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
    usable rules:
     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)
    interpretation:
     [d#](x0) = [0 0]x0 + [1],
     
     [r#](x0) = [1 1]x0 + [2],
     
     [a#](x0) = [0 0]x0 + [0],
     
               [0  0 ]     [0 ]
     [d](x0) = [0  -&]x0 + [-&],
     
               [0 0]     [0]
     [r](x0) = [0 0]x0 + [0],
     
               [0  0 ]     [0 ]
     [a](x0) = [0  -&]x0 + [-&],
     
               [1  1 ]     [2 ]
     [b](x0) = [-& -&]x0 + [-&]
    orientation:
     a#(b(x1)) = [1 1]x1 + [2] >= [1 1]x1 + [2] = r#(x1)
     
     r#(a(x1)) = [1 1]x1 + [2] >= [1 1]x1 + [2] = r#(x1)
     
     r#(a(x1)) = [1 1]x1 + [2] >= [0 0]x1 + [1] = d#(r(x1))
     
     r#(x1) = [1 1]x1 + [2] >= [0 0]x1 + [1] = d#(x1)
     
     d#(a(x1)) = [0 0]x1 + [1] >= [0 0]x1 + [1] = d#(x1)
     
     d#(a(x1)) = [0 0]x1 + [1] >= [0 0]x1 + [0] = a#(d(x1))
     
     d#(a(x1)) = [0 0]x1 + [1] >= [0 0]x1 + [0] = a#(a(d(x1)))
     
     d#(x1) = [0 0]x1 + [1] >= [0 0]x1 + [0] = a#(x1)
     
                [1 1]     [2]    [1  1 ]     [2 ]           
     a(b(x1)) = [1 1]x1 + [2] >= [-& -&]x1 + [-&] = b(r(x1))
     
                [0 0]     [0]    [0 0]     [0]           
     r(a(x1)) = [0 0]x1 + [0] >= [0 0]x1 + [0] = d(r(x1))
     
             [0 0]     [0]    [0  0 ]     [0 ]        
     r(x1) = [0 0]x1 + [0] >= [0  -&]x1 + [-&] = d(x1)
     
                [0 0]     [0]    [0 0]     [0]              
     d(a(x1)) = [0 0]x1 + [0] >= [0 0]x1 + [0] = 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)
      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)
    Restore Modifier:
     DPs:
      a#(b(x1)) -> r#(x1)
      r#(a(x1)) -> r#(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)
     EDG Processor:
      DPs:
       a#(b(x1)) -> r#(x1)
       r#(a(x1)) -> r#(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#(a(x1)) -> d#(x1)
       d#(a(x1)) -> d#(x1) -> d#(a(x1)) -> a#(d(x1))
       d#(a(x1)) -> d#(x1) -> d#(a(x1)) -> a#(a(d(x1)))
       d#(a(x1)) -> d#(x1) -> d#(x1) -> a#(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)) -> 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#(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)) -> a#(d(x1)) -> a#(b(x1)) -> r#(x1)
        d#(a(x1)) -> a#(a(d(x1))) -> a#(b(x1)) -> r#(x1)
       SCC Processor:
        #sccs: 0
        #rules: 0
        #arcs: 3/36