YES

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

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