YES

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

Proof:
 DP Processor:
  DPs:
   c#(c(c(c(x1)))) -> b#(x1)
   c#(c(c(c(x1)))) -> b#(b(x1))
   c#(c(c(c(x1)))) -> b#(b(b(x1)))
   c#(c(c(c(x1)))) -> b#(b(b(b(x1))))
   b#(b(x1)) -> c#(x1)
   b#(b(x1)) -> b#(c(x1))
   b#(b(x1)) -> c#(b(c(x1)))
  TRS:
   c(c(c(c(x1)))) -> b(b(b(b(x1))))
   b(b(x1)) -> x1
   b(b(x1)) -> c(b(c(x1)))
  TDG Processor:
   DPs:
    c#(c(c(c(x1)))) -> b#(x1)
    c#(c(c(c(x1)))) -> b#(b(x1))
    c#(c(c(c(x1)))) -> b#(b(b(x1)))
    c#(c(c(c(x1)))) -> b#(b(b(b(x1))))
    b#(b(x1)) -> c#(x1)
    b#(b(x1)) -> b#(c(x1))
    b#(b(x1)) -> c#(b(c(x1)))
   TRS:
    c(c(c(c(x1)))) -> b(b(b(b(x1))))
    b(b(x1)) -> x1
    b(b(x1)) -> c(b(c(x1)))
   graph:
    b#(b(x1)) -> b#(c(x1)) -> b#(b(x1)) -> c#(b(c(x1)))
    b#(b(x1)) -> b#(c(x1)) -> b#(b(x1)) -> b#(c(x1))
    b#(b(x1)) -> b#(c(x1)) -> b#(b(x1)) -> c#(x1)
    b#(b(x1)) -> c#(b(c(x1))) -> c#(c(c(c(x1)))) -> b#(b(b(b(x1))))
    b#(b(x1)) -> c#(b(c(x1))) -> c#(c(c(c(x1)))) -> b#(b(b(x1)))
    b#(b(x1)) -> c#(b(c(x1))) -> c#(c(c(c(x1)))) -> b#(b(x1))
    b#(b(x1)) -> c#(b(c(x1))) -> c#(c(c(c(x1)))) -> b#(x1)
    b#(b(x1)) -> c#(x1) -> c#(c(c(c(x1)))) -> b#(b(b(b(x1))))
    b#(b(x1)) -> c#(x1) -> c#(c(c(c(x1)))) -> b#(b(b(x1)))
    b#(b(x1)) -> c#(x1) -> c#(c(c(c(x1)))) -> b#(b(x1))
    b#(b(x1)) -> c#(x1) -> c#(c(c(c(x1)))) -> b#(x1)
    c#(c(c(c(x1)))) -> b#(b(b(b(x1)))) ->
    b#(b(x1)) -> c#(b(c(x1)))
    c#(c(c(c(x1)))) -> b#(b(b(b(x1)))) -> b#(b(x1)) -> b#(c(x1))
    c#(c(c(c(x1)))) -> b#(b(b(b(x1)))) -> b#(b(x1)) -> c#(x1)
    c#(c(c(c(x1)))) -> b#(b(b(x1))) -> b#(b(x1)) -> c#(b(c(x1)))
    c#(c(c(c(x1)))) -> b#(b(b(x1))) -> b#(b(x1)) -> b#(c(x1))
    c#(c(c(c(x1)))) -> b#(b(b(x1))) -> b#(b(x1)) -> c#(x1)
    c#(c(c(c(x1)))) -> b#(b(x1)) -> b#(b(x1)) -> c#(b(c(x1)))
    c#(c(c(c(x1)))) -> b#(b(x1)) -> b#(b(x1)) -> b#(c(x1))
    c#(c(c(c(x1)))) -> b#(b(x1)) -> b#(b(x1)) -> c#(x1)
    c#(c(c(c(x1)))) -> b#(x1) -> b#(b(x1)) -> c#(b(c(x1)))
    c#(c(c(c(x1)))) -> b#(x1) -> b#(b(x1)) -> b#(c(x1))
    c#(c(c(c(x1)))) -> b#(x1) -> b#(b(x1)) -> c#(x1)
   Arctic Interpretation Processor:
    dimension: 2
    usable rules:
     c(c(c(c(x1)))) -> b(b(b(b(x1))))
     b(b(x1)) -> x1
     b(b(x1)) -> c(b(c(x1)))
    interpretation:
     [b#](x0) = [-& -2]x0 + [2],
     
     [c#](x0) = [-2 -4]x0 + [0],
     
               [-4 -1]     [1]
     [b](x0) = [1  0 ]x0 + [3],
     
               [1  -2]     [2]
     [c](x0) = [0  -4]x0 + [2]
    orientation:
     c#(c(c(c(x1)))) = [1  -2]x1 + [2] >= [-& -2]x1 + [2] = b#(x1)
     
     c#(c(c(c(x1)))) = [1  -2]x1 + [2] >= [-1 -2]x1 + [2] = b#(b(x1))
     
     c#(c(c(c(x1)))) = [1  -2]x1 + [2] >= [-1 -2]x1 + [2] = b#(b(b(x1)))
     
     c#(c(c(c(x1)))) = [1  -2]x1 + [2] >= [-1 -2]x1 + [2] = b#(b(b(b(x1))))
     
     b#(b(x1)) = [-1 -2]x1 + [2] >= [-2 -4]x1 + [0] = c#(x1)
     
     b#(b(x1)) = [-1 -2]x1 + [2] >= [-2 -6]x1 + [2] = b#(c(x1))
     
     b#(b(x1)) = [-1 -2]x1 + [2] >= [-2 -5]x1 + [0] = c#(b(c(x1)))
     
                      [4 1]     [5]    [0  -1]     [2]                 
     c(c(c(c(x1)))) = [3 0]x1 + [4] >= [1  0 ]x1 + [3] = b(b(b(b(x1))))
     
                [0  -1]     [2]           
     b(b(x1)) = [1  0 ]x1 + [3] >= x1 = x1
     
                [0  -1]     [2]    [0  -3]     [2]              
     b(b(x1)) = [1  0 ]x1 + [3] >= [-1 -5]x1 + [2] = c(b(c(x1)))
    problem:
     DPs:
      c#(c(c(c(x1)))) -> b#(x1)
      c#(c(c(c(x1)))) -> b#(b(x1))
      c#(c(c(c(x1)))) -> b#(b(b(x1)))
      c#(c(c(c(x1)))) -> b#(b(b(b(x1))))
      b#(b(x1)) -> b#(c(x1))
     TRS:
      c(c(c(c(x1)))) -> b(b(b(b(x1))))
      b(b(x1)) -> x1
      b(b(x1)) -> c(b(c(x1)))
    Restore Modifier:
     DPs:
      c#(c(c(c(x1)))) -> b#(x1)
      c#(c(c(c(x1)))) -> b#(b(x1))
      c#(c(c(c(x1)))) -> b#(b(b(x1)))
      c#(c(c(c(x1)))) -> b#(b(b(b(x1))))
      b#(b(x1)) -> b#(c(x1))
     TRS:
      c(c(c(c(x1)))) -> b(b(b(b(x1))))
      b(b(x1)) -> x1
      b(b(x1)) -> c(b(c(x1)))
     EDG Processor:
      DPs:
       c#(c(c(c(x1)))) -> b#(x1)
       c#(c(c(c(x1)))) -> b#(b(x1))
       c#(c(c(c(x1)))) -> b#(b(b(x1)))
       c#(c(c(c(x1)))) -> b#(b(b(b(x1))))
       b#(b(x1)) -> b#(c(x1))
      TRS:
       c(c(c(c(x1)))) -> b(b(b(b(x1))))
       b(b(x1)) -> x1
       b(b(x1)) -> c(b(c(x1)))
      graph:
       b#(b(x1)) -> b#(c(x1)) -> b#(b(x1)) -> b#(c(x1))
       c#(c(c(c(x1)))) -> b#(b(b(b(x1)))) -> b#(b(x1)) -> b#(c(x1))
       c#(c(c(c(x1)))) -> b#(b(b(x1))) -> b#(b(x1)) -> b#(c(x1))
       c#(c(c(c(x1)))) -> b#(b(x1)) -> b#(b(x1)) -> b#(c(x1))
       c#(c(c(c(x1)))) -> b#(x1) -> b#(b(x1)) -> b#(c(x1))
      SCC Processor:
       #sccs: 1
       #rules: 1
       #arcs: 5/25
       DPs:
        b#(b(x1)) -> b#(c(x1))
       TRS:
        c(c(c(c(x1)))) -> b(b(b(b(x1))))
        b(b(x1)) -> x1
        b(b(x1)) -> c(b(c(x1)))
       Arctic Interpretation Processor:
        dimension: 2
        usable rules:
         c(c(c(c(x1)))) -> b(b(b(b(x1))))
         b(b(x1)) -> x1
         b(b(x1)) -> c(b(c(x1)))
        interpretation:
         [b#](x0) = [-3 1 ]x0 + [0],
         
                   [-2 1 ]     [1]
         [b](x0) = [-1 -&]x0 + [2],
         
                   [1  0 ]     [3]
         [c](x0) = [-2 -4]x0 + [0]
        orientation:
         b#(b(x1)) = [0  -2]x1 + [3] >= [-1 -3]x1 + [1] = b#(c(x1))
         
                          [4 3]     [6]    [0  -1]     [3]                 
         c(c(c(c(x1)))) = [1 0]x1 + [3] >= [-3 0 ]x1 + [2] = b(b(b(b(x1))))
         
                    [0  -1]     [3]           
         b(b(x1)) = [-3 0 ]x1 + [2] >= x1 = x1
         
                    [0  -1]     [3]    [0  -1]     [3]              
         b(b(x1)) = [-3 0 ]x1 + [2] >= [-3 -4]x1 + [0] = c(b(c(x1)))
        problem:
         DPs:
          
         TRS:
          c(c(c(c(x1)))) -> b(b(b(b(x1))))
          b(b(x1)) -> x1
          b(b(x1)) -> c(b(c(x1)))
        Qed