YES

We show the termination of the TRS R:

  f(X) -> g(n__h(n__f(X)))
  h(X) -> n__h(X)
  f(X) -> n__f(X)
  activate(n__h(X)) -> h(activate(X))
  activate(n__f(X)) -> f(activate(X))
  activate(X) -> X

-- SCC decomposition.

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

p1: activate#(n__h(X)) -> h#(activate(X))
p2: activate#(n__h(X)) -> activate#(X)
p3: activate#(n__f(X)) -> f#(activate(X))
p4: activate#(n__f(X)) -> activate#(X)

and R consists of:

r1: f(X) -> g(n__h(n__f(X)))
r2: h(X) -> n__h(X)
r3: f(X) -> n__f(X)
r4: activate(n__h(X)) -> h(activate(X))
r5: activate(n__f(X)) -> f(activate(X))
r6: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p2, p4}


-- Reduction pair.

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

p1: activate#(n__h(X)) -> activate#(X)
p2: activate#(n__f(X)) -> activate#(X)

and R consists of:

r1: f(X) -> g(n__h(n__f(X)))
r2: h(X) -> n__h(X)
r3: f(X) -> n__f(X)
r4: activate(n__h(X)) -> h(activate(X))
r5: activate(n__f(X)) -> f(activate(X))
r6: 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:
        activate#_A(x1) = ((1,1),(1,0)) x1
        n__h_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),(1,0)) x1
        n__h_A(x1) = ((1,1),(0,1)) 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),(1,1)) x1
        n__h_A(x1) = ((0,1),(1,1)) x1 + (1,1)
        n__f_A(x1) = ((1,1),(1,1)) x1 + (1,1)
    

The next rules are strictly ordered:

  p1, p2

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