YES

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

Proof:
 DP Processor:
  DPs:
   a#(d(x1)) -> b#(x1)
   a#(d(x1)) -> d#(b(x1))
   a#(x1) -> b#(x1)
   a#(x1) -> b#(b(x1))
   a#(x1) -> b#(b(b(x1)))
   b#(d(b(x1))) -> d#(x1)
   b#(d(b(x1))) -> a#(d(x1))
   b#(c(x1)) -> d#(x1)
   b#(c(x1)) -> d#(d(x1))
   a#(c(x1)) -> d#(x1)
   a#(c(x1)) -> b#(c(d(x1)))
   a#(c(x1)) -> b#(b(c(d(x1))))
  TRS:
   a(d(x1)) -> d(b(x1))
   a(x1) -> b(b(b(x1)))
   d(x1) -> x1
   a(x1) -> x1
   b(d(b(x1))) -> a(d(x1))
   b(c(x1)) -> c(d(d(x1)))
   a(c(x1)) -> b(b(c(d(x1))))
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [d#](x0) = 1,
    
    [b#](x0) = 12x0 + 4,
    
    [a#](x0) = 12x0 + 31,
    
    [c](x0) = 0,
    
    [b](x0) = x0 + 1,
    
    [a](x0) = x0 + 5,
    
    [d](x0) = 4x0
   orientation:
    a#(d(x1)) = 48x1 + 31 >= 12x1 + 4 = b#(x1)
    
    a#(d(x1)) = 48x1 + 31 >= 1 = d#(b(x1))
    
    a#(x1) = 12x1 + 31 >= 12x1 + 4 = b#(x1)
    
    a#(x1) = 12x1 + 31 >= 12x1 + 16 = b#(b(x1))
    
    a#(x1) = 12x1 + 31 >= 12x1 + 28 = b#(b(b(x1)))
    
    b#(d(b(x1))) = 48x1 + 52 >= 1 = d#(x1)
    
    b#(d(b(x1))) = 48x1 + 52 >= 48x1 + 31 = a#(d(x1))
    
    b#(c(x1)) = 4 >= 1 = d#(x1)
    
    b#(c(x1)) = 4 >= 1 = d#(d(x1))
    
    a#(c(x1)) = 31 >= 1 = d#(x1)
    
    a#(c(x1)) = 31 >= 4 = b#(c(d(x1)))
    
    a#(c(x1)) = 31 >= 16 = b#(b(c(d(x1))))
    
    a(d(x1)) = 4x1 + 5 >= 4x1 + 4 = d(b(x1))
    
    a(x1) = x1 + 5 >= x1 + 3 = b(b(b(x1)))
    
    d(x1) = 4x1 >= x1 = x1
    
    a(x1) = x1 + 5 >= x1 = x1
    
    b(d(b(x1))) = 4x1 + 5 >= 4x1 + 5 = a(d(x1))
    
    b(c(x1)) = 1 >= 0 = c(d(d(x1)))
    
    a(c(x1)) = 5 >= 2 = b(b(c(d(x1))))
   problem:
    DPs:
     
    TRS:
     a(d(x1)) -> d(b(x1))
     a(x1) -> b(b(b(x1)))
     d(x1) -> x1
     a(x1) -> x1
     b(d(b(x1))) -> a(d(x1))
     b(c(x1)) -> c(d(d(x1)))
     a(c(x1)) -> b(b(c(d(x1))))
   Qed