YES

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

Proof:
 DP Processor:
  DPs:
   f#(f(x1)) -> b#(x1)
   f#(f(x1)) -> b#(b(x1))
   f#(f(x1)) -> b#(b(b(x1)))
   a#(f(x1)) -> a#(x1)
   a#(f(x1)) -> a#(a(x1))
   a#(f(x1)) -> f#(a(a(x1)))
   b#(b(x1)) -> c#(x1)
   b#(b(x1)) -> a#(c(x1))
   b#(b(x1)) -> c#(a(c(x1)))
   b#(b(x1)) -> c#(c(a(c(x1))))
   d#(b(x1)) -> a#(b(x1))
   d#(b(x1)) -> d#(a(b(x1)))
   c#(c(x1)) -> d#(x1)
   c#(c(x1)) -> d#(d(x1))
   c#(c(x1)) -> d#(d(d(x1)))
   b#(d(x1)) -> b#(x1)
   b#(d(x1)) -> d#(b(x1))
   c#(d(d(x1))) -> f#(x1)
  TRS:
   f(f(x1)) -> b(b(b(x1)))
   a(f(x1)) -> f(a(a(x1)))
   b(b(x1)) -> c(c(a(c(x1))))
   d(b(x1)) -> d(a(b(x1)))
   c(c(x1)) -> d(d(d(x1)))
   b(d(x1)) -> d(b(x1))
   c(d(d(x1))) -> f(x1)
  TDG Processor:
   DPs:
    f#(f(x1)) -> b#(x1)
    f#(f(x1)) -> b#(b(x1))
    f#(f(x1)) -> b#(b(b(x1)))
    a#(f(x1)) -> a#(x1)
    a#(f(x1)) -> a#(a(x1))
    a#(f(x1)) -> f#(a(a(x1)))
    b#(b(x1)) -> c#(x1)
    b#(b(x1)) -> a#(c(x1))
    b#(b(x1)) -> c#(a(c(x1)))
    b#(b(x1)) -> c#(c(a(c(x1))))
    d#(b(x1)) -> a#(b(x1))
    d#(b(x1)) -> d#(a(b(x1)))
    c#(c(x1)) -> d#(x1)
    c#(c(x1)) -> d#(d(x1))
    c#(c(x1)) -> d#(d(d(x1)))
    b#(d(x1)) -> b#(x1)
    b#(d(x1)) -> d#(b(x1))
    c#(d(d(x1))) -> f#(x1)
   TRS:
    f(f(x1)) -> b(b(b(x1)))
    a(f(x1)) -> f(a(a(x1)))
    b(b(x1)) -> c(c(a(c(x1))))
    d(b(x1)) -> d(a(b(x1)))
    c(c(x1)) -> d(d(d(x1)))
    b(d(x1)) -> d(b(x1))
    c(d(d(x1))) -> f(x1)
   graph:
    d#(b(x1)) -> d#(a(b(x1))) -> d#(b(x1)) -> d#(a(b(x1)))
    d#(b(x1)) -> d#(a(b(x1))) -> d#(b(x1)) -> a#(b(x1))
    d#(b(x1)) -> a#(b(x1)) -> a#(f(x1)) -> f#(a(a(x1)))
    d#(b(x1)) -> a#(b(x1)) -> a#(f(x1)) -> a#(a(x1))
    d#(b(x1)) -> a#(b(x1)) -> a#(f(x1)) -> a#(x1)
    c#(d(d(x1))) -> f#(x1) -> f#(f(x1)) -> b#(b(b(x1)))
    c#(d(d(x1))) -> f#(x1) -> f#(f(x1)) -> b#(b(x1))
    c#(d(d(x1))) -> f#(x1) -> f#(f(x1)) -> b#(x1)
    c#(c(x1)) -> d#(d(d(x1))) -> d#(b(x1)) -> d#(a(b(x1)))
    c#(c(x1)) -> d#(d(d(x1))) -> d#(b(x1)) -> a#(b(x1))
    c#(c(x1)) -> d#(d(x1)) -> d#(b(x1)) -> d#(a(b(x1)))
    c#(c(x1)) -> d#(d(x1)) -> d#(b(x1)) -> a#(b(x1))
    c#(c(x1)) -> d#(x1) -> d#(b(x1)) -> d#(a(b(x1)))
    c#(c(x1)) -> d#(x1) -> d#(b(x1)) -> a#(b(x1))
    a#(f(x1)) -> a#(a(x1)) -> a#(f(x1)) -> f#(a(a(x1)))
    a#(f(x1)) -> a#(a(x1)) -> a#(f(x1)) -> a#(a(x1))
    a#(f(x1)) -> a#(a(x1)) -> a#(f(x1)) -> a#(x1)
    a#(f(x1)) -> a#(x1) -> a#(f(x1)) -> f#(a(a(x1)))
    a#(f(x1)) -> a#(x1) -> a#(f(x1)) -> a#(a(x1))
    a#(f(x1)) -> a#(x1) -> a#(f(x1)) -> a#(x1)
    a#(f(x1)) -> f#(a(a(x1))) -> f#(f(x1)) -> b#(b(b(x1)))
    a#(f(x1)) -> f#(a(a(x1))) -> f#(f(x1)) -> b#(b(x1))
    a#(f(x1)) -> f#(a(a(x1))) -> f#(f(x1)) -> b#(x1)
    b#(d(x1)) -> d#(b(x1)) -> d#(b(x1)) -> d#(a(b(x1)))
    b#(d(x1)) -> d#(b(x1)) -> d#(b(x1)) -> a#(b(x1))
    b#(d(x1)) -> b#(x1) -> b#(d(x1)) -> d#(b(x1))
    b#(d(x1)) -> b#(x1) -> b#(d(x1)) -> b#(x1)
    b#(d(x1)) -> b#(x1) -> b#(b(x1)) -> c#(c(a(c(x1))))
    b#(d(x1)) -> b#(x1) -> b#(b(x1)) -> c#(a(c(x1)))
    b#(d(x1)) -> b#(x1) -> b#(b(x1)) -> a#(c(x1))
    b#(d(x1)) -> b#(x1) -> b#(b(x1)) -> c#(x1)
    b#(b(x1)) -> c#(c(a(c(x1)))) -> c#(d(d(x1))) -> f#(x1)
    b#(b(x1)) -> c#(c(a(c(x1)))) -> c#(c(x1)) -> d#(d(d(x1)))
    b#(b(x1)) -> c#(c(a(c(x1)))) -> c#(c(x1)) -> d#(d(x1))
    b#(b(x1)) -> c#(c(a(c(x1)))) -> c#(c(x1)) -> d#(x1)
    b#(b(x1)) -> c#(a(c(x1))) -> c#(d(d(x1))) -> f#(x1)
    b#(b(x1)) -> c#(a(c(x1))) -> c#(c(x1)) -> d#(d(d(x1)))
    b#(b(x1)) -> c#(a(c(x1))) -> c#(c(x1)) -> d#(d(x1))
    b#(b(x1)) -> c#(a(c(x1))) -> c#(c(x1)) -> d#(x1)
    b#(b(x1)) -> c#(x1) -> c#(d(d(x1))) -> f#(x1)
    b#(b(x1)) -> c#(x1) -> c#(c(x1)) -> d#(d(d(x1)))
    b#(b(x1)) -> c#(x1) -> c#(c(x1)) -> d#(d(x1))
    b#(b(x1)) -> c#(x1) -> c#(c(x1)) -> d#(x1)
    b#(b(x1)) -> a#(c(x1)) -> a#(f(x1)) -> f#(a(a(x1)))
    b#(b(x1)) -> a#(c(x1)) -> a#(f(x1)) -> a#(a(x1))
    b#(b(x1)) -> a#(c(x1)) -> a#(f(x1)) -> a#(x1)
    f#(f(x1)) -> b#(b(b(x1))) -> b#(d(x1)) -> d#(b(x1))
    f#(f(x1)) -> b#(b(b(x1))) -> b#(d(x1)) -> b#(x1)
    f#(f(x1)) -> b#(b(b(x1))) -> b#(b(x1)) -> c#(c(a(c(x1))))
    f#(f(x1)) -> b#(b(b(x1))) -> b#(b(x1)) -> c#(a(c(x1)))
    f#(f(x1)) -> b#(b(b(x1))) -> b#(b(x1)) -> a#(c(x1))
    f#(f(x1)) -> b#(b(b(x1))) -> b#(b(x1)) -> c#(x1)
    f#(f(x1)) -> b#(b(x1)) -> b#(d(x1)) -> d#(b(x1))
    f#(f(x1)) -> b#(b(x1)) -> b#(d(x1)) -> b#(x1)
    f#(f(x1)) -> b#(b(x1)) -> b#(b(x1)) -> c#(c(a(c(x1))))
    f#(f(x1)) -> b#(b(x1)) -> b#(b(x1)) -> c#(a(c(x1)))
    f#(f(x1)) -> b#(b(x1)) -> b#(b(x1)) -> a#(c(x1))
    f#(f(x1)) -> b#(b(x1)) -> b#(b(x1)) -> c#(x1)
    f#(f(x1)) -> b#(x1) -> b#(d(x1)) -> d#(b(x1))
    f#(f(x1)) -> b#(x1) -> b#(d(x1)) -> b#(x1)
    f#(f(x1)) -> b#(x1) -> b#(b(x1)) -> c#(c(a(c(x1))))
    f#(f(x1)) -> b#(x1) -> b#(b(x1)) -> c#(a(c(x1)))
    f#(f(x1)) -> b#(x1) -> b#(b(x1)) -> a#(c(x1))
    f#(f(x1)) -> b#(x1) -> b#(b(x1)) -> c#(x1)
   Arctic Interpretation Processor:
    dimension: 1
    interpretation:
     [d#](x0) = 6x0,
     
     [c#](x0) = 8x0 + 0,
     
     [a#](x0) = 1x0 + 0,
     
     [b#](x0) = 11x0,
     
     [f#](x0) = 15x0,
     
     [d](x0) = 4x0 + 5,
     
     [c](x0) = 6x0 + 7,
     
     [a](x0) = x0,
     
     [b](x0) = 9x0 + 10,
     
     [f](x0) = 14x0 + 15
    orientation:
     f#(f(x1)) = 29x1 + 30 >= 11x1 = b#(x1)
     
     f#(f(x1)) = 29x1 + 30 >= 20x1 + 21 = b#(b(x1))
     
     f#(f(x1)) = 29x1 + 30 >= 29x1 + 30 = b#(b(b(x1)))
     
     a#(f(x1)) = 15x1 + 16 >= 1x1 + 0 = a#(x1)
     
     a#(f(x1)) = 15x1 + 16 >= 1x1 + 0 = a#(a(x1))
     
     a#(f(x1)) = 15x1 + 16 >= 15x1 = f#(a(a(x1)))
     
     b#(b(x1)) = 20x1 + 21 >= 8x1 + 0 = c#(x1)
     
     b#(b(x1)) = 20x1 + 21 >= 7x1 + 8 = a#(c(x1))
     
     b#(b(x1)) = 20x1 + 21 >= 14x1 + 15 = c#(a(c(x1)))
     
     b#(b(x1)) = 20x1 + 21 >= 20x1 + 21 = c#(c(a(c(x1))))
     
     d#(b(x1)) = 15x1 + 16 >= 10x1 + 11 = a#(b(x1))
     
     d#(b(x1)) = 15x1 + 16 >= 15x1 + 16 = d#(a(b(x1)))
     
     c#(c(x1)) = 14x1 + 15 >= 6x1 = d#(x1)
     
     c#(c(x1)) = 14x1 + 15 >= 10x1 + 11 = d#(d(x1))
     
     c#(c(x1)) = 14x1 + 15 >= 14x1 + 15 = d#(d(d(x1)))
     
     b#(d(x1)) = 15x1 + 16 >= 11x1 = b#(x1)
     
     b#(d(x1)) = 15x1 + 16 >= 15x1 + 16 = d#(b(x1))
     
     c#(d(d(x1))) = 16x1 + 17 >= 15x1 = f#(x1)
     
     f(f(x1)) = 28x1 + 29 >= 27x1 + 28 = b(b(b(x1)))
     
     a(f(x1)) = 14x1 + 15 >= 14x1 + 15 = f(a(a(x1)))
     
     b(b(x1)) = 18x1 + 19 >= 18x1 + 19 = c(c(a(c(x1))))
     
     d(b(x1)) = 13x1 + 14 >= 13x1 + 14 = d(a(b(x1)))
     
     c(c(x1)) = 12x1 + 13 >= 12x1 + 13 = d(d(d(x1)))
     
     b(d(x1)) = 13x1 + 14 >= 13x1 + 14 = d(b(x1))
     
     c(d(d(x1))) = 14x1 + 15 >= 14x1 + 15 = f(x1)
    problem:
     DPs:
      f#(f(x1)) -> b#(b(b(x1)))
      a#(f(x1)) -> f#(a(a(x1)))
      b#(b(x1)) -> c#(c(a(c(x1))))
      d#(b(x1)) -> d#(a(b(x1)))
      c#(c(x1)) -> d#(d(d(x1)))
      b#(d(x1)) -> d#(b(x1))
     TRS:
      f(f(x1)) -> b(b(b(x1)))
      a(f(x1)) -> f(a(a(x1)))
      b(b(x1)) -> c(c(a(c(x1))))
      d(b(x1)) -> d(a(b(x1)))
      c(c(x1)) -> d(d(d(x1)))
      b(d(x1)) -> d(b(x1))
      c(d(d(x1))) -> f(x1)
    EDG Processor:
     DPs:
      f#(f(x1)) -> b#(b(b(x1)))
      a#(f(x1)) -> f#(a(a(x1)))
      b#(b(x1)) -> c#(c(a(c(x1))))
      d#(b(x1)) -> d#(a(b(x1)))
      c#(c(x1)) -> d#(d(d(x1)))
      b#(d(x1)) -> d#(b(x1))
     TRS:
      f(f(x1)) -> b(b(b(x1)))
      a(f(x1)) -> f(a(a(x1)))
      b(b(x1)) -> c(c(a(c(x1))))
      d(b(x1)) -> d(a(b(x1)))
      c(c(x1)) -> d(d(d(x1)))
      b(d(x1)) -> d(b(x1))
      c(d(d(x1))) -> f(x1)
     graph:
      d#(b(x1)) -> d#(a(b(x1))) -> d#(b(x1)) -> d#(a(b(x1)))
      c#(c(x1)) -> d#(d(d(x1))) -> d#(b(x1)) -> d#(a(b(x1)))
      a#(f(x1)) -> f#(a(a(x1))) -> f#(f(x1)) -> b#(b(b(x1)))
      b#(d(x1)) -> d#(b(x1)) -> d#(b(x1)) -> d#(a(b(x1)))
      b#(b(x1)) -> c#(c(a(c(x1)))) -> c#(c(x1)) -> d#(d(d(x1)))
      f#(f(x1)) -> b#(b(b(x1))) -> b#(b(x1)) -> c#(c(a(c(x1))))
      f#(f(x1)) -> b#(b(b(x1))) -> b#(d(x1)) -> d#(b(x1))
     SCC Processor:
      #sccs: 1
      #rules: 1
      #arcs: 7/36
      DPs:
       d#(b(x1)) -> d#(a(b(x1)))
      TRS:
       f(f(x1)) -> b(b(b(x1)))
       a(f(x1)) -> f(a(a(x1)))
       b(b(x1)) -> c(c(a(c(x1))))
       d(b(x1)) -> d(a(b(x1)))
       c(c(x1)) -> d(d(d(x1)))
       b(d(x1)) -> d(b(x1))
       c(d(d(x1))) -> f(x1)
      Arctic Interpretation Processor:
       dimension: 2
       interpretation:
        [d#](x0) = [0 0]x0 + [0],
        
                  [0  -&]     [-&]
        [d](x0) = [2  1 ]x0 + [2 ],
        
                  [-& 0 ]     [0]
        [c](x0) = [2  2 ]x0 + [2],
        
                  [0  -&]     [0]
        [a](x0) = [0  -&]x0 + [0],
        
                  [-& 0 ]     [0]
        [b](x0) = [0  2 ]x0 + [2],
        
                  [3 1]     [3]
        [f](x0) = [2 3]x0 + [3]
       orientation:
        d#(b(x1)) = [0 2]x1 + [2] >= [-& 0 ]x1 + [0] = d#(a(b(x1)))
        
                   [6 4]     [6]    [2 4]     [4]              
        f(f(x1)) = [5 6]x1 + [6] >= [4 6]x1 + [6] = b(b(b(x1)))
        
                   [3 1]     [3]    [3  -&]     [3]              
        a(f(x1)) = [3 1]x1 + [3] >= [3  -&]x1 + [3] = f(a(a(x1)))
        
                   [0 2]     [2]    [-& 2 ]     [2]                 
        b(b(x1)) = [2 4]x1 + [4] >= [-& 4 ]x1 + [4] = c(c(a(c(x1))))
        
                   [-& 0 ]     [0]    [-& 0 ]     [0]              
        d(b(x1)) = [1  3 ]x1 + [3] >= [-& 2 ]x1 + [2] = d(a(b(x1)))
        
                   [2 2]     [2]    [0  -&]     [-&]              
        c(c(x1)) = [4 4]x1 + [4] >= [4  3 ]x1 + [4 ] = d(d(d(x1)))
        
                   [2 1]     [2]    [-& 0 ]     [0]           
        b(d(x1)) = [4 3]x1 + [4] >= [1  3 ]x1 + [3] = d(b(x1))
        
                      [3 2]     [3]    [3 1]     [3]        
        c(d(d(x1))) = [5 4]x1 + [5] >= [2 3]x1 + [3] = f(x1)
       problem:
        DPs:
         
        TRS:
         f(f(x1)) -> b(b(b(x1)))
         a(f(x1)) -> f(a(a(x1)))
         b(b(x1)) -> c(c(a(c(x1))))
         d(b(x1)) -> d(a(b(x1)))
         c(c(x1)) -> d(d(d(x1)))
         b(d(x1)) -> d(b(x1))
         c(d(d(x1))) -> f(x1)
       Qed