Y)
p3: activate#(n__s(X)) -> activate#(X)
p4: activate#(n__filter(X1,X2,X3)) -> activate#(X1)

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(n__filter(activate(Y),N,N)))
r5: nats(N) -> cons(N,n__nats(n__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: s(X) -> n__s(X)
r11: activate(n__filter(X1,X2,X3)) -> filter(activate(X1),activate(X2),activate(X3))
r12: activate(n__sieve(X)) -> sieve(activate(X))
r13: activate(n__nats(X)) -> nats(activate(X))
r14: activate(n__s(X)) -> s(activate(X))
r15: activate(X) -> X

The set of usable rules consists of

  r1, r2, r3, r4, r5, r7, r8, r9, r10, r11, r12, r13, r14, r15

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          activate#_A(x1) = x1
          n__filter_A(x1,x2,x3) = max{24, x1 + 10, x2 + 18, x3 + 19}
          filter#_A(x1,x2,x3) = max{x1 + 1, x2 - 6, x3 + 19}
          activate_A(x1) = max{5, x1}
          cons_A(x1,x2) = max{22, x2}
          s_A(x1) = max{20, x1 + 15}
          n__s_A(x1) = max{20, x1 + 15}
          filter_A(x1,x2,x3) = max{24, x1 + 10, x2 + 18, x3 + 19}
          |0|_A = 7
          sieve_A(x1) = 23
          n__sieve_A(x1) = 23
          nats_A(x1) = 22
          n__nats_A(x1) = 22
    
      precedence:
      
        activate# = activate = s > n__s = filter > filter# = sieve > n__sieve = nats > cons > n__filter = |0| = n__nats
      
      partial status:
    
        pi(activate#) = [1]
        pi(n__filter) = [1, 2]
        pi(filter#) = [1, 3]
        pi(activate) = [1]
        pi(cons) = [2]
        pi(s) = []
        pi(n__s) = [1]
        pi(filter) = []
        pi(|0|) = []
        pi(sieve) = []
        pi(n__sieve) = []
        pi(nats) = []
        pi(n__nats) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          activate#_A(x1) = max{61, x1}
          n__filter_A(x1,x2,x3) = x1 + 61
          filter#_A(x1,x2,x3) = max{60, x1 + 35}
          activate_A(x1) = max{23, x1 + 14}
          cons_A(x1,x2) = x2 + 162
          s_A(x1) = 104
          n__s_A(x1) = x1 + 4
          filter_A(x1,x2,x3) = 218
          |0|_A = 28
          sieve_A(x1) = 26
          n__sieve_A(x1) = 1
          nats_A(x1) = 24
          n__nats_A(x1) = 1
    
      precedence:
      
        n__filter > activate# = filter# = activate = cons = s = n__s = filter = |0| = sieve = n__sieve = nats = n__nats
      
      partial status:
    
        pi(activate#) = [1]
        pi(n__filter) = []
        pi(filter#) = []
        pi(activate) = [1]
        pi(cons) = [2]
        pi(s) = []
        pi(n__s) = [1]
        pi(filter) = []
        pi(|0|) = []
        pi(sieve) = []
        pi(n__sieve) = []
        pi(nats) = []
        pi(n__nats) = []
    

The next rules are strictly ordered:

  p1, p2, p3, p4

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