YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(x1)) -> c#(x1)
   a#(a(x1)) -> a#(b(c(x1)))
   c#(b(x1)) -> c#(x1)
   c#(b(x1)) -> a#(c(x1))
   c#(b(x1)) -> a#(a(c(x1)))
  TRS:
   a(a(x1)) -> a(b(c(x1)))
   c(a(x1)) -> x1
   c(b(x1)) -> a(a(c(x1)))
  TDG Processor:
   DPs:
    a#(a(x1)) -> c#(x1)
    a#(a(x1)) -> a#(b(c(x1)))
    c#(b(x1)) -> c#(x1)
    c#(b(x1)) -> a#(c(x1))
    c#(b(x1)) -> a#(a(c(x1)))
   TRS:
    a(a(x1)) -> a(b(c(x1)))
    c(a(x1)) -> x1
    c(b(x1)) -> a(a(c(x1)))
   graph:
    c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> a#(a(c(x1)))
    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(c(x1)))
    c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> c#(x1)
    c#(b(x1)) -> a#(a(c(x1))) -> a#(a(x1)) -> a#(b(c(x1)))
    c#(b(x1)) -> a#(a(c(x1))) -> a#(a(x1)) -> c#(x1)
    a#(a(x1)) -> c#(x1) -> c#(b(x1)) -> a#(a(c(x1)))
    a#(a(x1)) -> c#(x1) -> c#(b(x1)) -> a#(c(x1))
    a#(a(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1)
    a#(a(x1)) -> a#(b(c(x1))) -> a#(a(x1)) -> a#(b(c(x1)))
    a#(a(x1)) -> a#(b(c(x1))) -> a#(a(x1)) -> c#(x1)
   EDG Processor:
    DPs:
     a#(a(x1)) -> c#(x1)
     a#(a(x1)) -> a#(b(c(x1)))
     c#(b(x1)) -> c#(x1)
     c#(b(x1)) -> a#(c(x1))
     c#(b(x1)) -> a#(a(c(x1)))
    TRS:
     a(a(x1)) -> a(b(c(x1)))
     c(a(x1)) -> x1
     c(b(x1)) -> a(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)) -> c#(x1) -> c#(b(x1)) -> a#(a(c(x1)))
     c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> c#(x1)
     c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> a#(b(c(x1)))
     c#(b(x1)) -> a#(a(c(x1))) -> a#(a(x1)) -> c#(x1)
     c#(b(x1)) -> a#(a(c(x1))) -> a#(a(x1)) -> a#(b(c(x1)))
     a#(a(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1)
     a#(a(x1)) -> c#(x1) -> c#(b(x1)) -> a#(c(x1))
     a#(a(x1)) -> c#(x1) -> c#(b(x1)) -> a#(a(c(x1)))
    SCC Processor:
     #sccs: 1
     #rules: 4
     #arcs: 10/25
     DPs:
      c#(b(x1)) -> c#(x1)
      c#(b(x1)) -> a#(a(c(x1)))
      a#(a(x1)) -> c#(x1)
      c#(b(x1)) -> a#(c(x1))
     TRS:
      a(a(x1)) -> a(b(c(x1)))
      c(a(x1)) -> x1
      c(b(x1)) -> a(a(c(x1)))
     Arctic Interpretation Processor:
      dimension: 2
      usable rules:
       a(a(x1)) -> a(b(c(x1)))
       c(a(x1)) -> x1
       c(b(x1)) -> a(a(c(x1)))
      interpretation:
       [c#](x0) = [2 3]x0 + [0],
       
       [a#](x0) = [0 3]x0 + [0],
       
                 [2 1]     [2]
       [b](x0) = [1 0]x0 + [1],
       
                 [-& 0 ]     [0]
       [c](x0) = [0  -&]x0 + [0],
       
                 [-& 0 ]     [0]
       [a](x0) = [0  1 ]x0 + [1]
      orientation:
       c#(b(x1)) = [4 3]x1 + [4] >= [2 3]x1 + [0] = c#(x1)
       
       c#(b(x1)) = [4 3]x1 + [4] >= [4 3]x1 + [4] = a#(a(c(x1)))
       
       a#(a(x1)) = [3 4]x1 + [4] >= [2 3]x1 + [0] = c#(x1)
       
       c#(b(x1)) = [4 3]x1 + [4] >= [3 0]x1 + [3] = a#(c(x1))
       
                  [0 1]     [1]    [0 1]     [1]              
       a(a(x1)) = [1 2]x1 + [2] >= [1 2]x1 + [2] = a(b(c(x1)))
       
                  [0  1 ]     [1]           
       c(a(x1)) = [-& 0 ]x1 + [0] >= x1 = x1
       
                  [1 0]     [1]    [1 0]     [1]              
       c(b(x1)) = [2 1]x1 + [2] >= [2 1]x1 + [2] = a(a(c(x1)))
      problem:
       DPs:
        c#(b(x1)) -> c#(x1)
        c#(b(x1)) -> a#(a(c(x1)))
       TRS:
        a(a(x1)) -> a(b(c(x1)))
        c(a(x1)) -> x1
        c(b(x1)) -> a(a(c(x1)))
      Restore Modifier:
       DPs:
        c#(b(x1)) -> c#(x1)
        c#(b(x1)) -> a#(a(c(x1)))
       TRS:
        a(a(x1)) -> a(b(c(x1)))
        c(a(x1)) -> x1
        c(b(x1)) -> a(a(c(x1)))
       EDG Processor:
        DPs:
         c#(b(x1)) -> c#(x1)
         c#(b(x1)) -> a#(a(c(x1)))
        TRS:
         a(a(x1)) -> a(b(c(x1)))
         c(a(x1)) -> x1
         c(b(x1)) -> a(a(c(x1)))
        graph:
         c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> a#(a(c(x1)))
         c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1)
        CDG Processor:
         DPs:
          c#(b(x1)) -> c#(x1)
          c#(b(x1)) -> a#(a(c(x1)))
         TRS:
          a(a(x1)) -> a(b(c(x1)))
          c(a(x1)) -> x1
          c(b(x1)) -> a(a(c(x1)))
         graph:
          
         Qed