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
    usable rules:
     a(x1) -> x1
     a(x1) -> b(b(c(x1)))
     c(c(a(x1))) -> a(a(c(c(x1))))
    interpretation:
     [c#](x0) = [1 0]x0 + [0],
     
               [-& 0 ]     [0]
     [c](x0) = [0  -&]x0 + [0],
     
     [a#](x0) = [1 0]x0 + [0],
     
               [0  -&]     [0]
     [a](x0) = [2  0 ]x0 + [3],
     
               [-& -&]     [0]
     [b](x0) = [0  0 ]x0 + [0]
    orientation:
     a#(x1) = [1 0]x1 + [0] >= [1 0]x1 + [0] = c#(x1)
     
     c#(c(a(x1))) = [3 1]x1 + [4] >= [1 0]x1 + [0] = c#(x1)
     
     c#(c(a(x1))) = [3 1]x1 + [4] >= [0 1]x1 + [1] = c#(c(x1))
     
     c#(c(a(x1))) = [3 1]x1 + [4] >= [1 0]x1 + [1] = a#(c(c(x1)))
     
     c#(c(a(x1))) = [3 1]x1 + [4] >= [2 0]x1 + [3] = a#(a(c(c(x1))))
     
             [0  -&]     [0]           
     a(x1) = [2  0 ]x1 + [3] >= x1 = x1
     
             [0  -&]     [0]    [-& -&]     [0]              
     a(x1) = [2  0 ]x1 + [3] >= [0  0 ]x1 + [0] = b(b(c(x1)))
     
                   [0  -&]     [0]    [0  -&]     [0]                 
     c(c(a(x1))) = [2  0 ]x1 + [3] >= [2  0 ]x1 + [3] = a(a(c(c(x1))))
    problem:
     DPs:
      a#(x1) -> c#(x1)
      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))))
    Restore Modifier:
     DPs:
      a#(x1) -> c#(x1)
      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))))
     EDG Processor:
      DPs:
       a#(x1) -> c#(x1)
       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))))
      graph:
       c#(c(a(x1))) -> c#(c(x1)) -> c#(c(a(x1))) -> c#(c(x1))
       a#(x1) -> c#(x1) -> c#(c(a(x1))) -> c#(c(x1))
      SCC Processor:
       #sccs: 1
       #rules: 1
       #arcs: 2/4
       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
        usable rules:
         a(x1) -> x1
         a(x1) -> b(b(c(x1)))
         c(c(a(x1))) -> a(a(c(c(x1))))
        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