∀ {α : Type u} {β : Type v} {cmp : α → α → Ordering} {t : Std.TreeMap α β cmp} [Std.TransCmp cmp] [inst : Inhabited α]
  {k : α} {f : Option β → Option β},
  (t.alter k f).isEmpty = false →
    ((t.alter k f).minKey! = k ↔ (f t[k]?).isSome = true ∧ ∀ k' ∈ t, (cmp k k').isLE = true)