YES(?,O(n^1))

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

Proof:
 RT Transformation Processor:
  strict:
   a(a(x1)) -> b(c(x1))
   b(b(x1)) -> c(d(x1))
   c(c(x1)) -> d(d(d(x1)))
   d(d(d(x1))) -> a(c(x1))
  weak:
   
  Matrix Interpretation Processor:
   dimension: 1
   interpretation:
    [d](x0) = x0 + 6,
    
    [b](x0) = x0,
    
    [c](x0) = x0 + 19,
    
    [a](x0) = x0 + 20
   orientation:
    a(a(x1)) = x1 + 40 >= x1 + 19 = b(c(x1))
    
    b(b(x1)) = x1 >= x1 + 25 = c(d(x1))
    
    c(c(x1)) = x1 + 38 >= x1 + 18 = d(d(d(x1)))
    
    d(d(d(x1))) = x1 + 18 >= x1 + 39 = a(c(x1))
   problem:
    strict:
     b(b(x1)) -> c(d(x1))
     d(d(d(x1))) -> a(c(x1))
    weak:
     a(a(x1)) -> b(c(x1))
     c(c(x1)) -> d(d(d(x1)))
   Matrix Interpretation Processor:
    dimension: 1
    interpretation:
     [d](x0) = x0 + 14,
     
     [b](x0) = x0 + 19,
     
     [c](x0) = x0 + 21,
     
     [a](x0) = x0 + 20
    orientation:
     b(b(x1)) = x1 + 38 >= x1 + 35 = c(d(x1))
     
     d(d(d(x1))) = x1 + 42 >= x1 + 41 = a(c(x1))
     
     a(a(x1)) = x1 + 40 >= x1 + 40 = b(c(x1))
     
     c(c(x1)) = x1 + 42 >= x1 + 42 = d(d(d(x1)))
    problem:
     strict:
      
     weak:
      b(b(x1)) -> c(d(x1))
      d(d(d(x1))) -> a(c(x1))
      a(a(x1)) -> b(c(x1))
      c(c(x1)) -> d(d(d(x1)))
    Qed