YES

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

Proof:
 DP Processor:
  DPs:
   a#(d(x1)) -> b#(d(x1))
   a#(d(x1)) -> b#(c(b(d(x1))))
   a#(d(x1)) -> d#(b(c(b(d(x1)))))
   a#(x1) -> b#(x1)
   a#(x1) -> b#(b(x1))
   a#(x1) -> b#(f(b(b(x1))))
   a#(x1) -> b#(b(f(b(b(x1)))))
   b#(d(b(x1))) -> d#(x1)
   b#(d(b(x1))) -> a#(d(x1))
   d#(f(x1)) -> d#(x1)
   d#(f(x1)) -> b#(d(x1))
  TRS:
   a(d(x1)) -> d(b(c(b(d(x1)))))
   a(x1) -> b(b(f(b(b(x1)))))
   b(d(b(x1))) -> a(d(x1))
   d(f(x1)) -> b(d(x1))
  TDG Processor:
   DPs:
    a#(d(x1)) -> b#(d(x1))
    a#(d(x1)) -> b#(c(b(d(x1))))
    a#(d(x1)) -> d#(b(c(b(d(x1)))))
    a#(x1) -> b#(x1)
    a#(x1) -> b#(b(x1))
    a#(x1) -> b#(f(b(b(x1))))
    a#(x1) -> b#(b(f(b(b(x1)))))
    b#(d(b(x1))) -> d#(x1)
    b#(d(b(x1))) -> a#(d(x1))
    d#(f(x1)) -> d#(x1)
    d#(f(x1)) -> b#(d(x1))
   TRS:
    a(d(x1)) -> d(b(c(b(d(x1)))))
    a(x1) -> b(b(f(b(b(x1)))))
    b(d(b(x1))) -> a(d(x1))
    d(f(x1)) -> b(d(x1))
   graph:
    d#(f(x1)) -> d#(x1) -> d#(f(x1)) -> b#(d(x1))
    d#(f(x1)) -> d#(x1) -> d#(f(x1)) -> d#(x1)
    d#(f(x1)) -> b#(d(x1)) -> b#(d(b(x1))) -> a#(d(x1))
    d#(f(x1)) -> b#(d(x1)) -> b#(d(b(x1))) -> d#(x1)
    b#(d(b(x1))) -> d#(x1) -> d#(f(x1)) -> b#(d(x1))
    b#(d(b(x1))) -> d#(x1) -> d#(f(x1)) -> d#(x1)
    b#(d(b(x1))) -> a#(d(x1)) -> a#(x1) -> b#(b(f(b(b(x1)))))
    b#(d(b(x1))) -> a#(d(x1)) -> a#(x1) -> b#(f(b(b(x1))))
    b#(d(b(x1))) -> a#(d(x1)) -> a#(x1) -> b#(b(x1))
    b#(d(b(x1))) -> a#(d(x1)) -> a#(x1) -> b#(x1)
    b#(d(b(x1))) -> a#(d(x1)) -> a#(d(x1)) -> d#(b(c(b(d(x1)))))
    b#(d(b(x1))) -> a#(d(x1)) -> a#(d(x1)) -> b#(c(b(d(x1))))
    b#(d(b(x1))) -> a#(d(x1)) -> a#(d(x1)) -> b#(d(x1))
    a#(d(x1)) -> d#(b(c(b(d(x1))))) -> d#(f(x1)) -> b#(d(x1))
    a#(d(x1)) -> d#(b(c(b(d(x1))))) -> d#(f(x1)) -> d#(x1)
    a#(d(x1)) -> b#(c(b(d(x1)))) -> b#(d(b(x1))) -> a#(d(x1))
    a#(d(x1)) -> b#(c(b(d(x1)))) -> b#(d(b(x1))) -> d#(x1)
    a#(d(x1)) -> b#(d(x1)) -> b#(d(b(x1))) -> a#(d(x1))
    a#(d(x1)) -> b#(d(x1)) -> b#(d(b(x1))) -> d#(x1)
    a#(x1) -> b#(f(b(b(x1)))) -> b#(d(b(x1))) -> a#(d(x1))
    a#(x1) -> b#(f(b(b(x1)))) -> b#(d(b(x1))) -> d#(x1)
    a#(x1) -> b#(b(f(b(b(x1))))) -> b#(d(b(x1))) -> a#(d(x1))
    a#(x1) -> b#(b(f(b(b(x1))))) -> b#(d(b(x1))) -> d#(x1)
    a#(x1) -> b#(b(x1)) -> b#(d(b(x1))) -> a#(d(x1))
    a#(x1) -> b#(b(x1)) -> b#(d(b(x1))) -> d#(x1)
    a#(x1) -> b#(x1) -> b#(d(b(x1))) -> a#(d(x1))
    a#(x1) -> b#(x1) -> b#(d(b(x1))) -> d#(x1)
   EDG Processor:
    DPs:
     a#(d(x1)) -> b#(d(x1))
     a#(d(x1)) -> b#(c(b(d(x1))))
     a#(d(x1)) -> d#(b(c(b(d(x1)))))
     a#(x1) -> b#(x1)
     a#(x1) -> b#(b(x1))
     a#(x1) -> b#(f(b(b(x1))))
     a#(x1) -> b#(b(f(b(b(x1)))))
     b#(d(b(x1))) -> d#(x1)
     b#(d(b(x1))) -> a#(d(x1))
     d#(f(x1)) -> d#(x1)
     d#(f(x1)) -> b#(d(x1))
    TRS:
     a(d(x1)) -> d(b(c(b(d(x1)))))
     a(x1) -> b(b(f(b(b(x1)))))
     b(d(b(x1))) -> a(d(x1))
     d(f(x1)) -> b(d(x1))
    graph:
     d#(f(x1)) -> d#(x1) -> d#(f(x1)) -> d#(x1)
     d#(f(x1)) -> d#(x1) -> d#(f(x1)) -> b#(d(x1))
     d#(f(x1)) -> b#(d(x1)) -> b#(d(b(x1))) -> d#(x1)
     d#(f(x1)) -> b#(d(x1)) -> b#(d(b(x1))) -> a#(d(x1))
     b#(d(b(x1))) -> d#(x1) -> d#(f(x1)) -> d#(x1)
     b#(d(b(x1))) -> d#(x1) -> d#(f(x1)) -> b#(d(x1))
     b#(d(b(x1))) -> a#(d(x1)) -> a#(d(x1)) -> b#(d(x1))
     b#(d(b(x1))) -> a#(d(x1)) -> a#(d(x1)) -> b#(c(b(d(x1))))
     b#(d(b(x1))) -> a#(d(x1)) -> a#(d(x1)) -> d#(b(c(b(d(x1)))))
     b#(d(b(x1))) -> a#(d(x1)) -> a#(x1) -> b#(x1)
     b#(d(b(x1))) -> a#(d(x1)) -> a#(x1) -> b#(b(x1))
     b#(d(b(x1))) -> a#(d(x1)) -> a#(x1) -> b#(f(b(b(x1))))
     b#(d(b(x1))) -> a#(d(x1)) -> a#(x1) -> b#(b(f(b(b(x1)))))
     a#(d(x1)) -> b#(d(x1)) -> b#(d(b(x1))) -> d#(x1)
     a#(d(x1)) -> b#(d(x1)) -> b#(d(b(x1))) -> a#(d(x1))
     a#(x1) -> b#(b(x1)) -> b#(d(b(x1))) -> d#(x1)
     a#(x1) -> b#(b(x1)) -> b#(d(b(x1))) -> a#(d(x1))
     a#(x1) -> b#(x1) -> b#(d(b(x1))) -> d#(x1)
     a#(x1) -> b#(x1) -> b#(d(b(x1))) -> a#(d(x1))
    SCC Processor:
     #sccs: 1
     #rules: 7
     #arcs: 19/121
     DPs:
      d#(f(x1)) -> d#(x1)
      d#(f(x1)) -> b#(d(x1))
      b#(d(b(x1))) -> a#(d(x1))
      a#(x1) -> b#(b(x1))
      b#(d(b(x1))) -> d#(x1)
      a#(x1) -> b#(x1)
      a#(d(x1)) -> b#(d(x1))
     TRS:
      a(d(x1)) -> d(b(c(b(d(x1)))))
      a(x1) -> b(b(f(b(b(x1)))))
      b(d(b(x1))) -> a(d(x1))
      d(f(x1)) -> b(d(x1))
     Matrix Interpretation Processor: dim=1
      
      interpretation:
       [d#](x0) = 8x0 + 28,
       
       [b#](x0) = 2x0 + 1,
       
       [a#](x0) = 2x0 + 16,
       
       [f](x0) = x0 + 2,
       
       [c](x0) = 0,
       
       [b](x0) = x0 + 2,
       
       [a](x0) = x0 + 10,
       
       [d](x0) = 4x0 + 8
      orientation:
       d#(f(x1)) = 8x1 + 44 >= 8x1 + 28 = d#(x1)
       
       d#(f(x1)) = 8x1 + 44 >= 8x1 + 17 = b#(d(x1))
       
       b#(d(b(x1))) = 8x1 + 33 >= 8x1 + 32 = a#(d(x1))
       
       a#(x1) = 2x1 + 16 >= 2x1 + 5 = b#(b(x1))
       
       b#(d(b(x1))) = 8x1 + 33 >= 8x1 + 28 = d#(x1)
       
       a#(x1) = 2x1 + 16 >= 2x1 + 1 = b#(x1)
       
       a#(d(x1)) = 8x1 + 32 >= 8x1 + 17 = b#(d(x1))
       
       a(d(x1)) = 4x1 + 18 >= 16 = d(b(c(b(d(x1)))))
       
       a(x1) = x1 + 10 >= x1 + 10 = b(b(f(b(b(x1)))))
       
       b(d(b(x1))) = 4x1 + 18 >= 4x1 + 18 = a(d(x1))
       
       d(f(x1)) = 4x1 + 16 >= 4x1 + 10 = b(d(x1))
      problem:
       DPs:
        
       TRS:
        a(d(x1)) -> d(b(c(b(d(x1)))))
        a(x1) -> b(b(f(b(b(x1)))))
        b(d(b(x1))) -> a(d(x1))
        d(f(x1)) -> b(d(x1))
      Qed