YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(x1)) -> c#(x1)
   a#(a(x1)) -> a#(b(c(x1)))
   c#(b(x1)) -> a#(x1)
   c#(b(x1)) -> c#(a(x1))
   c#(b(x1)) -> a#(c(a(x1)))
  TRS:
   a(x1) -> x1
   a(a(x1)) -> a(b(c(x1)))
   c(b(x1)) -> a(c(a(x1)))
  TDG Processor:
   DPs:
    a#(a(x1)) -> c#(x1)
    a#(a(x1)) -> a#(b(c(x1)))
    c#(b(x1)) -> a#(x1)
    c#(b(x1)) -> c#(a(x1))
    c#(b(x1)) -> a#(c(a(x1)))
   TRS:
    a(x1) -> x1
    a(a(x1)) -> a(b(c(x1)))
    c(b(x1)) -> a(c(a(x1)))
   graph:
    c#(b(x1)) -> c#(a(x1)) -> c#(b(x1)) -> a#(c(a(x1)))
    c#(b(x1)) -> c#(a(x1)) -> c#(b(x1)) -> c#(a(x1))
    c#(b(x1)) -> c#(a(x1)) -> c#(b(x1)) -> a#(x1)
    c#(b(x1)) -> a#(c(a(x1))) -> a#(a(x1)) -> a#(b(c(x1)))
    c#(b(x1)) -> a#(c(a(x1))) -> a#(a(x1)) -> c#(x1)
    c#(b(x1)) -> a#(x1) -> a#(a(x1)) -> a#(b(c(x1)))
    c#(b(x1)) -> a#(x1) -> a#(a(x1)) -> c#(x1)
    a#(a(x1)) -> c#(x1) -> c#(b(x1)) -> a#(c(a(x1)))
    a#(a(x1)) -> c#(x1) -> c#(b(x1)) -> c#(a(x1))
    a#(a(x1)) -> c#(x1) -> c#(b(x1)) -> a#(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)) -> a#(x1)
     c#(b(x1)) -> c#(a(x1))
     c#(b(x1)) -> a#(c(a(x1)))
    TRS:
     a(x1) -> x1
     a(a(x1)) -> a(b(c(x1)))
     c(b(x1)) -> a(c(a(x1)))
    graph:
     c#(b(x1)) -> c#(a(x1)) -> c#(b(x1)) -> a#(x1)
     c#(b(x1)) -> c#(a(x1)) -> c#(b(x1)) -> c#(a(x1))
     c#(b(x1)) -> c#(a(x1)) -> c#(b(x1)) -> a#(c(a(x1)))
     c#(b(x1)) -> a#(c(a(x1))) -> a#(a(x1)) -> c#(x1)
     c#(b(x1)) -> a#(c(a(x1))) -> a#(a(x1)) -> a#(b(c(x1)))
     c#(b(x1)) -> a#(x1) -> a#(a(x1)) -> c#(x1)
     c#(b(x1)) -> a#(x1) -> a#(a(x1)) -> a#(b(c(x1)))
     a#(a(x1)) -> c#(x1) -> c#(b(x1)) -> a#(x1)
     a#(a(x1)) -> c#(x1) -> c#(b(x1)) -> c#(a(x1))
     a#(a(x1)) -> c#(x1) -> c#(b(x1)) -> a#(c(a(x1)))
    SCC Processor:
     #sccs: 1
     #rules: 4
     #arcs: 10/25
     DPs:
      c#(b(x1)) -> c#(a(x1))
      c#(b(x1)) -> a#(c(a(x1)))
      a#(a(x1)) -> c#(x1)
      c#(b(x1)) -> a#(x1)
     TRS:
      a(x1) -> x1
      a(a(x1)) -> a(b(c(x1)))
      c(b(x1)) -> a(c(a(x1)))
     Arctic Interpretation Processor:
      dimension: 2
      interpretation:
       [c#](x0) = [0 0]x0,
       
       [a#](x0) = [0 0]x0 + [0],
       
                 [2 2]     [0 ]
       [b](x0) = [2 0]x0 + [-&],
       
                 [0  -&]  
       [c](x0) = [0  -&]x0,
       
                 [0 0]     [-&]
       [a](x0) = [2 2]x0 + [0 ]
      orientation:
       c#(b(x1)) = [2 2]x1 + [0] >= [2 2]x1 + [0] = c#(a(x1))
       
       c#(b(x1)) = [2 2]x1 + [0] >= [0 0]x1 + [0] = a#(c(a(x1)))
       
       a#(a(x1)) = [2 2]x1 + [0] >= [0 0]x1 = c#(x1)
       
       c#(b(x1)) = [2 2]x1 + [0] >= [0 0]x1 + [0] = a#(x1)
       
               [0 0]     [-&]           
       a(x1) = [2 2]x1 + [0 ] >= x1 = x1
       
                  [2 2]     [0]    [2  -&]     [0]              
       a(a(x1)) = [4 4]x1 + [2] >= [4  -&]x1 + [2] = a(b(c(x1)))
       
                  [2 2]     [0]    [0 0]     [-&]              
       c(b(x1)) = [2 2]x1 + [0] >= [2 2]x1 + [0 ] = a(c(a(x1)))
      problem:
       DPs:
        c#(b(x1)) -> c#(a(x1))
        c#(b(x1)) -> a#(c(a(x1)))
        c#(b(x1)) -> a#(x1)
       TRS:
        a(x1) -> x1
        a(a(x1)) -> a(b(c(x1)))
        c(b(x1)) -> a(c(a(x1)))
      EDG Processor:
       DPs:
        c#(b(x1)) -> c#(a(x1))
        c#(b(x1)) -> a#(c(a(x1)))
        c#(b(x1)) -> a#(x1)
       TRS:
        a(x1) -> x1
        a(a(x1)) -> a(b(c(x1)))
        c(b(x1)) -> a(c(a(x1)))
       graph:
        c#(b(x1)) -> c#(a(x1)) -> c#(b(x1)) -> a#(c(a(x1)))
        c#(b(x1)) -> c#(a(x1)) -> c#(b(x1)) -> c#(a(x1))
        c#(b(x1)) -> c#(a(x1)) -> c#(b(x1)) -> a#(x1)
       SCC Processor:
        #sccs: 1
        #rules: 1
        #arcs: 3/9
        DPs:
         c#(b(x1)) -> c#(a(x1))
        TRS:
         a(x1) -> x1
         a(a(x1)) -> a(b(c(x1)))
         c(b(x1)) -> a(c(a(x1)))
        Arctic Interpretation Processor:
         dimension: 2
         interpretation:
          [c#](x0) = [0  -&]x0 + [0],
          
                    [1 1]     [1]
          [b](x0) = [0 0]x0 + [1],
          
                    [0  -&]     [0]
          [c](x0) = [0  -&]x0 + [0],
          
                    [0 0]     [0]
          [a](x0) = [1 1]x0 + [1]
         orientation:
          c#(b(x1)) = [1 1]x1 + [1] >= [0 0]x1 + [0] = c#(a(x1))
          
                  [0 0]     [0]           
          a(x1) = [1 1]x1 + [1] >= x1 = x1
          
                     [1 1]     [1]    [1  -&]     [1]              
          a(a(x1)) = [2 2]x1 + [2] >= [2  -&]x1 + [2] = a(b(c(x1)))
          
                     [1 1]     [1]    [0 0]     [0]              
          c(b(x1)) = [1 1]x1 + [1] >= [1 1]x1 + [1] = a(c(a(x1)))
         problem:
          DPs:
           
          TRS:
           a(x1) -> x1
           a(a(x1)) -> a(b(c(x1)))
           c(b(x1)) -> a(c(a(x1)))
         Qed