YES

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

Proof:
 Extended Uncurrying Processor:
  application symbol: a
  symbol table:
   g ==> g0/0 g1/1
   f ==> f0/0 f1/1
   
  uncurry-rules:
   a(f0(),x1) -> f1(x1)
   a(g0(),x3) -> g1(x3)
  eta-rules:
   
  problem:
   f1(f1(g1(g1(x)))) -> g1(g1(g1(f1(f1(f1(x))))))
   a(f0(),x1) -> f1(x1)
   a(g0(),x3) -> g1(x3)
  Matrix Interpretation Processor: dim=3
   
   interpretation:
               [1 0 0]     [0]
    [f1](x0) = [1 0 0]x0 + [1]
               [1 0 0]     [0],
    
           [0]
    [f0] = [1]
           [0],
    
           [1]
    [g0] = [0]
           [0],
    
               [1 0 0]  
    [g1](x0) = [1 0 0]x0
               [1 0 0]  ,
    
                  [1 0 0]     [1 1 1]  
    [a](x0, x1) = [0 1 0]x0 + [1 0 1]x1
                  [1 0 0]     [1 0 0]  
   orientation:
                        [1 0 0]    [0]    [1 0 0]                             
    f1(f1(g1(g1(x)))) = [1 0 0]x + [1] >= [1 0 0]x = g1(g1(g1(f1(f1(f1(x))))))
                        [1 0 0]    [0]    [1 0 0]                             
    
                 [1 1 1]     [0]    [1 0 0]     [0]         
    a(f0(),x1) = [1 0 1]x1 + [1] >= [1 0 0]x1 + [1] = f1(x1)
                 [1 0 0]     [0]    [1 0 0]     [0]         
    
                 [1 1 1]     [1]    [1 0 0]           
    a(g0(),x3) = [1 0 1]x3 + [0] >= [1 0 0]x3 = g1(x3)
                 [1 0 0]     [1]    [1 0 0]           
   problem:
    f1(f1(g1(g1(x)))) -> g1(g1(g1(f1(f1(f1(x))))))
    a(f0(),x1) -> f1(x1)
   Matrix Interpretation Processor: dim=3
    
    interpretation:
                  
     [f1](x0) = x0
                  ,
     
            [0]
     [f0] = [0]
            [0],
     
                [1 1 0]  
     [g1](x0) = [0 0 0]x0
                [1 0 0]  ,
     
                   [1 0 0]     [1 0 0]     [1]
     [a](x0, x1) = [0 0 0]x0 + [0 1 1]x1 + [0]
                   [0 0 0]     [0 1 1]     [0]
    orientation:
                         [1 1 0]     [1 1 0]                             
     f1(f1(g1(g1(x)))) = [0 0 0]x >= [0 0 0]x = g1(g1(g1(f1(f1(f1(x))))))
                         [1 1 0]     [1 1 0]                             
     
                  [1 0 0]     [1]               
     a(f0(),x1) = [0 1 1]x1 + [0] >= x1 = f1(x1)
                  [0 1 1]     [0]               
    problem:
     f1(f1(g1(g1(x)))) -> g1(g1(g1(f1(f1(f1(x))))))
    DP Processor:
     DPs:
      f{1,#}(f1(g1(g1(x)))) -> f{1,#}(x)
      f{1,#}(f1(g1(g1(x)))) -> f{1,#}(f1(x))
      f{1,#}(f1(g1(g1(x)))) -> f{1,#}(f1(f1(x)))
     TRS:
      f1(f1(g1(g1(x)))) -> g1(g1(g1(f1(f1(f1(x))))))
     Bounds Processor:
      bound: 3
      enrichment: match
      automaton:
       final states: {7,5,3,1}
       transitions:
        f{1,3}(103) -> 104*
        f{1,3}(111) -> 112*
        f{1,3}(109) -> 110*
        g{1,0}(9) -> 10*
        g{1,0}(10) -> 7*
        g{1,0}(8) -> 9*
        f{1,#,3}(99) -> 100*
        g{1,2}(96) -> 97*
        g{1,2}(97) -> 98*
        g{1,2}(95) -> 96*
        g{1,2}(68) -> 69*
        g{1,2}(70) -> 71*
        g{1,2}(69) -> 70*
        f{1,#,1}(35) -> 36*
        f{1,#,1}(11) -> 12*
        f{1,0}(4) -> 6*
        f{1,0}(6) -> 8*
        f{1,0}(2) -> 4*
        f{1,1}(15) -> 16*
        f{1,1}(21) -> 22*
        f{1,1}(23) -> 24*
        f{1,1}(50) -> 51*
        f{1,1}(39) -> 40*
        f{1,1}(41) -> 42*
        f{1,#,0}(2) -> 1*
        f{1,#,0}(4) -> 3*
        f{1,#,0}(6) -> 5*
        f{1,#,2}(55) -> 56*
        f{1,#,2}(79) -> 80*
        f80() -> 2*
        g{1,3}(112) -> 113*
        g{1,3}(113) -> 114*
        g{1,3}(114) -> 115*
        f{1,2}(83) -> 84*
        f{1,2}(59) -> 60*
        f{1,2}(94) -> 95*
        f{1,2}(85) -> 86*
        f{1,2}(67) -> 68*
        f{1,2}(65) -> 66*
        g{1,1}(25) -> 26*
        g{1,1}(51) -> 52*
        g{1,1}(52) -> 53*
        g{1,1}(26) -> 27*
        g{1,1}(24) -> 25*
        g{1,1}(53) -> 54*
        56 -> 12,5
        27 -> 8*
        24 -> 59,55
        115 -> 68*
        42 -> 50,11
        7 -> 4,6
        100 -> 56,12
        104 -> 109,99
        12 -> 5,1
        16 -> 21,11
        66 -> 67,55
        22 -> 23,11
        52 -> 83,79
        54 -> 22,11,23
        36 -> 12,5
        86 -> 94,55
        26 -> 39,35
        60 -> 65,55
        71 -> 42,11,50
        40 -> 41,11
        1 -> 3*
        110 -> 111,99
        5 -> 3*
        9 -> 15,11
        3 -> 1*
        84 -> 85,55
        96 -> 103,99
        98 -> 60,55,65
        80 -> 56,12
      problem:
       DPs:
        
       TRS:
        f1(f1(g1(g1(x)))) -> g1(g1(g1(f1(f1(f1(x))))))
      Qed