YES

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

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