YES

TRS:

  f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
  f(X) -> n__f(X)
  g(X) -> n__g(X)
  a() -> n__a()
  activate(n__f(X)) -> f(activate(X))
  activate(n__g(X)) -> g(activate(X))
  activate(n__a()) -> a()
  activate(X) -> X

linear polynomial interpretations on N:

  f_A(x1) = x1 + 5
  f#_A(x1) = x1 + 5
  a_A = 4
  a#_A = 4
  c_A(x1) = x1 + 1
  c#_A(x1) = x1 + 1
  n__f_A(x1) = x1 + 5
  n__f#_A(x1) = x1 + 5
  n__g_A(x1) = x1 + 1
  n__g#_A(x1) = x1 + 1
  n__a_A = 1
  n__a#_A = 1
  g_A(x1) = x1 + 1
  g#_A(x1) = x1 + 1
  activate_A(x1) = x1 + 4
  activate#_A(x1) = x1 + 4

precedence:

  a > n__a > c > activate > f > n__f > g > n__g