YES

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

Proof:
 DP Processor:
  DPs:
   a#(b(x1)) -> a#(x1)
   a#(b(x1)) -> a#(a(x1))
   a#(b(x1)) -> a#(a(a(x1)))
   a#(b(x1)) -> b#(a(a(a(x1))))
   b#(a(x1)) -> a#(a(x1))
   a#(a(x1)) -> b#(x1)
   a#(a(x1)) -> a#(c(b(x1)))
  TRS:
   a(b(x1)) -> b(a(a(a(x1))))
   b(a(x1)) -> a(a(x1))
   a(a(x1)) -> a(c(b(x1)))
  TDG Processor:
   DPs:
    a#(b(x1)) -> a#(x1)
    a#(b(x1)) -> a#(a(x1))
    a#(b(x1)) -> a#(a(a(x1)))
    a#(b(x1)) -> b#(a(a(a(x1))))
    b#(a(x1)) -> a#(a(x1))
    a#(a(x1)) -> b#(x1)
    a#(a(x1)) -> a#(c(b(x1)))
   TRS:
    a(b(x1)) -> b(a(a(a(x1))))
    b(a(x1)) -> a(a(x1))
    a(a(x1)) -> a(c(b(x1)))
   graph:
    b#(a(x1)) -> a#(a(x1)) -> a#(a(x1)) -> a#(c(b(x1)))
    b#(a(x1)) -> a#(a(x1)) -> a#(a(x1)) -> b#(x1)
    b#(a(x1)) -> a#(a(x1)) -> a#(b(x1)) -> b#(a(a(a(x1))))
    b#(a(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(a(a(x1)))
    b#(a(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(a(x1))
    b#(a(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(x1)
    a#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(a(x1))
    a#(a(x1)) -> a#(c(b(x1))) -> a#(a(x1)) -> a#(c(b(x1)))
    a#(a(x1)) -> a#(c(b(x1))) -> a#(a(x1)) -> b#(x1)
    a#(a(x1)) -> a#(c(b(x1))) -> a#(b(x1)) -> b#(a(a(a(x1))))
    a#(a(x1)) -> a#(c(b(x1))) -> a#(b(x1)) -> a#(a(a(x1)))
    a#(a(x1)) -> a#(c(b(x1))) -> a#(b(x1)) -> a#(a(x1))
    a#(a(x1)) -> a#(c(b(x1))) -> a#(b(x1)) -> a#(x1)
    a#(b(x1)) -> b#(a(a(a(x1)))) -> b#(a(x1)) -> a#(a(x1))
    a#(b(x1)) -> a#(a(a(x1))) -> a#(a(x1)) -> a#(c(b(x1)))
    a#(b(x1)) -> a#(a(a(x1))) -> a#(a(x1)) -> b#(x1)
    a#(b(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> b#(a(a(a(x1))))
    a#(b(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> a#(a(a(x1)))
    a#(b(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> a#(a(x1))
    a#(b(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> a#(x1)
    a#(b(x1)) -> a#(a(x1)) -> a#(a(x1)) -> a#(c(b(x1)))
    a#(b(x1)) -> a#(a(x1)) -> a#(a(x1)) -> b#(x1)
    a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> b#(a(a(a(x1))))
    a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(a(a(x1)))
    a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(a(x1))
    a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(x1)
    a#(b(x1)) -> a#(x1) -> a#(a(x1)) -> a#(c(b(x1)))
    a#(b(x1)) -> a#(x1) -> a#(a(x1)) -> b#(x1)
    a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> b#(a(a(a(x1))))
    a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(a(a(x1)))
    a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(a(x1))
    a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1)
   EDG Processor:
    DPs:
     a#(b(x1)) -> a#(x1)
     a#(b(x1)) -> a#(a(x1))
     a#(b(x1)) -> a#(a(a(x1)))
     a#(b(x1)) -> b#(a(a(a(x1))))
     b#(a(x1)) -> a#(a(x1))
     a#(a(x1)) -> b#(x1)
     a#(a(x1)) -> a#(c(b(x1)))
    TRS:
     a(b(x1)) -> b(a(a(a(x1))))
     b(a(x1)) -> a(a(x1))
     a(a(x1)) -> a(c(b(x1)))
    graph:
     b#(a(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(x1)
     b#(a(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(a(x1))
     b#(a(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(a(a(x1)))
     b#(a(x1)) -> a#(a(x1)) -> a#(b(x1)) -> b#(a(a(a(x1))))
     b#(a(x1)) -> a#(a(x1)) -> a#(a(x1)) -> b#(x1)
     b#(a(x1)) -> a#(a(x1)) -> a#(a(x1)) -> a#(c(b(x1)))
     a#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(a(x1))
     a#(b(x1)) -> b#(a(a(a(x1)))) -> b#(a(x1)) -> a#(a(x1))
     a#(b(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> a#(x1)
     a#(b(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> a#(a(x1))
     a#(b(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> a#(a(a(x1)))
     a#(b(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> b#(a(a(a(x1))))
     a#(b(x1)) -> a#(a(a(x1))) -> a#(a(x1)) -> b#(x1)
     a#(b(x1)) -> a#(a(a(x1))) -> a#(a(x1)) -> a#(c(b(x1)))
     a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(x1)
     a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(a(x1))
     a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(a(a(x1)))
     a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> b#(a(a(a(x1))))
     a#(b(x1)) -> a#(a(x1)) -> a#(a(x1)) -> b#(x1)
     a#(b(x1)) -> a#(a(x1)) -> a#(a(x1)) -> a#(c(b(x1)))
     a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1)
     a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(a(x1))
     a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(a(a(x1)))
     a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> b#(a(a(a(x1))))
     a#(b(x1)) -> a#(x1) -> a#(a(x1)) -> b#(x1)
     a#(b(x1)) -> a#(x1) -> a#(a(x1)) -> a#(c(b(x1)))
    SCC Processor:
     #sccs: 1
     #rules: 6
     #arcs: 26/49
     DPs:
      b#(a(x1)) -> a#(a(x1))
      a#(a(x1)) -> b#(x1)
      a#(b(x1)) -> b#(a(a(a(x1))))
      a#(b(x1)) -> a#(a(a(x1)))
      a#(b(x1)) -> a#(a(x1))
      a#(b(x1)) -> a#(x1)
     TRS:
      a(b(x1)) -> b(a(a(a(x1))))
      b(a(x1)) -> a(a(x1))
      a(a(x1)) -> a(c(b(x1)))
     Matrix Interpretation Processor: dim=3
      
      interpretation:
       [b#](x0) = [2 0 2]x0,
       
       [a#](x0) = [0 2 2]x0 + [1],
       
                 [0 0 1]     [1]
       [c](x0) = [0 0 0]x0 + [0]
                 [0 0 0]     [0],
       
                 [0 1 0]     [1]
       [a](x0) = [0 1 0]x0 + [0]
                 [1 0 1]     [0],
       
                 [0 3 0]     [3]
       [b](x0) = [0 3 0]x0 + [3]
                 [1 0 1]     [0]
      orientation:
       b#(a(x1)) = [2 2 2]x1 + [2] >= [2 2 2]x1 + [1] = a#(a(x1))
       
       a#(a(x1)) = [2 2 2]x1 + [1] >= [2 0 2]x1 = b#(x1)
       
       a#(b(x1)) = [2 6 2]x1 + [7] >= [2 6 2]x1 + [6] = b#(a(a(a(x1))))
       
       a#(b(x1)) = [2 6 2]x1 + [7] >= [2 4 2]x1 + [3] = a#(a(a(x1)))
       
       a#(b(x1)) = [2 6 2]x1 + [7] >= [2 2 2]x1 + [1] = a#(a(x1))
       
       a#(b(x1)) = [2 6 2]x1 + [7] >= [0 2 2]x1 + [1] = a#(x1)
       
                  [0 3 0]     [4]    [0 3 0]     [3]                 
       a(b(x1)) = [0 3 0]x1 + [3] >= [0 3 0]x1 + [3] = b(a(a(a(x1))))
                  [1 3 1]     [3]    [1 3 1]     [3]                 
       
                  [0 3 0]     [3]    [0 1 0]     [1]           
       b(a(x1)) = [0 3 0]x1 + [3] >= [0 1 0]x1 + [0] = a(a(x1))
                  [1 1 1]     [1]    [1 1 1]     [1]           
       
                  [0 1 0]     [1]    [0 0 0]     [1]              
       a(a(x1)) = [0 1 0]x1 + [0] >= [0 0 0]x1 + [0] = a(c(b(x1)))
                  [1 1 1]     [1]    [1 0 1]     [1]              
      problem:
       DPs:
        
       TRS:
        a(b(x1)) -> b(a(a(a(x1))))
        b(a(x1)) -> a(a(x1))
        a(a(x1)) -> a(c(b(x1)))
      Qed