YES

Problem:
 g(h(g(x))) -> g(x)
 g(g(x)) -> g(h(g(x)))
 h(h(x)) -> h(f(h(x),x))

Proof:
 DP Processor:
  DPs:
   g#(g(x)) -> h#(g(x))
   g#(g(x)) -> g#(h(g(x)))
   h#(h(x)) -> h#(f(h(x),x))
  TRS:
   g(h(g(x))) -> g(x)
   g(g(x)) -> g(h(g(x)))
   h(h(x)) -> h(f(h(x),x))
  TDG Processor:
   DPs:
    g#(g(x)) -> h#(g(x))
    g#(g(x)) -> g#(h(g(x)))
    h#(h(x)) -> h#(f(h(x),x))
   TRS:
    g(h(g(x))) -> g(x)
    g(g(x)) -> g(h(g(x)))
    h(h(x)) -> h(f(h(x),x))
   graph:
    h#(h(x)) -> h#(f(h(x),x)) -> h#(h(x)) -> h#(f(h(x),x))
    g#(g(x)) -> h#(g(x)) -> h#(h(x)) -> h#(f(h(x),x))
    g#(g(x)) -> g#(h(g(x))) -> g#(g(x)) -> g#(h(g(x)))
    g#(g(x)) -> g#(h(g(x))) -> g#(g(x)) -> h#(g(x))
   SCC Processor:
    #sccs: 2
    #rules: 2
    #arcs: 4/9
    DPs:
     g#(g(x)) -> g#(h(g(x)))
    TRS:
     g(h(g(x))) -> g(x)
     g(g(x)) -> g(h(g(x)))
     h(h(x)) -> h(f(h(x),x))
    CDG Processor:
     DPs:
      g#(g(x)) -> g#(h(g(x)))
     TRS:
      g(h(g(x))) -> g(x)
      g(g(x)) -> g(h(g(x)))
      h(h(x)) -> h(f(h(x),x))
     graph:
      
     Qed
    
    DPs:
     h#(h(x)) -> h#(f(h(x),x))
    TRS:
     g(h(g(x))) -> g(x)
     g(g(x)) -> g(h(g(x)))
     h(h(x)) -> h(f(h(x),x))
    EDG Processor:
     DPs:
      h#(h(x)) -> h#(f(h(x),x))
     TRS:
      g(h(g(x))) -> g(x)
      g(g(x)) -> g(h(g(x)))
      h(h(x)) -> h(f(h(x),x))
     graph:
      
     Qed