YES(?,O(n^1))

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:
 RT Transformation Processor:
  strict:
   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)
  weak:
   
  Matrix Interpretation Processor:
   dimension: 1
   interpretation:
    [g](x0) = x0 + 8,
    
    [f](x0) = x0 + 13,
    
    [d](x0) = x0 + 28,
    
    [b](x0) = x0 + 18,
    
    [c](x0) = x0 + 24,
    
    [a](x0) = x0 + 2
   orientation:
    a(a(x1)) = x1 + 4 >= x1 + 42 = b(c(x1))
    
    b(b(x1)) = x1 + 36 >= x1 + 52 = c(d(x1))
    
    b(x1) = x1 + 18 >= x1 + 2 = a(x1)
    
    c(c(x1)) = x1 + 48 >= x1 + 41 = d(f(x1))
    
    d(d(x1)) = x1 + 56 >= x1 + 39 = f(f(f(x1)))
    
    d(x1) = x1 + 28 >= x1 + 18 = b(x1)
    
    f(f(x1)) = x1 + 26 >= x1 + 10 = g(a(x1))
    
    g(g(x1)) = x1 + 16 >= x1 + 2 = a(x1)
   problem:
    strict:
     a(a(x1)) -> b(c(x1))
     b(b(x1)) -> c(d(x1))
    weak:
     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)
   Matrix Interpretation Processor:
    dimension: 1
    interpretation:
     [g](x0) = x0 + 6,
     
     [f](x0) = x0 + 7,
     
     [d](x0) = x0 + 17,
     
     [b](x0) = x0 + 15,
     
     [c](x0) = x0 + 12,
     
     [a](x0) = x0 + 8
    orientation:
     a(a(x1)) = x1 + 16 >= x1 + 27 = b(c(x1))
     
     b(b(x1)) = x1 + 30 >= x1 + 29 = c(d(x1))
     
     b(x1) = x1 + 15 >= x1 + 8 = a(x1)
     
     c(c(x1)) = x1 + 24 >= x1 + 24 = d(f(x1))
     
     d(d(x1)) = x1 + 34 >= x1 + 21 = f(f(f(x1)))
     
     d(x1) = x1 + 17 >= x1 + 15 = b(x1)
     
     f(f(x1)) = x1 + 14 >= x1 + 14 = g(a(x1))
     
     g(g(x1)) = x1 + 12 >= x1 + 8 = a(x1)
    problem:
     strict:
      a(a(x1)) -> b(c(x1))
     weak:
      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)
    Arctic Interpretation Processor:
     dimension: 2
     interpretation:
                [0 5]  
      [g](x0) = [1 6]x0,
      
                [0 2]  
      [f](x0) = [4 3]x0,
      
                [6 5]  
      [d](x0) = [4 1]x0,
      
                [6 4]  
      [b](x0) = [1 0]x0,
      
                [5 3]  
      [c](x0) = [0 3]x0,
      
                [6 4]  
      [a](x0) = [1 0]x0
     orientation:
                 [12 10]      [11 9 ]             
      a(a(x1)) = [7  5 ]x1 >= [6  4 ]x1 = b(c(x1))
      
                 [12 10]      [11 10]             
      b(b(x1)) = [7  5 ]x1 >= [7  5 ]x1 = c(d(x1))
      
              [6 4]      [6 4]          
      b(x1) = [1 0]x1 >= [1 0]x1 = a(x1)
      
                 [10 8 ]      [9 8]             
      c(c(x1)) = [5  6 ]x1 >= [5 6]x1 = d(f(x1))
      
                 [12 11]      [9  8 ]                
      d(d(x1)) = [10 9 ]x1 >= [10 9 ]x1 = f(f(f(x1)))
      
              [6 5]      [6 4]          
      d(x1) = [4 1]x1 >= [1 0]x1 = b(x1)
      
                 [6 5]      [6 5]             
      f(f(x1)) = [7 6]x1 >= [7 6]x1 = g(a(x1))
      
                 [6  11]      [6 4]          
      g(g(x1)) = [7  12]x1 >= [1 0]x1 = a(x1)
     problem:
      strict:
       
      weak:
       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)
     Qed