Precondition costs for structures built with Halicarnassus "build for free"?
There are no building prerequisites. The building names next to the cost indicate that you may construct the building for free, without having to fulfill the resource cost. From the rulebook, p. 4:


  Free Construction (Chains):
  Some structures in Age II and Age III have, to the right of their resource 
  cost, the name of a structure of a previous age. If the player has built the structure named there during a previous Age, that  player  can  build  the  structure  for  free,  meaning  without  having  to  fulfill the resource cost.


Since Halikarnassos allows you to construct any building in the discard pile for free, anyway, the "chained" card is irrelevant at that point. Nevertheless, you are still forbidden from constructing a duplicate card.