YES

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

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