Lemmas about filters and ordered rings.

theorem Filter.EventuallyLE.mul_le_mul {α : Type u} {β : Type v} [MulZeroClass β] [Preorder β] [PosMulMono β] [MulPosMono β] {l : Filter α} {f₁ f₂ g₁ 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 f₂ g₁ 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 f₂ g₁ 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 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} [OrderedAddCommGroup β] {l : Filter α} {f 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

• One or more equations did not get rendered due to their size.

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

Equations

• One or more equations did not get rendered due to their size.

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

Equations

• One or more equations did not get rendered due to their size.

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

Equations

• Filter.Germ.instOrderedCommSemiring = { toOrderedSemiring := Filter.Germ.instOrderedSemiring, mul_comm := ⋯ }

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

Equations

• Filter.Germ.instOrderedRing = { toRing := Filter.Germ.instRing, toPartialOrder := OrderedAddCommGroup.toPartialOrder, add_le_add_left := ⋯, zero_le_one := ⋯, mul_nonneg := ⋯ }

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

Equations

• Filter.Germ.instOrderedCommRing = { toOrderedRing := Filter.Germ.instOrderedRing, mul_comm := ⋯ }