YES

Problem:
 a(a(a(x1))) -> b(b(x1))
 b(b(b(x1))) -> c(x1)
 c(x1) -> d(d(x1))
 d(x1) -> a(a(x1))

Proof:
 Arctic Interpretation Processor:
  dimension: 1
  interpretation:
   [d](x0) = 6x0,
   
   [c](x0) = 12x0,
   
   [b](x0) = 4x0,
   
   [a](x0) = 3x0
  orientation:
   a(a(a(x1))) = 9x1 >= 8x1 = b(b(x1))
   
   b(b(b(x1))) = 12x1 >= 12x1 = c(x1)
   
   c(x1) = 12x1 >= 12x1 = d(d(x1))
   
   d(x1) = 6x1 >= 6x1 = a(a(x1))
  problem:
   b(b(b(x1))) -> c(x1)
   c(x1) -> d(d(x1))
   d(x1) -> a(a(x1))
  Arctic Interpretation Processor:
   dimension: 2
   interpretation:
              [1  -&]  
    [d](x0) = [-& 0 ]x0,
    
              [2  -&]  
    [c](x0) = [-& 0 ]x0,
    
              [2  0 ]  
    [b](x0) = [-& 0 ]x0,
    
              [0  -&]  
    [a](x0) = [-& -&]x0
   orientation:
                  [6  4 ]      [2  -&]          
    b(b(b(x1))) = [-& 0 ]x1 >= [-& 0 ]x1 = c(x1)
    
            [2  -&]      [2  -&]             
    c(x1) = [-& 0 ]x1 >= [-& 0 ]x1 = d(d(x1))
    
            [1  -&]      [0  -&]             
    d(x1) = [-& 0 ]x1 >= [-& -&]x1 = a(a(x1))
   problem:
    b(b(b(x1))) -> c(x1)
    c(x1) -> d(d(x1))
   Arctic Interpretation Processor:
    dimension: 3
    interpretation:
               [0  0  0 ]  
     [d](x0) = [0  0  -&]x0
               [0  -& 0 ]  ,
     
               [0 0 0]  
     [c](x0) = [0 0 0]x0
               [0 0 0]  ,
     
               [0  0  0 ]  
     [b](x0) = [1  0  -&]x0
               [0  0  0 ]  
    orientation:
                   [1 1 1]      [0 0 0]          
     b(b(b(x1))) = [2 1 1]x1 >= [0 0 0]x1 = c(x1)
                   [1 1 1]      [0 0 0]          
     
             [0 0 0]      [0 0 0]             
     c(x1) = [0 0 0]x1 >= [0 0 0]x1 = d(d(x1))
             [0 0 0]      [0 0 0]             
    problem:
     c(x1) -> d(d(x1))
    Arctic Interpretation Processor:
     dimension: 3
     interpretation:
                [0  0  0 ]  
      [d](x0) = [0  0  1 ]x0
                [0  -& 1 ]  ,
      
                [2 2 2]  
      [c](x0) = [2 1 3]x0
                [2 1 3]  
     orientation:
              [2 2 2]      [0 0 1]             
      c(x1) = [2 1 3]x1 >= [1 0 2]x1 = d(d(x1))
              [2 1 3]      [1 0 2]             
     problem:
      
     Qed