YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(x1)) -> b#(x1)
   a#(a(x1)) -> b#(b(x1))
   a#(a(x1)) -> b#(b(b(x1)))
   b#(x1) -> c#(d(x1))
   b#(x1) -> c#(c(d(x1)))
   b#(c(x1)) -> b#(x1)
   b#(c(x1)) -> c#(b(x1))
   b#(c(d(x1))) -> a#(x1)
  TRS:
   a(a(x1)) -> b(b(b(x1)))
   b(x1) -> c(c(d(x1)))
   c(x1) -> d(d(d(x1)))
   b(c(x1)) -> c(b(x1))
   b(c(d(x1))) -> a(x1)
  TDG Processor:
   DPs:
    a#(a(x1)) -> b#(x1)
    a#(a(x1)) -> b#(b(x1))
    a#(a(x1)) -> b#(b(b(x1)))
    b#(x1) -> c#(d(x1))
    b#(x1) -> c#(c(d(x1)))
    b#(c(x1)) -> b#(x1)
    b#(c(x1)) -> c#(b(x1))
    b#(c(d(x1))) -> a#(x1)
   TRS:
    a(a(x1)) -> b(b(b(x1)))
    b(x1) -> c(c(d(x1)))
    c(x1) -> d(d(d(x1)))
    b(c(x1)) -> c(b(x1))
    b(c(d(x1))) -> a(x1)
   graph:
    b#(c(d(x1))) -> a#(x1) -> a#(a(x1)) -> b#(b(b(x1)))
    b#(c(d(x1))) -> a#(x1) -> a#(a(x1)) -> b#(b(x1))
    b#(c(d(x1))) -> a#(x1) -> a#(a(x1)) -> b#(x1)
    b#(c(x1)) -> b#(x1) -> b#(c(d(x1))) -> a#(x1)
    b#(c(x1)) -> b#(x1) -> b#(c(x1)) -> c#(b(x1))
    b#(c(x1)) -> b#(x1) -> b#(c(x1)) -> b#(x1)
    b#(c(x1)) -> b#(x1) -> b#(x1) -> c#(c(d(x1)))
    b#(c(x1)) -> b#(x1) -> b#(x1) -> c#(d(x1))
    a#(a(x1)) -> b#(b(b(x1))) -> b#(c(d(x1))) -> a#(x1)
    a#(a(x1)) -> b#(b(b(x1))) -> b#(c(x1)) -> c#(b(x1))
    a#(a(x1)) -> b#(b(b(x1))) -> b#(c(x1)) -> b#(x1)
    a#(a(x1)) -> b#(b(b(x1))) -> b#(x1) -> c#(c(d(x1)))
    a#(a(x1)) -> b#(b(b(x1))) -> b#(x1) -> c#(d(x1))
    a#(a(x1)) -> b#(b(x1)) -> b#(c(d(x1))) -> a#(x1)
    a#(a(x1)) -> b#(b(x1)) -> b#(c(x1)) -> c#(b(x1))
    a#(a(x1)) -> b#(b(x1)) -> b#(c(x1)) -> b#(x1)
    a#(a(x1)) -> b#(b(x1)) -> b#(x1) -> c#(c(d(x1)))
    a#(a(x1)) -> b#(b(x1)) -> b#(x1) -> c#(d(x1))
    a#(a(x1)) -> b#(x1) -> b#(c(d(x1))) -> a#(x1)
    a#(a(x1)) -> b#(x1) -> b#(c(x1)) -> c#(b(x1))
    a#(a(x1)) -> b#(x1) -> b#(c(x1)) -> b#(x1)
    a#(a(x1)) -> b#(x1) -> b#(x1) -> c#(c(d(x1)))
    a#(a(x1)) -> b#(x1) -> b#(x1) -> c#(d(x1))
   SCC Processor:
    #sccs: 1
    #rules: 5
    #arcs: 23/64
    DPs:
     b#(c(d(x1))) -> a#(x1)
     a#(a(x1)) -> b#(x1)
     b#(c(x1)) -> b#(x1)
     a#(a(x1)) -> b#(b(x1))
     a#(a(x1)) -> b#(b(b(x1)))
    TRS:
     a(a(x1)) -> b(b(b(x1)))
     b(x1) -> c(c(d(x1)))
     c(x1) -> d(d(d(x1)))
     b(c(x1)) -> c(b(x1))
     b(c(d(x1))) -> a(x1)
    Arctic Interpretation Processor:
     dimension: 1
     interpretation:
      [b#](x0) = 2x0 + 0,
      
      [a#](x0) = 4x0,
      
      [c](x0) = 2x0 + -1,
      
      [d](x0) = x0,
      
      [b](x0) = 4x0 + 1,
      
      [a](x0) = 6x0 + 3
     orientation:
      b#(c(d(x1))) = 4x1 + 1 >= 4x1 = a#(x1)
      
      a#(a(x1)) = 10x1 + 7 >= 2x1 + 0 = b#(x1)
      
      b#(c(x1)) = 4x1 + 1 >= 2x1 + 0 = b#(x1)
      
      a#(a(x1)) = 10x1 + 7 >= 6x1 + 3 = b#(b(x1))
      
      a#(a(x1)) = 10x1 + 7 >= 10x1 + 7 = b#(b(b(x1)))
      
      a(a(x1)) = 12x1 + 9 >= 12x1 + 9 = b(b(b(x1)))
      
      b(x1) = 4x1 + 1 >= 4x1 + 1 = c(c(d(x1)))
      
      c(x1) = 2x1 + -1 >= x1 = d(d(d(x1)))
      
      b(c(x1)) = 6x1 + 3 >= 6x1 + 3 = c(b(x1))
      
      b(c(d(x1))) = 6x1 + 3 >= 6x1 + 3 = a(x1)
     problem:
      DPs:
       b#(c(d(x1))) -> a#(x1)
       a#(a(x1)) -> b#(b(b(x1)))
      TRS:
       a(a(x1)) -> b(b(b(x1)))
       b(x1) -> c(c(d(x1)))
       c(x1) -> d(d(d(x1)))
       b(c(x1)) -> c(b(x1))
       b(c(d(x1))) -> a(x1)
     EDG Processor:
      DPs:
       b#(c(d(x1))) -> a#(x1)
       a#(a(x1)) -> b#(b(b(x1)))
      TRS:
       a(a(x1)) -> b(b(b(x1)))
       b(x1) -> c(c(d(x1)))
       c(x1) -> d(d(d(x1)))
       b(c(x1)) -> c(b(x1))
       b(c(d(x1))) -> a(x1)
      graph:
       b#(c(d(x1))) -> a#(x1) -> a#(a(x1)) -> b#(b(b(x1)))
       a#(a(x1)) -> b#(b(b(x1))) -> b#(c(d(x1))) -> a#(x1)
      Arctic Interpretation Processor:
       dimension: 1
       interpretation:
        [b#](x0) = x0,
        
        [a#](x0) = 3x0,
        
        [c](x0) = 3x0,
        
        [d](x0) = 1x0,
        
        [b](x0) = 7x0,
        
        [a](x0) = 11x0
       orientation:
        b#(c(d(x1))) = 4x1 >= 3x1 = a#(x1)
        
        a#(a(x1)) = 14x1 >= 14x1 = b#(b(b(x1)))
        
        a(a(x1)) = 22x1 >= 21x1 = b(b(b(x1)))
        
        b(x1) = 7x1 >= 7x1 = c(c(d(x1)))
        
        c(x1) = 3x1 >= 3x1 = d(d(d(x1)))
        
        b(c(x1)) = 10x1 >= 10x1 = c(b(x1))
        
        b(c(d(x1))) = 11x1 >= 11x1 = a(x1)
       problem:
        DPs:
         a#(a(x1)) -> b#(b(b(x1)))
        TRS:
         a(a(x1)) -> b(b(b(x1)))
         b(x1) -> c(c(d(x1)))
         c(x1) -> d(d(d(x1)))
         b(c(x1)) -> c(b(x1))
         b(c(d(x1))) -> a(x1)
       EDG Processor:
        DPs:
         a#(a(x1)) -> b#(b(b(x1)))
        TRS:
         a(a(x1)) -> b(b(b(x1)))
         b(x1) -> c(c(d(x1)))
         c(x1) -> d(d(d(x1)))
         b(c(x1)) -> c(b(x1))
         b(c(d(x1))) -> a(x1)
        graph:
         
        Qed