YES

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

Proof:
 DP Processor:
  DPs:
   a#(d(x1)) -> b#(x1)
   a#(x1) -> b#(x1)
   a#(x1) -> b#(b(x1))
   a#(x1) -> b#(b(b(x1)))
   b#(d(b(x1))) -> c#(x1)
   b#(d(b(x1))) -> a#(c(x1))
  TRS:
   a(d(x1)) -> d(b(x1))
   a(x1) -> b(b(b(x1)))
   b(d(b(x1))) -> a(c(x1))
   c(x1) -> d(x1)
  TDG Processor:
   DPs:
    a#(d(x1)) -> b#(x1)
    a#(x1) -> b#(x1)
    a#(x1) -> b#(b(x1))
    a#(x1) -> b#(b(b(x1)))
    b#(d(b(x1))) -> c#(x1)
    b#(d(b(x1))) -> a#(c(x1))
   TRS:
    a(d(x1)) -> d(b(x1))
    a(x1) -> b(b(b(x1)))
    b(d(b(x1))) -> a(c(x1))
    c(x1) -> d(x1)
   graph:
    b#(d(b(x1))) -> a#(c(x1)) -> a#(x1) -> b#(b(b(x1)))
    b#(d(b(x1))) -> a#(c(x1)) -> a#(x1) -> b#(b(x1))
    b#(d(b(x1))) -> a#(c(x1)) -> a#(x1) -> b#(x1)
    b#(d(b(x1))) -> a#(c(x1)) -> a#(d(x1)) -> b#(x1)
    a#(d(x1)) -> b#(x1) -> b#(d(b(x1))) -> a#(c(x1))
    a#(d(x1)) -> b#(x1) -> b#(d(b(x1))) -> c#(x1)
    a#(x1) -> b#(b(b(x1))) -> b#(d(b(x1))) -> a#(c(x1))
    a#(x1) -> b#(b(b(x1))) -> b#(d(b(x1))) -> c#(x1)
    a#(x1) -> b#(b(x1)) -> b#(d(b(x1))) -> a#(c(x1))
    a#(x1) -> b#(b(x1)) -> b#(d(b(x1))) -> c#(x1)
    a#(x1) -> b#(x1) -> b#(d(b(x1))) -> a#(c(x1))
    a#(x1) -> b#(x1) -> b#(d(b(x1))) -> c#(x1)
   SCC Processor:
    #sccs: 1
    #rules: 5
    #arcs: 12/36
    DPs:
     b#(d(b(x1))) -> a#(c(x1))
     a#(d(x1)) -> b#(x1)
     a#(x1) -> b#(x1)
     a#(x1) -> b#(b(x1))
     a#(x1) -> b#(b(b(x1)))
    TRS:
     a(d(x1)) -> d(b(x1))
     a(x1) -> b(b(b(x1)))
     b(d(b(x1))) -> a(c(x1))
     c(x1) -> d(x1)
    Matrix Interpretation Processor: dim=1
     
     interpretation:
      [b#](x0) = x0,
      
      [a#](x0) = x0 + 3,
      
      [c](x0) = 6x0 + 1,
      
      [b](x0) = x0 + 1,
      
      [a](x0) = x0 + 6,
      
      [d](x0) = 6x0 + 1
     orientation:
      b#(d(b(x1))) = 6x1 + 7 >= 6x1 + 4 = a#(c(x1))
      
      a#(d(x1)) = 6x1 + 4 >= x1 = b#(x1)
      
      a#(x1) = x1 + 3 >= x1 = b#(x1)
      
      a#(x1) = x1 + 3 >= x1 + 1 = b#(b(x1))
      
      a#(x1) = x1 + 3 >= x1 + 2 = b#(b(b(x1)))
      
      a(d(x1)) = 6x1 + 7 >= 6x1 + 7 = d(b(x1))
      
      a(x1) = x1 + 6 >= x1 + 3 = b(b(b(x1)))
      
      b(d(b(x1))) = 6x1 + 8 >= 6x1 + 7 = a(c(x1))
      
      c(x1) = 6x1 + 1 >= 6x1 + 1 = d(x1)
     problem:
      DPs:
       
      TRS:
       a(d(x1)) -> d(b(x1))
       a(x1) -> b(b(b(x1)))
       b(d(b(x1))) -> a(c(x1))
       c(x1) -> d(x1)
     Qed