YES

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

Proof:
 DP Processor:
  DPs:
   a#(c(x1)) -> b#(c(c(x1)))
   b#(c(x1)) -> b#(x1)
   b#(c(x1)) -> a#(b(x1))
  TRS:
   a(b(x1)) -> x1
   a(c(x1)) -> c(b(c(c(x1))))
   b(c(x1)) -> a(b(x1))
  TDG Processor:
   DPs:
    a#(c(x1)) -> b#(c(c(x1)))
    b#(c(x1)) -> b#(x1)
    b#(c(x1)) -> a#(b(x1))
   TRS:
    a(b(x1)) -> x1
    a(c(x1)) -> c(b(c(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(c(x1)))
    a#(c(x1)) -> b#(c(c(x1))) -> b#(c(x1)) -> a#(b(x1))
    a#(c(x1)) -> b#(c(c(x1))) -> b#(c(x1)) -> b#(x1)
   Arctic Interpretation Processor:
    dimension: 1
    usable rules:
     a(b(x1)) -> x1
     a(c(x1)) -> c(b(c(c(x1))))
     b(c(x1)) -> a(b(x1))
    interpretation:
     [b#](x0) = -8x0 + 1,
     
     [a](x0) = 1x0 + 2,
     
     [a#](x0) = -6x0 + 1,
     
     [b](x0) = -1x0 + 1,
     
     [c](x0) = 1x0 + 8
    orientation:
     a#(c(x1)) = -5x1 + 2 >= -6x1 + 1 = b#(c(c(x1)))
     
     b#(c(x1)) = -7x1 + 1 >= -8x1 + 1 = b#(x1)
     
     b#(c(x1)) = -7x1 + 1 >= -7x1 + 1 = a#(b(x1))
     
     a(b(x1)) = x1 + 2 >= x1 = x1
     
     a(c(x1)) = 2x1 + 9 >= 2x1 + 9 = c(b(c(c(x1))))
     
     b(c(x1)) = x1 + 7 >= x1 + 2 = 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(b(c(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(b(c(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(b(c(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))
      SCC Processor:
       #sccs: 1
       #rules: 1
       #arcs: 2/4
       DPs:
        b#(c(x1)) -> b#(x1)
       TRS:
        a(b(x1)) -> x1
        a(c(x1)) -> c(b(c(c(x1))))
        b(c(x1)) -> a(b(x1))
       Usable Rule Processor:
        DPs:
         b#(c(x1)) -> b#(x1)
        TRS:
         
        Arctic Interpretation Processor:
         dimension: 4
         usable rules:
          
         interpretation:
          [b#](x0) = [0  -& -& -&]x0,
          
                    [1 0 0 0]     [0]
                    [0 0 0 0]     [0]
          [c](x0) = [1 0 0 0]x0 + [0]
                    [0 0 0 0]     [0]
         orientation:
          b#(c(x1)) = [1 0 0 0]x1 + [0] >= [0  -& -& -&]x1 = b#(x1)
         problem:
          DPs:
           
          TRS:
           
         Qed