YES

Problem:
 f(f(X)) -> f(g(f(g(f(X)))))
 f(g(f(X))) -> f(g(X))

Proof:
 DP Processor:
  DPs:
   f#(f(X)) -> f#(g(f(X)))
   f#(f(X)) -> f#(g(f(g(f(X)))))
   f#(g(f(X))) -> f#(g(X))
  TRS:
   f(f(X)) -> f(g(f(g(f(X)))))
   f(g(f(X))) -> f(g(X))
  EDG Processor:
   DPs:
    f#(f(X)) -> f#(g(f(X)))
    f#(f(X)) -> f#(g(f(g(f(X)))))
    f#(g(f(X))) -> f#(g(X))
   TRS:
    f(f(X)) -> f(g(f(g(f(X)))))
    f(g(f(X))) -> f(g(X))
   graph:
    f#(g(f(X))) -> f#(g(X)) -> f#(g(f(X))) -> f#(g(X))
    f#(f(X)) -> f#(g(f(g(f(X))))) -> f#(g(f(X))) -> f#(g(X))
    f#(f(X)) -> f#(g(f(X))) -> f#(g(f(X))) -> f#(g(X))
   SCC Processor:
    #sccs: 1
    #rules: 1
    #arcs: 3/9
    DPs:
     f#(g(f(X))) -> f#(g(X))
    TRS:
     f(f(X)) -> f(g(f(g(f(X)))))
     f(g(f(X))) -> f(g(X))
    Bounds Processor:
     bound: 1
     enrichment: match-dp
     automaton:
      final states: {4}
      transitions:
       f{#,0}(5) -> 4*
       g0(2) -> 2*
       g0(1) -> 2*
       g0(3) -> 5*
       f0(2) -> 1*
       f0(1) -> 1*
       f{#,1}(10) -> 11*
       g1(12) -> 13*
       g1(9) -> 10*
       1 -> 12,3
       2 -> 9,3
       11 -> 4*
       13 -> 10*
     problem:
      DPs:
       
      TRS:
       f(f(X)) -> f(g(f(g(f(X)))))
       f(g(f(X))) -> f(g(X))
     Qed