∀ {α : Type u} {β : Type v} {cmp : α → α → Ordering} {t : Std.TreeMap α β cmp} [Std.TransCmp cmp] {k : α} {v : β},
  (t.insertIfNew k v).maxKey? = some (t.maxKey?.elim k fun k' => if cmp k' k = Ordering.lt then k else k')