Bornology structure on products and subtypes

In this file we define Bornology and BoundedSpace instances on α × β, Π i, π i, and {x // p x}. We also prove basic lemmas about Bornology.cobounded and Bornology.IsBounded on these types.

instance Prod.instBornology {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] : Bornology (α × β)

Equations

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

instance Pi.instBornology {ι : Type u_3} {π : ι → Type u_4} [Finite ι] [(i : ι) → Bornology (π i)] : Bornology ((i : ι) → π i)

Equations

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

@[reducible]

def Bornology.induced {α : Type u_5} {β : Type u_6} [Bornology β] (f : α → β) : Bornology α

Inverse image of a bornology.

Equations

• Bornology.induced f = { cobounded' := Filter.comap f (Bornology.cobounded β), le_cofinite' := (_ : Filter.comap f (Bornology.cobounded β) ≤ Filter.cofinite) }

instance instBornologySubtype {α : Type u_1} [Bornology α] {p : α → Prop} : Bornology (Subtype p)

Equations

• instBornologySubtype = Bornology.induced Subtype.val

Bounded sets in α × β

theorem Bornology.cobounded_prod {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] : Bornology.cobounded (α × β) = Filter.coprod (Bornology.cobounded α) (Bornology.cobounded β)

theorem Bornology.isBounded_image_fst_and_snd {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] {s : Set (α × β)} : Bornology.IsBounded (Prod.fst '' s) ∧ Bornology.IsBounded (Prod.snd '' s) ↔ Bornology.IsBounded s

theorem Bornology.IsBounded.image_fst {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] {s : Set (α × β)} (hs : Bornology.IsBounded s) : Bornology.IsBounded (Prod.fst '' s)

theorem Bornology.IsBounded.image_snd {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] {s : Set (α × β)} (hs : Bornology.IsBounded s) : Bornology.IsBounded (Prod.snd '' s)

theorem Bornology.IsBounded.fst_of_prod {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] {s : Set α} {t : Set β} (h : Bornology.IsBounded (s ×ˢ t)) (ht : Set.Nonempty t) : Bornology.IsBounded s

theorem Bornology.IsBounded.snd_of_prod {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] {s : Set α} {t : Set β} (h : Bornology.IsBounded (s ×ˢ t)) (hs : Set.Nonempty s) : Bornology.IsBounded t

theorem Bornology.IsBounded.prod {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] {s : Set α} {t : Set β} (hs : Bornology.IsBounded s) (ht : Bornology.IsBounded t) : Bornology.IsBounded (s ×ˢ t)

theorem Bornology.isBounded_prod_of_nonempty {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] {s : Set α} {t : Set β} (hne : Set.Nonempty (s ×ˢ t)) : Bornology.IsBounded (s ×ˢ t) ↔ Bornology.IsBounded s ∧ Bornology.IsBounded t

theorem Bornology.isBounded_prod {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] {s : Set α} {t : Set β} : Bornology.IsBounded (s ×ˢ t) ↔ s = ∅ ∨ t = ∅ ∨ Bornology.IsBounded s ∧ Bornology.IsBounded t

theorem Bornology.isBounded_prod_self {α : Type u_1} [Bornology α] {s : Set α} : Bornology.IsBounded (s ×ˢ s) ↔ Bornology.IsBounded s

Bounded sets in Π i, π i

theorem Bornology.cobounded_pi {ι : Type u_3} {π : ι → Type u_4} [Finite ι] [(i : ι) → Bornology (π i)] : Bornology.cobounded ((i : ι) → π i) = Filter.coprodᵢ fun (i : ι) => Bornology.cobounded (π i)

theorem Bornology.forall_isBounded_image_eval_iff {ι : Type u_3} {π : ι → Type u_4} [Finite ι] [(i : ι) → Bornology (π i)] {s : Set ((i : ι) → π i)} : (∀ (i : ι), Bornology.IsBounded (Function.eval i '' s)) ↔ Bornology.IsBounded s

theorem Bornology.IsBounded.image_eval {ι : Type u_3} {π : ι → Type u_4} [Finite ι] [(i : ι) → Bornology (π i)] {s : Set ((i : ι) → π i)} (hs : Bornology.IsBounded s) (i : ι) : Bornology.IsBounded (Function.eval i '' s)

theorem Bornology.IsBounded.pi {ι : Type u_3} {π : ι → Type u_4} [Finite ι] [(i : ι) → Bornology (π i)] {S : (i : ι) → Set (π i)} (h : ∀ (i : ι), Bornology.IsBounded (S i)) : Bornology.IsBounded (Set.pi Set.univ S)

theorem Bornology.isBounded_pi_of_nonempty {ι : Type u_3} {π : ι → Type u_4} [Finite ι] [(i : ι) → Bornology (π i)] {S : (i : ι) → Set (π i)} (hne : Set.Nonempty (Set.pi Set.univ S)) : Bornology.IsBounded (Set.pi Set.univ S) ↔ ∀ (i : ι), Bornology.IsBounded (S i)

theorem Bornology.isBounded_pi {ι : Type u_3} {π : ι → Type u_4} [Finite ι] [(i : ι) → Bornology (π i)] {S : (i : ι) → Set (π i)} : Bornology.IsBounded (Set.pi Set.univ S) ↔ (∃ (i : ι), S i = ∅) ∨ ∀ (i : ι), Bornology.IsBounded (S i)

Bounded sets in {x // p x}

theorem Bornology.isBounded_induced {α : Type u_5} {β : Type u_6} [Bornology β] {f : α → β} {s : Set α} : Bornology.IsBounded s ↔ Bornology.IsBounded (f '' s)

theorem Bornology.isBounded_image_subtype_val {α : Type u_1} [Bornology α] {p : α → Prop} {s : Set { x : α // p x }} : Bornology.IsBounded (Subtype.val '' s) ↔ Bornology.IsBounded s

Bounded spaces

instance instBoundedSpaceProdInstBornology {α : Type u_1} {β : Type u_2} [Bornology α] [Bornology β] [BoundedSpace α] [BoundedSpace β] : BoundedSpace (α × β)

Equations

• (_ : BoundedSpace (α × β)) = (_ : BoundedSpace (α × β))

instance instBoundedSpaceForAllInstBornology {ι : Type u_3} {π : ι → Type u_4} [Finite ι] [(i : ι) → Bornology (π i)] [∀ (i : ι), BoundedSpace (π i)] : BoundedSpace ((i : ι) → π i)

Equations

• (_ : BoundedSpace ((i : ι) → π i)) = (_ : BoundedSpace ((i : ι) → π i))

theorem boundedSpace_induced_iff {α : Type u_5} {β : Type u_6} [Bornology β] {f : α → β} : BoundedSpace α ↔ Bornology.IsBounded (Set.range f)

theorem boundedSpace_subtype_iff {α : Type u_1} [Bornology α] {p : α → Prop} : BoundedSpace (Subtype p) ↔ Bornology.IsBounded {x : α | p x}

theorem boundedSpace_val_set_iff {α : Type u_1} [Bornology α] {s : Set α} : BoundedSpace ↑s ↔ Bornology.IsBounded s

theorem Bornology.IsBounded.boundedSpace_subtype {α : Type u_1} [Bornology α] {p : α → Prop} : Bornology.IsBounded {x : α | p x} → BoundedSpace (Subtype p)

Alias of the reverse direction of boundedSpace_subtype_iff.

theorem Bornology.IsBounded.boundedSpace_val {α : Type u_1} [Bornology α] {s : Set α} : Bornology.IsBounded s → BoundedSpace ↑s

Alias of the reverse direction of boundedSpace_val_set_iff.

instance instBoundedSpaceSubtypeInstBornologySubtype {α : Type u_1} [Bornology α] [BoundedSpace α] {p : α → Prop} : BoundedSpace (Subtype p)

Equations

• (_ : BoundedSpace (Subtype p)) = (_ : BoundedSpace { x : α // p x })

Additive, Multiplicative

The bornology on those type synonyms is inherited without change.

instance instBornologyAdditive {α : Type u_1} [Bornology α] : Bornology (Additive α)

Equations

• instBornologyAdditive = inst

instance instBornologyMultiplicative {α : Type u_1} [Bornology α] : Bornology (Multiplicative α)

Equations

• instBornologyMultiplicative = inst

instance instBoundedSpaceAdditiveInstBornologyAdditive {α : Type u_1} [Bornology α] [BoundedSpace α] : BoundedSpace (Additive α)

Equations

• (_ : BoundedSpace (Additive α)) = inst

instance instBoundedSpaceMultiplicativeInstBornologyMultiplicative {α : Type u_1} [Bornology α] [BoundedSpace α] : BoundedSpace (Multiplicative α)

Equations

• (_ : BoundedSpace (Multiplicative α)) = inst

Order dual

The bornology on this type synonym is inherited without change.

instance instBornologyOrderDual {α : Type u_1} [Bornology α] : Bornology αᵒᵈ

Equations

• instBornologyOrderDual = inst

instance instBoundedSpaceOrderDualInstBornologyOrderDual {α : Type u_1} [Bornology α] [BoundedSpace α] : BoundedSpace αᵒᵈ

Equations

• (_ : BoundedSpace αᵒᵈ) = inst