YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(a(x1))) -> b#(x1)
   a#(a(a(x1))) -> b#(b(x1))
   b#(b(b(x1))) -> d#(x1)
   b#(b(b(x1))) -> c#(d(x1))
   c#(x1) -> a#(x1)
   c#(x1) -> a#(a(x1))
   d#(x1) -> c#(x1)
  TRS:
   a(a(a(x1))) -> b(b(x1))
   b(b(b(x1))) -> c(d(x1))
   c(x1) -> a(a(x1))
   d(x1) -> c(x1)
  TDG Processor:
   DPs:
    a#(a(a(x1))) -> b#(x1)
    a#(a(a(x1))) -> b#(b(x1))
    b#(b(b(x1))) -> d#(x1)
    b#(b(b(x1))) -> c#(d(x1))
    c#(x1) -> a#(x1)
    c#(x1) -> a#(a(x1))
    d#(x1) -> c#(x1)
   TRS:
    a(a(a(x1))) -> b(b(x1))
    b(b(b(x1))) -> c(d(x1))
    c(x1) -> a(a(x1))
    d(x1) -> c(x1)
   graph:
    c#(x1) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(b(x1))
    c#(x1) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(x1)
    c#(x1) -> a#(x1) -> a#(a(a(x1))) -> b#(b(x1))
    c#(x1) -> a#(x1) -> a#(a(a(x1))) -> b#(x1)
    d#(x1) -> c#(x1) -> c#(x1) -> a#(a(x1))
    d#(x1) -> c#(x1) -> c#(x1) -> a#(x1)
    b#(b(b(x1))) -> c#(d(x1)) -> c#(x1) -> a#(a(x1))
    b#(b(b(x1))) -> c#(d(x1)) -> c#(x1) -> a#(x1)
    b#(b(b(x1))) -> d#(x1) -> d#(x1) -> c#(x1)
    a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(x1))) -> c#(d(x1))
    a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(x1))) -> d#(x1)
    a#(a(a(x1))) -> b#(x1) -> b#(b(b(x1))) -> c#(d(x1))
    a#(a(a(x1))) -> b#(x1) -> b#(b(b(x1))) -> d#(x1)
   Matrix Interpretation Processor: dim=1
    
    interpretation:
     [c#](x0) = 2x0 + 19,
     
     [d#](x0) = 2x0 + 20,
     
     [b#](x0) = 2x0 + 10,
     
     [a#](x0) = 2x0 + 2,
     
     [c](x0) = x0 + 16,
     
     [d](x0) = x0 + 17,
     
     [b](x0) = x0 + 11,
     
     [a](x0) = x0 + 8
    orientation:
     a#(a(a(x1))) = 2x1 + 34 >= 2x1 + 10 = b#(x1)
     
     a#(a(a(x1))) = 2x1 + 34 >= 2x1 + 32 = b#(b(x1))
     
     b#(b(b(x1))) = 2x1 + 54 >= 2x1 + 20 = d#(x1)
     
     b#(b(b(x1))) = 2x1 + 54 >= 2x1 + 53 = c#(d(x1))
     
     c#(x1) = 2x1 + 19 >= 2x1 + 2 = a#(x1)
     
     c#(x1) = 2x1 + 19 >= 2x1 + 18 = a#(a(x1))
     
     d#(x1) = 2x1 + 20 >= 2x1 + 19 = c#(x1)
     
     a(a(a(x1))) = x1 + 24 >= x1 + 22 = b(b(x1))
     
     b(b(b(x1))) = x1 + 33 >= x1 + 33 = c(d(x1))
     
     c(x1) = x1 + 16 >= x1 + 16 = a(a(x1))
     
     d(x1) = x1 + 17 >= x1 + 16 = c(x1)
    problem:
     DPs:
      
     TRS:
      a(a(a(x1))) -> b(b(x1))
      b(b(b(x1))) -> c(d(x1))
      c(x1) -> a(a(x1))
      d(x1) -> c(x1)
    Qed