YES(?,O(n^1))

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

Proof:
 RT Transformation Processor:
  strict:
   a(a(x1)) -> b(c(x1))
   b(b(x1)) -> c(d(x1))
   c(c(x1)) -> d(d(d(x1)))
   d(c(x1)) -> b(f(x1))
   d(d(d(x1))) -> a(c(x1))
   f(f(x1)) -> f(b(x1))
  weak:
   
  Matrix Interpretation Processor:
   dimension: 1
   interpretation:
    [f](x0) = x0 + 19,
    
    [d](x0) = x0,
    
    [b](x0) = x0 + 1,
    
    [c](x0) = x0 + 6,
    
    [a](x0) = x0 + 8
   orientation:
    a(a(x1)) = x1 + 16 >= x1 + 7 = b(c(x1))
    
    b(b(x1)) = x1 + 2 >= x1 + 6 = c(d(x1))
    
    c(c(x1)) = x1 + 12 >= x1 = d(d(d(x1)))
    
    d(c(x1)) = x1 + 6 >= x1 + 20 = b(f(x1))
    
    d(d(d(x1))) = x1 >= x1 + 14 = a(c(x1))
    
    f(f(x1)) = x1 + 38 >= x1 + 20 = f(b(x1))
   problem:
    strict:
     b(b(x1)) -> c(d(x1))
     d(c(x1)) -> b(f(x1))
     d(d(d(x1))) -> a(c(x1))
    weak:
     a(a(x1)) -> b(c(x1))
     c(c(x1)) -> d(d(d(x1)))
     f(f(x1)) -> f(b(x1))
   Matrix Interpretation Processor:
    dimension: 1
    interpretation:
     [f](x0) = x0 + 17,
     
     [d](x0) = x0,
     
     [b](x0) = x0 + 7,
     
     [c](x0) = x0,
     
     [a](x0) = x0 + 9
    orientation:
     b(b(x1)) = x1 + 14 >= x1 = c(d(x1))
     
     d(c(x1)) = x1 >= x1 + 24 = b(f(x1))
     
     d(d(d(x1))) = x1 >= x1 + 9 = a(c(x1))
     
     a(a(x1)) = x1 + 18 >= x1 + 7 = b(c(x1))
     
     c(c(x1)) = x1 >= x1 = d(d(d(x1)))
     
     f(f(x1)) = x1 + 34 >= x1 + 24 = f(b(x1))
    problem:
     strict:
      d(c(x1)) -> b(f(x1))
      d(d(d(x1))) -> a(c(x1))
     weak:
      b(b(x1)) -> c(d(x1))
      a(a(x1)) -> b(c(x1))
      c(c(x1)) -> d(d(d(x1)))
      f(f(x1)) -> f(b(x1))
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [f](x0) = x0 + 20,
      
      [d](x0) = x0 + 16,
      
      [b](x0) = x0 + 20,
      
      [c](x0) = x0 + 24,
      
      [a](x0) = x0 + 22
     orientation:
      d(c(x1)) = x1 + 40 >= x1 + 40 = b(f(x1))
      
      d(d(d(x1))) = x1 + 48 >= x1 + 46 = a(c(x1))
      
      b(b(x1)) = x1 + 40 >= x1 + 40 = c(d(x1))
      
      a(a(x1)) = x1 + 44 >= x1 + 44 = b(c(x1))
      
      c(c(x1)) = x1 + 48 >= x1 + 48 = d(d(d(x1)))
      
      f(f(x1)) = x1 + 40 >= x1 + 40 = f(b(x1))
     problem:
      strict:
       d(c(x1)) -> b(f(x1))
      weak:
       d(d(d(x1))) -> a(c(x1))
       b(b(x1)) -> c(d(x1))
       a(a(x1)) -> b(c(x1))
       c(c(x1)) -> d(d(d(x1)))
       f(f(x1)) -> f(b(x1))
     Arctic Interpretation Processor:
      dimension: 2
      interpretation:
                 [0 0]  
       [f](x0) = [6 6]x0,
       
                 [4  0 ]  
       [d](x0) = [-& -&]x0,
       
                 [6  0 ]  
       [b](x0) = [-& -&]x0,
       
                 [6  1 ]  
       [c](x0) = [-& 7 ]x0,
       
                 [6  1 ]  
       [a](x0) = [-& -&]x0
      orientation:
                  [10 7 ]      [6  6 ]             
       d(c(x1)) = [-& -&]x1 >= [-& -&]x1 = b(f(x1))
       
                     [12 8 ]      [12 8 ]             
       d(d(d(x1))) = [-& -&]x1 >= [-& -&]x1 = a(c(x1))
       
                  [12 6 ]      [10 6 ]             
       b(b(x1)) = [-& -&]x1 >= [-& -&]x1 = c(d(x1))
       
                  [12 7 ]      [12 7 ]             
       a(a(x1)) = [-& -&]x1 >= [-& -&]x1 = b(c(x1))
       
                  [12 8 ]      [12 8 ]                
       c(c(x1)) = [-& 14]x1 >= [-& -&]x1 = d(d(d(x1)))
       
                  [6  6 ]      [6  0 ]             
       f(f(x1)) = [12 12]x1 >= [12 6 ]x1 = f(b(x1))
      problem:
       strict:
        
       weak:
        d(c(x1)) -> b(f(x1))
        d(d(d(x1))) -> a(c(x1))
        b(b(x1)) -> c(d(x1))
        a(a(x1)) -> b(c(x1))
        c(c(x1)) -> d(d(d(x1)))
        f(f(x1)) -> f(b(x1))
      Qed