YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(x1)) -> c#(a(x1))
   a#(a(x1)) -> a#(c(a(x1)))
   a#(a(x1)) -> a#(b(a(c(a(x1)))))
   c#(b(x1)) -> c#(x1)
   c#(b(x1)) -> a#(c(x1))
  TRS:
   a(x1) -> x1
   a(a(x1)) -> a(b(a(c(a(x1)))))
   c(b(x1)) -> a(c(x1))
  TDG Processor:
   DPs:
    a#(a(x1)) -> c#(a(x1))
    a#(a(x1)) -> a#(c(a(x1)))
    a#(a(x1)) -> a#(b(a(c(a(x1)))))
    c#(b(x1)) -> c#(x1)
    c#(b(x1)) -> a#(c(x1))
   TRS:
    a(x1) -> x1
    a(a(x1)) -> a(b(a(c(a(x1)))))
    c(b(x1)) -> a(c(x1))
   graph:
    c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> a#(c(x1))
    c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1)
    c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> a#(b(a(c(a(x1)))))
    c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> a#(c(a(x1)))
    c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> c#(a(x1))
    a#(a(x1)) -> c#(a(x1)) -> c#(b(x1)) -> a#(c(x1))
    a#(a(x1)) -> c#(a(x1)) -> c#(b(x1)) -> c#(x1)
    a#(a(x1)) -> a#(b(a(c(a(x1))))) ->
    a#(a(x1)) -> a#(b(a(c(a(x1)))))
    a#(a(x1)) -> a#(b(a(c(a(x1))))) -> a#(a(x1)) -> a#(c(a(x1)))
    a#(a(x1)) -> a#(b(a(c(a(x1))))) -> a#(a(x1)) -> c#(a(x1))
    a#(a(x1)) -> a#(c(a(x1))) -> a#(a(x1)) -> a#(b(a(c(a(x1)))))
    a#(a(x1)) -> a#(c(a(x1))) -> a#(a(x1)) -> a#(c(a(x1)))
    a#(a(x1)) -> a#(c(a(x1))) -> a#(a(x1)) -> c#(a(x1))
   EDG Processor:
    DPs:
     a#(a(x1)) -> c#(a(x1))
     a#(a(x1)) -> a#(c(a(x1)))
     a#(a(x1)) -> a#(b(a(c(a(x1)))))
     c#(b(x1)) -> c#(x1)
     c#(b(x1)) -> a#(c(x1))
    TRS:
     a(x1) -> x1
     a(a(x1)) -> a(b(a(c(a(x1)))))
     c(b(x1)) -> a(c(x1))
    graph:
     c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1)
     c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> a#(c(x1))
     c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> c#(a(x1))
     c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> a#(c(a(x1)))
     c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> a#(b(a(c(a(x1)))))
     a#(a(x1)) -> c#(a(x1)) -> c#(b(x1)) -> c#(x1)
     a#(a(x1)) -> c#(a(x1)) -> c#(b(x1)) -> a#(c(x1))
     a#(a(x1)) -> a#(c(a(x1))) -> a#(a(x1)) -> c#(a(x1))
     a#(a(x1)) -> a#(c(a(x1))) -> a#(a(x1)) -> a#(c(a(x1)))
     a#(a(x1)) -> a#(c(a(x1))) -> a#(a(x1)) -> a#(b(a(c(a(x1)))))
    SCC Processor:
     #sccs: 1
     #rules: 4
     #arcs: 10/25
     DPs:
      c#(b(x1)) -> c#(x1)
      c#(b(x1)) -> a#(c(x1))
      a#(a(x1)) -> a#(c(a(x1)))
      a#(a(x1)) -> c#(a(x1))
     TRS:
      a(x1) -> x1
      a(a(x1)) -> a(b(a(c(a(x1)))))
      c(b(x1)) -> a(c(x1))
     Arctic Interpretation Processor:
      dimension: 2
      interpretation:
       [c#](x0) = [-& 0 ]x0 + [0],
       
       [a#](x0) = [0 1]x0 + [0],
       
                 [0 1]     [0]
       [b](x0) = [0 1]x0 + [1],
       
                 [-& 0 ]     [0]
       [c](x0) = [-& 0 ]x0 + [0],
       
                 [1 1]     [1]
       [a](x0) = [0 0]x0 + [0]
      orientation:
       c#(b(x1)) = [0 1]x1 + [1] >= [-& 0 ]x1 + [0] = c#(x1)
       
       c#(b(x1)) = [0 1]x1 + [1] >= [-& 1 ]x1 + [1] = a#(c(x1))
       
       a#(a(x1)) = [1 1]x1 + [1] >= [1 1]x1 + [1] = a#(c(a(x1)))
       
       a#(a(x1)) = [1 1]x1 + [1] >= [0 0]x1 + [0] = c#(a(x1))
       
               [1 1]     [1]           
       a(x1) = [0 0]x1 + [0] >= x1 = x1
       
                  [2 2]     [2]    [2 2]     [2]                    
       a(a(x1)) = [1 1]x1 + [1] >= [1 1]x1 + [1] = a(b(a(c(a(x1)))))
       
                  [0 1]     [1]    [-& 1 ]     [1]           
       c(b(x1)) = [0 1]x1 + [1] >= [-& 0 ]x1 + [0] = a(c(x1))
      problem:
       DPs:
        c#(b(x1)) -> a#(c(x1))
        a#(a(x1)) -> a#(c(a(x1)))
       TRS:
        a(x1) -> x1
        a(a(x1)) -> a(b(a(c(a(x1)))))
        c(b(x1)) -> a(c(x1))
      EDG Processor:
       DPs:
        c#(b(x1)) -> a#(c(x1))
        a#(a(x1)) -> a#(c(a(x1)))
       TRS:
        a(x1) -> x1
        a(a(x1)) -> a(b(a(c(a(x1)))))
        c(b(x1)) -> a(c(x1))
       graph:
        c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> a#(c(a(x1)))
        a#(a(x1)) -> a#(c(a(x1))) -> a#(a(x1)) -> a#(c(a(x1)))
       SCC Processor:
        #sccs: 1
        #rules: 1
        #arcs: 2/4
        DPs:
         a#(a(x1)) -> a#(c(a(x1)))
        TRS:
         a(x1) -> x1
         a(a(x1)) -> a(b(a(c(a(x1)))))
         c(b(x1)) -> a(c(x1))
        Arctic Interpretation Processor:
         dimension: 2
         interpretation:
          [a#](x0) = [0  -&]x0 + [0],
          
                    [0  1 ]     [0]
          [b](x0) = [-& 1 ]x0 + [1],
          
                    [-& 0 ]     [0]
          [c](x0) = [-& 0 ]x0 + [0],
          
                    [1 1]     [1]
          [a](x0) = [0 0]x0 + [0]
         orientation:
          a#(a(x1)) = [1 1]x1 + [1] >= [0 0]x1 + [0] = a#(c(a(x1)))
          
                  [1 1]     [1]           
          a(x1) = [0 0]x1 + [0] >= x1 = x1
          
                     [2 2]     [2]    [2 2]     [2]                    
          a(a(x1)) = [1 1]x1 + [1] >= [1 1]x1 + [1] = a(b(a(c(a(x1)))))
          
                     [-& 1 ]     [1]    [-& 1 ]     [1]           
          c(b(x1)) = [-& 1 ]x1 + [1] >= [-& 0 ]x1 + [0] = a(c(x1))
         problem:
          DPs:
           
          TRS:
           a(x1) -> x1
           a(a(x1)) -> a(b(a(c(a(x1)))))
           c(b(x1)) -> a(c(x1))
         Qed