Theorems · Theorem · real analysis

MonovaryOn.sum_comp_perm_smul_lt_sum_smul_iff

∀ {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] [PosSMulStrictMono α β] {s : Finset ι} {σ : Equiv.Perm ι} {f : ι → α} {g : ι → β},
  MonovaryOn f g ↑s → {x | σ x ≠ x} ⊆ ↑s → (∑ i ∈ s, f (σ i) • g i < ∑ i ∈ s, f i • g i ↔ ¬MonovaryOn (f ∘ ⇑σ) g ↑s)

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

Defined in

Mathlib.Algebra.Order.Rearrangement

Cited by

2 results in Mathlib

Foundations

Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringLinearOrderIsStrictOrderedRingExistsAddOfLEAddCommMonoidLinearOrderIsOrderedCancelAddMonoidModulePosSMulStrictMono