∀ {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : Std.DTreeMap α β cmp} [Std.TransCmp cmp]
  [inst : Std.LawfulEqCmp cmp] {k a : α} {fallback : β a} {v : β k},
  (t.insertIfNew k v).getD a fallback = if h : cmp k a = Ordering.eq ∧ k ∉ t then cast ⋯ v else t.getD a fallback