YES

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

Proof:
 DP Processor:
  DPs:
   a#(b(x1)) -> c#(x1)
   a#(b(x1)) -> a#(c(x1))
   a#(b(x1)) -> a#(a(c(x1)))
   a#(b(x1)) -> c#(a(a(c(x1))))
   c#(c(x1)) -> b#(x1)
   c#(c(x1)) -> b#(b(x1))
  TRS:
   a(x1) -> x1
   a(b(x1)) -> c(a(a(c(x1))))
   b(x1) -> x1
   c(c(x1)) -> b(b(x1))
  TDG Processor:
   DPs:
    a#(b(x1)) -> c#(x1)
    a#(b(x1)) -> a#(c(x1))
    a#(b(x1)) -> a#(a(c(x1)))
    a#(b(x1)) -> c#(a(a(c(x1))))
    c#(c(x1)) -> b#(x1)
    c#(c(x1)) -> b#(b(x1))
   TRS:
    a(x1) -> x1
    a(b(x1)) -> c(a(a(c(x1))))
    b(x1) -> x1
    c(c(x1)) -> b(b(x1))
   graph:
    a#(b(x1)) -> c#(a(a(c(x1)))) -> c#(c(x1)) -> b#(b(x1))
    a#(b(x1)) -> c#(a(a(c(x1)))) -> c#(c(x1)) -> b#(x1)
    a#(b(x1)) -> c#(x1) -> c#(c(x1)) -> b#(b(x1))
    a#(b(x1)) -> c#(x1) -> c#(c(x1)) -> b#(x1)
    a#(b(x1)) -> a#(c(x1)) -> a#(b(x1)) -> c#(a(a(c(x1))))
    a#(b(x1)) -> a#(c(x1)) -> a#(b(x1)) -> a#(a(c(x1)))
    a#(b(x1)) -> a#(c(x1)) -> a#(b(x1)) -> a#(c(x1))
    a#(b(x1)) -> a#(c(x1)) -> a#(b(x1)) -> c#(x1)
    a#(b(x1)) -> a#(a(c(x1))) -> a#(b(x1)) -> c#(a(a(c(x1))))
    a#(b(x1)) -> a#(a(c(x1))) -> a#(b(x1)) -> a#(a(c(x1)))
    a#(b(x1)) -> a#(a(c(x1))) -> a#(b(x1)) -> a#(c(x1))
    a#(b(x1)) -> a#(a(c(x1))) -> a#(b(x1)) -> c#(x1)
   CDG Processor:
    DPs:
     a#(b(x1)) -> c#(x1)
     a#(b(x1)) -> a#(c(x1))
     a#(b(x1)) -> a#(a(c(x1)))
     a#(b(x1)) -> c#(a(a(c(x1))))
     c#(c(x1)) -> b#(x1)
     c#(c(x1)) -> b#(b(x1))
    TRS:
     a(x1) -> x1
     a(b(x1)) -> c(a(a(c(x1))))
     b(x1) -> x1
     c(c(x1)) -> b(b(x1))
    graph:
     a#(b(x1)) -> c#(a(a(c(x1)))) -> c#(c(x1)) -> b#(x1)
     a#(b(x1)) -> c#(a(a(c(x1)))) -> c#(c(x1)) -> b#(b(x1))
     a#(b(x1)) -> a#(c(x1)) -> a#(b(x1)) -> c#(x1)
     a#(b(x1)) -> a#(c(x1)) -> a#(b(x1)) -> a#(c(x1))
     a#(b(x1)) -> a#(c(x1)) -> a#(b(x1)) -> a#(a(c(x1)))
     a#(b(x1)) -> a#(c(x1)) -> a#(b(x1)) -> c#(a(a(c(x1))))
     a#(b(x1)) -> a#(a(c(x1))) -> a#(b(x1)) -> c#(x1)
     a#(b(x1)) -> a#(a(c(x1))) -> a#(b(x1)) -> a#(c(x1))
     a#(b(x1)) -> a#(a(c(x1))) -> a#(b(x1)) -> a#(a(c(x1)))
     a#(b(x1)) -> a#(a(c(x1))) -> a#(b(x1)) -> c#(a(a(c(x1))))
    SCC Processor:
     #sccs: 1
     #rules: 2
     #arcs: 10/36
     DPs:
      a#(b(x1)) -> a#(c(x1))
      a#(b(x1)) -> a#(a(c(x1)))
     TRS:
      a(x1) -> x1
      a(b(x1)) -> c(a(a(c(x1))))
      b(x1) -> x1
      c(c(x1)) -> b(b(x1))
     Arctic Interpretation Processor:
      dimension: 2
      usable rules:
       a(x1) -> x1
       a(b(x1)) -> c(a(a(c(x1))))
       b(x1) -> x1
       c(c(x1)) -> b(b(x1))
      interpretation:
       [a#](x0) = [-& 0 ]x0 + [0],
       
                 [3  0 ]     [3]
       [c](x0) = [0  -&]x0 + [0],
       
                 [0  -&]     [0]
       [b](x0) = [3  0 ]x0 + [3],
       
                 [0  3 ]     [0]
       [a](x0) = [-& 0 ]x0 + [0]
      orientation:
       a#(b(x1)) = [3 0]x1 + [3] >= [0  -&]x1 + [0] = a#(c(x1))
       
       a#(b(x1)) = [3 0]x1 + [3] >= [0  -&]x1 + [0] = a#(a(c(x1)))
       
               [0  3 ]     [0]           
       a(x1) = [-& 0 ]x1 + [0] >= x1 = x1
       
                  [6 3]     [6]    [6 3]     [6]                 
       a(b(x1)) = [3 0]x1 + [3] >= [3 0]x1 + [3] = c(a(a(c(x1))))
       
               [0  -&]     [0]           
       b(x1) = [3  0 ]x1 + [3] >= x1 = x1
       
                  [6 3]     [6]    [0  -&]     [0]           
       c(c(x1)) = [3 0]x1 + [3] >= [3  0 ]x1 + [3] = b(b(x1))
      problem:
       DPs:
        
       TRS:
        a(x1) -> x1
        a(b(x1)) -> c(a(a(c(x1))))
        b(x1) -> x1
        c(c(x1)) -> b(b(x1))
      Qed