YES

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

Proof:
 DP Processor:
  DPs:
   a#(f(),a(f(),x)) -> a#(x,g())
   a#(x,g()) -> a#(f(),x)
   a#(x,g()) -> a#(g(),a(f(),x))
   a#(x,g()) -> a#(f(),a(g(),a(f(),x)))
  TRS:
   a(f(),a(f(),x)) -> a(x,g())
   a(x,g()) -> a(f(),a(g(),a(f(),x)))
  EDG Processor:
   DPs:
    a#(f(),a(f(),x)) -> a#(x,g())
    a#(x,g()) -> a#(f(),x)
    a#(x,g()) -> a#(g(),a(f(),x))
    a#(x,g()) -> a#(f(),a(g(),a(f(),x)))
   TRS:
    a(f(),a(f(),x)) -> a(x,g())
    a(x,g()) -> a(f(),a(g(),a(f(),x)))
   graph:
    a#(f(),a(f(),x)) -> a#(x,g()) -> a#(x,g()) -> a#(f(),x)
    a#(f(),a(f(),x)) -> a#(x,g()) -> a#(x,g()) -> a#(g(),a(f(),x))
    a#(f(),a(f(),x)) -> a#(x,g()) ->
    a#(x,g()) -> a#(f(),a(g(),a(f(),x)))
    a#(x,g()) -> a#(f(),a(g(),a(f(),x))) -> a#(f(),a(f(),x)) -> a#(x,g())
    a#(x,g()) -> a#(f(),x) -> a#(f(),a(f(),x)) -> a#(x,g())
    a#(x,g()) -> a#(f(),x) -> a#(x,g()) -> a#(f(),x)
    a#(x,g()) -> a#(f(),x) -> a#(x,g()) -> a#(g(),a(f(),x))
    a#(x,g()) -> a#(f(),x) -> a#(x,g()) -> a#(f(),a(g(),a(f(),x)))
   SCC Processor:
    #sccs: 1
    #rules: 3
    #arcs: 8/16
    DPs:
     a#(f(),a(f(),x)) -> a#(x,g())
     a#(x,g()) -> a#(f(),a(g(),a(f(),x)))
     a#(x,g()) -> a#(f(),x)
    TRS:
     a(f(),a(f(),x)) -> a(x,g())
     a(x,g()) -> a(f(),a(g(),a(f(),x)))
    Bounds Processor:
     bound: 0
     enrichment: match-dp
     automaton:
      final states: {4}
      transitions:
       a{#,0}(6,9) -> 4*
       f0() -> 6*
       g0() -> 8*
       f40() -> 5*
       a0(8,7) -> 9*
       a0(6,6) -> 7*
       a0(6,9) -> 7*
       a0(6,5) -> 7*
       a0(5,8) -> 7*
     problem:
      DPs:
       a#(f(),a(f(),x)) -> a#(x,g())
       a#(x,g()) -> a#(f(),x)
      TRS:
       a(f(),a(f(),x)) -> a(x,g())
       a(x,g()) -> a(f(),a(g(),a(f(),x)))
     Bounds Processor:
      bound: 0
      enrichment: match-dp
      automaton:
       final states: {1}
       transitions:
        a{#,0}(7,7) -> 1*
        a{#,0}(3,2) -> 1*
        a{#,0}(7,3) -> 1*
        f0() -> 7*
        g0() -> 2*
        f120() -> 3*
      problem:
       DPs:
        a#(x,g()) -> a#(f(),x)
       TRS:
        a(f(),a(f(),x)) -> a(x,g())
        a(x,g()) -> a(f(),a(g(),a(f(),x)))
      Bounds Processor:
       bound: 0
       enrichment: match-dp
       automaton:
        final states: {1}
        transitions:
         a{#,0}(3,2) -> 1*
         a{#,0}(3,3) -> 1*
         f0() -> 3*
         f140() -> 2*
       problem:
        DPs:
         
        TRS:
         a(f(),a(f(),x)) -> a(x,g())
         a(x,g()) -> a(f(),a(g(),a(f(),x)))
       Qed