Theorems · Theorem · real analysis

Antivary.sum_mul_lt_sum_mul_comp_perm_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 i • g (σ i) ↔ ¬Antivary f (g ∘ ⇑σ))

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 and g ∘ σ do not antivary together. Stated by permuting the entries of g.

Defined in

Mathlib.Algebra.Order.Rearrangement

Cited by

0 results in Mathlib

Foundations

Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringLinearOrderIsStrictOrderedRingExistsAddOfLEFintype