YES

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

Proof:
 DP Processor:
  DPs:
   b#(b(x1)) -> c#(x1)
   b#(b(x1)) -> c#(c(x1))
   b#(b(x1)) -> c#(c(c(x1)))
   b#(b(x1)) -> c#(c(c(c(x1))))
   b#(c(b(x1))) -> b#(b(x1))
   b#(c(b(x1))) -> b#(b(b(x1)))
  TRS:
   b(b(x1)) -> c(c(c(c(x1))))
   c(x1) -> x1
   b(c(b(x1))) -> b(b(b(x1)))
  TDG Processor:
   DPs:
    b#(b(x1)) -> c#(x1)
    b#(b(x1)) -> c#(c(x1))
    b#(b(x1)) -> c#(c(c(x1)))
    b#(b(x1)) -> c#(c(c(c(x1))))
    b#(c(b(x1))) -> b#(b(x1))
    b#(c(b(x1))) -> b#(b(b(x1)))
   TRS:
    b(b(x1)) -> c(c(c(c(x1))))
    c(x1) -> x1
    b(c(b(x1))) -> b(b(b(x1)))
   graph:
    b#(c(b(x1))) -> b#(b(b(x1))) -> b#(c(b(x1))) -> b#(b(b(x1)))
    b#(c(b(x1))) -> b#(b(b(x1))) -> b#(c(b(x1))) -> b#(b(x1))
    b#(c(b(x1))) -> b#(b(b(x1))) -> b#(b(x1)) -> c#(c(c(c(x1))))
    b#(c(b(x1))) -> b#(b(b(x1))) -> b#(b(x1)) -> c#(c(c(x1)))
    b#(c(b(x1))) -> b#(b(b(x1))) -> b#(b(x1)) -> c#(c(x1))
    b#(c(b(x1))) -> b#(b(b(x1))) -> b#(b(x1)) -> c#(x1)
    b#(c(b(x1))) -> b#(b(x1)) -> b#(c(b(x1))) -> b#(b(b(x1)))
    b#(c(b(x1))) -> b#(b(x1)) -> b#(c(b(x1))) -> b#(b(x1))
    b#(c(b(x1))) -> b#(b(x1)) -> b#(b(x1)) -> c#(c(c(c(x1))))
    b#(c(b(x1))) -> b#(b(x1)) -> b#(b(x1)) -> c#(c(c(x1)))
    b#(c(b(x1))) -> b#(b(x1)) -> b#(b(x1)) -> c#(c(x1))
    b#(c(b(x1))) -> b#(b(x1)) -> b#(b(x1)) -> c#(x1)
   SCC Processor:
    #sccs: 1
    #rules: 2
    #arcs: 12/36
    DPs:
     b#(c(b(x1))) -> b#(b(b(x1)))
     b#(c(b(x1))) -> b#(b(x1))
    TRS:
     b(b(x1)) -> c(c(c(c(x1))))
     c(x1) -> x1
     b(c(b(x1))) -> b(b(b(x1)))
    Arctic Interpretation Processor:
     dimension: 2
     interpretation:
      [b#](x0) = [0 0]x0,
      
                [0  -&]  
      [c](x0) = [2  0 ]x0,
      
                [0 0]  
      [b](x0) = [1 1]x0
     orientation:
      b#(c(b(x1))) = [2 2]x1 >= [2 2]x1 = b#(b(b(x1)))
      
      b#(c(b(x1))) = [2 2]x1 >= [1 1]x1 = b#(b(x1))
      
                 [1 1]      [0  -&]                   
      b(b(x1)) = [2 2]x1 >= [2  0 ]x1 = c(c(c(c(x1))))
      
              [0  -&]             
      c(x1) = [2  0 ]x1 >= x1 = x1
      
                    [2 2]      [2 2]                
      b(c(b(x1))) = [3 3]x1 >= [3 3]x1 = b(b(b(x1)))
     problem:
      DPs:
       b#(c(b(x1))) -> b#(b(b(x1)))
      TRS:
       b(b(x1)) -> c(c(c(c(x1))))
       c(x1) -> x1
       b(c(b(x1))) -> b(b(b(x1)))
     EDG Processor:
      DPs:
       b#(c(b(x1))) -> b#(b(b(x1)))
      TRS:
       b(b(x1)) -> c(c(c(c(x1))))
       c(x1) -> x1
       b(c(b(x1))) -> b(b(b(x1)))
      graph:
       b#(c(b(x1))) -> b#(b(b(x1))) -> b#(c(b(x1))) -> b#(b(b(x1)))
      Arctic Interpretation Processor:
       dimension: 2
       interpretation:
        [b#](x0) = [-& 0 ]x0 + [0],
        
                  [0  -&]     [0]
        [c](x0) = [2  0 ]x0 + [2],
        
                  [1 0]     [1 ]
        [b](x0) = [1 0]x0 + [-&]
       orientation:
        b#(c(b(x1))) = [3 2]x1 + [3] >= [2 1]x1 + [2] = b#(b(b(x1)))
        
                   [2 1]     [2]    [0  -&]     [0]                 
        b(b(x1)) = [2 1]x1 + [2] >= [2  0 ]x1 + [2] = c(c(c(c(x1))))
        
                [0  -&]     [0]           
        c(x1) = [2  0 ]x1 + [2] >= x1 = x1
        
                      [3 2]     [3]    [3 2]     [3]              
        b(c(b(x1))) = [3 2]x1 + [3] >= [3 2]x1 + [3] = b(b(b(x1)))
       problem:
        DPs:
         
        TRS:
         b(b(x1)) -> c(c(c(c(x1))))
         c(x1) -> x1
         b(c(b(x1))) -> b(b(b(x1)))
       Qed