∀ {α : Type u} {β : Type v} {cmp : α → α → Ordering} {t : Std.ExtTreeMap α β cmp} [inst : Std.TransCmp cmp] {k : α}
  {f : Option β → Option β},
  t.alter k f ≠ ∅ →
    ∀ {fallback : α}, (t.alter k f).minKeyD fallback = k ↔ (f t[k]?).isSome = true ∧ ∀ k' ∈ t, (cmp k k').isLE = true