English
Equality case for AntivaryOn f g: ∑ f i · g i = ∑ f i · g(σ i) iff AntivaryOn f (g ∘ σ) s.
Русский
Условие равенства для AntivaryOn: сумма равна тогда и только тогда, когда AntivaryOn f (g ∘ σ) s.
LaTeX
$$$\displaystyle \text{If } \operatorname{AntivaryOn}(f,g,s) \text{ then } \sum_{i\in s} f(i) \cdot g(i) = \sum_{i\in s} f(i) \cdot g(σ(i)) \iff \operatorname{AntivaryOn}(f, g \circ σ, s).$$$
Lean4
/-- **Equality case of the Rearrangement Inequality**: Pointwise multiplication of `f` and `g`,
which antivary together on `s`, is unchanged by a permutation if and only if `f` and `g ∘ σ`
antivary together on `s`. Stated by permuting the entries of `g`. -/
theorem sum_mul_eq_sum_mul_comp_perm_iff (hfg : AntivaryOn f g s) (hσ : {x | σ x ≠ x} ⊆ s) :
∑ i ∈ s, f i * g (σ i) = ∑ i ∈ s, f i * g i ↔ AntivaryOn f (g ∘ σ) s :=
hfg.sum_smul_comp_perm_eq_sum_smul_iff hσ
