∀ {ι : Type u_1} {α : Type u_2} [inst : Semiring α] [inst_1 : LinearOrder α] [IsStrictOrderedRing α] [ExistsAddOfLE α]
  {s : Finset ι} {σ : Equiv.Perm ι} {f g : ι → α},
  AntivaryOn f g ↑s → {x | σ x ≠ x} ⊆ ↑s → (∑ i ∈ s, f i * g i < ∑ i ∈ s, f (σ i) * g i ↔ ¬AntivaryOn (f ∘ ⇑σ) g ↑s)