∀ {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : Std.DTreeMap α β cmp} [Std.TransCmp cmp]
  [inst : Std.LawfulEqCmp cmp] {f : (a : α) → β a → Bool},
  (Std.DTreeMap.filter f t).isEmpty = true ↔ ∀ (k : α) (h : k ∈ t), f k (t.get k h) = false