YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(x1)) -> b#(x1)
   b#(c(x1)) -> a#(x1)
   c#(b(x1)) -> a#(x1)
   c#(b(x1)) -> c#(a(x1))
   c#(b(x1)) -> c#(c(a(x1)))
   c#(b(x1)) -> b#(c(c(a(x1))))
  TRS:
   a(a(x1)) -> b(x1)
   b(c(x1)) -> a(x1)
   c(b(x1)) -> b(c(c(a(x1))))
  TDG Processor:
   DPs:
    a#(a(x1)) -> b#(x1)
    b#(c(x1)) -> a#(x1)
    c#(b(x1)) -> a#(x1)
    c#(b(x1)) -> c#(a(x1))
    c#(b(x1)) -> c#(c(a(x1)))
    c#(b(x1)) -> b#(c(c(a(x1))))
   TRS:
    a(a(x1)) -> b(x1)
    b(c(x1)) -> a(x1)
    c(b(x1)) -> b(c(c(a(x1))))
   graph:
    c#(b(x1)) -> c#(c(a(x1))) -> c#(b(x1)) -> b#(c(c(a(x1))))
    c#(b(x1)) -> c#(c(a(x1))) -> c#(b(x1)) -> c#(c(a(x1)))
    c#(b(x1)) -> c#(c(a(x1))) -> c#(b(x1)) -> c#(a(x1))
    c#(b(x1)) -> c#(c(a(x1))) -> c#(b(x1)) -> a#(x1)
    c#(b(x1)) -> c#(a(x1)) -> c#(b(x1)) -> b#(c(c(a(x1))))
    c#(b(x1)) -> c#(a(x1)) -> c#(b(x1)) -> c#(c(a(x1)))
    c#(b(x1)) -> c#(a(x1)) -> c#(b(x1)) -> c#(a(x1))
    c#(b(x1)) -> c#(a(x1)) -> c#(b(x1)) -> a#(x1)
    c#(b(x1)) -> b#(c(c(a(x1)))) -> b#(c(x1)) -> a#(x1)
    c#(b(x1)) -> a#(x1) -> a#(a(x1)) -> b#(x1)
    b#(c(x1)) -> a#(x1) -> a#(a(x1)) -> b#(x1)
    a#(a(x1)) -> b#(x1) -> b#(c(x1)) -> a#(x1)
   SCC Processor:
    #sccs: 2
    #rules: 4
    #arcs: 12/36
    DPs:
     c#(b(x1)) -> c#(c(a(x1)))
     c#(b(x1)) -> c#(a(x1))
    TRS:
     a(a(x1)) -> b(x1)
     b(c(x1)) -> a(x1)
     c(b(x1)) -> b(c(c(a(x1))))
    Arctic Interpretation Processor:
     dimension: 3
     usable rules:
      a(a(x1)) -> b(x1)
      b(c(x1)) -> a(x1)
      c(b(x1)) -> b(c(c(a(x1))))
     interpretation:
                [0 0 1]     [1 ]
      [b](x0) = [0 0 0]x0 + [-&]
                [0 1 0]     [0 ],
      
      [c#](x0) = [0  -& 0 ]x0 + [0],
      
                [-& -& 0 ]     [0 ]
      [a](x0) = [-& -& 0 ]x0 + [-&]
                [0  0  1 ]     [1 ],
      
                [0  0  -&]     [0]
      [c](x0) = [0  -& 0 ]x0 + [0]
                [0  -& -&]     [0]
     orientation:
      c#(b(x1)) = [0 1 1]x1 + [1] >= [-& -& 0 ]x1 + [0] = c#(c(a(x1)))
      
      c#(b(x1)) = [0 1 1]x1 + [1] >= [0 0 1]x1 + [1] = c#(a(x1))
      
                 [0 0 1]     [1]    [0 0 1]     [1 ]        
      a(a(x1)) = [0 0 1]x1 + [1] >= [0 0 0]x1 + [-&] = b(x1)
                 [1 1 2]     [2]    [0 1 0]     [0 ]        
      
                 [1 0 0]     [1]    [-& -& 0 ]     [0 ]        
      b(c(x1)) = [0 0 0]x1 + [0] >= [-& -& 0 ]x1 + [-&] = a(x1)
                 [1 0 1]     [1]    [0  0  1 ]     [1 ]        
      
                 [0 0 1]     [1]    [0 0 1]     [1]                 
      c(b(x1)) = [0 1 1]x1 + [1] >= [0 0 1]x1 + [1] = b(c(c(a(x1))))
                 [0 0 1]     [1]    [0 0 1]     [1]                 
     problem:
      DPs:
       c#(b(x1)) -> c#(a(x1))
      TRS:
       a(a(x1)) -> b(x1)
       b(c(x1)) -> a(x1)
       c(b(x1)) -> b(c(c(a(x1))))
     Restore Modifier:
      DPs:
       c#(b(x1)) -> c#(a(x1))
      TRS:
       a(a(x1)) -> b(x1)
       b(c(x1)) -> a(x1)
       c(b(x1)) -> b(c(c(a(x1))))
      EDG Processor:
       DPs:
        c#(b(x1)) -> c#(a(x1))
       TRS:
        a(a(x1)) -> b(x1)
        b(c(x1)) -> a(x1)
        c(b(x1)) -> b(c(c(a(x1))))
       graph:
        c#(b(x1)) -> c#(a(x1)) -> c#(b(x1)) -> c#(a(x1))
       Usable Rule Processor:
        DPs:
         c#(b(x1)) -> c#(a(x1))
        TRS:
         a(a(x1)) -> b(x1)
         b(c(x1)) -> a(x1)
        Arctic Interpretation Processor:
         dimension: 1
         usable rules:
          a(a(x1)) -> b(x1)
          b(c(x1)) -> a(x1)
         interpretation:
          [b](x0) = 9x0 + 10,
          
          [c#](x0) = x0 + 9,
          
          [a](x0) = 8x0 + 8,
          
          [c](x0) = 7x0 + 3
         orientation:
          c#(b(x1)) = 9x1 + 10 >= 8x1 + 9 = c#(a(x1))
          
          a(a(x1)) = 16x1 + 16 >= 9x1 + 10 = b(x1)
          
          b(c(x1)) = 16x1 + 12 >= 8x1 + 8 = a(x1)
         problem:
          DPs:
           
          TRS:
           a(a(x1)) -> b(x1)
           b(c(x1)) -> a(x1)
         Qed
    
    DPs:
     a#(a(x1)) -> b#(x1)
     b#(c(x1)) -> a#(x1)
    TRS:
     a(a(x1)) -> b(x1)
     b(c(x1)) -> a(x1)
     c(b(x1)) -> b(c(c(a(x1))))
    Usable Rule Processor:
     DPs:
      a#(a(x1)) -> b#(x1)
      b#(c(x1)) -> a#(x1)
     TRS:
      
     Arctic Interpretation Processor:
      dimension: 1
      usable rules:
       
      interpretation:
       [b#](x0) = x0,
       
       [a#](x0) = x0,
       
       [a](x0) = x0 + 15,
       
       [c](x0) = 15x0 + 2
      orientation:
       a#(a(x1)) = x1 + 15 >= x1 = b#(x1)
       
       b#(c(x1)) = 15x1 + 2 >= x1 = a#(x1)
      problem:
       DPs:
        a#(a(x1)) -> b#(x1)
       TRS:
        
      Restore Modifier:
       DPs:
        a#(a(x1)) -> b#(x1)
       TRS:
        a(a(x1)) -> b(x1)
        b(c(x1)) -> a(x1)
        c(b(x1)) -> b(c(c(a(x1))))
       EDG Processor:
        DPs:
         a#(a(x1)) -> b#(x1)
        TRS:
         a(a(x1)) -> b(x1)
         b(c(x1)) -> a(x1)
         c(b(x1)) -> b(c(c(a(x1))))
        graph:
         
        SCC Processor:
         #sccs: 0
         #rules: 0
         #arcs: 0/1