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: 1
     enrichment: match-dp
     automaton:
      final states: {12}
      transitions:
       a{#,1}(22,25) -> 10,11
       g1() -> 24*
       f1() -> 22*
       a1(22,22) -> 32*
       a1(7,24) -> 23*
       a1(9,24) -> 23*
       a1(24,32) -> 7*
       a1(22,7) -> 23*
       a1(22,9) -> 23*
       a1(22,25) -> 23*
       a1(8,24) -> 23*
       a1(24,23) -> 25*
       a1(10,24) -> 23*
       a1(22,8) -> 23*
       a1(22,10) -> 23*
       a{#,0}(8,7) -> 10*
       a{#,0}(8,9) -> 10*
       a{#,0}(9,8) -> 10*
       a{#,0}(9,10) -> 10*
       a{#,0}(10,7) -> 10*
       a{#,0}(10,9) -> 10*
       a{#,0}(7,7) -> 10*
       a{#,0}(7,9) -> 10*
       a{#,0}(8,8) -> 10*
       a{#,0}(8,10) -> 10*
       a{#,0}(8,14) -> 12*
       a{#,0}(9,7) -> 10*
       a{#,0}(9,9) -> 10*
       a{#,0}(10,8) -> 10*
       a{#,0}(10,10) -> 10*
       a{#,0}(7,8) -> 10*
       a{#,0}(7,10) -> 10*
       f0() -> 8*
       a0(8,7) -> 7*
       a0(8,9) -> 7*
       a0(8,11) -> 13*
       a0(9,8) -> 7*
       a0(9,10) -> 7*
       a0(10,7) -> 7*
       a0(10,9) -> 13,7
       a0(7,7) -> 7*
       a0(7,9) -> 13,7
       a0(8,8) -> 7*
       a0(8,10) -> 7*
       a0(9,7) -> 7*
       a0(9,9) -> 13,7
       a0(9,13) -> 14*
       a0(10,8) -> 7*
       a0(10,10) -> 7*
       a0(7,8) -> 7*
       a0(7,10) -> 7*
       g0() -> 9*
       7 -> 11*
       8 -> 11*
       9 -> 11*
       10 -> 11*
     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: 1
      enrichment: match-dp
      automaton:
       final states: {6}
       transitions:
        a{#,1}(3,13) -> 6,4,5
        a{#,1}(16,2) -> 6,4
        a{#,1}(16,4) -> 6,4
        a{#,1}(16,16) -> 6,4,5
        a{#,1}(2,13) -> 6,4,5
        a{#,1}(4,13) -> 6,4,5
        a{#,1}(16,1) -> 6,4
        a{#,1}(16,3) -> 6,4
        a{#,1}(1,13) -> 6,4,5
        g1() -> 13*
        f1() -> 16*
        a{#,0}(3,1) -> 4*
        a{#,0}(3,3) -> 4*
        a{#,0}(3,5) -> 6*
        a{#,0}(4,2) -> 4*
        a{#,0}(4,4) -> 4*
        a{#,0}(5,1) -> 6*
        a{#,0}(1,2) -> 4*
        a{#,0}(1,4) -> 4*
        a{#,0}(2,1) -> 4*
        a{#,0}(2,3) -> 4*
        a{#,0}(3,2) -> 4*
        a{#,0}(3,4) -> 4*
        a{#,0}(4,1) -> 4*
        a{#,0}(4,3) -> 4*
        a{#,0}(1,1) -> 4*
        a{#,0}(1,3) -> 4*
        a{#,0}(2,2) -> 4*
        a{#,0}(2,4) -> 4*
        f0() -> 3*
        a0(3,1) -> 2*
        a0(3,3) -> 2*
        a0(4,2) -> 2*
        a0(4,4) -> 2*
        a0(1,2) -> 2*
        a0(1,4) -> 2*
        a0(2,1) -> 2*
        a0(2,3) -> 2*
        a0(3,2) -> 2*
        a0(3,4) -> 2*
        a0(4,1) -> 2*
        a0(4,3) -> 2*
        a0(1,1) -> 2*
        a0(1,3) -> 2*
        a0(2,2) -> 2*
        a0(2,4) -> 2*
        g0() -> 1*
        1 -> 5*
        2 -> 5*
        3 -> 5*
        4 -> 5*
      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)))
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [a#](x0, x1) = x0 + x1,
        
        [g] = 1,
        
        [a](x0, x1) = 0,
        
        [f] = 0
       orientation:
        a#(x,g()) = x + 1 >= x = a#(f(),x)
        
        a(f(),a(f(),x)) = 0 >= 0 = a(x,g())
        
        a(x,g()) = 0 >= 0 = a(f(),a(g(),a(f(),x)))
       problem:
        DPs:
         
        TRS:
         a(f(),a(f(),x)) -> a(x,g())
         a(x,g()) -> a(f(),a(g(),a(f(),x)))
       Qed