YES(?,O(n^1))

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

Proof:
 RT Transformation Processor:
  strict:
   a(a(a(x1))) -> b(b(x1))
   b(b(b(x1))) -> c(x1)
   c(x1) -> d(d(x1))
   d(x1) -> a(a(x1))
  weak:
   
  Matrix Interpretation Processor:
   dimension: 1
   interpretation:
    [d](x0) = x0 + 1,
    
    [c](x0) = x0,
    
    [b](x0) = x0 + 17,
    
    [a](x0) = x0
   orientation:
    a(a(a(x1))) = x1 >= x1 + 34 = b(b(x1))
    
    b(b(b(x1))) = x1 + 51 >= x1 = c(x1)
    
    c(x1) = x1 >= x1 + 2 = d(d(x1))
    
    d(x1) = x1 + 1 >= x1 = a(a(x1))
   problem:
    strict:
     a(a(a(x1))) -> b(b(x1))
     c(x1) -> d(d(x1))
    weak:
     b(b(b(x1))) -> c(x1)
     d(x1) -> a(a(x1))
   Matrix Interpretation Processor:
    dimension: 1
    interpretation:
     [d](x0) = x0 + 16,
     
     [c](x0) = x0,
     
     [b](x0) = x0,
     
     [a](x0) = x0 + 8
    orientation:
     a(a(a(x1))) = x1 + 24 >= x1 = b(b(x1))
     
     c(x1) = x1 >= x1 + 32 = d(d(x1))
     
     b(b(b(x1))) = x1 >= x1 = c(x1)
     
     d(x1) = x1 + 16 >= x1 + 16 = a(a(x1))
    problem:
     strict:
      c(x1) -> d(d(x1))
     weak:
      a(a(a(x1))) -> b(b(x1))
      b(b(b(x1))) -> c(x1)
      d(x1) -> a(a(x1))
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [d](x0) = x0 + 10,
      
      [c](x0) = x0 + 21,
      
      [b](x0) = x0 + 7,
      
      [a](x0) = x0 + 5
     orientation:
      c(x1) = x1 + 21 >= x1 + 20 = d(d(x1))
      
      a(a(a(x1))) = x1 + 15 >= x1 + 14 = b(b(x1))
      
      b(b(b(x1))) = x1 + 21 >= x1 + 21 = c(x1)
      
      d(x1) = x1 + 10 >= x1 + 10 = a(a(x1))
     problem:
      strict:
       
      weak:
       c(x1) -> d(d(x1))
       a(a(a(x1))) -> b(b(x1))
       b(b(b(x1))) -> c(x1)
       d(x1) -> a(a(x1))
     Qed