YES

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

Proof:
 DP Processor:
  DPs:
   b#(b(b(x1))) -> a#(b(x1))
   a#(a(x1)) -> b#(a(x1))
   a#(a(x1)) -> a#(b(a(x1)))
   a#(a(a(x1))) -> b#(x1)
   a#(a(a(x1))) -> b#(b(x1))
   a#(a(a(x1))) -> a#(b(b(x1)))
  TRS:
   b(b(b(x1))) -> a(b(x1))
   a(a(x1)) -> a(b(a(x1)))
   a(a(a(x1))) -> a(b(b(x1)))
  Arctic Interpretation Processor:
   dimension: 2
   interpretation:
    [a#](x0) = [2 0]x0 + [0],
    
    [b#](x0) = [-& 3 ]x0 + [0],
    
              [2  0 ]     [1 ]
    [a](x0) = [-& -&]x0 + [-&],
    
              [-& 2 ]     [0]
    [b](x0) = [1  -1]x0 + [0]
   orientation:
    b#(b(b(x1))) = [3 6]x1 + [4] >= [1 4]x1 + [2] = a#(b(x1))
    
    a#(a(x1)) = [4 2]x1 + [3] >= [0] = b#(a(x1))
    
    a#(a(x1)) = [4 2]x1 + [3] >= [3 1]x1 + [2] = a#(b(a(x1)))
    
    a#(a(a(x1))) = [6 4]x1 + [5] >= [-& 3 ]x1 + [0] = b#(x1)
    
    a#(a(a(x1))) = [6 4]x1 + [5] >= [4 2]x1 + [3] = b#(b(x1))
    
    a#(a(a(x1))) = [6 4]x1 + [5] >= [5 3]x1 + [4] = a#(b(b(x1)))
    
                  [2 5]     [3]    [1  4 ]     [2 ]           
    b(b(b(x1))) = [4 2]x1 + [3] >= [-& -&]x1 + [-&] = a(b(x1))
    
               [4  2 ]     [3 ]    [3  1 ]     [2 ]              
    a(a(x1)) = [-& -&]x1 + [-&] >= [-& -&]x1 + [-&] = a(b(a(x1)))
    
                  [6  4 ]     [5 ]    [5  3 ]     [4 ]              
    a(a(a(x1))) = [-& -&]x1 + [-&] >= [-& -&]x1 + [-&] = a(b(b(x1)))
   problem:
    DPs:
     
    TRS:
     b(b(b(x1))) -> a(b(x1))
     a(a(x1)) -> a(b(a(x1)))
     a(a(a(x1))) -> a(b(b(x1)))
   Qed