YES

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

Proof:
 DP Processor:
  DPs:
   a#(x1) -> c#(x1)
   c#(c(a(x1))) -> c#(x1)
   c#(c(a(x1))) -> c#(c(x1))
   c#(c(a(x1))) -> a#(c(c(x1)))
   c#(c(a(x1))) -> a#(a(c(c(x1))))
  TRS:
   a(x1) -> x1
   a(x1) -> b(b(c(x1)))
   c(c(a(x1))) -> a(a(c(c(x1))))
  TDG Processor:
   DPs:
    a#(x1) -> c#(x1)
    c#(c(a(x1))) -> c#(x1)
    c#(c(a(x1))) -> c#(c(x1))
    c#(c(a(x1))) -> a#(c(c(x1)))
    c#(c(a(x1))) -> a#(a(c(c(x1))))
   TRS:
    a(x1) -> x1
    a(x1) -> b(b(c(x1)))
    c(c(a(x1))) -> a(a(c(c(x1))))
   graph:
    c#(c(a(x1))) -> c#(c(x1)) -> c#(c(a(x1))) -> a#(a(c(c(x1))))
    c#(c(a(x1))) -> c#(c(x1)) -> c#(c(a(x1))) -> a#(c(c(x1)))
    c#(c(a(x1))) -> c#(c(x1)) -> c#(c(a(x1))) -> c#(c(x1))
    c#(c(a(x1))) -> c#(c(x1)) -> c#(c(a(x1))) -> c#(x1)
    c#(c(a(x1))) -> c#(x1) -> c#(c(a(x1))) -> a#(a(c(c(x1))))
    c#(c(a(x1))) -> c#(x1) -> c#(c(a(x1))) -> a#(c(c(x1)))
    c#(c(a(x1))) -> c#(x1) -> c#(c(a(x1))) -> c#(c(x1))
    c#(c(a(x1))) -> c#(x1) -> c#(c(a(x1))) -> c#(x1)
    c#(c(a(x1))) -> a#(c(c(x1))) -> a#(x1) -> c#(x1)
    c#(c(a(x1))) -> a#(a(c(c(x1)))) -> a#(x1) -> c#(x1)
    a#(x1) -> c#(x1) -> c#(c(a(x1))) -> a#(a(c(c(x1))))
    a#(x1) -> c#(x1) -> c#(c(a(x1))) -> a#(c(c(x1)))
    a#(x1) -> c#(x1) -> c#(c(a(x1))) -> c#(c(x1))
    a#(x1) -> c#(x1) -> c#(c(a(x1))) -> c#(x1)
   Arctic Interpretation Processor:
    dimension: 2
    interpretation:
     [c#](x0) = [0 1]x0 + [2],
     
     [a#](x0) = [1 2]x0 + [3],
     
               [0  -&]     [0]
     [b](x0) = [0  -&]x0 + [0],
     
               [-& 0 ]     [0]
     [c](x0) = [0  -&]x0 + [0],
     
               [0  1 ]     [2]
     [a](x0) = [-& 0 ]x0 + [1]
    orientation:
     a#(x1) = [1 2]x1 + [3] >= [0 1]x1 + [2] = c#(x1)
     
     c#(c(a(x1))) = [1 2]x1 + [3] >= [0 1]x1 + [2] = c#(x1)
     
     c#(c(a(x1))) = [1 2]x1 + [3] >= [1 0]x1 + [2] = c#(c(x1))
     
     c#(c(a(x1))) = [1 2]x1 + [3] >= [1 2]x1 + [3] = a#(c(c(x1)))
     
     c#(c(a(x1))) = [1 2]x1 + [3] >= [1 2]x1 + [3] = a#(a(c(c(x1))))
     
             [0  1 ]     [2]           
     a(x1) = [-& 0 ]x1 + [1] >= x1 = x1
     
             [0  1 ]     [2]    [-& 0 ]     [0]              
     a(x1) = [-& 0 ]x1 + [1] >= [-& 0 ]x1 + [0] = b(b(c(x1)))
     
                   [0  1 ]     [2]    [0  1 ]     [2]                 
     c(c(a(x1))) = [-& 0 ]x1 + [1] >= [-& 0 ]x1 + [1] = a(a(c(c(x1))))
    problem:
     DPs:
      c#(c(a(x1))) -> c#(c(x1))
      c#(c(a(x1))) -> a#(c(c(x1)))
      c#(c(a(x1))) -> a#(a(c(c(x1))))
     TRS:
      a(x1) -> x1
      a(x1) -> b(b(c(x1)))
      c(c(a(x1))) -> a(a(c(c(x1))))
    EDG Processor:
     DPs:
      c#(c(a(x1))) -> c#(c(x1))
      c#(c(a(x1))) -> a#(c(c(x1)))
      c#(c(a(x1))) -> a#(a(c(c(x1))))
     TRS:
      a(x1) -> x1
      a(x1) -> b(b(c(x1)))
      c(c(a(x1))) -> a(a(c(c(x1))))
     graph:
      c#(c(a(x1))) -> c#(c(x1)) -> c#(c(a(x1))) -> c#(c(x1))
      c#(c(a(x1))) -> c#(c(x1)) -> c#(c(a(x1))) -> a#(c(c(x1)))
      c#(c(a(x1))) -> c#(c(x1)) -> c#(c(a(x1))) -> a#(a(c(c(x1))))
     SCC Processor:
      #sccs: 1
      #rules: 1
      #arcs: 3/9
      DPs:
       c#(c(a(x1))) -> c#(c(x1))
      TRS:
       a(x1) -> x1
       a(x1) -> b(b(c(x1)))
       c(c(a(x1))) -> a(a(c(c(x1))))
      LPO Processor:
       argument filtering:
        pi(a) = [0]
        pi(c) = [0]
        pi(b) = []
        pi(c#) = 0
       precedence:
        c > a > c# ~ b
       problem:
        DPs:
         
        TRS:
         a(x1) -> x1
         a(x1) -> b(b(c(x1)))
         c(c(a(x1))) -> a(a(c(c(x1))))
       Qed