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: 2
     enrichment: roof
     automaton:
      final states: {7}
      transitions:
       g0(6) -> 6*
       f0(6,6) -> 6*
       g1(16) -> 6,19
       g1(6) -> 6,11
       h1(16) -> 6*
       h1(6) -> 10*
       a1() -> 16*
       f1(19,6) -> 6*
       f1(11,10) -> 6*
       f{#,1}(19,6) -> 7*
       f{#,1}(11,10) -> 7*
       g2(16) -> 11,6
       f{#,0}(6,6) -> 7*
       h2(20) -> 10*
       a0() -> 6*
       a2() -> 20*
       h0(6) -> 6*
       6 -> 10*
       16 -> 10,6
     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