Bug in ties over barlines

[Joseph Haig]

I have (I believe) found a bug in Lilypond, and I am fairly sure what
it is, but I would like to check with people who have better knowledge
of music theory than I before I submit it to the bug list. In the
following code:

  {
    \time 4/4
    aes'1( a')
    aes'~ aes'
    aes'( aes')
  }

I believe that the first and third ties are displayed incorrectly.
Specifically, the first tie should have a natural in front of the
second note, and the third tie should not have a flat in front of the
second note.The second and third ties should be identical, except for
the type of tie used, while the first one should be different. Am I
correct?

Thanks,

Joe