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:
 Arctic Interpretation Processor:
  dimension: 2
  interpretation:
             [3  -&]  
   [c](x0) = [2  -&]x0,
   
             [0  3 ]  
   [b](x0) = [-& 0 ]x0,
   
             [3 3]  
   [a](x0) = [0 0]x0,
   
             [0  -&]  
   [d](x0) = [0  3 ]x0
  orientation:
              [3 6]      [0 3]             
   a(d(x1)) = [0 3]x1 >= [0 3]x1 = d(b(x1))
   
           [3 3]      [0  3 ]                
   a(x1) = [0 0]x1 >= [-& 0 ]x1 = b(b(b(x1)))
   
           [0  -&]             
   d(x1) = [0  3 ]x1 >= x1 = x1
   
           [3 3]             
   a(x1) = [0 0]x1 >= x1 = x1
   
                 [3 6]      [3 6]             
   b(d(b(x1))) = [0 3]x1 >= [0 3]x1 = a(d(x1))
   
              [5  -&]      [3  -&]                
   b(c(x1)) = [2  -&]x1 >= [2  -&]x1 = c(d(d(x1)))
   
              [6  -&]      [5  -&]                   
   a(c(x1)) = [3  -&]x1 >= [2  -&]x1 = b(b(c(d(x1))))
  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)))
  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))
   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)))
   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))
    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)))
    graph:
     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#(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: 21/81
     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)))
     Arctic Interpretation Processor:
      dimension: 2
      usable rules:
       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)))
      interpretation:
       [b#](x0) = [-& 0 ]x0 + [0],
       
       [a#](x0) = [-& 0 ]x0 + [3],
       
                 [1]
       [c](x0) = [0],
       
                 [0  0 ]     [0]
       [b](x0) = [-& 0 ]x0 + [1],
       
                 [0  0 ]     [1]
       [a](x0) = [-& 0 ]x0 + [3],
       
                 [0  0 ]     [0]
       [d](x0) = [-& 2 ]x0 + [0]
      orientation:
       b#(d(b(x1))) = [-& 2 ]x1 + [3] >= [-& 2 ]x1 + [3] = a#(d(x1))
       
       a#(d(x1)) = [-& 2 ]x1 + [3] >= [-& 0 ]x1 + [0] = b#(x1)
       
       a#(x1) = [-& 0 ]x1 + [3] >= [-& 0 ]x1 + [0] = b#(x1)
       
       a#(x1) = [-& 0 ]x1 + [3] >= [-& 0 ]x1 + [1] = b#(b(x1))
       
       a#(x1) = [-& 0 ]x1 + [3] >= [-& 0 ]x1 + [1] = b#(b(b(x1)))
       
                  [0  2 ]     [1]    [0  0 ]     [1]           
       a(d(x1)) = [-& 2 ]x1 + [3] >= [-& 2 ]x1 + [3] = d(b(x1))
       
               [0  0 ]     [1]    [0  0 ]     [1]              
       a(x1) = [-& 0 ]x1 + [3] >= [-& 0 ]x1 + [1] = b(b(b(x1)))
       
               [0  0 ]     [0]           
       d(x1) = [-& 2 ]x1 + [0] >= x1 = x1
       
               [0  0 ]     [1]           
       a(x1) = [-& 0 ]x1 + [3] >= x1 = x1
       
                     [0  2 ]     [3]    [0  2 ]     [1]           
       b(d(b(x1))) = [-& 2 ]x1 + [3] >= [-& 2 ]x1 + [3] = a(d(x1))
       
                  [1]    [1]              
       b(c(x1)) = [1] >= [0] = c(d(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)))
      Restore Modifier:
       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)))
       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)))
        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
         
         usable rules:
          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)))
         interpretation:
          [b#](x0) = x0,
          
          [a#](x0) = x0 + 2,
          
          [c](x0) = 2,
          
          [b](x0) = x0 + 1,
          
          [a](x0) = x0 + 13/2,
          
          [d](x0) = 13/2x0 + 3/2
         orientation:
          b#(d(b(x1))) = 13/2x1 + 8 >= 13/2x1 + 7/2 = a#(d(x1))
          
          a#(x1) = x1 + 2 >= x1 = b#(x1)
          
          a#(x1) = x1 + 2 >= x1 + 1 = b#(b(x1))
          
          a#(x1) = x1 + 2 >= x1 + 2 = b#(b(b(x1)))
          
          a(d(x1)) = 13/2x1 + 8 >= 13/2x1 + 8 = d(b(x1))
          
          a(x1) = x1 + 13/2 >= x1 + 3 = b(b(b(x1)))
          
          d(x1) = 13/2x1 + 3/2 >= x1 = x1
          
          a(x1) = x1 + 13/2 >= x1 = x1
          
          b(d(b(x1))) = 13/2x1 + 9 >= 13/2x1 + 8 = a(d(x1))
          
          b(c(x1)) = 3 >= 2 = c(d(d(x1)))
         problem:
          DPs:
           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)))
         Restore Modifier:
          DPs:
           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)))
          EDG Processor:
           DPs:
            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)))
           graph:
            
           Qed