YES

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

Proof:
 DP Processor:
  DPs:
   a#(c(x1)) -> b#(c(x1))
   b#(c(x1)) -> b#(x1)
   b#(c(x1)) -> a#(b(x1))
  TRS:
   a(b(x1)) -> x1
   a(c(x1)) -> c(c(b(c(x1))))
   b(c(x1)) -> a(b(x1))
  TDG Processor:
   DPs:
    a#(c(x1)) -> b#(c(x1))
    b#(c(x1)) -> b#(x1)
    b#(c(x1)) -> a#(b(x1))
   TRS:
    a(b(x1)) -> x1
    a(c(x1)) -> c(c(b(c(x1))))
    b(c(x1)) -> a(b(x1))
   graph:
    b#(c(x1)) -> b#(x1) -> b#(c(x1)) -> a#(b(x1))
    b#(c(x1)) -> b#(x1) -> b#(c(x1)) -> b#(x1)
    b#(c(x1)) -> a#(b(x1)) -> a#(c(x1)) -> b#(c(x1))
    a#(c(x1)) -> b#(c(x1)) -> b#(c(x1)) -> a#(b(x1))
    a#(c(x1)) -> b#(c(x1)) -> b#(c(x1)) -> b#(x1)
   Arctic Interpretation Processor:
    dimension: 1
    usable rules:
     a(b(x1)) -> x1
     a(c(x1)) -> c(c(b(c(x1))))
     b(c(x1)) -> a(b(x1))
    interpretation:
     [b#](x0) = -8x0 + 7,
     
     [a#](x0) = -6x0 + 7,
     
     [c](x0) = 1x0 + 14,
     
     [a](x0) = 1x0 + 0,
     
     [b](x0) = -1x0 + 2
    orientation:
     a#(c(x1)) = -5x1 + 8 >= -7x1 + 7 = b#(c(x1))
     
     b#(c(x1)) = -7x1 + 7 >= -8x1 + 7 = b#(x1)
     
     b#(c(x1)) = -7x1 + 7 >= -7x1 + 7 = a#(b(x1))
     
     a(b(x1)) = x1 + 3 >= x1 = x1
     
     a(c(x1)) = 2x1 + 15 >= 2x1 + 15 = c(c(b(c(x1))))
     
     b(c(x1)) = x1 + 13 >= x1 + 3 = a(b(x1))
    problem:
     DPs:
      b#(c(x1)) -> b#(x1)
      b#(c(x1)) -> a#(b(x1))
     TRS:
      a(b(x1)) -> x1
      a(c(x1)) -> c(c(b(c(x1))))
      b(c(x1)) -> a(b(x1))
    Restore Modifier:
     DPs:
      b#(c(x1)) -> b#(x1)
      b#(c(x1)) -> a#(b(x1))
     TRS:
      a(b(x1)) -> x1
      a(c(x1)) -> c(c(b(c(x1))))
      b(c(x1)) -> a(b(x1))
     EDG Processor:
      DPs:
       b#(c(x1)) -> b#(x1)
       b#(c(x1)) -> a#(b(x1))
      TRS:
       a(b(x1)) -> x1
       a(c(x1)) -> c(c(b(c(x1))))
       b(c(x1)) -> a(b(x1))
      graph:
       b#(c(x1)) -> b#(x1) -> b#(c(x1)) -> b#(x1)
       b#(c(x1)) -> b#(x1) -> b#(c(x1)) -> a#(b(x1))
      CDG Processor:
       DPs:
        b#(c(x1)) -> b#(x1)
        b#(c(x1)) -> a#(b(x1))
       TRS:
        a(b(x1)) -> x1
        a(c(x1)) -> c(c(b(c(x1))))
        b(c(x1)) -> a(b(x1))
       graph:
        
       Qed