YES

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

Proof:
 DP Processor:
  DPs:
   f#(a(),x) -> g#(x)
   f#(a(),x) -> f#(g(x),x)
   h#(g(x)) -> h#(a())
   g#(h(x)) -> g#(x)
  TRS:
   f(a(),x) -> f(g(x),x)
   h(g(x)) -> h(a())
   g(h(x)) -> g(x)
   h(h(x)) -> x
  EDG Processor:
   DPs:
    f#(a(),x) -> g#(x)
    f#(a(),x) -> f#(g(x),x)
    h#(g(x)) -> h#(a())
    g#(h(x)) -> g#(x)
   TRS:
    f(a(),x) -> f(g(x),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(),x) -> g#(x) -> g#(h(x)) -> g#(x)
    f#(a(),x) -> f#(g(x),x) -> f#(a(),x) -> g#(x)
    f#(a(),x) -> f#(g(x),x) -> f#(a(),x) -> f#(g(x),x)
   SCC Processor:
    #sccs: 2
    #rules: 2
    #arcs: 4/16
    DPs:
     f#(a(),x) -> f#(g(x),x)
    TRS:
     f(a(),x) -> f(g(x),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:
       f0(6,6) -> 6*
       h0(6) -> 6*
       g1(12) -> 6*
       g1(6) -> 6,9
       h1(12) -> 6*
       a1() -> 12*
       f1(9,6) -> 6*
       f{#,1}(9,6) -> 7*
       g2(12) -> 9,6
       f{#,0}(6,6) -> 7*
       a0() -> 6*
       g0(6) -> 6*
       12 -> 6*
     problem:
      DPs:
       
      TRS:
       f(a(),x) -> f(g(x),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(),x) -> f(g(x),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(),x) -> f(g(x),x)
       h(g(x)) -> h(a())
       g(h(x)) -> g(x)
       h(h(x)) -> x
     Qed