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))
  ADG 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:
    g#(g(x)) -> h#(g(x)) -> h#(h(x)) -> h#(f(h(x),x))
    g#(g(x)) -> g#(h(g(x))) -> g#(g(x)) -> h#(g(x))
    g#(g(x)) -> g#(h(g(x))) -> g#(g(x)) -> g#(h(g(x)))
   Restore Modifier:
    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))
    SCC Processor:
     #sccs: 1
     #rules: 1
     #arcs: 3/9
     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))
     Matrix Interpretation Processor:
      dimension: 1
      interpretation:
       [g#](x0) = x0,
       
       [f](x0, x1) = 0,
       
       [h](x0) = 0,
       
       [g](x0) = 1
      orientation:
       g#(g(x)) = 1 >= 0 = g#(h(g(x)))
       
       g(h(g(x))) = 1 >= 1 = g(x)
       
       g(g(x)) = 1 >= 1 = g(h(g(x)))
       
       h(h(x)) = 0 >= 0 = h(f(h(x),x))
      problem:
       DPs:
        
       TRS:
        g(h(g(x))) -> g(x)
        g(g(x)) -> g(h(g(x)))
        h(h(x)) -> h(f(h(x),x))
      Qed