YES

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

Proof:
 DP Processor:
  DPs:
   f#(a(),h(x)) -> g#(x)
   f#(a(),h(x)) -> f#(g(x),h(x))
   h#(g(x)) -> h#(a())
   g#(h(x)) -> g#(x)
  TRS:
   f(a(),h(x)) -> f(g(x),h(x))
   h(g(x)) -> h(a())
   g(h(x)) -> g(x)
   h(h(x)) -> x
  EDG Processor:
   DPs:
    f#(a(),h(x)) -> g#(x)
    f#(a(),h(x)) -> f#(g(x),h(x))
    h#(g(x)) -> h#(a())
    g#(h(x)) -> g#(x)
   TRS:
    f(a(),h(x)) -> f(g(x),h(x))
    h(g(x)) -> h(a())
    g(h(x)) -> g(x)
    h(h(x)) -> x
   graph:
    g#(h(x)) -> g#(x) -> g#(h(x)) -> g#(x)
    f#(a(),h(x)) -> g#(x) -> g#(h(x)) -> g#(x)
    f#(a(),h(x)) -> f#(g(x),h(x)) -> f#(a(),h(x)) -> g#(x)
    f#(a(),h(x)) -> f#(g(x),h(x)) -> f#(a(),h(x)) -> f#(g(x),h(x))
   SCC Processor:
    #sccs: 2
    #rules: 2
    #arcs: 4/16
    DPs:
     f#(a(),h(x)) -> f#(g(x),h(x))
    TRS:
     f(a(),h(x)) -> f(g(x),h(x))
     h(g(x)) -> h(a())
     g(h(x)) -> g(x)
     h(h(x)) -> x
    Bounds Processor:
     bound: 0
     enrichment: top-dp
     automaton:
      final states: {6}
      transitions:
       g0(12) -> 14*
       g0(14) -> 14*
       g0(9) -> 14*
       g0(13) -> 14*
       f0(12,14) -> 12*
       f0(13,9) -> 12*
       f0(13,13) -> 12*
       f0(14,12) -> 12*
       f0(9,12) -> 12*
       f0(14,14) -> 12*
       f0(9,14) -> 12*
       f0(12,9) -> 12*
       f0(12,13) -> 12*
       f0(13,12) -> 12*
       f0(13,14) -> 12*
       f0(14,9) -> 12*
       f0(9,9) -> 12*
       f0(14,13) -> 12*
       f0(9,13) -> 12*
       f0(12,12) -> 12*
       f{#,0}(14,13) -> 6*
       a0() -> 9*
       h0(12) -> 13*
       h0(14) -> 13*
       h0(9) -> 13*
       h0(13) -> 13*
       9 -> 13*
       12 -> 13*
       14 -> 13*
     problem:
      DPs:
       
      TRS:
       f(a(),h(x)) -> f(g(x),h(x))
       h(g(x)) -> h(a())
       g(h(x)) -> g(x)
       h(h(x)) -> x
     Qed
    
    DPs:
     g#(h(x)) -> g#(x)
    TRS:
     f(a(),h(x)) -> f(g(x),h(x))
     h(g(x)) -> h(a())
     g(h(x)) -> g(x)
     h(h(x)) -> x
    Subterm Criterion Processor:
     simple projection:
      pi(g#) = 0
     problem:
      DPs:
       
      TRS:
       f(a(),h(x)) -> f(g(x),h(x))
       h(g(x)) -> h(a())
       g(h(x)) -> g(x)
       h(h(x)) -> x
     Qed