YES

We show the termination of the TRS R:

  from(X) -> cons(X,n__from(n__s(X)))
  sel(|0|(),cons(X,Y)) -> X
  sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
  from(X) -> n__from(X)
  s(X) -> n__s(X)
  activate(n__from(X)) -> from(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: sel#(s(X),cons(Y,Z)) -> sel#(X,activate(Z))
p2: sel#(s(X),cons(Y,Z)) -> activate#(Z)
p3: activate#(n__from(X)) -> from#(activate(X))
p4: activate#(n__from(X)) -> activate#(X)
p5: activate#(n__s(X)) -> s#(activate(X))
p6: activate#(n__s(X)) -> activate#(X)

and R consists of:

r1: from(X) -> cons(X,n__from(n__s(X)))
r2: sel(|0|(),cons(X,Y)) -> X
r3: sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
r4: from(X) -> n__from(X)
r5: s(X) -> n__s(X)
r6: activate(n__from(X)) -> from(activate(X))
r7: activate(n__s(X)) -> s(activate(X))
r8: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p1}
  {p4, p6}


-- Reduction pair.

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

p1: sel#(s(X),cons(Y,Z)) -> sel#(X,activate(Z))

and R consists of:

r1: from(X) -> cons(X,n__from(n__s(X)))
r2: sel(|0|(),cons(X,Y)) -> X
r3: sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
r4: from(X) -> n__from(X)
r5: s(X) -> n__s(X)
r6: activate(n__from(X)) -> from(activate(X))
r7: activate(n__s(X)) -> s(activate(X))
r8: activate(X) -> X

The set of usable rules consists of

  r1, r4, r5, r6, r7, r8

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. lexicographic path order with precedence:
    
      precedence:
      
        activate > from > s > n__s > n__from > cons > sel#
      
      argument filter:
    
        pi(sel#) = [1]
        pi(s) = [1]
        pi(cons) = []
        pi(activate) = [1]
        pi(from) = [1]
        pi(n__from) = [1]
        pi(n__s) = [1]
    
    2. matrix interpretations:
    
      carrier: N^4
      order: lexicographic order
      interpretations:
        sel#_A(x1,x2) = ((1,0,0,0),(0,0,0,0),(1,0,0,0),(0,1,0,0)) x1
        s_A(x1) = (1,1,2,2)
        cons_A(x1,x2) = (1,1,1,3)
        activate_A(x1) = ((1,0,0,0),(1,1,0,0),(0,0,0,0),(0,0,1,0)) x1 + (2,1,4,2)
        from_A(x1) = ((0,0,0,0),(1,0,0,0),(0,1,0,0),(0,0,0,0)) x1 + (4,2,2,2)
        n__from_A(x1) = ((0,0,0,0),(1,0,0,0),(0,1,0,0),(0,0,0,0)) x1 + (3,1,1,1)
        n__s_A(x1) = ((1,0,0,0),(1,1,0,0),(1,1,1,0),(0,0,0,0)) x1 + (2,2,1,1)
    

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: activate#(n__s(X)) -> activate#(X)
p2: activate#(n__from(X)) -> activate#(X)

and R consists of:

r1: from(X) -> cons(X,n__from(n__s(X)))
r2: sel(|0|(),cons(X,Y)) -> X
r3: sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
r4: from(X) -> n__from(X)
r5: s(X) -> n__s(X)
r6: activate(n__from(X)) -> from(activate(X))
r7: activate(n__s(X)) -> s(activate(X))
r8: activate(X) -> X

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. lexicographic path order with precedence:
    
      precedence:
      
        n__from > n__s > activate#
      
      argument filter:
    
        pi(activate#) = 1
        pi(n__s) = 1
        pi(n__from) = 1
    
    2. matrix interpretations:
    
      carrier: N^4
      order: lexicographic order
      interpretations:
        activate#_A(x1) = ((1,0,0,0),(1,1,0,0),(1,1,1,0),(1,1,1,1)) x1
        n__s_A(x1) = ((1,0,0,0),(1,1,0,0),(1,1,1,0),(0,0,1,1)) x1
        n__from_A(x1) = ((1,0,0,0),(1,1,0,0),(1,1,1,0),(1,1,1,1)) x1 + (1,1,1,1)
    

The next rules are strictly ordered:

  p2
  r1, r2, r3, r4, r5, r6, r7, r8

We remove them from the problem.

-- SCC decomposition.

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

p1: activate#(n__s(X)) -> activate#(X)

and R consists of:

  (no rules)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: activate#(n__s(X)) -> activate#(X)

and R consists of:

  (no rules)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. lexicographic path order with precedence:
    
      precedence:
      
        n__s > activate#
      
      argument filter:
    
        pi(activate#) = 1
        pi(n__s) = [1]
    
    2. matrix interpretations:
    
      carrier: N^4
      order: lexicographic order
      interpretations:
        activate#_A(x1) = ((1,0,0,0),(0,1,0,0),(0,0,1,0),(1,1,1,1)) x1
        n__s_A(x1) = ((1,0,0,0),(1,1,0,0),(1,1,1,0),(1,0,1,1)) x1 + (1,1,1,1)
    

The next rules are strictly ordered:

  p1

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