YES

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

Proof:
 DP Processor:
  DPs:
   a#(x1) -> b#(x1)
   a#(x1) -> b#(b(x1))
   a#(b(b(x1))) -> a#(x1)
   a#(b(b(x1))) -> c#(a(x1))
   a#(b(b(x1))) -> c#(c(a(x1)))
   a#(b(b(x1))) -> c#(c(c(a(x1))))
   a#(b(b(x1))) -> b#(c(c(c(a(x1)))))
   a#(b(b(x1))) -> b#(b(c(c(c(a(x1))))))
   b#(b(x1)) -> c#(x1)
   b#(b(x1)) -> c#(c(x1))
   b#(b(x1)) -> c#(c(c(x1)))
   c#(c(c(b(b(x1))))) -> a#(x1)
  TRS:
   a(x1) -> b(b(x1))
   a(b(b(x1))) -> b(b(c(c(c(a(x1))))))
   b(b(x1)) -> c(c(c(x1)))
   c(c(c(b(b(x1))))) -> a(x1)
  TDG Processor:
   DPs:
    a#(x1) -> b#(x1)
    a#(x1) -> b#(b(x1))
    a#(b(b(x1))) -> a#(x1)
    a#(b(b(x1))) -> c#(a(x1))
    a#(b(b(x1))) -> c#(c(a(x1)))
    a#(b(b(x1))) -> c#(c(c(a(x1))))
    a#(b(b(x1))) -> b#(c(c(c(a(x1)))))
    a#(b(b(x1))) -> b#(b(c(c(c(a(x1))))))
    b#(b(x1)) -> c#(x1)
    b#(b(x1)) -> c#(c(x1))
    b#(b(x1)) -> c#(c(c(x1)))
    c#(c(c(b(b(x1))))) -> a#(x1)
   TRS:
    a(x1) -> b(b(x1))
    a(b(b(x1))) -> b(b(c(c(c(a(x1))))))
    b(b(x1)) -> c(c(c(x1)))
    c(c(c(b(b(x1))))) -> a(x1)
   graph:
    c#(c(c(b(b(x1))))) -> a#(x1) ->
    a#(b(b(x1))) -> b#(b(c(c(c(a(x1))))))
    c#(c(c(b(b(x1))))) -> a#(x1) -> a#(b(b(x1))) -> b#(c(c(c(a(x1)))))
    c#(c(c(b(b(x1))))) -> a#(x1) -> a#(b(b(x1))) -> c#(c(c(a(x1))))
    c#(c(c(b(b(x1))))) -> a#(x1) -> a#(b(b(x1))) -> c#(c(a(x1)))
    c#(c(c(b(b(x1))))) -> a#(x1) -> a#(b(b(x1))) -> c#(a(x1))
    c#(c(c(b(b(x1))))) -> a#(x1) -> a#(b(b(x1))) -> a#(x1)
    c#(c(c(b(b(x1))))) -> a#(x1) -> a#(x1) -> b#(b(x1))
    c#(c(c(b(b(x1))))) -> a#(x1) -> a#(x1) -> b#(x1)
    b#(b(x1)) -> c#(c(c(x1))) -> c#(c(c(b(b(x1))))) -> a#(x1)
    b#(b(x1)) -> c#(c(x1)) -> c#(c(c(b(b(x1))))) -> a#(x1)
    b#(b(x1)) -> c#(x1) -> c#(c(c(b(b(x1))))) -> a#(x1)
    a#(b(b(x1))) -> c#(c(c(a(x1)))) -> c#(c(c(b(b(x1))))) -> a#(x1)
    a#(b(b(x1))) -> c#(c(a(x1))) -> c#(c(c(b(b(x1))))) -> a#(x1)
    a#(b(b(x1))) -> c#(a(x1)) -> c#(c(c(b(b(x1))))) -> a#(x1)
    a#(b(b(x1))) -> b#(c(c(c(a(x1))))) ->
    b#(b(x1)) -> c#(c(c(x1)))
    a#(b(b(x1))) -> b#(c(c(c(a(x1))))) -> b#(b(x1)) -> c#(c(x1))
    a#(b(b(x1))) -> b#(c(c(c(a(x1))))) -> b#(b(x1)) -> c#(x1)
    a#(b(b(x1))) -> b#(b(c(c(c(a(x1)))))) ->
    b#(b(x1)) -> c#(c(c(x1)))
    a#(b(b(x1))) -> b#(b(c(c(c(a(x1)))))) ->
    b#(b(x1)) -> c#(c(x1))
    a#(b(b(x1))) -> b#(b(c(c(c(a(x1)))))) -> b#(b(x1)) -> c#(x1)
    a#(b(b(x1))) -> a#(x1) -> a#(b(b(x1))) -> b#(b(c(c(c(a(x1))))))
    a#(b(b(x1))) -> a#(x1) -> a#(b(b(x1))) -> b#(c(c(c(a(x1)))))
    a#(b(b(x1))) -> a#(x1) -> a#(b(b(x1))) -> c#(c(c(a(x1))))
    a#(b(b(x1))) -> a#(x1) -> a#(b(b(x1))) -> c#(c(a(x1)))
    a#(b(b(x1))) -> a#(x1) -> a#(b(b(x1))) -> c#(a(x1))
    a#(b(b(x1))) -> a#(x1) -> a#(b(b(x1))) -> a#(x1)
    a#(b(b(x1))) -> a#(x1) -> a#(x1) -> b#(b(x1))
    a#(b(b(x1))) -> a#(x1) -> a#(x1) -> b#(x1)
    a#(x1) -> b#(b(x1)) -> b#(b(x1)) -> c#(c(c(x1)))
    a#(x1) -> b#(b(x1)) -> b#(b(x1)) -> c#(c(x1))
    a#(x1) -> b#(b(x1)) -> b#(b(x1)) -> c#(x1)
    a#(x1) -> b#(x1) -> b#(b(x1)) -> c#(c(c(x1)))
    a#(x1) -> b#(x1) -> b#(b(x1)) -> c#(c(x1))
    a#(x1) -> b#(x1) -> b#(b(x1)) -> c#(x1)
   Arctic Interpretation Processor:
    dimension: 1
    interpretation:
     [c#](x0) = x0 + 0,
     
     [b#](x0) = x0 + 0,
     
     [a#](x0) = 4x0 + 2,
     
     [c](x0) = x0 + 0,
     
     [b](x0) = 4x0 + 0,
     
     [a](x0) = 8x0 + 4
    orientation:
     a#(x1) = 4x1 + 2 >= x1 + 0 = b#(x1)
     
     a#(x1) = 4x1 + 2 >= 4x1 + 0 = b#(b(x1))
     
     a#(b(b(x1))) = 12x1 + 8 >= 4x1 + 2 = a#(x1)
     
     a#(b(b(x1))) = 12x1 + 8 >= 8x1 + 4 = c#(a(x1))
     
     a#(b(b(x1))) = 12x1 + 8 >= 8x1 + 4 = c#(c(a(x1)))
     
     a#(b(b(x1))) = 12x1 + 8 >= 8x1 + 4 = c#(c(c(a(x1))))
     
     a#(b(b(x1))) = 12x1 + 8 >= 8x1 + 4 = b#(c(c(c(a(x1)))))
     
     a#(b(b(x1))) = 12x1 + 8 >= 12x1 + 8 = b#(b(c(c(c(a(x1))))))
     
     b#(b(x1)) = 4x1 + 0 >= x1 + 0 = c#(x1)
     
     b#(b(x1)) = 4x1 + 0 >= x1 + 0 = c#(c(x1))
     
     b#(b(x1)) = 4x1 + 0 >= x1 + 0 = c#(c(c(x1)))
     
     c#(c(c(b(b(x1))))) = 8x1 + 4 >= 4x1 + 2 = a#(x1)
     
     a(x1) = 8x1 + 4 >= 8x1 + 4 = b(b(x1))
     
     a(b(b(x1))) = 16x1 + 12 >= 16x1 + 12 = b(b(c(c(c(a(x1))))))
     
     b(b(x1)) = 8x1 + 4 >= x1 + 0 = c(c(c(x1)))
     
     c(c(c(b(b(x1))))) = 8x1 + 4 >= 8x1 + 4 = a(x1)
    problem:
     DPs:
      a#(x1) -> b#(b(x1))
      a#(b(b(x1))) -> b#(b(c(c(c(a(x1))))))
      b#(b(x1)) -> c#(x1)
      b#(b(x1)) -> c#(c(x1))
      b#(b(x1)) -> c#(c(c(x1)))
     TRS:
      a(x1) -> b(b(x1))
      a(b(b(x1))) -> b(b(c(c(c(a(x1))))))
      b(b(x1)) -> c(c(c(x1)))
      c(c(c(b(b(x1))))) -> a(x1)
    EDG Processor:
     DPs:
      a#(x1) -> b#(b(x1))
      a#(b(b(x1))) -> b#(b(c(c(c(a(x1))))))
      b#(b(x1)) -> c#(x1)
      b#(b(x1)) -> c#(c(x1))
      b#(b(x1)) -> c#(c(c(x1)))
     TRS:
      a(x1) -> b(b(x1))
      a(b(b(x1))) -> b(b(c(c(c(a(x1))))))
      b(b(x1)) -> c(c(c(x1)))
      c(c(c(b(b(x1))))) -> a(x1)
     graph:
      a#(b(b(x1))) -> b#(b(c(c(c(a(x1)))))) ->
      b#(b(x1)) -> c#(x1)
      a#(b(b(x1))) -> b#(b(c(c(c(a(x1)))))) ->
      b#(b(x1)) -> c#(c(x1))
      a#(b(b(x1))) -> b#(b(c(c(c(a(x1)))))) -> b#(b(x1)) -> c#(c(c(x1)))
      a#(x1) -> b#(b(x1)) -> b#(b(x1)) -> c#(x1)
      a#(x1) -> b#(b(x1)) -> b#(b(x1)) -> c#(c(x1))
      a#(x1) -> b#(b(x1)) -> b#(b(x1)) -> c#(c(c(x1)))
     SCC Processor:
      #sccs: 0
      #rules: 0
      #arcs: 6/25