∀ {α : Type u} {β : Type v} {cmp : α → α → Ordering} {t : Std.TreeMap α β cmp} [inst : Std.TransCmp cmp] {k : α}
  {f : β → β} {he : (t.modify k f).isEmpty = false},
  (t.modify k f).minKey he = if cmp (t.minKey ⋯) k = Ordering.eq then k else t.minKey ⋯