YES

We show the termination of the TRS R:

  f(X) -> cons(X,n__f(n__g(X)))
  g(|0|()) -> s(|0|())
  g(s(X)) -> s(s(g(X)))
  sel(|0|(),cons(X,Y)) -> X
  sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
  f(X) -> n__f(X)
  g(X) -> n__g(X)
  activate(n__f(X)) -> f(activate(X))
  activate(n__g(X)) -> g(activate(X))
  activate(X) -> X

-- SCC decomposition.

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

p1: g#(s(X)) -> g#(X)
p2: sel#(s(X),cons(Y,Z)) -> sel#(X,activate(Z))
p3: sel#(s(X),cons(Y,Z)) -> activate#(Z)
p4: activate#(n__f(X)) -> f#(activate(X))
p5: activate#(n__f(X)) -> activate#(X)
p6: activate#(n__g(X)) -> g#(activate(X))
p7: activate#(n__g(X)) -> activate#(X)

and R consists of:

r1: f(X) -> cons(X,n__f(n__g(X)))
r2: g(|0|()) -> s(|0|())
r3: g(s(X)) -> s(s(g(X)))
r4: sel(|0|(),cons(X,Y)) -> X
r5: sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
r6: f(X) -> n__f(X)
r7: g(X) -> n__g(X)
r8: activate(n__f(X)) -> f(activate(X))
r9: activate(n__g(X)) -> g(activate(X))
r10: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p2}
  {p5, p7}
  {p1}


-- 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: f(X) -> cons(X,n__f(n__g(X)))
r2: g(|0|()) -> s(|0|())
r3: g(s(X)) -> s(s(g(X)))
r4: sel(|0|(),cons(X,Y)) -> X
r5: sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
r6: f(X) -> n__f(X)
r7: g(X) -> n__g(X)
r8: activate(n__f(X)) -> f(activate(X))
r9: activate(n__g(X)) -> g(activate(X))
r10: activate(X) -> X

The set of usable rules consists of

  r1, r2, r3, r6, r7, r8, r9, r10

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        sel#_A(x1,x2) = ((0,1),(0,0)) x1
        s_A(x1) = x1 + (1,1)
        cons_A(x1,x2) = (1,1)
        activate_A(x1) = ((1,1),(1,1)) x1 + (3,1)
        f_A(x1) = ((0,1),(1,0)) x1 + (4,1)
        n__f_A(x1) = ((0,1),(1,0)) x1 + (3,0)
        n__g_A(x1) = ((1,1),(1,1)) x1 + (3,2)
        g_A(x1) = ((1,1),(1,1)) x1 + (4,2)
        |0|_A() = (0,0)
    
    2. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        sel#_A(x1,x2) = (0,0)
        s_A(x1) = x1 + (0,4)
        cons_A(x1,x2) = (1,0)
        activate_A(x1) = ((0,0),(1,0)) x1 + (3,1)
        f_A(x1) = (4,7)
        n__f_A(x1) = (5,8)
        n__g_A(x1) = ((1,1),(0,1)) x1 + (4,2)
        g_A(x1) = ((0,0),(1,1)) x1 + (2,1)
        |0|_A() = (1,1)
    
    3. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        sel#_A(x1,x2) = (0,0)
        s_A(x1) = x1 + (0,1)
        cons_A(x1,x2) = (0,0)
        activate_A(x1) = (1,0)
        f_A(x1) = (2,1)
        n__f_A(x1) = (3,0)
        n__g_A(x1) = (3,1)
        g_A(x1) = (2,1)
        |0|_A() = (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__g(X)) -> activate#(X)
p2: activate#(n__f(X)) -> activate#(X)

and R consists of:

r1: f(X) -> cons(X,n__f(n__g(X)))
r2: g(|0|()) -> s(|0|())
r3: g(s(X)) -> s(s(g(X)))
r4: sel(|0|(),cons(X,Y)) -> X
r5: sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
r6: f(X) -> n__f(X)
r7: g(X) -> n__g(X)
r8: activate(n__f(X)) -> f(activate(X))
r9: activate(n__g(X)) -> g(activate(X))
r10: activate(X) -> X

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        activate#_A(x1) = ((1,1),(1,1)) x1
        n__g_A(x1) = ((1,1),(1,1)) x1 + (1,1)
        n__f_A(x1) = ((1,1),(1,1)) x1 + (1,1)
    
    2. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        activate#_A(x1) = ((1,1),(1,1)) x1
        n__g_A(x1) = ((1,1),(1,1)) x1 + (1,1)
        n__f_A(x1) = ((1,1),(1,1)) x1 + (1,1)
    
    3. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        activate#_A(x1) = ((1,1),(1,0)) x1
        n__g_A(x1) = ((1,1),(1,0)) x1 + (1,1)
        n__f_A(x1) = ((1,1),(1,1)) x1 + (1,1)
    

The next rules are strictly ordered:

  p1, p2
  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10

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: g#(s(X)) -> g#(X)

and R consists of:

r1: f(X) -> cons(X,n__f(n__g(X)))
r2: g(|0|()) -> s(|0|())
r3: g(s(X)) -> s(s(g(X)))
r4: sel(|0|(),cons(X,Y)) -> X
r5: sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
r6: f(X) -> n__f(X)
r7: g(X) -> n__g(X)
r8: activate(n__f(X)) -> f(activate(X))
r9: activate(n__g(X)) -> g(activate(X))
r10: activate(X) -> X

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        g#_A(x1) = ((1,1),(1,1)) x1
        s_A(x1) = ((1,1),(1,1)) x1 + (1,1)
    
    2. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        g#_A(x1) = x1
        s_A(x1) = x1 + (1,1)
    
    3. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        g#_A(x1) = ((0,1),(1,1)) x1
        s_A(x1) = ((0,1),(1,1)) x1 + (1,1)
    

The next rules are strictly ordered:

  p1

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