YES

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

Proof:
 String Reversal Processor:
  a(x1) -> x1
  a(x1) -> b(x1)
  b(x1) -> x1
  c(a(b(x1))) -> b(a(a(c(c(x1)))))
  DP Processor:
   DPs:
    a#(x1) -> b#(x1)
    c#(a(b(x1))) -> c#(x1)
    c#(a(b(x1))) -> c#(c(x1))
    c#(a(b(x1))) -> a#(c(c(x1)))
    c#(a(b(x1))) -> a#(a(c(c(x1))))
    c#(a(b(x1))) -> b#(a(a(c(c(x1)))))
   TRS:
    a(x1) -> x1
    a(x1) -> b(x1)
    b(x1) -> x1
    c(a(b(x1))) -> b(a(a(c(c(x1)))))
   TDG Processor:
    DPs:
     a#(x1) -> b#(x1)
     c#(a(b(x1))) -> c#(x1)
     c#(a(b(x1))) -> c#(c(x1))
     c#(a(b(x1))) -> a#(c(c(x1)))
     c#(a(b(x1))) -> a#(a(c(c(x1))))
     c#(a(b(x1))) -> b#(a(a(c(c(x1)))))
    TRS:
     a(x1) -> x1
     a(x1) -> b(x1)
     b(x1) -> x1
     c(a(b(x1))) -> b(a(a(c(c(x1)))))
    graph:
     c#(a(b(x1))) -> c#(c(x1)) -> c#(a(b(x1))) -> b#(a(a(c(c(x1)))))
     c#(a(b(x1))) -> c#(c(x1)) -> c#(a(b(x1))) -> a#(a(c(c(x1))))
     c#(a(b(x1))) -> c#(c(x1)) -> c#(a(b(x1))) -> a#(c(c(x1)))
     c#(a(b(x1))) -> c#(c(x1)) -> c#(a(b(x1))) -> c#(c(x1))
     c#(a(b(x1))) -> c#(c(x1)) -> c#(a(b(x1))) -> c#(x1)
     c#(a(b(x1))) -> c#(x1) -> c#(a(b(x1))) -> b#(a(a(c(c(x1)))))
     c#(a(b(x1))) -> c#(x1) -> c#(a(b(x1))) -> a#(a(c(c(x1))))
     c#(a(b(x1))) -> c#(x1) -> c#(a(b(x1))) -> a#(c(c(x1)))
     c#(a(b(x1))) -> c#(x1) -> c#(a(b(x1))) -> c#(c(x1))
     c#(a(b(x1))) -> c#(x1) -> c#(a(b(x1))) -> c#(x1)
     c#(a(b(x1))) -> a#(c(c(x1))) -> a#(x1) -> b#(x1)
     c#(a(b(x1))) -> a#(a(c(c(x1)))) -> a#(x1) -> b#(x1)
    SCC Processor:
     #sccs: 1
     #rules: 2
     #arcs: 12/36
     DPs:
      c#(a(b(x1))) -> c#(c(x1))
      c#(a(b(x1))) -> c#(x1)
     TRS:
      a(x1) -> x1
      a(x1) -> b(x1)
      b(x1) -> x1
      c(a(b(x1))) -> b(a(a(c(c(x1)))))
     Arctic Interpretation Processor:
      dimension: 2
      usable rules:
       a(x1) -> x1
       a(x1) -> b(x1)
       b(x1) -> x1
       c(a(b(x1))) -> b(a(a(c(c(x1)))))
      interpretation:
                 [0  1 ]  
       [b](x0) = [-& 0 ]x0,
       
       [c#](x0) = [0 3]x0,
       
                 [0 1]     [-&]
       [a](x0) = [0 0]x0 + [1 ],
       
                 [0  1 ]  
       [c](x0) = [-& 0 ]x0
      orientation:
       c#(a(b(x1))) = [3 4]x1 + [4] >= [0 3]x1 = c#(c(x1))
       
       c#(a(b(x1))) = [3 4]x1 + [4] >= [0 3]x1 = c#(x1)
       
               [0 1]     [-&]           
       a(x1) = [0 0]x1 + [1 ] >= x1 = x1
       
               [0 1]     [-&]    [0  1 ]          
       a(x1) = [0 0]x1 + [1 ] >= [-& 0 ]x1 = b(x1)
       
               [0  1 ]             
       b(x1) = [-& 0 ]x1 >= x1 = x1
       
                     [1 2]     [2]    [1 2]     [2]                    
       c(a(b(x1))) = [0 1]x1 + [1] >= [0 1]x1 + [1] = b(a(a(c(c(x1)))))
      problem:
       DPs:
        
       TRS:
        a(x1) -> x1
        a(x1) -> b(x1)
        b(x1) -> x1
        c(a(b(x1))) -> b(a(a(c(c(x1)))))
      Qed