YES

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

Proof:
 DP Processor:
  DPs:
   a#(b(b(x1))) -> a#(x1)
   a#(b(b(x1))) -> a#(a(x1))
  TRS:
   a(x1) -> x1
   a(x1) -> b(c(b(x1)))
   a(b(b(x1))) -> b(b(a(a(x1))))
  Arctic Interpretation Processor:
   dimension: 2
   interpretation:
    [a#](x0) = [2 0]x0 + [3],
    
              [-& 0 ]     [0 ]
    [c](x0) = [-& -&]x0 + [-&],
    
              [-& 3 ]     [0]
    [b](x0) = [0  -&]x0 + [0],
    
              [0  -&]     [0]
    [a](x0) = [1  0 ]x0 + [0]
   orientation:
    a#(b(b(x1))) = [5 3]x1 + [5] >= [2 0]x1 + [3] = a#(x1)
    
    a#(b(b(x1))) = [5 3]x1 + [5] >= [2 0]x1 + [3] = a#(a(x1))
    
            [0  -&]     [0]           
    a(x1) = [1  0 ]x1 + [0] >= x1 = x1
    
            [0  -&]     [0]    [-& -&]     [0]              
    a(x1) = [1  0 ]x1 + [0] >= [0  -&]x1 + [0] = b(c(b(x1)))
    
                  [3  -&]     [3]    [3  -&]     [3]                 
    a(b(b(x1))) = [4  3 ]x1 + [4] >= [4  3 ]x1 + [4] = b(b(a(a(x1))))
   problem:
    DPs:
     
    TRS:
     a(x1) -> x1
     a(x1) -> b(c(b(x1)))
     a(b(b(x1))) -> b(b(a(a(x1))))
   Qed