∀ {ι : Type u_1} {α : Type u_2} [inst : AddCommMonoid α] [inst_1 : LinearOrder α] [OrderBot α] [CanonicallyOrderedAdd α]
  {s : Finset ι} {f : ι → α} (hs : s.Nonempty), (s.sup' hs f = 0) = ∀ i ∈ s, f i = 0