MAYBE

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

Proof:
 Complexity Transformation Processor:
  strict:
   a(b(x1)) -> b(r(x1))
   r(a(x1)) -> d(r(x1))
   r(x1) -> d(x1)
   d(a(x1)) -> a(a(d(x1)))
   d(x1) -> a(x1)
  weak:
   
  Matrix Interpretation Processor:
   dimension: 1
   max_matrix:
    1
    interpretation:
     [d](x0) = x0,
     
     [r](x0) = x0 + 1,
     
     [a](x0) = x0,
     
     [b](x0) = x0 + 1
    orientation:
     a(b(x1)) = x1 + 1 >= x1 + 2 = b(r(x1))
     
     r(a(x1)) = x1 + 1 >= x1 + 1 = d(r(x1))
     
     r(x1) = x1 + 1 >= x1 = d(x1)
     
     d(a(x1)) = x1 >= x1 = a(a(d(x1)))
     
     d(x1) = x1 >= x1 = a(x1)
    problem:
     strict:
      a(b(x1)) -> b(r(x1))
      r(a(x1)) -> d(r(x1))
      d(a(x1)) -> a(a(d(x1)))
      d(x1) -> a(x1)
     weak:
      r(x1) -> d(x1)
    Matrix Interpretation Processor:
     dimension: 1
     max_matrix:
      1
      interpretation:
       [d](x0) = x0,
       
       [r](x0) = x0 + 1,
       
       [a](x0) = x0 + 1,
       
       [b](x0) = x0
      orientation:
       a(b(x1)) = x1 + 1 >= x1 + 1 = b(r(x1))
       
       r(a(x1)) = x1 + 2 >= x1 + 1 = d(r(x1))
       
       d(a(x1)) = x1 + 1 >= x1 + 2 = a(a(d(x1)))
       
       d(x1) = x1 >= x1 + 1 = a(x1)
       
       r(x1) = x1 + 1 >= x1 = d(x1)
      problem:
       strict:
        a(b(x1)) -> b(r(x1))
        d(a(x1)) -> a(a(d(x1)))
        d(x1) -> a(x1)
       weak:
        r(a(x1)) -> d(r(x1))
        r(x1) -> d(x1)
      Matrix Interpretation Processor:
       dimension: 1
       max_matrix:
        1
        interpretation:
         [d](x0) = x0,
         
         [r](x0) = x0,
         
         [a](x0) = x0 + 1,
         
         [b](x0) = x0
        orientation:
         a(b(x1)) = x1 + 1 >= x1 = b(r(x1))
         
         d(a(x1)) = x1 + 1 >= x1 + 2 = a(a(d(x1)))
         
         d(x1) = x1 >= x1 + 1 = a(x1)
         
         r(a(x1)) = x1 + 1 >= x1 = d(r(x1))
         
         r(x1) = x1 >= x1 = d(x1)
        problem:
         strict:
          d(a(x1)) -> a(a(d(x1)))
          d(x1) -> a(x1)
         weak:
          a(b(x1)) -> b(r(x1))
          r(a(x1)) -> d(r(x1))
          r(x1) -> d(x1)
        Open