Theorems · Theorem · real analysis

Antivary.sum_mul_lt_sum_comp_perm_mul_iff

∀ {ι : Type u_1} {α : Type u_2} [inst : Semiring α] [inst_1 : LinearOrder α] [IsStrictOrderedRing α] [ExistsAddOfLE α]
  {σ : Equiv.Perm ι} {f g : ι → α} [inst_4 : Fintype ι],
  Antivary f g → (∑ i, f i * g i < ∑ i, f (σ i) * g i ↔ ¬Antivary (f ∘ ⇑σ) g)

Strict inequality case of the Rearrangement Inequality: Pointwise multiplication of f and g, which antivary together, is strictly decreased by a permutation if and only if f ∘ σ and g do not antivary together. Stated by permuting the entries of f.

Defined in

Mathlib.Algebra.Order.Rearrangement

Cited by

0 results in Mathlib

Foundations

Depth 87 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringLinearOrderIsStrictOrderedRingExistsAddOfLEFintype