∀ {α : Type u} {β : Type v} {cmp : α → α → Ordering} {t : Std.ExtTreeMap α β cmp} [inst : Std.TransCmp cmp]
  [inst_1 : Inhabited β] {k a : α} {v : β}, (t.insert k v)[a]! = if cmp k a = Ordering.eq then v else t[a]!