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))
    Usable Rule Processor:
     DPs:
      f#(g(f(X))) -> f#(g(X))
     TRS:
      
     Bounds Processor:
      bound: 1
      enrichment: match
      automaton:
       final states: {6}
       transitions:
        f{#,1}(11) -> 12*
        g1(10) -> 11*
        g1(13) -> 14*
        f{#,0}(5) -> 6*
        f{#,0}(4) -> 6*
        g0(5) -> 4*
        g0(4) -> 4*
        f0(5) -> 5*
        f0(4) -> 5*
        4 -> 13*
        5 -> 10*
        12 -> 6*
        14 -> 11*
      problem:
       DPs:
        
       TRS:
        
      Qed