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: 0
     enrichment: top-dp
     automaton:
      final states: {6}
      transitions:
       f0(8,11) -> 10*
       f0(10,11) -> 10*
       f0(11,8) -> 10*
       f0(11,10) -> 10*
       f0(11,12) -> 10*
       f0(12,11) -> 10*
       f0(8,8) -> 10*
       f0(8,10) -> 10*
       f0(8,12) -> 10*
       f0(10,8) -> 10*
       f0(10,10) -> 10*
       f0(10,12) -> 10*
       f0(11,11) -> 10*
       f0(12,8) -> 10*
       f0(12,10) -> 10*
       f0(12,12) -> 10*
       h0(10) -> 11*
       h0(12) -> 11*
       h0(11) -> 11*
       h0(8) -> 11*
       f{#,0}(12,11) -> 6*
       f{#,0}(12,8) -> 6*
       f{#,0}(12,10) -> 6*
       f{#,0}(12,12) -> 6*
       a0() -> 8*
       g0(10) -> 12*
       g0(12) -> 12*
       g0(11) -> 12*
       g0(8) -> 12*
       8 -> 11*
       10 -> 11*
       12 -> 11*
     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