YES

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

Proof:
 DP Processor:
  DPs:
   a#(b(x1)) -> b#(b(x1))
   a#(b(x1)) -> c#(b(b(x1)))
   b#(x1) -> c#(x1)
   b#(x1) -> a#(c(x1))
   b#(x1) -> a#(a(c(x1)))
  TRS:
   a(x1) -> x1
   a(b(x1)) -> c(b(b(x1)))
   b(x1) -> a(a(c(x1)))
   c(c(x1)) -> x1
  TDG Processor:
   DPs:
    a#(b(x1)) -> b#(b(x1))
    a#(b(x1)) -> c#(b(b(x1)))
    b#(x1) -> c#(x1)
    b#(x1) -> a#(c(x1))
    b#(x1) -> a#(a(c(x1)))
   TRS:
    a(x1) -> x1
    a(b(x1)) -> c(b(b(x1)))
    b(x1) -> a(a(c(x1)))
    c(c(x1)) -> x1
   graph:
    b#(x1) -> a#(c(x1)) -> a#(b(x1)) -> c#(b(b(x1)))
    b#(x1) -> a#(c(x1)) -> a#(b(x1)) -> b#(b(x1))
    b#(x1) -> a#(a(c(x1))) -> a#(b(x1)) -> c#(b(b(x1)))
    b#(x1) -> a#(a(c(x1))) -> a#(b(x1)) -> b#(b(x1))
    a#(b(x1)) -> b#(b(x1)) -> b#(x1) -> a#(a(c(x1)))
    a#(b(x1)) -> b#(b(x1)) -> b#(x1) -> a#(c(x1))
    a#(b(x1)) -> b#(b(x1)) -> b#(x1) -> c#(x1)
   SCC Processor:
    #sccs: 1
    #rules: 3
    #arcs: 7/25
    DPs:
     b#(x1) -> a#(c(x1))
     a#(b(x1)) -> b#(b(x1))
     b#(x1) -> a#(a(c(x1)))
    TRS:
     a(x1) -> x1
     a(b(x1)) -> c(b(b(x1)))
     b(x1) -> a(a(c(x1)))
     c(c(x1)) -> x1
    Arctic Interpretation Processor:
     dimension: 2
     usable rules:
      a(x1) -> x1
      a(b(x1)) -> c(b(b(x1)))
      b(x1) -> a(a(c(x1)))
      c(c(x1)) -> x1
     interpretation:
      [b#](x0) = [1 0]x0 + [1],
      
      [a#](x0) = [-& 0 ]x0 + [0],
      
                [0  0 ]  
      [c](x0) = [0  -&]x0,
      
                [1 0]     [1]
      [b](x0) = [2 1]x0 + [3],
      
                [0  1 ]     [-&]
      [a](x0) = [-& 0 ]x0 + [0 ]
     orientation:
      b#(x1) = [1 0]x1 + [1] >= [0  -&]x1 + [0] = a#(c(x1))
      
      a#(b(x1)) = [2 1]x1 + [3] >= [2 1]x1 + [3] = b#(b(x1))
      
      b#(x1) = [1 0]x1 + [1] >= [0  -&]x1 + [0] = a#(a(c(x1)))
      
              [0  1 ]     [-&]           
      a(x1) = [-& 0 ]x1 + [0 ] >= x1 = x1
      
                 [3 2]     [4]    [3 2]     [4]              
      a(b(x1)) = [2 1]x1 + [3] >= [2 1]x1 + [3] = c(b(b(x1)))
      
              [1 0]     [1]    [1  0 ]     [1]              
      b(x1) = [2 1]x1 + [3] >= [0  -&]x1 + [0] = a(a(c(x1)))
      
                 [0 0]             
      c(c(x1)) = [0 0]x1 >= x1 = x1
     problem:
      DPs:
       a#(b(x1)) -> b#(b(x1))
      TRS:
       a(x1) -> x1
       a(b(x1)) -> c(b(b(x1)))
       b(x1) -> a(a(c(x1)))
       c(c(x1)) -> x1
     Restore Modifier:
      DPs:
       a#(b(x1)) -> b#(b(x1))
      TRS:
       a(x1) -> x1
       a(b(x1)) -> c(b(b(x1)))
       b(x1) -> a(a(c(x1)))
       c(c(x1)) -> x1
      EDG Processor:
       DPs:
        a#(b(x1)) -> b#(b(x1))
       TRS:
        a(x1) -> x1
        a(b(x1)) -> c(b(b(x1)))
        b(x1) -> a(a(c(x1)))
        c(c(x1)) -> x1
       graph:
        
       Qed