YES

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

Proof:
 Arctic Interpretation Processor:
  dimension: 2
  interpretation:
             [0  -&]  
   [a](x0) = [-& -&]x0,
   
             [0  1 ]  
   [f](x0) = [-& -&]x0,
   
             [0 1]  
   [g](x0) = [1 3]x0,
   
             [0 1]  
   [c](x0) = [1 3]x0,
   
             [0 0]  
   [d](x0) = [0 2]x0,
   
             [0 1]  
   [b](x0) = [2 3]x0
  orientation:
              [3 4]      [1 3]             
   b(b(x1)) = [5 6]x1 >= [3 5]x1 = c(d(x1))
   
              [2 4]      [2 4]                
   c(c(x1)) = [4 6]x1 >= [4 6]x1 = d(d(d(x1)))
   
           [0 1]      [0 1]          
   c(x1) = [1 3]x1 >= [1 3]x1 = g(x1)
   
              [0 2]      [0 1]             
   d(d(x1)) = [2 4]x1 >= [1 2]x1 = c(f(x1))
   
                 [2 4]      [2 4]             
   d(d(d(x1))) = [4 6]x1 >= [4 6]x1 = g(c(x1))
   
           [0  1 ]      [0  1 ]             
   f(x1) = [-& -&]x1 >= [-& -&]x1 = a(g(x1))
   
           [0 1]      [0 1]                
   g(x1) = [1 3]x1 >= [0 1]x1 = d(a(b(x1)))
   
              [2 4]      [2 4]             
   g(g(x1)) = [4 6]x1 >= [4 6]x1 = b(c(x1))
  problem:
   c(c(x1)) -> d(d(d(x1)))
   c(x1) -> g(x1)
   d(d(x1)) -> c(f(x1))
   d(d(d(x1))) -> g(c(x1))
   f(x1) -> a(g(x1))
   g(x1) -> d(a(b(x1)))
   g(g(x1)) -> b(c(x1))
  Arctic Interpretation Processor:
   dimension: 2
   interpretation:
              [0  -&]  
    [a](x0) = [-& -&]x0,
    
              [0  0 ]  
    [f](x0) = [-& 0 ]x0,
    
              [0 0]  
    [g](x0) = [1 1]x0,
    
              [0 0]  
    [c](x0) = [1 2]x0,
    
              [0 0]  
    [d](x0) = [0 1]x0,
    
              [0  -&]  
    [b](x0) = [0  -&]x0
   orientation:
               [1 2]      [1 2]                
    c(c(x1)) = [3 4]x1 >= [2 3]x1 = d(d(d(x1)))
    
            [0 0]      [0 0]          
    c(x1) = [1 2]x1 >= [1 1]x1 = g(x1)
    
               [0 1]      [0 0]             
    d(d(x1)) = [1 2]x1 >= [1 2]x1 = c(f(x1))
    
                  [1 2]      [1 2]             
    d(d(d(x1))) = [2 3]x1 >= [2 3]x1 = g(c(x1))
    
            [0  0 ]      [0  0 ]             
    f(x1) = [-& 0 ]x1 >= [-& -&]x1 = a(g(x1))
    
            [0 0]      [0  -&]                
    g(x1) = [1 1]x1 >= [0  -&]x1 = d(a(b(x1)))
    
               [1 1]      [0 0]             
    g(g(x1)) = [2 2]x1 >= [0 0]x1 = b(c(x1))
   problem:
    c(c(x1)) -> d(d(d(x1)))
    c(x1) -> g(x1)
    d(d(x1)) -> c(f(x1))
    d(d(d(x1))) -> g(c(x1))
    f(x1) -> a(g(x1))
    g(x1) -> d(a(b(x1)))
   Arctic Interpretation Processor:
    dimension: 2
    interpretation:
               [0  -&]  
     [a](x0) = [0  -&]x0,
     
               [0  -&]  
     [f](x0) = [0  -&]x0,
     
               [0  -&]  
     [g](x0) = [2  0 ]x0,
     
               [2 0]  
     [c](x0) = [2 3]x0,
     
               [0 0]  
     [d](x0) = [2 1]x0,
     
               [0  -&]  
     [b](x0) = [0  0 ]x0
    orientation:
                [4 3]      [3 2]                
     c(c(x1)) = [5 6]x1 >= [4 3]x1 = d(d(d(x1)))
     
             [2 0]      [0  -&]          
     c(x1) = [2 3]x1 >= [2  0 ]x1 = g(x1)
     
                [2 1]      [2  -&]             
     d(d(x1)) = [3 2]x1 >= [3  -&]x1 = c(f(x1))
     
                   [3 2]      [2 0]             
     d(d(d(x1))) = [4 3]x1 >= [4 3]x1 = g(c(x1))
     
             [0  -&]      [0  -&]             
     f(x1) = [0  -&]x1 >= [0  -&]x1 = a(g(x1))
     
             [0  -&]      [0  -&]                
     g(x1) = [2  0 ]x1 >= [2  -&]x1 = d(a(b(x1)))
    problem:
     c(x1) -> g(x1)
     d(d(x1)) -> c(f(x1))
     d(d(d(x1))) -> g(c(x1))
     f(x1) -> a(g(x1))
     g(x1) -> d(a(b(x1)))
    Arctic Interpretation Processor:
     dimension: 2
     interpretation:
                [0  -&]  
      [a](x0) = [0  -&]x0,
      
                [0  -&]  
      [f](x0) = [0  0 ]x0,
      
                [0  -&]  
      [g](x0) = [1  0 ]x0,
      
                [0 0]  
      [c](x0) = [1 1]x0,
      
                [0 0]  
      [d](x0) = [0 1]x0,
      
                [0  -&]  
      [b](x0) = [2  1 ]x0
     orientation:
              [0 0]      [0  -&]          
      c(x1) = [1 1]x1 >= [1  0 ]x1 = g(x1)
      
                 [0 1]      [0 0]             
      d(d(x1)) = [1 2]x1 >= [1 1]x1 = c(f(x1))
      
                    [1 2]      [0 0]             
      d(d(d(x1))) = [2 3]x1 >= [1 1]x1 = g(c(x1))
      
              [0  -&]      [0  -&]             
      f(x1) = [0  0 ]x1 >= [0  -&]x1 = a(g(x1))
      
              [0  -&]      [0  -&]                
      g(x1) = [1  0 ]x1 >= [1  -&]x1 = d(a(b(x1)))
     problem:
      c(x1) -> g(x1)
      d(d(x1)) -> c(f(x1))
      f(x1) -> a(g(x1))
      g(x1) -> d(a(b(x1)))
     Arctic Interpretation Processor:
      dimension: 2
      interpretation:
                 [0  -&]  
       [a](x0) = [0  -&]x0,
       
                 [0 0]  
       [f](x0) = [0 1]x0,
       
                 [0  -&]  
       [g](x0) = [1  -&]x0,
       
                 [1 0]  
       [c](x0) = [2 0]x0,
       
                 [0 0]  
       [d](x0) = [1 1]x0,
       
                 [0  -&]  
       [b](x0) = [2  0 ]x0
      orientation:
               [1 0]      [0  -&]          
       c(x1) = [2 0]x1 >= [1  -&]x1 = g(x1)
       
                  [1 1]      [1 1]             
       d(d(x1)) = [2 2]x1 >= [2 2]x1 = c(f(x1))
       
               [0 0]      [0  -&]             
       f(x1) = [0 1]x1 >= [0  -&]x1 = a(g(x1))
       
               [0  -&]      [0  -&]                
       g(x1) = [1  -&]x1 >= [1  -&]x1 = d(a(b(x1)))
      problem:
       d(d(x1)) -> c(f(x1))
       f(x1) -> a(g(x1))
       g(x1) -> d(a(b(x1)))
      Arctic Interpretation Processor:
       dimension: 2
       interpretation:
                  [0  0 ]  
        [a](x0) = [-& 0 ]x0,
        
                  [2 2]  
        [f](x0) = [2 2]x0,
        
                  [2 2]  
        [g](x0) = [2 2]x0,
        
                  [0  0 ]  
        [c](x0) = [-& -&]x0,
        
                  [0 2]  
        [d](x0) = [1 2]x0,
        
                  [0 1]  
        [b](x0) = [0 0]x0
       orientation:
                   [3 4]      [2  2 ]             
        d(d(x1)) = [3 4]x1 >= [-& -&]x1 = c(f(x1))
        
                [2 2]      [2 2]             
        f(x1) = [2 2]x1 >= [2 2]x1 = a(g(x1))
        
                [2 2]      [2 2]                
        g(x1) = [2 2]x1 >= [2 2]x1 = d(a(b(x1)))
       problem:
        f(x1) -> a(g(x1))
        g(x1) -> d(a(b(x1)))
       Arctic Interpretation Processor:
        dimension: 3
        interpretation:
                   [0  0  0 ]  
         [a](x0) = [-& 0  0 ]x0
                   [0  -& -&]  ,
         
                   [1 1 1]  
         [f](x0) = [1 1 1]x0
                   [1 1 1]  ,
         
                   [0 0 0]  
         [g](x0) = [0 0 0]x0
                   [0 0 0]  ,
         
                   [0  0  0 ]  
         [d](x0) = [0  0  0 ]x0
                   [-& -& 0 ]  ,
         
                   [0 0 0]  
         [b](x0) = [0 0 0]x0
                   [0 0 0]  
        orientation:
                 [1 1 1]      [0 0 0]             
         f(x1) = [1 1 1]x1 >= [0 0 0]x1 = a(g(x1))
                 [1 1 1]      [0 0 0]             
         
                 [0 0 0]      [0 0 0]                
         g(x1) = [0 0 0]x1 >= [0 0 0]x1 = d(a(b(x1)))
                 [0 0 0]      [0 0 0]                
        problem:
         g(x1) -> d(a(b(x1)))
        Arctic Interpretation Processor:
         dimension: 3
         interpretation:
                    [0  0  0 ]  
          [a](x0) = [0  0  0 ]x0
                    [0  0  -&]  ,
          
                    [2 1 3]  
          [g](x0) = [2 1 3]x0
                    [2 2 0]  ,
          
                    [0  0  0 ]  
          [d](x0) = [-& 0  -&]x0
                    [-& -& 0 ]  ,
          
                    [1  0  -&]  
          [b](x0) = [1  0  -&]x0
                    [-& 0  2 ]  
         orientation:
                  [2 1 3]      [1  0  2 ]                
          g(x1) = [2 1 3]x1 >= [1  0  2 ]x1 = d(a(b(x1)))
                  [2 2 0]      [1  0  -&]                
         problem:
          
         Qed