YES

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

Proof:
 DP Processor:
  DPs:
   a#(x1) -> b#(x1)
   b#(a(c(x1))) -> a#(x1)
   b#(a(c(x1))) -> a#(a(x1))
   b#(a(c(x1))) -> c#(a(a(x1)))
   b#(a(c(x1))) -> a#(c(a(a(x1))))
   b#(a(c(x1))) -> c#(a(c(a(a(x1)))))
  TRS:
   a(x1) -> b(x1)
   b(a(c(x1))) -> c(a(c(a(a(x1)))))
   c(c(x1)) -> x1
  TDG Processor:
   DPs:
    a#(x1) -> b#(x1)
    b#(a(c(x1))) -> a#(x1)
    b#(a(c(x1))) -> a#(a(x1))
    b#(a(c(x1))) -> c#(a(a(x1)))
    b#(a(c(x1))) -> a#(c(a(a(x1))))
    b#(a(c(x1))) -> c#(a(c(a(a(x1)))))
   TRS:
    a(x1) -> b(x1)
    b(a(c(x1))) -> c(a(c(a(a(x1)))))
    c(c(x1)) -> x1
   graph:
    b#(a(c(x1))) -> a#(c(a(a(x1)))) -> a#(x1) -> b#(x1)
    b#(a(c(x1))) -> a#(a(x1)) -> a#(x1) -> b#(x1)
    b#(a(c(x1))) -> a#(x1) -> a#(x1) -> b#(x1)
    a#(x1) -> b#(x1) -> b#(a(c(x1))) -> c#(a(c(a(a(x1)))))
    a#(x1) -> b#(x1) -> b#(a(c(x1))) -> a#(c(a(a(x1))))
    a#(x1) -> b#(x1) -> b#(a(c(x1))) -> c#(a(a(x1)))
    a#(x1) -> b#(x1) -> b#(a(c(x1))) -> a#(a(x1))
    a#(x1) -> b#(x1) -> b#(a(c(x1))) -> a#(x1)
   SCC Processor:
    #sccs: 1
    #rules: 4
    #arcs: 8/36
    DPs:
     b#(a(c(x1))) -> a#(c(a(a(x1))))
     a#(x1) -> b#(x1)
     b#(a(c(x1))) -> a#(x1)
     b#(a(c(x1))) -> a#(a(x1))
    TRS:
     a(x1) -> b(x1)
     b(a(c(x1))) -> c(a(c(a(a(x1)))))
     c(c(x1)) -> x1
    Arctic Interpretation Processor:
     dimension: 2
     usable rules:
      a(x1) -> b(x1)
      b(a(c(x1))) -> c(a(c(a(a(x1)))))
      c(c(x1)) -> x1
     interpretation:
      [b#](x0) = [1 0]x0 + [0],
      
                [-& 0 ]     [0 ]
      [b](x0) = [0  -&]x0 + [-&],
      
      [a#](x0) = [1 0]x0 + [0],
      
                [-& 0 ]     [0]
      [a](x0) = [0  -&]x0 + [0],
      
                [-& 0 ]     [0]
      [c](x0) = [0  2 ]x0 + [2]
     orientation:
      b#(a(c(x1))) = [1 3]x1 + [3] >= [0 2]x1 + [2] = a#(c(a(a(x1))))
      
      a#(x1) = [1 0]x1 + [0] >= [1 0]x1 + [0] = b#(x1)
      
      b#(a(c(x1))) = [1 3]x1 + [3] >= [1 0]x1 + [0] = a#(x1)
      
      b#(a(c(x1))) = [1 3]x1 + [3] >= [0 1]x1 + [1] = a#(a(x1))
      
              [-& 0 ]     [0]    [-& 0 ]     [0 ]        
      a(x1) = [0  -&]x1 + [0] >= [0  -&]x1 + [-&] = b(x1)
      
                    [-& 0 ]     [0]    [-& 0 ]     [0]                    
      b(a(c(x1))) = [0  2 ]x1 + [2] >= [0  2 ]x1 + [2] = c(a(c(a(a(x1)))))
      
                 [0 2]     [2]           
      c(c(x1)) = [2 4]x1 + [4] >= x1 = x1
     problem:
      DPs:
       a#(x1) -> b#(x1)
       b#(a(c(x1))) -> a#(x1)
      TRS:
       a(x1) -> b(x1)
       b(a(c(x1))) -> c(a(c(a(a(x1)))))
       c(c(x1)) -> x1
     Restore Modifier:
      DPs:
       a#(x1) -> b#(x1)
       b#(a(c(x1))) -> a#(x1)
      TRS:
       a(x1) -> b(x1)
       b(a(c(x1))) -> c(a(c(a(a(x1)))))
       c(c(x1)) -> x1
      EDG Processor:
       DPs:
        a#(x1) -> b#(x1)
        b#(a(c(x1))) -> a#(x1)
       TRS:
        a(x1) -> b(x1)
        b(a(c(x1))) -> c(a(c(a(a(x1)))))
        c(c(x1)) -> x1
       graph:
        b#(a(c(x1))) -> a#(x1) -> a#(x1) -> b#(x1)
        a#(x1) -> b#(x1) -> b#(a(c(x1))) -> a#(x1)
       Usable Rule Processor:
        DPs:
         a#(x1) -> b#(x1)
         b#(a(c(x1))) -> a#(x1)
        TRS:
         
        Arctic Interpretation Processor:
         dimension: 1
         usable rules:
          
         interpretation:
          [b#](x0) = x0,
          
          [a#](x0) = x0 + 0,
          
          [a](x0) = x0 + 0,
          
          [c](x0) = 1x0 + 8
         orientation:
          a#(x1) = x1 + 0 >= x1 = b#(x1)
          
          b#(a(c(x1))) = 1x1 + 8 >= x1 + 0 = a#(x1)
         problem:
          DPs:
           a#(x1) -> b#(x1)
          TRS:
           
         Restore Modifier:
          DPs:
           a#(x1) -> b#(x1)
          TRS:
           a(x1) -> b(x1)
           b(a(c(x1))) -> c(a(c(a(a(x1)))))
           c(c(x1)) -> x1
          EDG Processor:
           DPs:
            a#(x1) -> b#(x1)
           TRS:
            a(x1) -> b(x1)
            b(a(c(x1))) -> c(a(c(a(a(x1)))))
            c(c(x1)) -> x1
           graph:
            
           SCC Processor:
            #sccs: 0
            #rules: 0
            #arcs: 0/1