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))))
  TDG 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))))
   graph:
    b#(d(b(x1))) -> a#(d(x1)) -> a#(c(x1)) -> b#(b(c(d(x1))))
    b#(d(b(x1))) -> a#(d(x1)) -> a#(c(x1)) -> b#(c(d(x1)))
    b#(d(b(x1))) -> a#(d(x1)) -> a#(c(x1)) -> d#(x1)
    b#(d(b(x1))) -> a#(d(x1)) -> a#(x1) -> b#(b(b(x1)))
    b#(d(b(x1))) -> a#(d(x1)) -> a#(x1) -> b#(b(x1))
    b#(d(b(x1))) -> a#(d(x1)) -> a#(x1) -> b#(x1)
    b#(d(b(x1))) -> a#(d(x1)) -> a#(d(x1)) -> d#(b(x1))
    b#(d(b(x1))) -> a#(d(x1)) -> a#(d(x1)) -> b#(x1)
    a#(c(x1)) -> b#(c(d(x1))) -> b#(c(x1)) -> d#(d(x1))
    a#(c(x1)) -> b#(c(d(x1))) -> b#(c(x1)) -> d#(x1)
    a#(c(x1)) -> b#(c(d(x1))) -> b#(d(b(x1))) -> a#(d(x1))
    a#(c(x1)) -> b#(c(d(x1))) -> b#(d(b(x1))) -> d#(x1)
    a#(c(x1)) -> b#(b(c(d(x1)))) -> b#(c(x1)) -> d#(d(x1))
    a#(c(x1)) -> b#(b(c(d(x1)))) -> b#(c(x1)) -> d#(x1)
    a#(c(x1)) -> b#(b(c(d(x1)))) -> b#(d(b(x1))) -> a#(d(x1))
    a#(c(x1)) -> b#(b(c(d(x1)))) -> b#(d(b(x1))) -> d#(x1)
    a#(d(x1)) -> b#(x1) -> b#(c(x1)) -> d#(d(x1))
    a#(d(x1)) -> b#(x1) -> b#(c(x1)) -> d#(x1)
    a#(d(x1)) -> b#(x1) -> b#(d(b(x1))) -> a#(d(x1))
    a#(d(x1)) -> b#(x1) -> b#(d(b(x1))) -> d#(x1)
    a#(x1) -> b#(b(b(x1))) -> b#(c(x1)) -> d#(d(x1))
    a#(x1) -> b#(b(b(x1))) -> b#(c(x1)) -> d#(x1)
    a#(x1) -> b#(b(b(x1))) -> b#(d(b(x1))) -> a#(d(x1))
    a#(x1) -> b#(b(b(x1))) -> b#(d(b(x1))) -> d#(x1)
    a#(x1) -> b#(b(x1)) -> b#(c(x1)) -> d#(d(x1))
    a#(x1) -> b#(b(x1)) -> b#(c(x1)) -> d#(x1)
    a#(x1) -> b#(b(x1)) -> b#(d(b(x1))) -> a#(d(x1))
    a#(x1) -> b#(b(x1)) -> b#(d(b(x1))) -> d#(x1)
    a#(x1) -> b#(x1) -> b#(c(x1)) -> d#(d(x1))
    a#(x1) -> b#(x1) -> b#(c(x1)) -> d#(x1)
    a#(x1) -> b#(x1) -> b#(d(b(x1))) -> a#(d(x1))
    a#(x1) -> b#(x1) -> b#(d(b(x1))) -> d#(x1)
   EDG 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))))
    graph:
     b#(d(b(x1))) -> a#(d(x1)) -> a#(d(x1)) -> b#(x1)
     b#(d(b(x1))) -> a#(d(x1)) -> a#(d(x1)) -> d#(b(x1))
     b#(d(b(x1))) -> a#(d(x1)) -> a#(x1) -> b#(x1)
     b#(d(b(x1))) -> a#(d(x1)) -> a#(x1) -> b#(b(x1))
     b#(d(b(x1))) -> a#(d(x1)) -> a#(x1) -> b#(b(b(x1)))
     b#(d(b(x1))) -> a#(d(x1)) -> a#(c(x1)) -> d#(x1)
     b#(d(b(x1))) -> a#(d(x1)) -> a#(c(x1)) -> b#(c(d(x1)))
     b#(d(b(x1))) -> a#(d(x1)) -> a#(c(x1)) -> b#(b(c(d(x1))))
     a#(c(x1)) -> b#(c(d(x1))) -> b#(c(x1)) -> d#(x1)
     a#(c(x1)) -> b#(c(d(x1))) -> b#(c(x1)) -> d#(d(x1))
     a#(c(x1)) -> b#(b(c(d(x1)))) -> b#(d(b(x1))) -> d#(x1)
     a#(c(x1)) -> b#(b(c(d(x1)))) -> b#(d(b(x1))) -> a#(d(x1))
     a#(c(x1)) -> b#(b(c(d(x1)))) -> b#(c(x1)) -> d#(x1)
     a#(c(x1)) -> b#(b(c(d(x1)))) -> b#(c(x1)) -> d#(d(x1))
     a#(d(x1)) -> b#(x1) -> b#(d(b(x1))) -> d#(x1)
     a#(d(x1)) -> b#(x1) -> b#(d(b(x1))) -> a#(d(x1))
     a#(d(x1)) -> b#(x1) -> b#(c(x1)) -> d#(x1)
     a#(d(x1)) -> b#(x1) -> b#(c(x1)) -> d#(d(x1))
     a#(x1) -> b#(b(b(x1))) -> b#(d(b(x1))) -> d#(x1)
     a#(x1) -> b#(b(b(x1))) -> b#(d(b(x1))) -> a#(d(x1))
     a#(x1) -> b#(b(b(x1))) -> b#(c(x1)) -> d#(x1)
     a#(x1) -> b#(b(b(x1))) -> b#(c(x1)) -> d#(d(x1))
     a#(x1) -> b#(b(x1)) -> b#(d(b(x1))) -> d#(x1)
     a#(x1) -> b#(b(x1)) -> b#(d(b(x1))) -> a#(d(x1))
     a#(x1) -> b#(b(x1)) -> b#(c(x1)) -> d#(x1)
     a#(x1) -> b#(b(x1)) -> b#(c(x1)) -> d#(d(x1))
     a#(x1) -> b#(x1) -> b#(d(b(x1))) -> d#(x1)
     a#(x1) -> b#(x1) -> b#(d(b(x1))) -> a#(d(x1))
     a#(x1) -> b#(x1) -> b#(c(x1)) -> d#(x1)
     a#(x1) -> b#(x1) -> b#(c(x1)) -> d#(d(x1))
    CDG 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))))
     graph:
      b#(d(b(x1))) -> a#(d(x1)) -> a#(c(x1)) -> b#(b(c(d(x1))))
      b#(d(b(x1))) -> a#(d(x1)) -> a#(c(x1)) -> b#(c(d(x1)))
      b#(d(b(x1))) -> a#(d(x1)) -> a#(c(x1)) -> d#(x1)
      b#(d(b(x1))) -> a#(d(x1)) -> a#(x1) -> b#(b(b(x1)))
      b#(d(b(x1))) -> a#(d(x1)) -> a#(x1) -> b#(b(x1))
      b#(d(b(x1))) -> a#(d(x1)) -> a#(x1) -> b#(x1)
      b#(d(b(x1))) -> a#(d(x1)) -> a#(d(x1)) -> d#(b(x1))
      b#(d(b(x1))) -> a#(d(x1)) -> a#(d(x1)) -> b#(x1)
      a#(c(x1)) -> b#(c(d(x1))) -> b#(c(x1)) -> d#(d(x1))
      a#(c(x1)) -> b#(c(d(x1))) -> b#(c(x1)) -> d#(x1)
      a#(c(x1)) -> b#(b(c(d(x1)))) -> b#(c(x1)) -> d#(d(x1))
      a#(c(x1)) -> b#(b(c(d(x1)))) -> b#(c(x1)) -> d#(x1)
      a#(d(x1)) -> b#(x1) -> b#(c(x1)) -> d#(d(x1))
      a#(d(x1)) -> b#(x1) -> b#(c(x1)) -> d#(x1)
      a#(d(x1)) -> b#(x1) -> b#(d(b(x1))) -> a#(d(x1))
      a#(d(x1)) -> b#(x1) -> b#(d(b(x1))) -> d#(x1)
      a#(x1) -> b#(b(b(x1))) -> b#(c(x1)) -> d#(d(x1))
      a#(x1) -> b#(b(b(x1))) -> b#(c(x1)) -> d#(x1)
      a#(x1) -> b#(b(b(x1))) -> b#(d(b(x1))) -> a#(d(x1))
      a#(x1) -> b#(b(b(x1))) -> b#(d(b(x1))) -> d#(x1)
      a#(x1) -> b#(b(x1)) -> b#(c(x1)) -> d#(d(x1))
      a#(x1) -> b#(b(x1)) -> b#(c(x1)) -> d#(x1)
      a#(x1) -> b#(b(x1)) -> b#(d(b(x1))) -> a#(d(x1))
      a#(x1) -> b#(b(x1)) -> b#(d(b(x1))) -> d#(x1)
      a#(x1) -> b#(x1) -> b#(c(x1)) -> d#(d(x1))
      a#(x1) -> b#(x1) -> b#(c(x1)) -> d#(x1)
      a#(x1) -> b#(x1) -> b#(d(b(x1))) -> a#(d(x1))
      a#(x1) -> b#(x1) -> b#(d(b(x1))) -> d#(x1)
     SCC Processor:
      #sccs: 1
      #rules: 5
      #arcs: 28/144
      DPs:
       b#(d(b(x1))) -> a#(d(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)))
       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))))
      Arctic Interpretation Processor:
       dimension: 1
       interpretation:
        [b#](x0) = x0 + -16,
        
        [a#](x0) = x0 + -15,
        
        [c](x0) = 2,
        
        [b](x0) = x0 + -15,
        
        [a](x0) = x0 + 0,
        
        [d](x0) = 15x0 + -15
       orientation:
        b#(d(b(x1))) = 15x1 + 0 >= 15x1 + -15 = a#(d(x1))
        
        a#(d(x1)) = 15x1 + -15 >= x1 + -16 = b#(x1)
        
        a#(x1) = x1 + -15 >= x1 + -16 = b#(x1)
        
        a#(x1) = x1 + -15 >= x1 + -15 = b#(b(x1))
        
        a#(x1) = x1 + -15 >= x1 + -15 = b#(b(b(x1)))
        
        a(d(x1)) = 15x1 + 0 >= 15x1 + 0 = d(b(x1))
        
        a(x1) = x1 + 0 >= x1 + -15 = b(b(b(x1)))
        
        d(x1) = 15x1 + -15 >= x1 = x1
        
        a(x1) = x1 + 0 >= x1 = x1
        
        b(d(b(x1))) = 15x1 + 0 >= 15x1 + 0 = a(d(x1))
        
        b(c(x1)) = 2 >= 2 = c(d(d(x1)))
        
        a(c(x1)) = 2 >= 2 = b(b(c(d(x1))))
       problem:
        DPs:
         b#(d(b(x1))) -> a#(d(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)))
         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))))
       EDG Processor:
        DPs:
         b#(d(b(x1))) -> a#(d(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)))
         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))))
        graph:
         b#(d(b(x1))) -> a#(d(x1)) -> a#(x1) -> b#(x1)
         b#(d(b(x1))) -> a#(d(x1)) -> a#(x1) -> b#(b(x1))
         b#(d(b(x1))) -> a#(d(x1)) -> a#(x1) -> b#(b(b(x1)))
         a#(x1) -> b#(b(b(x1))) -> b#(d(b(x1))) -> a#(d(x1))
         a#(x1) -> b#(b(x1)) -> b#(d(b(x1))) -> a#(d(x1))
         a#(x1) -> b#(x1) -> b#(d(b(x1))) -> a#(d(x1))
        Matrix Interpretation Processor: dim=1
         
         interpretation:
          [b#](x0) = 6x0 + 3,
          
          [a#](x0) = 6x0 + 28,
          
          [c](x0) = 6,
          
          [b](x0) = x0 + 2,
          
          [a](x0) = x0 + 10,
          
          [d](x0) = 4x0 + 2
         orientation:
          b#(d(b(x1))) = 24x1 + 63 >= 24x1 + 40 = a#(d(x1))
          
          a#(x1) = 6x1 + 28 >= 6x1 + 3 = b#(x1)
          
          a#(x1) = 6x1 + 28 >= 6x1 + 15 = b#(b(x1))
          
          a#(x1) = 6x1 + 28 >= 6x1 + 27 = b#(b(b(x1)))
          
          a(d(x1)) = 4x1 + 12 >= 4x1 + 10 = d(b(x1))
          
          a(x1) = x1 + 10 >= x1 + 6 = b(b(b(x1)))
          
          d(x1) = 4x1 + 2 >= x1 = x1
          
          a(x1) = x1 + 10 >= x1 = x1
          
          b(d(b(x1))) = 4x1 + 12 >= 4x1 + 12 = a(d(x1))
          
          b(c(x1)) = 8 >= 6 = c(d(d(x1)))
          
          a(c(x1)) = 16 >= 10 = 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