YES

Problem:
 a__f(f(a())) -> a__f(g(f(a())))
 mark(f(X)) -> a__f(X)
 mark(a()) -> a()
 mark(g(X)) -> g(mark(X))
 a__f(X) -> f(X)

Proof:
 DP Processor:
  DPs:
   a__f#(f(a())) -> a__f#(g(f(a())))
   mark#(f(X)) -> a__f#(X)
   mark#(g(X)) -> mark#(X)
  TRS:
   a__f(f(a())) -> a__f(g(f(a())))
   mark(f(X)) -> a__f(X)
   mark(a()) -> a()
   mark(g(X)) -> g(mark(X))
   a__f(X) -> f(X)
  TDG Processor:
   DPs:
    a__f#(f(a())) -> a__f#(g(f(a())))
    mark#(f(X)) -> a__f#(X)
    mark#(g(X)) -> mark#(X)
   TRS:
    a__f(f(a())) -> a__f(g(f(a())))
    mark(f(X)) -> a__f(X)
    mark(a()) -> a()
    mark(g(X)) -> g(mark(X))
    a__f(X) -> f(X)
   graph:
    mark#(g(X)) -> mark#(X) -> mark#(g(X)) -> mark#(X)
    mark#(g(X)) -> mark#(X) -> mark#(f(X)) -> a__f#(X)
    mark#(f(X)) -> a__f#(X) -> a__f#(f(a())) -> a__f#(g(f(a())))
    a__f#(f(a())) -> a__f#(g(f(a()))) -> a__f#(f(a())) -> a__f#(g(f(a())))
   EDG Processor:
    DPs:
     a__f#(f(a())) -> a__f#(g(f(a())))
     mark#(f(X)) -> a__f#(X)
     mark#(g(X)) -> mark#(X)
    TRS:
     a__f(f(a())) -> a__f(g(f(a())))
     mark(f(X)) -> a__f(X)
     mark(a()) -> a()
     mark(g(X)) -> g(mark(X))
     a__f(X) -> f(X)
    graph:
     mark#(g(X)) -> mark#(X) -> mark#(f(X)) -> a__f#(X)
     mark#(g(X)) -> mark#(X) -> mark#(g(X)) -> mark#(X)
     mark#(f(X)) -> a__f#(X) -> a__f#(f(a())) -> a__f#(g(f(a())))
    CDG Processor:
     DPs:
      a__f#(f(a())) -> a__f#(g(f(a())))
      mark#(f(X)) -> a__f#(X)
      mark#(g(X)) -> mark#(X)
     TRS:
      a__f(f(a())) -> a__f(g(f(a())))
      mark(f(X)) -> a__f(X)
      mark(a()) -> a()
      mark(g(X)) -> g(mark(X))
      a__f(X) -> f(X)
     graph:
      
     Qed