YES

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

Proof:
 DP Processor:
  DPs:
   a#(b(x1)) -> d#(x1)
   a#(b(x1)) -> b#(d(x1))
   a#(c(x1)) -> d#(x1)
   a#(c(x1)) -> d#(d(x1))
   a#(c(x1)) -> d#(d(d(x1)))
   b#(d(x1)) -> b#(x1)
   b#(d(x1)) -> c#(b(x1))
   b#(d(x1)) -> a#(c(b(x1)))
   c#(f(x1)) -> c#(x1)
   c#(f(x1)) -> d#(c(x1))
   c#(f(x1)) -> d#(d(c(x1)))
   d#(d(x1)) -> f#(x1)
   f#(f(x1)) -> a#(x1)
  TRS:
   a(b(x1)) -> b(d(x1))
   a(c(x1)) -> d(d(d(x1)))
   b(d(x1)) -> a(c(b(x1)))
   c(f(x1)) -> d(d(c(x1)))
   d(d(x1)) -> f(x1)
   f(f(x1)) -> a(x1)
  TDG Processor:
   DPs:
    a#(b(x1)) -> d#(x1)
    a#(b(x1)) -> b#(d(x1))
    a#(c(x1)) -> d#(x1)
    a#(c(x1)) -> d#(d(x1))
    a#(c(x1)) -> d#(d(d(x1)))
    b#(d(x1)) -> b#(x1)
    b#(d(x1)) -> c#(b(x1))
    b#(d(x1)) -> a#(c(b(x1)))
    c#(f(x1)) -> c#(x1)
    c#(f(x1)) -> d#(c(x1))
    c#(f(x1)) -> d#(d(c(x1)))
    d#(d(x1)) -> f#(x1)
    f#(f(x1)) -> a#(x1)
   TRS:
    a(b(x1)) -> b(d(x1))
    a(c(x1)) -> d(d(d(x1)))
    b(d(x1)) -> a(c(b(x1)))
    c(f(x1)) -> d(d(c(x1)))
    d(d(x1)) -> f(x1)
    f(f(x1)) -> a(x1)
   graph:
    f#(f(x1)) -> a#(x1) -> a#(c(x1)) -> d#(d(d(x1)))
    f#(f(x1)) -> a#(x1) -> a#(c(x1)) -> d#(d(x1))
    f#(f(x1)) -> a#(x1) -> a#(c(x1)) -> d#(x1)
    f#(f(x1)) -> a#(x1) -> a#(b(x1)) -> b#(d(x1))
    f#(f(x1)) -> a#(x1) -> a#(b(x1)) -> d#(x1)
    c#(f(x1)) -> c#(x1) -> c#(f(x1)) -> d#(d(c(x1)))
    c#(f(x1)) -> c#(x1) -> c#(f(x1)) -> d#(c(x1))
    c#(f(x1)) -> c#(x1) -> c#(f(x1)) -> c#(x1)
    c#(f(x1)) -> d#(c(x1)) -> d#(d(x1)) -> f#(x1)
    c#(f(x1)) -> d#(d(c(x1))) -> d#(d(x1)) -> f#(x1)
    b#(d(x1)) -> c#(b(x1)) -> c#(f(x1)) -> d#(d(c(x1)))
    b#(d(x1)) -> c#(b(x1)) -> c#(f(x1)) -> d#(c(x1))
    b#(d(x1)) -> c#(b(x1)) -> c#(f(x1)) -> c#(x1)
    b#(d(x1)) -> b#(x1) -> b#(d(x1)) -> a#(c(b(x1)))
    b#(d(x1)) -> b#(x1) -> b#(d(x1)) -> c#(b(x1))
    b#(d(x1)) -> b#(x1) -> b#(d(x1)) -> b#(x1)
    b#(d(x1)) -> a#(c(b(x1))) -> a#(c(x1)) -> d#(d(d(x1)))
    b#(d(x1)) -> a#(c(b(x1))) -> a#(c(x1)) -> d#(d(x1))
    b#(d(x1)) -> a#(c(b(x1))) -> a#(c(x1)) -> d#(x1)
    b#(d(x1)) -> a#(c(b(x1))) -> a#(b(x1)) -> b#(d(x1))
    b#(d(x1)) -> a#(c(b(x1))) -> a#(b(x1)) -> d#(x1)
    d#(d(x1)) -> f#(x1) -> f#(f(x1)) -> a#(x1)
    a#(c(x1)) -> d#(d(d(x1))) -> d#(d(x1)) -> f#(x1)
    a#(c(x1)) -> d#(d(x1)) -> d#(d(x1)) -> f#(x1)
    a#(c(x1)) -> d#(x1) -> d#(d(x1)) -> f#(x1)
    a#(b(x1)) -> b#(d(x1)) -> b#(d(x1)) -> a#(c(b(x1)))
    a#(b(x1)) -> b#(d(x1)) -> b#(d(x1)) -> c#(b(x1))
    a#(b(x1)) -> b#(d(x1)) -> b#(d(x1)) -> b#(x1)
    a#(b(x1)) -> d#(x1) -> d#(d(x1)) -> f#(x1)
   Arctic Interpretation Processor:
    dimension: 1
    interpretation:
     [f#](x0) = x0,
     
     [c#](x0) = x0 + 0,
     
     [b#](x0) = 1x0 + 4,
     
     [d#](x0) = x0,
     
     [a#](x0) = x0,
     
     [f](x0) = x0,
     
     [c](x0) = x0 + 0,
     
     [d](x0) = x0 + 0,
     
     [a](x0) = x0,
     
     [b](x0) = 1x0 + 4
    orientation:
     a#(b(x1)) = 1x1 + 4 >= x1 = d#(x1)
     
     a#(b(x1)) = 1x1 + 4 >= 1x1 + 4 = b#(d(x1))
     
     a#(c(x1)) = x1 + 0 >= x1 = d#(x1)
     
     a#(c(x1)) = x1 + 0 >= x1 + 0 = d#(d(x1))
     
     a#(c(x1)) = x1 + 0 >= x1 + 0 = d#(d(d(x1)))
     
     b#(d(x1)) = 1x1 + 4 >= 1x1 + 4 = b#(x1)
     
     b#(d(x1)) = 1x1 + 4 >= 1x1 + 4 = c#(b(x1))
     
     b#(d(x1)) = 1x1 + 4 >= 1x1 + 4 = a#(c(b(x1)))
     
     c#(f(x1)) = x1 + 0 >= x1 + 0 = c#(x1)
     
     c#(f(x1)) = x1 + 0 >= x1 + 0 = d#(c(x1))
     
     c#(f(x1)) = x1 + 0 >= x1 + 0 = d#(d(c(x1)))
     
     d#(d(x1)) = x1 + 0 >= x1 = f#(x1)
     
     f#(f(x1)) = x1 >= x1 = a#(x1)
     
     a(b(x1)) = 1x1 + 4 >= 1x1 + 4 = b(d(x1))
     
     a(c(x1)) = x1 + 0 >= x1 + 0 = d(d(d(x1)))
     
     b(d(x1)) = 1x1 + 4 >= 1x1 + 4 = a(c(b(x1)))
     
     c(f(x1)) = x1 + 0 >= x1 + 0 = d(d(c(x1)))
     
     d(d(x1)) = x1 + 0 >= x1 = f(x1)
     
     f(f(x1)) = x1 >= x1 = a(x1)
    problem:
     DPs:
      a#(b(x1)) -> b#(d(x1))
      a#(c(x1)) -> d#(x1)
      a#(c(x1)) -> d#(d(x1))
      a#(c(x1)) -> d#(d(d(x1)))
      b#(d(x1)) -> b#(x1)
      b#(d(x1)) -> c#(b(x1))
      b#(d(x1)) -> a#(c(b(x1)))
      c#(f(x1)) -> c#(x1)
      c#(f(x1)) -> d#(c(x1))
      c#(f(x1)) -> d#(d(c(x1)))
      d#(d(x1)) -> f#(x1)
      f#(f(x1)) -> a#(x1)
     TRS:
      a(b(x1)) -> b(d(x1))
      a(c(x1)) -> d(d(d(x1)))
      b(d(x1)) -> a(c(b(x1)))
      c(f(x1)) -> d(d(c(x1)))
      d(d(x1)) -> f(x1)
      f(f(x1)) -> a(x1)
    EDG Processor:
     DPs:
      a#(b(x1)) -> b#(d(x1))
      a#(c(x1)) -> d#(x1)
      a#(c(x1)) -> d#(d(x1))
      a#(c(x1)) -> d#(d(d(x1)))
      b#(d(x1)) -> b#(x1)
      b#(d(x1)) -> c#(b(x1))
      b#(d(x1)) -> a#(c(b(x1)))
      c#(f(x1)) -> c#(x1)
      c#(f(x1)) -> d#(c(x1))
      c#(f(x1)) -> d#(d(c(x1)))
      d#(d(x1)) -> f#(x1)
      f#(f(x1)) -> a#(x1)
     TRS:
      a(b(x1)) -> b(d(x1))
      a(c(x1)) -> d(d(d(x1)))
      b(d(x1)) -> a(c(b(x1)))
      c(f(x1)) -> d(d(c(x1)))
      d(d(x1)) -> f(x1)
      f(f(x1)) -> a(x1)
     graph:
      f#(f(x1)) -> a#(x1) -> a#(b(x1)) -> b#(d(x1))
      f#(f(x1)) -> a#(x1) -> a#(c(x1)) -> d#(x1)
      f#(f(x1)) -> a#(x1) -> a#(c(x1)) -> d#(d(x1))
      f#(f(x1)) -> a#(x1) -> a#(c(x1)) -> d#(d(d(x1)))
      c#(f(x1)) -> c#(x1) -> c#(f(x1)) -> c#(x1)
      c#(f(x1)) -> c#(x1) -> c#(f(x1)) -> d#(c(x1))
      c#(f(x1)) -> c#(x1) -> c#(f(x1)) -> d#(d(c(x1)))
      c#(f(x1)) -> d#(c(x1)) -> d#(d(x1)) -> f#(x1)
      c#(f(x1)) -> d#(d(c(x1))) -> d#(d(x1)) -> f#(x1)
      b#(d(x1)) -> c#(b(x1)) -> c#(f(x1)) -> c#(x1)
      b#(d(x1)) -> c#(b(x1)) -> c#(f(x1)) -> d#(c(x1))
      b#(d(x1)) -> c#(b(x1)) -> c#(f(x1)) -> d#(d(c(x1)))
      b#(d(x1)) -> b#(x1) -> b#(d(x1)) -> b#(x1)
      b#(d(x1)) -> b#(x1) -> b#(d(x1)) -> c#(b(x1))
      b#(d(x1)) -> b#(x1) -> b#(d(x1)) -> a#(c(b(x1)))
      b#(d(x1)) -> a#(c(b(x1))) -> a#(b(x1)) -> b#(d(x1))
      b#(d(x1)) -> a#(c(b(x1))) -> a#(c(x1)) -> d#(x1)
      b#(d(x1)) -> a#(c(b(x1))) -> a#(c(x1)) -> d#(d(x1))
      b#(d(x1)) -> a#(c(b(x1))) -> a#(c(x1)) -> d#(d(d(x1)))
      d#(d(x1)) -> f#(x1) -> f#(f(x1)) -> a#(x1)
      a#(c(x1)) -> d#(d(d(x1))) -> d#(d(x1)) -> f#(x1)
      a#(c(x1)) -> d#(d(x1)) -> d#(d(x1)) -> f#(x1)
      a#(c(x1)) -> d#(x1) -> d#(d(x1)) -> f#(x1)
      a#(b(x1)) -> b#(d(x1)) -> b#(d(x1)) -> b#(x1)
      a#(b(x1)) -> b#(d(x1)) -> b#(d(x1)) -> c#(b(x1))
      a#(b(x1)) -> b#(d(x1)) -> b#(d(x1)) -> a#(c(b(x1)))
     Matrix Interpretation Processor: dim=1
      
      interpretation:
       [f#](x0) = 2x0 + 17,
       
       [c#](x0) = 2x0 + 23,
       
       [b#](x0) = 8x0 + 8,
       
       [d#](x0) = 2x0 + 14,
       
       [a#](x0) = 2x0 + 24,
       
       [f](x0) = x0 + 4,
       
       [c](x0) = x0,
       
       [d](x0) = x0 + 2,
       
       [a](x0) = x0 + 8,
       
       [b](x0) = 4x0
      orientation:
       a#(b(x1)) = 8x1 + 24 >= 8x1 + 24 = b#(d(x1))
       
       a#(c(x1)) = 2x1 + 24 >= 2x1 + 14 = d#(x1)
       
       a#(c(x1)) = 2x1 + 24 >= 2x1 + 18 = d#(d(x1))
       
       a#(c(x1)) = 2x1 + 24 >= 2x1 + 22 = d#(d(d(x1)))
       
       b#(d(x1)) = 8x1 + 24 >= 8x1 + 8 = b#(x1)
       
       b#(d(x1)) = 8x1 + 24 >= 8x1 + 23 = c#(b(x1))
       
       b#(d(x1)) = 8x1 + 24 >= 8x1 + 24 = a#(c(b(x1)))
       
       c#(f(x1)) = 2x1 + 31 >= 2x1 + 23 = c#(x1)
       
       c#(f(x1)) = 2x1 + 31 >= 2x1 + 14 = d#(c(x1))
       
       c#(f(x1)) = 2x1 + 31 >= 2x1 + 18 = d#(d(c(x1)))
       
       d#(d(x1)) = 2x1 + 18 >= 2x1 + 17 = f#(x1)
       
       f#(f(x1)) = 2x1 + 25 >= 2x1 + 24 = a#(x1)
       
       a(b(x1)) = 4x1 + 8 >= 4x1 + 8 = b(d(x1))
       
       a(c(x1)) = x1 + 8 >= x1 + 6 = d(d(d(x1)))
       
       b(d(x1)) = 4x1 + 8 >= 4x1 + 8 = a(c(b(x1)))
       
       c(f(x1)) = x1 + 4 >= x1 + 4 = d(d(c(x1)))
       
       d(d(x1)) = x1 + 4 >= x1 + 4 = f(x1)
       
       f(f(x1)) = x1 + 8 >= x1 + 8 = a(x1)
      problem:
       DPs:
        a#(b(x1)) -> b#(d(x1))
        b#(d(x1)) -> a#(c(b(x1)))
       TRS:
        a(b(x1)) -> b(d(x1))
        a(c(x1)) -> d(d(d(x1)))
        b(d(x1)) -> a(c(b(x1)))
        c(f(x1)) -> d(d(c(x1)))
        d(d(x1)) -> f(x1)
        f(f(x1)) -> a(x1)
      EDG Processor:
       DPs:
        a#(b(x1)) -> b#(d(x1))
        b#(d(x1)) -> a#(c(b(x1)))
       TRS:
        a(b(x1)) -> b(d(x1))
        a(c(x1)) -> d(d(d(x1)))
        b(d(x1)) -> a(c(b(x1)))
        c(f(x1)) -> d(d(c(x1)))
        d(d(x1)) -> f(x1)
        f(f(x1)) -> a(x1)
       graph:
        b#(d(x1)) -> a#(c(b(x1))) -> a#(b(x1)) -> b#(d(x1))
        a#(b(x1)) -> b#(d(x1)) -> b#(d(x1)) -> a#(c(b(x1)))
       Matrix Interpretation Processor: dim=2
        
        interpretation:
         [b#](x0) = [0 3]x0,
         
         [a#](x0) = [0 1]x0 + [2],
         
                   [1 0]     [2]
         [f](x0) = [2 0]x0 + [3],
         
                   [1 0]  
         [c](x0) = [2 0]x0,
         
                   [1 0]     [1]
         [d](x0) = [2 0]x0 + [1],
         
                   [1 0]     [3]
         [a](x0) = [2 0]x0 + [7],
         
                   [3 0]     [0]
         [b](x0) = [6 0]x0 + [1]
        orientation:
         a#(b(x1)) = [6 0]x1 + [3] >= [6 0]x1 + [3] = b#(d(x1))
         
         b#(d(x1)) = [6 0]x1 + [3] >= [6 0]x1 + [2] = a#(c(b(x1)))
         
                    [3 0]     [3]    [3 0]     [3]           
         a(b(x1)) = [6 0]x1 + [7] >= [6 0]x1 + [7] = b(d(x1))
         
                    [1 0]     [3]    [1 0]     [3]              
         a(c(x1)) = [2 0]x1 + [7] >= [2 0]x1 + [5] = d(d(d(x1)))
         
                    [3 0]     [3]    [3 0]     [3]              
         b(d(x1)) = [6 0]x1 + [7] >= [6 0]x1 + [7] = a(c(b(x1)))
         
                    [1 0]     [2]    [1 0]     [2]              
         c(f(x1)) = [2 0]x1 + [4] >= [2 0]x1 + [3] = d(d(c(x1)))
         
                    [1 0]     [2]    [1 0]     [2]        
         d(d(x1)) = [2 0]x1 + [3] >= [2 0]x1 + [3] = f(x1)
         
                    [1 0]     [4]    [1 0]     [3]        
         f(f(x1)) = [2 0]x1 + [7] >= [2 0]x1 + [7] = a(x1)
        problem:
         DPs:
          a#(b(x1)) -> b#(d(x1))
         TRS:
          a(b(x1)) -> b(d(x1))
          a(c(x1)) -> d(d(d(x1)))
          b(d(x1)) -> a(c(b(x1)))
          c(f(x1)) -> d(d(c(x1)))
          d(d(x1)) -> f(x1)
          f(f(x1)) -> a(x1)
        EDG Processor:
         DPs:
          a#(b(x1)) -> b#(d(x1))
         TRS:
          a(b(x1)) -> b(d(x1))
          a(c(x1)) -> d(d(d(x1)))
          b(d(x1)) -> a(c(b(x1)))
          c(f(x1)) -> d(d(c(x1)))
          d(d(x1)) -> f(x1)
          f(f(x1)) -> a(x1)
         graph:
          
         Qed