YES

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

Proof:
 DP Processor:
  DPs:
   a#(x1) -> c#(b(x1))
   c#(c(b(x1))) -> c#(x1)
   c#(c(b(x1))) -> c#(c(x1))
   c#(c(b(x1))) -> a#(c(c(x1)))
  TRS:
   a(x1) -> b(c(b(x1)))
   a(b(x1)) -> x1
   c(c(b(x1))) -> a(c(c(x1)))
  TDG Processor:
   DPs:
    a#(x1) -> c#(b(x1))
    c#(c(b(x1))) -> c#(x1)
    c#(c(b(x1))) -> c#(c(x1))
    c#(c(b(x1))) -> a#(c(c(x1)))
   TRS:
    a(x1) -> b(c(b(x1)))
    a(b(x1)) -> x1
    c(c(b(x1))) -> a(c(c(x1)))
   graph:
    c#(c(b(x1))) -> c#(c(x1)) -> c#(c(b(x1))) -> a#(c(c(x1)))
    c#(c(b(x1))) -> c#(c(x1)) -> c#(c(b(x1))) -> c#(c(x1))
    c#(c(b(x1))) -> c#(c(x1)) -> c#(c(b(x1))) -> c#(x1)
    c#(c(b(x1))) -> c#(x1) -> c#(c(b(x1))) -> a#(c(c(x1)))
    c#(c(b(x1))) -> c#(x1) -> c#(c(b(x1))) -> c#(c(x1))
    c#(c(b(x1))) -> c#(x1) -> c#(c(b(x1))) -> c#(x1)
    c#(c(b(x1))) -> a#(c(c(x1))) -> a#(x1) -> c#(b(x1))
    a#(x1) -> c#(b(x1)) -> c#(c(b(x1))) -> a#(c(c(x1)))
    a#(x1) -> c#(b(x1)) -> c#(c(b(x1))) -> c#(c(x1))
    a#(x1) -> c#(b(x1)) -> c#(c(b(x1))) -> c#(x1)
   EDG Processor:
    DPs:
     a#(x1) -> c#(b(x1))
     c#(c(b(x1))) -> c#(x1)
     c#(c(b(x1))) -> c#(c(x1))
     c#(c(b(x1))) -> a#(c(c(x1)))
    TRS:
     a(x1) -> b(c(b(x1)))
     a(b(x1)) -> x1
     c(c(b(x1))) -> a(c(c(x1)))
    graph:
     c#(c(b(x1))) -> c#(c(x1)) -> c#(c(b(x1))) -> c#(x1)
     c#(c(b(x1))) -> c#(c(x1)) -> c#(c(b(x1))) -> c#(c(x1))
     c#(c(b(x1))) -> c#(c(x1)) -> c#(c(b(x1))) -> a#(c(c(x1)))
     c#(c(b(x1))) -> c#(x1) -> c#(c(b(x1))) -> c#(x1)
     c#(c(b(x1))) -> c#(x1) -> c#(c(b(x1))) -> c#(c(x1))
     c#(c(b(x1))) -> c#(x1) -> c#(c(b(x1))) -> a#(c(c(x1)))
     c#(c(b(x1))) -> a#(c(c(x1))) -> a#(x1) -> c#(b(x1))
    SCC Processor:
     #sccs: 1
     #rules: 2
     #arcs: 7/16
     DPs:
      c#(c(b(x1))) -> c#(c(x1))
      c#(c(b(x1))) -> c#(x1)
     TRS:
      a(x1) -> b(c(b(x1)))
      a(b(x1)) -> x1
      c(c(b(x1))) -> a(c(c(x1)))
     Arctic Interpretation Processor:
      dimension: 2
      usable rules:
       a(x1) -> b(c(b(x1)))
       a(b(x1)) -> x1
       c(c(b(x1))) -> a(c(c(x1)))
      interpretation:
       [c#](x0) = [-& 0 ]x0 + [0],
       
                 [1  0 ]     [1]
       [b](x0) = [0  -&]x0 + [0],
       
                 [1  0 ]     [1]
       [a](x0) = [0  -&]x0 + [0],
       
                 [-& 0 ]     [0]
       [c](x0) = [0  -&]x0 + [0]
      orientation:
       c#(c(b(x1))) = [1 0]x1 + [1] >= [0  -&]x1 + [0] = c#(c(x1))
       
       c#(c(b(x1))) = [1 0]x1 + [1] >= [-& 0 ]x1 + [0] = c#(x1)
       
               [1  0 ]     [1]    [1  0 ]     [1]              
       a(x1) = [0  -&]x1 + [0] >= [0  -&]x1 + [0] = b(c(b(x1)))
       
                  [2 1]     [2]           
       a(b(x1)) = [1 0]x1 + [1] >= x1 = x1
       
                     [1  0 ]     [1]    [1  0 ]     [1]              
       c(c(b(x1))) = [0  -&]x1 + [0] >= [0  -&]x1 + [0] = a(c(c(x1)))
      problem:
       DPs:
        c#(c(b(x1))) -> c#(x1)
       TRS:
        a(x1) -> b(c(b(x1)))
        a(b(x1)) -> x1
        c(c(b(x1))) -> a(c(c(x1)))
      Restore Modifier:
       DPs:
        c#(c(b(x1))) -> c#(x1)
       TRS:
        a(x1) -> b(c(b(x1)))
        a(b(x1)) -> x1
        c(c(b(x1))) -> a(c(c(x1)))
       EDG Processor:
        DPs:
         c#(c(b(x1))) -> c#(x1)
        TRS:
         a(x1) -> b(c(b(x1)))
         a(b(x1)) -> x1
         c(c(b(x1))) -> a(c(c(x1)))
        graph:
         c#(c(b(x1))) -> c#(x1) -> c#(c(b(x1))) -> c#(x1)
        Usable Rule Processor:
         DPs:
          c#(c(b(x1))) -> c#(x1)
         TRS:
          
         Arctic Interpretation Processor:
          dimension: 1
          usable rules:
           
          interpretation:
           [c#](x0) = x0,
           
           [b](x0) = 1x0 + 13,
           
           [c](x0) = x0 + 0
          orientation:
           c#(c(b(x1))) = 1x1 + 13 >= x1 = c#(x1)
          problem:
           DPs:
            
           TRS:
            
          Qed