YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(x1)) -> c#(x1)
   a#(a(x1)) -> b#(c(x1))
   b#(b(x1)) -> d#(x1)
   b#(b(x1)) -> c#(d(x1))
   c#(c(x1)) -> d#(x1)
   c#(c(x1)) -> d#(d(x1))
   c#(c(x1)) -> d#(d(d(x1)))
   d#(d(d(x1))) -> c#(x1)
   d#(d(d(x1))) -> a#(c(x1))
  TRS:
   a(a(x1)) -> b(c(x1))
   b(b(x1)) -> c(d(x1))
   c(c(x1)) -> d(d(d(x1)))
   d(d(d(x1))) -> a(c(x1))
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [d#](x0) = x0 + 13,
    
    [b#](x0) = x0 + 17,
    
    [c#](x0) = x0 + 20,
    
    [a#](x0) = x0 + 18,
    
    [d](x0) = x0 + 12,
    
    [b](x0) = x0 + 16,
    
    [c](x0) = x0 + 18,
    
    [a](x0) = x0 + 18
   orientation:
    a#(a(x1)) = x1 + 36 >= x1 + 20 = c#(x1)
    
    a#(a(x1)) = x1 + 36 >= x1 + 35 = b#(c(x1))
    
    b#(b(x1)) = x1 + 33 >= x1 + 13 = d#(x1)
    
    b#(b(x1)) = x1 + 33 >= x1 + 32 = c#(d(x1))
    
    c#(c(x1)) = x1 + 38 >= x1 + 13 = d#(x1)
    
    c#(c(x1)) = x1 + 38 >= x1 + 25 = d#(d(x1))
    
    c#(c(x1)) = x1 + 38 >= x1 + 37 = d#(d(d(x1)))
    
    d#(d(d(x1))) = x1 + 37 >= x1 + 20 = c#(x1)
    
    d#(d(d(x1))) = x1 + 37 >= x1 + 36 = a#(c(x1))
    
    a(a(x1)) = x1 + 36 >= x1 + 34 = b(c(x1))
    
    b(b(x1)) = x1 + 32 >= x1 + 30 = c(d(x1))
    
    c(c(x1)) = x1 + 36 >= x1 + 36 = d(d(d(x1)))
    
    d(d(d(x1))) = x1 + 36 >= x1 + 36 = a(c(x1))
   problem:
    DPs:
     
    TRS:
     a(a(x1)) -> b(c(x1))
     b(b(x1)) -> c(d(x1))
     c(c(x1)) -> d(d(d(x1)))
     d(d(d(x1))) -> a(c(x1))
   Qed