YES

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

Proof:
 String Reversal Processor:
  a(a(x1)) -> c(b(x1))
  b(b(x1)) -> d(c(x1))
  b(x1) -> a(x1)
  c(c(x1)) -> f(d(x1))
  d(d(x1)) -> f(f(f(x1)))
  d(x1) -> b(x1)
  f(f(x1)) -> a(g(x1))
  g(g(x1)) -> a(x1)
  Matrix Interpretation Processor: dim=3
   
   interpretation:
              [1 1 0]     [0]
    [g](x0) = [0 1 1]x0 + [1]
              [0 0 0]     [0],
    
              [1 1 0]  
    [f](x0) = [0 0 0]x0
              [0 0 0]  ,
    
              [1 0 1]  
    [d](x0) = [0 0 0]x0
              [0 1 0]  ,
    
              [1 0 1]  
    [b](x0) = [0 0 0]x0
              [0 0 0]  ,
    
              [1 0 1]  
    [c](x0) = [0 0 0]x0
              [0 0 0]  ,
    
              [1 0 1]  
    [a](x0) = [0 0 0]x0
              [0 0 0]  
   orientation:
               [1 0 1]      [1 0 1]             
    a(a(x1)) = [0 0 0]x1 >= [0 0 0]x1 = c(b(x1))
               [0 0 0]      [0 0 0]             
    
               [1 0 1]      [1 0 1]             
    b(b(x1)) = [0 0 0]x1 >= [0 0 0]x1 = d(c(x1))
               [0 0 0]      [0 0 0]             
    
            [1 0 1]      [1 0 1]          
    b(x1) = [0 0 0]x1 >= [0 0 0]x1 = a(x1)
            [0 0 0]      [0 0 0]          
    
               [1 0 1]      [1 0 1]             
    c(c(x1)) = [0 0 0]x1 >= [0 0 0]x1 = f(d(x1))
               [0 0 0]      [0 0 0]             
    
               [1 1 1]      [1 1 0]                
    d(d(x1)) = [0 0 0]x1 >= [0 0 0]x1 = f(f(f(x1)))
               [0 0 0]      [0 0 0]                
    
            [1 0 1]      [1 0 1]          
    d(x1) = [0 0 0]x1 >= [0 0 0]x1 = b(x1)
            [0 1 0]      [0 0 0]          
    
               [1 1 0]      [1 1 0]             
    f(f(x1)) = [0 0 0]x1 >= [0 0 0]x1 = a(g(x1))
               [0 0 0]      [0 0 0]             
    
               [1 2 1]     [1]    [1 0 1]          
    g(g(x1)) = [0 1 1]x1 + [2] >= [0 0 0]x1 = a(x1)
               [0 0 0]     [0]    [0 0 0]          
   problem:
    a(a(x1)) -> c(b(x1))
    b(b(x1)) -> d(c(x1))
    b(x1) -> a(x1)
    c(c(x1)) -> f(d(x1))
    d(d(x1)) -> f(f(f(x1)))
    d(x1) -> b(x1)
    f(f(x1)) -> a(g(x1))
   Arctic Interpretation Processor:
    dimension: 1
    interpretation:
     [g](x0) = 5x0,
     
     [f](x0) = 8x0,
     
     [d](x0) = 12x0,
     
     [b](x0) = 12x0,
     
     [c](x0) = 10x0,
     
     [a](x0) = 11x0
    orientation:
     a(a(x1)) = 22x1 >= 22x1 = c(b(x1))
     
     b(b(x1)) = 24x1 >= 22x1 = d(c(x1))
     
     b(x1) = 12x1 >= 11x1 = a(x1)
     
     c(c(x1)) = 20x1 >= 20x1 = f(d(x1))
     
     d(d(x1)) = 24x1 >= 24x1 = f(f(f(x1)))
     
     d(x1) = 12x1 >= 12x1 = b(x1)
     
     f(f(x1)) = 16x1 >= 16x1 = a(g(x1))
    problem:
     a(a(x1)) -> c(b(x1))
     c(c(x1)) -> f(d(x1))
     d(d(x1)) -> f(f(f(x1)))
     d(x1) -> b(x1)
     f(f(x1)) -> a(g(x1))
    Arctic Interpretation Processor:
     dimension: 2
     interpretation:
                [2  -&]  
      [g](x0) = [1  -&]x0,
      
                [2  -&]  
      [f](x0) = [-& -&]x0,
      
                [3 1]  
      [d](x0) = [0 3]x0,
      
                [0  -&]  
      [b](x0) = [-& -&]x0,
      
                [3  0 ]  
      [c](x0) = [-& 2 ]x0,
      
                [2  3 ]  
      [a](x0) = [-& -&]x0
     orientation:
                 [4  5 ]      [3  -&]             
      a(a(x1)) = [-& -&]x1 >= [-& -&]x1 = c(b(x1))
      
                 [6  3 ]      [5  3 ]             
      c(c(x1)) = [-& 4 ]x1 >= [-& -&]x1 = f(d(x1))
      
                 [6 4]      [6  -&]                
      d(d(x1)) = [3 6]x1 >= [-& -&]x1 = f(f(f(x1)))
      
              [3 1]      [0  -&]          
      d(x1) = [0 3]x1 >= [-& -&]x1 = b(x1)
      
                 [4  -&]      [4  -&]             
      f(f(x1)) = [-& -&]x1 >= [-& -&]x1 = a(g(x1))
     problem:
      c(c(x1)) -> f(d(x1))
      d(d(x1)) -> f(f(f(x1)))
      f(f(x1)) -> a(g(x1))
     Arctic Interpretation Processor:
      dimension: 2
      interpretation:
                 [0  -&]  
       [g](x0) = [0  0 ]x0,
       
                 [0  0 ]  
       [f](x0) = [-& -&]x0,
       
                 [0 0]  
       [d](x0) = [0 0]x0,
       
                 [2  0 ]  
       [c](x0) = [-& 0 ]x0,
       
                 [0  0 ]  
       [a](x0) = [-& -&]x0
      orientation:
                  [4  2 ]      [0  0 ]             
       c(c(x1)) = [-& 0 ]x1 >= [-& -&]x1 = f(d(x1))
       
                  [0 0]      [0  0 ]                
       d(d(x1)) = [0 0]x1 >= [-& -&]x1 = f(f(f(x1)))
       
                  [0  0 ]      [0  0 ]             
       f(f(x1)) = [-& -&]x1 >= [-& -&]x1 = a(g(x1))
      problem:
       d(d(x1)) -> f(f(f(x1)))
       f(f(x1)) -> a(g(x1))
      Arctic Interpretation Processor:
       dimension: 2
       interpretation:
                  [0  -&]  
        [g](x0) = [0  -&]x0,
        
                    
        [f](x0) = x0,
        
                  [3 0]  
        [d](x0) = [2 0]x0,
        
                  [0  0 ]  
        [a](x0) = [-& -&]x0
       orientation:
                   [6 3]                      
        d(d(x1)) = [5 2]x1 >= x1 = f(f(f(x1)))
        
                         [0  -&]             
        f(f(x1)) = x1 >= [-& -&]x1 = a(g(x1))
       problem:
        f(f(x1)) -> a(g(x1))
       Arctic Interpretation Processor:
        dimension: 3
        interpretation:
                   [0  -& 0 ]  
         [g](x0) = [0  -& 2 ]x0
                   [0  -& 2 ]  ,
         
                   [0  0  2 ]  
         [f](x0) = [-& 0  2 ]x0
                   [1  -& 1 ]  ,
         
                   [0  0  0 ]  
         [a](x0) = [-& 0  0 ]x0
                   [0  0  0 ]  
        orientation:
                    [3 0 3]      [0  -& 2 ]             
         f(f(x1)) = [3 0 3]x1 >= [0  -& 2 ]x1 = a(g(x1))
                    [2 1 3]      [0  -& 2 ]             
        problem:
         
        Qed