YES

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

Proof:
 DP Processor:
  DPs:
   a#(c(x1)) -> a#(x1)
   a#(c(x1)) -> b#(c(c(a(x1))))
   b#(c(x1)) -> b#(x1)
   b#(c(x1)) -> a#(b(x1))
  TRS:
   a(b(x1)) -> x1
   a(c(x1)) -> b(c(c(a(x1))))
   b(c(x1)) -> a(b(x1))
  TDG Processor:
   DPs:
    a#(c(x1)) -> a#(x1)
    a#(c(x1)) -> b#(c(c(a(x1))))
    b#(c(x1)) -> b#(x1)
    b#(c(x1)) -> a#(b(x1))
   TRS:
    a(b(x1)) -> x1
    a(c(x1)) -> b(c(c(a(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(c(a(x1))))
    b#(c(x1)) -> a#(b(x1)) -> a#(c(x1)) -> a#(x1)
    a#(c(x1)) -> b#(c(c(a(x1)))) -> b#(c(x1)) -> a#(b(x1))
    a#(c(x1)) -> b#(c(c(a(x1)))) -> b#(c(x1)) -> b#(x1)
    a#(c(x1)) -> a#(x1) -> a#(c(x1)) -> b#(c(c(a(x1))))
    a#(c(x1)) -> a#(x1) -> a#(c(x1)) -> a#(x1)
   Arctic Interpretation Processor:
    dimension: 1
    interpretation:
     [b#](x0) = 4x0 + 0,
     
     [a#](x0) = 12x0 + 0,
     
     [c](x0) = 4x0 + 8,
     
     [a](x0) = 4x0 + -16,
     
     [b](x0) = -4x0 + 0
    orientation:
     a#(c(x1)) = 16x1 + 20 >= 12x1 + 0 = a#(x1)
     
     a#(c(x1)) = 16x1 + 20 >= 16x1 + 16 = b#(c(c(a(x1))))
     
     b#(c(x1)) = 8x1 + 12 >= 4x1 + 0 = b#(x1)
     
     b#(c(x1)) = 8x1 + 12 >= 8x1 + 12 = a#(b(x1))
     
     a(b(x1)) = x1 + 4 >= x1 = x1
     
     a(c(x1)) = 8x1 + 12 >= 8x1 + 8 = b(c(c(a(x1))))
     
     b(c(x1)) = x1 + 4 >= x1 + 4 = a(b(x1))
    problem:
     DPs:
      a#(c(x1)) -> b#(c(c(a(x1))))
      b#(c(x1)) -> a#(b(x1))
     TRS:
      a(b(x1)) -> x1
      a(c(x1)) -> b(c(c(a(x1))))
      b(c(x1)) -> a(b(x1))
    EDG Processor:
     DPs:
      a#(c(x1)) -> b#(c(c(a(x1))))
      b#(c(x1)) -> a#(b(x1))
     TRS:
      a(b(x1)) -> x1
      a(c(x1)) -> b(c(c(a(x1))))
      b(c(x1)) -> a(b(x1))
     graph:
      b#(c(x1)) -> a#(b(x1)) -> a#(c(x1)) -> b#(c(c(a(x1))))
      a#(c(x1)) -> b#(c(c(a(x1)))) -> b#(c(x1)) -> a#(b(x1))
     Matrix Interpretation Processor: dim=1
      
      interpretation:
       [b#](x0) = x0,
       
       [a#](x0) = 4x0 + 1/2,
       
       [c](x0) = 2x0 + 4,
       
       [a](x0) = 2x0 + 1,
       
       [b](x0) = 1/2x0
      orientation:
       a#(c(x1)) = 8x1 + 33/2 >= 8x1 + 16 = b#(c(c(a(x1))))
       
       b#(c(x1)) = 2x1 + 4 >= 2x1 + 1/2 = a#(b(x1))
       
       a(b(x1)) = x1 + 1 >= x1 = x1
       
       a(c(x1)) = 4x1 + 9 >= 4x1 + 8 = b(c(c(a(x1))))
       
       b(c(x1)) = x1 + 2 >= x1 + 1 = a(b(x1))
      problem:
       DPs:
        
       TRS:
        a(b(x1)) -> x1
        a(c(x1)) -> b(c(c(a(x1))))
        b(c(x1)) -> a(b(x1))
      Qed