YES

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

Proof:
 DP Processor:
  DPs:
   c#(c(b(x1))) -> c#(x1)
   c#(c(b(x1))) -> c#(c(x1))
   c#(c(b(x1))) -> a#(c(c(x1)))
   c#(c(b(x1))) -> c#(a(c(c(x1))))
  TRS:
   a(x1) -> b(x1)
   a(b(x1)) -> x1
   c(c(b(x1))) -> b(c(a(c(c(x1)))))
  TDG Processor:
   DPs:
    c#(c(b(x1))) -> c#(x1)
    c#(c(b(x1))) -> c#(c(x1))
    c#(c(b(x1))) -> a#(c(c(x1)))
    c#(c(b(x1))) -> c#(a(c(c(x1))))
   TRS:
    a(x1) -> b(x1)
    a(b(x1)) -> x1
    c(c(b(x1))) -> b(c(a(c(c(x1)))))
   graph:
    c#(c(b(x1))) -> c#(c(x1)) -> c#(c(b(x1))) -> c#(a(c(c(x1))))
    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#(a(c(c(x1)))) ->
    c#(c(b(x1))) -> c#(a(c(c(x1))))
    c#(c(b(x1))) -> c#(a(c(c(x1)))) -> c#(c(b(x1))) -> a#(c(c(x1)))
    c#(c(b(x1))) -> c#(a(c(c(x1)))) -> c#(c(b(x1))) -> c#(c(x1))
    c#(c(b(x1))) -> c#(a(c(c(x1)))) -> c#(c(b(x1))) -> c#(x1)
    c#(c(b(x1))) -> c#(x1) -> c#(c(b(x1))) -> c#(a(c(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)
   SCC Processor:
    #sccs: 1
    #rules: 3
    #arcs: 12/16
    DPs:
     c#(c(b(x1))) -> c#(c(x1))
     c#(c(b(x1))) -> c#(x1)
     c#(c(b(x1))) -> c#(a(c(c(x1))))
    TRS:
     a(x1) -> b(x1)
     a(b(x1)) -> x1
     c(c(b(x1))) -> b(c(a(c(c(x1)))))
    Arctic Interpretation Processor:
     dimension: 2
     usable rules:
      a(x1) -> b(x1)
      a(b(x1)) -> x1
      c(c(b(x1))) -> b(c(a(c(c(x1)))))
     interpretation:
      [c#](x0) = [0 2]x0 + [0],
      
                [-& 0 ]     [1]
      [c](x0) = [0  -&]x0 + [0],
      
                [1  0 ]     [3]
      [b](x0) = [0  -&]x0 + [2],
      
                [1  0 ]     [3]
      [a](x0) = [0  -&]x0 + [2]
     orientation:
      c#(c(b(x1))) = [3 2]x1 + [5] >= [2 0]x1 + [2] = c#(c(x1))
      
      c#(c(b(x1))) = [3 2]x1 + [5] >= [0 2]x1 + [0] = c#(x1)
      
      c#(c(b(x1))) = [3 2]x1 + [5] >= [2 0]x1 + [4] = c#(a(c(c(x1))))
      
              [1  0 ]     [3]    [1  0 ]     [3]        
      a(x1) = [0  -&]x1 + [2] >= [0  -&]x1 + [2] = b(x1)
      
                 [2 1]     [4]           
      a(b(x1)) = [1 0]x1 + [3] >= x1 = x1
      
                    [1  0 ]     [3]    [1  0 ]     [3]                    
      c(c(b(x1))) = [0  -&]x1 + [2] >= [0  -&]x1 + [2] = b(c(a(c(c(x1)))))
     problem:
      DPs:
       c#(c(b(x1))) -> c#(x1)
      TRS:
       a(x1) -> b(x1)
       a(b(x1)) -> x1
       c(c(b(x1))) -> b(c(a(c(c(x1)))))
     Restore Modifier:
      DPs:
       c#(c(b(x1))) -> c#(x1)
      TRS:
       a(x1) -> b(x1)
       a(b(x1)) -> x1
       c(c(b(x1))) -> b(c(a(c(c(x1)))))
      EDG Processor:
       DPs:
        c#(c(b(x1))) -> c#(x1)
       TRS:
        a(x1) -> b(x1)
        a(b(x1)) -> x1
        c(c(b(x1))) -> b(c(a(c(c(x1)))))
       graph:
        c#(c(b(x1))) -> c#(x1) -> c#(c(b(x1))) -> c#(x1)
       CDG Processor:
        DPs:
         c#(c(b(x1))) -> c#(x1)
        TRS:
         a(x1) -> b(x1)
         a(b(x1)) -> x1
         c(c(b(x1))) -> b(c(a(c(c(x1)))))
        graph:
         
        Qed