∀ {α : Type u} {β : Type v} {cmp : α → α → Ordering} {t : Std.ExtTreeMap α β cmp} [inst : Std.TransCmp cmp] {k a : α}
  {v : β}, (t.insertIfNew k v).getKey? a = if cmp k a = Ordering.eq ∧ k ∉ t then some k else t.getKey? a