YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(x1)) -> b#(x1)
   b#(a(x1)) -> b#(x1)
   b#(a(x1)) -> a#(b(x1))
   b#(b(c(x1))) -> a#(x1)
   b#(b(c(x1))) -> c#(a(x1))
   b#(b(x1)) -> a#(x1)
   b#(b(x1)) -> a#(a(x1))
   b#(b(x1)) -> a#(a(a(x1)))
   c#(a(x1)) -> c#(x1)
   c#(a(x1)) -> a#(c(x1))
   c#(a(x1)) -> b#(a(c(x1)))
  TRS:
   a(a(x1)) -> b(x1)
   b(a(x1)) -> a(b(x1))
   b(b(c(x1))) -> c(a(x1))
   b(b(x1)) -> a(a(a(x1)))
   c(a(x1)) -> b(a(c(x1)))
  TDG Processor:
   DPs:
    a#(a(x1)) -> b#(x1)
    b#(a(x1)) -> b#(x1)
    b#(a(x1)) -> a#(b(x1))
    b#(b(c(x1))) -> a#(x1)
    b#(b(c(x1))) -> c#(a(x1))
    b#(b(x1)) -> a#(x1)
    b#(b(x1)) -> a#(a(x1))
    b#(b(x1)) -> a#(a(a(x1)))
    c#(a(x1)) -> c#(x1)
    c#(a(x1)) -> a#(c(x1))
    c#(a(x1)) -> b#(a(c(x1)))
   TRS:
    a(a(x1)) -> b(x1)
    b(a(x1)) -> a(b(x1))
    b(b(c(x1))) -> c(a(x1))
    b(b(x1)) -> a(a(a(x1)))
    c(a(x1)) -> b(a(c(x1)))
   graph:
    c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> b#(a(c(x1)))
    c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> a#(c(x1))
    c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> c#(x1)
    c#(a(x1)) -> b#(a(c(x1))) -> b#(b(x1)) -> a#(a(a(x1)))
    c#(a(x1)) -> b#(a(c(x1))) -> b#(b(x1)) -> a#(a(x1))
    c#(a(x1)) -> b#(a(c(x1))) -> b#(b(x1)) -> a#(x1)
    c#(a(x1)) -> b#(a(c(x1))) -> b#(b(c(x1))) -> c#(a(x1))
    c#(a(x1)) -> b#(a(c(x1))) -> b#(b(c(x1))) -> a#(x1)
    c#(a(x1)) -> b#(a(c(x1))) -> b#(a(x1)) -> a#(b(x1))
    c#(a(x1)) -> b#(a(c(x1))) -> b#(a(x1)) -> b#(x1)
    c#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> b#(x1)
    b#(b(c(x1))) -> c#(a(x1)) -> c#(a(x1)) -> b#(a(c(x1)))
    b#(b(c(x1))) -> c#(a(x1)) -> c#(a(x1)) -> a#(c(x1))
    b#(b(c(x1))) -> c#(a(x1)) -> c#(a(x1)) -> c#(x1)
    b#(b(c(x1))) -> a#(x1) -> a#(a(x1)) -> b#(x1)
    b#(b(x1)) -> a#(a(a(x1))) -> a#(a(x1)) -> b#(x1)
    b#(b(x1)) -> a#(a(x1)) -> a#(a(x1)) -> b#(x1)
    b#(b(x1)) -> a#(x1) -> a#(a(x1)) -> b#(x1)
    b#(a(x1)) -> b#(x1) -> b#(b(x1)) -> a#(a(a(x1)))
    b#(a(x1)) -> b#(x1) -> b#(b(x1)) -> a#(a(x1))
    b#(a(x1)) -> b#(x1) -> b#(b(x1)) -> a#(x1)
    b#(a(x1)) -> b#(x1) -> b#(b(c(x1))) -> c#(a(x1))
    b#(a(x1)) -> b#(x1) -> b#(b(c(x1))) -> a#(x1)
    b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(b(x1))
    b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(x1)
    b#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) -> b#(x1)
    a#(a(x1)) -> b#(x1) -> b#(b(x1)) -> a#(a(a(x1)))
    a#(a(x1)) -> b#(x1) -> b#(b(x1)) -> a#(a(x1))
    a#(a(x1)) -> b#(x1) -> b#(b(x1)) -> a#(x1)
    a#(a(x1)) -> b#(x1) -> b#(b(c(x1))) -> c#(a(x1))
    a#(a(x1)) -> b#(x1) -> b#(b(c(x1))) -> a#(x1)
    a#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(b(x1))
    a#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(x1)
   Arctic Interpretation Processor:
    dimension: 1
    interpretation:
     [c#](x0) = 1x0,
     
     [b#](x0) = x0,
     
     [a#](x0) = x0,
     
     [c](x0) = 1x0,
     
     [b](x0) = x0,
     
     [a](x0) = x0
    orientation:
     a#(a(x1)) = x1 >= x1 = b#(x1)
     
     b#(a(x1)) = x1 >= x1 = b#(x1)
     
     b#(a(x1)) = x1 >= x1 = a#(b(x1))
     
     b#(b(c(x1))) = 1x1 >= x1 = a#(x1)
     
     b#(b(c(x1))) = 1x1 >= 1x1 = c#(a(x1))
     
     b#(b(x1)) = x1 >= x1 = a#(x1)
     
     b#(b(x1)) = x1 >= x1 = a#(a(x1))
     
     b#(b(x1)) = x1 >= x1 = a#(a(a(x1)))
     
     c#(a(x1)) = 1x1 >= 1x1 = c#(x1)
     
     c#(a(x1)) = 1x1 >= 1x1 = a#(c(x1))
     
     c#(a(x1)) = 1x1 >= 1x1 = b#(a(c(x1)))
     
     a(a(x1)) = x1 >= x1 = b(x1)
     
     b(a(x1)) = x1 >= x1 = a(b(x1))
     
     b(b(c(x1))) = 1x1 >= 1x1 = c(a(x1))
     
     b(b(x1)) = x1 >= x1 = a(a(a(x1)))
     
     c(a(x1)) = 1x1 >= 1x1 = b(a(c(x1)))
    problem:
     DPs:
      a#(a(x1)) -> b#(x1)
      b#(a(x1)) -> b#(x1)
      b#(a(x1)) -> a#(b(x1))
      b#(b(c(x1))) -> c#(a(x1))
      b#(b(x1)) -> a#(x1)
      b#(b(x1)) -> a#(a(x1))
      b#(b(x1)) -> a#(a(a(x1)))
      c#(a(x1)) -> c#(x1)
      c#(a(x1)) -> a#(c(x1))
      c#(a(x1)) -> b#(a(c(x1)))
     TRS:
      a(a(x1)) -> b(x1)
      b(a(x1)) -> a(b(x1))
      b(b(c(x1))) -> c(a(x1))
      b(b(x1)) -> a(a(a(x1)))
      c(a(x1)) -> b(a(c(x1)))
    EDG Processor:
     DPs:
      a#(a(x1)) -> b#(x1)
      b#(a(x1)) -> b#(x1)
      b#(a(x1)) -> a#(b(x1))
      b#(b(c(x1))) -> c#(a(x1))
      b#(b(x1)) -> a#(x1)
      b#(b(x1)) -> a#(a(x1))
      b#(b(x1)) -> a#(a(a(x1)))
      c#(a(x1)) -> c#(x1)
      c#(a(x1)) -> a#(c(x1))
      c#(a(x1)) -> b#(a(c(x1)))
     TRS:
      a(a(x1)) -> b(x1)
      b(a(x1)) -> a(b(x1))
      b(b(c(x1))) -> c(a(x1))
      b(b(x1)) -> a(a(a(x1)))
      c(a(x1)) -> b(a(c(x1)))
     graph:
      c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> c#(x1)
      c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> a#(c(x1))
      c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> b#(a(c(x1)))
      c#(a(x1)) -> b#(a(c(x1))) -> b#(a(x1)) -> b#(x1)
      c#(a(x1)) -> b#(a(c(x1))) -> b#(a(x1)) -> a#(b(x1))
      c#(a(x1)) -> b#(a(c(x1))) -> b#(b(c(x1))) -> c#(a(x1))
      c#(a(x1)) -> b#(a(c(x1))) -> b#(b(x1)) -> a#(x1)
      c#(a(x1)) -> b#(a(c(x1))) -> b#(b(x1)) -> a#(a(x1))
      c#(a(x1)) -> b#(a(c(x1))) -> b#(b(x1)) -> a#(a(a(x1)))
      c#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> b#(x1)
      b#(b(c(x1))) -> c#(a(x1)) -> c#(a(x1)) -> c#(x1)
      b#(b(c(x1))) -> c#(a(x1)) -> c#(a(x1)) -> a#(c(x1))
      b#(b(c(x1))) -> c#(a(x1)) -> c#(a(x1)) -> b#(a(c(x1)))
      b#(b(x1)) -> a#(a(a(x1))) -> a#(a(x1)) -> b#(x1)
      b#(b(x1)) -> a#(a(x1)) -> a#(a(x1)) -> b#(x1)
      b#(b(x1)) -> a#(x1) -> a#(a(x1)) -> b#(x1)
      b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(x1)
      b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(b(x1))
      b#(a(x1)) -> b#(x1) -> b#(b(c(x1))) -> c#(a(x1))
      b#(a(x1)) -> b#(x1) -> b#(b(x1)) -> a#(x1)
      b#(a(x1)) -> b#(x1) -> b#(b(x1)) -> a#(a(x1))
      b#(a(x1)) -> b#(x1) -> b#(b(x1)) -> a#(a(a(x1)))
      b#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) -> b#(x1)
      a#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(x1)
      a#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(b(x1))
      a#(a(x1)) -> b#(x1) -> b#(b(c(x1))) -> c#(a(x1))
      a#(a(x1)) -> b#(x1) -> b#(b(x1)) -> a#(x1)
      a#(a(x1)) -> b#(x1) -> b#(b(x1)) -> a#(a(x1))
      a#(a(x1)) -> b#(x1) -> b#(b(x1)) -> a#(a(a(x1)))
     Matrix Interpretation Processor: dim=1
      
      interpretation:
       [c#](x0) = 3x0,
       
       [b#](x0) = x0 + 1,
       
       [a#](x0) = x0,
       
       [c](x0) = 3x0,
       
       [b](x0) = x0 + 2,
       
       [a](x0) = x0 + 1
      orientation:
       a#(a(x1)) = x1 + 1 >= x1 + 1 = b#(x1)
       
       b#(a(x1)) = x1 + 2 >= x1 + 1 = b#(x1)
       
       b#(a(x1)) = x1 + 2 >= x1 + 2 = a#(b(x1))
       
       b#(b(c(x1))) = 3x1 + 3 >= 3x1 + 3 = c#(a(x1))
       
       b#(b(x1)) = x1 + 3 >= x1 = a#(x1)
       
       b#(b(x1)) = x1 + 3 >= x1 + 1 = a#(a(x1))
       
       b#(b(x1)) = x1 + 3 >= x1 + 2 = a#(a(a(x1)))
       
       c#(a(x1)) = 3x1 + 3 >= 3x1 = c#(x1)
       
       c#(a(x1)) = 3x1 + 3 >= 3x1 = a#(c(x1))
       
       c#(a(x1)) = 3x1 + 3 >= 3x1 + 2 = b#(a(c(x1)))
       
       a(a(x1)) = x1 + 2 >= x1 + 2 = b(x1)
       
       b(a(x1)) = x1 + 3 >= x1 + 3 = a(b(x1))
       
       b(b(c(x1))) = 3x1 + 4 >= 3x1 + 3 = c(a(x1))
       
       b(b(x1)) = x1 + 4 >= x1 + 3 = a(a(a(x1)))
       
       c(a(x1)) = 3x1 + 3 >= 3x1 + 3 = b(a(c(x1)))
      problem:
       DPs:
        a#(a(x1)) -> b#(x1)
        b#(a(x1)) -> a#(b(x1))
        b#(b(c(x1))) -> c#(a(x1))
       TRS:
        a(a(x1)) -> b(x1)
        b(a(x1)) -> a(b(x1))
        b(b(c(x1))) -> c(a(x1))
        b(b(x1)) -> a(a(a(x1)))
        c(a(x1)) -> b(a(c(x1)))
      EDG Processor:
       DPs:
        a#(a(x1)) -> b#(x1)
        b#(a(x1)) -> a#(b(x1))
        b#(b(c(x1))) -> c#(a(x1))
       TRS:
        a(a(x1)) -> b(x1)
        b(a(x1)) -> a(b(x1))
        b(b(c(x1))) -> c(a(x1))
        b(b(x1)) -> a(a(a(x1)))
        c(a(x1)) -> b(a(c(x1)))
       graph:
        b#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) -> b#(x1)
        a#(a(x1)) -> b#(x1) -> b#(b(c(x1))) -> c#(a(x1))
        a#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(b(x1))
       SCC Processor:
        #sccs: 1
        #rules: 2
        #arcs: 3/9
        DPs:
         b#(a(x1)) -> a#(b(x1))
         a#(a(x1)) -> b#(x1)
        TRS:
         a(a(x1)) -> b(x1)
         b(a(x1)) -> a(b(x1))
         b(b(c(x1))) -> c(a(x1))
         b(b(x1)) -> a(a(a(x1)))
         c(a(x1)) -> b(a(c(x1)))
        Matrix Interpretation Processor: dim=1
         
         interpretation:
          [b#](x0) = 4x0 + 6,
          
          [a#](x0) = 4x0,
          
          [c](x0) = 3x0,
          
          [b](x0) = x0 + 4,
          
          [a](x0) = x0 + 5/2
         orientation:
          b#(a(x1)) = 4x1 + 16 >= 4x1 + 16 = a#(b(x1))
          
          a#(a(x1)) = 4x1 + 10 >= 4x1 + 6 = b#(x1)
          
          a(a(x1)) = x1 + 5 >= x1 + 4 = b(x1)
          
          b(a(x1)) = x1 + 13/2 >= x1 + 13/2 = a(b(x1))
          
          b(b(c(x1))) = 3x1 + 8 >= 3x1 + 15/2 = c(a(x1))
          
          b(b(x1)) = x1 + 8 >= x1 + 15/2 = a(a(a(x1)))
          
          c(a(x1)) = 3x1 + 15/2 >= 3x1 + 13/2 = b(a(c(x1)))
         problem:
          DPs:
           b#(a(x1)) -> a#(b(x1))
          TRS:
           a(a(x1)) -> b(x1)
           b(a(x1)) -> a(b(x1))
           b(b(c(x1))) -> c(a(x1))
           b(b(x1)) -> a(a(a(x1)))
           c(a(x1)) -> b(a(c(x1)))
         EDG Processor:
          DPs:
           b#(a(x1)) -> a#(b(x1))
          TRS:
           a(a(x1)) -> b(x1)
           b(a(x1)) -> a(b(x1))
           b(b(c(x1))) -> c(a(x1))
           b(b(x1)) -> a(a(a(x1)))
           c(a(x1)) -> b(a(c(x1)))
          graph:
           
          Qed