fact: Definition alphabet_pos a := match a with a_ => one | b_ => xL one | c_ => xR one | d_ => xL (xL one) | e_ => xR (xL one) | f_ => xL (xR one) | g_ => xR (xR one) | h_ => xL (xL (xL one)) | i_ => xR (xL (xL one)) | j_ => xL (xR (xL one)) | k_ => xR (xR (xL one)) | l_ => xL (xL (xR one)) | m_ => xR (xL (xR one)) | n_ => xL (xR (xR one)) | o_ => xR (xR (xR one)) | p_ => xL (xL (xL (xL one))) | q_ => xR (xL (xL (xL one))) | r_ => xL (xR (xL (xL one))) | s_ => xR (xR (xL (xL one))) | t_ => xL (xL (xR (xL one))) | u_ => xR (xL (xR (xL one))) | v_ => xL (xR (xR (xL one))) | w_ => xR (xR (xR (xL one))) | x_ => xL (xL (xL (xR one))) | y_ => xR (xL (xL (xR one))) | z_ => xL (xR (xL (xR one))) end.