YES

Problem:
 0(0(0(0(x1)))) -> 0(1(1(1(x1))))
 1(0(0(1(x1)))) -> 0(0(0(0(x1))))

Proof:
 String Reversal Processor:
  0(0(0(0(x1)))) -> 1(1(1(0(x1))))
  1(0(0(1(x1)))) -> 0(0(0(0(x1))))
  DP Processor:
   DPs:
    0#(0(0(0(x1)))) -> 1#(0(x1))
    0#(0(0(0(x1)))) -> 1#(1(0(x1)))
    0#(0(0(0(x1)))) -> 1#(1(1(0(x1))))
    1#(0(0(1(x1)))) -> 0#(x1)
    1#(0(0(1(x1)))) -> 0#(0(x1))
    1#(0(0(1(x1)))) -> 0#(0(0(x1)))
    1#(0(0(1(x1)))) -> 0#(0(0(0(x1))))
   TRS:
    0(0(0(0(x1)))) -> 1(1(1(0(x1))))
    1(0(0(1(x1)))) -> 0(0(0(0(x1))))
   TDG Processor:
    DPs:
     0#(0(0(0(x1)))) -> 1#(0(x1))
     0#(0(0(0(x1)))) -> 1#(1(0(x1)))
     0#(0(0(0(x1)))) -> 1#(1(1(0(x1))))
     1#(0(0(1(x1)))) -> 0#(x1)
     1#(0(0(1(x1)))) -> 0#(0(x1))
     1#(0(0(1(x1)))) -> 0#(0(0(x1)))
     1#(0(0(1(x1)))) -> 0#(0(0(0(x1))))
    TRS:
     0(0(0(0(x1)))) -> 1(1(1(0(x1))))
     1(0(0(1(x1)))) -> 0(0(0(0(x1))))
    graph:
     1#(0(0(1(x1)))) -> 0#(0(0(0(x1)))) ->
     0#(0(0(0(x1)))) -> 1#(1(1(0(x1))))
     1#(0(0(1(x1)))) -> 0#(0(0(0(x1)))) ->
     0#(0(0(0(x1)))) -> 1#(1(0(x1)))
     1#(0(0(1(x1)))) -> 0#(0(0(0(x1)))) ->
     0#(0(0(0(x1)))) -> 1#(0(x1))
     1#(0(0(1(x1)))) -> 0#(0(0(x1))) ->
     0#(0(0(0(x1)))) -> 1#(1(1(0(x1))))
     1#(0(0(1(x1)))) -> 0#(0(0(x1))) ->
     0#(0(0(0(x1)))) -> 1#(1(0(x1)))
     1#(0(0(1(x1)))) -> 0#(0(0(x1))) -> 0#(0(0(0(x1)))) -> 1#(0(x1))
     1#(0(0(1(x1)))) -> 0#(0(x1)) -> 0#(0(0(0(x1)))) -> 1#(1(1(0(x1))))
     1#(0(0(1(x1)))) -> 0#(0(x1)) -> 0#(0(0(0(x1)))) -> 1#(1(0(x1)))
     1#(0(0(1(x1)))) -> 0#(0(x1)) -> 0#(0(0(0(x1)))) -> 1#(0(x1))
     1#(0(0(1(x1)))) -> 0#(x1) -> 0#(0(0(0(x1)))) -> 1#(1(1(0(x1))))
     1#(0(0(1(x1)))) -> 0#(x1) -> 0#(0(0(0(x1)))) -> 1#(1(0(x1)))
     1#(0(0(1(x1)))) -> 0#(x1) -> 0#(0(0(0(x1)))) -> 1#(0(x1))
     0#(0(0(0(x1)))) -> 1#(1(1(0(x1)))) ->
     1#(0(0(1(x1)))) -> 0#(0(0(0(x1))))
     0#(0(0(0(x1)))) -> 1#(1(1(0(x1)))) ->
     1#(0(0(1(x1)))) -> 0#(0(0(x1)))
     0#(0(0(0(x1)))) -> 1#(1(1(0(x1)))) ->
     1#(0(0(1(x1)))) -> 0#(0(x1))
     0#(0(0(0(x1)))) -> 1#(1(1(0(x1)))) -> 1#(0(0(1(x1)))) -> 0#(x1)
     0#(0(0(0(x1)))) -> 1#(1(0(x1))) ->
     1#(0(0(1(x1)))) -> 0#(0(0(0(x1))))
     0#(0(0(0(x1)))) -> 1#(1(0(x1))) ->
     1#(0(0(1(x1)))) -> 0#(0(0(x1)))
     0#(0(0(0(x1)))) -> 1#(1(0(x1))) -> 1#(0(0(1(x1)))) -> 0#(0(x1))
     0#(0(0(0(x1)))) -> 1#(1(0(x1))) -> 1#(0(0(1(x1)))) -> 0#(x1)
     0#(0(0(0(x1)))) -> 1#(0(x1)) -> 1#(0(0(1(x1)))) -> 0#(0(0(0(x1))))
     0#(0(0(0(x1)))) -> 1#(0(x1)) -> 1#(0(0(1(x1)))) -> 0#(0(0(x1)))
     0#(0(0(0(x1)))) -> 1#(0(x1)) -> 1#(0(0(1(x1)))) -> 0#(0(x1))
     0#(0(0(0(x1)))) -> 1#(0(x1)) -> 1#(0(0(1(x1)))) -> 0#(x1)
    Arctic Interpretation Processor:
     dimension: 1
     interpretation:
      [1#](x0) = 1x0,
      
      [0#](x0) = 1x0,
      
      [1](x0) = 1x0,
      
      [0](x0) = 1x0
     orientation:
      0#(0(0(0(x1)))) = 4x1 >= 2x1 = 1#(0(x1))
      
      0#(0(0(0(x1)))) = 4x1 >= 3x1 = 1#(1(0(x1)))
      
      0#(0(0(0(x1)))) = 4x1 >= 4x1 = 1#(1(1(0(x1))))
      
      1#(0(0(1(x1)))) = 4x1 >= 1x1 = 0#(x1)
      
      1#(0(0(1(x1)))) = 4x1 >= 2x1 = 0#(0(x1))
      
      1#(0(0(1(x1)))) = 4x1 >= 3x1 = 0#(0(0(x1)))
      
      1#(0(0(1(x1)))) = 4x1 >= 4x1 = 0#(0(0(0(x1))))
      
      0(0(0(0(x1)))) = 4x1 >= 4x1 = 1(1(1(0(x1))))
      
      1(0(0(1(x1)))) = 4x1 >= 4x1 = 0(0(0(0(x1))))
     problem:
      DPs:
       0#(0(0(0(x1)))) -> 1#(1(1(0(x1))))
       1#(0(0(1(x1)))) -> 0#(0(0(0(x1))))
      TRS:
       0(0(0(0(x1)))) -> 1(1(1(0(x1))))
       1(0(0(1(x1)))) -> 0(0(0(0(x1))))
     EDG Processor:
      DPs:
       0#(0(0(0(x1)))) -> 1#(1(1(0(x1))))
       1#(0(0(1(x1)))) -> 0#(0(0(0(x1))))
      TRS:
       0(0(0(0(x1)))) -> 1(1(1(0(x1))))
       1(0(0(1(x1)))) -> 0(0(0(0(x1))))
      graph:
       1#(0(0(1(x1)))) -> 0#(0(0(0(x1)))) ->
       0#(0(0(0(x1)))) -> 1#(1(1(0(x1))))
       0#(0(0(0(x1)))) -> 1#(1(1(0(x1)))) -> 1#(0(0(1(x1)))) -> 0#(0(0(0(x1))))
      Arctic Interpretation Processor:
       dimension: 4
       interpretation:
        [1#](x0) = [-& 0  -& -&]x0 + [0],
        
        [0#](x0) = [-& -& 0  -&]x0 + [0],
        
                  [-& -& 0  0 ]     [0]
                  [-& -& -& 0 ]     [0]
        [1](x0) = [0  0  0  0 ]x0 + [0]
                  [0  0  0  -&]     [0],
        
                  [-& 0  0  -&]     [0 ]
                  [0  0  0  0 ]     [-&]
        [0](x0) = [0  -& -& -&]x0 + [0 ]
                  [0  0  1  0 ]     [-&]
       orientation:
        0#(0(0(0(x1)))) = [0 0 0 0]x1 + [0] >= [0 0 0 0]x1 + [0] = 1#(1(1(0(x1))))
        
        1#(0(0(1(x1)))) = [1 1 1 1]x1 + [1] >= [0 0 0 0]x1 + [0] = 0#(0(0(0(x1))))
        
                         [1 0 1 0]     [1]    [0 0 1 0]     [0]                 
                         [1 1 1 0]     [1]    [0 0 1 0]     [0]                 
        0(0(0(0(x1)))) = [0 0 1 0]x1 + [0] >= [0 0 1 0]x1 + [0] = 1(1(1(0(x1))))
                         [1 1 1 1]     [1]    [0 0 1 0]     [0]                 
        
                         [1 1 1 1]     [1]    [1 0 1 0]     [1]                 
                         [1 1 1 1]     [1]    [1 1 1 0]     [1]                 
        1(0(0(1(x1)))) = [1 1 1 1]x1 + [1] >= [0 0 1 0]x1 + [0] = 0(0(0(0(x1))))
                         [1 1 1 1]     [1]    [1 1 1 1]     [1]                 
       problem:
        DPs:
         0#(0(0(0(x1)))) -> 1#(1(1(0(x1))))
        TRS:
         0(0(0(0(x1)))) -> 1(1(1(0(x1))))
         1(0(0(1(x1)))) -> 0(0(0(0(x1))))
       EDG Processor:
        DPs:
         0#(0(0(0(x1)))) -> 1#(1(1(0(x1))))
        TRS:
         0(0(0(0(x1)))) -> 1(1(1(0(x1))))
         1(0(0(1(x1)))) -> 0(0(0(0(x1))))
        graph:
         
        Qed