NO

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

Proof:
 DP Processor:
  DPs:
   a#(a(a(x1))) -> b#(x1)
   a#(a(a(x1))) -> a#(b(x1))
   b#(c(x1)) -> a#(x1)
   b#(c(x1)) -> a#(a(x1))
   b#(c(x1)) -> a#(a(a(x1)))
  TRS:
   a(a(a(x1))) -> a(b(x1))
   b(c(x1)) -> c(c(a(a(a(x1)))))
  TDG Processor:
   DPs:
    a#(a(a(x1))) -> b#(x1)
    a#(a(a(x1))) -> a#(b(x1))
    b#(c(x1)) -> a#(x1)
    b#(c(x1)) -> a#(a(x1))
    b#(c(x1)) -> a#(a(a(x1)))
   TRS:
    a(a(a(x1))) -> a(b(x1))
    b(c(x1)) -> c(c(a(a(a(x1)))))
   graph:
    b#(c(x1)) -> a#(a(a(x1))) -> a#(a(a(x1))) -> a#(b(x1))
    b#(c(x1)) -> a#(a(a(x1))) -> a#(a(a(x1))) -> b#(x1)
    b#(c(x1)) -> a#(a(x1)) -> a#(a(a(x1))) -> a#(b(x1))
    b#(c(x1)) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(x1)
    b#(c(x1)) -> a#(x1) -> a#(a(a(x1))) -> a#(b(x1))
    b#(c(x1)) -> a#(x1) -> a#(a(a(x1))) -> b#(x1)
    a#(a(a(x1))) -> b#(x1) -> b#(c(x1)) -> a#(a(a(x1)))
    a#(a(a(x1))) -> b#(x1) -> b#(c(x1)) -> a#(a(x1))
    a#(a(a(x1))) -> b#(x1) -> b#(c(x1)) -> a#(x1)
    a#(a(a(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> a#(b(x1))
    a#(a(a(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> b#(x1)
   Arctic Interpretation Processor:
    dimension: 2
    usable rules:
     a(a(a(x1))) -> a(b(x1))
     b(c(x1)) -> c(c(a(a(a(x1)))))
    interpretation:
     [b#](x0) = [0 0]x0 + [0],
     
     [a#](x0) = [0  -&]x0 + [0],
     
               [-& -&]     [0]
     [c](x0) = [0  0 ]x0 + [2],
     
               [-& -&]     [0 ]
     [b](x0) = [0  0 ]x0 + [-&],
     
               [0 0]     [0]
     [a](x0) = [0 0]x0 + [2]
    orientation:
     a#(a(a(x1))) = [0 0]x1 + [2] >= [0 0]x1 + [0] = b#(x1)
     
     a#(a(a(x1))) = [0 0]x1 + [2] >= [0] = a#(b(x1))
     
     b#(c(x1)) = [0 0]x1 + [2] >= [0  -&]x1 + [0] = a#(x1)
     
     b#(c(x1)) = [0 0]x1 + [2] >= [0 0]x1 + [0] = a#(a(x1))
     
     b#(c(x1)) = [0 0]x1 + [2] >= [0 0]x1 + [2] = a#(a(a(x1)))
     
                   [0 0]     [2]    [0 0]     [0]           
     a(a(a(x1))) = [0 0]x1 + [2] >= [0 0]x1 + [2] = a(b(x1))
     
                [-& -&]     [0]    [-& -&]     [0]                    
     b(c(x1)) = [0  0 ]x1 + [2] >= [0  0 ]x1 + [2] = c(c(a(a(a(x1)))))
    problem:
     DPs:
      a#(a(a(x1))) -> b#(x1)
      b#(c(x1)) -> a#(x1)
      b#(c(x1)) -> a#(a(x1))
      b#(c(x1)) -> a#(a(a(x1)))
     TRS:
      a(a(a(x1))) -> a(b(x1))
      b(c(x1)) -> c(c(a(a(a(x1)))))
    Restore Modifier:
     DPs:
      a#(a(a(x1))) -> b#(x1)
      b#(c(x1)) -> a#(x1)
      b#(c(x1)) -> a#(a(x1))
      b#(c(x1)) -> a#(a(a(x1)))
     TRS:
      a(a(a(x1))) -> a(b(x1))
      b(c(x1)) -> c(c(a(a(a(x1)))))
     EDG Processor:
      DPs:
       a#(a(a(x1))) -> b#(x1)
       b#(c(x1)) -> a#(x1)
       b#(c(x1)) -> a#(a(x1))
       b#(c(x1)) -> a#(a(a(x1)))
      TRS:
       a(a(a(x1))) -> a(b(x1))
       b(c(x1)) -> c(c(a(a(a(x1)))))
      graph:
       b#(c(x1)) -> a#(a(a(x1))) -> a#(a(a(x1))) -> b#(x1)
       b#(c(x1)) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(x1)
       b#(c(x1)) -> a#(x1) -> a#(a(a(x1))) -> b#(x1)
       a#(a(a(x1))) -> b#(x1) -> b#(c(x1)) -> a#(x1)
       a#(a(a(x1))) -> b#(x1) -> b#(c(x1)) -> a#(a(x1))
       a#(a(a(x1))) -> b#(x1) -> b#(c(x1)) -> a#(a(a(x1)))
      Loop Processor:
       loop length: 17
       terms:
        a(a(a(b(c(a(c(c(a(a(x1))))))))))
        a(b(b(c(a(c(c(a(a(x1)))))))))
        a(b(c(c(a(a(a(a(c(c(a(a(x1))))))))))))
        a(c(c(a(a(a(c(a(a(a(a(c(c(a(a(x1)))))))))))))))
        a(c(c(a(b(c(a(a(a(a(c(c(a(a(x1))))))))))))))
        a(c(c(a(b(c(a(a(b(c(c(a(a(x1)))))))))))))
        a(c(c(a(b(c(a(a(c(c(a(a(a(c(a(a(x1))))))))))))))))
        a(c(c(a(b(c(a(a(c(c(a(b(c(a(a(x1)))))))))))))))
        a(c(c(a(b(c(a(a(c(c(a(c(c(a(a(a(a(a(x1))))))))))))))))))
        a(c(c(a(b(c(a(a(c(c(a(c(c(a(a(b(a(x1)))))))))))))))))
        a(c(c(a(c(c(a(a(a(a(a(c(c(a(c(c(a(a(b(a(x1))))))))))))))))))))
        a(c(c(a(c(c(a(a(a(b(c(c(a(c(c(a(a(b(a(x1)))))))))))))))))))
        a(c(c(a(c(c(a(a(a(c(c(a(a(a(c(a(c(c(a(a(b(a(x1))))))))))))))))))))))
        a(c(c(a(c(c(a(b(c(c(a(a(a(c(a(c(c(a(a(b(a(x1)))))))))))))))))))))
        a(c(c(a(c(c(a(b(c(c(a(b(c(a(c(c(a(a(b(a(x1))))))))))))))))))))
        a(c(c(a(c(c(a(c(c(a(a(a(c(a(b(c(a(c(c(a(a(b(a(x1)))))))))))))))))))))))
        a(c(c(a(c(c(a(c(c(a(b(c(a(b(c(a(c(c(a(a(b(a(x1))))))))))))))))))))))
       context: a(c(c(a(c(c(a(c(c(a(c(c(a([])))))))))))))
       substitution:
        x1 -> b(a(x1))
       Qed