English
If f and g are AntivaryOn on s, then the strict inequality ∑ f(i)•g(i) < ∑ f(i)•g(σ(i)) holds exactly when f ∘ σ and g are not antivary on s.
Русский
Если f и g антиварианты на s, то строгая неравенство \sum_{i∈s} f(i)g(i) < \sum_{i∈s} f(i)g(σ(i)) происходит тогда и только тогда, когда f ∘ σ и g не антивариантны на s.
LaTeX
$$$\displaystyle \text{If } \operatorname{AntivaryOn}(f,g,s) \text{ and } \{x | σx \neq x\} \subseteq s, \\ \sum_{i} f(i) \cdot g(i) < \sum_{i} f(i) \cdot g(σ(i)) \iff \neg \operatorname{AntivaryOn}(f, g \circ σ, s).$$$
Lean4
/-- **Strict inequality case of the Rearrangement Inequality**: Pointwise scalar multiplication of
`f` and `g`, which antivary together on `s`, is strictly decreased by a permutation if and only if
`f ∘ σ` and `g` do not antivary together on `s`. Stated by permuting the entries of `f`. -/
theorem sum_smul_lt_sum_comp_perm_smul_iff (hfg : AntivaryOn f g s) (hσ : {x | σ x ≠ x} ⊆ s) :
∑ i ∈ s, f i • g i < ∑ i ∈ s, f (σ i) • g i ↔ ¬AntivaryOn (f ∘ σ) g s := by
simp [← hfg.sum_comp_perm_smul_eq_sum_smul_iff hσ, eq_comm, lt_iff_le_and_ne, hfg.sum_smul_le_sum_comp_perm_smul hσ]
