{--------------------------------------------------------------------
  Utility functions that maintain the balance properties of the tree.
  All constructors assume that all values in [l] < [x] and all values
  in [r] > [x], and that [l] and [r] are valid trees.

  In order of sophistication:
    [Bin sz x l r]    The type constructor.
    [bin x l r]       Maintains the correct size, assumes that both [l]
                      and [r] are balanced with respect to each other.
    [balance x l r]   Restores the balance and size.
                      Assumes that the original tree was balanced and
                      that [l] or [r] has changed by at most one element.
    [join x l r]      Restores balance and size.

  Furthermore, we can construct a new tree from two trees. Both operations
  assume that all values in [l] < all values in [r] and that [l] and [r]
  are valid:
    [glue l r]        Glues [l] and [r] together. Assumes that [l] and
                      [r] are already balanced with respect to each other.
    [merge l r]       Merges two trees and restores balance.

  Note: in contrast to Adam's paper, we use (<=) comparisons instead
  of (<) comparisons in [join], [merge] and [balance].
  Quickcheck (on [difference]) showed that this was necessary in order
  to maintain the invariants. It is quite unsatisfactory that I haven't
  been able to find out why this is actually the case! Fortunately, it
  doesn't hurt to be a bit more conservative.
--------------------------------------------------------------------}

{--------------------------------------------------------------------
  Join
--------------------------------------------------------------------}
join :: a -> Set a -> Set a -> Set a
join x Tip r  = insertMin x r
join :: a -> Set a -> Set a -> Set a
join x Tip r  = insertMin x r
join x Tip r  = insertMin x r