YES

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

Proof:
 DP Processor:
  DPs:
   a#(b(x1)) -> a#(x1)
   a#(b(x1)) -> a#(a(x1))
   a#(b(x1)) -> b#(a(a(x1)))
   a#(b(x1)) -> c#(b(a(a(x1))))
   b#(x1) -> c#(x1)
   c#(c(x1)) -> b#(x1)
  TRS:
   a(x1) -> x1
   a(b(x1)) -> c(b(a(a(x1))))
   b(x1) -> c(x1)
   c(c(x1)) -> b(x1)
  TDG Processor:
   DPs:
    a#(b(x1)) -> a#(x1)
    a#(b(x1)) -> a#(a(x1))
    a#(b(x1)) -> b#(a(a(x1)))
    a#(b(x1)) -> c#(b(a(a(x1))))
    b#(x1) -> c#(x1)
    c#(c(x1)) -> b#(x1)
   TRS:
    a(x1) -> x1
    a(b(x1)) -> c(b(a(a(x1))))
    b(x1) -> c(x1)
    c(c(x1)) -> b(x1)
   graph:
    c#(c(x1)) -> b#(x1) -> b#(x1) -> c#(x1)
    b#(x1) -> c#(x1) -> c#(c(x1)) -> b#(x1)
    a#(b(x1)) -> c#(b(a(a(x1)))) -> c#(c(x1)) -> b#(x1)
    a#(b(x1)) -> b#(a(a(x1))) -> b#(x1) -> c#(x1)
    a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> c#(b(a(a(x1))))
    a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> b#(a(a(x1)))
    a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(a(x1))
    a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(x1)
    a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> c#(b(a(a(x1))))
    a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> b#(a(a(x1)))
    a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(a(x1))
    a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1)
   SCC Processor:
    #sccs: 2
    #rules: 4
    #arcs: 12/36
    DPs:
     a#(b(x1)) -> a#(a(x1))
     a#(b(x1)) -> a#(x1)
    TRS:
     a(x1) -> x1
     a(b(x1)) -> c(b(a(a(x1))))
     b(x1) -> c(x1)
     c(c(x1)) -> b(x1)
    Arctic Interpretation Processor:
     dimension: 2
     usable rules:
      a(x1) -> x1
      a(b(x1)) -> c(b(a(a(x1))))
      b(x1) -> c(x1)
      c(c(x1)) -> b(x1)
     interpretation:
                [0 0]     [0]
      [b](x0) = [1 1]x0 + [1],
      
      [a#](x0) = [-& 0 ]x0 + [0],
      
                [0 0]     [-&]
      [a](x0) = [0 0]x0 + [0 ],
      
                [0 0]     [0]
      [c](x0) = [1 0]x0 + [0]
     orientation:
      a#(b(x1)) = [1 1]x1 + [1] >= [0 0]x1 + [0] = a#(a(x1))
      
      a#(b(x1)) = [1 1]x1 + [1] >= [-& 0 ]x1 + [0] = a#(x1)
      
              [0 0]     [-&]           
      a(x1) = [0 0]x1 + [0 ] >= x1 = x1
      
                 [1 1]     [1]    [1 1]     [1]                 
      a(b(x1)) = [1 1]x1 + [1] >= [1 1]x1 + [1] = c(b(a(a(x1))))
      
              [0 0]     [0]    [0 0]     [0]        
      b(x1) = [1 1]x1 + [1] >= [1 0]x1 + [0] = c(x1)
      
                 [1 0]     [0]    [0 0]     [0]        
      c(c(x1)) = [1 1]x1 + [1] >= [1 1]x1 + [1] = b(x1)
     problem:
      DPs:
       
      TRS:
       a(x1) -> x1
       a(b(x1)) -> c(b(a(a(x1))))
       b(x1) -> c(x1)
       c(c(x1)) -> b(x1)
     Qed
    
    DPs:
     c#(c(x1)) -> b#(x1)
     b#(x1) -> c#(x1)
    TRS:
     a(x1) -> x1
     a(b(x1)) -> c(b(a(a(x1))))
     b(x1) -> c(x1)
     c(c(x1)) -> b(x1)
    Usable Rule Processor:
     DPs:
      c#(c(x1)) -> b#(x1)
      b#(x1) -> c#(x1)
     TRS:
      
     Matrix Interpretation Processor: dim=5
      
      interpretation:
       [b#](x0) = [0 0 1 0 1]x0,
       
       [c#](x0) = [0 0 1 0 1]x0,
       
                 [0 0 0 0 0]     [0]
                 [0 0 0 0 0]     [0]
       [c](x0) = [0 0 0 0 0]x0 + [1]
                 [0 0 0 0 0]     [0]
                 [0 0 1 0 1]     [1]
      orientation:
       c#(c(x1)) = [0 0 1 0 1]x1 + [2] >= [0 0 1 0 1]x1 = b#(x1)
       
       b#(x1) = [0 0 1 0 1]x1 >= [0 0 1 0 1]x1 = c#(x1)
      problem:
       DPs:
        b#(x1) -> c#(x1)
       TRS:
        
      Restore Modifier:
       DPs:
        b#(x1) -> c#(x1)
       TRS:
        a(x1) -> x1
        a(b(x1)) -> c(b(a(a(x1))))
        b(x1) -> c(x1)
        c(c(x1)) -> b(x1)
       EDG Processor:
        DPs:
         b#(x1) -> c#(x1)
        TRS:
         a(x1) -> x1
         a(b(x1)) -> c(b(a(a(x1))))
         b(x1) -> c(x1)
         c(c(x1)) -> b(x1)
        graph:
         
        SCC Processor:
         #sccs: 0
         #rules: 0
         #arcs: 0/1