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: 2
      interpretation:
       [c#](x0) = [0 0]x0 + [0],
       
                 [0  -&]     [0]
       [c](x0) = [0  0 ]x0 + [0],
       
                 [1 1]     [3]
       [b](x0) = [0 0]x0 + [2],
       
                 [0 1]     [0]
       [a](x0) = [0 1]x0 + [3]
      orientation:
       c#(b(x1)) = [1 1]x1 + [3] >= [0 0]x1 + [0] = c#(c(x1))
       
       c#(b(x1)) = [1 1]x1 + [3] >= [0 0]x1 + [0] = c#(x1)
       
               [0 1]     [0]           
       a(x1) = [0 1]x1 + [3] >= x1 = x1
       
                  [1 2]     [4]    [1 1]     [3]           
       a(a(x1)) = [1 2]x1 + [4] >= [1 1]x1 + [3] = a(b(x1))
       
               [1 1]     [3]           
       b(x1) = [0 0]x1 + [2] >= x1 = x1
       
                  [1 1]     [3]    [1 1]     [3]                 
       c(b(x1)) = [1 1]x1 + [3] >= [1 1]x1 + [3] = a(b(c(c(x1))))
      problem:
       DPs:
        
       TRS:
        a(x1) -> x1
        a(a(x1)) -> a(b(x1))
        b(x1) -> x1
        c(b(x1)) -> a(b(c(c(x1))))
      Qed