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: 1
      interpretation:
       [b#](x0) = x0 + -16,
       
       [a#](x0) = x0 + -14,
       
       [c](x0) = 8,
       
       [b](x0) = x0 + -14,
       
       [a](x0) = x0 + 0,
       
       [d](x0) = 14x0 + -14
      orientation:
       b#(d(b(x1))) = 14x1 + 0 >= 14x1 + -14 = a#(d(x1))
       
       a#(d(x1)) = 14x1 + -14 >= x1 + -16 = b#(x1)
       
       a#(x1) = x1 + -14 >= x1 + -16 = b#(x1)
       
       a#(x1) = x1 + -14 >= x1 + -14 = b#(b(x1))
       
       a#(x1) = x1 + -14 >= x1 + -14 = b#(b(b(x1)))
       
       a(d(x1)) = 14x1 + 0 >= 14x1 + 0 = d(b(x1))
       
       a(x1) = x1 + 0 >= x1 + -14 = b(b(b(x1)))
       
       d(x1) = 14x1 + -14 >= x1 = x1
       
       a(x1) = x1 + 0 >= x1 = x1
       
       b(d(b(x1))) = 14x1 + 0 >= 14x1 + 0 = a(d(x1))
       
       b(c(x1)) = 8 >= 8 = 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)))
      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
        
        interpretation:
         [b#](x0) = 8x0,
         
         [a#](x0) = 8x0 + 16,
         
         [c](x0) = 16,
         
         [b](x0) = x0 + 1,
         
         [a](x0) = x0 + 3,
         
         [d](x0) = 2x0
        orientation:
         b#(d(b(x1))) = 16x1 + 16 >= 16x1 + 16 = a#(d(x1))
         
         a#(x1) = 8x1 + 16 >= 8x1 = b#(x1)
         
         a#(x1) = 8x1 + 16 >= 8x1 + 8 = b#(b(x1))
         
         a#(x1) = 8x1 + 16 >= 8x1 + 16 = b#(b(b(x1)))
         
         a(d(x1)) = 2x1 + 3 >= 2x1 + 2 = d(b(x1))
         
         a(x1) = x1 + 3 >= x1 + 3 = b(b(b(x1)))
         
         d(x1) = 2x1 >= x1 = x1
         
         a(x1) = x1 + 3 >= x1 = x1
         
         b(d(b(x1))) = 2x1 + 3 >= 2x1 + 3 = a(d(x1))
         
         b(c(x1)) = 17 >= 16 = c(d(d(x1)))
        problem:
         DPs:
          b#(d(b(x1))) -> a#(d(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#(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#(b(b(x1)))
          a#(x1) -> b#(b(b(x1))) -> b#(d(b(x1))) -> a#(d(x1))
         Matrix Interpretation Processor: dim=3
          
          interpretation:
           [b#](x0) = [0 0 1]x0,
           
           [a#](x0) = [0 1 0]x0 + [1],
           
                     [1]
           [c](x0) = [1]
                     [1],
           
                     [0 1 0]     [0]
           [b](x0) = [0 0 1]x0 + [0]
                     [1 0 0]     [1],
           
                          [1]
           [a](x0) = x0 + [1]
                          [1],
           
                     [1 1 1]     [0]
           [d](x0) = [1 1 1]x0 + [0]
                     [1 1 1]     [1]
          orientation:
           b#(d(b(x1))) = [1 1 1]x1 + [2] >= [1 1 1]x1 + [1] = a#(d(x1))
           
           a#(x1) = [0 1 0]x1 + [1] >= [0 1 0]x1 + [1] = b#(b(b(x1)))
           
                      [1 1 1]     [1]    [1 1 1]     [1]           
           a(d(x1)) = [1 1 1]x1 + [1] >= [1 1 1]x1 + [1] = d(b(x1))
                      [1 1 1]     [2]    [1 1 1]     [2]           
           
                        [1]         [1]              
           a(x1) = x1 + [1] >= x1 + [1] = b(b(b(x1)))
                        [1]         [1]              
           
                   [1 1 1]     [0]           
           d(x1) = [1 1 1]x1 + [0] >= x1 = x1
                   [1 1 1]     [1]           
           
                        [1]           
           a(x1) = x1 + [1] >= x1 = x1
                        [1]           
           
                         [1 1 1]     [1]    [1 1 1]     [1]           
           b(d(b(x1))) = [1 1 1]x1 + [2] >= [1 1 1]x1 + [1] = a(d(x1))
                         [1 1 1]     [2]    [1 1 1]     [2]           
           
                      [1]    [1]              
           b(c(x1)) = [1] >= [1] = c(d(d(x1)))
                      [2]    [1]              
          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)))
          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