YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(x1)) -> b#(x1)
   a#(a(x1)) -> b#(b(x1))
   a#(a(x1)) -> b#(b(b(x1)))
   b#(x1) -> c#(d(x1))
   b#(x1) -> c#(c(d(x1)))
   b#(c(x1)) -> b#(x1)
   b#(c(x1)) -> c#(b(x1))
   b#(c(d(x1))) -> a#(x1)
  TRS:
   a(a(x1)) -> b(b(b(x1)))
   b(x1) -> c(c(d(x1)))
   c(x1) -> d(d(d(x1)))
   b(c(x1)) -> c(b(x1))
   b(c(d(x1))) -> a(x1)
  TDG Processor:
   DPs:
    a#(a(x1)) -> b#(x1)
    a#(a(x1)) -> b#(b(x1))
    a#(a(x1)) -> b#(b(b(x1)))
    b#(x1) -> c#(d(x1))
    b#(x1) -> c#(c(d(x1)))
    b#(c(x1)) -> b#(x1)
    b#(c(x1)) -> c#(b(x1))
    b#(c(d(x1))) -> a#(x1)
   TRS:
    a(a(x1)) -> b(b(b(x1)))
    b(x1) -> c(c(d(x1)))
    c(x1) -> d(d(d(x1)))
    b(c(x1)) -> c(b(x1))
    b(c(d(x1))) -> a(x1)
   graph:
    b#(c(d(x1))) -> a#(x1) -> a#(a(x1)) -> b#(b(b(x1)))
    b#(c(d(x1))) -> a#(x1) -> a#(a(x1)) -> b#(b(x1))
    b#(c(d(x1))) -> a#(x1) -> a#(a(x1)) -> b#(x1)
    b#(c(x1)) -> b#(x1) -> b#(c(d(x1))) -> a#(x1)
    b#(c(x1)) -> b#(x1) -> b#(c(x1)) -> c#(b(x1))
    b#(c(x1)) -> b#(x1) -> b#(c(x1)) -> b#(x1)
    b#(c(x1)) -> b#(x1) -> b#(x1) -> c#(c(d(x1)))
    b#(c(x1)) -> b#(x1) -> b#(x1) -> c#(d(x1))
    a#(a(x1)) -> b#(b(b(x1))) -> b#(c(d(x1))) -> a#(x1)
    a#(a(x1)) -> b#(b(b(x1))) -> b#(c(x1)) -> c#(b(x1))
    a#(a(x1)) -> b#(b(b(x1))) -> b#(c(x1)) -> b#(x1)
    a#(a(x1)) -> b#(b(b(x1))) -> b#(x1) -> c#(c(d(x1)))
    a#(a(x1)) -> b#(b(b(x1))) -> b#(x1) -> c#(d(x1))
    a#(a(x1)) -> b#(b(x1)) -> b#(c(d(x1))) -> a#(x1)
    a#(a(x1)) -> b#(b(x1)) -> b#(c(x1)) -> c#(b(x1))
    a#(a(x1)) -> b#(b(x1)) -> b#(c(x1)) -> b#(x1)
    a#(a(x1)) -> b#(b(x1)) -> b#(x1) -> c#(c(d(x1)))
    a#(a(x1)) -> b#(b(x1)) -> b#(x1) -> c#(d(x1))
    a#(a(x1)) -> b#(x1) -> b#(c(d(x1))) -> a#(x1)
    a#(a(x1)) -> b#(x1) -> b#(c(x1)) -> c#(b(x1))
    a#(a(x1)) -> b#(x1) -> b#(c(x1)) -> b#(x1)
    a#(a(x1)) -> b#(x1) -> b#(x1) -> c#(c(d(x1)))
    a#(a(x1)) -> b#(x1) -> b#(x1) -> c#(d(x1))
   SCC Processor:
    #sccs: 1
    #rules: 5
    #arcs: 23/64
    DPs:
     b#(c(d(x1))) -> a#(x1)
     a#(a(x1)) -> b#(x1)
     b#(c(x1)) -> b#(x1)
     a#(a(x1)) -> b#(b(x1))
     a#(a(x1)) -> b#(b(b(x1)))
    TRS:
     a(a(x1)) -> b(b(b(x1)))
     b(x1) -> c(c(d(x1)))
     c(x1) -> d(d(d(x1)))
     b(c(x1)) -> c(b(x1))
     b(c(d(x1))) -> a(x1)
    Matrix Interpretation Processor: dim=1
     
     interpretation:
      [b#](x0) = 2x0 + 1,
      
      [a#](x0) = 2x0 + 8,
      
      [c](x0) = x0 + 3,
      
      [d](x0) = x0 + 1,
      
      [b](x0) = x0 + 7,
      
      [a](x0) = x0 + 11
     orientation:
      b#(c(d(x1))) = 2x1 + 9 >= 2x1 + 8 = a#(x1)
      
      a#(a(x1)) = 2x1 + 30 >= 2x1 + 1 = b#(x1)
      
      b#(c(x1)) = 2x1 + 7 >= 2x1 + 1 = b#(x1)
      
      a#(a(x1)) = 2x1 + 30 >= 2x1 + 15 = b#(b(x1))
      
      a#(a(x1)) = 2x1 + 30 >= 2x1 + 29 = b#(b(b(x1)))
      
      a(a(x1)) = x1 + 22 >= x1 + 21 = b(b(b(x1)))
      
      b(x1) = x1 + 7 >= x1 + 7 = c(c(d(x1)))
      
      c(x1) = x1 + 3 >= x1 + 3 = d(d(d(x1)))
      
      b(c(x1)) = x1 + 10 >= x1 + 10 = c(b(x1))
      
      b(c(d(x1))) = x1 + 11 >= x1 + 11 = a(x1)
     problem:
      DPs:
       
      TRS:
       a(a(x1)) -> b(b(b(x1)))
       b(x1) -> c(c(d(x1)))
       c(x1) -> d(d(d(x1)))
       b(c(x1)) -> c(b(x1))
       b(c(d(x1))) -> a(x1)
     Qed