Lemmas about filters and ordered rings.

theorem Filter.EventuallyLE.mul_le_mul {α : Type u} {β : Type v} [MulZeroClass β] [PartialOrder β] [PosMulMono β] [MulPosMono β] {l : Filter α} {f₁ : α → β} {f₂ : α → β} {g₁ : α → β} {g₂ : α → β} (hf : f₁ ≤ᶠ[l] f₂) (hg : g₁ ≤ᶠ[l] g₂) (hg₀ : 0 ≤ᶠ[l] g₁) (hf₀ : 0 ≤ᶠ[l] f₂) : f₁ * g₁ ≤ᶠ[l] f₂ * g₂

theorem Filter.EventuallyLE.mul_le_mul' {α : Type u} {β : Type v} [Mul β] [Preorder β] [MulLeftMono β] [MulRightMono β] {l : Filter α} {f₁ : α → β} {f₂ : α → β} {g₁ : α → β} {g₂ : α → β} (hf : f₁ ≤ᶠ[l] f₂) (hg : g₁ ≤ᶠ[l] g₂) : f₁ * g₁ ≤ᶠ[l] f₂ * g₂

theorem Filter.EventuallyLE.add_le_add {α : Type u} {β : Type v} [Add β] [Preorder β] [AddLeftMono β] [AddRightMono β] {l : Filter α} {f₁ : α → β} {f₂ : α → β} {g₁ : α → β} {g₂ : α → β} (hf : f₁ ≤ᶠ[l] f₂) (hg : g₁ ≤ᶠ[l] g₂) : f₁ + g₁ ≤ᶠ[l] f₂ + g₂

theorem Filter.EventuallyLE.mul_nonneg {α : Type u} {β : Type v} [OrderedSemiring β] {l : Filter α} {f : α → β} {g : α → β} (hf : 0 ≤ᶠ[l] f) (hg : 0 ≤ᶠ[l] g) : 0 ≤ᶠ[l] f * g

theorem Filter.eventually_sub_nonneg {α : Type u} {β : Type v} [OrderedRing β] {l : Filter α} {f : α → β} {g : α → β} : 0 ≤ᶠ[l] g - f ↔ f ≤ᶠ[l] g

instance Filter.Germ.instOrderedCommGroup {α : Type u} {β : Type v} {l : Filter α} [OrderedCommGroup β] : OrderedCommGroup (l.Germ β)

Equations

• Filter.Germ.instOrderedCommGroup = OrderedCommGroup.mk ⋯

instance Filter.Germ.instOrderedAddCommGroup {α : Type u} {β : Type v} {l : Filter α} [OrderedAddCommGroup β] : OrderedAddCommGroup (l.Germ β)

Equations

• Filter.Germ.instOrderedAddCommGroup = OrderedAddCommGroup.mk ⋯

instance Filter.Germ.instOrderedSemiring {α : Type u} {β : Type v} {l : Filter α} [OrderedSemiring β] : OrderedSemiring (l.Germ β)

Equations

• Filter.Germ.instOrderedSemiring = OrderedSemiring.mk ⋯ ⋯ ⋯ ⋯

instance Filter.Germ.instOrderedCommSemiring {α : Type u} {β : Type v} {l : Filter α} [OrderedCommSemiring β] : OrderedCommSemiring (l.Germ β)

Equations

• Filter.Germ.instOrderedCommSemiring = OrderedCommSemiring.mk ⋯

instance Filter.Germ.instOrderedRing {α : Type u} {β : Type v} {l : Filter α} [OrderedRing β] : OrderedRing (l.Germ β)

Equations

• Filter.Germ.instOrderedRing = OrderedRing.mk ⋯ ⋯ ⋯

instance Filter.Germ.instOrderedCommRing {α : Type u} {β : Type v} {l : Filter α} [OrderedCommRing β] : OrderedCommRing (l.Germ β)

Equations

• Filter.Germ.instOrderedCommRing = OrderedCommRing.mk ⋯