YES(?,O(n^2))

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:
 RT Transformation Processor:
  strict:
   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))
  weak:
   
  Matrix Interpretation Processor:
   dimension: 1
   interpretation:
    [a](x0) = x0 + 5,
    
    [f](x0) = x0,
    
    [g](x0) = x0 + 1,
    
    [c](x0) = x0,
    
    [d](x0) = x0 + 18,
    
    [b](x0) = x0 + 1
   orientation:
    b(b(x1)) = x1 + 2 >= x1 + 18 = c(d(x1))
    
    c(c(x1)) = x1 >= x1 + 54 = d(d(d(x1)))
    
    c(x1) = x1 >= x1 + 1 = g(x1)
    
    d(d(x1)) = x1 + 36 >= x1 = c(f(x1))
    
    d(d(d(x1))) = x1 + 54 >= x1 + 1 = g(c(x1))
    
    f(x1) = x1 >= x1 + 6 = a(g(x1))
    
    g(x1) = x1 + 1 >= x1 + 24 = d(a(b(x1)))
    
    g(g(x1)) = x1 + 2 >= x1 + 1 = b(c(x1))
   problem:
    strict:
     b(b(x1)) -> c(d(x1))
     c(c(x1)) -> d(d(d(x1)))
     c(x1) -> g(x1)
     f(x1) -> a(g(x1))
     g(x1) -> d(a(b(x1)))
    weak:
     d(d(x1)) -> c(f(x1))
     d(d(d(x1))) -> g(c(x1))
     g(g(x1)) -> b(c(x1))
   Matrix Interpretation Processor:
    dimension: 1
    interpretation:
     [a](x0) = x0 + 17,
     
     [f](x0) = x0 + 3,
     
     [g](x0) = x0 + 4,
     
     [c](x0) = x0 + 1,
     
     [d](x0) = x0 + 4,
     
     [b](x0) = x0 + 3
    orientation:
     b(b(x1)) = x1 + 6 >= x1 + 5 = c(d(x1))
     
     c(c(x1)) = x1 + 2 >= x1 + 12 = d(d(d(x1)))
     
     c(x1) = x1 + 1 >= x1 + 4 = g(x1)
     
     f(x1) = x1 + 3 >= x1 + 21 = a(g(x1))
     
     g(x1) = x1 + 4 >= x1 + 24 = d(a(b(x1)))
     
     d(d(x1)) = x1 + 8 >= x1 + 4 = c(f(x1))
     
     d(d(d(x1))) = x1 + 12 >= x1 + 5 = g(c(x1))
     
     g(g(x1)) = x1 + 8 >= x1 + 4 = b(c(x1))
    problem:
     strict:
      c(c(x1)) -> d(d(d(x1)))
      c(x1) -> g(x1)
      f(x1) -> a(g(x1))
      g(x1) -> d(a(b(x1)))
     weak:
      b(b(x1)) -> c(d(x1))
      d(d(x1)) -> c(f(x1))
      d(d(d(x1))) -> g(c(x1))
      g(g(x1)) -> b(c(x1))
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [a](x0) = x0 + 1,
      
      [f](x0) = x0 + 26,
      
      [g](x0) = x0 + 20,
      
      [c](x0) = x0,
      
      [d](x0) = x0 + 16,
      
      [b](x0) = x0 + 20
     orientation:
      c(c(x1)) = x1 >= x1 + 48 = d(d(d(x1)))
      
      c(x1) = x1 >= x1 + 20 = g(x1)
      
      f(x1) = x1 + 26 >= x1 + 21 = a(g(x1))
      
      g(x1) = x1 + 20 >= x1 + 37 = d(a(b(x1)))
      
      b(b(x1)) = x1 + 40 >= x1 + 16 = c(d(x1))
      
      d(d(x1)) = x1 + 32 >= x1 + 26 = c(f(x1))
      
      d(d(d(x1))) = x1 + 48 >= x1 + 20 = g(c(x1))
      
      g(g(x1)) = x1 + 40 >= x1 + 20 = b(c(x1))
     problem:
      strict:
       c(c(x1)) -> d(d(d(x1)))
       c(x1) -> g(x1)
       g(x1) -> d(a(b(x1)))
      weak:
       f(x1) -> a(g(x1))
       b(b(x1)) -> c(d(x1))
       d(d(x1)) -> c(f(x1))
       d(d(d(x1))) -> g(c(x1))
       g(g(x1)) -> b(c(x1))
     Matrix Interpretation Processor:
      dimension: 1
      interpretation:
       [a](x0) = x0,
       
       [f](x0) = x0 + 4,
       
       [g](x0) = x0 + 4,
       
       [c](x0) = x0,
       
       [d](x0) = x0 + 2,
       
       [b](x0) = x0 + 1
      orientation:
       c(c(x1)) = x1 >= x1 + 6 = d(d(d(x1)))
       
       c(x1) = x1 >= x1 + 4 = g(x1)
       
       g(x1) = x1 + 4 >= x1 + 3 = d(a(b(x1)))
       
       f(x1) = x1 + 4 >= x1 + 4 = a(g(x1))
       
       b(b(x1)) = x1 + 2 >= x1 + 2 = c(d(x1))
       
       d(d(x1)) = x1 + 4 >= x1 + 4 = c(f(x1))
       
       d(d(d(x1))) = x1 + 6 >= x1 + 4 = g(c(x1))
       
       g(g(x1)) = x1 + 8 >= x1 + 1 = b(c(x1))
      problem:
       strict:
        c(c(x1)) -> d(d(d(x1)))
        c(x1) -> g(x1)
       weak:
        g(x1) -> d(a(b(x1)))
        f(x1) -> a(g(x1))
        b(b(x1)) -> c(d(x1))
        d(d(x1)) -> c(f(x1))
        d(d(d(x1))) -> g(c(x1))
        g(g(x1)) -> b(c(x1))
      Matrix Interpretation Processor:
       dimension: 2
       interpretation:
                  [1 0]  
        [a](x0) = [0 0]x0,
        
                  [1  15]     [7]
        [f](x0) = [0  0 ]x0 + [0],
        
                  [1  15]     [6]
        [g](x0) = [0  0 ]x0 + [1],
        
                  [1  15]     [8]
        [c](x0) = [0  0 ]x0 + [1],
        
                  [1  15]     [0]
        [d](x0) = [0  0 ]x0 + [1],
        
                  [1  15]     [4]
        [b](x0) = [0  0 ]x0 + [1]
       orientation:
                   [1  15]     [31]    [1  15]     [30]              
        c(c(x1)) = [0  0 ]x1 + [1 ] >= [0  0 ]x1 + [1 ] = d(d(d(x1)))
        
                [1  15]     [8]    [1  15]     [6]        
        c(x1) = [0  0 ]x1 + [1] >= [0  0 ]x1 + [1] = g(x1)
        
                [1  15]     [6]    [1  15]     [4]              
        g(x1) = [0  0 ]x1 + [1] >= [0  0 ]x1 + [1] = d(a(b(x1)))
        
                [1  15]     [7]    [1  15]     [6]           
        f(x1) = [0  0 ]x1 + [0] >= [0  0 ]x1 + [0] = a(g(x1))
        
                   [1  15]     [23]    [1  15]     [23]           
        b(b(x1)) = [0  0 ]x1 + [1 ] >= [0  0 ]x1 + [1 ] = c(d(x1))
        
                   [1  15]     [15]    [1  15]     [15]           
        d(d(x1)) = [0  0 ]x1 + [1 ] >= [0  0 ]x1 + [1 ] = c(f(x1))
        
                      [1  15]     [30]    [1  15]     [29]           
        d(d(d(x1))) = [0  0 ]x1 + [1 ] >= [0  0 ]x1 + [1 ] = g(c(x1))
        
                   [1  15]     [27]    [1  15]     [27]           
        g(g(x1)) = [0  0 ]x1 + [1 ] >= [0  0 ]x1 + [1 ] = b(c(x1))
       problem:
        strict:
         
        weak:
         c(c(x1)) -> d(d(d(x1)))
         c(x1) -> g(x1)
         g(x1) -> d(a(b(x1)))
         f(x1) -> a(g(x1))
         b(b(x1)) -> c(d(x1))
         d(d(x1)) -> c(f(x1))
         d(d(d(x1))) -> g(c(x1))
         g(g(x1)) -> b(c(x1))
       Qed