English
Equality case for MonovaryOn sum_comp_perm_mul_eq_sum_mul_iff: ∑ f(i)·g(σ i) = ∑ f(i)·g(i) iff MonovaryOn f (g ∘ σ) s.
Русский
Условие равенства для MonovaryOn: сумма ∑ f(i)·g(σ i) равна ∑ f(i)·g(i) тогда и только тогда, когда MonovaryOn f (g ∘ σ) s.
LaTeX
$$$\displaystyle \text{If } \operatorname{MonovaryOn}(f,g,s) \text{ then } \sum_{i\in s} f(i) \cdot g(σ(i)) = \sum_{i\in s} f(i) \cdot g(i) \iff \operatorname{MonovaryOn}(f, g \circ σ, s).$$$
Lean4
/-- **Strict inequality case of the Rearrangement Inequality**: Pointwise multiplication of
`f` and `g`, which antivary together on `s`, is strictly decreased by a permutation if and only if
`f` and `g ∘ σ` do not antivary together on `s`. Stated by permuting the entries of `g`. -/
theorem sum_mul_lt_sum_mul_comp_perm_iff (hfg : AntivaryOn f g s) (hσ : {x | σ x ≠ x} ⊆ s) :
∑ i ∈ s, f i * g i < ∑ i ∈ s, f i * g (σ i) ↔ ¬AntivaryOn f (g ∘ σ) s :=
hfg.sum_smul_lt_sum_smul_comp_perm_iff hσ
