YES

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

Proof:
 DP Processor:
  DPs:
   a#(x1) -> c#(x1)
   c#(b(b(b(x1)))) -> a#(x1)
   c#(b(b(b(x1)))) -> a#(a(x1))
  TRS:
   a(x1) -> x1
   a(x1) -> b(c(x1))
   c(b(b(b(x1)))) -> b(b(a(a(x1))))
  TDG Processor:
   DPs:
    a#(x1) -> c#(x1)
    c#(b(b(b(x1)))) -> a#(x1)
    c#(b(b(b(x1)))) -> a#(a(x1))
   TRS:
    a(x1) -> x1
    a(x1) -> b(c(x1))
    c(b(b(b(x1)))) -> b(b(a(a(x1))))
   graph:
    c#(b(b(b(x1)))) -> a#(a(x1)) -> a#(x1) -> c#(x1)
    c#(b(b(b(x1)))) -> a#(x1) -> a#(x1) -> c#(x1)
    a#(x1) -> c#(x1) -> c#(b(b(b(x1)))) -> a#(a(x1))
    a#(x1) -> c#(x1) -> c#(b(b(b(x1)))) -> a#(x1)
   Arctic Interpretation Processor:
    dimension: 4
    usable rules:
     a(x1) -> x1
     a(x1) -> b(c(x1))
     c(b(b(b(x1)))) -> b(b(a(a(x1))))
    interpretation:
     [c#](x0) = [0  0  0  -&]x0,
     
     [a#](x0) = [0 0 0 0]x0 + [0],
     
               [0  0  -& 0 ]     [-&]
               [0  0  -& 0 ]     [0 ]
     [b](x0) = [1  0  0  0 ]x0 + [1 ]
               [0  0  0  0 ]     [0 ],
     
               [0  -& -& -&]     [0 ]
               [0  -& -& -&]     [-&]
     [c](x0) = [-& -& -& 0 ]x0 + [-&]
               [0  -& -& -&]     [0 ],
     
               [0  -& -& -&]     [0]
               [0  0  -& -&]     [0]
     [a](x0) = [1  0  0  0 ]x0 + [1]
               [0  0  -& 0 ]     [0]
    orientation:
     a#(x1) = [0 0 0 0]x1 + [0] >= [0  0  0  -&]x1 = c#(x1)
     
     c#(b(b(b(x1)))) = [1 1 1 1]x1 + [1] >= [0 0 0 0]x1 + [0] = a#(x1)
     
     c#(b(b(b(x1)))) = [1 1 1 1]x1 + [1] >= [1 0 0 0]x1 + [1] = a#(a(x1))
     
             [0  -& -& -&]     [0]           
             [0  0  -& -&]     [0]           
     a(x1) = [1  0  0  0 ]x1 + [1] >= x1 = x1
             [0  0  -& 0 ]     [0]           
     
             [0  -& -& -&]     [0]    [0  -& -& -&]     [0]           
             [0  0  -& -&]     [0]    [0  -& -& -&]     [0]           
     a(x1) = [1  0  0  0 ]x1 + [1] >= [1  -& -& 0 ]x1 + [1] = b(c(x1))
             [0  0  -& 0 ]     [0]    [0  -& -& 0 ]     [0]           
     
                      [1 0 0 0]     [1]    [1 0 0 0]     [1]                 
                      [1 0 0 0]     [1]    [1 0 0 0]     [1]                 
     c(b(b(b(x1)))) = [1 1 0 1]x1 + [1] >= [1 1 0 1]x1 + [1] = b(b(a(a(x1))))
                      [1 0 0 0]     [1]    [1 0 0 0]     [1]                 
    problem:
     DPs:
      a#(x1) -> c#(x1)
      c#(b(b(b(x1)))) -> a#(a(x1))
     TRS:
      a(x1) -> x1
      a(x1) -> b(c(x1))
      c(b(b(b(x1)))) -> b(b(a(a(x1))))
    Restore Modifier:
     DPs:
      a#(x1) -> c#(x1)
      c#(b(b(b(x1)))) -> a#(a(x1))
     TRS:
      a(x1) -> x1
      a(x1) -> b(c(x1))
      c(b(b(b(x1)))) -> b(b(a(a(x1))))
     EDG Processor:
      DPs:
       a#(x1) -> c#(x1)
       c#(b(b(b(x1)))) -> a#(a(x1))
      TRS:
       a(x1) -> x1
       a(x1) -> b(c(x1))
       c(b(b(b(x1)))) -> b(b(a(a(x1))))
      graph:
       c#(b(b(b(x1)))) -> a#(a(x1)) -> a#(x1) -> c#(x1)
       a#(x1) -> c#(x1) -> c#(b(b(b(x1)))) -> a#(a(x1))
      Arctic Interpretation Processor:
       dimension: 3
       usable rules:
        a(x1) -> x1
        a(x1) -> b(c(x1))
        c(b(b(b(x1)))) -> b(b(a(a(x1))))
       interpretation:
        [c#](x0) = [0 1 1]x0 + [0],
        
        [a#](x0) = [0 1 1]x0 + [0],
        
                  [-& 1  0 ]     [0 ]
        [b](x0) = [-& 0  0 ]x0 + [0 ]
                  [0  0  0 ]     [-&],
        
                  [-& 0  0 ]     [0 ]
        [c](x0) = [-& -& 0 ]x0 + [0 ]
                  [-& 0  0 ]     [-&],
        
                  [0  0  1 ]     [1]
        [a](x0) = [-& 0  0 ]x0 + [0]
                  [-& 0  0 ]     [0]
       orientation:
        a#(x1) = [0 1 1]x1 + [0] >= [0 1 1]x1 + [0] = c#(x1)
        
        c#(b(b(b(x1)))) = [1 2 2]x1 + [2] >= [0 1 1]x1 + [1] = a#(a(x1))
        
                [0  0  1 ]     [1]           
        a(x1) = [-& 0  0 ]x1 + [0] >= x1 = x1
                [-& 0  0 ]     [0]           
        
                [0  0  1 ]     [1]    [-& 0  1 ]     [1]           
        a(x1) = [-& 0  0 ]x1 + [0] >= [-& 0  0 ]x1 + [0] = b(c(x1))
                [-& 0  0 ]     [0]    [-& 0  0 ]     [0]           
        
                         [0 1 1]     [1]    [0 1 1]     [1]                 
        c(b(b(b(x1)))) = [0 1 1]x1 + [1] >= [0 1 1]x1 + [1] = b(b(a(a(x1))))
                         [0 1 1]     [1]    [0 1 1]     [1]                 
       problem:
        DPs:
         a#(x1) -> c#(x1)
        TRS:
         a(x1) -> x1
         a(x1) -> b(c(x1))
         c(b(b(b(x1)))) -> b(b(a(a(x1))))
       Restore Modifier:
        DPs:
         a#(x1) -> c#(x1)
        TRS:
         a(x1) -> x1
         a(x1) -> b(c(x1))
         c(b(b(b(x1)))) -> b(b(a(a(x1))))
        EDG Processor:
         DPs:
          a#(x1) -> c#(x1)
         TRS:
          a(x1) -> x1
          a(x1) -> b(c(x1))
          c(b(b(b(x1)))) -> b(b(a(a(x1))))
         graph:
          
         Qed