YES

Problem:
 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

Proof:
 DP Processor:
  DPs:
   activate#(n__f(X)) -> activate#(X)
   activate#(n__f(X)) -> f#(activate(X))
   activate#(n__g(X)) -> activate#(X)
   activate#(n__g(X)) -> g#(activate(X))
   activate#(n__a()) -> a#()
  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
  TDG Processor:
   DPs:
    activate#(n__f(X)) -> activate#(X)
    activate#(n__f(X)) -> f#(activate(X))
    activate#(n__g(X)) -> activate#(X)
    activate#(n__g(X)) -> g#(activate(X))
    activate#(n__a()) -> a#()
   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
   graph:
    activate#(n__g(X)) -> activate#(X) ->
    activate#(n__a()) -> a#()
    activate#(n__g(X)) -> activate#(X) ->
    activate#(n__g(X)) -> g#(activate(X))
    activate#(n__g(X)) -> activate#(X) ->
    activate#(n__g(X)) -> activate#(X)
    activate#(n__g(X)) -> activate#(X) ->
    activate#(n__f(X)) -> f#(activate(X))
    activate#(n__g(X)) -> activate#(X) ->
    activate#(n__f(X)) -> activate#(X)
    activate#(n__f(X)) -> activate#(X) ->
    activate#(n__a()) -> a#()
    activate#(n__f(X)) -> activate#(X) ->
    activate#(n__g(X)) -> g#(activate(X))
    activate#(n__f(X)) -> activate#(X) ->
    activate#(n__g(X)) -> activate#(X)
    activate#(n__f(X)) -> activate#(X) ->
    activate#(n__f(X)) -> f#(activate(X))
    activate#(n__f(X)) -> activate#(X) -> activate#(n__f(X)) -> activate#(X)
   CDG Processor:
    DPs:
     activate#(n__f(X)) -> activate#(X)
     activate#(n__f(X)) -> f#(activate(X))
     activate#(n__g(X)) -> activate#(X)
     activate#(n__g(X)) -> g#(activate(X))
     activate#(n__a()) -> a#()
    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
    graph:
     
    Qed