Theorems · Theorem · real analysis

AntivaryOn.sum_smul_le_sum_comp_perm_smul

∀ {ι : Type u_1} {α : Type u_2} {β : Type u_3} [inst : Semiring α] [inst_1 : LinearOrder α] [IsStrictOrderedRing α]
  [ExistsAddOfLE α] [inst_4 : AddCommMonoid β] [inst_5 : LinearOrder β] [IsOrderedCancelAddMonoid β]
  [inst_7 : Module α β] [PosSMulMono α β] {s : Finset ι} {σ : Equiv.Perm ι} {f : ι → α} {g : ι → β},
  AntivaryOn f g ↑s → {x | σ x ≠ x} ⊆ ↑s → ∑ i ∈ s, f i • g i ≤ ∑ i ∈ s, f (σ i) • g i

Rearrangement Inequality: Pointwise scalar multiplication of f and g is minimized when f and g antivary together on s. Stated by permuting the entries of f.

Defined in

Mathlib.Algebra.Order.Rearrangement

Cited by

3 results in Mathlib

Foundations

Depth 83 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringLinearOrderIsStrictOrderedRingExistsAddOfLEAddCommMonoidLinearOrderIsOrderedCancelAddMonoidModulePosSMulMono