YES(?,O(n^1))

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

Proof:
 RT Transformation Processor:
  strict:
   a(a(x1)) -> b(b(b(x1)))
   b(b(x1)) -> c(c(c(x1)))
   c(c(x1)) -> d(d(d(x1)))
   b(x1) -> d(d(x1))
   c(d(d(x1))) -> a(x1)
  weak:
   
  Matrix Interpretation Processor:
   dimension: 1
   interpretation:
    [d](x0) = x0 + 9,
    
    [c](x0) = x0 + 1,
    
    [b](x0) = x0 + 9,
    
    [a](x0) = x0 + 8
   orientation:
    a(a(x1)) = x1 + 16 >= x1 + 27 = b(b(b(x1)))
    
    b(b(x1)) = x1 + 18 >= x1 + 3 = c(c(c(x1)))
    
    c(c(x1)) = x1 + 2 >= x1 + 27 = d(d(d(x1)))
    
    b(x1) = x1 + 9 >= x1 + 18 = d(d(x1))
    
    c(d(d(x1))) = x1 + 19 >= x1 + 8 = a(x1)
   problem:
    strict:
     a(a(x1)) -> b(b(b(x1)))
     c(c(x1)) -> d(d(d(x1)))
     b(x1) -> d(d(x1))
    weak:
     b(b(x1)) -> c(c(c(x1)))
     c(d(d(x1))) -> a(x1)
   Matrix Interpretation Processor:
    dimension: 1
    interpretation:
     [d](x0) = x0 + 6,
     
     [c](x0) = x0 + 5,
     
     [b](x0) = x0 + 11,
     
     [a](x0) = x0 + 17
    orientation:
     a(a(x1)) = x1 + 34 >= x1 + 33 = b(b(b(x1)))
     
     c(c(x1)) = x1 + 10 >= x1 + 18 = d(d(d(x1)))
     
     b(x1) = x1 + 11 >= x1 + 12 = d(d(x1))
     
     b(b(x1)) = x1 + 22 >= x1 + 15 = c(c(c(x1)))
     
     c(d(d(x1))) = x1 + 17 >= x1 + 17 = a(x1)
    problem:
     strict:
      c(c(x1)) -> d(d(d(x1)))
      b(x1) -> d(d(x1))
     weak:
      a(a(x1)) -> b(b(b(x1)))
      b(b(x1)) -> c(c(c(x1)))
      c(d(d(x1))) -> a(x1)
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [d](x0) = x0 + 8,
      
      [c](x0) = x0 + 11,
      
      [b](x0) = x0 + 17,
      
      [a](x0) = x0 + 26
     orientation:
      c(c(x1)) = x1 + 22 >= x1 + 24 = d(d(d(x1)))
      
      b(x1) = x1 + 17 >= x1 + 16 = d(d(x1))
      
      a(a(x1)) = x1 + 52 >= x1 + 51 = b(b(b(x1)))
      
      b(b(x1)) = x1 + 34 >= x1 + 33 = c(c(c(x1)))
      
      c(d(d(x1))) = x1 + 27 >= x1 + 26 = a(x1)
     problem:
      strict:
       c(c(x1)) -> d(d(d(x1)))
      weak:
       b(x1) -> d(d(x1))
       a(a(x1)) -> b(b(b(x1)))
       b(b(x1)) -> c(c(c(x1)))
       c(d(d(x1))) -> a(x1)
     Matrix Interpretation Processor:
      dimension: 1
      interpretation:
       [d](x0) = x0 + 5,
       
       [c](x0) = x0 + 8,
       
       [b](x0) = x0 + 12,
       
       [a](x0) = x0 + 18
      orientation:
       c(c(x1)) = x1 + 16 >= x1 + 15 = d(d(d(x1)))
       
       b(x1) = x1 + 12 >= x1 + 10 = d(d(x1))
       
       a(a(x1)) = x1 + 36 >= x1 + 36 = b(b(b(x1)))
       
       b(b(x1)) = x1 + 24 >= x1 + 24 = c(c(c(x1)))
       
       c(d(d(x1))) = x1 + 18 >= x1 + 18 = a(x1)
      problem:
       strict:
        
       weak:
        c(c(x1)) -> d(d(d(x1)))
        b(x1) -> d(d(x1))
        a(a(x1)) -> b(b(b(x1)))
        b(b(x1)) -> c(c(c(x1)))
        c(d(d(x1))) -> a(x1)
      Qed