YES

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

Proof:
 DP Processor:
  DPs:
   a#(d(x1)) -> b#(x1)
   a#(x1) -> b#(x1)
   a#(x1) -> b#(b(x1))
   a#(x1) -> b#(b(b(x1)))
   b#(d(b(x1))) -> c#(x1)
   b#(d(b(x1))) -> a#(c(x1))
  TRS:
   a(d(x1)) -> d(b(x1))
   a(x1) -> b(b(b(x1)))
   b(d(b(x1))) -> a(c(x1))
   c(x1) -> d(x1)
  TDG Processor:
   DPs:
    a#(d(x1)) -> b#(x1)
    a#(x1) -> b#(x1)
    a#(x1) -> b#(b(x1))
    a#(x1) -> b#(b(b(x1)))
    b#(d(b(x1))) -> c#(x1)
    b#(d(b(x1))) -> a#(c(x1))
   TRS:
    a(d(x1)) -> d(b(x1))
    a(x1) -> b(b(b(x1)))
    b(d(b(x1))) -> a(c(x1))
    c(x1) -> d(x1)
   graph:
    b#(d(b(x1))) -> a#(c(x1)) -> a#(x1) -> b#(b(b(x1)))
    b#(d(b(x1))) -> a#(c(x1)) -> a#(x1) -> b#(b(x1))
    b#(d(b(x1))) -> a#(c(x1)) -> a#(x1) -> b#(x1)
    b#(d(b(x1))) -> a#(c(x1)) -> a#(d(x1)) -> b#(x1)
    a#(d(x1)) -> b#(x1) -> b#(d(b(x1))) -> a#(c(x1))
    a#(d(x1)) -> b#(x1) -> b#(d(b(x1))) -> c#(x1)
    a#(x1) -> b#(b(b(x1))) -> b#(d(b(x1))) -> a#(c(x1))
    a#(x1) -> b#(b(b(x1))) -> b#(d(b(x1))) -> c#(x1)
    a#(x1) -> b#(b(x1)) -> b#(d(b(x1))) -> a#(c(x1))
    a#(x1) -> b#(b(x1)) -> b#(d(b(x1))) -> c#(x1)
    a#(x1) -> b#(x1) -> b#(d(b(x1))) -> a#(c(x1))
    a#(x1) -> b#(x1) -> b#(d(b(x1))) -> c#(x1)
   SCC Processor:
    #sccs: 1
    #rules: 5
    #arcs: 12/36
    DPs:
     b#(d(b(x1))) -> a#(c(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)))
     b(d(b(x1))) -> a(c(x1))
     c(x1) -> d(x1)
    Arctic Interpretation Processor:
     dimension: 1
     interpretation:
      [b#](x0) = x0,
      
      [a#](x0) = x0,
      
      [c](x0) = 4x0,
      
      [b](x0) = x0,
      
      [a](x0) = x0,
      
      [d](x0) = 4x0
     orientation:
      b#(d(b(x1))) = 4x1 >= 4x1 = a#(c(x1))
      
      a#(d(x1)) = 4x1 >= x1 = b#(x1)
      
      a#(x1) = x1 >= x1 = b#(x1)
      
      a#(x1) = x1 >= x1 = b#(b(x1))
      
      a#(x1) = x1 >= x1 = b#(b(b(x1)))
      
      a(d(x1)) = 4x1 >= 4x1 = d(b(x1))
      
      a(x1) = x1 >= x1 = b(b(b(x1)))
      
      b(d(b(x1))) = 4x1 >= 4x1 = a(c(x1))
      
      c(x1) = 4x1 >= 4x1 = d(x1)
     problem:
      DPs:
       b#(d(b(x1))) -> a#(c(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)))
       b(d(b(x1))) -> a(c(x1))
       c(x1) -> d(x1)
     EDG Processor:
      DPs:
       b#(d(b(x1))) -> a#(c(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)))
       b(d(b(x1))) -> a(c(x1))
       c(x1) -> d(x1)
      graph:
       b#(d(b(x1))) -> a#(c(x1)) -> a#(x1) -> b#(x1)
       b#(d(b(x1))) -> a#(c(x1)) -> a#(x1) -> b#(b(x1))
       b#(d(b(x1))) -> a#(c(x1)) -> a#(x1) -> b#(b(b(x1)))
       a#(x1) -> b#(b(b(x1))) -> b#(d(b(x1))) -> a#(c(x1))
       a#(x1) -> b#(b(x1)) -> b#(d(b(x1))) -> a#(c(x1))
       a#(x1) -> b#(x1) -> b#(d(b(x1))) -> a#(c(x1))
      Matrix Interpretation Processor: dim=4
       
       interpretation:
        [b#](x0) = [0 0 0 1]x0,
        
        [a#](x0) = [0 0 0 1]x0,
        
                  [1 0 0 0]  
                  [0 1 1 0]  
        [c](x0) = [1 1 1 0]x0
                  [1 1 1 0]  ,
        
                  [0 0 0 0]     [1]
                  [0 0 1 0]     [0]
        [b](x0) = [1 1 0 0]x0 + [0]
                  [0 0 0 1]     [0],
        
                  [0 0 0 0]     [1]
                  [0 0 1 0]     [1]
        [a](x0) = [1 1 0 0]x0 + [1]
                  [0 0 0 1]     [1],
        
                  [1 0 0 0]  
                  [0 1 1 0]  
        [d](x0) = [1 1 1 0]x0
                  [1 1 1 0]  
       orientation:
        b#(d(b(x1))) = [1 1 1 0]x1 + [1] >= [1 1 1 0]x1 = a#(c(x1))
        
        a#(x1) = [0 0 0 1]x1 >= [0 0 0 1]x1 = b#(x1)
        
        a#(x1) = [0 0 0 1]x1 >= [0 0 0 1]x1 = b#(b(x1))
        
        a#(x1) = [0 0 0 1]x1 >= [0 0 0 1]x1 = b#(b(b(x1)))
        
                   [0 0 0 0]     [1]    [0 0 0 0]     [1]           
                   [1 1 1 0]     [1]    [1 1 1 0]     [0]           
        a(d(x1)) = [1 1 1 0]x1 + [1] >= [1 1 1 0]x1 + [1] = d(b(x1))
                   [1 1 1 0]     [1]    [1 1 1 0]     [1]           
        
                [0 0 0 0]     [1]    [0 0 0 0]     [1]              
                [0 0 1 0]     [1]    [0 0 1 0]     [1]              
        a(x1) = [1 1 0 0]x1 + [1] >= [1 1 0 0]x1 + [1] = b(b(b(x1)))
                [0 0 0 1]     [1]    [0 0 0 1]     [0]              
        
                      [0 0 0 0]     [1]    [0 0 0 0]     [1]           
                      [1 1 1 0]     [1]    [1 1 1 0]     [1]           
        b(d(b(x1))) = [1 1 1 0]x1 + [1] >= [1 1 1 0]x1 + [1] = a(c(x1))
                      [1 1 1 0]     [1]    [1 1 1 0]     [1]           
        
                [1 0 0 0]      [1 0 0 0]          
                [0 1 1 0]      [0 1 1 0]          
        c(x1) = [1 1 1 0]x1 >= [1 1 1 0]x1 = d(x1)
                [1 1 1 0]      [1 1 1 0]          
       problem:
        DPs:
         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)))
         b(d(b(x1))) -> a(c(x1))
         c(x1) -> d(x1)
       EDG Processor:
        DPs:
         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)))
         b(d(b(x1))) -> a(c(x1))
         c(x1) -> d(x1)
        graph:
         
        Qed