YES

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

Proof:
 DP Processor:
  DPs:
   a#(b(x1)) -> d#(x1)
   b#(a(x1)) -> b#(x1)
   b#(a(x1)) -> a#(b(x1))
   d#(c(x1)) -> b#(c(x1))
   d#(c(x1)) -> b#(b(c(x1)))
   d#(c(x1)) -> a#(b(b(c(x1))))
   d#(f(x1)) -> b#(x1)
   d#(f(x1)) -> a#(b(x1))
   a#(f(x1)) -> a#(x1)
  TRS:
   a(b(x1)) -> d(x1)
   b(a(x1)) -> a(b(x1))
   d(c(x1)) -> f(a(b(b(c(x1)))))
   d(f(x1)) -> f(a(b(x1)))
   a(f(x1)) -> a(x1)
  TDG Processor:
   DPs:
    a#(b(x1)) -> d#(x1)
    b#(a(x1)) -> b#(x1)
    b#(a(x1)) -> a#(b(x1))
    d#(c(x1)) -> b#(c(x1))
    d#(c(x1)) -> b#(b(c(x1)))
    d#(c(x1)) -> a#(b(b(c(x1))))
    d#(f(x1)) -> b#(x1)
    d#(f(x1)) -> a#(b(x1))
    a#(f(x1)) -> a#(x1)
   TRS:
    a(b(x1)) -> d(x1)
    b(a(x1)) -> a(b(x1))
    d(c(x1)) -> f(a(b(b(c(x1)))))
    d(f(x1)) -> f(a(b(x1)))
    a(f(x1)) -> a(x1)
   graph:
    b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(b(x1))
    b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(x1)
    b#(a(x1)) -> a#(b(x1)) -> a#(f(x1)) -> a#(x1)
    b#(a(x1)) -> a#(b(x1)) -> a#(b(x1)) -> d#(x1)
    d#(f(x1)) -> b#(x1) -> b#(a(x1)) -> a#(b(x1))
    d#(f(x1)) -> b#(x1) -> b#(a(x1)) -> b#(x1)
    d#(f(x1)) -> a#(b(x1)) -> a#(f(x1)) -> a#(x1)
    d#(f(x1)) -> a#(b(x1)) -> a#(b(x1)) -> d#(x1)
    d#(c(x1)) -> b#(c(x1)) -> b#(a(x1)) -> a#(b(x1))
    d#(c(x1)) -> b#(c(x1)) -> b#(a(x1)) -> b#(x1)
    d#(c(x1)) -> b#(b(c(x1))) -> b#(a(x1)) -> a#(b(x1))
    d#(c(x1)) -> b#(b(c(x1))) -> b#(a(x1)) -> b#(x1)
    d#(c(x1)) -> a#(b(b(c(x1)))) -> a#(f(x1)) -> a#(x1)
    d#(c(x1)) -> a#(b(b(c(x1)))) -> a#(b(x1)) -> d#(x1)
    a#(f(x1)) -> a#(x1) -> a#(f(x1)) -> a#(x1)
    a#(f(x1)) -> a#(x1) -> a#(b(x1)) -> d#(x1)
    a#(b(x1)) -> d#(x1) -> d#(f(x1)) -> a#(b(x1))
    a#(b(x1)) -> d#(x1) -> d#(f(x1)) -> b#(x1)
    a#(b(x1)) -> d#(x1) -> d#(c(x1)) -> a#(b(b(c(x1))))
    a#(b(x1)) -> d#(x1) -> d#(c(x1)) -> b#(b(c(x1)))
    a#(b(x1)) -> d#(x1) -> d#(c(x1)) -> b#(c(x1))
   EDG Processor:
    DPs:
     a#(b(x1)) -> d#(x1)
     b#(a(x1)) -> b#(x1)
     b#(a(x1)) -> a#(b(x1))
     d#(c(x1)) -> b#(c(x1))
     d#(c(x1)) -> b#(b(c(x1)))
     d#(c(x1)) -> a#(b(b(c(x1))))
     d#(f(x1)) -> b#(x1)
     d#(f(x1)) -> a#(b(x1))
     a#(f(x1)) -> a#(x1)
    TRS:
     a(b(x1)) -> d(x1)
     b(a(x1)) -> a(b(x1))
     d(c(x1)) -> f(a(b(b(c(x1)))))
     d(f(x1)) -> f(a(b(x1)))
     a(f(x1)) -> a(x1)
    graph:
     b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(x1)
     b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(b(x1))
     b#(a(x1)) -> a#(b(x1)) -> a#(b(x1)) -> d#(x1)
     b#(a(x1)) -> a#(b(x1)) -> a#(f(x1)) -> a#(x1)
     d#(f(x1)) -> b#(x1) -> b#(a(x1)) -> b#(x1)
     d#(f(x1)) -> b#(x1) -> b#(a(x1)) -> a#(b(x1))
     d#(f(x1)) -> a#(b(x1)) -> a#(b(x1)) -> d#(x1)
     d#(f(x1)) -> a#(b(x1)) -> a#(f(x1)) -> a#(x1)
     d#(c(x1)) -> a#(b(b(c(x1)))) -> a#(b(x1)) -> d#(x1)
     a#(f(x1)) -> a#(x1) -> a#(b(x1)) -> d#(x1)
     a#(f(x1)) -> a#(x1) -> a#(f(x1)) -> a#(x1)
     a#(b(x1)) -> d#(x1) -> d#(c(x1)) -> b#(c(x1))
     a#(b(x1)) -> d#(x1) -> d#(c(x1)) -> b#(b(c(x1)))
     a#(b(x1)) -> d#(x1) -> d#(c(x1)) -> a#(b(b(c(x1))))
     a#(b(x1)) -> d#(x1) -> d#(f(x1)) -> b#(x1)
     a#(b(x1)) -> d#(x1) -> d#(f(x1)) -> a#(b(x1))
    CDG Processor:
     DPs:
      a#(b(x1)) -> d#(x1)
      b#(a(x1)) -> b#(x1)
      b#(a(x1)) -> a#(b(x1))
      d#(c(x1)) -> b#(c(x1))
      d#(c(x1)) -> b#(b(c(x1)))
      d#(c(x1)) -> a#(b(b(c(x1))))
      d#(f(x1)) -> b#(x1)
      d#(f(x1)) -> a#(b(x1))
      a#(f(x1)) -> a#(x1)
     TRS:
      a(b(x1)) -> d(x1)
      b(a(x1)) -> a(b(x1))
      d(c(x1)) -> f(a(b(b(c(x1)))))
      d(f(x1)) -> f(a(b(x1)))
      a(f(x1)) -> a(x1)
     graph:
      b#(a(x1)) -> a#(b(x1)) -> a#(f(x1)) -> a#(x1)
      b#(a(x1)) -> a#(b(x1)) -> a#(b(x1)) -> d#(x1)
      d#(f(x1)) -> a#(b(x1)) -> a#(f(x1)) -> a#(x1)
      d#(f(x1)) -> a#(b(x1)) -> a#(b(x1)) -> d#(x1)
      d#(c(x1)) -> a#(b(b(c(x1)))) -> a#(b(x1)) -> d#(x1)
      a#(f(x1)) -> a#(x1) -> a#(f(x1)) -> a#(x1)
      a#(f(x1)) -> a#(x1) -> a#(b(x1)) -> d#(x1)
     SCC Processor:
      #sccs: 1
      #rules: 1
      #arcs: 7/81
      DPs:
       a#(f(x1)) -> a#(x1)
      TRS:
       a(b(x1)) -> d(x1)
       b(a(x1)) -> a(b(x1))
       d(c(x1)) -> f(a(b(b(c(x1)))))
       d(f(x1)) -> f(a(b(x1)))
       a(f(x1)) -> a(x1)
      Matrix Interpretation Processor: dim=3
       
       interpretation:
        [a#](x0) = [0 0 1]x0 + [1],
        
                  [0 0 0]     [0]
        [f](x0) = [0 0 0]x0 + [0]
                  [1 1 1]     [1],
        
                  [0 0 0]     [0]
        [c](x0) = [0 0 0]x0 + [1]
                  [0 0 1]     [0],
        
                  [0 0 0]  
        [d](x0) = [0 0 0]x0
                  [0 1 1]  ,
        
                  [0 0 0]  
        [a](x0) = [0 0 0]x0
                  [1 1 1]  ,
        
                  [0 1 0]  
        [b](x0) = [0 0 0]x0
                  [0 0 1]  
       orientation:
        a#(f(x1)) = [1 1 1]x1 + [2] >= [0 0 1]x1 + [1] = a#(x1)
        
                   [0 0 0]      [0 0 0]          
        a(b(x1)) = [0 0 0]x1 >= [0 0 0]x1 = d(x1)
                   [0 1 1]      [0 1 1]          
        
                   [0 0 0]      [0 0 0]             
        b(a(x1)) = [0 0 0]x1 >= [0 0 0]x1 = a(b(x1))
                   [1 1 1]      [0 1 1]             
        
                   [0 0 0]     [0]    [0 0 0]     [0]                    
        d(c(x1)) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = f(a(b(b(c(x1)))))
                   [0 0 1]     [1]    [0 0 1]     [1]                    
        
                   [0 0 0]     [0]    [0 0 0]     [0]              
        d(f(x1)) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = f(a(b(x1)))
                   [1 1 1]     [1]    [0 1 1]     [1]              
        
                   [0 0 0]     [0]    [0 0 0]          
        a(f(x1)) = [0 0 0]x1 + [0] >= [0 0 0]x1 = a(x1)
                   [1 1 1]     [1]    [1 1 1]          
       problem:
        DPs:
         
        TRS:
         a(b(x1)) -> d(x1)
         b(a(x1)) -> a(b(x1))
         d(c(x1)) -> f(a(b(b(c(x1)))))
         d(f(x1)) -> f(a(b(x1)))
         a(f(x1)) -> a(x1)
       Qed