YES

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

Proof:
 DP Processor:
  DPs:
   a#(b(x1)) -> r#(x1)
   r#(a(x1)) -> r#(x1)
   r#(a(x1)) -> d#(r(x1))
   r#(x1) -> d#(x1)
   d#(a(x1)) -> d#(x1)
   d#(a(x1)) -> a#(d(x1))
   d#(a(x1)) -> a#(a(d(x1)))
   d#(x1) -> a#(x1)
  TRS:
   a(b(x1)) -> b(r(x1))
   r(a(x1)) -> d(r(x1))
   r(x1) -> d(x1)
   d(a(x1)) -> a(a(d(x1)))
   d(x1) -> a(x1)
  TDG Processor:
   DPs:
    a#(b(x1)) -> r#(x1)
    r#(a(x1)) -> r#(x1)
    r#(a(x1)) -> d#(r(x1))
    r#(x1) -> d#(x1)
    d#(a(x1)) -> d#(x1)
    d#(a(x1)) -> a#(d(x1))
    d#(a(x1)) -> a#(a(d(x1)))
    d#(x1) -> a#(x1)
   TRS:
    a(b(x1)) -> b(r(x1))
    r(a(x1)) -> d(r(x1))
    r(x1) -> d(x1)
    d(a(x1)) -> a(a(d(x1)))
    d(x1) -> a(x1)
   graph:
    d#(a(x1)) -> d#(x1) -> d#(x1) -> a#(x1)
    d#(a(x1)) -> d#(x1) -> d#(a(x1)) -> a#(a(d(x1)))
    d#(a(x1)) -> d#(x1) -> d#(a(x1)) -> a#(d(x1))
    d#(a(x1)) -> d#(x1) -> d#(a(x1)) -> d#(x1)
    d#(a(x1)) -> a#(d(x1)) -> a#(b(x1)) -> r#(x1)
    d#(a(x1)) -> a#(a(d(x1))) -> a#(b(x1)) -> r#(x1)
    d#(x1) -> a#(x1) -> a#(b(x1)) -> r#(x1)
    r#(a(x1)) -> d#(r(x1)) -> d#(x1) -> a#(x1)
    r#(a(x1)) -> d#(r(x1)) -> d#(a(x1)) -> a#(a(d(x1)))
    r#(a(x1)) -> d#(r(x1)) -> d#(a(x1)) -> a#(d(x1))
    r#(a(x1)) -> d#(r(x1)) -> d#(a(x1)) -> d#(x1)
    r#(a(x1)) -> r#(x1) -> r#(x1) -> d#(x1)
    r#(a(x1)) -> r#(x1) -> r#(a(x1)) -> d#(r(x1))
    r#(a(x1)) -> r#(x1) -> r#(a(x1)) -> r#(x1)
    r#(x1) -> d#(x1) -> d#(x1) -> a#(x1)
    r#(x1) -> d#(x1) -> d#(a(x1)) -> a#(a(d(x1)))
    r#(x1) -> d#(x1) -> d#(a(x1)) -> a#(d(x1))
    r#(x1) -> d#(x1) -> d#(a(x1)) -> d#(x1)
    a#(b(x1)) -> r#(x1) -> r#(x1) -> d#(x1)
    a#(b(x1)) -> r#(x1) -> r#(a(x1)) -> d#(r(x1))
    a#(b(x1)) -> r#(x1) -> r#(a(x1)) -> r#(x1)
   Arctic Interpretation Processor:
    dimension: 2
    interpretation:
     [d#](x0) = [1  -&]x0 + [-4],
     
     [r#](x0) = [1  -&]x0 + [-4],
     
     [a#](x0) = [0  -&]x0 + [-4],
     
               [0  0 ]     [0 ]
     [d](x0) = [-& 0 ]x0 + [-&],
     
               [0  0 ]     [0 ]
     [r](x0) = [-4 0 ]x0 + [-&],
     
               [0  0 ]     [0 ]
     [a](x0) = [-& 0 ]x0 + [-&],
     
               [1  1 ]     [0]
     [b](x0) = [-4 0 ]x0 + [1]
    orientation:
     a#(b(x1)) = [1 1]x1 + [0] >= [1  -&]x1 + [-4] = r#(x1)
     
     r#(a(x1)) = [1 1]x1 + [1] >= [1  -&]x1 + [-4] = r#(x1)
     
     r#(a(x1)) = [1 1]x1 + [1] >= [1 1]x1 + [1] = d#(r(x1))
     
     r#(x1) = [1  -&]x1 + [-4] >= [1  -&]x1 + [-4] = d#(x1)
     
     d#(a(x1)) = [1 1]x1 + [1] >= [1  -&]x1 + [-4] = d#(x1)
     
     d#(a(x1)) = [1 1]x1 + [1] >= [0 0]x1 + [0] = a#(d(x1))
     
     d#(a(x1)) = [1 1]x1 + [1] >= [0 0]x1 + [0] = a#(a(d(x1)))
     
     d#(x1) = [1  -&]x1 + [-4] >= [0  -&]x1 + [-4] = a#(x1)
     
                [1  1 ]     [1]    [1  1 ]     [1]           
     a(b(x1)) = [-4 0 ]x1 + [1] >= [-4 0 ]x1 + [1] = b(r(x1))
     
                [0  0 ]     [0 ]    [0  0 ]     [0 ]           
     r(a(x1)) = [-4 0 ]x1 + [-4] >= [-4 0 ]x1 + [-&] = d(r(x1))
     
             [0  0 ]     [0 ]    [0  0 ]     [0 ]        
     r(x1) = [-4 0 ]x1 + [-&] >= [-& 0 ]x1 + [-&] = d(x1)
     
                [0  0 ]     [0 ]    [0  0 ]     [0 ]              
     d(a(x1)) = [-& 0 ]x1 + [-&] >= [-& 0 ]x1 + [-&] = a(a(d(x1)))
     
             [0  0 ]     [0 ]    [0  0 ]     [0 ]        
     d(x1) = [-& 0 ]x1 + [-&] >= [-& 0 ]x1 + [-&] = a(x1)
    problem:
     DPs:
      a#(b(x1)) -> r#(x1)
      r#(a(x1)) -> r#(x1)
      r#(a(x1)) -> d#(r(x1))
      r#(x1) -> d#(x1)
      d#(a(x1)) -> d#(x1)
      d#(x1) -> a#(x1)
     TRS:
      a(b(x1)) -> b(r(x1))
      r(a(x1)) -> d(r(x1))
      r(x1) -> d(x1)
      d(a(x1)) -> a(a(d(x1)))
      d(x1) -> a(x1)
    EDG Processor:
     DPs:
      a#(b(x1)) -> r#(x1)
      r#(a(x1)) -> r#(x1)
      r#(a(x1)) -> d#(r(x1))
      r#(x1) -> d#(x1)
      d#(a(x1)) -> d#(x1)
      d#(x1) -> a#(x1)
     TRS:
      a(b(x1)) -> b(r(x1))
      r(a(x1)) -> d(r(x1))
      r(x1) -> d(x1)
      d(a(x1)) -> a(a(d(x1)))
      d(x1) -> a(x1)
     graph:
      d#(a(x1)) -> d#(x1) -> d#(a(x1)) -> d#(x1)
      d#(a(x1)) -> d#(x1) -> d#(x1) -> a#(x1)
      d#(x1) -> a#(x1) -> a#(b(x1)) -> r#(x1)
      r#(a(x1)) -> d#(r(x1)) -> d#(a(x1)) -> d#(x1)
      r#(a(x1)) -> d#(r(x1)) -> d#(x1) -> a#(x1)
      r#(a(x1)) -> r#(x1) -> r#(a(x1)) -> r#(x1)
      r#(a(x1)) -> r#(x1) -> r#(a(x1)) -> d#(r(x1))
      r#(a(x1)) -> r#(x1) -> r#(x1) -> d#(x1)
      r#(x1) -> d#(x1) -> d#(a(x1)) -> d#(x1)
      r#(x1) -> d#(x1) -> d#(x1) -> a#(x1)
      a#(b(x1)) -> r#(x1) -> r#(a(x1)) -> r#(x1)
      a#(b(x1)) -> r#(x1) -> r#(a(x1)) -> d#(r(x1))
      a#(b(x1)) -> r#(x1) -> r#(x1) -> d#(x1)
     Arctic Interpretation Processor:
      dimension: 1
      interpretation:
       [d#](x0) = x0,
       
       [r#](x0) = 2x0,
       
       [a#](x0) = x0,
       
       [d](x0) = x0,
       
       [r](x0) = x0,
       
       [a](x0) = x0,
       
       [b](x0) = 2x0
      orientation:
       a#(b(x1)) = 2x1 >= 2x1 = r#(x1)
       
       r#(a(x1)) = 2x1 >= 2x1 = r#(x1)
       
       r#(a(x1)) = 2x1 >= x1 = d#(r(x1))
       
       r#(x1) = 2x1 >= x1 = d#(x1)
       
       d#(a(x1)) = x1 >= x1 = d#(x1)
       
       d#(x1) = x1 >= x1 = a#(x1)
       
       a(b(x1)) = 2x1 >= 2x1 = b(r(x1))
       
       r(a(x1)) = x1 >= x1 = d(r(x1))
       
       r(x1) = x1 >= x1 = d(x1)
       
       d(a(x1)) = x1 >= x1 = a(a(d(x1)))
       
       d(x1) = x1 >= x1 = a(x1)
      problem:
       DPs:
        a#(b(x1)) -> r#(x1)
        r#(a(x1)) -> r#(x1)
        d#(a(x1)) -> d#(x1)
        d#(x1) -> a#(x1)
       TRS:
        a(b(x1)) -> b(r(x1))
        r(a(x1)) -> d(r(x1))
        r(x1) -> d(x1)
        d(a(x1)) -> a(a(d(x1)))
        d(x1) -> a(x1)
      EDG Processor:
       DPs:
        a#(b(x1)) -> r#(x1)
        r#(a(x1)) -> r#(x1)
        d#(a(x1)) -> d#(x1)
        d#(x1) -> a#(x1)
       TRS:
        a(b(x1)) -> b(r(x1))
        r(a(x1)) -> d(r(x1))
        r(x1) -> d(x1)
        d(a(x1)) -> a(a(d(x1)))
        d(x1) -> a(x1)
       graph:
        d#(a(x1)) -> d#(x1) -> d#(x1) -> a#(x1)
        d#(a(x1)) -> d#(x1) -> d#(a(x1)) -> d#(x1)
        d#(x1) -> a#(x1) -> a#(b(x1)) -> r#(x1)
        r#(a(x1)) -> r#(x1) -> r#(a(x1)) -> r#(x1)
        a#(b(x1)) -> r#(x1) -> r#(a(x1)) -> r#(x1)
       CDG Processor:
        DPs:
         a#(b(x1)) -> r#(x1)
         r#(a(x1)) -> r#(x1)
         d#(a(x1)) -> d#(x1)
         d#(x1) -> a#(x1)
        TRS:
         a(b(x1)) -> b(r(x1))
         r(a(x1)) -> d(r(x1))
         r(x1) -> d(x1)
         d(a(x1)) -> a(a(d(x1)))
         d(x1) -> a(x1)
        graph:
         d#(a(x1)) -> d#(x1) -> d#(x1) -> a#(x1)
        SCC Processor:
         #sccs: 0
         #rules: 0
         #arcs: 1/16