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)))
    Root-Labeling Processor:
     DPs:
      b{#,(f4)}(f4(c)(c(b)(b(f4)(x1)))) -> b{#,(f4)}(f4(b)(b(b)(b(f4)(x1))))
      b{#,(f4)}(f4(c)(c(b)(b(b)(x1)))) -> b{#,(f4)}(f4(b)(b(b)(b(b)(x1))))
      b{#,(f4)}(f4(c)(c(b)(b(c)(x1)))) -> b{#,(f4)}(f4(b)(b(b)(b(c)(x1))))
      b{#,(f4)}(f4(c)(c(b)(b(f4)(x1)))) -> b{#,(f4)}(f4(b)(b(f4)(x1)))
      b{#,(f4)}(f4(c)(c(b)(b(b)(x1)))) -> b{#,(f4)}(f4(b)(b(b)(x1)))
      b{#,(f4)}(f4(c)(c(b)(b(c)(x1)))) -> b{#,(f4)}(f4(b)(b(c)(x1)))
     TRS:
      f4(b)(b(b)(b(f4)(x1))) -> f4(c)(c(c)(c(c)(c(c)(c(f4)(x1)))))
      f4(b)(b(b)(b(b)(x1))) -> f4(c)(c(c)(c(c)(c(c)(c(b)(x1)))))
      f4(b)(b(b)(b(c)(x1))) -> f4(c)(c(c)(c(c)(c(c)(c(c)(x1)))))
      b(b)(b(b)(b(f4)(x1))) -> b(c)(c(c)(c(c)(c(c)(c(f4)(x1)))))
      b(b)(b(b)(b(b)(x1))) -> b(c)(c(c)(c(c)(c(c)(c(b)(x1)))))
      b(b)(b(b)(b(c)(x1))) -> b(c)(c(c)(c(c)(c(c)(c(c)(x1)))))
      c(b)(b(b)(b(f4)(x1))) -> c(c)(c(c)(c(c)(c(c)(c(f4)(x1)))))
      c(b)(b(b)(b(b)(x1))) -> c(c)(c(c)(c(c)(c(c)(c(b)(x1)))))
      c(b)(b(b)(b(c)(x1))) -> c(c)(c(c)(c(c)(c(c)(c(c)(x1)))))
      f4(c)(c(f4)(x1)) -> f4(f4)(x1)
      f4(c)(c(b)(x1)) -> f4(b)(x1)
      f4(c)(c(c)(x1)) -> f4(c)(x1)
      b(c)(c(f4)(x1)) -> b(f4)(x1)
      b(c)(c(b)(x1)) -> b(b)(x1)
      b(c)(c(c)(x1)) -> b(c)(x1)
      c(c)(c(f4)(x1)) -> c(f4)(x1)
      c(c)(c(b)(x1)) -> c(b)(x1)
      c(c)(c(c)(x1)) -> c(c)(x1)
      b(c)(c(b)(b(f4)(x1))) -> b(b)(b(b)(b(f4)(x1)))
      b(c)(c(b)(b(b)(x1))) -> b(b)(b(b)(b(b)(x1)))
      b(c)(c(b)(b(c)(x1))) -> b(b)(b(b)(b(c)(x1)))
     Polynomial Interpretation Processor:
      dimension: 1
      interpretation:
       [b(f4)](x0) = x0 + 1,
       
       [b{#,(f4)}](x0) = x0 + 1,
       
       [b(b)](x0) = x0,
       
       [c(b)](x0) = x0,
       
       [f4(f4)](x0) = x0,
       
       [c(c)](x0) = x0,
       
       [b(c)](x0) = x0,
       
       [f4(b)](x0) = x0 + 1,
       
       [c(f4)](x0) = x0 + 1,
       
       [f4(c)](x0) = x0 + 1
      orientation:
       b{#,(f4)}(f4(c)(c(b)(b(f4)(x1)))) = x1 + 3 >= x1 + 3 = b{#,(f4)}(f4(b)(b(b)(b(f4)(x1))))
       
       b{#,(f4)}(f4(c)(c(b)(b(b)(x1)))) = x1 + 2 >= x1 + 2 = b{#,(f4)}(f4(b)(b(b)(b(b)(x1))))
       
       b{#,(f4)}(f4(c)(c(b)(b(c)(x1)))) = x1 + 2 >= x1 + 2 = b{#,(f4)}(f4(b)(b(b)(b(c)(x1))))
       
       b{#,(f4)}(f4(c)(c(b)(b(f4)(x1)))) = x1 + 3 >= x1 + 3 = b{#,(f4)}(f4(b)(b(f4)(x1)))
       
       b{#,(f4)}(f4(c)(c(b)(b(b)(x1)))) = x1 + 2 >= x1 + 2 = b{#,(f4)}(f4(b)(b(b)(x1)))
       
       b{#,(f4)}(f4(c)(c(b)(b(c)(x1)))) = x1 + 2 >= x1 + 2 = b{#,(f4)}(f4(b)(b(c)(x1)))
       
       f4(b)(b(b)(b(f4)(x1))) = x1 + 2 >= x1 + 2 = f4(c)(c(c)(c(c)(c(c)(c(f4)(x1)))))
       
       f4(b)(b(b)(b(b)(x1))) = x1 + 1 >= x1 + 1 = f4(c)(c(c)(c(c)(c(c)(c(b)(x1)))))
       
       f4(b)(b(b)(b(c)(x1))) = x1 + 1 >= x1 + 1 = f4(c)(c(c)(c(c)(c(c)(c(c)(x1)))))
       
       b(b)(b(b)(b(f4)(x1))) = x1 + 1 >= x1 + 1 = b(c)(c(c)(c(c)(c(c)(c(f4)(x1)))))
       
       b(b)(b(b)(b(b)(x1))) = x1 >= x1 = b(c)(c(c)(c(c)(c(c)(c(b)(x1)))))
       
       b(b)(b(b)(b(c)(x1))) = x1 >= x1 = b(c)(c(c)(c(c)(c(c)(c(c)(x1)))))
       
       c(b)(b(b)(b(f4)(x1))) = x1 + 1 >= x1 + 1 = c(c)(c(c)(c(c)(c(c)(c(f4)(x1)))))
       
       c(b)(b(b)(b(b)(x1))) = x1 >= x1 = c(c)(c(c)(c(c)(c(c)(c(b)(x1)))))
       
       c(b)(b(b)(b(c)(x1))) = x1 >= x1 = c(c)(c(c)(c(c)(c(c)(c(c)(x1)))))
       
       f4(c)(c(f4)(x1)) = x1 + 2 >= x1 = f4(f4)(x1)
       
       f4(c)(c(b)(x1)) = x1 + 1 >= x1 + 1 = f4(b)(x1)
       
       f4(c)(c(c)(x1)) = x1 + 1 >= x1 + 1 = f4(c)(x1)
       
       b(c)(c(f4)(x1)) = x1 + 1 >= x1 + 1 = b(f4)(x1)
       
       b(c)(c(b)(x1)) = x1 >= x1 = b(b)(x1)
       
       b(c)(c(c)(x1)) = x1 >= x1 = b(c)(x1)
       
       c(c)(c(f4)(x1)) = x1 + 1 >= x1 + 1 = c(f4)(x1)
       
       c(c)(c(b)(x1)) = x1 >= x1 = c(b)(x1)
       
       c(c)(c(c)(x1)) = x1 >= x1 = c(c)(x1)
       
       b(c)(c(b)(b(f4)(x1))) = x1 + 1 >= x1 + 1 = b(b)(b(b)(b(f4)(x1)))
       
       b(c)(c(b)(b(b)(x1))) = x1 >= x1 = b(b)(b(b)(b(b)(x1)))
       
       b(c)(c(b)(b(c)(x1))) = x1 >= x1 = b(b)(b(b)(b(c)(x1)))
      problem:
       DPs:
        b{#,(f4)}(f4(c)(c(b)(b(f4)(x1)))) -> b{#,(f4)}(f4(b)(b(b)(b(f4)(x1))))
        b{#,(f4)}(f4(c)(c(b)(b(b)(x1)))) -> b{#,(f4)}(f4(b)(b(b)(b(b)(x1))))
        b{#,(f4)}(f4(c)(c(b)(b(c)(x1)))) -> b{#,(f4)}(f4(b)(b(b)(b(c)(x1))))
        b{#,(f4)}(f4(c)(c(b)(b(f4)(x1)))) -> b{#,(f4)}(f4(b)(b(f4)(x1)))
        b{#,(f4)}(f4(c)(c(b)(b(b)(x1)))) -> b{#,(f4)}(f4(b)(b(b)(x1)))
        b{#,(f4)}(f4(c)(c(b)(b(c)(x1)))) -> b{#,(f4)}(f4(b)(b(c)(x1)))
       TRS:
        f4(b)(b(b)(b(f4)(x1))) -> f4(c)(c(c)(c(c)(c(c)(c(f4)(x1)))))
        f4(b)(b(b)(b(b)(x1))) -> f4(c)(c(c)(c(c)(c(c)(c(b)(x1)))))
        f4(b)(b(b)(b(c)(x1))) -> f4(c)(c(c)(c(c)(c(c)(c(c)(x1)))))
        b(b)(b(b)(b(f4)(x1))) -> b(c)(c(c)(c(c)(c(c)(c(f4)(x1)))))
        b(b)(b(b)(b(b)(x1))) -> b(c)(c(c)(c(c)(c(c)(c(b)(x1)))))
        b(b)(b(b)(b(c)(x1))) -> b(c)(c(c)(c(c)(c(c)(c(c)(x1)))))
        c(b)(b(b)(b(f4)(x1))) -> c(c)(c(c)(c(c)(c(c)(c(f4)(x1)))))
        c(b)(b(b)(b(b)(x1))) -> c(c)(c(c)(c(c)(c(c)(c(b)(x1)))))
        c(b)(b(b)(b(c)(x1))) -> c(c)(c(c)(c(c)(c(c)(c(c)(x1)))))
        f4(c)(c(b)(x1)) -> f4(b)(x1)
        f4(c)(c(c)(x1)) -> f4(c)(x1)
        b(c)(c(f4)(x1)) -> b(f4)(x1)
        b(c)(c(b)(x1)) -> b(b)(x1)
        b(c)(c(c)(x1)) -> b(c)(x1)
        c(c)(c(f4)(x1)) -> c(f4)(x1)
        c(c)(c(b)(x1)) -> c(b)(x1)
        c(c)(c(c)(x1)) -> c(c)(x1)
        b(c)(c(b)(b(f4)(x1))) -> b(b)(b(b)(b(f4)(x1)))
        b(c)(c(b)(b(b)(x1))) -> b(b)(b(b)(b(b)(x1)))
        b(c)(c(b)(b(c)(x1))) -> b(b)(b(b)(b(c)(x1)))
      Polynomial Interpretation Processor:
       dimension: 1
       interpretation:
        [b(f4)](x0) = x0 + 1,
        
        [b{#,(f4)}](x0) = x0,
        
        [b(b)](x0) = x0 + 1,
        
        [c(b)](x0) = x0 + 1,
        
        [c(c)](x0) = x0,
        
        [b(c)](x0) = x0 + 1,
        
        [f4(b)](x0) = x0,
        
        [c(f4)](x0) = x0 + 1,
        
        [f4(c)](x0) = x0 + 1
       orientation:
        b{#,(f4)}(f4(c)(c(b)(b(f4)(x1)))) = x1 + 3 >= x1 + 2 = b{#,(f4)}(f4(b)(b(b)(b(f4)(x1))))
        
        b{#,(f4)}(f4(c)(c(b)(b(b)(x1)))) = x1 + 3 >= x1 + 2 = b{#,(f4)}(f4(b)(b(b)(b(b)(x1))))
        
        b{#,(f4)}(f4(c)(c(b)(b(c)(x1)))) = x1 + 3 >= x1 + 2 = b{#,(f4)}(f4(b)(b(b)(b(c)(x1))))
        
        b{#,(f4)}(f4(c)(c(b)(b(f4)(x1)))) = x1 + 3 >= x1 + 1 = b{#,(f4)}(f4(b)(b(f4)(x1)))
        
        b{#,(f4)}(f4(c)(c(b)(b(b)(x1)))) = x1 + 3 >= x1 + 1 = b{#,(f4)}(f4(b)(b(b)(x1)))
        
        b{#,(f4)}(f4(c)(c(b)(b(c)(x1)))) = x1 + 3 >= x1 + 1 = b{#,(f4)}(f4(b)(b(c)(x1)))
        
        f4(b)(b(b)(b(f4)(x1))) = x1 + 2 >= x1 + 2 = f4(c)(c(c)(c(c)(c(c)(c(f4)(x1)))))
        
        f4(b)(b(b)(b(b)(x1))) = x1 + 2 >= x1 + 2 = f4(c)(c(c)(c(c)(c(c)(c(b)(x1)))))
        
        f4(b)(b(b)(b(c)(x1))) = x1 + 2 >= x1 + 1 = f4(c)(c(c)(c(c)(c(c)(c(c)(x1)))))
        
        b(b)(b(b)(b(f4)(x1))) = x1 + 3 >= x1 + 2 = b(c)(c(c)(c(c)(c(c)(c(f4)(x1)))))
        
        b(b)(b(b)(b(b)(x1))) = x1 + 3 >= x1 + 2 = b(c)(c(c)(c(c)(c(c)(c(b)(x1)))))
        
        b(b)(b(b)(b(c)(x1))) = x1 + 3 >= x1 + 1 = b(c)(c(c)(c(c)(c(c)(c(c)(x1)))))
        
        c(b)(b(b)(b(f4)(x1))) = x1 + 3 >= x1 + 1 = c(c)(c(c)(c(c)(c(c)(c(f4)(x1)))))
        
        c(b)(b(b)(b(b)(x1))) = x1 + 3 >= x1 + 1 = c(c)(c(c)(c(c)(c(c)(c(b)(x1)))))
        
        c(b)(b(b)(b(c)(x1))) = x1 + 3 >= x1 = c(c)(c(c)(c(c)(c(c)(c(c)(x1)))))
        
        f4(c)(c(b)(x1)) = x1 + 2 >= x1 = f4(b)(x1)
        
        f4(c)(c(c)(x1)) = x1 + 1 >= x1 + 1 = f4(c)(x1)
        
        b(c)(c(f4)(x1)) = x1 + 2 >= x1 + 1 = b(f4)(x1)
        
        b(c)(c(b)(x1)) = x1 + 2 >= x1 + 1 = b(b)(x1)
        
        b(c)(c(c)(x1)) = x1 + 1 >= x1 + 1 = b(c)(x1)
        
        c(c)(c(f4)(x1)) = x1 + 1 >= x1 + 1 = c(f4)(x1)
        
        c(c)(c(b)(x1)) = x1 + 1 >= x1 + 1 = c(b)(x1)
        
        c(c)(c(c)(x1)) = x1 >= x1 = c(c)(x1)
        
        b(c)(c(b)(b(f4)(x1))) = x1 + 3 >= x1 + 3 = b(b)(b(b)(b(f4)(x1)))
        
        b(c)(c(b)(b(b)(x1))) = x1 + 3 >= x1 + 3 = b(b)(b(b)(b(b)(x1)))
        
        b(c)(c(b)(b(c)(x1))) = x1 + 3 >= x1 + 3 = b(b)(b(b)(b(c)(x1)))
       problem:
        DPs:
         
        TRS:
         f4(b)(b(b)(b(f4)(x1))) -> f4(c)(c(c)(c(c)(c(c)(c(f4)(x1)))))
         f4(b)(b(b)(b(b)(x1))) -> f4(c)(c(c)(c(c)(c(c)(c(b)(x1)))))
         f4(c)(c(c)(x1)) -> f4(c)(x1)
         b(c)(c(c)(x1)) -> b(c)(x1)
         c(c)(c(f4)(x1)) -> c(f4)(x1)
         c(c)(c(b)(x1)) -> c(b)(x1)
         c(c)(c(c)(x1)) -> c(c)(x1)
         b(c)(c(b)(b(f4)(x1))) -> b(b)(b(b)(b(f4)(x1)))
         b(c)(c(b)(b(b)(x1))) -> b(b)(b(b)(b(b)(x1)))
         b(c)(c(b)(b(c)(x1))) -> b(b)(b(b)(b(c)(x1)))
       Qed